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DAMPING CHARACTERISTICS OF REINFORCED AND
PRESTRESSED NORMAL- AND HIGH-STRENGTH
CONCRETE BEAMS
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy
by
Angela Salzmann BEng (Hons1), J.P.
from
School of Engineering
Faculty of Engineering and Information Technology
GRIFFITH UNIVERSITY
GOLD COAST CAMPUS
November 2002
To My Parents
Declaration i
Declaration
This work has not been previously submitted for a degree or diploma in any university.
To the best of my knowledge and belief, the thesis contains no material previously
published or written by another person except where due reference is made in the thesis
itself.
________________________
Angela Salzmann
November 2002
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Acknowledgements ii
Acknowledgements
The research from which this thesis has been composed was undertaken at the School of
Engineering, Griffith University Gold Coast Campus under the supervision of Dr. Sam
Fragomeni and Professor Yew-Chaye Loo. The author is greatly indebted to Dr.
Fragomeni and Professor Loo, whose continued support, encouragement, inspiration
and technical contributions helped to guide and shape the research effort. In particular,
the provision of significant guidance regarding the scope of the experimental work
provided invaluable assistance.
A special thankyou is given to all the technical staff and in particular to Charles Allport
without whose help, the experimental work could not have been possible. Grateful
thanks are also extended to the administrative staff of the School who provided constant
encouragement and to many final year students who enthusiastically assisted in the
laboratory testing tasks.
The author also wishes to thank the Australian Government Australian Postgraduate
Award (APA) Scholarship Scheme and the School of Engineering for providing the
financial assistance which allowed the research to be completed in this form.
Finally, her deep heartfelt thanks goes to her partner, Peter, and her parents, Gus and
Fiona and her sister Monique for their constant encouragement, understanding, financial
assistance, and belief. The completion of this research is but a small gift for their
efforts and great expectations.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Publications iii
List of Publications
Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002a) Damping behaviour of reinforced
concrete beams – Review and new developments, 17th Australasian Conference
on the Mechanics of Structures and Materials, 12-14 June 2002, Gold Coast,
Australia.
Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002b) The damping analysis of
experimental concrete beams under free-vibration, Advances in Structural
Engineering – An International Journal, Accepted for Publication.
Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002c) Estimation of the free-vibration
damping characteristics of untested reinforced concrete beams, Electronic Journal
of Structural Engineering, Submitted for Publication.
Salzmann, A., Fragomeni, S. and Loo, Y.C. (2001a) Verification of damping formulae
using experimental results from full-scale concrete beams reinforced with 500
MPA steel, The Australian Structural Engineering Conference, Surfers Paradise
Marriott Resort, Gold Coast, Australia, 29 April – 2 May 2001, pp. 95-102.
Salzmann, A., Fragomeni, S. and Loo, Y.C. (2001b) Investigation of damping in high-
strength prestressed concrete beams, The Eighth East Asia-Pacific Conference on
Structural Engineering & Construction, 5-7 December 2001, Singapore.
Salzmann, A. and Fragomeni, S. (2000) Experimental determination of damping from
full-scale reinforced and prestressed concrete beams, Civil Engineering
Challenges in the 21st Century, Queensland Civil Engineering Postgraduate
Conference, December 12-13, 2000, Physical Infrastructure Centre Queensland
University of Technology, pp 75-84.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Synopsis iv
Synopsis
In the last few decades there has been a significant increase in the design strength and
performance of different building materials. In particular, new methods, materials
and admixtures for the production of concrete have allowed for strengths as high as
100 MPa to be readily available. In addition, the standard manufactured yield
strength of reinforcing steel in Australia has increased from 400 MPa to 500 MPa.
A perceived design advantage of higher-strength materials is that structural elements
can have longer spans and be more slender than previously possible. An emerging
problem with slender concrete members is that they can be more vulnerable to loading
induced vibration. The damping capacity is an inherent fundamental quantity of all
structural concrete members that affects their vibrational response. It is defined as the
rate at which a structural member can dissipate the vibrational energy imparted to it.
Generally damping capacity measurements, to indicate the integrity of structural
members, are taken once the structure is in service. This type of non-destructive testing
has been the subject of much research. The published non-destructive testing research
on damping capacity is conflicting and a unified method to describe the effect of
damage on damping capacity has not yet been proposed.
Significantly, there is not one method in the published literature or national design
codes, including the Australian Standard AS 3600-2001, available to predict the
damping capacity of concrete beam members at the design stage. Further, little
research has implemented full-scale testing with a view to developing damping
capacity design equations, which is the primary focus of this thesis.
To examine the full-range damping behaviour of concrete beams, two categories of
testing were proposed. The categories are the ‘untested’ and ‘tested’ beam states.
These beam states have not been separately investigated in previous work and are
considered a major shortcoming of previous research on the damping behaviour of
concrete beams.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Synopsis v An extensive experimental programme was undertaken to obtain residual deflection and
damping capacity data for thirty-one reinforced and ten prestressed concrete beams.
The concrete beams had compressive strengths ranging between 23.1 MPa and 90.7
MPa, reinforcement with yield strengths of 400 MPa or 500 MPa, and tensile
reinforcement ratios between 0.76% and 2.90%. The full- and half-scale beams tested
had lengths of 6.0 m and 2.4 m, respectively. The testing regime consisted of a series of
on-off load increments, increasing until failure, designed to induce residual deflections
with increasing amounts of internal damage at which damping capacity (logarithmic
decrement) was measured.
The inconsistencies that were found between the experimental damping capacity of the
beams and previous research prompted an initial investigation into the data obtained.
It was found that the discrepancies were due to the various interpretations of the
method used to extract damping capacity from the free-vibration decay curve.
Therefore, a logarithmic decrement calculation method was proposed to ensure
consistency and accuracy of the extracted damping capacity data to be used in the
subsequent analytical research phase.
The experimental test data confirmed that the ‘untested’ damping capacity of reinforced
concrete beams is dependent upon the beam reinforcement ratio and distribution. This
quantity was termed the total longitudinal reinforcement distribution. For the
prestressed concrete beams, the ‘untested’ damping capacity was shown to be
proportional to the product of the prestressing force and prestressing eccentricity.
Separate ‘untested’ damping capacity equations for reinforced and prestressed concrete
beams were developed to reflect these quantities.
To account for the variation in damping capacity due to damage in ‘tested’ beams, a
residual deflection mechanism was utilised. The proposed residual deflection
mechanism estimates the magnitude of permanent deformation in the beam and
attempts to overcome traditional difficulties in calculating the damping capacity during
low loading levels. Residual deflection equations, based on the instantaneous
deflection data for the current experimental programme, were proposed for both the
reinforced and prestressed concrete beams, which in turn were utilised with the
proposed ‘untested’ damping equation to calculate the total damping capacity.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Synopsis vi
The proposed ‘untested’ damping, residual deflection and total damping capacity
equations were compared to published test data and an additional series of test beams.
These verification investigations have shown that the proposed equations are reliable
and applicable for a range of beam designs, test setups, constituent materials and
loading regimes.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table of Contents vii
Table of Contents
Declaration ....................................................................................................... i Acknowledgements .......................................................................................... ii List of Publications .......................................................................................... iii Synopsis ............................................................................................................ iv Table of Contents ............................................................................................ vii List of Figures .................................................................................................. xi List of Tables ................................................................................................... xv List of Plates .................................................................................................... xvi Notation ............................................................................................................ xvii
CHAPTER
1. Introduction ..................................................................................................... 1-1 1.1 General Remarks ................................................................................... 1-1 1.2 Research Objectives .............................................................................. 1-2 1.3 Research Methodology ......................................................................... 1-2 1.4 Layout of the Thesis .............................................................................. 1-3 1.5 Summary ............................................................................................... 1-4
2. Damping in Concrete ....................................................................................... 2-1 2.1 General Remarks ................................................................................... 2-1 2.2 Undamped Systems ............................................................................... 2-2 2.2.1 Single-DOF and multi-DOF structures .................................. 2-2 2.3 Damped Systems ................................................................................... 2-4 2.3.1 The idealized MDOF system ................................................. 2-4 2.3.2 Viscous damping .................................................................... 2-5 2.3.3 Coulomb damping .................................................................. 2-11 2.3.4 Hysteretic damping ................................................................ 2-12 2.3.5 Equivalent viscous damping .................................................. 2-14 2.4 Experimental Determination of Damping ............................................. 2-15 2.4.1 Free-vibration damping .......................................................... 2-15 2.4.2 Forced excitation damping ..................................................... 2-16 2.4.2.1 Half-power (bandwidth) method ............................... 2-17 2.4.2.2 Resonant amplification ............................................. 2-17 2.4.2.3 Energy loss per cycle ................................................ 2-18 2.5 Literature Review of Damping in Concrete ............................................ 2-21 2.5.1 Material damping ..................................................................... 2-21 2.5.2 Member damping ..................................................................... 2-24 2.5.3 Structural damping................................................................... 2-32 2.6 Summary ............................................................................................... 2-34
3. Theoretical Consideration ............................................................................. 3-1 3.1 General Remarks ................................................................................... 3-1 3.2 Pilot Study............................................................................................... 3-1 3.2.1 Verifying the accuracy of logdec ........................................... 3-2
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table of Contents viii
3.2.2 Logdec versus stage of testing ............................................... 3-2 3.2.3 Damage mechanisms in concrete beams ................................ 3-3 3.2.4 Residual deflection ................................................................. 3-4 3.3 The Total Damping Capacity Equation .................................................. 3-4 3.3.1 ‘Untested’ beams .................................................................... 3-5 3.3.2 ‘Tested’ beams ....................................................................... 3-6 3.4 Summary ............................................................................................... 3-7 4. Experimental Programme ............................................................................. 4-1 4.1 General Remarks ................................................................................... 4-1 4.2 Design of Beam Test Specimens ........................................................... 4-1 4.2.1 Geometrical and mechanical details ........................................ 4-2 4.2.2 Primary test variables .............................................................. 4-2 4.3 Materials.................................................................................................. 4-2 4.3.1 Concrete .................................................................................. 4-2 4.3.2 Reinforcement ......................................................................... 4-9 4.4 Fabrication .............................................................................................. 4-13 4.4.1 Reinforced concrete beams .................................................... 4-13 4.4.2 Prestressed concrete beams ..................................................... 4-13 4.4.3 Curing ...................................................................................... 4-15 4.5 Test Set-Up ........................................................................................... 4-15 4.5.1 Beam support system ............................................................... 4-15 4.5.2 Loading beam width (LBW) ..................................................... 4-15 4.5.3 Hammer excitation position (HEP).......................................... 4-15 4.5.4 Hammer weight (HW) .............................................................. 4-18 4.6 Test Procedures ..................................................................................... 4-18 4.7 Instrumentation ..................................................................................... 4-18 4.7.1 Damping ................................................................................. 4-19 4.7.2 Crack width ............................................................................ 4-21 4.7.3 Crack Patterns ......................................................................... 4-22 4.7.4 Deflection ............................................................................... 4-22 4.8 Summary ............................................................................................... 4-23
5. A Method for Extracting Damping Capacity ............................................... 5-1 5.1 General Remarks ................................................................................... 5-1 5.2 Experimental Techniques ...................................................................... 5-1 5.3 Analytical Technique for Calculating Logdec ...................................... 5-3 5.4 Applying the TLT and DCM Techniques .............................................. 5-4 5.4.1 Differences between the TLT and DCM output ...................... 5-6 5.4.2 Proposed rules for calculating logdec .................................... 5-7 5.5 Effect of Experimental Test Variables .................................................. 5-10 5.5.1 Hammer weight (HW) ............................................................ 5-10 5.5.2 Hammer excitation position (HEP) ........................................ 5-10 5.6 Summary ............................................................................................... 5-11
6. Damping Prediction in ‘Untested’ Concrete Beams .................................... 6-1 6.1 General Remarks ................................................................................... 6-1 6.2 Damping in ‘Untested’ Reinforced Concrete Beams ............................ 6-1 6.2.1 Historical review .................................................................... 6-1 6.2.2 Experimental effect of fcm and fsy ........................................... 6-3 6.2.3 Proposed damping equation ................................................... 6-3
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table of Contents ix
6.3 Damping in ‘Untested’ Prestressed Concrete Beams ............................. 6-7 6.3.1 Historical review .................................................................... 6-7 6.3.2 Experimental observations ..................................................... 6-8 6.3.3 Hop’s prestressed equation .................................................... 6-8 6.3.4 Proposed damping equation ..................................................... 6-10 6.4 Verification ............................................................................................ 6-11 6.4.1 Original beam data .................................................................. 6-11 6.4.2 F-Series Beams ....................................................................... 6-11 6.4.3 Neild’s Beam ........................................................................... 6-13 6.5 Summary ............................................................................................... 6-14
7. Residual Deflection Mechanisms in Concrete Beams ................................. 7-1 7.1 General Remarks ................................................................................... 7-1 7.2 Residual Deflection in Reinforced Concrete Beams .............................. 7-1 7.2.1 Effect of fsy .............................................................................. 7-1 7.2.2 Effect of fcm .............................................................................. 7-2 7.2.3 Effect of ρt................................................................................ 7-2 7.2.4 Effect of loading conditions ..................................................... 7-2 7.2.5 Summary of effects ................................................................. 7-2 7.2.6 The proposed equation ............................................................ 7-7 7.3 Residual Deflection in Prestressed Concrete Beams ............................. 7-7 7.3.1 Effect of fcm and e ..................................................................... 7-9 7.3.2 Effect of H ............................................................................... 7-9 7.3.3 Effect of H and e ...................................................................... 7-10 7.3.4 Summary of effects ................................................................. 7-10 7.3.5 The proposed equation ............................................................. 7-10 7.4 Verification ............................................................................................ 7-14 7.4.1 Original beam data .................................................................. 7-14 7.4.2 F-Series Beams ....................................................................... 7-16 7.4.3 James’ Beams........................................................................... 7-17 7.5 Summary ............................................................................................... 7-20
8. Damping in ‘Tested’ Concrete Beams............................................................. 8-1 8.1 General Remarks ................................................................................... 8-1 8.2 Development of Total Damping Equations ........................................... 8-1 8.3 Verification ........................................................................................... 8-7 8.3.1 F-Series Beams ....................................................................... 8-7 8.3.2 Chowdhury’s beams ............................................................... 8-8 8.4 Advantages of Proposed Residual Deflection Equations ...................... 8-10 8.5 Summary ............................................................................................... 8-12
9. Conclusions and Recommendations .............................................................. 9-1 9.1 General Remarks ................................................................................... 9-1 9.2 Research Objectives and Outcomes ...................................................... 9-1 9.2.1 Summary of test results ......................................................... 9-2 9.2.2 Verification of the proposed methods ................................... 9-4 9.3 Recommendations and Scope for Future Research ............................... 9-4 9.4 Closure ................................................................................................ 9-5
References .................................................................................................................Re-1
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table of Contents x
Bibliography .............................................................................................................Bi-1 Appendix A Literature Review Summary Tabulations ........................................... A-1 Appendix B RC and PSC Beam Calculations ......................................................... B-1 Appendix C Beam Crack Pattern Photographs ....................................................... C-1 Appendix D Logdec Comparative Graphs .............................................................. D-1 Appendix E Damping Tabulations ......................................................................... E-1 Appendix F Serviceability Curves ......................................................................... F-1
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Figures xi
List of Figures
Page
Figure 2.1 The undamped free-vibration response (Clough and Penzien, 1975)
2-3
Figure 2.2 Freely vibrating body in the x,y,z direction (Fertis, 1995) 2-3Figure 2.3 Example of a structure modelled as a SDOF system (Fertis,
1995) 2-4
Figure 2.4 Beam system (Fertis, 1995): (a) MDOF beam (b) Idealized SDOF mass-spring system
2-3
Figure 2.5 Decay of free-vibration under the assumption of viscous damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics
2-6
Figure 2.6 Free-vibration response of critical and overdamped systems (Clough and Penzien, 1975)
2-8
Figure 2.7 Free-vibration response of an underdamped system (Fertis, 1995)
2-9
Figure 2.8 Decay of free-vibration under the assumption of Coulomb damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics
2-11
Figure 2.9 Decay of vibration under the assumption of hystetic damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics
2-13
Figure 2.10 Force-displacement Hysteresis loop (Fertis, 1995) 2-13Figure 2.11 Frequency response curve for moderately damped system
(Clough and Penzien, 1975) 2-18
Figure 2.12 Actual and equivalent damping energy per cycle (Clough and Penzien, 1975)
2-19
Figure 2.13 Elastic stiffness and strain energy (Clough and Penzien, 1975)
2-20
Figure 2.14 Damping for the (Dieterle and Bachman, 1981): (a) Uncracked (b) Cracked beam sections
2-27
Figure 2.15 Damping ratio as a function of the relative steel stress (Dieterle and Bachman, 1981)
2-27
Figure 2.16 SDOF RC cantilever beam element during one loading cycle (Flesch, 1981)
2-28
Figure 2.17 Damping in beams without load as a function of the maximum load (Wang et al., 1998)
2-30
Figure 3.1 Schematic residual load-deflection curve for concrete beams 3-5
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Figures xii Figure 4.1 Geometric detailing for B-, CS, and F-Series test beams 4-3Figure 4.2 Geometric detailing for PS-Series test beams 4-3Figure 4.3 B-Series – Primary test variables 4-5Figure 4.4 PS-Series – Primary test variables 4-6Figure 4.5 CS-Series– Primary test variables 4-7Figure 4.6 F-Series – Primary test variables 4-8Figure 4.7 Stress-strain curve for 400 MPa reinforcing steel (BHP
Laboratory, Brisbane, Australia) 4-11
Figure 4.8 Stress-strain curve for 500 MPa reinforcing steel (BHP Laboratory, Brisbane, Australia)
4-12
Figure 4.9 Stress-strain curve for prestressing tendons (BHP Laboratory, Brisbane, Australia)
4-12
Figure 4.10 Rig used for prestressing the tendons 4-14Figure 4.11 Diagram of beam test set-up 4-16Figure 4.12 Location of beam testing equipment 4-19Figure 4.13 Impact energy frequency spectrum induced by hammer
excitation (Døssing, 1988b) 4-20
Figure 5.1 Experimental beams and HEP test variables 5-2Figure 5.2 Analytical algorithm used by the DCM 5-4Figure 5.3 Definition of data lengths using the TLT and DCM 5-5Figure 5.4 Calculation of logdec (TLT) using cycle number (n) 5-6Figure 5.5 Calculation of logdec (DCM) using NDP 5-6Figure 5.6 Variation of logdec (TLT) with n 5-8Figure 5.7 Plot of natural logarithm of An versus cycle number 5-8Figure 5.8 Example calculation of:
(a) Peak ratio (b) “Optimal peak ratio” curves
5-9
Figure 5.9 Effect of HW on logdec (TLT) 5-10Figure 5.10 Effect of HEP on logdec (TLT) 5-11 Figure 6.1 Classification of historical ‘untested’ damping research 6-2Figure 6.2 Variation of logdec with steel yield strength – B-Series
Beams 6-4
Figure 6.3 Variation of logdec with concrete compressive strength (a) B-Series (b) CS-Series beams
6-5
Figure 6.4 Dependence of ‘untested’ logdec on LRD in RC beams: (a) Separate trendlines (b) Unified prediction equation
6-6
Figure 6.5 Prestressing force versus ‘untested’ logdec for PS-Series beams
6-9
Figure 6.6 Prestressing eccentricity versus ‘untested’ logdec for PS-Series beams
6-9
Figure 6.7 ‘Untested’ logdec versus logdec prediction using Hop (1991) 6-10Figure 6.8 ‘Untested’ logdec versus initial prestress in beam 6-11Figure 6.9 Comparison between experimental logdec and logdec
calculated by Equation 6.2 and 6.3 6-12
Figure 6.10 Observed versus predicted ‘untested’ logdec for the F-Series beams
6-12
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Figures xiii Figure 6.11 Details of test beams and testing arrangement of Neild
(2001) 6-13
Figure 6.12 ‘Optimal peak ratio’ analysis of Neild’s (2001) free-decay curve
6-14
Figure 7.1 Effect of reinforcement yield strength for B-Series beams on
residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment
7-3
Figure 7.2 Effect of concrete compressive strength for B-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment
7-4
Figure 7.3 Effect of tensile reinforcement ratio for B-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment
7-5
Figure 7.4 Effect of LBW for CS-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment
7-6
Figure 7.5 Effect of reinforcement ratio on the instantaneous versus residual deflection relationship: (a) B-Series (b) CS-Series
7-8
Figure 7.6 Selection of curve coefficient, αrc for the calculation of residual deflection
7-9
Figure 7.7 Residual deflection for PS3, PS6, PS8 and PS9 versus: (a) and (b) Concrete compressive strength and prestress eccentricity (c) Normalised mid-span bending moment
7-11
Figure 7.8 Residual deflection for PS9 and PS10 versus: (a) Prestressing force (b) Normalised mid-span bending moment
7-12
Figure 7.9 Residual deflection for PS5 and PS6 versus: (a) Prestressing force and prestressing eccentricity (b) Normalised mid-span bending moment
7-13
Figure 7.10 Correlation between instantaneous and residual deflection for PSC beams
7-14
Figure 7.11 Experimental versus calculated residual deflection for: (a) B-Series (b) CS-Series (c) PS-Series test beams
7-15
Figure 7.12 Experimental versus calculated residual deflection for F-Series test beams
7-16
Figure 7.13 Experimental versus calculated residual deflection for James’ beams
7-19
Figure 8.1 Logdec versus residual deflection for B-Series:
(a) BI-, BI-3 and BII-4 (b) BI-5 to BII-8
8-2
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Figures xiv
(c) BI-9 to BII-12 Figure 8.2 Logdec versus residual deflection for CS-Series:
(a) CS1 to CS3 (b) CS4 to CS6 (c) CS7 to CS9
8-3
Figure 8.3 Logdec versus residual deflection for PS-Series: (a) PS3 to PS6 (b) PS7 to PS10
8-4
Figure 8.4 Dependence of D-R slope on concrete compressive strength: (a) B- and PS-Series beams (b) CS-Series beams
8-6
Figure 8.5 Calculated versus experimental logdec – F-Series beams 8-7Figure 8.6 Details of test beams and testing arrangement of Chowdhury
(1999) 8-8
Figure 8.7 Chowdhury’s experimental versus calculated logdec (Equation 8.2)
8-11
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Tables xv
List of Tables
Page
Table 4.1 Geometrical and reinforcement details – RC beams 4-4Table 4.2 Geometrical and reinforcement details – PSC beams 4-4Table 4.3 Concrete technical data – Materials (CSR Construction
Materials) 4-9
Table 4.4 Concrete technical data – Mix design (CSR Construction Materials)
4-9
Table 4.5 Details of test beams - Concrete 4-10Table 4.6 Technical design data for 400 and 500 MPa reinforcing steel
(Patrick, 1999) 4-11
Table 4.7 Details of test beams – Reinforcing bars 4-13Table 4.8 Loading beam width specifications 4-17Table 4.9 Specifications for ICP® impulse-force hammer (PCB ®
Piezotronics, 1992) 4-20
Table 6.1 Experimental ‘untested’ damping data – RC beams 6-4Table 6.2 Experimental prestressed ‘untested’ damping data 6-8 Table 7.1 Details of James’ (1997) reinforced concrete box beams 7-17Table 7.2 Deflection data for beam 5 (James, 1997) 7-17Table 7.3 Deflection data for beam 7 (James, 1997) 7-18Table 7.4 Deflection data for beam 16 (James, 1997) 7-18Table 7.5 Deflection data for beam 17 (James, 1997) 7-18Table 7.6 Deflection data for beam 18 (James, 1997) 7-19 Table 8.1 Details of Chowdhury’s (1997) reinforced concrete box beams 8-8Table 8.2 Deflection versus damping data for beam 5 (James, 1997) 8-9Table 8.3 Deflection versus damping data for beam 7 (James, 1997) 8-9Table 8.4 Deflection versus damping data for beam 16 (James, 1997) 8-9Table 8.5 Deflection versus damping data for beam 17 (James, 1997) 8-10Table 8.6 Deflection versus damping data for beam 18 (James, 1997) 8-10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
List of Plates xvi
List of Plates
Page
Plate 4.1 Photograph of beam test set-up 4-16Plate 4.2 Beam support system:
(a) Roller (b) Knife supports
4-17
Plate 4.3 Test set-up of oscilloscope during experimentation 4-22Plate 4.4 Crack width microscope 4-23
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Notation xvii
Notation
a = Shear span during testing (mm)
A0 = Initial displacement amplitude, or zero-frequency displacement
A1,2,3 = Displacement amplitude at the first, second and third cycle, respectively
A = Displacement amplitude, or displacement amplitude under forced-
vibration, or Proportionality coefficient (Dieterle and Bachmann, 1981)
Ac = Area of concrete in the tensile zone (mm2)
Agt = Uniform elongation of reinforcement sample (elongation at max. stress)
Amax = Maximum amplitude on the harmonic-response curve
An = Displacement amplitude after n number of cycles
Ap = Area of steel in the tensile zone (mm2)
Ast = Area of tension reinforcement (mm2)
Asc = Area of compression reinforcement (mm2)
b = Width of rectangular beam section (mm)
C = Viscous damping matrix in linear equation of motion
c = Viscous damping coefficient of a system, or concrete cover (mm)
ccr = Critical damping coefficient
co = Dimensionless material constant for hysteresis damping
ce = Equivalent viscous damping coefficient (considering hysteresis damping)
ceq = Equivalent viscous damping coefficient
d = Effective depth to the centroid of the tensile steel of a rectangular beam
section (mm), or specific damping coefficient (MPa) (Dieterle and
Bachmann, 1981)
D = Total depth of the rectangular beam section (mm)
Dm = Dynamic magnification factor evaluated at maximum amplitude (Amax)
e = Eccentricity of the prestressing tendons (mm)
Ec = Young’s modulus of elasticity of concrete (MPa)
ED = Dynamic modulus of elasticity of concrete (MPa)
Em = Young’s modulus of elasticity of mortar (MPa)
Ep = Dynamic modulus of elasticity (MPa) (Cole and Spooner, 1968)
Es = Young’s modulus of elasticity of steel (GPa)
f = Vibration frequency defined by the number of cycles per time unit, in Hz
(= 1/τ)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Notation xviii f1 = Fundamental natural frequency of vibration (Hz)
f’c = Characteristic compressive strength of concrete at 28 days (MPa)
f’cf = Flexural stress in tension (MPa)
fcm = Concrete compressive strength on the day of testing (MPa)
fD = Damping force developed during energy loss per cycle test
fD,max = Maximum damping force developed during energy loss per cycle test
fs = Stress in the steel (MPa), or static force applied during energy loss per
cycle forced vibration test
fs,max = Maximum static force applied during energy loss per cycle forced
vibration test
fsy = Nominal reinforcement yield strength (MPa)
F = Resisting force developed in a body due to hysteresis damping
Ff = Frictional damping force produced between a vibrating mass and surface
during Coulomb damping
H = Prestressing force in the member (kN)
I = Moment of inertia of a section (mm4)
Icr = Fully-cracked moment of inertia of the section (mm4)
Ief = Effective moment of inertia of a section (mm4)
Ig = Gross moment of inertia of the uncracked section (mm4)
k = Stiffness (also termed the spring force)
ku = Compressive stress block parameter defining the effective depth of the
member
K = Stiffness matrix in linear equation of motion, stiffness (Flesch, 1981)
L = Total beam length (mm)
m = Lumped mass of structure in an idealized system
M = Mass matrix in linear equation of motion, or age of beam specimen
(Cole, 1966)
Mcr = Bending moment at which first cracking is observed (kNm)
Ms = Service bending moment (kNm)
Mu = Ultimate bending moment capacity of the beam (kNm)
n = Cycle number from a free-vibration decay record
P = General load applied to the structure
0P = Maximum force amplitude at resonant frequency (Flesch, 1981)
Pcr = Load causing first cracking in the member (kN)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Notation xix Ps = Service load (kN)
Py = Load causing yielding of the tensile steel (kN)
Re = Yield strength of reinforcing steel (MPa) (Patrick, 1999)
Rm = Tensile strength of reinforcing steel (MPa) (Patrick, 1999)
s = Spacing of the reinforcing bars (mm)
st = Spacing of the tension reinforcing bars (mm)
sc = Spacing of the compression reinforcing bars (mm)
S = Represents a surface
t = Time (sec)
U = Steel peak stress on stress-strain curve (MPa)
vo = Initial velocity given to a mass
vmax = Maximum displacement during forced vibration
v& max = Maximum velocity during forced vibration
V = Absolute volume fraction of coarse aggregates (Swamy and Rigby, 1971)
wD = Area under the force-displacement curve during energy loss per cycle
forced vibration test
wi = Average instantaneous crack width (mm)
ws = Area under the static-force-displacement during energy loss per cycle
forced vibration test
Wr = Average residual crack width (mm)
x,y,z = Generalised response co-ordinates defining displacement
X = Evaporable water content (Cole, 1966)
0y = Initial displacement given to a body
0y& = Initial velocity given to a body
0y&& = Initial acceleration given to a body
yt = Distance between the neutral axis and the extreme fibres in tension of the
uncracked section (mm)
Y = Steel off-yield stress on stress-strain curve (MPa)
y& = Velocity of motion as a function of time
y&& = Acceleration of motion as a function of time
∆ = Generalised deflection (mm)
∆i = Mid-span instantaneous deflection (mm)
∆r = Mid-span residual deflection (mm)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Notation xx x = Mean of the data set
∆U = Area within a hysteresis loop representing the amount of energy
dissipated
Φ = Diameter of reinforcing bar (mm)
α = Compressive stress block parameter defining the effective width of the
member
αrc = Residual deflection curve coefficient for RC beams
αp = Loading constant for deflection calculation
αps = Residual deflection curve coefficient for PSC beams
αw = Loading constant for deflection calculation
β = Frequency response during a forced vibration experiment
β1, β2 = Frequencies at which the harmonic-response curve has reduced to 1/√2
βfl = Flexural damage function for the calculation of the ‘tested’ logdec
δ = Logarithmic decrement (logdec)
δcr = ‘Cracked’ damping capacity
δm = Damping capacity of the mortar
δtest = Damping capacity of the ‘tested’ concrete beam
δtotal = Total damping capacity for RC and PSC beams (in terms of logdec)
δuncr = ‘Uncracked’ damping capacity
δuntest = Damping capacity of the ‘untested’ concrete beam
εc = Ultimate compressive strain in the concrete at failure
φ = Phase angle of motion
γ = Effective depth of the compressive stress block
η = Effective prestressing coefficient
ϕ = Initial camber (upwards deflection) in a section due to prestressing (mm)
ρ = Density of concrete
ρt = Tensile reinforcement ratio (=Ast/bd) sometimes defined using, ρ
ρc = Compression reinforcement ratio (=Asc/bd)
σn-1 = Standard deviation of the data set
σe,max = Steel stress (Dieterle and Bachmann, 1981)
σp = Stress in the prestressing tendon (daN/cm2) (Hop, 1991)
σpi = Stress in the prestressing tendon immediately after transfer (MPa)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Notation xxi σpu = Ultimate stress in the prestressing tendon (MPa)
σbp = Flexural stress provided by prestressing (MPa)
τ = Period of time required to complete one cycle of motion
τd = Damped period of motion
ω = Undamped natural circular frequency or angular velocity of motion
(=2π f )
ωd = Damped circular frequency of motion
ξ = Damping ratio
ξVD = Viscous damping ratio (Dieterle and Bachmann, 1981)
ξFD = Friction (Coulomb) damping ratio (Dieterle and Bachmann, 1981)
ξ cr = Damping ratio for a cracked beam (Dieterle and Bachmann, 1981)
ξ un = Damping ratio for an uncracked beam (Dieterle and Bachmann, 1981)
ξe = Equivalent damping ratio for a hysteretically damped system
ξs = Material damping of concrete (Flesch, 1981)
ξv = Slip damping (Flesch,1981)
ξtotal = Total damping of an uncracked beam (Flesch, 1981)
ζ = Equivalent viscous damping ratio
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 1: Introduction 1-1
CHAPTER 1
Introduction
1.1 General Remarks
“All structures exhibit vibration damping, but despite a large literature
on the subject damping remains one of the least well-understood
aspects of general vibration analysis” Woodhouse (1998).
Damping is a broad topic that has been the subject of a wide variety of research efforts
over the years. Pioneering work was undertaken by Coulomb in 1784, who speculated
on the micro-structural mechanisms of damping in his ‘Memoir on Torsion’. Between
the period of 1784 and 1968 well over 2500 articles had been published with a scientific
or engineering interest in the damping of polymers, elastomers and other non-metallic
materials, with approximately four papers specifically devoted to concrete (Lazan,
1968). Since 1968, there has been a significant amount of concrete specific damping
research examining a very wide range of damping topics.
The introduction of high-strength concrete (currently in AS3600-2001 the compressive
strength is > 65MPa, but is constantly being revised with each new publication) and
steel reinforcement (a nominal yield strength of 500 MPa was introduced in Australia in
2000) has resulted in the design of more slender concrete members. In concrete column
design for example, significant savings in cross-sectional dimensions have been
achieved using high strength concrete whilst maintaining the same load capacity. Even
though reduced sections are seen as more cost effective some related side effects occur.
High-strength concrete beams with reduced cross-sections may be more susceptibility to
vibration and serviceability problems. This particular vibration response has not been
studied adequately by the literature and is therefore a focus of this thesis.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 1: Introduction 1-2 It is well known that damping capacity is, in some way, dependent on the level of
damage that exists in a concrete beam (Van Den Abeele and De Visscher, 2000).
Therefore, it has been common for researchers to adopt an expression relating damping
capacity to the cracking in a concrete beam. Dieterle and Bachman (1981) based
damping capacity on the level of stress in the reinforcing bars (derived from cracking
theory), whilst Chowdhury (1999) utilised a direct measure of residual crack width to
calculate damping. These previous studies are not, however, able to fully explain all the
experimentally observed damping characteristics. They do not indicate how damping
should be calculated prior to the beam cracking. This is particularly important for
prestressed beams that commence cracking at 70% of their ultimate capacity. Another
omission in previous work is the calculation of the basic inherent damping capacity of a
beam when it is first cast. This has been termed the ‘untested’ damping capacity and
along with ‘tested’ damping, forms the major focus of this research.
1.2 Research Objectives
There are two interrelated core objectives within this thesis, the investigation of the (1)
damping; and (2) residual deflection characteristics of reinforced and prestressed
concrete beams. To achieve both of these research objectives the following four items
will be addressed:
(a) The importance of technique for accurately extracting the logarithmic decrement
(a measure of damping capacity) from the experimental data;
(b) The identification and quantification of the experimental variables for the
development of a damping prediction formula for ‘untested’ reinforced and
prestressed concrete beams;
(c) The examination of the residual deflection characteristics of reinforced and
prestressed concrete beams in order to calculate ‘tested’ damping capacity. Even
though residual deflection implies the consideration of long-term effects, crack
patterns and stress relaxation etc., for the purposes of this thesis it relates
specifically to the experimental regime implemented; and
(d) Using residual deflection along with ‘untested’ damping predictions as a predictor
of total damping capacity for beams in-service.
1.3 Research Methodology
To achieve the aims and objectives, multiple stages of research are required. The initial
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 1: Introduction 1-3 stage is comprised of an extensive literature review, used primarily to highlight the
major gaps and omissions in previous damping research so that the framework for the
research could be established.
The experimental programme stage developed in response to the desired outcomes of
the research, which includes the development of the design equations in which a
substantial experimental database is required.
Following the experimental programme, an extensive analytical investigation is
undertaken. In this phase, a review of the difference between the damping capacities of
the various tested beam types is made and the proposed damping models incorporating
residual deflection developed and verified.
1.4 Layout of the Thesis
Following the introduction presented here, Chapter 2 presents a brief overview of
vibration and damping theory and also a literature review of the published concrete
damping research. Chapter 3 presents the development of the theoretical framework for
the prediction of the total damping capacity of concrete beams, that includes the
‘untested’ and ‘tested’ damping components.
Chapter 4 presents details of the extensive experimental programme carried out to
investigate the damping and deflection characteristics of reinforced and
prestressed concrete beams. A total of forty-one beams were tested over a period of
three years. The complete data on the geometrical and mechanical details of these test
beams, supports and loading systems, and test set-up are also given.
Chapter 5 examines how the logarithmic decrement is calculated from the experimental
free-vibration decay records. This Chapter arose because of the initial difficulties found
in calculating logarithmic decrement consistently and accurately.
Utilising the findings of Chapter 5, Chapter 6 is concerned with the determination of
the damping capacity of ‘untested’ concrete beams. The resulting ‘untested’ damping
capacity equation defines the starting point from which to calculate the total damping
capacity of a beam at any stage of its service life. This Chapter examines the dependent
variables for both reinforced and prestressed beams and presents proposed damping
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 1: Introduction 1-4 prediction equations.
Chapter 7 evaluates the calculation of the residual deflection characteristics of
reinforced and prestressed concrete beams. This is necessary because the prediction of
residual deflection is not well understood. Initial verification of the proposed equation
for the calculation of residual deflection is made using the experimental test results of
James (1997).
Chapter 8 presents the proposed equation for the calculation of the total damping
capacity for beams in-service using residual deflection along with ‘untested’ damping
capacity predictions. Verification of the equation, using test data and the published
experimental damping test data of Chowdhury (1999), is conducted.
The final Chapter, Chapter 9, summarises the main findings of the research, draws
conclusions and identifies the shortcomings. Recommendations are then made as to the
anticipated applications of the current findings and suggestions are made for further
research studies.
1.5 Summary
As initially highlighted by Woodhouse (1998) at the beginning of this Chapter, the
quantification of damping is perhaps one of the most vexing problems in structural
engineering. Unlike the unique physical properties of a structural system, such as
inertial and stiffness properties, which can be related to deflection and cracking,
damping is dependent upon a variety of features, such as component materials and
external influences. Damping has traditionally been treated as a relatively unknown
quantity, simply because it has been difficult to define and quantify.
This thesis provides essential experimental data and analytical proposals that act as the
foundation for further knowledge that will contribute to the meagre existing database on
the damping characteristics of reinforced and prestressed concrete beams. Such
proposals will benefit many areas of damping knowledge, including:
The development of a set of rules that will allow researchers to fully compare
their research to existing published research. Up to this point, this has not been
feasible because the style in which damping data has been reported is inconsistent;
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 1: Introduction 1-5 The development of a unique technique to simply and easily determine the
damping capacity of a structural concrete member. This is currently not available
to structural and civil engineers (Wilyman and Ranzi, 2001).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-1
CHAPTER 2
Damping in Concrete
2.1 General Remarks
All materials, members and structures, regardless of shape or function, are governed by
the same basic fundamental laws of motion. For vibration damping, these fundamental
theories have been well researched. Nevertheless, despite the extensive amount of
literature available on the subject, damping remains one of the least understood aspects
of general vibration analysis. The major reason proposed for this is “the absence of a
universal mathematical model to represent damping forces” (Woodhouse, 1998).
A discretely damped vibrating system with N degrees of freedom, executing small
vibrations about a stable equilibrium position, obeys the governing linear equation of
motion:
xyKyCyM =++ &&& (2.1)
where M, C, and K are the mass, damping and stiffness matrices respectively, y is the
vector of generalised response co-ordinates, and x is the vector of generalised forces
driving the vibration. Also, y and x are functions of time.
The modelling of the mass and stiffness matrices in Equation 2.1 is well established in
the literature, where, for example, the mass matrix for a highrise building would
conprise of the weights of each floor (Béliveau, 1976). However, as discussed by
Woodhouse (1998), in mathematically constructing the linear damping matrix required
by Equation 2.1, it is still not clear which variables the damping forces will depend on
i.e. viscous or Coulomb damping which will be discussed presently (Mo, 1994; Kareem
and Gurley, 1996). Furthermore, Kana (1981) discussed the difficulty in assigning
damping values for individual components in different parts of the structural system.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-2
Nonlinear behaviour is generally considered in the analysis of structures subject to
motions well into the inelastic range, such as during earthquakes, and as such is not
considered here (further information may be found in Tilley, 1986; Economou et al.,
1993; Fajfar et al., 1993; Jeary, 1996; Kunnath et al., 1997; Xiao and Ma, 1998).
Initially, vibration concepts will be introduced, as an understanding of these
fundamental laws is important to the development of the thesis. A discussion of the
methods of experimental determination of fundamental vibration and damping
quantities, is also presented. The Chapter concludes with a review of selected available
published concrete damping research. The bibliography provides a complete listing.
2.2 Undamped Systems
In an undamped single-degree-of-freedom (SDOF) vibrating system, the displacements
of motion about equilibrium are time-dependent and in the absence of attenuating
forces, will continue on forever. If these vibration repetitions are at regular time
intervals, the motion is called periodic, where a period, τ, is the amount of time required
to complete one cycle of motion. The frequency of vibration, f, is the number of cycles
per time unit (f = 1/τ). The frequency of vibration may also be discussed in terms of
number of radians per unit time, called the circular frequency or angular velocity of
motion, ω (where ω = 2π f). These various definitions may be seen in Figure 2.1. In
many structural engineering cases, motion is harmonic. Harmonic motions are naturally
periodic and may be described in terms of sine and cosine functions which allows for
mathematical simplicity.
2.2.1 Single-DOF and Multi-DOF Structures
The degree-of-freedoms (DOF’s) of a freely vibrating body are defined as the number
of independent co-ordinates that are required to identify it’s displacement configuration
during vibration. Each possibility of free-vibration is defined as a mode of vibration.
For example, the rigid block in Figure 2.2 may vibrate in six different ways: three
displacement translations (x, y and z coordinates) and three angular displacement
rotations (x, y and z axes). If the rigid block now moves only in the vertical direction
(SDOF), a single vertical coordinate y(t) is sufficient to completely define the position
of the mass m, with stiffness k, during vertical vibration.
Most structures, regardless of complexity, may be reduced to a form that readily allows
computations. Without this simplification, it would be practically impossible for an
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-3
engineer to conduct a dynamic analysis. Therefore, for an elastic body, such as the
simply supported beam modelled as a SDOF structure, shown in Figure 2.3, there are an
infinite number of modes of vibration and, subsequently, an infinite number of degrees
of freedom. That is, there are an infinite number of particles requiring an infinite
number of coordinates to determine the position of each particle during vibration. Thus,
in general, even the simplest of structures, such as the simply-supported beam, are in
reality multi-degree-of-freedom (MDOF) systems with an infinite number of DOF’s.
An important assumption involved in any SDOF approach is that one mode of vibration
dominates. This would be inappropriate for structures with high modal density, but for
the present beam problem it is satisfactory (Fahey and Pratt, 1998a).
τ = 2π/ω = 1/f
A
DisplacementResponse
y(t)
Time (t)
Motion described by:y(t)= A cos (ω t - φ )
φ/ω
φ =Phase Angle
φ
Figure 2.1: The Undamped Free-Vibration Response (Clough and Penzien, 1975)
kz
xm
y(t)y
kkz
xm
y(t)y
Figure 2.2: Freely Vibrating Body in the X, Y, Z Direction (Fertis, 1995)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-4
Deflected Shape = Fundamental Mode Shape= Fundamental Mode of Vibration
Figure 2.3: Example of a Structure Modelled as a SDOF System (Fertis, 1995)
The natural frequencies of a freely vibrating body are equal in number to its DOF’s and
there is a mode shape associated with each frequency. One important mode of vibration
is at the lowest frequency and is called the fundamental frequency of vibration. This
correlates to the fundamental mode of vibration, as shown in Figure 2.3. The
fundamental mode of vibration has the same shape as it’s deflected shape. In many
structural problems, the fundamental mode of vibration is of particular importance
because the amplitudes of vibration are the largest, and are often used to define the
rigidity of the structure. In a study of the response of tall buildings, with a structural
system consisting of frames, it has been shown that the fundamental mode contributes
about 80% of the total response (Penelis and Kappos, 1997). Since the rigidity of a
structure is a function of its free frequency of vibration, as stiffness (k) increases,
frequency (f) of vibration also increases.
2.3 Damped Systems
Damping is a collective term describing the non-conservative forces that act upon
bodies to resist motion. The amplitudes of a freely vibrating damped body are reduced
by the resisting force that is developed during the period of free-vibration. This
resisting force dissipates energy and in time the vibrations die out. Damping is present
to some degree in all structural systems, but its nature and magnitude are generally not
well understood (Irvine, 1986). If damping were not present, vibrations would never die
out, but continue on forever.
2.3.1 The Idealized MDOF System
When undertaking a dynamic analysis of beam and frame MDOF systems with
continuous mass and elasticity, such as that shown in Figure 2.4a, sufficient accuracy
may be obtained by using a simpler SDOF model, such as that shown in Figure 2.4b.
This is commonly termed the idealized system.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-5
P = Applied Force
EI
L/2
m
P = Applied Force
kc
a) b)L/2
m
∆
Figure 2.4: Beam System: a) MDOF Beam, b) Idealized SDOF Mass-Spring System
(Fertis, 1995)
In Figure 2.4b, the spring constant k represents the stiffness of the simply-supported
beam at its centre and c represents the damping force. The deflection, ∆ of the beam is
given by:
EIPL
48
3
=∆ (2.2)
where E is the Young’s modulus for the beam material and I is the moment of inertia of
the beam’s cross-sectional area about the neutral axis.
The equivalent stiffness, k, of the beam is the ratio of the applied load to the deflection
at the point of application of the load, giving:
3
48LEIkP
==∆
(2.3)
where P is the vertical load which produces a vertical displacement ∆ equal to unity.
The damping of the system c, in Figure 2.4b, may be mathematically modelled by one,
or a combination of the three primary types of damping: viscous, Coulomb and
hysteretic. Generally, one form dominates, thus allowing a reasonable analysis to be
undertaken (Tedesco et al., 1999). Each different type of damping is detailed below.
2.3.2 Viscous Damping
Viscous damping is considered to be proportional to the velocity of the oscillatory
motion. The equation of motion can be written by summing the forces shown in Figure
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-6
2.5a in the x-direction. Figure 2.5b is representative of the decaying vibratory
oscillations under viscous damping action. The resulting summation produces the
differential equation that describes the motion of this viscously damped system as:
0=++ kyycym &&& (2.4)
where and y are the acceleration, velocity and displacement of the body,
respectively.
yy &&& ,
The solution y(t) to Equation 2.4 takes the form of an exponential function, given by:
pteAty =)( (2.5)
where A and p are constants.
By substituting Equation 2.5 into Equation 2.4, followed by the cancellation of common
factors, the characteristic equation that describes the system is arrived at, namely:
02 =++ kcpmp (2.6)
The solution to this quadratic equation is given by the following two roots:
mk
mc
mcp −⎟
⎠⎞
⎜⎝⎛±−=
2
2,1 22 (2.7)
k m
a) Mechanical Models b) Decay Curve Characteristics
Exponential rate of decayAmplitude
ofVibration
Time
Viscous Damping
c
Displacement output, measuredfrom neutral position of spring
y
Force input= F
Figure 2.5: Decay of Free-Vibration Under the Assumption of Viscous Damping: a)
Mechanical Model; and b) Decay Curve Characteristics (Newland, 1989)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-7
The general solution to Equation 2.5 is, therefore, given by the superposition of the two
possible solutions given by Equation 2.8, namely:
tptp eAeAty 2121)( += (2.8)
where A1 and A2 are constants determined from the initial conditions of the vibratory
motion.
For example, solving for constants A1 and A2, at time t=0, with an initial displacement
of A0 and initial velocity of , gives the following solution: 0y&
)()( /)2/(2
/)2/(
1)2/(
22 tmkmctmkmctmc eAeAety ⎥⎦⎤
⎢⎣⎡ −−⎥⎦
⎤⎢⎣⎡ −− += (2.9)
where the factor e-(c/2m)t is an exponentially decaying function of time, which shows that
the damped vibratory motion has an exponentially decaying amplitude with time.
The final form of Equation 2.6 is dependent upon the sign of the expression under the
radical sign in Equation 2.7. It may either be zero, positive or negative. Where it is
zero, the case is called critical damping. This case will be considered first.
Critical damping is the value of the damping coefficient for which the system will not
oscillate when disturbed initially, but will simply return to the equilibrium position. A
useful definition is that it is the smallest amount of damping for which no oscillation
occurs in the free response (Clough and Penzien, 1975). This condition does not usually
occur in practice (Fertis, 1995).
The critical damping value, ccr is defined as the value of c in Equation 2.7 that makes
the algebraic sum of the terms under the radical equal to zero. Thus:
ωω kmkmccr
222 === (2.10)
where ω is the undamped natural frequency of vibration of the spring-mass system
(ω = km ), and the damping of the spring-mass system may now be specified in terms
of c and the damping ratio, ξ, given by (Buchholdt, 1997):
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-8
crcc
=ξ (2.11)
which may be further expressed by the following expressions (Fertis, 1995):
ωξ=mc
2 (2.12)
and
( 222
12
ξω −=⎟⎠⎞
⎜⎝⎛−
mc
mk ) (2.13)
In an overdamped system, the damping coefficient (c) is greater than the critical
damping (ccr), i.e. c>ccr. For this case, the term under the radical sign in Equation 2.7 is
positive and the solution may be determined directly from Equation 2.7. The motion of
critically damped and overdamped systems is not oscillatory. Figure 2.6 depicts,
graphically, the motion for both the critically and overdamped system. In the case of
the critically system, the curve would return to the neutral position more quickly
(Meirovitch, 1975).
y
Time, t
y(t)y&
Overdamped andCriticallyDamped SystemResponse
Figure 2.6: Free-Vibration Response of Critical and Overdamped Systems (Clough and
Penzien, 1975)
A graphical record of the response of an underdamped system, with initial displacement
A0 and zero initial velocity ( =0) and phase angle, φ, is shown in Figure 2.7. In an 0y&
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-9
underdamped system, motion is oscillatory, but not periodic, and the amplitude of
vibration is not constant, but decreases exponentially for each successive cycle. The
oscillations occur at equal intervals of time. This is termed the damped period of
vibration (τd) and is defined by:
21
22
ξω
πω
πτ−
==d
d (2.14)
where ωd is the damped circular frequency.
t = time
Amplitude ofVibration
τd
A0 e -ξ ω t
A2
A1
t1 t2+τd
A0
A0 sin φ
φ
Figure 2.7: Free-Vibration Response of an Underdamped System (Fertis, 1995)
In an underdamped system, the damping coefficient (c) is less than the critical damping
(ccr), i.e. c<ccr. For this case, the term under the radical sign in Equation 2.7 is negative
and the solution is given by complex conjugates, so that:
2
2,1 22⎟⎠⎞
⎜⎝⎛−±−=
mc
mki
mcp (2.15)
where i = √-1 is the imaginary root.
Substituting the roots from Equation 2.15 into Equation 2.8 will give the general
solution of the displacement of the underdamped system, given by:
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-10
)sincos()( )2/( tBtAety ddtmc ωω += − (2.16)
where A and B are constants of integration, and the damped frequency of the system
(ωd) is given by:
2
2⎟⎠⎞
⎜⎝⎛−=
mc
mk
d ωω (2.17)
Substituting the expression for the undamped natural frequency (Equation 2.13) gives:
21 ξωω −=d (2.18)
In real structures, the damping coefficient (c) is generally much less than the critical
damping coefficient (ccr), thus indicating an underdamped condition, usually of the
order of 2 to 10% of critical damping. At 10% of critical (i.e. ξ = 0.10), Equation 2.18,
gives the following:
ωd = 0.995ω (2.19)
Thus Equation 2.19 indicates that the frequency of vibration for a system with as much
as 10% of critical damping is essentially equal to the undamped natural frequency.
Therefore in practice, the natural frequency of a damped system, ωd is taken as being
equal to the undamped natural frequency, ω (Paz, 1997).
In practice, the effect of damping on the natural frequency is ignored, and the
underdamped motion is described by the following (Irvine, 1986):
⎟⎠⎞
⎜⎝⎛=
δπξ 1ln
21n
(2.20)
where δ is the logarithmic decrement (an alternative measure of damping discussed in
Section 2.4.1) and n is the number of free vibration cycles.
For example, if after 10 cycles of free vibration (n = 10), the effect of damping has
reduced the peak displacement to 50% of its original height, the percentage of critical
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-11
damping is:
( ) %1011.02ln20
1≈==
πξ (2.21)
2.3.3 Coulomb Damping
Coulomb damping, also known as dry friction damping, is the result of rubbing and
sliding between vibrating dry surfaces. It assumes a frictional damping force, Ff is
produced between the mass m and the surface S (see Figure 2.8a). It is constant in
magnitude but changes sign according to the sign of the vibrational velocity.
A freely vibrating system subject to pure Coulomb damping may be diagrammatically
represented by Figure 2.8, which shows the linearly decaying amplitude of a Coulomb
damped system. The ‘linear-decay’ property of vibrating systems exhibiting pure
Coulomb damping was first found by Lorenz (1924), and Malushte and Singh (1987)
discussed the mechanism as found in a single storey frame building.
Coulomb / Friction Damping
k m
a) Mechanical Model b) Decay Curve Characteristics
Linear rate of decay
Amplitudeof
Vibration
Time, t
t=2π/ω
4Ff /k
Ff /k
y
Ao
Ff
S
A0
Figure 2.8: Decay of Free-Vibration under the Assumption of Coulomb Damping: a)
Mechanical Model; and b) Decay Curve Characteristics (Newland, 1989)
In Coulomb damping, motion ceases when the amplitude is less than Ff/k and the spring
force k is no longer able to overcome the static friction force. Coulomb damping is
harmonic, and the frequency of oscillation (f) is the same as the free undamped
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-12
frequency of the spring-mass system. The natural period of vibration, τ is unchanged by
this form of damping. The equation of motion of the free body diagram in Figure 2.8a
is given by:
00 >−=+<=+ yFykymandyFykym ff &&&&&& (2.22)
The mathematical solution to Equation 2.22, that describes the displacement y at time
t=2π /ω, is given by (Figure 2.8b):
kF
Ay ft
40/2 −== ωπ (2.23)
where A0 is the initial displacement given to the mass m, k is the spring stiffness, and Ff
is the resisting force which is produced from the friction between the mass m, and the
surface S (assumed to remain constant).
2.3.4 Hysteretic Damping
Hysteretic damping, also known as solid or structural damping, is generally attributed to
internal material friction created during motion. These frictional forces develop
between material matrix planes that slip, relative to one another during oscillatory
motion. This type of damping is considered to be independent of the frequency of
vibration, but approximately proportional to the amplitude of the deformed elastic body.
Figure 2.9 shows the time-history of an oscillating, purely hysteretically damped
system: the rate of reduction of its amplitude, A, depends on the size of the area within
the hysteresis loop (Figure 2.10).
Hysteresis damping causes a reduction in amplitude depending on the size of the area
within the hysteresis loop. Rubber materials have loops containing a much larger area
when compared to metallic materials, like steel. This is the reason why artificial
dampers use rubber-like materials.
Considering one cycle of motion of a vibrating single-degree-of-freedom spring-mass
system (Figure 2.9) and plotting the force-displacement diagram shown in Figure 2.10
(F is the resisting force developed in the body due to hysteresis damping and A is the
amplitude of displacement), the area within the loop represents the amount of energy,
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-13
∆U, transformed into heat, per cycle of motion, due to the internal friction in the
material.
Experiments have shown that ∆U (from Figure 2.10) could be obtained from the
approximate expression (Fertis, 1995):
2AckU oπ=∆ (2.24)
where co is the dimensionless constant of the material for solid damping, and k is the
force required to deflect the spring by an amount equal to unity.
k m
F
a) Mechanical Model b) Decay Curve CharacteristicsAmplitude
of Vibration
Time, t
Hysteretic Damping
2π
A1A2
A3
Figure 2.9: Decay of Free-Vibration under the Assumption of Hysteretic Damping: a)
Mechanical Model; and b) Decay Curve Characteristics (Newland, 1989)
O
A A
F =Force
y(t)
∆U is areawithin hysteresisloop
Figure 2.10: Force-Displacement Hysteresis Loop (Fertis, 1995)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-14
The amount of hysterestis damping in engineering structures is generally very small and
is usually neglected in an analysis (Fertis, 1995). However, if hysterestic damping is to
be considered then the damping, ξ, is termed the equivalent viscous damping ratio for
hysteretically damped systems, ξe. For the SDOF mass-spring system shown in Figure
2.9a, the equivalent damping ratio, ξe is given by the expression:
2o
ec
=ξ (2.25)
where co is the hysteresis damping coefficient, and for small hysteresis damping is
defined by the approximate expression (refer to Figure 2.9b):
ocAA
π+≈ 13
1 (2.26)
The equivalent viscous damping coefficient, ce, may be determined from:
ωξ
kckmccc o
oece === (2.27)
and the time-displacement response y(t) of a hysteretically damped system is:
( )[ ]tAety ete 21sin)( ξωωξ −= − (2.28)
where it is possible to assume, without appreciable loss of accuracy, that the undamped
and damped frequencies are the same (ω = ωd = (k/m)½) (Fertis, 1995).
2.3.5 Equivalent Viscous Damping
The true damping characteristics of typical structural systems are very complex and
difficult to define. The original theory (based on hysteretic damping discussed in
Section 2.3.4) to describe internal damping was first proposed by Kelvin (1865). It was
later shown that the theory was not universally applicable to all solids and, since then,
researchers have been unable to develop a theory to describe solid damping for general
applications (James et al., 1964). This is because, even though structural systems
exhibit a combination of linear (damping independent of amplitude) and non-linear
damping elements, (damping amplitude dependent, i.e. damping increases as motion-
generated stress levels increase) (Irvine, 1986)), it is material type, material history,
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-15
environment and test conditions that provide enormous complications to the
development of a ‘general’ theory to describe the dynamic response of vibrating
materials. In other words, all vibrating structures cannot be modelled by a single
mathematical equation.
In general, viscous damping is the most conducive to mathematical formulation (due to
its velocity dependence) and it is generally employed in a vibration analysis. Even
when researchers know that viscous damping is not operating, i.e. hysteresis or
Coulomb damping actually exists, an ‘equivalent’ viscous damping is assumed
(Tedesco et al., 1999). This is when the linear equations for viscous damping are
simply adopted for a vibration analysis, and thus the ‘equivalent viscous damping’
concept was introduced to allow the evaluation of the internal damping (of any kind) of
a member. Under these circumstances, structural damping is defined as the equivalent
viscous damping ratio, ζ (Sun and Lu, 1995).
It has been discussed within the literature that researchers tend to use damping
terminology interchangeably and indiscriminately without specifying whether pure
viscous damping exists, or whether equivalent viscous damping has been assumed. A
significant amount of confusion has thus been created, and has contributed to some of
the general ‘misdirection’ with regards to damping research (Jeary, 1996).
2.4 Experimental Determination of Damping
In undertaking a dynamic response analysis of a SDOF structure it is assumed that the
physical properties of the system (mass, stiffness and damping) are known. In the
majority of cases, structural mass and stiffness can be evaluated rather easily, usually
through simple generalized mathematical equations. On the other hand, the basic
energy-loss mechanisms that exist in practical structures are seldom fully understood.
Consequently it has not been traditionally feasible to determine the damping coefficient
by means of a generalized mathematical damping expression (Clough and Penzien,
1975). For this reason, the damping in most structural systems must be evaluated
directly from experimental tests. As these measures will be discussed within the
following Chapter, a brief survey of the principal procedures for evaluating damping
from experimental measurements follows.
2.4.1 Free-Vibration Damping
Logarithmic decrement, δ (generally shortened to logdec), is a measure of the rate of
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-16
decay of free-vibration. It is probably the simplest and most frequently used
experimental damping technique (Clough and Penzien, 1975; Kummer et al., 1981).
The original idea of utilising the free-vibration decay of a viscously damped system was
proposed by Helmholtz (1877), who used it to determine frequency information from
musical tones. It was, however, Rayleigh (1945) who coined the term ‘logarithmic
decrement’ for the estimation of the damping. The equation that analytically describes
the damped free-vibrating system represented by the response in Figure 2.7 is:
( )φωωξ += − teAty dt sin)( (2.29)
where Equation 2.29 represents pseudoharmonic motion with an exponentially decaying
amplitude (Ae-ξωt) and a phase angle (φ). The damped period of vibration (τd) may be
obtained from Equation 2.14. Since the amplitude (Ae-ξωt) depends on the damping
ratio (ξ), the rate of decay of the amplitude therefore depends on the amount of damping
in the system.
Using the resultant of Equation 2.29, logdec (δ) can be calculated from the ratio of
amplitudes several cycles apart (refer to Figure 2.7). If A2 is the amplitude n number of
cycles after the initial amplitude A1, logdec (δ) may be determined experimentally from
the traditional logdec technique (TLT) equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2
1ln1AA
nδ (2.30)
A major advantage of the traditional logdec technique (TLT) is that equipment and
instrumentation requirements are minimal; free-vibrations can be initiated by many
convenient and appropriate methods, i.e. simple vibration hammer or from ambient
excitations (traffic) and only relative displacements need to be measured and recorded
(Ibrahim and Mikulcik, 1977). Furthermore, when the structure is set into free-vibration
by a shock load, the fundamental mode dominates since all the higher modes are
damped out quite quickly. It is not usually possible to excite any mode other than the
fundamental mode using this method (Beards, 1996).
2.4.2 Forced Excitation Damping
Forced vibration techniques rely on observations of the steady-state harmonic response
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-17
behaviour of a SDOF structure, and thus require a means of applying harmonic
excitations to the structure at specific frequencies and amplitudes. A frequency-
response curve for the structure can be constructed from the application of a harmonic
load at a specified sequence of frequencies (that span the resonant frequency response
range), whereby the resulting displacement amplitudes can be plotted as a function of
the applied frequencies (as shown in Figure 2.11). The three main methods of harmonic
excitation are described below.
2.4.2.1 Half-Power (Bandwidth) Method
One of the most convenient and utilised methods is the half-power, or bandwidth
method, whereby the damping is determined from the frequencies at which the
maximum response (Amax) of the general harmonic-response curve shown in Figure
2.11, is reduced to (1/√2). These points correlating to β1 and β2 are indicated in Figure
2.11. Provided that the damping is small, and the natural frequencies are sufficiently
separated, and if the excitation frequency is close to a natural frequency, only one mode
will dominate the response (assuming that the point of excitation does not happen to be
close to a node for the mode in question) (Newland, 1989).
The equivalent viscous damping ratio is given by half the difference between the half-
power frequencies:
( 1221 ββζ −= ) (2.32)
This technique helps to avoid the need for determining the static response, but it does
require the response curve to be plotted accurately in the half-power range and at
resonance (Figure 2.11).
2.4.2.2 Resonant Amplification
From Figure 2.11, the equivalent viscous damping ratio (ζ) may also be evaluated from
the expression:
m
o
DAA
21
21
max
=≅ζ (2.33)
where Dm is the dynamic magnification factor at resonance, Amax (Clough and Penzien,
1975).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-18
Ao
2Ao
3Ao
Amax
√ 2
β2 - β1 = 2ζ
Amax= Ao 2ζ
0 1 2β2β1
Frequency ratio, β
Har
mon
ic re
spon
se a
mpl
itude
~
.Ao = Zero-frequency or static displacement
Figure 2.11: Frequency Response Curve for Moderately Damped System (Clough and
Penzien, 1975)
In practice, the equivalent viscous damping ratio, ζ, is determined from the dynamic
magnification factor Dm evaluated at maximum amplitude (Amax), namely:
0
max
AA
Dm = (2.34)
where Amax is the maximum response amplitude, and A0 is the zero-frequency or static
displacement (see Figure 2.11).
This technique has the advantage of requiring only simple instrumentation capable of
vibrating a structure over a range of frequencies that span the resonant frequency, and a
simple transducer capable of measuring relative displacement amplitudes. However,
evaluation of the static displacement can be difficult because many types of loading
systems cannot operate at zero frequency and it can be difficult, in practice, to apply a
static lateral load to a structure (Paz, 1997).
2.4.2.3 Energy Loss per Cycle
If instrumentation is available to measure the phase relationship between the input force
and the resulting displacements, damping can be evaluated from tests run only at
resonance, and there is no need to construct the frequency-response curve as given in
Figure 2.11. The procedure involves establishing resonance by adjusting the input
frequency until the response is 90o out of phase with the applied loading. Then the
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-19
damping force (FD) exactly balances the applied load, so that if the relationship between
the applied load and the resulting displacements is plotted for one loading cycle, as
shown in Figure 2.12, the result can be interpreted as the damping-force-displacement
diagram.
DampingForce =fD
Area=wD
Ellipse (viscous damping)(Equivalent area=wD)
Displacement
Amax
Figure 2.12: Actual and Equivalent Damping Energy Per Cycle (Clough and Penzien,
1975)
If the structure has linear viscous damping, the curve will be an ellipse, as shown by the
dashed line in Figure 2.12. In this case, the damping coefficient can be determined
directly from the ratio of the maximum damping force (fD,max) to the maximum velocity
( ) where the maximum velocity is given by the product of the frequency, ω and
displacement amplitude, A:
maxv&
max
max,
vf
c D
&= (2.35)
If the damping is not linear viscous, the shape of the force-displacement diagram will
not be elliptical, such as the solid line as shown in Figure 2.12, and the equivalent
viscous-damping coefficient is then given by:
2maxv
wc D
eq&πω
= (2.36)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-20
In most cases, it is more convenient to define damping in terms of the critical damping
ratio rather than the damping coefficient. For this purpose, it is necessary to define a
measure of the critical damping coefficient of the structure. This is expressed in terms
of stiffness (k) and frequency (f):
ωk
ccr2
= (2.37)
This expression is more convenient because the stiffness of the structure can be
measured by the same instrumentation used to measure the damping energy loss per
cycle, merely by operating the system very slowly at essentially static conditions. The
static-force-displacement diagram, obtained in this way, will be of the form shown in
Figure 2.13, if the structure is linearly elastic, then the slope of the curve represents the
stiffness (Clough and Penzien, 1975).
1k
Area=ws
Displacement
StaticForce
fs
fs,max
Amax
Figure 2.13: Elastic Stiffness and Strain Energy (Clough and Penzien, 1975)
Alternatively, the stiffness may be expressed by the area under the force-displacement
diagram, ws, as follows:
2
2Aw
k s= (2.38)
Thus the equivalent viscous damping ratio, ζ, can be obtained by combining Equations
2.36 to 2.38:
s
D
c ww
cc
πζ
4== (2.39)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-21
2.5 Literature Review of Damping in Concrete
In the past it has been extremely difficult to develop equations to predict the level of
damping in real-life structures, and so designers were generally only able to conduct a
dynamic analysis once the structure had been built using non-destructive testing (NDT).
Despite the wealth of damping research undertaken in a huge variety of research areas,
there remains to be found: (a) a consensus regarding basic damping levels in reinforced
and prestressed concrete beams; (b) clarification of the mechanisms that affect in-
service damping; and (c) an accurate prediction methodology to estimate real-life
damping levels in concrete beams at any stage of their service life. In this review,
concrete damping research has been broadly classified into the three main categories of
material, member and structure damping.
2.5.1 Material Damping
Jones (1957) investigated the effect forced frequency of vibration has on the dynamic
modulus and damping coefficient of concrete cylinders. The results implied that
damping arises from the interfacial boundaries in concrete. Contrary to Kesler and
Higuchi (1953), damping was found not to depend on the frequency of vibration.
Cole and Spooner (1965) also examined the effect of forced vibration at very low
frequencies on the damping capacity of small rectangular cement paste beams and found
that damping capacity is influenced by variations in both the frequency and stress
amplitudes of vibration.
Cole (1966) measured the damping capacity of small rectangular cement paste beams
vibrated at low stress amplitudes where the variation of logdec was attributed primarily
to changes in age and water content. An approximate value for the logarithmic
decrement, δ, was given as
δ = a + bX – (c + dX) loge M (2.40)
where X is the evaporable water content, M the age of the specimen in months, and the
constants a, b, c and d have values of, a = 0.026, b = 0.022, c = 0.0039 and d = 0.0030.
Jones and Welch (1967) described a series of free-vibration decay experiments to
determine the vibrational damping coefficient of three types of plain concrete and
corresponding sand/cement mortars. No appreciable difference in damping could be
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-22
detected with change in frequency, size of specimen or method of measurement (i.e.
forced versus free-vibration experimental techniques).
Swamy and Rigby (1971) experimentally investigated the dynamic moduli (Young’s
Modulus) and damping properties of hardened pastes, mortars and concrete prisms.
Equation 2.47 was proposed for the determination of logdec for concrete in the dry
state, undergoing forced flexural vibration.
cmmc
VEE
0000913.0265.010640.01113.00174.0 −+−+= δδ (2.41)
where Ec and Em are the dynamic modulus of concrete and mortar (psi) respectively, δm
is the logdec of the mortar matrix, and Vc is the absolute volume fraction of coarse
aggregates.
Spooner and Dougill (1975) developed an analytical technique to quantitatively describe
the extent of damage experienced by a concrete specimen in compression. It was found
that, at low strains, logdec is approximately 0.09 and that the energy dissipated in
damage correlates to the change in initial elastic modulus during initial loading.
Ashbee et al. (1976) described an experimental method which purported to overcome
the inherent problem of frequency and load control commonly encountered in the
experimental study of damping. The logdec of concrete was found to range from 0.03
to 0.15, with an average value of about 0.05.
Spooner et al. (1976) suggested that damping depends specifically on the strain range
and that it is independent of the degree of damage exhibited by the specimen. This
result supported the findings of Swamy (1970). Thus, for cement paste prisms, the
relative contribution to damping by sliding friction between cracks (degree of damage)
and movement of capillary water was insignificant.
Using forced vibration techniques, Jordan (1980) studied the effects of stress (amplitude
and rate of vibration), frequency of vibration, and curing regime and age, upon the
damping of concrete cylinders. The presence of microcracks was found to be of great
importance in determining the damping capacity.
Sri Ravindrarajah and Tam (1985) investigated the use of recycled concrete as coarse
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-23
aggregates in concrete. For both normal and recycled aggregate concrete, damping
increased with a decrease in compressive strength. For all grades of concrete, those
with recycled concrete aggregates had higher levels of damping capacity.
Fu and Chung (1996) undertook experimental investigations into the effect of latex,
methylcellulose, silica fume and short fibres on the damping capacity of cement pastes.
It was found that the damping capacity of cement pastes is increased by the addition of
silica fume, latex, methyl-cellulose or short fibres.
The dependence on frequency of the damping capacity of hardened cement paste in the
temperature region of –900C and forced vibration frequency was studied by Xu and
Setzer (1997). The temperature dependence of damping was attributed to the interaction
between the pore ice and internal solid surface.
Furthering the experimental work of Fu and Chung (1996), Fu et al. (1998) investigated
the use of additives to improve the vibration damping capacity of cement. Silica fume
and latex were found to increase damping capacity by up to 390%. Methylcellulose was
found to only marginally affect the damping capacity. Li and Chung (1998) found that
the damping capacity at all temperatures and frequencies was increased by the surface
treatment of the silica fume. This increase was up to 300% in some cases.
Wang and Chung (1998) studied the effect of the addition of sand and silica fume on the
damping capacity of cement mortars. It was found that the loss modulus was decreased
greatly by the addition of sand, but the addition of silica fume increased the loss
modulus to a level comparable to, or exceeding that, of plain cement paste.
Orak (2000) investigated the damping capacity of polymer concrete. The damping
capacity of polymer concrete was found to be four to seven times larger than a
corresponding cast iron sample (it was not compared to that of normal concrete).
From the preceding discussion, it can be seen that a wide range of very different
damping variables have been identified and explored within the literature. For this
reason a summary table has been developed to attempt to extricate the most important
topics (Table A.1 may be found in Appendix A). The complete summary table is not
presented here due to its complexity, rather a point summary of the most important
conclusions, are outlined below.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-24
(1) State of Material: In the early weeks of a beam’s life, damping is affected
considerably by hydration. Hydration has, within the literature, been termed as
moisture content, degree of hydration, curing conditions and age. Typically, it has
been well established that as the beam dries out, damping is reduced. After 28 days,
damping becomes stable. These findings do not hold significant implications for the
current experiments and indeed for general concrete construction because concrete
specimens are generally subjected to 28 day minimum curing periods. Interestingly,
recent work by Almansa et al. (1993), who investigated the effect of the early
unforming of concrete beams found that damping was not significantly affected by
unforming age (between 2 and 90 days).
(2) Concrete Composition: Generally, with respect to concrete strength (f’c) and the
elastic or dynamic modulus of elasticity (ED and Ec, respectively), an inverse
relationship with damping is suggested, but not conclusively (Kesler and Higuchi,
1953; Jones and Welch, 1967; Swamy and Rigby, 1971; Sri Ravindrarajah and Tam,
1985). For the effect of the various aggregates and additives, previous research has
not been conclusive as to the nature of their effect on damping.
(3) Testing Procedures: An extremely wide variety of testing procedures has been
employed. For this reason, there has been an equally wide variety of testing results
and conclusions. The interdependency between frequency and damping has been
discussed, with tests showing damping as ‘very dependent on frequency’ (Kesler
and Highuchi, 1953; Cole and Spooner, 1965) to ‘not dependent on frequency’
(Jones and Welch, 1967; Jordan, 1980).
(4) Cracking Mechanisms: The effect of cracking on damping capacity has been of
interest to researchers as it gives an indication of the internal structure. Jones and
Welch (1967), Swamy (1970) and Spooner et al. (1976) have stated that from tests,
sliding friction within the solid gel structure (hysteretic damping) is unimportant,
whereas the primary energy loss is due to moisture movement within the solid
structure (viscous damping). Jordan (1980) strongly indicated that the component of
damping from microcracking (Coulomb damping) is by far the most important
damping mechanism. Also the bonding between the different types of aggregate-
cement pastes under load was investigated by Sri Ravindrarajah and Tam (1985)
who examined the suitability and resiliance of recycled-aggregate concretes.
2.5.2 Member Damping
The following discussion reviews damping research on reinforced and prestressed
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-25
concrete members. A complete summary of published research on concrete member
damping is presented in Tables A.2 and A.3 (Appendix A).
Bock (1942) undertook forced vibrations experimental tests and found that damping
was independent of forced vibration amplitude and was higher in beams not containing
reinforcement.
James et al. (1964) tested beams that were subjected to forced vibrations at various
stages of cracking in order to examine flexural rigidity and damping properties. They
found that the dynamic response of reinforced concrete beams was significantly affected
by test history. James et al. (1964) also undertook forced vibration experimental tests
on six prestressed concrete beams. It was found that the damping characteristics of
prestressed and reinforced concrete beams were generally the same. Test history or
cracking did not affect the dynamic response characteristics.
In his reinforced concrete experimental investigation, Penzien (1964) found that for
both forced and free vibration, damping increased with an increase in cracking. Penzien
(1964) also undertook free-vibration experiments on prestressed concrete beams and
found that damping was affected primarily by the cracked state of the beam, thus the
degree and type of prestress only indirectly influenced damping by influencing the
cracking of the beam.
In 1974, Wills’ finding that the value of logdec regularly being used for chimney design
only applies to small amplitudes of vibration whilst for large vibration amplitudes,
tensile stresses are induced and the damping increases by a factor of five to ten. From
an experimental investigation, Jordan (1977) found that the material damping did not
increase as tensile stresses were induced.
Dieterle and Bachmann (1981) developed theoretical models to explain the damping
behaviour exhibited by reinforced concrete beams in the uncracked and cracked states.
The theoretical damping models are shown in Figure 2.14. Forced vibration
experimental investigations were undertaken from which the curves given in Figure
2.15 were developed. The curves show that the damping, as a result of cracking, can
tend towards a very small value after an initial growth. For the uncracked beam the
damping due to viscous damping was described by
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-26
⎥⎦
⎤⎢⎣
⎡= 2
02 cm
cunVD fC
Edπ
ξ (2.42)
( )c
ect E
EnH
hH
HhnC =
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛ −−+=
2
0
2
'231 ρρ (2.43)
For a cracked beam element, total damping, ξ cr
was specified as the sum of two
components, viscous damping, ξVD, and friction damping, ξFD
⎥⎥⎦
⎤
⎢⎢⎣
⎡×+⎥
⎦
⎤⎢⎣
⎡=+= 2
max,
max,
2
2
21
62 e
emvtc
cm
ccrFD
unVD
cr VBhCAjEn
fCEd
σσφτρ
πξξξ (2.44)
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛ −+= 3
2
3
2
1''331
Na
Nac
Na
Nat x
hxhnxxhhnC ρρ (2.45)
( )
2
2
3
2'
3 ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−++
−=
Na
Nact
Na
Na
xhhxnn
xhhxC ρρ (2.46)
where fcm is concrete compressive strength (MPa), d is a specific damping coefficient
(MPa) that depends on the material and must be determined experimentally, σe,max is the
steel stress (MPa), Ec is the modulus of elasticity for concrete (MPa), A is a
proportionality coefficient, and h, H, h’, xNa, are defined in Figure 2.14.
The reinforcement ratio is considered in the coefficient Co, but its influence on damping
appears to be inversely proportional to damping capacity. Interestingly, despite the
development of Equation 2.42, the viscous damping component (damping ratio, ) of
normal concrete beams was established as 0.006 (δ = 0.0377) in virtually all test data.
This suggests that the development of a prediction formula was unnecessary.
unVDξ
Equations 2.42 and 2.44 are the only known attempt to theoretically derive a damping
prediction equation for the uncracked beam state. However, they are complicated and
would be difficult for practitioners to use, also their validity was not proven.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-27
a)
UNCRACKED BENDING ELEMENT
h
h’
HM M
Fe’=µ’ b h
Fe=µ b h
s
Zb
D
Ze
Bending Element Model
TensionZone
CompressionZone
k
mz
ViscousDamping
b)
CRACKED BENDING ELEMENT
xNA
h’
h
M M
Fe’=µ’ b h
Fe=µ b h
s
D
Z
Bending Element Model
CompressionZone
k
mz
ViscousDamping
Friction
Damping
)(,
eFNµ
Figure 2.14: Damping for the: a) Uncracked; and b) Cracked Beam Sections (Dieterle
and Bachmann, 1981)
crFD
unVD
cr ξξξ +=
ξ
σe, relTP1: Initial stressing & crackingTP2: After cracking & repeat of test
ξunVD
=0.0
06
TP1=Test Phase 1Viscous Damping (ξun
VD) Friction Damping (ξcr
FD)
TP1TP1
TP1
TP2
TP2
ξπδ 2=
Figure 2.15: Damping Ratio as a Function of the Relative Steel Stress (Dieterle and
Bachmann, 1981)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-28
Flesch (1981) developed a mathematical model to predict the damping capacity of a
reinforced concrete cantilever beam element during one cycle of loading for both the
uncracked and cracked condition. This cantilever element is shown in Figure 2.16. The
total damping of the uncracked beam, ξtotal may be calculated from
ξtotal = ξv + ξs (2.47)
where ξv = slip damping and ξs = material damping of concrete (material damping of
steel neglected below yield point).
In addition to material damping, the hysteretic damping caused by the slip between steel
and concrete was derived using the equivalent viscous damping mechanism for the
uncracked section, it is given by the iterative formula
02
48 20
2
1
6
513
6
32 =⎟⎟
⎠
⎞⎜⎜⎝
⎛+− −−− n
nn
c
n Pclc
hh
hbEhl ξπξ (2.48)
where l, b and h are defined in Figure 2.16 and are in m, Ec is the modulus of elasticity
for concrete (kN/cm2), n = material constant evaluated from tests, Po = maximum force
amplitude at resonant frequency, and h5 , h6 , c1 and c2 are constants described by
additional equations.
my tP ωsin0
h
l
AA
xE E
Cross-Section A-A
Uncracked Cracked
h
εc
b
εc
εs
y
b
Figure 2.16: SDOF RC Cantilever Beam During One Loading Cycle (Flesch, 1981)
The prediction equation for the equivalent damping ratio with cracks is given by
Equation 2.49:
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-29
312
2
23
23
/44
vvpvK
pv
pv
Kp
vp
ooooo −⎥
⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −−⎟⎠
⎞⎜⎝
⎛ −=ππ
ξ (2.49)
where K = stiffness given by 3EI/l3 and I = bh3/12 and v1, v2 and v3 are constants
described by additional equations.
The equations of Flesch (1981) are similar in concept to Dieterle and Bachmann (1981)
and as such, are complicated and would be difficult for the practitioner to implement.
Furthermore, the equations by Flesch (1981) were not verified using experimental tests,
rather the equations were utilised to undertake a parametric study.
Askegaard and Langsœ (1986) experimentally investigated damping measurements to
determine the extent of damage and deterioration in concrete caused by the freeze-thaw
cycle. Using both free- and forced-vibration, it was found that the formation of cracks
in the test beams approximately doubled the damping capacity of the freeze-thaw
resistant beams from 0.04, to 0.07 in the fully-cracked beam.
Hop (1991) found that an increase in prestressing, σp (daN/cm2), caused a considerable
decrease in the logarithmic decrement, δ, according to the following
325 00109.01316.023.18945010 ppp σσσδ −+−= (2.50)
Almansa et al. (1993) undertook free-vibration experimental tests to determine the
effectiveness of using natural frequency and modal damping factors to determine the
degree of cracking of early stripped reinforced concrete beams. Most significantly, they
found that damping was not influenced by age of stripping or by the level of cracking
found in the beams.
Wang et al. (1998) studied the free-vibration dynamic behaviour of reinforced concrete
beams and found that when initial cracks first develop resonant frequency could be 20 –
25 % greater than the original value and damping (dB/ms) could change by a factor of
four. The empirical formula to estimate the experimentally observed damping
behaviour of the progressively cracked beam in Figure 2.17 was given by
( ) una
crcr M
M δααδβ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 122 (2.51)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-30
where δcr is the damping of a cracked beam; α2 = 4; the constant β is between 0.7 to 0.8
but was not described; δun is the decay rate of an uncracked beam; Mcr and Ma are the
cracking and applied bending moments in Nm.
The primary difficulty with Equation 2.51 is that it does not describe how to determine
the uncracked damping capacity (δun) and would therefore be difficult to use for design
purposes.
D
ecay
rate
in d
B/m
sec
0
0.2
0.4
0.6
0.8
0 50 100 150 200Bending moment in Nm
Proposed curve to fitexperimental damping data(δcr)
δcr is the decay rate ofthe beam
Figure 2.17: Damping in Beams Without Load as a Function of the Maximum Load
(Wang et al., 1998)
Chowdhury (1999) undertook free-vibration experiments on full-scale box beams. The
research suggested that the level of cracking that exists in a reinforced or partially
prestressed concrete beam is of primary significance in the damping response of that
beam according to
rW205.010075.0 ×=δ (2.52)
where Wr is the average residual crack width in mm.
According to Chowdhury (1999), at low or no residual crack widths, the measured
damping values varied widely for partially-prestressed beams of similar geometric
properties. He suggests that this phenomena may be due to the variation in internal
cracking due to different levels of prestressing in the beams. Similar to Wang et al.’s
(1998) equation, Equation 2.52 also does not provide a method to calculate the
uncracked logdec (δun), but rather ‘suggests’ a base logdec value of 0.075 for the
uncracked condition.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-31
Ndambi et al. (2000) found from concrete beam tests that damping increased with
excitation amplitude, and that modal damping ratios were highly influenced by non-
linear effects, and are, therefore, highly subjective and difficult to estimate.
Damping capacity experiments of small plain mortar beams were undertaken by Wen
and Chung (2000). It was found that damping increased three-fold with the embedment
of steel reinforcing bars into the mortar. Damping was also increased by at least two
orders of magnitude with the addition of silica fume.
Yan et al. (2000a) found that damping was doubled in the polyolefin fibre-reinforced
concrete as compared to plain reinforced concrete specimens. Further work of Yan et
al. (2000b) found that fibres with a crimped or wavy surface produced a significant
increase in damping and a discernible decrease in response frequency.
Shield (1997) undertook experimental tests on two prestressed concrete beams to assess
the practice of using acoustic emissions as an indicator of the relative ‘health’ of the
beams. The test results clearly showed that the formation and propagation of cracks in
concrete is preceded by a significant increase in acoustic emission activity (AEA) rate
(i.e. a measurement similar to damping).
The primary damping variables identified from the above literature review can be
classed into three groups: beam constituent materials, testing procedures, and the effect
of cracking. Significantly, there is a lack of research into the prediction of damping
capacity prior to the construction of the beam for design purposes. That is, the initial
damping value of the ‘untested’ beam needs to be established. In summary, the
following overall conclusions on member damping can be made:
(1) Beam Constituent Materials: The literature did not fully investigate the effect of
age, concrete compressive strength (fcm), percentage of tensile reinforcement (ρt), or
beam dimensions on damping capacity, which are the fundamental features defining
the basic damping capacity of a member. Of the limited research on the influence of
the amount of steel, Flesch (1981) found that damping may decrease slightly, or
increase strongly, depending on the state of cracking, whereas Wen and Chung
(2000) found that damping decreased by three orders of magnitude with the addition
of reinforcing bars.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-32
(2) Testing Procedures: In terms of testing procedures, test results are also
inconclusive. For example, James et al (1964) found that damping (δ) is
independent of vibration amplitude, whilst Yan et al. (2000a,b) found that damping
increases as the maximum response amplitude increases. Penzien (1964) found that
when undertaking both forced- and free-vibrations on the same beam, the damping
value for free-vibration was much greater than that for forced.
(3) Effects of Cracking: An interrelationship exists between damping and cracking,
however the exact nature of this relationship has still not been confirmed in the
literature. Dieterle and Bachmann (1981) found that damping can sink to a lower
value than the initial damping capacity with cracking present, whilst Wang et al.
(1998) found the opposite.
2.5.3 Structural Damping
The most common full-scale experiments on damping in concrete structures have been
conducted on bridges and buildings. Generally, the intent is to develop damping
databases that assist structural designers in assigning nominal damping values in a
dynamic analysis. Assigning damping values is not easy to achieve because the
fundamental knowledge about essential damping mechanisms of individual structural
components is not yet fully understood.
Jeary (1974) took frequency and damping measurements on full-scale reinforced
concrete multi-flue chimneys using accelerometers. It was found that the value of
logdec = 0.06, commonly adopted in industry for multi-flue chimney design was
excessive and a more appropriate value of δ = 0.03 was suggested.
Leonard and Eyre (1975) took damping and frequency measurements from eight full-
scale box girder bridges. It was concluded that the damping of bridges is significantly
influenced by construction conditions therefore damping values cannot be assigned until
the mechanisms of damping for individual bridge components are fully understood.
Douglas et al. (1981) undertook dynamic field tests on a reinforced concrete bridge,
producing transverse vibrations by quick-release pullback using tractors, vertical
vibrations by truck traffic and by dropping a sand-laden truck onto the bridge. The
modal data was useful for assessing whether the concrete bridge had been overloaded or
damaged at any stage during its lifetime.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-33
From experimental studies of twenty-one full-scale footbridges under pedestrian
excitation, Wheeler (1982) found that logdec was affected primarily by the first mode of
frequency (f1) rather than the construction materials. The variation of damping with
frequency was organised into categories that aid the design of footbridges to remain free
from damaging vibrations induced by pedestrian traffic.
Jeary (1986) proposed two formulae for the calculation of damping for a particular
building, vibrating at a specific amplitude. The equations were divided into two parts:
a) those buildings with low-amplitude damping; and b) those whose damping increases
with amplitude.
Lagomarsino (1993) found that the mechanisms which produce dissipation during
oscillation in a building are varied in nature and include: (a) damping intrinsic to the
structural material; (b) damping due to friction in the structural joints and between
structural and non-structural elements; (c) energy dissipated in the foundation soil; (d)
aerodynamical damping; and (e) passive and active dissipation systems.
Brownjohn (1994) examined published suspension bridge damping databases in order to
determine the primary factors contributing to the damping characteristics exhibited by
suspension bridges. In practice, this is extremely difficult because a suspension bridge
is comprised of friction and bearing joints, construction joints, hysteresis damping in
wire hangers, aero-dynamic damping and foundation damping. Therefore, determining
the overall damping cannot be defined by a small set of certain factors, such as material
or dimensions alone.
Lutes and Sarkani (1995) assessed the effect that building foundations have on the
damping characteristics of an entire structure in a parametric study. It was suggested
that the two separate dynamic characteristics of both the structure and foundation should
be merged to obtain a mathematical expression for the whole system.
Denoon and Kwok (1996) and Glanville et al. (1996), developed and installed
equipment for the measurement of the dynamical response of an 84 m high reinforced
concrete control tower. The focus of the investigation was on the effects that different
wind turbulence regimes, natural frequencies, mode shapes and structural damping,
have on the code specified serviceability criteria for human comfort.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-34
Full-scale experimental data was collected by Suda et al. (1996) on the dynamic
properties of 123 steel structure buildings and 66 reinforced concrete buildings in Japan.
It was found that the damping ratio is dependent, to varying degrees, on building height,
foundation height, building usage, vibration amplitude and type of damping evaluation
method. Despite the large amount of collected data, the authors were unable to develop
any relationship between damping and these variables.
Using data collected from a 78-storey high-rise building in China, Fang et al. (1998)
developed an empirical equation to predict the value of damping at high amplitudes of
vibration in reinforced concrete buildings (using Jeary’s (1986) proposal that damping
is highly amplitude dependent).
Boroschek and Yáñez (2000) obtained strong earthquake motion and ambient vibration
records of 22-storey high Chilean buildings. The data was used to determine damping
ratios for the whole structure and to add to the database used for the dynamic modelling
of a Chilean building’s response to seismic actions.
Pagnini and Solari (2001) found that the availability of experimental data on the
damping of steel poles and tubular towers is extremely rare. In response to this, full-
scale testing was conducted, where it was found that damping (in terms of logdec),
increased considerably with motion amplitude.
From this overview of structural damping, it may be concluded that a wide range of
different concrete structures have been studied by many different researchers. Table A.4
in Appendix A presents a summary of selected structural damping studies. Most
significantly the studies indicate that it is difficult to undertake a meaningful dynamic
analysis without in-depth knowledge about the damping of the individual structural
components.
2.6 Summary
In this Chapter, published research on the damping behaviour of concrete materials,
members and structures has been briefly reviewed. From the discussions it was deduced
that:
• free-vibration excitation is an appropriate experimental technique for the
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 2: Damping in Concrete 2-35
determination of damping;
• there is no published research focussing on the ‘untested’ damping capacity;
• even though it has been established that damping is affected by the damage in the
‘tested’ beam, a method to describe the relationship has yet to be proposed;
• For both ‘untested’ and ‘tested’ concrete beam damping, variables such as the effect
of constituent materials (i.e. high-strength versus normal-strength concrete and 400
MPa versus 500 MPa reinforcing steel) on the damping capacity has not been
adequately examined; and
• the calculation of member damping needs to be better understood before structure
damping can be examined.
These points form the basis of the pilot study presented in Chapter 3.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-1
CHAPTER 3
Theoretical Considerations
3.1 General Remarks
It is evident that there have been many damping studies undertaken. However, many of
the studies are conflicting and have yet to determine the primary variables responsible
for the observed damping mechanisms in concrete beams. In view of this, a
comprehensive experimental programme designed to examine all of the possible
variables should be conducted. Before attempting the comprehensive experimental
programme, a pilot study aided in identifying the major variables.
3.2 Pilot Study
A pilot test programme, involving six full-scale reinforced concrete test beams (BI-1 to
BII-6 in Table 4.1), was carried out to investigate the literature review findings
presented in Chapter 2. It was found that:
(a) previous studies cannot verify the accuracy of their reported logdec making it
difficult to compare to the current test data;
(b) logdec is dependent upon the stage of testing from which is taken. Previous
methods are not able to calculate the logdec of the concrete beams at any specific
stage of testing; and
(c) all beams experience different damage mechanisms according to their
construction, constituent materials, test set-up and loading regime, which will
affect logdec and cannot be explained by direct measures of cracking.
These findings formed the basis of the remaining experimental and analytical
investigations and are described in more depth in the following sections.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-2
3.2.1 Verifying the Accuracy of Logdec
The traditional logarithmic decrement technique (TLT), may be employed for any type
of vibrational response experiment (Section 2.4.1). The TLT utilises the free-vibration
decay curve for the extraction of the inherent damping capacity, logdec, of that member
(Cole and Spooner, 1968; Leonard and Eyre, 1975). Generally, for systems exhibiting
less than 0.5% of critical damping, which is generally so in concrete structures, this time
domain technique is vastly superior to frequency domain techniques, such as the
Bandwidth method (Fahey and Pratt, 1998a,b).
Despite the use of the TLT method by previous researchers, it was difficult to make
meaningful comparisons between the pilot study test data and various published
damping results. Table 3.1 shows some conflicting basic data within the literature that
describes damping levels of both the ‘uncracked’ and ‘cracked’ conditions for
reinforced and prestressed concrete beams. Note that for the ‘uncracked’ condition
there is up to a ten-fold difference in the experimental damping levels between Hop’s
(1991) and Dieterle & Bachman’s (1981) results. For the ‘cracked’ condition similar
large discrepancies were found. Plunkett (1960), identified a similar problem, by
investigating which damping result would be most correct amongst different results for
the same material. Chapter 5 is devoted to resolving this issue.
Table 3.1. Summary of Historical Experimental Damping Research ‘Uncracked’ ‘Cracked’
Researcher Beam Type (δuncr)* Eqn. ? (δcr)* Eqn. ?
Dieterle & Bachman (1981) RC 0.038 Y 0.06 Y
Hop (1991) RC 0.11-0.33 N ~0.224 N
Wang et al. (1998) RC 0.02-0.04 N ~0.032 Y
Chowdhury (1999) RC 0.075 N ≥ 0.075 Y
* All damping values expressed as logarithmic decrement (logdec) in the uncracked (δuncr) or cracked condition (δcr)
3.2.2 Logdec versus Stage of Testing
Another difficulty identified with previous damping research is that they focussed on
individual stages of testing and did not seem to link all the stages. For instance, Hop
(1991) reported on the effect of prestressing force with respect to damping capacity,
however, it was unclear whether the beams were ‘tested’, ‘untested’, ‘cracked’ or
‘uncracked’.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-3
It is therefore proposed in this thesis that damping falls into one of two categories,
‘untested’ and ‘tested’. The ‘tested’ category further contains the ‘uncracked’ and
‘cracked’ sub-categories.
The ‘untested’ beam classification encompasses beams that have not yet received any
service load. It is essential to ascertain the “start” value of damping in a beam based on
its material properties. This category would utilise material damping research
information on concrete.
The ‘uncracked’ beam sub-category covers those beams that have been in service but
have not yet received significant loading to ‘crack’ them. This sub-category is the most
ambiguous, because beams can be in service for many years without any outward signs
of cracking but contain micro-cracking or some form of internal micro-damage.
The ‘cracked’ beam category is self-explanatory. The presence of cracks can suggest
that the beam has experienced a significantly damaging event in the past or, the beam
could be suffering from corrosion, poor concrete quality, etc.
3.2.3 Damage Mechanisms in Concrete Beams
Although some previous work (Chowdhury, 1999) advocated cracking as being the
most appropriate method by which to calculate logdec, the following highlight why this
is not advocated:
(1) Significant disagreements exist regarding the nature of the crack-dependent
damping mechanism. For instance, Dieterle and Bachman (1981) suggest that,
after cracking, damping can sink to a value lower than that to begin with, whilst
various other researchers suggest the opposite (Askegaard and Langsœ, 1986;
Almansa et al., 1993);
(2) Crack-dependent damping does not fully explain the behaviour of many beams
without obvious cracking, such as prestressed beams that may never exhibit
obvious in-service cracking or damage, yet they increase in damping capacity
(Hop, 1991); and
(3) The selection of an appropriate crack-width equation is difficult practically when
many different methods and types of loading can be used to calculate crack
width.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-4
3.2.4 Residual Deflection
This thesis focuses primarily on the use of residual deflection to predict damping. It is
considered that residual deflection can be used for the ‘tested’ sub-categories of
‘uncracked’ and ‘cracked’ stress stages more adequately than crack width alone. This
mechanism has not previously been proposed as a method for calculating logdec.
In many practical situations, certain structural members will be subjected to repeated,
damage inducing impact or fatigue loadings (Dexter and Fisher, 1997). The
accumulated damage that results from repetitive loads is reflected in a flexural member
in the form of non-recoverable deflection. This condition is termed here as residual
deflection and as briefly discussed in Chapter 1, for the current testing regime the term
residual deflection does not include a study of the effects of long-term loading, cyclic
loading, creep and shrinkage effects or stress relaxation in the prestressed beams.
Furthermore, it does not consider the special case of load-reversal, which can occur
during events such as earthquakes.
For further definition of residual deflection refer to the generalised residual load-
deflection (L-D) curve for a two-point loaded simply-supported rectangular reinforced
concrete beam in Figure 3.1. The solid line traces the load deflection history from
linear to non-linear behaviour where the beam becomes cracked. If the load is released
from point C, the load path will return along the dotted line to point D. A residual
deflection would remain (Distance A-D) due to incomplete crack closure. It has also
been suggested that the amount of recovery of deflection is a direct indication of the
integrity of the bond between concrete and steel (Swamy and Anand, 1974).
It is proposed here that the easiest method by which to estimate the residual deflection is
to correlate it to the instantaneous deflection using the relationship in Equation 3.1.
∆r = f1 (∆i) (3.1)
where the residual deflection ∆r is a function of the instantaneous deflection ∆i for the
experimental programme described in this thesis. This relationship is established
experimentally in Chapter 7.
3.3 The Total Damping Capacity Equation
The proposed equation to calculate the total logdec of a concrete beam at any stage of
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-5
its service life will be the sum of the ‘untested’ and ‘tested’ components.
δtotal = δuntest + δtest (3.2)
The presentation of the theory for the δuntest and δtest components are presented below.
.
Py = Yield LoadPs = Service LoadPcr = Cracking Load
Fully Cracked
Load
, PPy
Ps
Pcr
Deflection, ∆
. ..A
B
C
D
Residualdeflection, ∆r
PartiallyCracked
Figure 3.1: Schematic Residual Load-Deflection Curve for Concrete Beams
3.3.1 ‘Untested’ Beams
One of the most elementary studies on material damping is that by Lazan (1968) who
described reinforced concrete as being a composite material and stated that the
properties, number and location of the individual reinforcing bars in the concrete must
be known before the damping properties of the reinforced concrete can be determined.
This particular statement has not been researched further in any subsequent studies.
Lazan (1968) further stated that the material damping of any whole composite may be
determined by summing the damping contribution of each component part or member.
Even though a method for doing so was not provided, the research of Lazan (1968)
does, however, provide a starting point for the development of the theory for the
calculation of the ‘untested’ damping capacity (δuntest) for the present research.
Following the findings of Lazan (1968), the damping capacity, in terms of logdec (δ), of
an ‘untested’ concrete beam, is expected to incorporate
δuntest = f2 (ρ, fcm) (3.3)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-6
where ρ is the reinforcement ratio and fcm mean concrete compressive strength.
The relative contributions of each of these variables, if any, are to be determined by
experimentation in Chapter 6.
3.3.2 ‘Tested’ Beams
Residual deflection is proposed as the theoretical foundation for the calculation of the
damping capacity for the ‘tested’ beam sub-categories of ‘uncracked’ and ‘cracked’.
Residual deflection will reflect the unseen micro-cracking in an ‘uncracked’ beam,
something that cracking models do not.
Crack closure in reinforced concrete is a complex mechanism where, once a crack has
opened up, small amounts of solid particles become dislodged and prevent the crack
from closing (Neild et al., 2002). At a given level of damage, concrete exhibits energy
dissipation due to the frictional sliding between the crack lips (due to crack closure)
(Ragueneau et al., 2000). This mechanism operates even at very low loads, which are
insufficient to cause visible cracking, for instance in the ‘uncracked’ beam (Zhang and
Wu, 1997; Zhang, 1998). This mechanism also helps to explain why damping increases
from the instance it is first loaded. Additionally, this provides justification for the use
of residual deflection to estimate damping capacity, as opposed to explicit
measurements of crack width.
It has also been shown that the presence of cracks increases the flexibility of a structure.
Thus the experimentally observed changes in the flexural stiffness rigidity can be
interpreted as an indicator of damage in the structure (Jerath and Shibani, 1985; Carr
and Tabuchi, 1993; Pandey and Biswas, 1994; Zhao and DeWolf, 1999). This is why
non-destructive testing is used extensively in industry as an on-site monitoring tool to
detect the changes, and thus an extensive amount of research has been conducted in this
area (Johns and Belanger, 1981; Stephens and Yao, 1987; Ahlborn et al., 1997; Ohtsu et
al., 1998; Wang et al., 1998; Wahab and De Roeck, 1999; Kisa and Brandon, 2000;
Maeck et al., 2000; Ravi and Liew, 2000; Van Den Abeele and De Visscher, 2000;
Dems and Mróz, 2001; Ohtsu and Watanabe, 2001; Razak and Choi, 2001; Tan et al.,
2001; Capozucca and Cerri, 2002; Ndambi et al., 2002).
The relationship that would result between logdec δ and residual deflection ∆r, in the
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 3: Theoretical Considerations 3-7
‘tested’ beam is represented by the following equation:
δtest = f3 (∆r) (3.4)
where the function defining the relationship is determined by experimentation in
Chapter 8.
3.4 Summary
Using the pilot study test results, this Chapter has presented a collection of concepts and
general equations used to develop a unified method for the determination of the
damping capacity of both reinforced and prestressed concrete beams.
The first concept proposed is that damping capacity, as reported in the literature and
observed in the pilot study, can very enormously even though they have been obtained
using the same technique. Chapter 5 presents an in-depth investigation to solve this
problem.
It has also been proposed that damping capacity measurements fall within the ‘untested’
and ‘tested’ stages of beam testing. This division is important as it allows for research
efforts to be compared. The divisions have not been previously proposed.
For the ‘untested’ damping capacity, Chapter 6 presents the quantification of the
contribution by concrete and reinforcement to damping capacity. To investigate the
total damping capacity in Chapter 8, the residual deflection is examined and quantified
in Chapter 7.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-1
CHAPTER 4
Experimental Programme
4.1 General Remarks
An extensive experimental programme was undertaken to investigate the damping and
deflection behaviour of reinforced and prestressed concrete beams containing normal-
and high-strength concrete and/or 400 MPa or 500 MPa reinforcement. This Chapter
describes the geometrical and mechanical details of the beams, the primary test
variables and details of the constituent materials, the testing regime and the
instrumentation employed. A total of forty-one reinforced and prestressed concrete
beams were tested. These beams were divided into four test series as follows:
B-Series Test Beams: 12 reinforced concrete beams, 6-metres in length;
PS-Series Test Beams: 10 prestressed concrete beams, 6-metres in length;
CS-Series Test Beams: 9 reinforced concrete beams, 2.4-metres in length;
F-Series Test Beams: 10 reinforced concrete beams, 2.4-metres in length, beams
with small self-weight induced stresses.
The beams are grouped so that comparisons can be made between each test series. For
instance, comparisons between the B- and CS-Series beams provides valuable
information on the size effect in reinforced concrete beams, whilst comparisons between
the CS- and F-Series highlight the effect of small self-weight induced stresses.
4.2 Design of Beam Test Specimens
The reinforced and prestressed concrete beams were generally under-reinforced and
designed to fail in flexure, a condition preferred by most national codes of practice.
Some beams were designed to fail in shear and were included for comparative purposes
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-2
and also to provide a wider range of different test results. The primary test variables
selected for investigation in the reinforced concrete beams (B-Series, CS-Series and F-
Series) were (note that all test variables implemented are in SI Units):
Concrete compressive strength on day of testing (fcm in MPa);
Nominal reinforcement yield strength (fsy in MPa); and
Tensile reinforcement ratio (ρt = Ast/bd), bar spacing (s in mm) and bar diameter
(Φ in mm).
For the prestressed concrete beams (PS-Series), the primary test variables selected were:
Prestressing force (H in kN);
Prestressing eccentricity (e in mm); and
Concrete compressive strength on day of testing (fcm in MPa).
4.2.1 Geometrical and Mechanical Details
The geometric details of the reinforced and prestressed concrete beams are given in
Figures 4.1 and 4.2. Tables 4.1 and 4.2 present the complete tabulated variables for
each reinforced and prestressed concrete beam, respectively. The calculation of the
section moment of inertias (Ig) and the ultimate moment capacity (Mu) for the design of
each beam may be found in Appendix B.
4.2.2 Primary Test Variables
Figures 4.3 to 4.6 give a diagrammatic summary of primary variables and comparisons
to be investigated within each test series.
4.3 Materials
The concrete was supplied by a local ready-mix supply company (CSR Construction
Materials), and the reinforcing steel was supplied by BHP OneSteel. All materials are
currently in widespread use within the Australian construction industry.
4.3.1 Concrete
The normal- and high-strength concrete mixes ordered from the supplier for the test
beams were required to be 32 MPa and 80 MPa, respectively. The technical data for the
ready-mix concrete, as provided by the manufacturer, is presented in Tables 4.3 and 4.4.
Due to variations in slump values and test dates, a range of normal and high-strength
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-3
compressive concrete strengths were obtained (20.0 MPa<fcm<90.7 MPa). Excessive
slump variations are attributed to overzealous truck drivers introducing extra water to
the mix on delivery.
CR SR
TR100100
All dimensions are in mm Indicates reinforcing bar
Variables Defined in Table 4.1:Tension Reinforcement (TR) Compression Reinforcement (CR)Shear Reinforcement (SR)Beam Effective Depth (d)TR Spacing (st) and CR Spacing (sc)Concrete Cover (c) = 20 mm
Section A-A
b
D=300 mm for B-SeriesD= 250 mm for CS- and F-Series
c
d
c
A
AL = 6000 mm for B-SeriesL = 2400 mm for CS- and F-Series
st
sc
D
b = 200 mm for B-Seriesb = 150 mm for CS- and F-Series
Figure 4.1: Geometric Detailing for B-, CS- and F-Series Test Beams
SR
TR100100
All dimensions are in mm Indicates prestressing tendon
Additional Variables Defined in Table 4.2:
Prestressing Force from Tendons (H)Prestressing Eccentricity (e)Concrete Cover (c) = 20 mm
Section A-A
b = 200 mm
D =
300
mm
c
c
A
AL = 6000 mm
150
NA
e
Horizontal spacing between tendons = 40mmVertical spacing between tendons = 20mm
Figure 4.2: Geometric Detailing for PS-Series Test Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-4
Table 4.1. Geometrical and Reinforcement Details – RC Beams Beam d (mm) TR CR SR TR Spacing
st, (mm) CR Spacing
sc, (mm) BI-1 264 4Y20 2Y12 R6@90mm 22.7 124 BII-2 264 3N20 2N12 R6@90 44.0 124 BI-3 262 3Y24 2Y12 R6@90 38.0 124 BII-4 262 2N24 2N12 R6@90 100.0 124 BII-5 264 4N20 2N12 R6@90 22.7 124 BII-6 264 3N20 2N12 R6@90 44.0 124 BI-7 266 2Y16 2Y12 R6@90 116.0 124 BII-8 266 2N16 2N12 R6@90 116.0 124 BI-9 262 2Y24 2Y12 R6@90 100.0 124 BII-10 262 2N24 2N12 R6@90 100.0 124 BII-11 264 3N20 2N12 R6@90 44.0 124 BII-12 264 4N20 2N12 R6@90 22.7 124 CS1 214 3N20 2N12 R6@90 14 74 CS2 214 3N20 2N12 R6@90 14 74 CS3 210 3N20 2N12 R10@125 13 66 CS4 213 2N24 None None 52 - CS5 213 2N24 None None 52 - CS6 213 2N24 None None 52 - CS7 212 2N24 2N12 R6@90 40 74 CS8 212 2N24 2N12 R6@90 40 74 CS9 208 2N24 2N12 R10@125 32 66 F1 220 3Y20 None None 25 - F2 216 2Y16 1Y6 R6@150 66 98 F3 210 2Y20 2Y10 R10@110 50 70 F4 222 3Y16 None None 31 - F5 220 2N20 None None 35 - F6 208 2N24 None None 31 - F7 220 3N20 None None 25 - F8 208 2N24 2N12 R10@125 42 66 F9 210 2N20 2N12 R10@125 50 66 F10 214 2N20 2N12 R6@110 58 74
Table 4.2. Geometrical and Reinforcement Details – PSC Beams
Beam
Prestressing Force
H (kN)
Prestressing Eccentricity
e (mm)
Initial Camber due to Prestress
ϕ (mm)
d (mm) TR # CR SR
PS1 239 111.7 8 261.7 9HS5 None R10@225mm PS2 293 110.5 9 260.5 11HS5 None R10@225 PS3 346 97.0 9 247.0 13HS5 None R10@225 PS4 585 80.5 14 230.5 22HS5 4Y12 R10@225 PS5 612 60.0 15 210.0 23HS5 None R10@225 PS6 400 99.7 11 249.7 15HS5 None R10@225 PS7 450 97.5 14 247.5 13HS5 None R6@200 PS8 400 80.0 13 230.0 15HS5 None R6@200 PS9 346 90.0 7 240.0 13HS5 None R6@200 PS10 480 90.0 17.5 240.0 18HS5 None R6@200
# The first number refers to the number of tendons, the last number refers to the diameter in mm of the tendon (see Figure 4.4 for tendon layout details).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-5
Variables : ρt (reinforcement ratio)φ (bar diameter) s (bar spacing) f’c (concrete strength (E)) fsy (steel strength)
Investigate:f’c: BII-2 vs BII-6 f’c, fsy : BI-1 vs BII-5 BII-2 vs BII-11 BI-1 vs BII-12 BII-4 vs BII-10 BI-4 vs BII-9 BII-5 vs BII-12 fsy : BI-7 vs BII-8 BII-6 vs BII-11 BI-9 vs BII-10
30.0 MPa400 MPa
4Y20= 1257 mm2
2Y12’s
30.0 MPa500 MPa
3N20 = 942 mm2
2N12’s
23.1 MPa400 MPa
3Y24= 1357 mm2
2Y12’s
23.1 MPa500 MPa
2N24 = 905 mm2
2N12’s
41.5 MPa500 MPa
3N20 = 942 mm2
2N12’s
41.5 MPa500 MPa
4N20 = 1257mm2
2N12’s
BI-1 BII-2
BI-3 BII-4
BII-5 BII-6
BII-5Comparefsy, , f’c
BII-6Compare
f’c
BI-9Compare
fsy , f’c
BII-10Compare
f’c
BII-12Compare
f’c
BII-11Compare
f’c
Compare fsy
64.5 MPa400 MPa
2Y16 = 402 mm2
2Y12’s
64.5 MPa500 MPa
2N16 = 402 mm2
2N12’s
53.0 MPa400 MPa
2Y24 = 905 mm2
2Y12’s
53.0 MPa500 MPa
2N24 = 905 mm2
2N12’s
90.7 MPa500 MPa
3N20 = 942 mm2
2N12’s
90.7 MPa500 MPa
4N20 =1257 mm2
2N12’s
BI-7 BII-8
BI-9 BII-10
BII-11 BII-12
Compare fsy BII-4
Compare f’c
BII-5Compare
f’c
BI-1Compare
f’c, fsy
BII-6Compare
f’c
BII-2Compare
f’c
BII-4Compare
f’c, fsy
BII-12Comparefsy, , f’c
BII-11Compare
f’c
BII-1Compare
fsy , f’c
BII-2Compare
f’c
Compare ρt
Compare ρt
Compare ρt
Figure 4.3: B-Series – Primary Test Variables
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-6
69.82 MPaH = 612 kNe = 60.0 mm
23 HS5 =452 mm2
69.82 MPaH = 400.0 kNe = 99.7 mm
15 HS5 =295 mm2
PS1 PS2
PS5 PS6
60.63 MPaH = 239 kN
e = 111.7 mm
9 HS5 =177 mm2
60.63 MPaH = 293 kN
e = 110.5 mm
11 HS5 = 216 mm2
PS3 PS4
60.18 MPaH = 585 kNe = 80.5 mm
22 HS5=432 mm2
4Y12’s60.18 MPaH = 346 kNe = 97.0 mm
13 HS5=255 mm2
CompareH
CompareH, e
PS7Compare
H
PS7 PS8
f’c = 52.5 MPaH = 450 kNe = 97.5 mm
13 HS5 =255 mm2
f’c = 52.5 MPaH = 400 kNe =80.0 mm
15 HS5 =295 mm2
PS9 PS10
f’c = 83.5 MPaH = 480 kNe = 90 mm
18 HS5 =353 mm2
f’c = 83.5 MPaH = 346 kNe = 90.0 mm
13 HS5 =255 mm2
PS3Compare
H
CompareH, e, f’c,
Comparef’c ,e
Investigate: H: PS1 vs PS2 H , e: PS5 vs PS6 PS3 vs PS7 PS7 vs PS8
f’c, e : PS3 vs PS9 PS6 vs PS8
Variables: H (prestressing force) e (eccentricity) f’c (concrete strength (E))Secondary Variables: ρt (tensile reinforcement ratio)ρc (compression reinforcement ratio)
PS3Compare
f’c, e
PS9Compare
f’c, e
PS9Compare
f’c, e
CompareH
CompareH , e, ρc
Figure 4.4: PS-Series – Primary Test Variables
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-7
Variables : fsy (steel strength)ρ (reinforcement ratio) φ (bar diameter)s (bar spacing) f’c (concrete strength (E))Loading Conditions
22.5 MPa400 MPa
3N20 = 942 mm2
2N12
CS2
32.0 MPa500 MPa
2N24 = 905 mm2
CS4
31.5 MPa500 MPa
2N24 = 905 mm2
2N12
CS7
315 MPa500 MPa
2N24 = 905 mm2
2N12
CS8
31.5 MPa500 MPa
2N24 = 905 mm2
2N12
CS9
22.5 MPa400 MPa
2N12
CS1
3N20 = 942 mm2
22.5 MPa400 MPa
2N12
CS3
3N20 = 942 mm2
32.0 MPa500 MPa
CS5
2N24 = 905 mm2
32.0 MPa500 MPa
CS6
2N24 = 905 mm2
Beam Length
Load
Beam Length
Load400 mm
Beam Length
Load800 mm
Beam Length
Load
Beam Length
Load500 mm
Beam Length
Load700 mm
Beam Length
Load
Beam Length
Load500 mm
Beam Length
Load700 mm
FLEXURALCS1,2,3CompareLoading
Conditions
SHEARCS4,5,6CompareLoading
Conditions
FLEXURALCS7,8,9CompareLoading
Conditions
Figure 4.5: CS-Series – Primary Test Variables
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-8
Variables : ρc (compression reinforcement ratio)ρt (tensile reinforcement ratio)s (bar spacing) f’c (concrete strength (E))φ (bar diameter)
SHEAR32.0 MPa500 MPa
1N20 =314 mm2
F1
FLEXURAL 32.0 MPa500 MPa
F2
2N16 =402 mm2
1N6
FLEXURAL32.0 MPa500 MPa
F3
2N20 =628 mm2
2N10
SHEAR32.0 MPa500 MPa
F4
3N16 =603 mm2
SHEAR72.0 MPa500 MPa
2N20 =628 mm2
F5
SHEAR72.0 MPa500 MPa
2N24 =905 mm2
F6
SHEAR72.0 MPa500 MPa
3N20 =942 mm2
F7
FLEXURAL72.0 MPa500 MPa
2N24 = 905 mm2
2N12
F8
FLEXURAL72.0 MPa500 MPa
2N20 = 628 mm2
2N12
F9
FLEXURAL72.0 MPa500 MPa
2N20 = 628 mm2
2N12
F10
F4Compare
ρt
F5,6,7Compare
f’c, ρt
F5,6Compare
ρt
F1,4Compare
f’c, ρt
F6,7Compare
ρt
F1,4Compare
f’c, ρt
F5,7Compare
ρt
F1,4Compare
f’c, ρt
F5,6,7Compare
ρt
F1Compare
f’c, ρt
F3,8,9,10Compare
ρc, ρt
F2Compareρc, ρt
F8Compareρc, ρt, f’c
F9,10Compareρc, f’c
F2,3Compareρc, ρt, f’c
F9,10Compare
ρt
F2,3Compareρc, ρt, f’c
F8Compare
ρc
F2,3Compareρc, ρt, f’c
F8Compare
ρc
Figure 4.6: F-Series – Primary Test Variables
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-9
During the casting of each beam specimen, six compression cylinders (200 mm depth
and 100 mm diameter) were cast for each and every beam specimen. The cylinders
were cured under exactly the same conditions as the beams. This involved the cylinders
being continuously kept moist with wet hessian sacking. On the day of testing, these
cylinders were used to determine an average for the concrete compressive strength at
testing, fcm (see Table 4.5). The testing of the concrete cylinders was carried out using
the DMG Denison Type 7640 with automatic printout.
4.3.2 Reinforcement
Table 4.6 presents the manufacturers specifications for 400 MPa and 500 MPa
reinforcing steel used. In determining the actual strength and ductility properties of the
three different types of reinforcing bars/tendons used in the current test beams, stress-
strain tests were undertaken by the BHP Laboratories in Brisbane, Australia. The
resulting stress-strain curves for the 400 and 500 MPa reinforcement bars are given in
Figures 4.7 and 4.8 respectively, and for the prestressing steel, Figure 4.9. Each figure
contains two different curves: a) the entire stress-strain curve where the steel peak stress
U may be observed; and, b) a close up of the yield portion of the curve where the yield
strength is denoted by Y. A minimum of ten samples of each bar type was tested. This
approach was used to obtain averages as detailed in Table 4.7.
Table 4.3. Concrete Technical Data – Materials (CSR Construction Materials)
Material No Supplier Works, Quarry, Pit Specification
Type GP Cement 1 QCL Bulwer Island AS3972 Fly Ash 2 Pozzolanic Calide AS3582.1 Silica Fume 3 10 mm Aggregate 4 CSR Readymix Beenleigh AS2758.1 Coarse Sand 5 CSR Readymix Oxenford AS2758.1 Fine Sand 6 CSR Readymix Oxenford AS2758.1 Water Reducing Agent 10 WR Grace Archerfield AS1478/79 Superplasticiser 11 WR Grace Archerfield AS1478/79
Table 4.4. Concrete Technical Data – Mix Design (CSR Construction Materials)
Mass of Materials (kg/m3) Admixtures
Mix Name Slump (mm)
Water (l/m3)
1 2 3 4 5 6 10
(ml/100kg cement)
11 (ml/m3)
N32/10mm 120 115 235 75 - 990 620 310 400 100
S80/10mm 120 100 550 40 60 1060 540 180 400 9000 Note: Coarse and fine sand moisture contents are at 6% and 8%, respectively.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-10
Table 4.5. Details of Test Beams – Concrete
Beam Age at Testing (days)
Date of Testing
Maximum Aggregate Size (mm)
Concrete Density (kg/m3)
Nominal
Concrete Compressive
Strength at Testing fcm (MPa)
BI-1 28 17/5/00 10 2400 30.0 BII-2 28 17/5/00 10 2400 30.0 BI-3 30 14/7/00 10 2400 23.1 BII-4 30 14/7/00 10 2400 23.1 BII-5 28 24/7/00 10 2400 41.5 BII-6 28 24/7/00 10 2400 41.5 BI-7 63 15/5/01 10 2400 64.5 BII-8 63 15/5/01 10 2400 64.5 BI-9 61 29/5/01 10 2400 53.0 BII-10 61 29/5/01 10 2400 53.0 BII-11 90 9/7/01 10 2500 90.7 BII-12 90 9/7/01 10 2500 90.7
PS1 36 18/9/00 10 2400 60.6 PS2 36 18/9/00 10 2400 60.6 PS3 28 19/9/00 10 2400 60.2 PS4 28 19/9/00 10 2400 60.2 PS5 35 21/9/00 10 2500 69.8 PS6 35 21/9/00 10 2500 69.8 PS7 45 17/8/01 10 2400 52.5 PS8 45 21/8/01 10 2400 52.5 PS9 60 10/10/01 10 2500 83.5 PS10 60 9/10/01 10 2500 83.5
CS1 35 18/9/00 10 2400 22.5 CS2 35 18/9/00 10 2400 22.5 CS3 35 18/9/00 10 2400 22.5 CS4 28 1/10/01 10 2400 32.0 CS5 28 1/10/01 10 2400 32.0 CS6 28 1/10/01 10 2400 32.0 CS7 35 8/10/01 10 2400 31.5 CS8 35 8/10/01 10 2400 31.5 CS9 35 8/10/01 10 2400 31.5
F1 ~ 4 yrs 10/4/02 10 2400 32.0 F2 ~ 4 yrs 10/4/02 10 2400 32.0 F3 ~ 4 yrs 10/4/02 10 2400 32.0 F4 ~ 4 yrs 10/4/02 10 2400 32.0 F5 28 17/4/02 10 2500 72.0 F6 28 17/4/02 10 2500 72.0 F7 28 17/4/02 10 2500 72.0 F8 28 17/4/02 10 2500 72.0 F9 28 17/4/02 10 2500 72.0 F10 28 17/4/02 10 2500 72.0
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-11
Table 4.6. Technical Design Data for 400 and 500 MPa Reinforcing Steel (Patrick, 1999)
Ductility Properties Rebar Type Yield Strength
(MPa) #Ratio Rm/Re Elongation (Agt%)
OneSteel TEMPCORE/Microalloy 400Y 400 minimum 1.10 min
16% min on gauge length of 5d at
fracture fsy,L 500 Class L
Low Ductility fsy,U 750 1.03 1.5
fsy,L 500 OneSteel 500PLUS
Class N Normal Ductility fsy,U 650
1.08 6
# fsy,L and fsy,U are the lower and upper characteristics yield strengths.
a)
U
b)
Figure 4.7: Stress-Strain Curve for 400 MPa Reinforcement (BHP Laboratory,
Brisbane, Australia)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-12
a)
b)
Figure 4.8: Stress-Strain Curve for 500 MPa Reinforcement (BHP Laboratory,
Brisbane, Australia)
Figure 4.9: Stress-Strain Curve for Prestressing Tendons (BHP Laboratory, Brisbane,
Australia)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-13
Table 4.7. Details of Test Beams – Reinforcing Bars
Bar Type
Nominal Steel Yield
Strength fsy (MPa)
Steel Peak Stress U (MPa)
Steel Off-Yield Stress
Y (MPa)
Modulus of Elasticity for Steel
Es (GPa)
N 400 526.2 423.5 203.4
Y 500 642.8 569.4 196.8
Tendon 1710 1470.4 1461.5 227.0
4.4 Fabrication
This section contains complete details regarding the construction, casting and curing of
each beam specimen.
4.4.1 Reinforced Concrete Beams
The smooth inner surface of the beam moulds were oiled with laboratory grease to
produce a smooth finish to all sides of the beam specimen. The reinforcement cages
were constructed and carefully placed into the beam moulds (using 20 mm chairs). If
required, small mortar wedges were placed down the sides of the mould to ensure that
the reinforcement cage would stay positioned exactly in the centre of the mould during
pouring.
Concrete was then poured and, using a poker vibrator, vibrated to ensure the concrete
mix was distributed evenly in the formwork. A rubber mallet was pounded along the
exterior mould to ensure that all bubbles were removed from the concrete mix. The top
of the beam was trowelled to a smooth finish. This was to ensure a uniform surface for
the damping hammer impacts.
4.4.2 Prestressed Concrete Beams
In the prestressed beam preparation, the primary steps involved positioning and
prestressing the tendons, fixing the shear reinforcement, pouring the concrete and then
cutting the tendons. The prestressing test rig is diagrammatically shown in Figure 4.10.
The beam casting moulds were first positioned between two prestressing buttresses.
Each prestressing buttress was bolted to a two-metre thick concrete strong floor, 7000
mm apart. Twenty-millimetre thick plywood plates were attached to the ends of the
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-14
beams. They contained holes with the number, diameter and spacing corresponding to
the strand arrangements as detailed in Figures 4.2 and 4.4. After positioning the
tendons in the beam moulds, each tendon was anchored to one buttress through male
and female cones at the passive end. Each prestressing buttress that was set-up had a
shearing resistance of 30 tonnes, with the entire prestressing rig able to withstand 60
tonnes of prestressing force. At the active end, the tendons were tensioned by a
hydraulic jack.
Anchor bolts
7.0 metres
1.0 metre thickconcrete strong floor
ButtressSteel reinforced beam mould6.0m length, 300mm deep200mm wide
Male/Femalecones
ElectricHydraulicJackingSystem
Hydraulic jack
PASSIVEEND
ACTIVEEND
Figure 4.10: Rig Used for Prestressing the Tendons
To implement the required extension needed to reach the required prestress force (H), a
Vernier calliper was used. Hook’s Law was used to calculate the required wire
extension, using Young’s Modulus of Elasticity as determined from the steel stress-
strain tests (Table 4.7). Each tendon was prestressed individually in a symmetrical
pattern to ensure that the beam was evenly stressed around its longitudinal axis
eliminating excessive longitudinal eccentricity. The complete calculation details used
for the prestressing extension process are given in Appendix B.
Stirrups intended to limit shear failure were then fixed to the prestressing tendons at the
required spacing and the concrete was cast. Following the minimum concrete curing
period required to achieve minimum concrete compressive strength (see Appendix B for
calculations), the tendons were cut individually using an oxy-acetylene gas flame. This
type of cutting method ensures the gradual transfer of compressive forces to the
concrete. Compression cylinders were tested periodically to monitor when the
minimum compressive strength was reached. Following the tendon cutting, the beams
were ready for testing.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-15
4.4.3 Curing
Immediately following the pouring of the concrete the beams, and their respective
cylinders, were covered with wet hessian sacking. They remained under this condition
for the entirety of the curing period. As indicated in Table 4.5, the concrete beams were
tested between 28 and 90 days. The variable test days were due to the unavailability of
the laboratory facilities or extra curing time to reach the desired strength.
4.5 Test Set-Up
All experiments conducted on the beams used the third point loading test set-up as seen
in Figure 4.11 and Plate 4.1. This kind of loading system is the most common type of
loading arrangement and is favoured for laboratory experiments because it has the
advantage of offering a substantial region of nearly uniform moment coupled with very
small shears (Bungey and Millard, 1996). All of the beams were simply-supported, and
subject to static concentrated point loads of various magnitudes. The load was applied
by a calibrated hydraulic jack with a capacity of 100 tonnes, and transmitted through a
load-cell and thick metal plate to the loading beam, which in turn produced two point
loads on the test beam.
4.5.1 Beam Support System
As shown in Figure 4.11, each concrete beam was simply-supported at each end using
roller bars and angles, with the concrete supports affixed firmly to the floor. This type
of support provides minimum friction and restraining moments, and helps to isolate the
damping measurements to the beam itself. In addition, the steel bearing plates were
grouted to the supports, this was to ensure that the load was uniformly distributed over
the beam, thereby avoiding bearing failure at the supports. Detailed photos of the beam
support system may be seen in Plate 4.2.
4.5.2 Loading Beam Width (LBW)
As shown in Figure 4.11, the LBW’s were designed to vary between each test series so
as to provide information of the effect of bending moment distribution on the
instantaneous and residual deflection. The LBW’s along with shear span length for
every test beam are presented in Table 4.8.
4.5.3 Hammer Excitation Position (HEP)
The test variable of hammer excitation position (HEP) was part of the investigations
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-16
because ensuring consistency during experimental testing was always of great concern
in this research. Previous literature has indicated that concrete beam damping should be
relatively constant, regardless of the hammer excitation position (HEP) (Chowdhury,
1999). A portion of the current study however did focus on examining and verifying the
impact of HEP on logdec. Each beam was impacted during testing at various locations
along its length, as indicated in Figure 4.11. Chapter 5 presents the conclusions drawn
on the effect of HEP on damping measurements.
Base plate boltedto floor with 100
tonne capacityeach plate
CL
Hammer Excitation Position(HEP)
Variable
Loading cell = 18.5 kg (All beams)Weight of loading beam= 221 kg for B-and PS-Series Beams
= 70.5 kg for CS- and F-Series Beams
100 tonne capacity cylinder
Loading Beam Width (LBW)
Hammer Weight(HW)
Shear Span (a)
Figure 4.11. Diagram of Beam Test Set-Up
Plate 4.1. Photograph of Beam Test Set-Up
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-17
a)
b)
Plate 4.2. Beam Support System a) Roller and b) Knife Supports
Table 4.8. Loading Beam Width Specifications Beam LBW (mm) a# (mm) All B-Series Beams 2000 1900 All PS-Series Beams 2000 1900 CS1 0 1150 CS2 400 950 CS3 800 750 CS4 0 1050 CS5 500 800 CS6 700 700 CS7 0 1050 CS8 500 800 CS9 700 700 All F-Series Beams 700 700
# Represents the shear span – the distance between the support and the loading beam (Figure 4.13).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-18
4.5.4 Hammer Weight (HW)
In the current experimental tests, two different hammer weights were used (HW1=
163.1 g and HW2= 239.5 g). The primary purpose of this was to examine the effect
hammer weight has on the vibration decay curve, and the subsequent calculation of
logdec. According to French (1999) the imparted frequency spectrum and force
duration is affected by hammer tip (i.e. steel versus rubber) and hammer weight.
A discussion on the effect of hammer weight on logdec will also be presented in
Chapter 5.
4.6 Test Procedures
Static load testing of the beams was induced by using the servo-controlled 100 ton
capacity, Enerpac hydraulic jacks. The loading regime followed a set cycle of ‘load on’
and ‘load off’ increments, increasing at each step until beam failure. The ‘load-on’
testing phase was necessary in order to induce various stages of damage to the beam,
which corresponded to in-service loading damage. Secondly, the ‘load-off’ phase
allowed residual deflection and logdec measurements to be taken from the beam.
The loading regime consisted of a series of ‘load-on’ and ‘load-off’ increments. Each
increment was approximately 1/10th of the anticipated failure load. The load was
released at the same rate as it was applied. Between pre-load and first crack the
increments were smaller so that first crack could be accurately identified.
A rest period of approximately one-minute was given prior to the taking of
measurements at each ‘load-on’ and ‘load-off’ position. This was a sufficient amount of
time that would account for the primary phase of creep in the beams immediately
following initial loading (Penzien and Hansen, 1954).
At each ‘load-on’ position, deflection, crack widths by crack microscope and
reinforcement strains were obtained. At each ‘load-off’ position, deflection, crack
widths by crack microscope, reinforcement strains and free-vibration decay curve
signatures were obtained. The regime was repeated at ever increasing intervals until the
beam failed. All of the raw data taken from the beam is presented in Appendix F.
4.7 Instrumentation
The current testing programme involved a number of types of measurements.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-19
Generally, these measurements were of two categories: dynamic and static. The
dynamic measurements were of the free-vibration decay curves for damping calculation,
and the static measurements included deflections, crack widths and reinforcement
strains. Figure 4.12 presents a diagram with the locations of all the relevant testing
equipment.
4.7.1 Damping
One of the most popular excitation techniques used for modal damping analysis is
impact, or hammer excitation. The waveform, produced by an impact, is a transient (or
short duration) energy transfer event. Figure 4.13 shows a common type of energy
frequency spectrum produced by a short duration impact. The duration, and thus the
shape of the spectrum of an impact, is determined by the mass and stiffness of both the
impactor and the structure (Døssing, 1988a,b). As outlined previously, care was taken
to ensure the relative stiffness of the beam testing structure was much greater than that
of the beam itself, helping isolate vibration to within the beam. Also, it is extremely
important that the impact technique is as consistent as possible, so to ensure the
repeatability of the acquired signatures (Bhuvanagiri and Swartz, 2000).
L/3 L/3
Beam Vibration DetectedBy Accelerometer
Model 353A Quartz ShearMode Accelerometer(PCB® Piezotronics)
Vibration signalrecorded onoscilloscope
TDS 460A DigitisingOscilloscope (Tektronix)
Vibration decay curvenow ready for analysis
on PC(i.e. TLT or DCM)
Dial Gauge Deflection
LVDT Mid-Span Deflection
ICP® Impulse-Force Hammer (PCB® Piezotronics)
Crack Width
Microscope Levelof TR
Figure 4.12. Location of Beam Testing Equipment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-20
TDuration
time
F(t)
Figure 4.13. Impact Energy Frequency Spectrum Induced by Hammer Excitation
(Døssing, 1988b)
As discussed in Section 4.5.4, the current experimentation included a research
component that examined the effect of the hammer weight on the quality and accuracy
of the free-vibration decay curves. In general, massive structures, with lower stiffness,
require the use of the extender (added weight) and soft impact tip (also employed) to
adequately excite low frequency resonance (Døssing, 1988b). For the current
experiments, an ICP® Impulse-Force Hammer (PCB® Piezotronics brand) was used to
excite the test beam into free-vibration. The technical specifications, for this hammer,
as provided by the manufacturer, are shown in Table 4.9.
Table 4.9. Specifications for ICP® Impulse-Force Hammer (PCB® Piezotronics, 1992) Model No: 086CO4Hammer Range: 8 kHz (approx)Hammer Range: 4400N (5V output) Hammer Sensitivity: 1.2 mV/N (approx) Resonant Frequency: 31 kHzHammer Mass (excluding counterweight): 163.1 gMass of Hammer Counterweight: 76.4 g
This particular type of hammer is commonly employed in industrial applications
because of its simple implementation and suitability for on-site monitoring. The impact
hammer supplies a short impulse, with a large frequency range, including the first
several resonance frequencies of the beam. Usually, in vibration monitoring, the first
several modes of the beam are most important, since their energy content is
comparatively large when the beam vibrates. A Fast Fourier Transform (FFT) of the
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-21
vibration decay curve signature, is used to convert the time signal into its component
frequencies, and confirms whether a single dominant 1st modal frequency response has
been induced in the beam. Details of the procedure for doing this will be presented in
Section 5.3.
Figure 4.13 shows the frequency response of a beam, excited by a hammer impact,
which is detected by a Model 353A Quartz Shear Mode Accelerometer (PCB®
Piezotronics). These accelerometers convert mechanical accelerations into proportional
electrical signals for measurement and analysis. The electrical signal that is generated,
is the result of a force imposed upon the quartz crystal by a mass, which undergoes
acceleration.
As shown previously in Figure 4.12, the TDS 460A Digitizing Oscilloscope (Tektronix,
1995) captured the acquisition of the vibration response. The waveform acquisition
details for the current experiments included:
A sampling rate of 5 kilosamples/second (5000 Hz) and a high-resolution
sampling mode. The sampling rate of the oscilloscope is, according to the
Shannon Sampling Theorem, required to be at least twice that of the fundamental
frequency (McClellan et al., 1999). For the current test beams the fundamental
frequency of the test beams was approximately 800 Hz, therefore the sampling
frequency of 5000 Hz more than satisfied this criterion (>1600 Hz).
The waveform record length selected was 450 data points; and
Edge triggering, was the waveform acquisition method used, and it is through this
method that the oscilloscope knows when to begin acquiring and displaying a
waveform.
Shown in Plate 4.3, is a photograph of the oscilloscope set-up during experimentation.
4.7.2 Crack Width
Both the instantaneous and residual crack widths were measured using a crack
microscope (correlating to the ‘load-on’ and ‘load-off’ testing phases). The crack
widths were measured at the level of the tension steel to maintain consistency in the
readings. For the prestressed beams, crack widths were measured at a similar distance
from the bottom beam edge as for the RC beams. The crack width microscope has an
accuracy of 0.005 mm.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-22
Plate 4.3. Test Set-Up of Oscilloscope during Experimentation
The measurement of crack widths, using the crack microscope, required the identifying
of a minimum of ten cracks per beam face, located within the constant moment region.
Each of these cracks were marked and measured at every ‘load on’ and ‘load off’
position. This approach ensured consistency and accuracy in the test measurements.
The experimental use of the crack microscope is shown in Plate 4.4.
4.7.3 Crack Patterns
A number of photographs were taken of the beams following failure. The primary
reason for this was to demonstrate the exact type of failure they experienced, to provide
information on the cracking pattern and to ensure that both beam sides experienced
symmetrical crack patterns. The photographs may be found in Appendix C.
4.7.4 Deflection
Deflection measurements were taken at the third points of the beam as shown in Figure
4.12. A linear variable differential transducer (LVDT) recorded the mid-span deflection
of the beam. The deflection at the third points under the loading beam nodes, was
monitored with dial gauges. This monitoring was to ensure that beam loading and
deflection was symmetrical about the longitudinal x- and y-axes.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 4: Experimental Programme 4-23
Plate 4.4. Crack Width Microscope
4.8 Summary
This Chapter described the extensive experimental programme for the investigation into
the damping, deflection and cracking behaviour of the reinforced and prestressed
concrete test beams. Geometrical and mechanical details of the experimental beams and
constituent materials were outlined, and the beam test set-up was explained.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-1
CHAPTER 5
A Method for Extracting Damping Capacity
5.1 General Remarks
This Chapter initially presents a detailed review of the experimental implementation of
the logdec technique (TLT) and describes the shortcomings of the use of the TLT by
previous researchers. To investigate these identified shortcomings, another analytical
technique based on the free-vibration decay curve, the “Decay Curve Method” (DCM),
was devised.
The next portion of the Chapter establishes a ‘set of rules’ governing the determination
and reporting of damping capacity to ensure confidence in the ‘accuracy’ of the
presented data.
Finally, the effect of experimental variables such as hammer weight and hammer
excitation position are also explored to determine if they themselves, affect the
calculation of logdec.
5.2 Experimental Techniques
A series of beam vibration tests, using a wide variety of beam types, was undertaken.
Four types of beams, impacted as various HEP, were used for the experiments as shown
in Figure 5.1. The different beam materials, shapes and sizes, were necessary to ensure
that the test findings were consistent and repeatable.
In this verification stage, all of the damping measurements were taken immediately
prior to testing, when the beam was in its final test set-up position.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-2
150250
100 100L=2400
CL Concrete Beam
RC Beam - Series CS
RC Beam - Series BL=6000
200300
100 100CL
Concrete Beam
200
PS Concrete Beam - Series PS
300
100 100L=6000
CL
Concrete Beam
Beam and Experimental Variables
+++ B1 to B640011001800AC B
B/15B/6B/3
+++ B760010001300AC B
B/10B/6B/4.6
+++ B863011001300AC B
B/10B/6B/4.6
+++ B9 & B10100012501500
AC B
B/6B/5B/4
+++ B11 & B12100012501500BD C
B/6B/5B/4
+750A
B/8
+ PS3 to PS61150
A
B/5.22
+++ PS7 to PS1075011501500AC B
B/8B/5.22B/4
+ CS1 to CS3450A
B/ 5
++ CS4450650AB
B/ 5 B/32/3
Steel Beam - Series S
200
100 100L=2000
CL Steel I-Beam130
105
Accelerometer
Hammer Weight(HW)
Hammer ExcitationPosition (HEP)
+ CS5 & CS 6450A
B/ 5
++ CS7450650AB
B/ 5 B/32/3
+ CS8 & CS9450A
B/ 5
SB1+++500600800B/4B/3B/2
HEP (distance in mm) from end of beam
HEP as a fractionof Total BeamLength (FTL)
KEY
+++75011501500
AC B
B/8B/5.22B/4
HEPReferenceNode
End ofBeam
Note: All dimensions are in mm
Figure 5.1: Experimental Beams and HEP Test Variables
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-3
It is well known from in-field damping experiments that the support conditions can
contribute greatly to the damping of the system as a whole (Leonard and Eyre, 1975).
Consequently, care was taken to ensure that there was minimal contact between the
testing rig and the concrete beam and this was generally why small-amplitude free-
vibrations were initiated when no static load was on the beams. Furthermore, the
supports were made significantly stiffer than the beam itself and this was achieved by
ensuring that the solid concrete supports were fixed to the floor.
The testing regime for this component of research consisted of a minimum of 10
hammer impacts, for each HEP/HW combination, in order to obtain high-quality free-
vibration decay signatures for one single logdec value. This approach was used
because, up to 10-20% of the hammer ‘hits’ can be spoiled due to the rattling
phenomenon from hard hits, or from ambient influences (Reynolds and Pavic, 2000).
Jones and Welch (1967) undertook a similar regime, taking the mean value of the
lowest set of logdec values because it was thought that these readings would be least
influenced by external influences.
5.3 Analytical Technique for Calculating Logdec
Equation 2.30 and Figure 2.7 presented the TLT experimental method for extracting the
logdec from a free-vibration decay curve. It is based on the assumption that the free-
vibration decay is purely viscous or exponential (Swamy, 1970). The TLT method is
very easy to use because viscous damping introduces a linear term into the equation of
motion, thus making the system very easy to analyse (Kelly, 1993).
It is, however, well known that in real-life one pure type of damping does not exist,
rather it is a combination of viscous, frictional or hysteretic damping (Bachmann et al.,
1995; Jeary, 1997a,b). However, the definition of logdec, as given by Equation 2.30,
holds true for whatever type of damping is present (Newland, 1989). However, some
conditions are required when using Equation 2.30 including:
That only one dominant mode of vibration is operating, meaning that there must
be a single-frequency response; and
That the decay of vibration must be strictly exponential, otherwise logdec, δ,
becomes a function of the absolute time at which it is measured and the systems
damping cannot be described by a single parameter (Newland, 1989).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-4
As mentioned in the introduction, a program called the Decay Curve Method (DCM),
based on free-vibration theory, was devised to compare and verify the results obtained
by the TLT. Complete details of the “Decay Curve Method” (DCM), are given in
Appendix D, but a summary of the DCM analytical algorithm is shown in Figure 5.2. A
flowchart of the output produced by the DCM is shown in Figure D.1.
Decay Curve Method
a) Obtain raw vibration curve data
b) Apply a FFT* to remove ‘noise’ to obtain fundamental frequency, f
c) Plot a line through a plot of the natural log of oscillation peaks. The
slope of this line
= -2π f δ (since ln(A× exp(-δ ωt)) = ln(A)+ -δ ωt)
d) Use equation of Decay Curve Envelope to extract damping ratio ξ
(= A e -ξω t). Then use logdec formula to determine logdec: δ = 2 π ξ
* FFT: The Fast Fourier Transform algorithm computes the frequency spectrum from the
time domain signal and returns the fundamental frequency.
Figure 5.2: Analytical Algorithm used by the DCM
Note that the focus here is on first modal damping, as much of the literature has justified
that first modal bending is of primary importance for dynamic calculations undertaken
during the design process (Jeary, 1974). Furthermore, first modal bending, as calculated
from tests, tends to be on the lower end of the range of damping, and therefore
conservative (Wheeler, 1982).
5.4 Applying the TLT and DCM Techniques
Every recorded free-vibration decay curve obtained from the oscilloscope was of a
standard 450 data points in length. To each free-vibration decay curve both the TLT and
DCM were applied, thus giving two logdec values for each curve. Both the TLT and
DCM involve the calculation of logdec at nominal points on the free-vibration decay
curve. These nominal data lengths (DL) are defined for the TLT in terms of the number
of cycles n that are contained within 50/450, 100/450, 150/450, 200/450, 250/450 and 300/450 data
point windows. Equation 2.30 requires the height of the nth peak, and to make sure that
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-5
the TLT and DCM outcomes were comparable, therefore the DL’s of both methods
needed to be roughly equal. Measurements ceased beyond the DL of 300/450 because of
the difficulty associated with reading low amplitude oscillations (Penzien, 1964). For
the DCM, the data length’s (DL) processed by the DCM are called the number of data
points or NDP contained in windows of 50/450, 100/450, 150/450, 200/450, 250/450 and 300/450 data
point lengths. These definitions are shown diagrammatically in Figure 5.3.
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Time (s)
Vib
ratio
n Am
plitu
de (
mV
)
50/450
100/450
150/450
200/450
250/450
300/450For the DCM:
Data Length (DL) isdefined in terms of
Number of Data Pointsor NDP
For TLT n=7(in 50/450 NDP)δ = 1/7 ln A1/An
A1
A7 For the TLT: Data Length (DL)is defined in terms
of Number ofCycles or n
For DCMNDP = 50/450
Figure 5.3: Definition of the Data Lengths Used by the TLT and DCM
Examples of the calculated logdec using the TLT and DCM for selected beams are
presented in Figures 5.4 and 5.5. Complete curves for every test beam may be observed
in Appendix D, Section D.2 and D.3, respectively and also Section D.4.
From Figure 5.4 it may be seen that there are significant variations in calculated logdec
depending on the number of cycles used. As the decay curve sample gets longer,
moving along the x-axis, the calculated logdec becomes smaller. Again, Figure 5.5
indicates there are variations in the calculated logdec, depending on the NDP used.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-6
11
1
1 11
77
7
7 7
7
B
B
BB B B
f
f
f
f f f
Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2SB1CS1CS7BII-2PS6
17Bf
Figure 5.4: Calculation of Logdec (TLT) using Cycle Number (n)
1
11
11
17 77
77 7
B
BB
BB B
f
f ff
ff
Number of Data Points (NDP)
Logd
ec(D
CM
)
0 100 200 3000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24 SB1CS1CS7BII-2PS6
17Bf
Figure 5.5: Calculation of Logdec (DCM) using NDP
5.4.1 Differences Between the TLT and DCM Output
Figures 5.4 and 5.5 indicate that the early portion of the curve produces significantly
elevated logdec values, as compared to the latter portions. This “NDP reduction effect”
appears to be more pronounced for the TLT than the DCM. The most obvious
assumption for the “NDP reduction effect”, is that the curve does not decay in a strictly
exponential manner. This is illustrated for a typical free-vibration decay curve in
Figure 5.6 where using the TLT, logdecs for every successive peak of the free-vibration
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-7
decay curve are calculated. It seems that logdec is not strictly exponential in the initial
portion but begins to decay exponentially after an initial period as shown in Figure 5.7.
When a beam is subjected to free vibration, the fundamental (800 Hz) and harmonic
modes (1200 Hz, 1600 Hz etc) are all excited. This has an effect on the initial portion
of Figure 5.7, because the higher harmonic modes override the fundamental mode. The
region where logdec becomes linear is the region where the fundamental frequency is
predominant, which occurs after the ‘non-exponential’ portion (i.e. when n > 10 in
Figure 5.7) of the curve.
In the literature, researchers using the TLT took their single logdec measurement from
somewhere (usually unknown) along the free-vibration decay curve. This location is
occasionally specified, but they vary enormously from researcher to researcher.
Penzien (1964) used n=4, whilst Cole and Spooner (1965) calculated logdec with n very
much greater than 10 (with the total number of oscillations specified as at least 10).
Cole (1966) measured An over ‘at least 40 cycles’. Leonard and Eyre (1975) used n=10
cycles, and despite observing that the plot of loge An versus decay cycle n did not
produce a straight line (i.e. non-viscous damping), they argued that for practical
purposes, an ‘average’ value of logdec is an adequate way of presenting damping
information. From Figure 5.6, a huge range of logdec values could be extracted from
the same curve, the question is: Which one is correct?
5.4.2 Proposed Rules for Calculating Logdec
In Figure 5.8a, the method of calculating the ratio between the initial (A1) and
subsequent peak (An) is shown. In Figure 5.8b, the effect of the peak ratio (An/A1) on the
calculated logdec for various test beams (using the TLT) is shown.
In Figure 5.8b, the last three nodes for each curve indicate where the decay becomes
exponential (according to the fundamental frequency), and it is at this point where
logdec would be most appropriately and consistently measured (see Figure 5.7). This
region, is termed the “optimal peak ratio An/A1” and produces a conservative, stable
estimate of damping capacity.
The “optimal peak ratio An/A1” region for each experimental specimen does vary for each
specimen, but nevertheless, it was found to range between a peak ratio of approximately
10 to 15% for all beam specimens. The “optimal peak ratio An/A1” curves for all the test
beams may be found in Appendix D, Section D.5, Figures D.9 to D.15.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-8
Time (ms)
Vib
ratio
nA
mpl
itude
(mV
)
0 0.01 0.02 0.03
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2 Logdec, δ = 1/n ln A1/An
δ=0.
2284
A1 = 0.222844
δ=0.2
401
δ=0.2
539
δ=0.0
999
δ=0.116
7
δ=0.110
6
δ=0.1
450
δ=0.101
3
δ=0.107
1
δ=0.1
119
δ=0.126
3δ=0.1
064δ=
0.1479δ=
0.1028
δ=0.1
039
δ=0.119
0
Non-Exponential Decay
Figure 5.6: Variation of Logdec (TLT) with n
Cycle Number, n
Nat
ural
Loga
rithm
ofA
mpl
itude
(lnA
n)
0 10 20 30-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Region of stableexponential decay
A1= ln(0.222844)
Non-exponentialdecay
Figure 5.7: Plot of Natural Logarithm of An versus Cycle Number
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-9
a)
Time, (ms)V
ibra
tion
Am
plitu
de, A . Vibration
Amplitudeat n=4
A4 = 0.5.
MaximumVibrationAmplitude
A1 = 1
A4
A1Peak Ratio =
= 0.5 1 = 0.5
= 50%
b) Peak Ratio An/A1 (%)
Logd
ec(T
LT)
0102030400
0.025
0.05
0.075
0.1
0.125
0.15
0.175SB1CS6BII-11PS9
DCM 300NDP
DCM 50NDP
DCM 100NDP
DCM 150NDP
DCM 200NDP
DCM 250NDP
Figure 5.8: Example Calculation of: a) Peak Ratio; and b) “Optimal Peak Ratio” Curves
Finally, it should be noted that Figure 5.8b would allow all types of logdec research to
be comparable. Thus, it will help avoid the current identified situation where it is
difficult to effectively utilise published data. The TLT has been employed for this
thesis.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-10
5.5 Effect Of Experimental Test Variables
Two primary variables were examined: a) hammer weight HW, and b) hammer
excitation position HEP. A summary of their effect is presented below.
5.5.1 Hammer Weight (HW)
In the current experimental tests, two different hammer weights were used (HW1=
163.1 g and HW2= 239.5 g). Figure 5.9 shows that logdec is unaffected by hammer
weight. It should be noted that by using different hammer weights, the actual excitation
force and response amplitude varied with each hammer impact. Thus, it can be
concluded that minor variations in impact weights during small-amplitude damping
laboratory experiments do not appear to affect the obtained values of damping.
o
oo o o o
v
v
v
v vv
A
A
A
AA
A
*
*
*
** *
Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 1000.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18SB1-With CWSB1-No CWCS6-With CWCS6-No CWPS10-With CWPS10-No CW
o
vA*
Figure 5.9: Effect of HW on Logdec (TLT)
5.5.2 Hammer Excitation Position (HEP)
Previous studies suggest that concrete beam damping should be relatively constant,
regardless of the hammer impact position (Chowdhury, 1999). It is difficult to
consistently impact the beam in the exact same positions, therefore researchers need to
be confident that the selection of hammer impact position will not affect the obtained
damping values. In Figure 5.10, it should be noted that the HEP’s are specified in
terms of FTL, these are described in Figure 5.1.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-11
Figure 5.10 shows all data points from selected beams that have been extracted from the
free-vibration decay curve at their “optimal peak ratio An/A1”. For Beams SB1, BII-11
and PS9 the optimal peak ratios are at 14%, 17% and 16%, respectively.
For each beam in Figure 5.10, the average logdec for the three HEP’s given.
Accompanying the average logdec is the range of all three points from the average
logdec. For beams SB1, BII-1 and PS9 all data is contained within 15%, 13 % and 15%
of the average, respectively. This is acceptable.
Hammer Excitation Position, HEP (FTL)
Logd
ec(T
LT)
0 2 4 6 8 100
0.02
0.04
0.06
0.08
0.1SB1- 200NDPBII-11- 200NDPPS9- 200NDP
L/L/ L/L/L/L/
PS9Average Logdec0.0489 ± 15%Optimal Peak Ratio= 16%
SB1Average Logdec=0.0336 ± 15%Optimal Peak Ratio= 14%
BII-11Average Logdec0.0494 ± 13%Optimal Peak Ratio= 17%
Figure 5.10: Effect of HEP on Logdec (TLT)
5.6 Summary
The experimental test results show that free-vibration decay in concrete beams is not
strictly exponential, as is required by the TLT logdec equation (Equation 2.30). The
literature has acknowledged this peculiarity of damping calculation, however no
meaningful research has been conducted to assist researchers in overcoming these
difficulties.
The non-exponential decay effect has been demonstrated well, herein, whereby logdec
decreases for increasing cycles. However, logdec starts to stabilise and become
consistent at some region within the free-vibration decay curve. This ‘stabilisation’
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 5: A Method for Extracting Damping Capacity 5-12
region is different for every beam, and it is recommended that it be determined from
trial and error solutions. Commonly, stabilisation occurs when the peaks have been
reduced to about 10-15% of their original height and this has been termed herein as the
“optimal peak ratio An/A1”.
Regardless of experimental logdec technique used it is extremely important to report
full and complete details. A diagram such as Figure 5.8b would allow researchers to
both understand the published damping data and be able to make meaningful
comparisons. As mentioned, it is extremely difficult for researchers to make
comparative reviews when presented with the ‘logdec’ of a specimen.
The test results showed that the experimental variables of hammer weight and hammer
excitation position did not produce any detectable impact on calculated damping. It is
recommended to excite the beam at reasonable distances from the support and loading
nodes and that a number of measurements be taken from various HEP locations with
different HW’s to verify that the obtained results are consistent and repeatable.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-1
CHAPTER 6
Damping Prediction in ‘Untested’ Concrete Beams
6.1 General Remarks
The purpose of this Chapter is to utilise the experimental ‘untested’ logdec test data to
derive an equation to predict the ‘untested’ damping capacity. Establishing the
‘untested’ damping capacity is important, as it is the initial component in the calculation
of the total damping capacity.
6.2 Damping In ‘Untested’ Reinforced Concrete Beams
In the ‘untested’ phase, the concrete beam has not yet received any service loading and
is considered to be completely undamaged.
6.2.1 Historical Review
A majority of early damping research was concerned with the damping capacity of the
concrete material itself, utilising tests on concrete cylinders. Figure 6.1 outlines
examples of four such research programs. These studies tried to ascertain if damping
could be used to determine concrete compressive strength or other material properties.
Kesler and Higuchi (1953) found that the concrete strength, fcm, could not be estimated
merely by determining the damping capacity of a cylinder.
Cole and Spooner (1968) established that the measurement of damping capacity is not
suitable for the quality control of concrete, particularly in view of Akashi’s (1960)
equation that the compressive strength, fcm, equals:
410346.079.1 −×⎟⎠⎞
⎜⎝⎛ +×=
δpcm Ef ±30% (6.1)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-2
where Ep is the dynamic modulus; δ is the logarithmic decrement, regardless of age,
water content and mix design. Cole and Spooner (1968) state that “assessing the
compressive strength to ±30% is not a particularly practical advantage for a
measurement of this complexity”.
Swamy and Rigby (1971) developed an equation to predict the damping capacity of
concrete cylinders based on the properties of the concrete constituents (see Equation
3.2), but would be difficult to apply to structural elements other than cylinders.
Sri Ravindrarajah and Tam (1985) looked at the variation of logdec with different types
of aggregates. For the same composition, logdec was found to increase with a decrease
in compressive strength, in the case of recycled aggregates, logdec increased by up to
27%. No prediction equation was developed because of insufficient data trends.
Evidently, it is difficult to utilise damping equations developed for concrete cylinders
for full-scale beams as they do not consider the effect of more important variables, such
as steel reinforcement, dimensions or support conditions.
Figure 6.1 also highlights the work of Dieterle & Bachman (1981) and Flesch (1981),
who developed equations to model damping in ‘untested’ beams. As discussed in
Chapter 2, their equations rely on constants that need to be established by experimental
investigations. This therefore defeats the purpose of providing practitioners with a
simple yet accurate means of modelling material damping prior to construction.
CYLINDERSCompare Dampingand fcm or Concrete
Constituents Swamy and Rigby (1971)δc=0.0174-0.1131/Ec-0.06401/Em+0.265 δm-0.0000913Vc
Sri Ravindrarajah and Tam (1985)δ increased for decrease in fcm (Aggregate type)
⎥⎦
⎤⎢⎣
⎡= 2
02 cm
cunVD fC
Edπ
ξ
Dieterle and Bachman (1981)BEAMS
Measure Beam Constituents to
Predict DampingFlesch (1981)
Kesler and Higuchi (1953)Measuringδ alone cannot predict fcm
Cole and Spooner (1968)E cannot be used to calculateδ
Figure 6.1: Classification Of Historical ‘Untested’ Damping Research
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-3
6.2.2 Experimental Effect of fcm and fsy
The effect of concrete compressive strength, fcm, and reinforcement yield strength, fsy, on
the ‘untested’ damping capacity are discussed here. Table 6.1 gives the experimentally
determined logdec of each specimen in the ‘untested’ condition along with relevant
beam variables, fcm, fsy and total longitudinal reinforcement distribution, LRD.
The LRD variable gives an indication of not only the amount of total longitudinal
reinforcement, but also of how it is distributed. Also, by creating the primary variable
of LRD, a normalised cross-sectional area is produced, allowing beams with different
cross-sections (b×d) and lengths (L) to be compared such as the B- and CS-Series
beams.
Note that logdec values in Table 6.1 range from 0.0361 to 0.0700, where Appendix E
gives full details of the logdec values obtained according to the optimal peak ratio
method as described in Section 5.4.2.
The effect of steel yield strength on damping capacity appears to be negligible as also
shown in Figure 6.2. Considering the beam pairs of BI-7/BII-8 and BI-9/BII-10, which
contain the same LRD and concrete compressive strength but differ in steel yield
strength, only slight increases in logdec have occurred for the higher strength
reinforcement.
In Figures 6.3a and 6.3b, the ‘untested’ logdec versus the longitudinal reinforcement
distribution (LRD) curve for the B- and CS-Series beams are presented. It may be seen
in both figures, that when the beam is in the ‘untested’ state, the damping capacity
appears not to be affected by concrete compressive strength, fcm, alone. Despite a
natural expected variation in concrete constituents and casting variability, damping
capacity remains relatively constant for each LRD group.
6.2.3 Proposed Damping Equation
Figure 6.4 shows the variation in logdec versus the total longitudinal reinforcement
distribution for all the ‘untested’ RC beams.
In Figure 6.4a, a general power-fit trend for both sets of data shows that damping
capacity increases as LRD increases (i.e. with the bar spacing getting smaller).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-4
Table 6.1. Experimental ‘Untested’ Damping Data – RC Beams
Beam Code
Logdec of ‘Untested’ Beam,
δuntest Appendix E
Concrete Compressive
Strength fcm (MPa)
Reinforcement Yield
Strength fsy (MPa)
Total Longitudinal Reinforcement
LRD #
(ρt /st)+ (ρt /sc)
BI-1 0.0648 30.0 400 0.00108 BII-2 0.0548 30.0 500 0.00044 BI-3 0.0585 23.1 400 0.00072 BII-4 0.0500 23.1 500 0.00021 BII-5 0.0573 41.5 500 0.00108 BII-6 0.0534 41.5 500 0.00044 BI-7 0.0361 64.5 400 0.00010 BII-8 0.0388 64.5 500 0.00010 BI-9 0.0444 53.0 400 0.00021 BII-10 0.0457 53.0 500 0.00021 BII-11 0.0518 90.7 500 0.00044 BII-12 0.0566 90.7 500 0.00108 CS1 0.0700 22.5 400 0.00222 CS2 0.0693 22.5 400 0.00222 CS4 0.0495 32.0 500 0.00054 CS5 0.0501 32.0 500 0.00054 CS6 0.0519 32.0 500 0.00054 CS7 0.0631 31.5 500 0.00082 CS8 0.0591 31.5 500 0.00082 CS9 0.0627 31.5 500 0.00101
# Total longitudinal reinforcement distribution is defined in Section 7.2.3 where ρt = Ast/bd and ρc = Asc/bd
GH
IJ
Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)
'Unt
este
d'Lo
gdec
,δun
test
0 0.0005 0.001 0.0015 0.002 0.00250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)
GHIJ
BI-9fsy = 400 MPa
BII-10fsy = 500 MPa
BI-7fsy = 400 MPa
BII-8fsy = 500 MPa
Figure 6.2: Variation of Logdec with Steel Yield Strength – B-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-5
a)
B EFK
L
Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)
'Unt
este
d'Lo
gdec
,δun
test
0 0.0005 0.001 0.0015 0.002 0.00250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
BII-2 (B)BII-5 (E)BII-6 (F)BII-11 (K)BII-12 (L)
BEFKL
BII-2fcm = 30.0 MPa
BII-6fcm = 41.5 MPa
BII-11fcm = 90.7 MPa
BII-5fcm = 41.5 MPa
BII-12fcm = 90.7 MPa
b)
12
456
78
9
Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)
'Unt
este
d'Lo
gdec
,δun
test
0 0.0005 0.001 0.0015 0.002 0.00250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
CS1CS2CS4CS5CS6CS7CS8CS9
12456789
CS4fcm = 32.0 MPa
CS5fcm = 32.0 MPa
CS6fcm = 32.0 MPa
CS1fcm = 22.5 MPa
CS2fcm = 22.5 MPa
Figure 6.3: Variation of Logdec with Concrete Compressive Strength: a) B-Series; b)
CS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-6
a)
A
AA
A
AA
AA
AA
A
A
BB
BBB
BB
B
Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)
'Unt
este
dLo
gdec
',δ u
ntes
t
0 0.0005 0.001 0.0015 0.002 0.00250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08RC BeamsCS Beams
AB
For CS-Series Beams
Y = e-1.2811458 * X 0.22185349
For B-Series Beams
Y = e-1.5140317 * X 0.18673056
b)
A
AA
A
AA
AA
AA
A
A
BB
BBB
BB
B
Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)
'Unt
este
d'Lo
gdec
,δun
test
0 0.0005 0.001 0.0015 0.002 0.00250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
RC BeamsCS BeamsEquation 6.2
AB
For RC Beams
δuntest = 0.223 * (ρt /st+ρc /sc)0.19
R2 = 0.90
Figure 6.4: Dependence of ‘Untested’ Logdec on LRD in RC Beams: a) Separate
Trendlines; b) Unified Prediction Equation
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-7
In other words, for an equivalent LRD, large amounts of smaller bars give greater
damping capacities than fewer bigger bars (c.f. CS1 versus CS8 or BII-2 versus BII-4).
The R2 values of 0.89 and 0.87 for the B- and CS-Series beams respectively, indicates a
good measure of correlation. The identical nature of the two curves for the B- and CS-
Series in Figure 6.4a suggests that total beam length does not affect damping capacity.
Since the two trendlines in Figures 6.4a are reasonably similar, and independent of
length, a single equation was fitted to both sets of data as shown in Figure 6.4b. The
single equation for estimating the damping capacity of ‘untested’ reinforced concrete
beams is
19.0
223.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛+×=
c
c
t
tuntest ss
ρρδ (6.2)
where st and sc are the tension and compression reinforcement spacings, respectively in
mm and are given in Table 4.1; ρt = Ast/bd and ρc = Asc/bd. The equation is valid for
LRD distributions between 0.0001 and 0.0023 (see Table 6.1).
The relationship of Equation 6.2 to the current data set may be observed in Figure 6.4b.
As discussed previously, comparing this prediction equation to existing published
damping data for verification would be difficult. However, an attempt at doing so is
made later within this Chapter where further verification is undertaken using the F-
Series beam tests and comparisons to published experimental data.
6.3 Damping in ‘Untested’ Prestressed Concrete Beams
The following section will discuss the ‘untested’ damping characteristics of the PS-
Series, 6-metre prestressed concrete beams.
6.3.1 Historical Review
Only three researchers have examined damping in full-prestressed concrete (PSC)
beams. James et al. (1964) and Penzien (1964) did not examine damping prestressed
concrete beams in the ‘untested’ state at all, whilst Hop (1991) found that an increase in
prestressing caused a decrease in the beam’s logdec, δ as illustrated by his Equation 3.9.
The equation did not relate logdec to any particular beam state and it was unclear
whether the equation related to the ‘untested’ or ‘tested’ beam. This equation will be
examined in more depth in Section 6.3.3.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-8
6.3.2 Experimental Observations
In Table 6.2, the experimentally determined logdec of each PSC specimen in the
‘untested’ condition is given along with relevant beam variables, fcm, H and e. Appendix
E gives full details of the logdec values obtained. Note that logdec values range from
0.0412 to 0.0547.
In Figures 6.5 and 6.6, the relationships between logdec and prestressing force, H, and
logdec versus prestressing eccentricity, e, respectively, may be observed. From both
figures it seems that neither prestressing force or eccentricity ‘alone’ provides a
consistent trend to model the ‘untested’ logdec. It is proposed in Section 6.3.4 that the
total initial prestress, He, gives a better prediction of logdec.
6.3.3 Hop’s Prestressed Equation
Figure 6.7 gives a plot of the ‘untested’ logdec values from the current experiments
(using the initial prestress value, H) compared to predicted logdec values using Hop’s
(1991) Equation 3.9. The location of the data points in Figure 6.7 indicates the
following:
Equation 3.9 consistently overestimates the damping values of the current beams at
the pre-test stage; and
Equation 3.9 suggests that beams with a higher prestressing force give lower
amounts of damping. This was not found for the current tests as indicated in Table
6.2 and Figure 6.5.
Table 6.2. Experimental Prestressed ‘Untested’ Damping Data
Beam Code
Logdec of ‘Untested’
Beam, δuntest Appendix E
Concrete Compressive
Strength fcm (MPa)
Table 4.5
Prestressing Force
H (kN)
Prestressing Eccentricity
e (mm)
Total Initial Prestress
He (kNmm)
PS3 0.0420 60.2 346 97.0 33,562.0
PS4 0.0478 60.2 585 80.5 47,092.5
PS5 0.0430 69.8 612 60.0 36,720.0
PS6 0.0467 69.8 400 99.7 39,880.0
PS7 0.0565 52.5 450 97.5 43,875.0
PS8 0.0412 52.5 400 80.0 32,000.0
PS9 0.0424 83.5 346 90.0 31,140.0
PS10 0.0547 83.5 480 90.0 43,200.0
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-9
x
x
x
x
x
xx
x
Prestressing Force, H (kN)
'Unt
este
d'Lo
gdec
,δun
test
200 300 400 500 600 700 8000.03
0.04
0.05
0.06
All PS Beamsx
Figure 6.5: Prestressing Force versus ‘Untested’ Logdec for PS-Series Beams
x
x
x
x
x
xx
x
Prestressing Eccentricity, e (mm)
'Unt
este
d'Lo
gdec
,δun
test
50 75 100 1250.03
0.04
0.05
0.06
All PS Beamsx
PS7
PS10
PS4
PS6
PS9PS3
PS8
PS5
Figure 6.6: Prestressing Eccentricity versus ‘Untested’ Logdec for PS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-10
The current evaluation has demonstrated that, contrary to the findings of Hop (1991), a
damping prediction equation for the ‘untested’ beam appears not to be dependent upon
the single variable of prestressing force, H alone. In view of this, the development of a
more appropriate logdec prediction equation incorporating the eccentricity of prestress,
e, along with prestressing force, H, is undertaken in Section 6.3.4.
x
xx
x
x
xx
x
Untested Logdec By Hop (1991), δ
'Unt
este
d'Lo
gdec
,δun
test
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1All PS Series Beamsx
Figure 6.7: ‘Untested’ Logdec versus Logdec Prediction using Hop (1991)
6.3.4 Proposed Damping Equation
For the current prestressed test beams in the ‘untested’ state, the most obvious
distinguishing feature is the overall location and amount of prestressing that exists in
the beam. That is, it is not only the amount of prestressing force, H, significant, but
how it is located within the beam (i.e. its eccentricity, e). Shown in Figure 6.8, this
variable, He, is plotted against the ‘untested’ logdec of each beam. This variable He
gives an excellent correlation to the ‘untested’ logdec (R2 = 0.99), as shown by the
second order polynomial curve shown in Figure 6.8, and given by
δuntest = 1.4(×10-10)He2 – 9.4(×10-6)He + 0.2 (R2 = 1.0) (6.3)
where Equation 6.3 is valid for He between 30,000 and 45,000 (kNmm).
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-11
xx
x
x
xx
x
Total Initial Prestress, He (kNmm)
'Unt
este
d'Lo
gdec
,δun
test
30000 35000 40000 45000 500000.03
0.04
0.05
0.06
All PS Beamsx
Figure 6.8: ‘Untested’ Logdec versus Initial Prestress in Beam
6.4 Verification
Three verification checks of the proposed reinforced concrete logdec equation are
presented here using, a) the original beam data, b) the test data from the additional F-
Series beams. The final check, c) uses a published free-decay curve from Nield (2001).
6.4.1 Original Beam Data
The overall performance of both Equations 6.2 and 6.3 may be represented by the
scattergram shown in Figure 6.9. As may be seen, all values fall within the ±20%
envelope, and in fact close to the 45o line, indicating a very good prediction.
6.4.2 F-Series Beams
As the F-Series beams are of reinforced concrete only, Equation 6.2 is investigated here.
Shown in Figure 6.10 are the experimental versus calculated ‘untested’ logdec values
for the F-Series reinforced concrete beams. As seen, the ‘untested’ logdec values
predicted by Equation 6.2 are very good, with all the values within ±20%. A slightly
higher damping is given by Equation 6.2, indicating a conservative prediction.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-12
Calculated 'Untested' Logdec
Expe
rimen
tal'
Unt
este
d'Lo
gdec
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
B-SeriesCS-SeriesPS-Series
-20%
+20%
Figure 6.9: Comparison between Experimental Logdec and Logdec Calculated by
Equation 6.2 and 6.3
Calculated 'Untested' Logdec
Expe
rimen
tal'
Unt
este
d'Lo
gdec
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
All F-Series Beams
-20%
+20%Prediction Equationδuntest=0.223x(ρt/st +ρc/sc)
0.19
Figure 6.10: Observed versus Predicted ‘Untested’ Logdec for the F-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-13
6.4.3 Neild’s Beam
The use of non-linear vibration techniques, for the detection of damage in a reinforced
concrete beam, under low-amplitude, cyclic loading was reported by Neild (2001).
Free-vibration experiments were performed at incremental levels of damage loading,
where the impact excitation was applied 70 mm away from mid-span using an
instrumented (5.45 kg) sledge hammer dropped from a height of approximately 50 mm
above the beam. The cross-sectional details of Neild’s (2001) test beam, and test set-up
are given in Figure 6.11. Using Equation 6.2, the ‘untested’ damping capacity may be
established for the single beam as:
19.0
4.1380027.0
2.570161.0223.0 ⎟
⎠⎞
⎜⎝⎛ +×=untestδ
∴ δuntest,calc = 0.0478
In the free-vibration experiments, Neild (2001) did not actually calculate the damping
capacity, but did present a mid-span, free-vibration decay record for the beam prior to
testing. This type of published information is very rare. Using this free-vibration decay
record, the ‘optimal peak ratio’ technique was applied, the results of which are shown
below in Figure 6.12. From the figure, the δuntest,exp, can be established as being
between 0.043 and 0.049 and therefore the ratio between δuntest,calc/δuntest,exp is 1.11 and
0.98, respectively. This is more than acceptable.
200 mm
105 mm 3@12mm fsy = 410 MPa
2@6mm fsy = 240 MPa
Pin Roller
L = 3000 mm
l = 2800 mmst = 57.2 mm
sc = 138.4 mmStirrups =
4.8@80mm
f’c = 30 MPa
Three-point loading, withdamage levels in steps of
10% of failure load
Brüel and Kjær 4382 accelerometersat half and quarter span points
Figure 6.11: Details of Test Beam and Testing Arrangement of Neild (2001)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-14
Peak Ratio An/A1 (%)
Logd
ec,δ
(TLT
)
01020304050600
0.02
0.04
0.06
0.08
0.1
0.12
Neild (2001)
Region ofOptimal Peak Ratio
δ = 0.043
δ = 0.049
Figure 6.12: ‘Optimal Peak Ratio’ Analysis of Neild’s (2001) Free-Decay Curve
6.5 Summary
A critical review of the pertinent existing literature on concrete beam damping capacity
has highlighted a major omission in available ‘untested’ logdec computation methods
for both reinforced and prestressed concrete beams. From an analysis and design point
of view, the initial establishment of the ‘untested’ damping capacity is extremely
important.
From the extensive experimental investigation, an equation has been proposed that
illustrates a strong correlation between the damping capacity of ‘untested’ reinforced
concrete beams and the quantity and distribution of the longitudinal reinforcement of
the beam. The developed equation incorporates both tension and compression
longitudinal reinforcement ratios and respective spacing values. The effects of concrete
compressive strength and reinforcement yield strength were found to exert negligible
impact on the ‘untested’ damping capacity for the current series of reinforced concrete
beam tests.
For the prestressed concrete beams, the prediction of the ‘untested’ logdec is related to
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-15
the initial total prestress in the beam, He by a second order polynomial equation.
Although a damping equation was proposed, it is expected that future research will
allow further improvement of the developed equation.
Verification of the reinforced concrete ‘untested’ damping prediction equation using the
additional F-Series experimental test data and published damping data of Nield (2001)
has shown the proposed equation to be reliable.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-1
CHAPTER 7
Residual Deflection Mechanisms in Concrete Beams
7.1 General Remarks
This Chapter develops and verifies residual deflection equations for both the reinforced
and prestressed concrete beams. The prediction of the ‘tested’ damping capacity is
based upon the calculation of the residual deflection as developed for the current
experimental programme. As discussed in Section 3.2.4, the computation of the residual
deflection is to be directly related to the instantaneous deflection. Sections F.1 and F.2
of Appendix F present respectively, the raw instantaneous and residual deflection
experimental data. Figures 7.1 to 7.4 were produced using this raw data.
7.2 Residual Deflection in Reinforced Concrete Beams
This section considers the effect of a) fsy, b) fcm, c) ρt and d) loading conditions on the
residual deflection of concrete beams.
7.2.1 Effect of fsy
Figure 7.1a demonstrates the relationship between the experimental instantaneous
deflection, ∆i,exp, and the experimental residual deflection, ∆r,exp, and highlights the
effects of the two different grades of reinforcing steel. As indicated in Table 4.1, each
of the beam pairs contained identical geometrical and mechanical details. As shown in
Figure 7.1a, the initial linear elastic portion for each beam pair was very similar. In
Figure 7.1b it may be seen that the 500 MPa beams attained greater residual deflection
at failure.
As discussed previously, because 500 MPa reinforcing steel is the primary reinforcing
steel available in the current Australian market, the remaining discussion will focus on
the characteristics of this steel.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-2
7.2.2 Effect of fcm
The curves in Figure 7.2a show similar instantaneous versus residual deflection
characteristics suggesting that fcm alone does not impact significantly on the residual
deflection characteristics of the beam. Examining the normalised1 moment versus
residual deflection curve in Figure 7.2b, it may be seen that for the portion of each
curve up until about 80% of the ultimate load the curves are very similar, again
indicating that the effect of concrete compressive strength alone is minimal.
7.2.3 Effect of ρt
Figure 7.3a, compares beams containing different percentages of tensile reinforcement
(BII-5 versus BII-6 and BII-11 and BII-12). It shows that for a particular instantaneous
deflection, the beam containing lower amounts of reinforcing steel exhibit higher levels
of residual deflection. Interestingly however, the total residual deflection at failure, for
each beam pair was very similar. This can be verified in Figure 7.3b, where at an
equivalent normalised moment level, the curves are remarkably similar.
7.2.4 Effect of Loading Conditions
The CS-Series flexural beams were tested to investigate the variable of loading beam
width, LBW (as given in Table 4.8). The tensile and compressive reinforcement ratios,
and the concrete compressive strengths were very similar, with the tensile reinforcement
yield strengths being either 400 MPa or 500 MPa for these beams.
At first glance, Figure 7.4a seems to indicate that varying LBW does not affect the
instantaneous versus residual deflection characteristics. However, Figure 7.4b clearly
indicates that at an equivalent bending moment below the region of yielding say around
0.5 or 50% of ultimate the loading condition does impact on the residual deflection
characteristics of the CS-Series flexural beams. It can be concluded therefore, that the
calculation of the instantaneous deflection incorporates the effect of loading conditions.
This further demonstrates that the calculation of instantaneous deflection is an excellent
method for the prediction of the residual deflection characteristics.
7.2.5 Summary of Effects
In conclusion, it was found that ρt and LBW affect instantaneous and residual deflection.
1 Normalised moment equals the mid-span service moment divided by the failure moment.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-3
a)
G
GGGGG
G
G
HH
HHHHHHH
H
H
I
I
I
I
I
I
J
JJJJ
JJJ
J
J
J
J
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 30 40 50 600
25
50
75
100
125
BI-7 (fsy = 400 MPa)BII-8 (fsy = 500 MPa)BI-9 (fsy = 400 MPa)BII-10 (fsy = 500 MPa)
GHIJ
Variable = fsy
b)
G
G
G
G
G
G
G
G
H
HH
HHH
H
H
H
H
H
I
I
I
I
I
I
J
J
J
J
J
J
J
J
J
JJ J
Mid-Span Residual Deflection (mm)
Mi-S
pan
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)
GHIJ
Figure 7.1: Effect of Reinforcement Yield Strength for B-Series Beams On Residual
Deflection Versus: a) Instantaneous Deflection; b) Normalised Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-4
a)
B
BB
BBBBB
BB
B
D
D
D
D
D
D
D
D
F
F
F
F
FF
FF
J
JJJJ
JJJ
J
J
J
J
K
K
KK
K
K
K
K
K
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 30 40 50 600
25
50
75
100
125
BII-2 (ρ =1.78%, fcm=30.0 MPa)BII-4 (ρ =1.73%, fcm=23.0 MPa)BII-6 (ρ =1.78%, fcm=41.5 MPa)BII-10 (ρ =1.73%, fcm=53.0 MPa)BII-11 (ρ =1.78%, fcm=90.7 MPa)
BDFJK
Variable = fcm
500 MPaSteel Only
b)
B
B
B
B
B
B
B
B
B
B
B
F
F
F
F
F
FF
F
K
K
K
K
K
K
K
K
K
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BII-2 (B)BII-6 (F)BII-11 (K)
BFK
90.7 MPa
30.0 MPa 41.5 MPa
Figure 7.2: Effect of Concrete Compressive Strength for B-Series Beams On Residual
Deflection Versus: a) Instantaneous Deflection; b) Normalised Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-5
a)
E
E
E
E
E
E
E
F
F
F
F
FF
FF
K
K
KK
K
K
K
K
K
L
LLLLLL
LL
L
L
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 30 40 50 600
25
50
75
100
125
BII-5 (ρt = 2.38%)BII-6 (ρt =1.78%)BII-11 (ρt =1.78%)BII-12 (ρt =2.38%)
EFKL
Variable = ρt
2.38%90.7 MPa
1.78%90.7 MPa
2.38%41.5 MPa
1.78%41.5 MPa
500 MPaSteel Only
b)
E
E
E
E
E
EE
F
F
F
F
F
FF
F
K
K
K
K
K
K
K
K
K
L
L
L
L
L
L
L
L
L
LL
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BII-5 (E)BII-6 (F)BII-11 (K)BII-12 (L)
EFKL
Figure 7.3: Effect of Tensile Reinforcement Ratio for B-Series Beams On Residual
Deflection Versus: a) Instantaneous Deflection; b) Normalised Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-6
a)
1
1
1
2
2
2
3
3
3
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
9
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 2 4 6 8 10 120
5
10
15
20
25
30
CS1 (a=1150mm)CS2 (a=950mm)CS3 (a=750mm)CS7 (a=1050mm)CS8 (a=800mm)CS9 (a=700mm)
123789
b)
1
1
2
2
3
3
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
9
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
Figure 7.4: Effect of LBW for CS-Series Beams On Residual Deflection Versus: a)
Instantaneous Deflection; b) Normalised Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-7
7.2.6 The Proposed Equation
As discussed in Section 3.2.4, there is no established method by which to calculate the
residual deflection, ∆r for the current series of test beams.
A plot of the experimental instantaneous ∆i,exp versus residual deflection ∆r,exp is given
in Figures 7.5a and 7.5b, for all flexural B-Series and CS-Series test beams (with 500
MPa reinforcement only). The main variable influencing the slope of the curve is the
tensile reinforcement ratio, ρt. Note that the data is only plotted prior to the yielding
moment, after which it is not considered linear. In Figures 7.5a and 7.5b, each beam’s
instantaneous verus residual deflection curve was fitted to the linear equation x = αrc ⋅ y,
where y and x are the instantaneous and residual deflections, respectively. The value of
αrc is the residual deflection curve coefficient for reinforced concrete beams.
In Figure 7.6 the αrc values from Figure 7.5 are plotted against the tensile
reinforcement ratio. Even though beams of different sizes are shown, i.e. the B-Series
beam are 6.0 m in length and the CS-Series beams are 2.4 m, a reasonably correlated
linear relationship has been fitted to all data points, thereby indicating that beam size
does not significantly affect the calculation of the curve coefficient, αrc.
Therefore prior to yielding, an estimate of the residual deflection of a flexural reinforced
concrete beam having a length between 2.4 m and 6.0 m, reinforcement yield strength
of 500 MPa and variable concrete compressive strength, may be found from the
instantaneous deflection of that beam by the relationship
∆r,calc = αrc ×∆i,exp (7.1)
where αrc is the curve coefficient found from Equation 7.2 and derived from Figure 7.6
and ∆r,calc and ∆i,exp are in mm.
αrc = -0.08ρt + 0.39 (7.2)
where Equation 7.2 is valid for 0.76% < ρt < 3.0% .
7.3 Residual Deflection in Prestressed Concrete Beams
The prestressed beam pair comparisons, shown in Figures 7.7, 7.8 and 7.9, examine the
effect of the beam test variables, fcm, H and e on the residual deflection characteristics.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-8
The figures utilised the raw data presented in Table 4.2 and Sections F.1 and F.2 of
Appendix F.
a)
B2x=0.22y
R2 = 0.95
B4x=0.20y
R2 = 0.98
B5x=0.17y
R2 = 0.99
B6x=0.23y
R2 = 0.95
B8x=0.37y
R2 = 0.90
B10x=0.24y
R2 = 0.89
B11x=0.23y
R2 = 0.87
B12x=0.19y
R2 = 0.95
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
0 2 4 6 8 10 12 14 16 18 20
Mid-Span Residual Deflection (x)
Mid
-Spa
n In
stan
tane
ous
Def
lect
ion
(y)
BII-2
BII-4
BII-5
BII-6
BII-8
BII-10
BII-11
BII-12
Linear (BII-2)
Linear (BII-4)
Linear (BII-5)
Linear (BII-6)
Linear (BII-8)
Linear (BII-10)
Linear (BII-11)
Linear (BII-12)
Linear (BII-12)
b)
CS8x=0.19y
R2 = 0.99
CS9x=0.18y
R2 = 0.99
CS7x=0.12y
R2 = 0.93
0
5
10
15
20
25
30
0 1 2 3 4 5 6 7 8
Mid-Span Residual Deflection (x)
Mid
-Spa
n In
stan
tane
ous
Def
lect
ion
(y)
CS7
CS8
CS9
Linear (CS8)
Linear (CS9)
Linear (CS7)
Slope of Curve = αrc
Figure 7.5: Effect of Reinforcement Ratio on the Instantaneous versus Residual
Deflection Relationship: a) B-Series; b) CS-Series
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-9
BD
E
F
H
JK
L
7
89
Tensile Reinforcement Ratio, ρt
Cur
veC
oeff
icie
nt,α
rc
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
BII-2BII-4BII-5BII-6BII-8BII-10BII-11BII-12CS7CS8CS9αrc = - 0.08ρt + 0.39
BDEFHJKL789
R2 = 0.78
(%)
Figure 7.6: Selection of Curve Coefficient, αrc for the Calculation of Residual
Deflection
7.3.1 Effect of fcm and e
The first examination, made in Figures 7.7a and 7.7b, is of the combined effect of fcm
and prestress eccentricity, e, on the residual versus instantaneous deflection relationship.
Figures 7.7a and 7.7b indicate that the residual deflection trend for prestressed concrete
beams with the same H and varying combined effect of fcm and e is inconclusive.
Figure 7.7c shows similar normalised mid-span bending moment versus residual
deflection trends, prior to the occurrence of yielding, for the four beams shown.
7.3.2 Effect of H
Figure 7.8a shows that the beam with less prestressing force, H, only (PS9) exhibits
marginally less residual deflection after an equivalent instantaneous deflection. It
should be noted here that only that portion of the curve prior to the occurrence of
yielding is shown. This effect is confirmed in Figure 7.8b, where at a normalised
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-10
bending moment of say 0.5 (50% of ultimate), beam PS9 exhibits less residual
deflection.
7.3.3 Effect of H and e
Figure 7.9a and 7.9b shows the beam with less overall total prestressing, He (PS5)
exhibits more residual deflection than PS6. This is confirmed by the normalised mid-
span bending moment versus residual deflection plot in Figure 7.9b
7.3.4 Summary of Effects
Clearly, there is insufficient data (or significant trends) to identify if residual deflection
is directly impacted on by any particular prestressing variable. Consequently, all PS-
Series beams will be grouped together for the determination of the curve coefficient that
relates the residual deflection to the instantaneous deflection.
7.3.5 The Proposed Equation
A plot containing the instantaneous versus residual deflection curves for all PSC beams
(except PS4, which contained compression reinforcement and is omitted from these
calculations) is given in Figure 7.10.
The form of the equation to the right of the figure is x = αps y, where y and x are the
instantaneous and residual mid-span deflections, respectively. The value of αps is the
residual deflection curve coefficient for prestressed concrete beams.
Using the coefficients obtained in Figure 7.10, the average of αps is found to be 0.09
with a standard deviation of 0.02. This average value process is used as more
supporting experimental data is required before αps can be conclusively associated to a
particularly beam variable (as was done in Section 7.2.6 for the reinforced concrete
beams).
Thus, for the current PSC beams, the residual deflection may be approximated from the
instantaneous deflection by
∆r = 0.09 ∆i (7.3)
where ∆r and ∆i are the residual and instantaneous deflections in mm, at a given service
loading condition, respectively.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-11
a)
cc
c
c
i ii
i
i
i
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 300
25
50
75
100
125
PS3PS9
ci
PS3H=346 kNe=97.0 mmfcm= 60.2 MPa
PS9H=346 kNe=90.0 mmfcm= 83.5 MPa
Compare fcm and e
b)
f
f
f
hhh h
hhhh
hh
h
h
h
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 300
25
50
75
100
125
PS6PS8
fh
Compare fcm and ePS6H=400 kNe=99.7 mmfcm=69.8 MPa PS8
H=400 kNe=80.0 mmfcm= 52.5 MPa
c) c
c
c
c
f
f
f
h
h
h
h
h
hhh
hh
hh h
i
i
i
i
i
i
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PS3 (c)PS6 (f)PS8 (h)PS9 (i)
cfhi
Figure 7.7: Residual Deflection for PS3, PS6, PS8 and PS9 Versus: a) and b) Concrete
Compressive Strength and Prestress Eccentricity; c) Normalised Mid-Span Bending
Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-12
a)
i ii
i
i
i
j jjj j
jjjjj
j
j
j
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 300
25
50
75
100
125
PS9PS10
ij
PS9H=346 kNe=90.0 mmfcm= 83.5 MPa
PS10H=480 kNe=90.0 mmfcm= 83.5 MPa
Compare H
b)
i
i
i
i
i
i
j
j
j
jj
jjj
jj
j
jj
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PS9 (i)PS10 (j)
ij
Figure 7.8: Residual Deflection for PS9 and PS10 Versus: a) Prestressing Force; b)
Normalised Mid-Span Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-13
a)
e
ee
e
f
f
f
Mid-Span Residual Deflection (mm)
Mid
-Spa
nIn
stan
tane
ous
Def
lect
ion
(mm
)
0 10 20 300
25
50
75
100
125
PS5PS6
ef
PS6He = 39,880fcm= 69.8 MPa
PS5He = 36,720fcm= 69.8 MPa
Compare H and e
b)
e
e
e
e
f
f
f
Mid-Span Residual Deflection (mm)
Mid
-Spa
nB
endi
ngM
omen
t(N
orm
alis
ed)
0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PS5 (e)PS6 (f)
ef
Figure 7.9: Residual Deflection for PS5 and PS6 Versus: a) Prestressing Force and
Prestressing Eccentricity; b) Normalised Mid-Span Bending Moment
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-14
PS3x=0.08y
R2 = 0.90
PS5x=0.11y
R2 = 0.99
PS6x=0.06y
R2 = 0.99
PS7x=0.08y
R2 = 0.99
PS8x=0.10y
R2 = 0.98
PS9x=0.09y
R2 = 0.97
PS10x=0.12y
R2 = 0.950
10
20
30
40
50
60
70
80
90
0 2 4 6 8 10 12
Mid-Span Residual Deflection (x)
Mid
-Spa
n In
stan
tane
ous
Def
lect
ion
(y)
PS3
PS5
PS6
PS7
PS8
PS9
PS10
Linear (PS3)
Linear (PS5)
Linear (PS6)
Linear (PS7)
Linear (PS8)
Linear (PS9)
Linear (PS10)
Figure 7.10: Correlation between Instantaneous and Residual Deflection for PSC Beams
7.4 Verification
Three verification checks of the reinforced concrete residual deflection prediction
equation are presented using, a) the original beam data to recheck that the developed
equations are satisfactory, b) the test data from the F-Series beams, and c) the published
experimental data of James (1997).
7.4.1 Original Beam Data
The overall performance of Equations 7.1 and 7.3 may be represented by the
scattergram shown in Figure 7.11. The following can be observed:
84% of the RC beam deflection data points in Figures 7.11a and 7.11b fall within
the ±20% envelope indicating satisfactory prediction of the reinforced concrete
residual form;
In Figure 7.11c, 68.4% of the PSC beam data falls within the envelope. This is
reasonable considering that the residual deflection equation (Equation 7.3) was
made on limited data and it was recommended that more measurements were
required.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-15
a)
A
A
AA
AA
AA
A
A
A
A
B
B
B
B
BB
BB
B
C
C
C
C
C
C
C
D
D
D
D
E
E
E
E
F
F
F
F
F
G
G
GG
GG
H
H
H
HH
HH
I
I
I
I
J
J
JJ
J
J
J
K
K
KK
K
K
L
L LL
L
Calculated Residual Deflection (mm)
Expe
rimen
talR
esid
ualD
efle
ctio
n(m
m)
0 2 4 6 8 10 12 140
2
4
6
8
10
12
14
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
r,calc = αrc i,exp (7.1)
αrc = -0.08ρt + 0.39 (7.2)
20%
20%
b)
1
1
2
2
3
3
4
4
4
4
5
55
6
6
6
7
7
7
7
8
8
8
8
9
9
9
9
Calculated Residual Deflection (mm)
Expe
rimen
talR
esid
ualD
efle
ctio
n(m
m)
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
r,calc = αrc i,exp (7.1)
αrc = -0.08ρt + 0.39 (7.2)
20%
20%
c)
cc
cc
e
e
e
e
f
f
f
gg
gg
gg
g
g
hh
h
h
h
hh h
h
h
h
h
i
ii
i
i
j
jj
jj
jj
j
j
Calculated Residual Deflection (mm)
Expe
rimen
talR
esid
ualD
efle
ctio
n(m
m)
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
PS3PS5PS6PS7PS8PS9PS10
cefghij
r,calc = 0.09 i,exp (7.3)
20%
20%
Figure 7.11: Experimental versus Calculated Residual Deflection for a) B-Series; b) CS-
Series; and c) PS-Series Test Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-16
7.4.2 F-Series Beams
In Figure 7.12 the experimental versus calculated mid-span residual deflection
scattergram for the additional F-Series reinforced concrete beams, using Equation 7.1
are presented. It seems that at low residual deflections (< 0.5 mm) the scatter of the
data suggests the measurements may have larger margins of error.
Two general observations can be made regarding Figures 7.11 and 7.12:
The residual deflection around the region of first cracking (i.e. at small residual
deflections) was either over- or under-predicted. In spite of this, and considering
the expected variation in deflection measurements at smaller loads, the linear
relationships proposed appear overall to be more than acceptable; and
The prediction of the residual deflection post-cracking was good, where almost all
data values fall within the ±20% envelope. This was generally after 0.5 mm
residual deflection for all beams.
q
qqqq
rr
rr
r
rr
rr
rr
r r
r
r
s
ss
s
ss s
s
ss
s
s
ttt
tt
tt
tt t
t tt
uuuuuu
uu
u u uu
uu
v
vvv
vv
vv
w
ww w
w ww
x
x
xx
xx
x
x
x
y
y
y
y
y
y
y
z
z
z
z
z
z
Calculated Residual Deflection (mm)
Expe
rimen
talR
esid
ualD
efle
ctio
n(m
m)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
F1F2F3F4F5F6F7F8F9F10
qrstuvwxyz
r,calc = αrc i,exp (7.1)
αrc = -0.08ρt + 0.39 (7.2)
20%
20%
Figure 7.12: Experimental versus Calculated Residual Deflection for F-Series Test
Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-17
7.4.3 James’ Beams
James (1997) undertook an extensive study of the calculation of the instantaneous
deflection using the AS3600-1994 Concrete Structures Code. Details of James’ (1997)
reinforced concrete box beams tested are presented in Table 7.1.
Tables 7.2 to 7.6 present the instantaneous and residual experimental test results of
James (1997) and details of the calculated residual deflection, at every incremental
residual load level, using proposed Equations 7.1 and 7.2. Obtaining tabulated residual
deflection data conducted under similar testing regimes is very rare.
The average ratios ∆r,calc/∆r,exp are significantly close to unity, with acceptable variations,
indicating that Equations 7.1 and 7.2 give very good predictions. To further emphasise
this fact, Figure 7.13 has been produced, showing that most values fall within the ±20%
envelope.
Table 7.1. Details of James’ (1997) Reinforced Concrete Box Beams
Spacing of Reinforcement, s (mm)
Reinforcement Beam# fcm
(MPa) fsy
(MPa) Tension, st Comp., sc Tension Comp. Shear
Beam Type*
5 37.7 400 80 100 3Y20 2R6 R6@300 SS 7 32.4 400 28 100 6Y20 2R6 R6@125 SS 16 34.1 400 80 100 3Y20 2R10 R6@125 C 17 34.2 400 28 100 6Y20 2Y20 R10@130 C 18 30.6 400 55 100 4Y24 2Y24 R10@120 C * Beam types are SS – Simply Supported (Total length 6 metres); and C – Two–Span Continuous (Total length 12 metres). # All beams had b = 300 mm, D = 300 mm and internal void of b = 180 mm, D = 180 mm. Table 7.2. Deflection Data for Beam 5 (James, 1997)
Experimental In-Service
Instantaneous Deflection ∆i,exp (mm)
Curve Coefficient
αrc
Calculated In-Service Residual
Deflection ∆r,calc (mm)
Experimental In-Service Residual
Deflection ∆r,exp (mm)
Load (kN)
(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)
∆ r,calc
∆ r,exp
0 0 0 0 - 24.5 3.38 0.98 1.22 0.80 34.3 9.81 2.84 2.30 1.23 44.9 14.18 4.11 3.27 1.26 54.0 18.54 5.38 3.93 1.37 64.4 22.61 6.56 4.47 1.47 74.4 26.63
ρt = 1.2% ∴ αrc = 0.29
7.72 4.89 1.58 Mean, x = 1.29
Standard Deviation, σn-1 = 0.27
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-18
Table 7.3. Deflection Data for Beam 7 (James, 1997) Experimental
In-Service Instantaneous
Deflection ∆i,exp (mm)
Curve Coefficient
αrc
Calculated In-Service Residual
Deflection ∆r,calc (mm)
Experimental In-Service Residual
Deflection ∆r,exp (mm)
Load (kN)
(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)
∆ r,calc
∆ r,exp
0 0 0 0 - 24.5 3.62 0.76 0.89 0.85 39.2 6.71 1.41 1.83 0.77 54.2 10.05 2.1 2.42 0.87 68.7 13.27 2.8 3.00 0.93 83.4 16.51 3.47 3.36 1.03 117.7 24.46 5.14 4.58 1.12 157.0 33.83
ρt = 2.3% ∴ αrc = 0.21
7.10 5.88 1.21 Mean, x = 0.97
Standard Deviation, σn-1 = 0.16
Table 7.4. Deflection Data for Beam 16 (James, 1997) Experimental
In-Service Instantaneous
Deflection ∆i,exp (mm)
Curve Coefficient
αrc
Calculated In-Service Residual
Deflection ∆r,calc (mm)
Experimental In-Service Residual
Deflection ∆r,exp (mm)
Load (kN)
(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)
∆ r,calc
∆ r,exp
0 0 0 0 - 29.4 2.04 0.59 0.66 0.89 40.2 4.82 1.40 1.25 1.12 49.8 6.70 1.94 1.66 1.17 59.8 8.69 2.52 2.01 1.25 80.0 12.99 3.77 2.98 1.27 90.3 15.12
ρt = 1.2% ∴ αrc = 0.29
4.38 3.33 1.32 Mean, x = 1.17
Standard Deviation, σn-1 = 0.15
Table 7.5. Deflection Data for Beam 17 (James, 1997) Experimental
In-Service Instantaneous
Deflection ∆i,exp (mm)
Curve Coefficient
αrc
Calculated In-Service Residual
Deflection ∆r,calc (mm)
Experimental In-Service Residual
Deflection ∆r,exp (mm)
Load (kN)
(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)
∆ r,calc
∆ r,exp
0 0 0 0 - 21.0 1.82 0.33 0.38 0.87 44.7 4.4 0.79 0.95 0.83 68.9 7.66 1.38 1.52 0.91 118.9 15.23 2.74 2.74 1.0 167.0 22.65 4.08 3.79 1.08 216.4 30.29
ρt = 2.3% ∴ αrc = 0.21
5.45 4.87 1.12 Mean, x = 0.97
Standard Deviation, σn-1 = 0.12
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-19
Table 7.6. Deflection Data for Beam 18 (James, 1997) Experimental
In-Service Instantaneous
Deflection ∆i,exp (mm)
Curve Coefficient
αrc
Calculated In-Service Residual
Deflection ∆r,calc (mm)
Experimental In-Service Residual
Deflection ∆r,exp (mm)
Load (kN)
(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)
∆ r,calc
∆ r,exp
0 0 0 0 - 21.8 1.74 0.37 0.42 0.87 50.4 5.09 1.07 1.26 0.85 80.4 9.36 1.97 2.06 0.95 108.7 13.73 2.88 2.84 1.02 138.1 18.88 3.96 3.90 1.02 196.6 28.68
ρt = 2.2% ∴ αrc = 0.21
6.02 5.77 1.04 Mean, x = 0.96
Standard Deviation, σn-1 = 0.08
⊕
⊕
⊕
⊕
⊕⊕
⊕
⊕
∞
∞
∞
∞∞
∞∞
•
•
•
•
•
•
•
Calculated Residual Deflection (mm)
Jam
es'E
xper
imen
talR
esid
ualD
efle
ctio
n(m
m)
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
James 5James 7James 16James 17James 18
⊕∞
•
r,calc = αrc i,exp (7.1)
αrc = -0.08ρt + 0.39 (7.2)
20%
20%
Figure 7.13: Experimental versus Calculated Residual Deflection for James’ Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-20
The following general conclusions can be noted from Tables 7.2 to 7.6 and Figure 7.13:
All James’ test beams were constructed as hollow box beams. Furthermore, they
all contained 400 MPa reinforcement. Thus, Equations 7.1 and 7.2 give
remarkably good residual deflection predictions for these beams;
Beams 5 and 7 were simply-supported (single span beams), whilst beams 16, 17
and 18 were two-span continuous beams. Therefore Equations 7.1 and 7.2 seem
to be equally applicable for a range of beam and test setups;
A good range of tensile reinforcement ratios, between 1.2% and 2.3%, as used to
verify Equations 7.1 and 7.2, also the compression and shear reinforcement
arrangements varied, as shown in Table 7.1; and
The average ratios ∆r,calc/∆r,exp were 1.29, 0.97, 1.17, 0.97 and 0.96 for beams 5, 7,
16, 17 and 18, respectively. They are significantly close to unity, with acceptable
variations, indicating that Equations 7.1 and 7.2 give very good predictions.
7.5 Summary
The residual deflection characteristics of the reinforced concrete beams were modelled
using a linear relationship to instantaneous deflection, and selected according to the
tensile reinforcement ratio, ρt. The extensive verifications of the reinforced concrete
residual deflection equation indicate that:
The prediction of residual deflection for shear beams is good, however, Equations
7.1 and 7.2 tend to overpredict values particularly at low load levels as shown in
Figure 7.1;
The equations are considered valid for concrete compressive strengths between
22.5 and 90.7 MPa and reinforcement of either 400 or 500 MPa.
For the PS-Series beams a generic linear relationship for prestressed beams was
suggested. It gives reasonable prediction of residual deflections as shown in Figure
7.11c but as indicated in Section 7.3.5, more supporting data is required so a better
estimation of αps is derived.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-1
CHAPTER 8
Total Damping in Concrete Beams
8.1 General Remarks
As presented in Section 3.3, the equation for computing the total damping capacity of a
concrete beam at any stage of its service life is the sum of the contributions by the
‘untested’ and ‘tested’ logdec components. To estimate the ‘untested’ logdec
component for reinforced and prestressed concrete beams, Equations 6.2 and 6.3 were
developed and verified in Sections 6.2.3 and 6.3.4, respectively.
To estimate the ‘tested’ logdec component, residual deflection will be used. Sections
3.2.3 and 3.2.4, highlighted reasons why residual deflection is considered superior to
explicit measures of residual crack width.
Section 8.2 makes general observations of the experimental ‘tested’ logdec data versus
the residual deflection, and develops an equation to model the relationship. Chapter 7
focused on the development of residual deflection utilising instantaneous deflection.
Finally, in Section 8.3, verification of the developed total logdec prediction equation is
made using the F-Series beams and the data of Chowdhury (1999).
8.2 Development of Total Damping Equations
Presented in Figures 8.1, 8.2 and 8.3, are the experimental logdec versus mid-span
residual deflections for all the B-, CS- and PS-Series test beams, respectively. It should
be noted here that only the residual deflection prior to yielding is shown. Also, the y-
intercept is the experimental ‘untested’ logdec. In the figures, both flexural and shear
beams are presented, varying in concrete compressive strength and reinforcement yield
strength.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-2
a)
B4 = 0.0008x + 0.05
R2 = 0.74
B3 = 0.001x + 0.0585
R2 = 0.6104
B1 = 0.0011x + 0.0648
R2 = 0.8257
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0 10 20 30
Mid-Span Residual Deflection (mm)
Logd
ec
B1
B3
B4
Linear (B4)
Linear (B3)
Linear (B1)
δ
δ
δ
δ
b)
B6 = 0.0013x + 0.0534
R2 = 0.3593
B8 = 0.0019x + 0.0388
R2 = 0.5983
B7 = 0.0019x + 0.0361
R2 = 0.5885
B5 = 0.0016x + 0.0573
R2 = 0.8474
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0 5 10 15 20 25 30
Mid-Span Residual Deflection (mm)
Logd
ec
B5B6B7B8Linear (B6)Linear (B8)Linear (B7)Linear (B5)
δ
δ
δ
δ
δ
c)
B10 = 0.0015x + 0.0457
R2 = 0.6378
B12 = 0.0022x + 0.0566
R2 = 0.6238
B11 = 0.0033x + 0.0518
R2 = 0.4417
B9 = 0.0023x + 0.0444
R2 = 0.7741
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0 5 10 15 20 25 30
Mid-Span Residual Deflection (mm)
Logd
ec
B9
B10B11
B12
Linear (B10)Linear (B12)
Linear (B11)Linear (B9)
δ
δ
δ
δ
δ
Figure 8.1: Logdec versus Residual Deflection for B-Series: a) BI-1, BI-3, BII-4; b) BI-
5 to BII-8; and c) BI-9 to BII-12
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-3
a)
CS2 = 0.0098x + 0.0693
R2 = 1
CS1 = 0.0108x + 0.07
R2 = 1
CS3 = 0.0175x + 0.059
R2 = 1
0.045
0.055
0.065
0.075
0.085
0.095
0.105
0.115
0 2 4 6
Mid-Span Residual Deflection (mm)
Logd
ecCS1
CS2
CS3
Linear (CS2)
Linear (CS1)
Linear (CS3)
δ
δ
δ
δ
b)
CS6 = 0.0318x + 0.0519
R2 = 0.8931
CS5 = 0.018x + 0.0501
R2 = 0.9721
CS4 = 0.0292x + 0.0495
R2 = 0.9741
0.045
0.055
0.065
0.075
0.085
0.095
0.105
0.115
0 2 4 6Mid-Span Residual Deflection (mm)
Logd
ec
CS4
CS5
CS6
Linear (CS6)
Linear (CS5)
Linear (CS4)
δ
δ
δ
δ
c)
CS9 = 0.0188x + 0.0627
R2 = 0.8544
CS8 = 0.0191x + 0.0591
R2 = 0.8458
CS7 = 0.0173x + 0.0631
R2 = 0.7841
0.0450.0550.0650.0750.0850.0950.1050.1150.1250.1350.145
0 2 4 6
Mid-Span Residual Deflection (mm)
Logd
ec
CS7
CS8
CS9
Linear (CS9)
Linear (CS8)
Linear (CS7)
δ
δ
δ
δ
Figure 8.2: Logdec versus Residual Deflection for CS-Series: a) CS1 to CS3; b) CS4 to
CS6; and c) CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-4
a)
PS5 = 0.0018x + 0.043
R2 = 1
PS3 = 0.0019x + 0.042
R2 = 1
PS6 = 0.0029x + 0.0467
R2 = 0.8302
0.040.045
0.050.055
0.060.065
0.070.075
0.080.085
0.09
0 5 10 15Mid-Span Residual Deflection (mm)
Logd
ec
PS3
PS5
PS6
Linear (PS5)
Linear (PS3)
Linear (PS6)
δ
δ
δ
δ
b)
PS8 = 0.0027x + 0.0412
R2 = 0.9354
PS7 = 0.0019x + 0.042
R2 = 1
PS9 = 0.005x + 0.0424
R2 = 0.253
PS10 = 0.0027x + 0.0547
R2 = 0.8571
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 5 10 15Mid-Span Residual Deflection (mm)
Logd
ec
PS7PS8PS9PS10Linear (PS8)Linear (PS7)Linear (PS9)Linear (PS10)
δ
δ
δ
δ
δ
Figure 8.3: Logdec versus Residual Deflection for: a) PS3 to PS6; b) PS7 to PS10
For each beam data presented in Figures 8.1, 8.2 and 8.3, the line-of-best-fit has been
used and consists of two components, the last number in each equation shown is the y-
intercept which is in fact the ‘untested’ logdec, δuntest. The first number defines the
slope of the curve, and is termed here the damping-residual deflection (D-R) slope, βfl,
for concrete beams.
The D-R Slopes, βfl, for each test beam have been plotted against the concrete
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-5
compressive strength, fcm in Figure 8.4. In Figure 8.4a, both the B- and PS-Series
flexural beams having a length of 6.0 m and the same cross-sectional dimensions have
been plotted together. For these beams a relationship was found as
βfl = 0.0007e0.018fcm (8.1)
where fcm ranges between 22.5 MPa and 90.7 MPa.
In Figure 8.4b, all the CS-Series flexural and shear beams have been plotted. There is
some difference between the flexural and shear beams. In the absence of a significant
number of flexural test beams, the D-R Slope, βfl for beams CS1 to CS3 and CS7 to
CS9 has been averaged to 0.016.
Finally, the equation to predict the total logdec, δtotal, in reinforced and prestressed
flexural beams is given by
δtotal = βfl ∆r + δuntest (8.2)
where δuntest is the relevant ‘untested’ damping capacity as given in Chapter 6; ∆r is the
calculated residual deflection of the beam in mm for any service loading level as
detailed in Chapter 7; and βfl is calculated from Equation 8.1 for full-scale reinforced
and prestressed concrete beams similar to the B- and PS-Series beams, and equals 0.016
for half-scale concrete beams similar to the CS-Series beams.
From the above discussion and formulation, the following points may be noted:
The establishment of the ‘untested’ logdec is very important because the
calculation of the logdec in-service is based on this quantity;
It should be noted here, that even though the PSC beams do not exhibit visual
cracking until they reach 60% of failure, internal damage begins immediately, and
this is reflected by the presence of residual deflection, and thus an increase in
damping capacity;
The D-R Slope, βfl, defining the relationship between damping capacity and
residual deflection appears to be affected by beam size. This is concluded by the
separate plots of B- and CS-Series beams; and
Interestingly, the D-R Slope, βfl, was very similar for both the reinforced and
prestressed concrete beams.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-6
a) Concrete Compressive Strength, fcm (MPa)
D-R
Slop
e-F
lexu
ralB
eam
s(β
fl)
0 20 40 60 80 1000
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
0.0055
0.006
B-SeriesPS-Series(β fl ) = 0.0007e0.018 fcm
(β fl ) = 0.0007e0.018 fcm
R2 = 0.68
b)
xx
x xxx
Concrete Compressive Strength, fcm (MPa)
D-R
Slop
e-F
lexu
ralB
eam
s(β
fl)
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
CS-Series (Flexural Only)CS-Series (Shear Only)
x
Figure 8.4: Dependence of D-R Slope on Concrete Compressive Strength: a) B- and PS-
Series Beams; b) CS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-7
8.3 Verification
In order to verify the proposed total damping prediction equation, comparisons are
made with the F-Series test beams and Chowdhury’s (1999) test data. As mentioned
through this thesis, it is extremely difficult to obtain published work that is acceptable
for comparison due to the fact that they lack the necessary beam details, material
properties, tabulated loading history, logdec and residual deflection details. This is why
the additional F-Series was developed.
8.3.1 F-Series Beams
In Figure 8.5, a comparison of the experimentally observed F-Series damping test data
versus that predicted using Equation 8.2 is given. The scattergram shows that:
In general, the proposed equation is satisfactory for both flexural and shear beams
at all residual deflections, as most of the calculated and experimental logdecs are
within 20% of a perfect correlation;
For some flexural beams (F2, F3, F8 and F9), there appears to be a slight
underestimation of damping towards failure. This is most likely due to the fact
that the total damping equation was developed using pre-yield data, where the
data included in Figure 8.5 is for all load levels.
Overall however, the correlation is excellent.
qq qqq qqqq
r rrrr
r rr rrr rrr
r
ss s
ss sss
ssss
t tt ttt t t tttt
t
uu uuu uuuu
vv vv
vv
ww ww
w ww
xx
xxxxx
x
x
y
yy
yy
y
y
zz
zzz
z
z
Experimental Logdec, δtotal
Cal
cula
ted
Logd
ec,δ
tota
l
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
F1F2F3F4F5F6F7F8F9F10
qrstuvwxyz
+ 20%
- 20%F-Series Beams
Figure 8.5: Calculated versus Experimental Logdec – F-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-8
8.3.2 Chowdhury’s Beams
Chowdhury (1999) performed free-vibration experiments at incremental levels of
damage loading. A typical cross-sectional detail of Chowdhury’s (1999) test beams,
and test set-up are given in Figure 8.6. Table 8.1 presents details for each test beam.
Pin Roller
L varies according to beam typeSS = 6 metresC = 12 metres
l = 2000 mmfor all beams
PCB Piezotronics Model 353AAccelerometer at the mid point
Embeddedpolystyrene
void
21
112
30
12030 120 30
60 60
1446
180
60
180CR
SR
TR
Figure 8.6: Details of Test Beams and Testing Arrangement of Chowdhury (1999)
Table 8.1. Details of Chowdhury’s (1999) Reinforced Concrete Box Beams Spacing of
Reinforcement, s (mm) Reinforcement
Beam# fcm(MPa)
fsy(MPa)
Tension, st Comp., scTension
TR Comp.
CR Shear SR
Beam Type*
5 37.7 400 80 100 3Y20 2R6 R6@300 SS 7 32.4 400 28 100 6Y20 2R6 R6@125 SS 16 34.1 400 80 100 3Y20 2R10 R6@125 C 17 34.2 400 28 100 6Y20 2Y20 R10@130 C 18 30.6 400 55 100 4Y24 2Y24 R10@120 C * Beam types are SS – Simply Supported (Total length 6 metres); and C – Two–Span Continuous (Total length 12 metres). # All beams had b = 300 mm, D = 300 mm and internal void of b = 180 mm, D = 180 mm.
Tables 8.2 to 8.6 presents a comparison of the calculated ‘tested’ damping results (using
Equation 9.2), to the experimental damping test results for the beams of Chowdhury
(1999). The tables show that the ratio between the experimental and calculated total
logdec values of Chowdhury’s (1999) beams indicates a very good correlation, being
0.95, 0.97, 0.94, 1.04 and 1.18 for beams 5, 7, 16, 17 and 18, respectively.
Figure 8.7 shows that an excellent correlation exists between all ‘tested’ calculated and
experimental logdec values for Chowdhury’s (1999) beams where 95% of all points lie
within ±20% of a perfect correlation.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-9
Table 8.2. Deflection versus Damping Data for Beam 5 (Chowdhury, 1999) Experimental
In-Service Logdec δexp
Experimental In-Service
Residual Deflection ∆r,exp (mm)
Calculated In-Service
δtotal(Equation 8.2)
Load (kN) #
(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest
δtotal = 0.0014∆r + 0.073
δtotal,exp
δtotal,calc
0 0.073 0 0.073 (δuntest) 1.00 24.5 ** 0.074 1.22 0.075 0.99 34.3 0.075 2.30 0.076 1.0 44.9 0.071 3.27 0.078 0.91 54.0 0.069 3.93 0.079 0.87 64.4 0.074 4.47 0.080 0.93 74.4 0.074 4.89 0.080 0.93
Mean, x = 0.95 Standard Deviation, σn-1 = 0.05
** This is the load at which the beam first cracked. # The maximum (failure) load was 105.0 kN. Table 8.3. Deflection versus Damping Data for Beam 7 (Chowdhury, 1999)
Experimental In-Service
Logdec δexp
Experimental In-Service
Residual Deflection ∆r,exp (mm)
Calculated In-Service
δtotal(Equation 8.2)
Load (kN) #
(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest
δtotal = 0.0013∆r + 0.073
δtotal,exp
δtotal,calc
0 0.073 0 0.073 (δuntest) 1.00 24.5 0.070 0.89 0.074 0.95 39.2 ** 0.075 1.83 0.075 1.00 54.2 0.079 2.42 0.076 1.04 68.7 0.073 3.00 0.077 0.95 83.4 0.074 3.36 0.077 0.96 117.7 0.076 4.58 0.079 0.96 157.0 0.075 5.88 0.081 0.92
Mean, x = 0.97 Standard Deviation, σn-1 = 0.04
** This is the load at which the beam first cracked. # The maximum (failure) load was 230.0 kN. Table 8.4. Deflection versus Damping Data for Beam 16 (Chowdhury, 1999)
Experimental In-Service
Logdec δexp
Experimental In-Service
Residual Deflection ∆r,exp (mm)
Calculated In-Service
δtotal(Equation 8.2)
Load (kN) #
(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest
δtotal = 0.0013∆r + 0.082
δtotal,exp
δtotal,calc
0 0.082 0 0.082 (δuntest) 1.00 29.4 ** 0.081 0.66 0.083 0.98 40.2 0.078 1.25 0.084 0.93 49.8 0.078 1.66 0.084 0.93 59.8 0.077 2.01 0.085 0.91 80.0 0.074 2.98 0.086 0.86 90.3 0.083 3.33 0.086 0.97
Mean, x = 0.94 Standard Deviation, σn-1 = 0.05
** This is the load at which the beam first cracked. # The maximum (failure) load was 160.0 kN.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-10
Table 8.5. Deflection versus Damping Data for Beam 17 (Chowdhury, 1999) Experimental
In-Service Logdec δexp
Experimental In-Service
Residual Deflection ∆r,exp (mm)
Calculated In-Service
δtotal(Equation 8.2)
Load (kN) #
(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest
δtotal = 0.0013∆r + 0.076
δtotal,exp
δtotal,calc
0 0.076 0 0.076 (δuntest) 1.00 21.0** 0.084 0.38 0.077 1.09 44.7 0.076 0.95 0.077 0.99 68.9 0.088 1.52 0.078 1.13 118.9 0.083 2.74 0.080 1.04 167.0 0.079 3.79 0.081 0.98 216.4 0.086 4.87 0.082 1.05
Mean, x = 1.04 Standard Deviation, σn-1 = 0.06
** This is the load at which the beam first cracked. # The maximum (failure) load was 295.0 kN.
Table 8.6. Deflection versus Damping Data for Beam 18 (Chowdhury, 1999) Experimental
In-Service Logdec δexp
Experimental In-Service
Residual Deflection ∆r,exp (mm)
Calculated In-Service
δtotal(Equation 8.2)
Load (kN) #
(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest
δtotal = 0.0012∆r + 0.071
δtotal,exp
δtotal,calc
0 0.071 0 0.071 (δuntest) 1.00 21.8 ** 0.086 0.42 0.072 1.19 50.4 0.083 1.26 0.073 1.14 80.4 0.098 2.06 0.073 1.34 108.7 0.088 2.84 0.074 1.19 138.1 0.085 3.90 0.076 1.12 196.6 0.099 5.77 0.078 1.27
Mean, x = 1.18 Standard Deviation, σn-1 = 0.11
** This is the load at which the beam first cracked. # The maximum (failure) load was 260.0 kN.
8.4 Advantages of Proposed Residual Deflection Equations
From the discussions above, the advantages of the proposed total damping model are as
follows:
For prestressed beams: it has been demonstrated that damping will increase
during its service life. It is common for these beams to not exhibit visual cracking
and thus, residual deflection is an excellent way of predicting the in-service
damping levels;
For shear beams: flexural cracking could not be properly measured. However,
similar to the prestressed beams, damping still increased during the experiments
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-11
and thus the use of residual deflection was found to be very effective;
For flexural beams: the residual deflection mechanism can be reliably predicted
from a simply equation;
Concrete compressive strength: was found to affect the damping versus residual
deflection relationship for the flexural beams (the D-R Slope, βfl). Therefore,
concrete compressive strength (and associated beam constituents) most likely
affects the internal beam damage, thus affecting damping capacity;
For simply-supported and continuous beams: the calculation of residual
deflection and damping capacity using the proposed equation was excellent for
both types of test beams (using Chowdhury’s, 1999 test beams);
For concrete box beams: the calculation of total logdec in Chowdhury’s (1999)
hollow box beams was excellent; and
Establishing the ‘untested’ damping capacity: is of primary importance before
modelling the in-service, total damping capacity.
PPPPP
P
P
∅
∅
∅
∅
∅∅
∅
Total Logdec, δtotal (Calculated from Equation 8.2)
Tota
lLog
dec,δ t
otal
(Cho
wdh
ury'
sEx
perim
enta
lDat
a)
0.06 0.08 0.1 0.12
0.06
0.08
0.1
0.12
Beam 5Beam 7Beam 16Beam 17Beam 18
P∅
-20%
+20%
Figure 8.7: Chowdhury’s Experimental versus Calculated using Equation 8.2
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 8: Total Damping in Concrete Beams 8-12
8.5 Summary
A method by which to calculate the residual deflection of a beam, subject to transient
damage causing loads, was proposed in this Chapter. The experimental data from the
current series of test beams were used to develop a relationship, for the current testing
regime, between the experimental residual deflection and logdec values. Verifications
using the additional F-Series beams and Chowdhury’s (1999) test beams have shown
the proposed equation to be very good.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-1
CHAPTER 9
Conclusions and Recommendations
9.1 General Remarks
This thesis focussed on the problems associated with a current world-wide trend
towards optimised structures, in which structural members are increasingly longer,
lighter and more slender. One negative effect of these seemingly cost effective trends is
their vulnerability to vibrations causing damage. Following an extensive review of the
literature on damping, the omission of two broad areas were identified as follows:
There is an extreme paucity of damping data available;
There is no method currently available to predict the damping capacity of
reinforced and prestressed concrete beams at the design stage.
In response to these findings, an extensive experimental programme was devised to
investigate and provide the experimental data necessary to elucidate the necessary
information. These test results were used to evaluate the damping research presented
within the literature and to assist in formulating a new technique to analyse and
calculate the damping capacity of concrete beams over the full loading regime. The
developed equations were constructed using the extensive experimental data and
verified by: a) an additional series of test beams to provide supporting data; and b)
comparative experimental data extracted from the literature.
9.2 Research Objectives and Outcomes
In Section 1.2, a number of main research objectives were identified. Each of the
objectives have been addressed in depth within the thesis and the following outcomes
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-2
have been derived:
(a) A damping analysis technique was created for the extraction of the logdec from
the raw free-vibration decay curve. The proposed method is a generic technique
allowing researchers to interpret and compare experimental damping data
satisfactorily;
(b) Identification of damping categories to describe the damping characteristics of a
beam at any stage of its service life (i.e. ‘untested’ versus ‘tested’). Previous work
has used damping terminology inconsistently and interchangeably, making
damping data comparisons exceedingly difficult;
(c) The calculation of the ‘untested’ logdec has not been adequately addressed by the
literature. Equations to determine the ‘untested’ damping capacity of reinforced
and prestressed concrete beams were proposed using experimental results and
verified with supporting data;
(d) A residual deflection calculation method has been proposed and is based on the
instantaneous deflection characteristics of reinforced and prestressed concrete
beams for the current experimental programme. Verifications using additional
experimental data and data from previous research showed the proposed equation
to be very versatile; and
(e) A total damping capacity equation has been developed to predict the full-range
damping behaviour of reinforced and prestressed concrete beams. The proposed
equation has been verified using additional experimental data and data from
previous research.
9.2.1 Summary of Test Results
In order to achieve the research objectives, a large amount of experimental test data was
generated. Using the data, the following was found:
(a) Damping Analysis Technique
Free-vibration decay in reinforced and fully-prestressed concrete beams is not
strictly exponential in the initial portion of decay, as required by the theory, but
becomes exponential as the free-vibration stabilises;
The point from which logdec is calculated on the free-vibration decay curve
cannot be defined by a cycle number as traditionally suggested. It is proposed
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-3
here that the ‘optimal peak ratio’ (as defined in Section 5.4.2) be used to define
the point where logdec is extracted from decay curves;
The weight and location of the impact hammer does not affect the resulting
calculation of logdec of concrete beams.
(b) Damping Prediction in ‘Untested’ Concrete Beams
The damping capacity of an ‘untested’ beam was found to be the integral
component of any damping model, as it serves as the initial starting point in
calculating damping, a fact not previously considered or developed;
In an ‘untested’ reinforced concrete beam, the concrete compressive strength and
reinforcement yield strength did not affect damping capacity, but longitudinal
reinforcement ratio (LRD) did;
A damping prediction equation for ‘untested’ reinforced concrete beams, δuntest,
was thus proposed (Equation 6.2) utilising LRD as the primary variable. This
equation normalises the cross-section dimensions by utilising the reinforcement
ratio, and it was also found to be equally applicable for beams of length 2.4 m and
6.0 m. It is reproduced here:
19.0
223.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛+×=
c
c
t
tuntest ss
ρρδ (6.2)
where st and sc are the tension and compression reinforcement spacings,
respectively in mm and are given in Table 4.1; ρt = Ast/bd and ρc = Asc/bd. The
equation is valid for LRD distributions between 0.0001 and 0.0023 (see Table
6.1).
An ‘untested’ damping prediction equation for fully-prestressed beams was also
proposed (Equation 6.3) utilising the overall amount of prestress in a beam (He)
as the primary variable. It is reproduced here:
δuntest = 1.4×10-10He2 - 9.4×10-6He + 0.2 (6.3)
where Equation 6.3 is valid for He between 30,000 and 45,000 (kNmm).
(c) Calculation of Residual Deflection
In reinforced concrete beams the instantaneous versus residual deflection
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-4
relationship were affected primarily by the tensile reinforcement ratio, ρt;
A residual deflection equation, based on the instantaneous deflection was
proposed (Equations 7.1 and 7.2). They are reproduced here:
∆r = αrc ×∆i (7.1)
where αrc is the curve coefficient found from Equation 7.2 and derived from
Figure 7.6 and ∆r and ∆i are in mm.
αrc = -0.08ρt + 0.39 (7.2)
where Equation 7.2 is valid for 0.76% < ρt < 3.0% .
For prestressed concrete beams a residual deflection equation was also proposed
(Equation 7.3). It is reproduced here:
∆r = 0.09 ∆i (7.3)
where ∆r and ∆i are the residual and instantaneous deflections in mm, at a given
service loading condition, respectively.
(d) Total Damping in Concrete Beams
The equation to predict the total logdec, δtotal, in reinforced and prestressed
concrete beams (Equation 8.2), is the sum of the contribution of the ‘untested’
damping capacity, δuntest, and the ‘tested’ damping capacity defined by the residual
deflection relationship, βfl ∆r. It is reproduced here:
δtotal = βfl ∆r + δuntest (8.2)
For full-scale reinforced and prestressed concrete beams, βfl is the correlative
function, it may be found from Equation 8.1. It is reproduced here:
βfl = 0.0007e0.018fcm (8.1)
where δuntest is the relevant ‘untested’ damping capacity as given in Chapter 6; ∆r
is the calculated residual deflection of the beam in mm for any service loading
level as detailed in Chapter 7; and βfl is calculated from Equation 8.1 for full-
scale reinforced and prestressed concrete beams similar to the B- and PS-Series
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-5
beams, and equals 0.016 for half-scale concrete beams similar to the CS-Series
beams.
For half-scale reinforced concrete beams similar to the CS-Series, βfl is equal to
0.016.
9.2.2 Verification of the Proposed Methods
As mentioned throughout this thesis, the paucity of available and useful damping data
made verification of the proposed methods difficult. Therefore, it was necessary to
devise an additional test series (F-Series), along with the data of James (1997),
Chowdhury (1999) and Neild (2001) for validation purposes. Verification of the
proposed methods was undertaken in the appropriate sections of Chapters 6, 7, and 8.
These comparative investigations have shown that the proposed equations are reliable
and applicable for a range of beam designs, test set-ups, constituent materials and
loading regimes.
9.3 Recommendations and Scope for Future Research
A number of prediction equations have been presented. These equations appear to offer
promising methods of damping prediction for a range of variables. Further research
should be directed towards continually improving the prediction methods by
investigating:
Other damage sources: the effect of damage on the residual deflection/damping
capacity due to corrosion of reinforcement or spalling of concrete;
Other loading regimes: by examining damage caused by other cyclic and impact
loads and also load-reversal effects such as that created during earthquakes ; and
Structure damping: the effect of different types of joints and connections, such as
beam-column or beam-panel connections, on total damping.
9.4 Closure
The need for a user friendly damping calculation method has been addressed within this
thesis. In particular, a number of important damping considerations identified as being
absent from the literature were investigated.
This thesis has addressed the perceived gaps in the literature by presenting details and
results of a series of extensive experiments that will be an invaluable source of reference
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Chapter 9: Conclusions and Recommendations 9-6
not only to damping researchers but structural designers and practitioners. The
subsequent extensive analysis has resulted in a straightforward methodology for
deriving and calculating the total damping capacity of reinforced and fully-prestressed
concrete beams.
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
References Re-1
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APPENDIX A: Literature Review Summary Tabulations A-1
APPENDIX A
Literature Review Summary Tabulations
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table A.1. Material Damping Literature Review Summary Effect on Concrete Material Damping (Part 1)
Author’s
Moisture or Water
Content (mc) (wc)
Size,Shape, Type
Aggregate (Recycled -
r/a)
Stress/Strain Amplitude
or State of Stress
Degree of Hydration
or Age
Interfacial Damping /
Microcracking
Frequency (F) or
Vibration Amplitude
(A)
Curing Condition
Water/ Cement Ratio (w/c)
Aggregate/ Cement
Ratio (a/c)
Vibration Mode (Flexural-
Longitudinal- Torsional Fl-Lo-To
Concrete Compressive or Flexural Strength f’c
Air Voids or
Additives Added
Dynamic (ED) or
Elastic (Ec) Modulus
Testing/ Measurement Methods or
Errors
Kesler and Higuchi (1953)
δ as the mc of the
specimen δ with
in age δ is dependent on F
δ less dep.on w/c as mc
Measuring δ cannot predict
f’c
Measuring δ
cannot predict Ec
Jones (1957) δ is interfacial
or frictional
No ∆ in δ with F 70-10,000 c/s
δ same for Fl, Lo & To
Cole and Spooner (1965)
δ linearly with max.
strain amplitude
F below 2.5 cps: δ with
F
Cole (1966) δ very
sensitive to mc (Until dry)
δ with
age and mc.
δ in cement paste, is not interfacial
For small specimens, no ∆ in δ with ∆
in A
No difference in δ btw
concrete or paste
For sml specimens, δ
loss in mountings is
v.sml
Jones and Welch (1967)
Influences δ. Is a complex interaction.
‘Q’ value increases
(∴δ ) as specimen dries out
Not signif. Similar to Kesler and
Higuchi (1953)
δ as coarse
aggregate content
δ influenced by max. agg. size. Difficult
to define
Hard to define.
Disagrees with Kesler
and Highuchi (1953)
Influences δ. More
pronounced for higher f’c.
Swamy (1970)
Highly dependent. δ
is related to % of water-filled
pores
agg. size δ.
Angular agg’s δ.
Reactive agg.s infl. δ
For omplex stresses, δ . Infl. of stress
state unknown
δ for age
(&∴ mc)
Microcracks friction type δ
At very low frequency, δ
with a in freq
δ for Lo. vibration is highest and To
lowest
Air voids do not affect δ
Swamy and Rigby (1971)
Important – can
overshadow aggregate effects.
δ with
mc.
δ lge specif. area.
δ for crushed
δ smooth δ for sml δ
for lge
The more complex the state of stress the smaller the
δ
Ageing effect less
than drying effect. δ
significant in first 28
days
δ as aggregate content . Drying has
greater influence on δ
than microcracks
Not completely
established, δ may depend on frequency
Drying δ, but rapidly stabilised with time
in % water filled pores δ
More aggregates
δ. δ occurs in
matrix, less in agg. &
agg./matrix interface
Lo. vibration highest δ capacity, then Fl, then To
Not conclusive but δ gradually with in f’c
Adds little to δ
δ with an in dynamic
modulus. Not conclusive
Estimates accuracy of
experiment δ to be approx. ±
6%. Thought to be satisfactory
Effect on Concrete Material Damping (Part 2)
Author’s
Moisture or Water
Content (mc) (wc)
Size,Shape, Type
Aggregate (Recycled -
r/a)
Stress/Strain Amplitude
or State of Stress
Degree of Hydration
or Age
Interfacial Damping /
Microcracking
Frequency (F) or
Vibration Amplitude
(A)
Curing Condition
Water/ Cement Ratio (w/c)
Aggregate/ Cement
Ratio (a/c)
Vibration Mode (Flexural-
Longitudinal- Torsional Fl-Lo-To
Concrete Compressive or Flexural Strength f’c
Air Voids or
Additives Added
Dynamic (ED) or
Elastic (Ec) Modulus
Testing/ Measurement Methods or
Errors
Spooner and Dougill (1975)
δ by another process both during an
or of strain
Majority of δ from one
mechanism during 1st load
With strain & initial Ec
δ
Ashbee et al. (1976)
Shear stress plays a larger role in δ than
(mc)
δ dependent on mean &
dynamic load histories
δ depend. on age.
Not quantified
or discussed
Non-linear effects from
shrinkage cracks
δ dependent on curing
history
δ cyclic dependent. (ie. δ
with the no. of cycles )
Uses steel specimens to reduce errors and calibrate
Spooner et al. (1976)
Strain range applied to a specimen is
VIP in determining δ
δ of ‘solid’ material
independ. of age
δ is independ. of degree of damage
(cracking) of a specimen
δ of ‘solid’ material
independ. of w/c
For mortar specimens δ is ∝ to quantity
of ‘solid’ material
Jordan (1980)
Signif. in δ
for in dynamic stress
Age is signif. 20%
in δ with age
Micro-cracking is VIP Not signif.
δ of dry greater than
wet: cracking
w/c with a/c ratio,
δ
a/c gave δ
Sri Ravindraraj-ah and Tam (1985)
δ due to a higher wc in
recycled-aggregate (r-a)
concretes.
For all grades of concrete,
r/a had a δ.
δ due to a larger amount of micro-cracks in the r-a concrete.
δ with a in f’c,
possibly due to the in
total porosity
Fu and Chung (1996)
Latex andsilica fume
δ
Xu and Setzer (1997)
At verylow temps. δ is temp. dependent
At very low temps. δ is F
dependent
Fu et al. (1998)
Latex andsilica fume
δ by 390%
Effect on Concrete Material Damping (Part 3)
Author’s
Moisture or Water
Content (mc) (wc)
Size,Shape, Type
Aggregate (Recycled -
r/a)
Stress/Strain Amplitude
or State of Stress
Degree of Hydration
or Age
Interfacial Damping /
Microcracking
Frequency (F) or
Vibration Amplitude
(A)
Curing Condition
Water/ Cement Ratio (w/c)
Aggregate/ Cement
Ratio (a/c)
Vibration Mode (Flexural-
Longitudinal- Torsional Fl-Lo-To
Concrete Compressive or Flexural Strength f’c
Air Voids or
Additives Added
Dynamic (ED) or
Elastic (Ec) Modulus
Testing/ Measurement Methods or
Errors
Li and Chung (1998)
in δ because of high tensile ductility of silica fume
concrete
Latex and silica fume
δ by 300% at all F & Temps.
Wang and Chung (1998)
Aggregatesany shape or
size degrade δ
Addition ofsand δ
Aggregate+ Silica Fume =
large δ
Orak (2000)
ymer Poconcrete δ
is 4-7 times higher than
for steel
l
Table A.2 Member Damping Literature Review Summary
Effect on Plain RC Member Damping (Part 1)
Author’s
Concrete Strength/ Cracking/
Type
Type or Percentage of
Reinforcement
Amplitude (A) or Mode of Vibration
Test History (Repeated
Loading) or Test Method
Steel Stresses or Strains in
Member
Size Effects or Beam
Dimensions (h, b)
Steel Area
Elastic Modulus of Concrete
(Ec)
Material constant, n (Es/Ec)
Beam Displacement
/Support Influence
Crack spacing,
s Other
Bock (1942)
Beams without reinforcement had δ than those with reinforcement
Damping not affected by (A)
Penzien and Hansen (1954)
δ in RCB’s subject to
dynamic forces reduces max. strains signif.
James et al. (1964)
δ rapidly with the mode of vibration. δ
independent of (A)
Loading history has a significant
effect on δ.
δ not viscous for sml amplitudes, but viscous for lge
amplitudes. Unconfirmed that as
temp. δ
Penzien (1964)
Significant increases in δ with
cracking
Jordan (1977)
δ only by 25% when
severe cracking appeared
No evidence to suggest that
material δ as tensile stresses
are induced Un-
Cracked δ for reinf. amount δ till cracks
fully formed Dieterle and Bachman (1981) Cracked δ to nearly
zero at fsy
Small infl. on δ
ratio
Un-Cracked δ slightly δ slightly for
increasing h,b δ slightly δ slightly VIP Es neglected Flesch
(1981) Cracked δ strongly δ strongly for
increasing b,h δ strongly Strong in δ for displ.
Effect on Plain RC Member Damping (Part 2)
Author’s
Concrete Strength/ Cracking/
Type
Type or Percentage of
Reinforcement
Amplitude (A) or Mode of Vibration
Test History (Repeated
Loading) or Test Method
Steel Stresses or Strains in
Member
Size Effects or Beam
Dimensions (h, b)
Steel Area Elastic
Modulus of Concrete (Ec)
Material constant, n (Es/Ec)
Beam Displacement
/Support Influence
Crack spacing
s Other
Un-cracked
After frost- exposure δ
30-40%
Askegaard and Langsæ (1986)
Cracked
Well- developed cracking alters δ
Type of testing regimes Effect ons δ can be up
to 100%
Humidity changes may affect
operational δ significantly. Not
quantified
δ is dependent on mc. mc from 0 to
4% δ by 80%. Humidity and
temperature affects δ signif.
Almansa et al. (1993)
δ factors had no influence wrt cracking
levels
δ factors were found to exhibit no influence wrt the age of beam
un-forming
Wang et al. (1998)
On first cracking δ
by a factor of 4
δ is a function of the maximum load
to which it has been subjected
Chowdhury (1999) Effect of
cracking on δ is VIP
Ndambi et al. (2000)
δ with the excitation
amplitude up to 30%.
δ very sensitive to excitation method
(ie. impact vs shaker.
Found non-linear behaviour
introduced into system by impact
hammer excitation.
Weng and Chung (2000)
Addition of silica fume δ by two
or more orders of
magnitude
Sandblasted rebars δ 91% over
plain rebars. Considered to be
signif. for practical use
Concrete with rebars
δ (by 3 orders of
magnitude) than no rebar
mortar. Additional
reo δ more
Considers the high damping capacity of
rebars to be responsible for high damping capacities
of RC
Yan et al. (2000a/b) δ as max. response A
FRC exhibit δ with an in
number of vibration cycles
Concluded that δ is strain dependent
δ significantly with wavy fibres
Table A.3 Prestressed Damping Literature Review Summary
Effect on Prestressed Concrete Member Damping Initial Prestress
Condition Type of Vibration
Author’s
Cracked State of Concrete
Test History (Repeated / Past
Loading)
Magnitude/Degree of Prestressing
Type of Prestress Uniform
Axial Eccentric (triangle)
Steady-State Free-Decay
Strength of Concrete f’c
Age Amplitude of Vibration
Frequency of Vibration
Effect of Support Damping
James et al (1964) Little effect on δ
δ not viscous for sml amplitudes,
but viscous for lge amplitudes
Penzien (1964)
VIP. δ with tension crack
development. δ may be 3-6% of critical for sig.
cracking and 1% for uncracked
δ depends on load/stress level
history and initial amplitude
displacements that may cause cracking
Only indirect influence on δ as it
influences cracking
Only indirect influence on δ
as it influences cracking which ∴ affects δ
δ as degree of prestress is
(0.75-1.5% of critical)
Mag. of prestress
did not sig. influence δ (0.5-1% of
critical)
δ greater for free
vibration than steady-
state
Effect is entirely
masked by effect of cracking
δ with amplitude of
oscillation for all cases
Important, however not
studied or quantified
VIP, effect higher in free vibration than steady-state. Magnitude of influence not
quantified
Hop (1991)
δ is approx. 35% higher after
cracking than prior to the onset
of cracking
of axial & eccentric
prestressing δ signif.
in degree of axial
causes large in δ
in degree of eccentric
causes large in δ
δ with age. At 20
yrs δ 40% - 75% for
beams with high E
Shield (1997)
Prior to formation of cracks, AEA
activity increased
Table A.4 Structural Damping Literature Review Summary
Effect on RC Structural Damping (Part 1)
Author’s
Concrete Strength/ Cracking/ Type
(Crack Spacing/Width)
Type or Percentage (Steel Area) of Reinforcement
(Stresses/ Strains)
Amplitude or Mode or
Frequency of Vibration
Test History (Repeated
Loading) or Test Method
Structure Interaction
Effects
Structure Size Effects, Type or
Dimensions
Beam Displacement
/Support Influence
Applicability or Usefulness of
Damping Results
Seismic Aspects Type of δ Other
Jeary (1974)
All energy dissipation in
chimney’s in 1st mode of bending
Using acceleration response, signif.
Inaccuracies when δ reaches 0.1
Leonard and Eyre (1975)
δ with higher frequencies and
higher amplitudes of
vibration
RC bridges had δ with
structural movement
Determined that abutment
interaction accounted for
2/3rds of damping value
Roller supports do not contribute to δ.
Therefore overall δ from superstructure and movement of supporting piers
Suggested δ differences btw bridges due to
support conditions
δ results from one bridge can only be
used for predicting δof an identical
bridge. Even then, not with confidence
Deduced that RC bridge
damping was viscous damping
Douglas et al. (1981)
Free-vibration tests very effective in determining if
structure has ever been overloaded
during it's lifetime
Free-Vibration tests very
effective in determining δ
Soil structure interaction VIP
Wheeler (1982)
Single test pedestrian
adequate for footbridge studies
Proposes methods for design of
damping devices in footbridges
Jeary (1986)
δ predictor btw building base
dimension
Shears (1989)
In Oil Platform dynamic
modelling, soil-structure
interaction VIP
Longspan lightweight
floors have low δ, ∴ induce
signif. dynamic response
Lagomars-ino (1993)
δ highly dependent upon the size and
quantity of cracking
δ influenced by the stress state of the
structural components
Damping highly dependent on fundamental
vibration period
Initial damping highly
correlated to joint slippage
Effect on RC Structural Damping (Part 2)
Author’s
Concrete Strength/ Cracking/ Type
(Crack Spacing/Width)
Type or Percentage (Steel Area) of Reinforcement
(Stresses/ Strains)
Amplitude or Mode or
Frequency of Vibration
Test History (Repeated
Loading) or Test Method
Structure Interaction
Effects
Structure Size Effects, Type or
Dimensions
Beam Displacement
/Support Influence
Applicability or Usefulness of
Damping Results
Seismic Aspects Type of δ Other
Brownjohn (1994)
Suspensionbridge hangers may provide
hysteresis δ if inclined
The bridge in suspension bridges
influences higher mode δ
Deck bearings may provide
significant δ at low amplitudes
Suggests that for design calcs
only structural δ be considered
Farrar et al. (1994)
δ distortion affects will however, be more pronounced when the structure cracks and δ
δ will be in scale models compared to full-size structures. thought not to affect prediction of elastic dynamic response of
RC structures
Scaling δ from prototype tests
to full-size structures
affected by δ mechanism
Lutes and Sarkani (1995)
Using a fixed-base model in
soil-structure (s-s) interaction
studies will give sig. δ than a
free-based model
Wrong choice of fixed or free-
base may attribute
structural δ to s-s interaction, when actually
due to choice of δ
Denoon and Kwok (1996)
δ dependent on
vibration amplitude
δ dependent on type of test
method
δ dependent on building and
foundation height δ dependent on
building usage
Suda et al. (1996)
δ ratio with natural
frequency
δ ratios are in the longer
direction than shorter direction for office blocks
δ ratios of hotels or apartments (non-structural members) are
larger than office buildings
δ ratios as buildings become taller
δ ratios for pile foundation
buildings are than spread
foundations
Fang et al. (1998)
δ is amplitude dependent (buildings)
Li et al. (2000)
δ is amplitude dependent (buildings)
Table A.5. High-Strength Concrete and Reinforcement Specific Research Author’s Location Research Focus and Conclusions Drawn
Sparrow (1989) Connell Group High strength concrete in the Melbourne Central Project HSC 65 and 70 MPa usage described in the Melbourne Central office tower.
Collins, Mitchell and MacGregor (1993)
Canada Structural design considerations for high-strength concrete. - Questions and investigates the applicability of traditional design procedures for use with HSC.
Jensen (1994) Norway
Structural aspects of high strength concretes -Presents results of recent HSC research (overview) is given -Comments on HSC (or lack of it) in concrete design codes, Eurocode EC2 (up to 60 MPa), Norwegian Standard (recently changed from 65 to 105 MPa.
Park (1995) University of Canterbury, NZ
Opportunities in New Zealand for high strength reinforcing steel - Discusses HSC columns in particular. - Discusses NZS 3101 and its provisions for HSC up to 70 and 100 MPa concrete - Discusses NZS 3101 wrt 500 MPa steel.
Rasmussen and Baker (1995)
University of Queensland
Torsion in reinforced normal and high-strength concrete beams - Examines HSC beams subject to pure torsion (30, 50, 70 and 110 MPa). - Uses HS Danish Steel – Grade 550 MPa.
Foster and Gilbert (1996)
University of NSW
The design of nonflexural members with normal and high-strength concretes - Investigates concretes ranging btw 20 to 100 MPa for analysis techniques of nonflexural members (strut and tie model, plastic truss model). - Also looks at the main failure modes of nonflexural members.
Macchi, Pinto and Sanpaolesi (1996)
University of (Italy)
Ductility requirements for reinforcement under Eurocodes - Experimental research is reported in which cyclic tests showed that the currently accepted properties of reinforcing steel do not provide sufficient local ductility for the highest ductility class of structures envisaged by the Eurocodes. New requirements are proposed for reinforcing steel to be used in seismic regions, particularly with reference to uniform elongation at maximum load, and the ratio btw ultimate stress and yield stress.
Pendyala, Mendis and Patnaikuni (1996)
University of Melbourne
Full-range behaviour of high-strength concrete flexural members: Comparison of ductility parameters of high and normal-strength concrete members - Compares HSC flexural members against NSC in the 3 ductility parameters of hinge lengths, softening slopes and hinge rotation capacities. - The implications of designing with HSC are discussed.
Shah and Ouyang (1996)
Northwestern University
Tensile response of reinforced high strength concrete members - Effect of various parameters (reinforcement ratio and distribution of steel bars) were experimentally examined for HSC tensile members (fcm = 99 MPa). - Also uses 500 MPa steel.
Gilbert (1997a) University of NSW
Anchorage of reinforcement in high strength concrete - A comparison is made of the provisions in several international codes of practice for the anchorage of reinforcement bars in concrete with compressive strength up to 100 MPa. - Presents a case study on development lengths with f’c = 25, 32, 50,70 and 100 MPa. Also steel yield stresses fsy of 400 and 500 MPa.
Gilbert (1997b) University of NSW
High strength reinforcement in concrete structures: Serviceability implications - Serviceability of reinforced concrete beams and slabs is investigated for elements designed using Grade 500 reinforcement. - The short and long term behaviour of beams and slabs are considered.
Hoff (1997) Texas, US
The Hibernia offshore concrete platform - Case study of this project which is, to date, the largest single use of HSC (80 MPa). Additionally 500 MPa steel was used, as well as 400 steel.
Kong and Rangan (1997)
Curtin University
Reinforced high strength concrete (HSC) beams in shear - f’c ranged from 60 – 90 MPa. - Proposal to modify the value of minimum shear reinforcement given in AS 3600.
Mansur, Chin and Wee (1997)
University of Singapore
Flexural behaviour of high-strength concrete beams - 11 reinforced HSC beams tested in flexure - Yield strength of bars = 550 MPa - f’c ranged from 50 to 100 MPa.
Pendyala, Mendis and Bajaj (1997)
University of Melbourne
Design of high-strength concrete members - Ductility of beams and columns - Rectangular stress block for HSC - Shear design for HSC - Bond and anchorage for reinforcement in HSC members.
Pendyala, Mendis and Baweja (1997)
University of Melbourne
Towards the development of new codes and standards to increase the field application of high performance concretes - Investigates and discusses the broad attribues of HPC. - Reviews AS 3600 and summarises current research.
Attard and Stewart (1998)
Uni NSW – Uni Newcastle
A two parameter stress block for high-strength concrete - Looks at the applicability of the ACI rectangular stress block parameters to high-strength concretes. - Strengths btw 20 and 120 MPa.
Foster and Gilbert (1998)
University of NSW
Experimental studies on high-strength concrete deep beams - Presents the results for 16 HSC deep beams tested to destruction. Variables considered were shear-span to depth ratio, concrete strength (50 to 120 MPa) and the provision of secondary reinforcement.
Lin and Restrepo (1998)
University of Canterbury, NZ
Experimental verification of the concrete structures standard recommendations for the design of beam-column joints - Determining the shear strength of beam-column joints under seismic loading. - Provides experimental verification of the seismic performance of frames built using Grade 500 (threaded) longitudinal reinforcement (concrete 30 Mpa).
Lorrain, Maurel and Seffo (1998)
University of France
Cracking behaviour of reinforced high-strength concrete tension ties - The cracking behaviour of NSC and HSC tension ties under short term load was investigated experimentally. - The mechanical strength of the concrete, the reinforcement ratio, and the yield strength of deformed steel bars were taken as test parameters - f’c = 40 – 100 MPa - fsy = 620 and 830 MPa.
Park (1998) University of Canterbury, NZ
Some current and future aspects of design and construction of structural concrete for earthquake resistance - Investigates HSC (up to 100 MPA) and HSS (500 MPa) wrt ductile performance of a structure during an earthquake (specifically columns, precast floors and frames).
Rangan (1998) Curtin University
Suggestions for design of high performance high strength concrete (HPHSC structural members) - Proposes design rules of HPHSC beams, columns and walls in the range 20 – 100 MPa. - Briefly mentions 500 MPa steel in an example on calculating the max. tensile steel ratio for the flexural strength of HPHSC beams.
Sanjayan and Jeevanayagam (1998)
Monash University
Long term deflection of high strength concrete beams - Compares predicted results against experimental results for long term deflection of HSC beams. - Also develops a relationship btw basic creep factor and
compressive strength of concrete, for strengths up to 100 MPa.
Teng, Ma, Tan and Kong (1998)
Nanyang, Singapore
Fatigue tests of reinforced concrete deep beams - f’c = 50 MPa - fsy = 577 MPa - 3 different types of web reinforcement were investigated, and the tests revealed that web reinforcement has significant influence on the structural response of deep beams under fatigue loading
Adams, Walsh, Marsden, Patrick (1999)
BHP – University of Newcastle (Walsh)
Factors affecting the ductility of stiffened rafts - Theoretical and experimental study to assess the likely impact that steel ductility has on the behaviour of stiffened rafts detailed in accordance with AS 2870. - Uses Grade 500 reinforcing mesh.
Chick, Patrick and Wong (1999)
BHP – University of Adelaide
Ductility of reinforced-concrete beams and slabs, and AS3600 design requirements - Uses 500 MPa steel to study the overload behaviour of concrete beams and slabs. - In particular, the effect that low-ductility reinforcing steel used in welded mesh has on the load-carrying capacity of these types of flexural members is examined. Implications for AS 3600 are discussed.
Esfahani and Rangan (1999)
Curtin University
Evaluation of proposed revisions to AS3600 bond strength provisions - Presents a comparison of the proposed AS3600 bond strength provisions for NSC and HSC.
Gilbert (1999) University of NSW
Flexural crack control for reinforced concrete beams and slabs: An evaluation of design procedures - The advent of high strength reinforcing steels will inevitably lead to higher steel stresses under in-service conditions, thereby exacerbating the problem of crack control. - In the paper, the current flexural crack control provisions of AS3600 are presented and the crack width calculation procedure in several of the major international codes, ACI318, EC2 are assessed.
Gilbert, Patrick and Adams (1999)
University of NSW – BHP
Evaluation of crack control design rules for reinforced concrete beams and slabs - Recommendations are made as to a suitable crack control model for inclusion in an amendment to AS 3600 to allow the design yield strength to increase to 500 MPa.
Gravina and Warner (1999)
University of Adelaide
Modelling of high-moment plastification regions in concrete structures - A review of recent research into the rotation capacity of reinforced concrete flexural members shows renewed interest in the topic has been generated by the introduction, in various countries, of high-strength reinforcing steel with limited uniform elongation. This has caused concern over the possibility of steel fracture in high-moment regions - Grade 500 steel, 10% uniform elongation, - f’c = 30 MPa.
Panagopoulos, Mendis and Portella (1999)
University of Melbourne
Seismic performance of frame structures with high-strength concrete and 500 MPa steel - f’c = 50 and 100 MPa - grade 400 and 500 MPa
Patrick (1999) BHP Research
Australian 500 MPa reinforcing steels and new AS 3600 ductility design provisions - Discusses experimental and analytical studies of the overload behaviour of concrete beams and slabs. - In particular, the effect that low-ductility reinforcing steel used in welded mesh has on the load-carrying capacity of these types of flexural members is examined.
Sanders (1999) SRIA Advances in fire design for reinforced concrete structures- Moving to more rational design methods
- grade 400 and 500 MPa
Turner (1999) SRIA
Introduction of 500 MPa reinforcing steel and its effect on AS 3600 - Discusses mainly the new AS/NZS xxxx proposed reinforcing standard and its interaction with AS 3600. Reviews major technical changes esp. wrt ductility
Young, Fenwick and Lawley (1999)
University of Auckland, NZ
Mechanical threaded rebar couplers and plate anchors in seismic resistant concrete frames - Assessment of how couplers perform in plastic hinge zones under seismic loading, particularly beam-column connections. - Uses Grade 500 threaded bars, concrete Grade 35 MPa.
APPENDIX B: RC and PSC Beam Calculations B-1
APPENDIX B
RC and PSC Beam Calculations
B.1 General Remarks
If the stress for a reinforced concrete member, subjected to a bending moment causing
deflection, has never exceeded its tensile strength, the member is free from cracks
(Ghali and Favre, 1986). In this case, the reinforcement and concrete undergo similar
strains; this region is defined herein as ‘uncracked’. The analytical equation to calculate
the bending moment that causes first cracking, Mcr in kNm, is based on the flexural
strength of concrete in tension and the beam’s cross-sectional dimensions. It is defined
as Equation B.1 (AS3600-2001)
t
gcfcr y
IfM '= (B.1)
where f’cf is the characteristic flexural tensile strength of the concrete in MPa (f’cf =
0.6√fcm, AS3600-2001); yt is the distance between the neutral axis and extreme fibres in
tension of the uncracked section in mm; Ig is the gross moment of inertia of the
uncracked section in mm4 (determined here using Ig = bD3/12).
The ultimate moment capacity, Mu,calc in kNm of a doubly-reinforced (under-reinforced)
concrete beam is found from
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −= c
u
cscsystcalcu d
dkd
AdfAM2
16002,
αα (B.2)
where Ast is the cross-sectional area of reinforcing steel in mm2; d is the depth to the
centroid of the reinforcing steel in mm; and α and ku are the compressive stress block
parameters.
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-2
The cracking moment, Mcr (kNm) of a prestressed concrete beam may be calculated
from the following formula:
Mcr = (f’cf + σbp) I/yB (B.3)
where f’cf is the characteristic flexural tensile strength of concrete in MPa; and σbp is the
flexural stress provided by the prestress in MPa.
Similarly, the ultimate moment capacity may be calculated from the following:
Mu = [σpu Apt dp + fsy Ast ds - fsy Asc dsc - (0.85 f’c b(γ ku d)2/2)] (B.4)
Where γ is defined in Figure B.1; σpu, Apt and dp are the ultimate stress (MPa), cross-
section area (CSA) in mm2, and depth to the prestressing tendons (mm), respectively; fsy
Ast and ds are the yield stress (MPa), area of tension steel (mm2), and depth to the
tension reinforcement (mm), respectively; and Asc and dsc are the CSA (mm2) and depth
to the compression reinforcement (mm), respectively.
The formula for the calculation of the instantaneous static deflection in mm (∆i) of a 2-
point loaded reinforced concrete and prestressed concrete beams is given by
IElw
IElP
wpi
43
αα +=∆ (B.5)
where P is the applied load in kN, l is the effective span in m, w is the self-weight of the
beam in kN/m, and αp and αw are constants that depend on the loading conditions. The effective moment of inertia Ief (the second moment of area) of the member,
incorporating tension stiffening (see Bažant and Oh, 1984), is calculated using
Branson’s formula:
Ief = Icr + (Ig – Icr)(Mcr/Ms)3 ≤ Ig (B.6)
where Ief is defined by the following limits:
Icr ≤ Ief ≤ Ig (B.7)
and Ig is the second moment of area of the gross concrete cross section in mm4 about the
centroidal axis as discussed previously; Icr is the second moment of area of a cracked
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-3
section in mm4 with the reinforcement transformed to an equivalent area of concrete;
Mcr is the cracking moment at the section in kNm; and Ms is the applied bending
moment in kNm at the section for the loading increment being considered.
The Australian Concrete Structures Design Code (AS3600-1994), previously allowed,
as a further simplification (for rectangular RC members only with a width of b in mm
and effective depth to the centroid of the tensile steel in mm), that:
Ief = 0.045bd3 (B.8)
As a simplification (for RC members only), Ief is now calculated using the following
equation (AS3600-2001: Clause 8.5.3.1):
Ief = (0.02+(2.5ρt))bd3 (B.9)
where ρt is the tensile reinforcement ratio.
T
C = 0.85 f’c γ ku bd
0.85 f’c for NSCα f’c for HSC
γ kudkud
NA
γ kud/2
η f’c
C = αηf’cku bd
β kud
For NSC (AS3600-2001): γ = 0.85 for f’c ≤ 28 MPa andγ = 0.85 – 0.007(f’c-28) forf’c ≥ 28 MPa where 0.65 ≤ γ ≤ 0.85 for f’c ≤ 65 MPa.
For HSC (Mendis, 2002):γ = 0.65 – 0.00125(f’c-60)
60 MPa ≤ f’c ≤ 100 MPa
α = 0.85 – 0.0025(f’c-60)
60 MPa ≤ f’c ≤ 100 MPa
Figure B.1: Compressive Block Parameters
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-4
B.2 Calculations for B-Series Beams
Beam is under-reinforced
pt = Ast/bd = 1257/200×264 = 0.0238pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0.01933(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.0194(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.1618υ = 600 pc
0.85 γ f’c
= 0.1174
200
300
20
264
2044
Beam BII-2
30.0 MPa500 MPa
3N20’s = 942 mm2
2N12’s = 226 mm2
dc = 32 mmdt = 264 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.836
Beam is under-reinforced
pt = Ast/bd = 942/200×264 = 0.0179pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0.01343(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.03101(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.1477υ = 600 pc
0.85 γ f’c
= 0.1174
200
300
20
262
2038
Beam BI-3
23.1 MPa400 MPa
3Y24’s = 1357 mm2
2Y12’s = 226 mm2
dc = 32 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.85
Beam is under-reinforced
pt = Ast/bd = 1357/200×262 = 0.0259pc = Asc/bd = 226/200×264 = 0.00431pt - pc = 0.02160(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.01529(pt - pc)lim < pt - pc
∴ Asc will yield_________________________a = (Ast - Asc) fsy
0.85 f’c b = 115.1 mm
ku = η + √ η2 + υ (dc/d) = 0.3628a = γ ku d = 80.1 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 111.8 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
ku = η + √ η2 + υ (dc/d) = 0.3376a = γ ku d = 74.5 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 105.9 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
Mu = Astfsy(d-a/2) + Ascfsy (a/2 - dc) = 112.7 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
200
300
20
dt = 264
20 22.7
Beam BI-1
30.0 MPa400 MPa
4Y20’s = 1257 mm2
2Y12’s = 226 mm2
dc = 32 mmdt = 264 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.836
dc
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-5
Beam BII-5
41.5 MPa500 MPa
4N20’s = 1257 mm 2
2N12’s = 226 mm 2
200
300
20
264
22.720
Beam BII-4
23.1 MPa500 MPa
2N24’s = 905 mm2
2N12’s = 226 mm2
200
300
20
262
10020dc = 32 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.85
dc = 32 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.756
Beam BII-6
200
300
20
264
2044
41.54 MPa500 MPa
3N20’s = 942 mm 2
2N12’s = 226 mm 2
dc = 32 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.756
Beam is under-reinforced
pt = Ast/bd = 905/200×262 = 0.0173pc = Asc/bd = 226/200×262 = 0.00431pt - pc = 0.012989(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.02446(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.1818υ = 600 pc
0.85 γ f’c
= 0.1510
Beam is under-reinforced
pt = Ast/bd = 1257/200×264 = 0.0238pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0.01949(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.0388(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.1749υ = 600 pc
0.85 γ f’c
= 0.0938
Beam is under-reinforced
pt = Ast/bd = 942/200×264 = 0.0178pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0.01344(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.0388(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.1182υ = 600 pc
0.85 γ f’c
= 0.0938
ku = η + √ η2 + υ (dc/d) = 0.4087a = γ ku d = 91.0 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 98.7 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
ku = η + √ η2 + υ (dc/d) = 0.3797a = γ ku d = 75.8 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 112.1 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
ku = η + √ η2 + υ (dc/d) = 0.2774a = γ ku d = 55.4 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 109.6 kNm
Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-6
η = (ptfsy - 600pc) (2α) γ f’c
= 0.01791 Normal = 0.01874 HSυ = 600 pc
α γ f’c
= 0.0697 Normal = 0.07297 HSku = η + √ η2 + υ (dc/d) = 0.1112 Normal = 0.1143 HSa = γ ku d = 19.2 Normal = 19.8 HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 51.0 kNm Normal = 51.1 HS
a = (Ast - Asc) fsy
0.85 f’c b = 6.6 mm
Mu = Astfsy(d-a/2) + Ascfsy (a/2 - dc) = 39.5 kNm
For Both BI-7 and BII-8Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
Beam BI-7200
300
20
266
20 116
64.5 MPa400 MPa
2Y16’s = 402 mm 2
2Y12’s
dc = 32 mm
For Normalγ = 0.85 - 0.007(f’c - 28) = 0.5945 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.812pt = Ast/bd = 402/200×266 = 0.00756pc = Asc/bd = 226/200×266 = 0.00425pt - pc = 0.00338(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.03215 Normal
For Normal (pt - pc)lim < pt - pc
∴ Asc will yield
γ = 0.65 - 0.00125(f’c - 57)= 0.6406 Pendyala and Mendis (1997)
= 0.574 Rangan (1998)γ = 0.85 - 0.008(f’c - 30)
α = 0.85 - 0.0025(f’c - 57)= 0.8313 Pendyala and Mendis (1997)
= 0.812 Rangan (1998)α = 0.85 - 0.004(f’c - 55)
Beam BII-8200
300
20
266
116
64.5 MPa500 MPa
2N16’s = 402 mm 2
2N12’s = 226 mm 2
For Normalγ = 0.85 - 0.007(f’c - 28) = 0.5945 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.812pt = Ast/bd = 402/200×266 = 0. 00756pc = Asc/bd = 226/200×266 = 0. 00425pt - pc = 0. 00338(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.04705 Normal = 0.05144 HSFor Normal (pt - pc)lim > pt - pc
∴ Asc will not yield
Beam BI-9200
300
20
262
20 100
53.0 MPa400 MPa
2Y24’s = 905 mm2
2Y12’s = 226 mm 2
dc = 32 mmγ = 0.85 - 0.007(f’c - 28) = 0.675α = 0.85
For Normalpt = Ast/bd = 905/200×262 = 0.0173pc = Asc/bd = 226/200×262 = 0.00431pt - pc = 0.01298(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.04860(pt - pc)lim > pt - pc
∴ Asc will not yieldη = (ptfsy - 600pc) (2α) γ f’c
= 0.07156υ = 600 pc
α γ f’c
= 0.08287
ku = η + √ η2 + υ (dc/d) = 0.1950a = γ ku d = 34.5 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 87.4 kNm
For Both BI-9 and BII-10Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-7
Beam BII-10
200
300
20
262
100
53.0 MPa500 MPa
2N24’s = 905 mm2
2N12’s = 226 mm2
Beam BII-11200
300
20
264
2044
90.7 MPa500 MPa
3N20’s = 942 mm2
2N12’s = 226 mm 2
Beam BII-12200
300
20
264
20 22.7
80 MPa500 MPa
4N20’s = 1257 mm2
2N12’s
dc = 32 mmγ = 0.85 - 0.007(f’c - 28) = 0.675α = 0.85
For Normalpt = Ast/bd = 905/200×262 = 0.0173pc = Asc/bd = 226/200×262 = 0.00431pt - pc = 0.01298(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.04457(pt - pc)lim > pt - pc
∴ Asc will not yield
η = (ptfsy - 600pc) (2α) γ f’c
= 0.0998
υ = 600 pc
α γ f’c
= 0.0829ku = η + √ η2 + υ (dc/d) = 0.2415a = γ ku d = 42.7 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 107.6 kNm
γ = 0.65 - 0.00125(f’c - 57)= 0.6079 Pendyala and Mendis (1997)
= 0.3644 Rangan (1998)γ = 0.85 - 0.008(f’c - 30)
α = 0.85 - 0.0025(f’c - 57)= 0.7658 Pendyala and Mendis (1997)
= 0.7072 Rangan (1998)α = 0.85 - 0.004(f’c - 55)
For Normalγ = 0.85 - 0.007(f’c - 28) = 0.4111 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.7072pt = Ast/bd = 942/200×264 = 0. 0178pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0. 01344(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.0729 Normal = 0.0729 HSFor Normal (pt - pc)lim > pt - pc
∴ Asc will not yieldη = (ptfsy - 600pc) (2α) γ f’c
= 0.07870 Normal = 0.09459 HS
For Normalγ = 0.85 - 0.007(f’c - 28) = 0.4111 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.7072pt = Ast/bd = 1257/200×264 = 0. 0238pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0. 01931(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.0729 Normal = 0.0729 HSFor Normal (pt - pc)lim > pt - pc
∴ Asc will not yieldη = (ptfsy - 600pc) (2α) γ f’c
= 0.1080 Normal = 0.1298 HS
υ = 600 pc
α γ f’c
= 0.04993 Normal = 0.06001 HSku = η + √ η2 + υ (dc/d) = 0.2411 Normal = 0.2851 HSa = γ ku d = 41.4 mm Normal = 48.9 HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 150.1 kNm Normal = 150.1 kNm HS
υ = 600 pc
α γ f’c
= 0.04993 Normal = 0.06001 HSku = η + √ η2 + υ (dc/d) = 0.1894 Normal = 0.2220 HSa = γ ku d = 32.5 mm Normal = 38.1 mm HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 114.5 kNm Normal = 114.5 kNm HS
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-8
B.3 Calculations for PS-Series Beams
Beam PS1
60.6 MPa
200
300235
NA
e=111.67 mm
1. Computation of the Effective Prestress Coefficient, η
Is determined from the sum of:a) Elastic shortening of concrete;b) Shrinkage of concrete, and;c) Relaxation of steel wires.Each of these will be considered in turn.
a) Loss of prestress due to elastic shortening of concrete =
where σcp = Stress in concrete at the time of transfer; Ecp = Young’s Modulus of concrete at the time of transfer = 32093 MPa Ep = Young’s Modulus of steel = 227000 MPa (From BHP tests)
For beam PS1, the total number of tendons was 9, each transferring a force of26344.77N to the concrete.
Thus, the initial stress transferred by the wires to the concrete section = σcp = H / Ac = 9×26344.77/60,000 = 3.95 MPa.
∴ Loss of prestress due to elastic shortening of concrete =
= 28.0 MPa
% Loss of stress due to elastic shortening of concrete = 28.0 / 1432.4 (σpi)×100 = 2.50%
b) Loss of prestress due to shrinkage of concrete = ∆ σshrinkage = Ep εcs
where εcs = k1 εcs.b; εcs.b = Basic shrinkage factor, for high-strength concrete = 750×10-6; k1 = 0.21.
Thus, loss of stress due to shrinkage of concrete is = ∆ σshrinkage = 227000×0.21×750×10-6 = 35.8 MPa.% Loss of stress due to shrinkage of concrete = 32.70 / 1432.4 ×100
= 2.47%
c) (%) Loss of prestress due to relaxation of steel wires =
where R = Design relaxation of the tendon and is given by = k4 k5 k6 Rb; Rb = Basic relaxation of tendon = 1% for low relaxation wire; k4 = log[5.4(j)1/6] where j is the number of days after prestressing (36) = 0.992; k5 = 1.5; k6 = 1.00 for an average temperature of 200C.∴ R = 0.992 × 1.5 ×1.00 = 1.488
% Loss of stress due to relaxation of steel wires = 1.488[1-35.8/ 1432.4 ] = 1.45%
Thus, loss of stress due to relaxation of steel wires = (1.45/100) × 1432.4 = 20.8 MPa.______________________________________________________________________________________Total loss of stress at the day of testing the beam (i.e. 36 days after casting)
= Loss due to (Elastic shortening of concrete+Shrinkage of concrete+Creep of concrete+relaxation of tendon)= 28.0 + 35.8 + 0 + 20.8 = 84.6 MPa.Total loss of stress at the day of beam testing = 84.6/ 1432.4 ×100 = 5.91%
∴ The effective prestress coefficient, η to be used for design and analysis of PS1 = (100-6.34)/100 = 0.94.Thus, in summary for all prestressed beams:
cpc
pp E
Eσσ ×=∆
95.332093227000 ×=∆
pσ
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ∆+∆−=∆
pi
creepshrinkagerelaxation R
σσσ
σ 1
∆σp
∆σshrinkage
∆σrelaxation
η
PS191.95%28.0 MPa
2.50%35.8 MPa
1.45%20.8 MPa
0.94
PS2112.4%34.2
2.50%35.8
1.45%20.8
0.94
PS3132.82%40.4
2.50%35.8
1.45%20.8
0.93
PS4224.8%68.3
2.50%35.8
1.45%20.8
0.91
PS5234.98%71.4
2.50%35.8
1.45%20.8
0.91
PS6153.3%46.6
2.50%35.8
1.45%20.8
0.93
PS7132.82%40.4
2.50%35.8
1.45%20.8
0.93
PS8153.3%46.6
2.50%35.8
1.45%20.8
0.93
PS9132.82%40.4
2.50%35.8
1.43%20.8
0.93
PS1018
3.9%55.9
2.50%35.8
1.45%20.8
0.92
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-9
2. Flexural and Shear Design of Beam Test Specimens
The design of a fully prestressed concrete member, implies that cracking in not allowed during service. The main method ofensuring the serviceability requirements of structures are met, is by limiting the tensile and compressive stresses in concrete bothat transfer and under full service loads. The way in which these tensile and compressive stresses as controlled is through the safedesign of the prestressing force and prestressing eccentricity.
The main differences of prestressed design to that of reinforced concrete design include:a) Checking load transfer stresses;b) Limit state design at service loading;c) Limit state design at failure loading.These checks help to control for short and long term effects, cracking and deflection at service loads.
Graphical Representation of StressesThe following diagram shows the stresses that occur in the extreme fibers of a section when subject to various stresses that occurin prestressed concrete members.
H/A fct
+ +
+
+
+
+
+-
--
H eB ytI
H eB ytI
Mw ytI
= OR
i) Stresses due to prestress force
ii) Stresses due to moment created by eccentricity of prestressing force
iii) Stresses due to moment created by applied live anddead loads
iv) Combined effects of prestressing forces and applied loads
IyM
IyeH
AH twtB +−
IyM
IyeH
AH twtB −⎟
⎠⎞
⎜⎝⎛ +
IyM
IyeH
AH twtB +⎟
⎠⎞
⎜⎝⎛ −η
IyM
IyeH
AH twtB −⎟
⎠⎞
⎜⎝⎛ +η
General Equations for Bending DesignEquations describing the stresses at the top and bottom fibers under working stressed are defined as follows.
At Transfer: fCF = Equation 1.1
fCB = Equation 1.2
After Prestress Losses: fCF = Equation 1.3
fCB = Equation 1.4
Types of Prestressing CasesOne of two analysis and design methods can be used, depending on the type of moments the beam will be subject to during its life.They are Case A and Case B prestressing.
Case A PrestressingThis case applies if both the minimum moment (M1) and maximum moment (M2) are positive.
Case B PrestressingThis case applies if the minimum moment (M1) is negative and the maximum moment (M2) is positive.
For simply-supported beams, M1 is the bending moment due to beam self-weight, and M2 is the moment caused by the sum of thedead and live loads (it is always positive), an example follows.
Due to g = M1 = g L2 / 8 = 5.86 kNm
Due to Q = M2 = (Q L / 6) + (g L2 / 8)
g = self-weight
Q
gL2/8
QL/6
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-10
If we examine the prestress force alone, the stress in the top fibre (fct) is given by:
fct =
But I / A = k2 where k = = the radius of gyration
Therefore, fct = NB
Hence, fCT is negative (in tension) if This is referred to as Case A Prestressing
The positive fCT due to M1 should therefore be counteracted by a negative prestressing force.
Consequently, if fCT is positive (in compression) if This is referred to as Case B Prestressing
The negative fCT due to M1 should therefore be counteracted by a positive prestressing force.
Critical Stress State Equations for Design (Case A Only Exists in Current Investigations)
Case A Prestressing i.e.
When subject to M1, the following conditions must be satisfied:
Top fibre stress (fCT): Equation A1
Bottom fibre stress (fCB): Equation A2
When subject to M1, the following conditions must be satisfied:
Top fibre stress (fCT): Equation A3
Bottom fibre stress (fCB): Equation A4
:
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
AIye
AH
IyeH
AH tBtB 1
tB y
ke2
>
AI
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
t
BtB
yke
AH
kyeH
AH
22 1
tB y
ke2
<
6
2 Dyk
t
=
+
+
+-
-
M1 yTI
⎟⎟⎠
⎞⎜⎜⎝
⎛−
t
B
yke
AH
21
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
t
B
yke
AH
21η
⎟⎟⎠
⎞⎜⎜⎝
⎛+
t
B
yke
AH
21
M1 yBI
NANA
+
-
=
≤ C
≤ Ct or ≥ -Ct
tB y
ke2
>
tTTB C
IyM
IyeH
AH
−≥+− 1
CIyM
IyeH
AH BBB ≤−+ 1
tBBB C
IyM
IyeH
AH
−≥+⎥⎦⎤
⎢⎣⎡ − 2η
CI
yMI
yeHAH TTB ≤−⎥⎦
⎤⎢⎣⎡ + 2η
+
+
+-
-
M2 yT
I
⎟⎟⎠
⎞⎜⎜⎝
⎛−
T
B
yke
AH
21
NANA
+
-
=
≥ -Ct
≤ C
η
η
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-11
If we examine the prestress force alone, the stress in the top fibre (fct) is given by:
fct =
But I / A = k2 where k = = the radius of gyration
Therefore, fct = N
Hence, fCT is negative (in tension) if This is referred to as Case A Prestressing
The positive fCT due to M1 should therefore be counteracted by a negative prestressing force.
Consequently, if fCT is positive (in compression) if This is referred to as Case B Prestressing
The negative fCT due to M1 should therefore be counteracted by a positive prestressing force.
Critical Stress State Equations for Design (Case A Only Exists in Current Investigations)
Under M1
Top fibre stress (fCT): Equation A1
Bottom fibre stress (fCB): Equation A2
Under M2
Top fibre stress (fCT): Equation A3
Bottom fibre stress (fCB): Equation A4
3. Example Design of Prestressed Beam PS1
B:
a) Calculating maximum service moment allowable for the section:
Combining EquationsA1 and A3, we obtain: Equation A1.1
For PS1 = 102.8×106 Nmm ∴ M2 ≤ 102.8 kNm
Combining EquationsA2 and A4, we obtain: Equation A1.2
For PS1 = 98.3×106 Nmm ∴ M2 ≤ 98.3 kNm
b) Obtain maximum 1/H and maximum eB:
We rearrange to obtain the following equations to obtain max. H and max. eB (whilst ensuring concrete cover ismaintained):
Solving Equations A1 and A4 simultaneously, we obtain a min. H of 714.1 kN and min. eB of 76.57 mm.
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
AIye
AH
IyeH
AH tBtB 1
tB y
ke2
>
AI
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
t
BtB
yke
AH
kyeH
AH
22 1
tB y
ke2
<
6
2 Dyk
t
=
tTTB C
IyM
IyeH
AH
−≥+− 1
CIyM
IyeH
AH BBB ≤−+ 1
tBBB C
IyM
IyeH
AH
−≥+⎥⎦⎤
⎢⎣⎡ − 2η
CI
yMI
yeHAH TTB ≤−⎥⎦
⎤⎢⎣⎡ + 2η
tT
T CCMMZ
yI
ηη
+−
≥= 12
tB
B CCMMZ
yI
+−
≥=η
η 12
45.4937.0253.281086.5937.0103
626
×+××−
≥×M
45.4253.28937.01086.5937.0103
626
+×××−
≥×M
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-12
If we examine the prestress force alone, the stress in the top fibre (fct) is given by:
fct =
But I / A = k2 where k = = the radius of gyration
Therefore, fct = N
Hence, fCT is negative (in tension) if This is referred to as Case A Prestressing
The positive fCT due to M1 should therefore be counteracted by a negative prestressing force.
Consequently, if fCT is positive (in compression) if This is referred to as Case B Prestressing
The negative fCT due to M1 should therefore be counteracted by a positive prestressing force.
Critical Stress State Equations for Design (Case A Only Exists in Current Investigations)
Under M1
Top fibre stress (fCT): Equation A1
Bottom fibre stress (fCB): Equation A2
Under M2
Top fibre stress (fCT): Equation A3
Bottom fibre stress (fCB): Equation A4
3. Example Design of Prestressed Beam PS1
B:
a) Calculating maximum service moment allowable for the section:
Combining EquationsA1 and A3, we obtain: Equation A1.1
For PS1 = 102.8×106 Nmm ∴ M2 ≤ 102.8 kNm
Combining EquationsA2 and A4, we obtain: Equation A1.2
For PS1 = 98.3×106 Nmm ∴ M2 ≤ 98.3 kNm
b) Obtain maximum 1/H and maximum eB:
We rearrange to obtain the following equations to obtain max. H and max. eB (whilst ensuring concrete cover ismaintained):
Solving Equations A1 and A4 simultaneously, we obtain a min. H of 714.1 kN and min. eB of 76.57 mm.
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
AIye
AH
IyeH
AH tBtB 1
tB y
ke2
>
AI
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
t
BtB
yke
AH
kyeH
AH
22 1
tB y
ke2
<
6
2 Dyk
t
=
tTTB C
IyM
IyeH
AH
−≥+− 1
CIyM
IyeH
AH BBB ≤−+ 1
tBBB C
IyM
IyeH
AH
−≥+⎥⎦⎤
⎢⎣⎡ − 2η
CI
yMI
yeHAH TTB ≤−⎥⎦
⎤⎢⎣⎡ + 2η
tT
T CCMMZ
yI
ηη
+−
≥= 12
tB
B CCMMZ
yI
+−
≥=η
η 12
45.4937.0253.281086.5937.0103
626
×+××−
≥×M
45.4253.28937.01086.5937.0103
626
+×××−
≥×M
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-13
4. Example Computation of Maximum Bending Moment and Shear Capacity of Prestressed Beam PS1
Task 1: Ensure a tension failure by checking that ku≤0.4 (AS 3600-1994: Clause 8.1.3) by using:
(ku d) = [σpy Apt + fsy (Ast - Asc)] / (0.85 f’c γ b)
where σpu = ultimate stress in bonded tendons (Clause 8.1.5) = fp (1 - k1 k2/γ ) = 1710 (1-(0.4×0.0971/0.65)) = 1608.0 MPa
where k1 = 0.4 for fpy/fp < 0.9 = 0.28 for fpy/fp ≥ 0.9 k2 = [fp Apt + fsy (Ast - Asc)] / (bef dp f’c)
and γ = for HSC (Rangan, 1988) For NSC: γ = AS3600-1994 Clause 8.1.2.2(b) = 0.85 - 0.008(f’c-30), where 0.65≤ γ ≤0.85 = 0.85 - 0.007(f’c-28) = 0.6052 = 0.6218∴ γ = 0.65 = 0.65
Finally ku = 0.162 which is ≤ 0.4, ∴ means the beam is under-reinforced and will fail in tension.
Task 2: Calculate Ultimate Moment Capacity of under-reinforced beam PS1:
Mu = [σpu Apt dp + fsy Ast ds - fsy Asc dsc - (0.85 f’c b(γ ku d)2/2)] Conditions satisfied
Mu = [(1608×176.7 ×261.7)+0-0-((0.85 ×60.6 ×200 ×(0.65 ×0.162 ×261.7)2)/2)] = 70.5 kNm
Task 3: Calculate Cracking Moment of under-reinforced beam PS1:
Mcr = (f’cf + σbp) I/yB
where f’cf = 0.6√f’c = 4.7 MPa
where σbp = ηH(1/A + eB yB/I) = 0.94×239000 (1/60000 + (111.7×150/450×106)) = 12.51 MPa
∴ Mcr = (4.7+12.51)(450×10^6/150) = 51.6 MPa.
5. Example Computation of Initial Camber of Prestressed Beam PS1
Initial camber of the beam is given by: ϕc = (H eB L2 )/8EcIg
= 239000 × 111.7 × 60002
8 × 32093 × 450×106
= 8.3 mm (upwards deflection).
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-14
Computation of Initial Prestress Force, H (For One Wire)
AS3600-1994 Clause 6.3.1 -b(ii)Yield strength of tendons, fpy = 0.85 fp.
∴ fpy = 0.85 × 1710 = 1453.5 MPa
Allowable tensile strength in steel wires (Clause 19.3.4.6): = 0.80 fp = 0.80 × 1710 = 1368 MPa
Allowable strain in the wire: =Allowable Stress Young’s Modulus (BHP Tests) = 1368 227000 = 0.00603
Elastic Strain = ∆ L LThus the allowable extension in the wire = Elastic Strain × L
= 0.00603 × 7000 = 42.2 mm
Average slip in wire through the gripping cones (2 mm observed during stretching)∴ Net Extension = 44.2 mm
Strain in wire after releasing jack = 44.2/7000 = 0.00631Stress in wire after loosening jack = 0.00631 × 227000
= 1432.4Initial prestress force in each wire just after transfer = fpi × Ap
= 1432.4 ×19.63/1000 = 28118 kN
5. Example Computation of Moment of Inertias
Ief = Icr + [(I - Icr)(Mcr/Ms)3] ≤ Ig (Gross Moment of Inertia)
Icr = bd3 (4k3 + 12 ρ n (1-k)2)
whereModular Ratio, n = Es/ Ec
Es = Young’s Modulus of Steel = 227 Gpa
Ec = Young’s Modulus of Concrete on Day of Testing = 32093 MPa used for all calculations
Steel Ratio, ρ = Apt/bd
k = √ (ρ n)2 + 2ρ n - ρ n
Ms = Maximum Moment = Mu (Measured Ultimate Moment Capacity of Test Beam)
Mcr = Measured Cracking Moment of Test Beam
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-15
9 HS 5 = 177 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.94C = 0.5 fcm = 0.5 × 60.6 = 30.3 MPaCt = 0.6√fcm = 0.6 √60.6 = 4.67 MPa__________________________________________
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
3.3010450150
104501507.111239000
6000023900094.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 6.10472 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
67.410450150
104501507.111239000
6000023900094.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 3.502 ≤
11 HS 5 = 216 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.94C = 0.5 f’cp = 0.5 × 60.6 = 30.3 MPaCt = 0.6√f’c = 0.6 √60.6 = 4.67 MPa__________________________________________
tTTB C
IyM
IyeH
AH
−≥+− 1
67.410450
1501086.510450
1505.11029300060000293000
6
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
3.3010450
1501086.510450
1505.11029300060000293000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
67.496.3 −≥−
3.307.13 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
3.3010450150
104501505.110293000
6000029300094.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 0.10762 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
67.410450150
104501505.110293000
6000029300094.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 2.582 ≤
σpu = 1608 MPaku = 0.162γ = 0.65d = 261.7 mm
Mu = 70.5 kNmMcr = 51.6 kNmInitial Camber ϕ = 8.3 mm
Icr = 63.9 × 106 mm4
Ief = 285.9 × 106 mm4
___________________________
σpu = 1608 MPaku = 0.133γ = 0.65d = 260.5 mm
Mu = 84.6 kNmMcr = 58.3 kNmInitial Camber ϕ = 10.1 mm
Icr = 75.9 × 106 mm4
Ief = 150.1 × 106 mm4
___________________________
Beam PS2
fcm = 60.6 MPaH = 293.0 kN
200
300 NA
eB =110.5 mm
226
Beam PS1
200
300227
NA
eB =111.7 mm
fcm = 60.6 MPaH = 239.0 kN
tTTB C
IyM
IyeH
AH
−≥+− 1
4.6710450
1501086.510450
150111.72390006
6
6−≥
×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
3.3010450
1501086.510450
1507.11123900060000239000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
67.496.2 −≥−
3.309.10 ≤
60000239000
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-16
Beam PS3
200
fcm = 60.2 MPaH = 346 kN
300222
NA
eB =97.0 mm
tTTB C
IyM
IyeH
AH
−≥+− 1
66.410450
1501086.510450
1509734600060000
3460006
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
2010450
1501086.510450
1509734600060000
3460006
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
66.447.3 −≥−
1.300.15 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
1.3010450150
1045015097346000
6000034600093.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 2.10542 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
66.410450150
1045015097346000
6000034600093.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 3.612 ≤
tTTB C
IyM
IyeH
AH
−≥+− 1
66.410450
1501086.510450
1505.8058500060000
5850006
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
1.3010450
1501086.510450
1505.8058500060000
5850006
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
66.499.3 −≥−
1.305.23 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
1.3010450150
104501505.80585000
6000058500091.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 4.10652 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
66.410450150
104501505.80585000
6000058500091.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 5.832 ≤
22 HS 5 = 432 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.91C = 0.5 f’cp = 0.5 × 60.2 = 30.1 MPaCt = 0.6√f’c = 0.6 √60.2 = 4.66 MPa__________________________________________
σpu = 1608 MPaku = 0.304γ = 0.65d = 230.5 mm
Mu = 130.9 kNmMcr = 82.7 kNmInitial Camber ϕ = 14.7 mm
Icr = 98.0 × 106 mm4
Ief = 210 × 106 mm4
___________________________
13 HS 5 = 255 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.93C = 0.5 f’cp = 0.5 × 60.2 = 30.1 MPaCt = 0.6√f’c = 0.6 √60.2 = 4.66 MPa__________________________________________
σpu = 1608 MPaku = 0.191γ = 0.65d = 247.0 mm
Mu = 93.2 kNmMcr = 54.6 kNmInitial Camber ϕ = 8.9 mm
Icr = 77.7 × 106 mm4
Ief = 193.1 × 106 mm4
___________________________
Beam PS4
fcm = 60.2 MPaH = 585.0 kN
200
300 NAeB = 80.5 mm
172
4Y12’s = 440 mm2
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-17
tTTB C
IyM
IyeH
AH
−≥+− 1
01.510450
1501086.510450
1506061200060000
6120006
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
9.3410450
1501086.510450
1506061200060000
6120006
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
01.5614.3 −≥−
9.345.20 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
9.3410450150
1045015060612000
6000061200091.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 7.11022 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
01.510450150
1045015060612000
6000061200091.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 3.762 ≤
tTTB C
IyM
IyeH
AH
−≥+− 1
01.510450
1501086.510450
1507.9940000060000400000
6
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
9.3410450
1501086.510450
1507.9940000060000400000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
01.567.4 −≥−
9.340.18 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
9.3410450150
104501507.99400000
6000040000093.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 9.12312 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
01.510450150
104501507.99400000
6000040000093.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 7.702 ≤
Beam PS6
fcm = 69.8 MPaH = 400.0 kN
200
300 NA
eB = 99.7 mm
207
15 HS 5 = 295 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.91C = 0.5 f’cp = 0.5 × 69.8 = 34.9 MPaCt = 0.6√f’c = 0.6 √69.8 = 5.01 MPa__________________________________________
σpu = 1608 MPaku = 0.246γ = 0.65d = 249.7 mm
Mu = 109.0 kNmMcr = 69.5 kNmInitial Camber ϕ = 12.4 mm
Icr = 89.7 × 106 mm4
Ief = 245.2 × 106 mm4
___________________________
22 HS 5 = 452 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.91C = 0.5 f’cp = 0.5 × 69.8 = 34.9 MPaCt = 0.6√f’c = 0.6 √69.8 = 5.01 MPa__________________________________________
σpu = 1608 MPaku = 0.447γ = 0.65d = 210.0 mm
Mu = 130.5 kNmMcr = 76.3 kNmInitial Camber ϕ = 11.4 mm
Icr = 85.7 × 106 mm4
Ief = 173.1 × 106 mm4
___________________________
Beam PS5
200
fcm = 69.8 MPaH = 612 kN
300160
NAeB = 60.0 mm
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-18
Beam PS7
200
fcm = 52.5 MPaH = 450 kN
300223
NA
eB = 97.5 mm
tTTB C
IyM
IyeH
AH
−≥+− 1
35.410450
1501086.510450
1505.9745000060000450000
6
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
3.2610450
1501086.510450
1505.9745000060000450000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
35.417.5 −≥−
3.2617.20 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
3.2610450150
104501505.97450000
6000045000093.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 8.982 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
35.410450150
104501505.97450000
6000045000093.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 8.742 ≤
tTTB C
IyM
IyeH
AH
−≥+− 1
35.410450
1501086.510450
1508040000060000400000
6
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
3.2610450
1501086.510450
1508040000060000400000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
35.405.2 −≥−
3.2638.15 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
3.2610450150
1045015080400000
6000040000093.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 1.902 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
35.410450150
1045015080400000
6000040000093.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 4.612 ≤
Beam PS8
fcm = 52.5 MPaH = 400.0 kNe = 80.0 mm
200
300 NA
eB = 80 mm
193
15 HS 5 = 295 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.93C = 0.5 f’cp = 0.5 × 52.5 = 26.3 MPaCt = 0.6√f’c = 0.6 √52.5 = 4.35 MPa__________________________________________
σpu = 1608 MPaku = 0.345γ = 0.67d = 236 mm
Mu = 96.5 kNmMcr = 61.4 kNmInitial Camber ϕ = 10.0 mm
Icr = 79.3 × 106 mm4
Ief = 109.9 × 106 mm4
___________________________
13 HS 5 = 295 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.93C = 0.5 f’cp = 0.5 × 52.5 = 26.3 MPaCt = 0.6√f’c = 0.6 √52.5 = 4.35 MPa__________________________________________
σpu = 1608 MPaku = 0.277γ = 0.67d = 247.5 mm
Mu = 92.1 kNmMcr = 74.8 kNmInitial Camber ϕ = 13.7 mm
Icr = 78.0 × 106 mm4
Ief = 174.4 × 106 mm4
___________________________
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-19
Beam PS9
200
fcm = 83.5 MPaH = 346 kN
300215
NA
eB = 90.0 mm
tTTB C
IyM
IyeH
AH
−≥+− 1
48.510450
1501086.510450
1509034600060000
3460006
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
8.4110450
1501086.510450
1509034600060000
3460006
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
48.502.3 −≥−
8.4156.14 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
8.4110450150
1045015090346000
6000034600093.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 7.13822 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
48.510450150
1045015090346000
6000034600093.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 5.612 ≤
tTTB C
IyM
IyeH
AH
−≥+− 1
48.510450
1501086.510450
1509048000060000480000
6
6
6 −≥×××
+×
××−
CIyM
IyeH
AH BBB ≤−+ 1
8.4110450
1501086.510450
1509048000060000480000
6
6
6 ≤×××
−×
××+
1. Ensure Strength at Tendon Transfer
For Top Fibre Stress, Satisfy Equation A1:
For Bottom Fibre Stress, Satisfy Equation A2:
48.545.4 −≥−
8.415.20 ≤
CI
yMI
yeHAH TTB ≤+⎥⎦
⎤⎢⎣⎡ − 2η
8.4110450150
1045015090480000
6000048000092.0 6
26 ≤
××
+⎥⎦⎤
⎢⎣⎡
×××
−M
2. Find Ultimate Moment Capacity of Beam
Satisfy Equation A3:
Satisfy Equation A4:kNmM 6.14302 ≤
tBBB C
IyM
IyeH
AH
≥−⎥⎦⎤
⎢⎣⎡ + 2η
48.510450150
1045015090480000
6000048000092.0 6
26 −≥
××
−⎥⎦⎤
⎢⎣⎡
×××
+M
kNmM 3.782 ≤
Beam PS10
fcm = 83.5 MPaH = 480 kN
200
300 NAeB = 90.0 mm
203
18 HS 5 = 353 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.93C = 0.5 f’cp = 0.5 × 83.5 = 41.8 MPaCt = 0.6√f’c = 0.6 √83.5 = 5.48 MPa__________________________________________
σpu = 1608 MPaku = 0.256γ = 0.65d = 240.0 mm
Mu = 124.9 kNmMcr = 78.9 kNmInitial Camber ϕ = 13.5 mm
Icr = 95.0 × 106 mm4
Ief = 126.2 × 106 mm4
___________________________
13 HS 5 = 295 mm2
Density of Concrete 2328 kg/m3
Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m
Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm
η = 0.93C = 0.5 f’cp = 0.5 × 83.5 = 41.8 MPaCt = 0.6√f’c = 0.6 √83.5 = 5.48 MPa__________________________________________
σpu = 1608 MPaku = 0.185γ = 0.65d = 240.0 mm
Mu = 92.5 kNmMcr = 61.5 kNmInitial Camber ϕ = 9.7 mm
Icr = 83.6 × 106 mm4
Ief = 144.0 × 106 mm4
___________________________
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX B: RC and PSC Beam Calculations B-20
B.4 Calculations for CS-Series Beams
dc = 32 mmdt = 214 mm
γ = 0.85 - 0.007(f’c - 28)
≈ 0.85
20
150
250
20
214
14
2Y12’s=226 mm2
22.5 MPa400 MPa
CS1 and CS2 and CS3
3N20 = 942 mm2
Beam is under-reinforced
pt = Ast/bd = 942/150×214 = 0.0293pc = Asc/bd = 226/150×214 = 0.00704pt - pc = 0.02212(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.01823(pt - pc)lim < pt - pc
∴ Asc will yield_________________________a = (Ast - Asc) fsy
0.85 f’c b = 99.0 mm
Mu = Astfsy(d-a/2) + Ascfsy (a/2 - dc) = 62.7 kNm
Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4
32.0 MPa500 MPa
2N24 = 905 mm2
CS4 and CS5 and CS6
dc = 32 mmdt = 213 mm
γ = 0.85 - 0.007(f’c - 28)
= 0.822
213
52
Under-Reinforced (Bending)pt = Ast/bd = 905/150×213 = 0.0283pB = 0.85 f’c γ kuB
fsy
= 0.03354pt < pB
∴ Under-Reinforced_________________________Mu = Astfsyd [1-(0.6×Ast/bd×
fsy/f’c) = 60.5 kNm
Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4
ku = pt fsy / 0.85 γ f’c
= 0.630
Shear Capacity
Vuc = β1β2β3 bwd0 Ast f’c
bwd0
= 47.1 kN
3
31.5 MPa500 MPa
2N24 = 905 mm2
2Y12’s
CS7 and CS8 and CS9
20
150
250
20
210
40/32
Beam is under-reinforced
pt = Ast/bd = 905/150×210 = 0.0287pc = Asc/bd = 226/150×210 = 0.00717pt - pc = 0.02159(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy
= 0.04044(pt - pc)lim > pt - pc
∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c
= 0.2283υ = 600 pc
0.85 γ f’c
= 0.1894
dc = 32 mmdt = 213 mm
γ = 0.85 - 0.007(f’c - 28)= 0.826
ku = η + √ η2 + υ (dc/d) = 0.5129a = γ ku d = 89.0 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 75.6 kNm
Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4
Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams
APPENDIX C: Beam Crack Pattern Photographs C-1
APPENDIX C
Beam Crack Pattern Photographs
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-1
APPENDIX D
LOGDEC COMPARATIVE GRAPHS
D.1 Analytical Decay Curve Method Implemented Using Matlab
______________________________________________________________________ home; clear; close; fname = 'beam I-1 00 F.txt'; datfle = load(fname); t = datfle(:,1); v = datfle(:,2); plot(t,v); axis([0 max(t) min(v) max(v)]); Stage 1
Replotting Excel file of original vibration decay waveform detected by oscilloscope.
axis([min(t) max(t) -max(v) max(v)]); title('CONCRETE BEAM DAMPING SIGNAL'); xlabel('Time (s)'); ylabel('Amplitude (mV)'); grid on; disp('PRESS A KEY TO CONTINUE'); pause; % Find when time > 0 sec pos = 1; tmp = 1; while (pos==1)&(tmp<length(t)) if (t(tmp)>0) pos = tmp; end tmp = tmp+1; end if (pos>1) tmp = length(t); Stage 2
Replot Stage 1 plot by removing negative portion (of time) from the waveform record.
t = t(pos:tmp); v = v(pos:tmp); end plot(t,v); axis([0 max(t) min(v) max(v)]); axis([min(t) max(t) -max(v) max(v)]); title('CONCRETE BEAM DAMPING SIGNAL'); xlabel('Time (s)'); ylabel('Amplitude (mV)'); grid on; disp('PRESS A KEY TO CONTINUE'); pause;
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-2
% Power Spectrum % Get the Frequency data, and calibrate it, and graph it P = abs(fft(v)); dt = t(2)-t(1); % Time between each sample fmax = 1/dt; % Max freq = 1 / ( time between each sample ) freq = t/max(t) * fmax; % Scale freq from 0 to fmax Stage 3
Replotting Excel file of original vibration decay waveform detected by oscilloscope.
bar(freq,P); axis([0 max(freq)/2 0 max(P)]); % Graph is mirrored so only wanthalf grid on; title('SPECTRUM OF COMMUNCATION SIGNAL'); xlabel('Frequency (Hz)'); ylabel('Amplitude (mV)'); pause; % Choose freq between 800 - 1200 Hz tmp1 = 1; while (freq(tmp1)<800) % 800 Hz tmp1 = tmp1 + 1; end; tmp2 = 1; while (freq(tmp2)<1200) % 1200 Hz tmp2 = tmp2 + 1; end; P = P(tmp1:tmp2); freq = freq(tmp1:tmp2); % Show the new Freq range of data bar(freq,P); axis([min(freq) max(freq) 0 max(P)]); % Graph is mirrored so only wanthalf grid on; title('SPECTRUM OF COMMUNCATION SIGNAL'); xlabel('Frequency (Hz)'); ylabel('Amplitude (mV)'); % find the frequency of the peak position = min(find(P==max(P))); % The Position of the first peak of the fft graph
Stage 4 Undertaking the FFT.
Stage 4 As above
fpeak = freq(position); % The frequency of the peak of the fft graph disp(['The frequency peaks at: ' num2str(fpeak) ' Hz']); text(fpeak,max(P)/1.1,['Peak = ' num2str(fpeak) 'Hz']); disp('PRESS A KEY TO CONTINUE'); pause; % Remove negative parts of amplitude - time graph v = abs(v); % Get the peaks of the wave & Take the natural log num = 1; for tmp = 2:size(t) if ((v(tmp-1)<v(tmp))&(v(tmp-1)<v(tmp))&(v(tmp)>0)) tn(num) = t(tmp); vn(num) = log(v(tmp)); num = num + 1; Stage 5
Plotting the natural logarithm of each of the peaks (with respect to amplitude).
end end % Plot Log Graph plot(tn,vn,'o'); %axis([0 0.05 0 max(vn)]); title('LOG GRAPH OF DAMPING'); xlabel('Time (s)'); ylabel('ln(Amplitude)'); hold on; % Fit Line p = polyfit(tn,vn,1); lnV = p(2)+p(1)*tn; plot(tn,lnV,'-r'); text(max(tn)/2,max(lnV)-0.1,['Slope = ' num2str(p(1))]); Stage 6
Calculating slope, and logdec.
% Display Answer disp([ 'Intercept = ' num2str(p(2)) ]); disp([ 'Slope = ' num2str(p(1)) ]); w = 2*pi*fpeak; disp([ 'w = ' num2str(w) ]); s = -p(1)/w; disp([ 'Decay - s = ' num2str(s) ]); disp([ 'Log Dec = ' num2str(s*2*pi) ]);
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-3
1 2 3 4 5 6 7 8 9
x 10- 3
-0. 1
-0. 05
0
0. 05
0. 1
CONCRETE BEAM DAMP ING S IGNAL
Time (s )
Am
plitu
de
(mV
)
500 1 000 1 500 2 0000
0.2
0.4
0.6
0.8
1
1.2
SP ECTRUM O F CO MMUNCATION S IGNAL
F re quency (Hz)
Am
plitu
de (m
V)
Pea k = 819. 64 52Hz
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0. 01-3.6
-3.4
-3.2
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8LOG GRAP H O F DAMP IN G
Time (s )
ln(A
mpl
itude
)
S lope = -98 .5 093
The frequency peaks at: 820 HzIntercept = -2.2Slope = -98.5w = 5150.0Decay - s = 0.019Log Dec = 0.120
0. 005 0.01 0. 015 0.02 0. 025 0.03 0. 035
-0.1
-0.05
0
0.05
0.1
CO NCRETE BEAM DAMPING S IG NAL
Time (s)
Am
plitu
de (m
V)
0 500 1 000 1 500 2 000 2 5000
0.5
1
1.5
2
2.5
S P ECTRUM OF CO MMUNCATION S IG NAL
F re quency (Hz)
Am
plitu
de (m
V)
0 0. 005 0.01 0. 015 0.02 0. 025 0.03 0. 035 0.04-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5LOG GRAP H O F DAMP IN G
Time (s)
ln(A
mpl
itude
)
S lope = -61 .6 302
The frequency peaks at: 780 HzIntercept = -2.4Slope = -61.6w = 4899.1Decay - s = 0.013Log Dec = 0.079
CS6-200-01CS6-50-01
a) e)
b) f)
c)
d) h)
g)
Figure D.1: Diagrammatic Flowchart of the DCM: a) and e) Original Signal for 50 and
200 NDP (see Section 5.4); b) and f) FFT of Time Spectrum; c) and g) Checking logeAn
versus n; d) and h) DCM Algorithm Extracting Logdec
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-4
D.2 Interrelationship between Logdec (TLT) and Cycle Number, n
a) Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18SB1SB2
b)
11
1
1 11
2
2
2
2 22
33
3
3 3 3
4
4
4
4 4 4
5
5
5 55 5
66
6
6
66
77
7
7 7
7
8
8
8
8 8 8
9
9
99
9 9
Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c)
A
A
A
AAA
B
B
BB
B B
C
C C
CC C
D
D
DD
D D
E
E
E
E E E
F
F
F
F F F
G
G
G G G G
H
H
H H H H
I
II
I II
J
J
JJ
JJ
K
K
K
KK K
L
L
L
LL L
Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
d)
c
c
c
cc
c
d
d
dd
dd
e
e
e
ee e
f
f
f
ff f
g
g
g
g
g g
hh
h
hh h
i i
i
ii
i
j
j
j
j
jj
Cycle Number (n)
Logd
ec(T
LT)
0 25 50 75 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS3PS4PS5PS6PS7PS8PS9PS10
cdefghij
Figure D.2: Logdec (TLT) versus Cycle Number (n) for: a) S-Series; b) CS-Series; c) B-
Series; d) PS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-5
D.3 Interrelationship between Logdec (DCM) and Number of Data Points
(NDP)
a)
A
AA
A
AA
B
B
B
BB B
C
CC
CC C
D
D D D
D D
E
EE
EE
E
FF F
F FFG
G
G
GG
G
H
HH
H HH
I
I II I I
J
J
JJ J J
K
K K
K K K
L
L L L L L
Number of Data Points (NDP)
Logd
ec(D
CM
)
0 100 200 3000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b)
1
11
11
1
2
22
2 2
2
3
3
33 3
3
4
44
44
4
5
5
5 5 55
66
66
66
7 77
77 7
8
8
8
8 88
9
9
9
99 9
Number of Data Points (NDP)Lo
gdec
(DC
M)
0 100 200 3000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24 CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c)
A
AA
A
AA
B
B
B
BB B
C
CC
CC C
D
D D D
D D
E
EE
EE
E
FF F
F FFG
G
G
GG
G
H
HH
H HH
I
I II I I
J
J
JJ J J
K
K K
K K K
L
L L L L L
Number of Data Points (NDP)
Logd
ec(D
CM
)
0 100 200 3000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
d)
c
c
c cc c
d
d d d d d
e
e ee
ee
f
ff
ff
f
g
g
g
g
gg
h
hh
hh h
i
i
i
i
i i
j
j jj j
j
Number of Data Points (NDP)
Logd
ec(D
CM
)
0 100 200 3000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS3PS4PS5PS6PS7PS8PS9PS10
cdefghij
Figure D.3: Logdec (DCM) versus Number of Data Points (NDP) for: a) S-Series; b)
CS-Series; c) B-Series; d) PS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-6
a) Number of Data Points (NDP)
Free
-Vib
ratio
nFr
eque
ncy
(Hz)
0 100 200 3000
200
400
600
800
1000
1200
1400
1600
1800
2000SB1SB2
b)
11 1 1 1 1
2 2 2 2 2 23 3 3
3 3 34 4 4 4 4 4
55 5 5 5 5
66 6 6 6 688
8 8 8 89 9 9 9 9 9
Number of Data Points (NDP)
Free
-Vib
ratio
nFr
eque
ncy
(Hz)
0 100 200 3000
250
500
750
1000
1250
1500
1750
2000CS1CS2CS3CS4CS5CS6CS8CS9
12345689
c)
A AA A
A A
B B B B B
BC
C C C C C
D
D D D D D
EE
E
E E E
F F F F F F
G
G G
G
G GH HH H H H
I II I I I
J J
J J JJ
K K K K KK
LL L L L L
Number of Data Points (NDP)
Free
-Vib
ratio
nFr
eque
ncy
(Hz)
0 100 200 3000
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
d)
c
c c c c c
d
dd
d d d
ee
e e e ef
f
f f f f
gg g
g g g
h
h hh h hi
i
i
i i
i
jj j
j jj
Number of Data Points (NDP)
Free
-Vib
ratio
nFr
eque
ncy
(Hz)
0 100 200 3000
250
500
750
1000
1250
1500
1750
2000
2250
2500PS3PS4PS5PS6PS7PS8PS9PS10
cdefghij
Figure D.4: Frequency (Hz) versus Number of Data Points (NDP) for: a) S-Series; b)
CS-Series; c) B-Series; d) PS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-7
D.4 Correlation between Logdec (TLT) and Logdec (DCM) Techniques
a) Logdec (DCM)
Logd
ec(T
LT)
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
SB1
b) Logdec (DCM)
Logd
ec(T
LT)
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
SB2
Figure D.5: Logdec (TLT) versus Logdec (DCM) for: a) SB1 and b) SB2
a)
1 11
111
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS11
b)
2
2
2
222
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS22
c)
33
3333
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS33
d)
4
4
4
444
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS44
Figure D.6: Logdec (TLT) versus Logdec (DCM) for: a) CS1, b) CS2, c) CS3, d) CS4,
e) CS5, f) CS6, g) CS7, h) CS8, and i) CS9 (Continued Overleaf)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-8
e)
5
5
5555
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS55
f)
66
6
66
6
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS66
g)
77
7
777
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS77
h)
8
8
8
888
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS88
i)
9
9
99
99
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
CS99
Figure D.6: Logdec (TLT) versus Logdec (DCM) for: a) CS1, b) CS2, c) CS3, d) CS4,
e) CS5, f) CS6, g) CS7, h) CS8, and i) CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-9
a)
AA
AA
AA
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BI-1A
b)
B
B
BBBB
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-2B
c)
C
C CC
CC
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BI-3C
d)
D
D
DDDD
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-4D
e)
E
E
EEEE
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-5E
f)
F
F
FFFF
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-6F
Figure D.7: Logdec (TLT) versus Logdec (DCM) for: a) BI-1, b) BII-2, c) BI-3, d) BII-
4, e) BII-5, f) BII-6, g) BI-7, h) BII-8, i) BI-9, j) BII-10, k) BII-11, and l) BII-12
(Continued Overleaf)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-10
g)
G
G
GG GG
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BI-7G
h)
H
H
HHH H
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-8H
i)
I
II
III
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BI-9I
j)
J
J
JJ
JJ
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-10J
k)
K
KK
KKK
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-11K
l)
L
L
LLLL
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
BII-12L
Figure D.7: Logdec (TLT) versus Logdec (DCM) for: a) BI-1, b) BII-2, c) BI-3, d) BII-
4, e) BII-5, f) BII-6, g) BI-7, h) BII-8, i) BI-9, j) BII-10, k) BII-11, and l) BII-12
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-11
a)
c
c
cccc
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS3c
b)
d
ddddd
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS4d
c)
e
e
ee
ee
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS5e
d)
ff
f
fff
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS6f
e)
g
g
gg
gg
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS7g
f)
hh
hhhh
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS8h
Figure D.8: Logdec (TLT) versus Logdec (DCM) for: a) PS3, b) PS4, c) PS5, d) PS6, e)
PS7, f) PS8, g) PS9, and h) PS10 (Continued Overleaf)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-12
g)
ii
iiii
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS9i
h)
j
j
jjj
j
Logdec (DCM)
Logd
ec(T
LT)
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
PS10j
Figure D.8: Logdec (TLT) versus Logdec (DCM) for: a) PS3, b) PS4, c) PS5, d) PS6, e)
PS7, f) PS8, g) PS9, and h) PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-13
D.5 ‘Optimal Peak Ratio’ Curves for All Beams
Peak Ratio An/A1 (%)
Logd
ec(T
LT)
01020304050600
0.025
0.05
0.075
0.1
0.125
0.15
0.175SB1SB2
DCM 300NDP
DCM 50NDP
DCM 100NDP
DCM 150NDPDCM 200NDP
DCM 250NDP
Figure D.9: Optimal Peak Ratio (An/A1 %) Curves for All S-Series Beams
A
A
A
AA
B
B
B
BB
C
C
C CC
D
D
DD
D
E
E
E
E E
F
F
F
FF
G
G
G G G
H
H
H HH
I
II I I
J
J
J J
J
K
K
KKK
L L
L
LL
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
17%> An/A1>10%
Figure D.10: Optimal Peak Ratio (An/A1 %) Curves for All B-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-14
a)
A
A
A
AA
B
B
B
BB
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
BI-1BII-2
AB
17%> An/A1>10%
BII-2TLT (150)
BI-1TLT (150)
b)
C
C
C CC
D
D
DD
D
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
BI-3BII-4
CD
17%> An/A1>10%
BI-3TLT (150)
BII-4TLT (200)
c)
E
E
E
E E
F
F
F
FF
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16BII-5BII-6
EF
17%> An/A1>10%
BII-5TLT (150)
BII-6TLT (150)
d)
G
G
G G G
H
H
H HH
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16BI-7BII-8
GH
17%> An/A1>10%BI-7
TLT (200)
BII-8TLT (200)
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-15
e)
I
II I I
J
JJ J
J
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
BI-9BII-10
IJ
17%> An/A1>10%
BII-10TLT (150)
BI-9TLT (150)
f)
K
K
KKK
L L
L
LL
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
BII-11BII-12
KL
17%> An/A1>10%BII-12
TLT (150)
BII-11TLT (150)
Figure D.11: Optimal Peak Ratio (An/A1 %) Curves for: a) BI-1 and BII-2, b) BI-3 and
BII-4, c) BII-5 and BII-6, d) BI-7 and BII-8, e) BI-9 and BII-10, f) BII-11 and BII-12
1 11
1 1
2
2
2
22
33
33
3
4
4
4
4 4
5
5
5
5 5
6
6
6
6 6
7
7
7
7
8
8
8
88
9
9
99
9 9
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
05101520253035404550550
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
17%> An/A1>8%
Figure D.12: Optimal Peak Ratio (An/A1 %) Curves for All CS-Series Beams
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-16
a)
1 11
1 1
2
2
2
22
33
33
3
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
05101520253035404550550
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26CS1CS2CS3
123
17%> An/A1>8%
CS2TLT (200)
CS1TLT (200)
CS3TLT (200)
b)
4
4
4
4 4
5
5
5
5 5
6
6
6
6 6
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
05101520253035404550550
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26CS4CS5CS6
456
17%> An/A1>8%
CS4TLT (200)
CS5TLT (200)
CS6TLT (200)
Figure D.13: Optimal Peak Ratio (An/A1 %) Curves for: a) CS1, CS2 and CS3, b) CS4,
CS5 and CS6, c) CS7, CS8 and CS9 (Continued Overleaf)
c)
7
7
7
7
8
8
8
88
9
9
99
9 9
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
05101520253035404550550
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26CS7CS8CS9
789
17%> An/A1>8%
CS8TLT (200)
CS9TLT (250)
CS7TLT (200)
Figure D.13: Optimal Peak Ratio (An/A1 %) Curves for: a) CS1, CS2 and CS3, b) CS4,
CS5 and CS6, c) CS7, CS8 and CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-17
a
a
a
a a
b
b
bb
b
c
c
c
cc
d
d
d
dd
e
e
e
ee
ff
f
f f
g g
g
g g
h
h
h
h
h
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS3PS4PS5PS6PS7PS8PS9PS10
abcdefgh
20%> An/A1>10%
Figure D.14: Optimal Peak Ratio (An/A1 %) Curves for All PS-Series Beams
a)
a
a
a
a a
b
b
bb
b
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS3PS4
ab
20%> An/A1>10%
PS3TLT (200)
PS4TLT (200)
b)
c
c
c
cc
d
d
d
dd
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS5PS6
cd
PS6TLT (200)
PS5TLT (200)
20%> An/A1>10%
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Appendix D: Logdec Comparative Graphs D-18
c)
e
e
e
ee
ff
f
f f
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS7PS8
ef
PS8TLT (200)
PS7TLT (200)
20%> An/A1>10%
d)
g g
g
g g
h
h
h
h
h
An/A1 (%)
'Unt
este
d'Lo
garit
hmic
Dec
rem
ent(
TLT)
051015202530354045500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16PS9PS10
gh
PS9TLT (200)
PS10TLT (200)
20%> An/A1>10%
Figure D.15: Optimal Peak Ratio (An/A1 %) Curves for: a) PS3 and PS4, b) PS5 and PS6,
c) PS7 and PS8, d) PS9 and PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX E: Damping Tabulations E-1
APPENDIX E
Damping Tabulations
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
Table E.1. Optimal Peak Ratio Damping Tabulations for Beam BI-1
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1024 0.0600 0.0953 1012 0.0843 0.0821 975 0.0776 0.0648 981 0.0554 0.0512 714 0.0727 0.044918.6 0.16 2.38 0.005 1120 0.0778 0.0828 975 0.0649 0.0661 28.3 0.27 3.42 0.02 1025 0.0987 0.1812 969 0.0752 0.0923 980 0.0592 0.0677 956 0.0532 0.0515 945 0.0453 0.052437.4 0.37 4.24 0.022 969 0.0702 0.0847 969 0.0605 0.0696 46.6 0.42 5.05 0.025 1045 0.0944 0.1537 968 0.0613 0.0827 946 0.0542 0.0729 961 0.0496 0.0575 949 0.0479 0.053665.3 0.57 6.6 0.03 1047 0.0996 0.1643 962 0.0559 0.0882 941 0.0542 0.0724 937 0.0469 0.0598 949 0.0451 0.051074.6 0.62 7.37 0.03 969 0.0597 0.0893 946 0.0523 0.0714 83.9 0.70 8.51 0.032 1068 0.0731 0.1237 975 0.0575 0.0807 950 0.0508 0.0712 942 0.0477 0.0590 933 0.0452 0.050893.2 1.26 9.61 0.034 961 0.0555 0.0956 940 0.0513 0.0740
102.5 1.93 10.81 0.041 1023 0.0669 0.1398 1097 0.0417 0.0931 915 0.0464 0.0790 905 0.0504 0.0562 904 0.0482 0.0485Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 113.0 kNm. Date beam tested: 17 May 2000 Age of beam at testing: 28 days
Table E.2. Optimal Peak Ratio Damping Tabulations for Beam BII-2
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1584 0.083 0.0916 1550 0.047 0.0702 1533 0.0339 0.0548 1483 0.0295 0.0466 1477 0.0247 0.042618.6 0.55 2.87 0.016 1432 0.0452 0.0952 1198 0.0519 0.0711 28.1 0.71 4.13 0.019 1038 0.1304 0.1156 969 0.0807 0.0992 945 0.0613 0.0739 931 0.0693 0.0599 935 0.0626 0.057437.2 0.84 5.43 0.025 976 0.0548 0.0946 951 0.0618 0.0685 47.1 0.94 6.44 0.026 1023 0.119 0.0965 962 0.0578 0.0917 941 0.071 0.0748 930 0.0667 0.0665 831 0.0642 0.064656.0 1.02 7.02 0.027 960 0.0543 0.0855 940 0.0584 0.068 65.3 1.07 7.73 0.028 1022 0.1072 0.0928 961 0.0698 0.0842 940 0.0731 0.0752 930 0.0645 0.0669 924 0.0602 0.062974.6 1.13 8.55 0.03 962 0.0637 0.0842 941 0.0695 0.0688 83.9 1.20 9.60 0.031 1023 0.0974 0.1256 961 0.0626 0.1057 941 0.079 0.089 930 0.0647 0.0786 830 0.0607 0.069993.2 1.28 10.95 0.034 961 0.0591 0.1107 941 0.0568 0.0735 98.0 2.70 13.0 0.041 1025 0.0684 0.1315 962 0.0509 0.0798 908 0.058 0.0699 906 0.0505 0.0576 904 0.0519 0.0514
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 149.6 kNm. Date beam tested: 17 May 2000 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations2
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.3. Optimal Peak Ratio Damping Tabulations for Beam BI-3
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -# 0 0 1038 0.0695 0.1563 922 0.054 0.0695 922 0.0631 0.0585 911 0.0471 0.059 894 0.0425 0.051632.6 -# 1.88 0.005 918 0.0449 0.0731 914 0.0502 0.0583 51.3 -# 3.45 0.014 1022 0.0828 0.1247 909 0.0345 0.0726 906 0.0441 0.0623 886 0.0494 0.0518 984 0.0517 0.04969.9 -# 5.43 0.02 1553 0.0845 0.1311 922 0.0541 0.0757 898 0.0565 0.0642 886 0.0604 0.0619 889 0.0546 0.048788.5 -# 7.6 0.026 1075 0.045 0.1312 912 0.0503 0.0807 875 0.0504 0.0627 881 0.0513 0.0527 865 0.0505 0.049197.9 -# 9.44 0.030 919 0.0523 0.0800 879 0.0516 0.0721
107.2 -# 11.91 0.038 1022 0.0724 0.1412 960 0.0578 0.0978 962 0.0454 0.0789 855 0.0491 0.0609 844 0.0481 0.0593116.5 -# 16.65 0.045 1097 0.0575 0.1281 1150 0.0553 0.0923 935 0.0555 0.0693 836 0.056 0.061 833 0.0509 0.0545
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 116.5 kNm. -# Strain gauge was faulty on this beam Date beam tested: 14 July 2000 Age of beam at testing: 30 days
Table E.4. Optimal Peak Ratio Damping Tabulations for Beam BII-4
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1178 0.0273 0.1544 912 0.0535 0.0740 908 0.0504 0.0574 906 0.0484 0.0500 905 0.0428 0.045632.6 - 4.87 0.006 908 0.0545 0.0679 891 0.0593 0.062 51.3 0.77 7.01 0.011 1126 0.0686 0.1407 911 0.0625 0.0714 874 0.0634 0.0736 881 0.0622 0.0635 864 0.0572 0.05769.9 0.92 9.35 0.013 995 0.057 0.0772 873 0.057 0.0677 83.9 0.95 11.21 0.018 1125 0.0594 0.1021 1163 0.0563 0.0845 874 0.0615 0.0745 855 0.0588 0.0599 844 0.0519 0.05391.3 2.51 16.8 0.03 938 0.0676 0.0919 872 0.0623 0.0822 848 0.0497 0.0778 837 0.0489 0.0610 827 0.0484 0.047498.4 6.3 22.13 0.046 1047 0.0361 0.1420 973 0.0433 0.0950 949 0.0447 0.0824 811 0.0517 0.0643 939 0.0441 0.059398.8 10.84 28.86 0.075 1241 0.0419 0.1171 901 0.0778 0.0923 807 0.0653 0.0664 805 0.0604 0.0749 784 0.0555 0.0583
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 98.8 kNm. Date beam tested: 14 July 2000 Age of beam at testing: 30 days
Appendix E: Dam
ping Tabulations3
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.5. Optimal Peak Ratio Damping Tabulations for Beam BII-5
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1149 0.0231 0.1285 1074 0.0499 0.1052 1419 0.0463 0.0573 1062 0.0533 0.0447 1050 0.0432 0.045146.6 0.20 4.33 0.005 1241 0.0313 0.1152 1387 0.0786 0.1018 1369 0.0665 0.0865 1360 0.0566 0.0577 1469 0.0401 0.055574.6 0.21 6.5 0.005 1330 0.0672 0.156 1264 0.0782 0.1075 1175 0.0761 0.0892 1223 0.0584 0.0581 1219 0.044 0.054993.2 0.22 8.05 0.01 1241 0.0591 0.1354 1254 0.0743 0.0886 1183 0.0789 0.0756 1118 0.0605 0.0565 1160 0.0478 0.0503
111.8 0.80 10.41 0.02 936 0.0743 0.1129 1069 0.0589 0.079 1057 0.063 0.0606 848 0.0655 0.0618 839 0.0541 0.0501121.1 0.85 15.70 0.037 1090 0.06 0.1301 867 0.1001 0.1007 844 0.0815 0.0789 997 0.0663 0.0611 991 0.0549 0.0572128.6 - 35.60 0.044 0.1315 0.0943
Failure Mode: Beam failed in compression (crushing of concrete) at a Bending Moment (BM) of 128.6 kNm. Date beam tested: 24 July 2000 Age of beam at testing: 28 days
Table E.6. Optimal Peak Ratio Damping Tabulations for Beam BII-6
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1229 0.0488 0.1188 1181 0.0604 0.0929 1154 0.0523 0.0534 1149 0.043 0.0419 1139 0.0402 0.039737.2 0.60 4.85 - 1130 0.0479 0.0809 1120 0.0438 0.0703 56.0 0.60 6.5 0.005 1545 0.0451 0.0877 1137 0.0461 0.0638 1125 0.0423 0.0618 1110 0.0448 0.0462 1101 0.045 0.045674.6 0.60 8.03 0.01 1175 0.059 0.0964 1137 0.053 0.0761 1125 0.0557 0.0658 1102 0.0519 0.0485 987 0.0531 0.046593.2 0.60 10.15 0.01 1159 0.0546 0.0944 1112 0.0595 0.0783 1086 0.0631 0.065 1089 0.0597 0.0541 1199 0.046 0.0536
102.5 1.00 14.36 0.01 1159 0.0674 0.1286 1112 0.0627 0.0861 1086 0.0573 0.0697 1089 0.0538 0.0567 1078 0.0481 0.0551120.4 11.0 36.24 0.02 1045 0.0825 0.1011 1073 0.0506 0.0788 1065 0.0505 0.0746 1061 0.0571 0.0564 909 0.055 0.0498
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete) at a Bending Moment (BM) of 120.4 kNm. Date beam tested: 24 July 2000 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations4
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.7. Optimal Peak Ratio Damping Tabulations for Beam BI-7
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1545 0.0277 0.1122 1119 0.0328 0.0490 1113 0.0356 0.0364 1100 0.0227 0.0361 1088 0.0298 0.034615.8 1.55 6.31 0.051 923 0.0509 0.0917 1076 0.0495 0.0505 1050 0.0446 0.0396 813 0.0398 0.055 704 0.0439 0.047220.5 1.74 8.3 0.053 1067 0.0472 0.0401 1044 0.0373 0.0324 25.2 1.89 9.05 0.055 1139 0.0842 0.1402 1012 0.0525 0.0603 1008 0.0457 0.0458 1009 0.0371 0.0541 1007 0.0367 0.045528.0 1.94 9.65 0.057 1035 0.0919 0.104 1017 0.0703 0.0537 1011 0.0527 0.0512 1009 0.0407 0.0465 987 0.039 0.043332.6 2.05 10.30 0.063 1024 0.0846 0.1152 1011 0.0584 0.0638 1008 0.0516 0.0598 991 0.0415 0.0503 985 0.0406 0.048635.4 4.40 11.48 0.065 937 0.0629 0.1153 1011 0.075 0.0579 974 0.0655 0.0501 925 0.046 0.0652 867 0.05 0.0543
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 39.0 kNm. Date beam tested: 15 May 2001 Age of beam at testing: 63 days
Table E.8. Optimal Peak Ratio Damping Tabulations for Beam BII-8
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1128 0.0475 0.1043 1170 0.0288 0.0593 1091 0.0337 0.0401 1119 0.0316 0.0388 1106 0.0311 0.035615.8 1.50 6.43 0.050 1068 0.0768 0.0765 1062 0.0796 0.0691 0.0589 18.8 1.60 7.36 0.053 1140 0.0686 0.1357 1069 0.0415 0.0637 1046 0.038 0.0625 1035 0.0366 0.0489 1028 0.0366 0.043221.4 1.72 8.26 0.056 1069 0.0426 0.0651 1024 0.0412 0.0536 0.0603 24.4 1.80 9.01 0.060 1140 0.045 0.1411 1069 0.0414 0.0632 912 0.0399 0.0533 1010 0.0373 0.0489 1008 0.0368 0.046328.0 1.86 9.73 0.069 1125 0.0547 0.1451 910 0.0436 0.0577 1008 0.0358 0.0554 1006 0.0382 0.0550 1005 0.037 0.045832.6 1.94 10.4 0.069 1126 0.0667 0.1391 1011 0.0441 0.0624 1008 0.0344 0.0567 1006 0.0343 0.0577 1005 0.0359 0.048937.3 2.05 10.9 0.070 1140 0.0738 0.1435 1019 0.0436 0.0569 1013 0.0414 0.0548 1010 0.0378 0.0635 994 0.0378 0.0498
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 48.6 kNm. Date beam tested: 15 May 2001 Age of beam at testing: 63 days
Appendix E: Dam
ping Tabulations5
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.9. Optimal Peak Ratio Damping Tabulations for Beam BI-9
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1294 0.0973 0.1195 1255 0.0354 0.0502 1111 0.0426 0.0444 1107 0.0389 0.0439 1095 0.0370 0.042128.0 0.91 6.13 0.040 1278 0.0972 0.141 1239 0.0639 0.0622 1041 0.0561 0.0571 1220 0.041 0.0522 1216 0.0373 0.048442.0 1.07 7.5 0.045 1279 0.0841 0.1323 1239 0.0607 0.0628 1209 0.0528 0.0480 1207 0.0493 0.0563 1196 0.0437 0.05 56.0 1.20 8.59 0.050 1370 0.0812 0.1188 983 0.0749 0.0931 1267 0.0461 0.0609 942 0.0634 0.0609 940 0.0608 0.052369.9 1.27 9.34 0.060 1715 0.0817 0.1607 1136 0.0848 0.0923 1135 0.0616 0.0697 1127 0.0544 0.068 1128 0.0485 0.055976.4 13.15 15.13 0.070 922 0.1397 0.1889 1011 0.1042 0.0876 873 0.0832 0.0869 679 0.0919 0.0809 677 0.0741 0.0757
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 77.0 kNm. Date beam tested: 29 May 2001 Age of beam at testing: 61 days
Table E.10. Optimal Peak Ratio Damping Tabulations for Beam BII-10
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1410 0.0879 0.1031 1469 0.0261 0.0536 1240 0.0337 0.0457 1245 0.0285 0.0442 1256 0.0281 0.034718.6 0.79 5.14 0.052 948 0.0733 0.0817 928 0.0629 0.0654 27.9 1.00 7.33 0.058 1199 0.0656 0.0742 1031 0.0618 0.0577 37.2 1.12 7.97 0.058 1330 0.0236 0.109 1483 0.0461 0.0773 1231 0.0454 0.0532 1458 0.0343 0.0454 1219 0.0367 0.041946.6 1.19 8.57 0.067 1487 0.0368 0.0731 1234 0.0435 0.0534 56.0 1.27 9.65 0.071 1328 0.0407 0.0906 1484 0.0424 0.0771 1232 0.0473 0.0591 1223 0.044 0.0474 1446 0.0315 0.041765.3 1.34 10.61 0.079 1585 0.0224 0.0865 1295 0.0467 0.0789 1261 0.0426 0.0617 1169 0.0418 0.0512 1156 0.0379 0.044174.6 1.42 11.35 0.088 1585 0.0254 0.1061 1025 0.0666 0.0808 1158 0.0487 0.0633 1509 0.0301 0.0511 1156 0.0407 0.046783.9 1.62 13.69 0.091 1532 0.0281 0.1194 1146 0.0606 0.0841 1197 0.0519 0.0628 1144 0.0429 0.0517 1486 0.0282 0.039993.2 2.35 17.55 0.108 1026 0.0668 0.1449 1022 0.0809 0.0865 853 0.0686 0.0761 956 0.051 0.0579 965 0.0424 0.049999.8 2.76 25.90 0.196 1024 0.0776 0.1368 840 0.0696 0.0853 819 0.0518 0.0786 939 0.0496 0.0725 830 0.0477 0.0597
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 106.9 kNm. Date beam tested: 29 May 2001 Age of beam at testing: 61 days
Appendix E: Dam
ping Tabulations6
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.11. Optimal Peak Ratio Damping Tabulations for Beam BII-11
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1021 0.0702 0.1182 1049 0.0481 0.0931 1025 0.0521 0.0518 1012 0.037 0.0489 999 0.0343 0.044223.3 0.74 4.83 0.027 990 0.0664 0.0943 993 0.0584 0.0794 37.2 1.16 6.19 0.03 885 0.1931 0.1165 943 0.0795 0.1232 928 0.063 0.0795 804 0.0709 0.0604 904 0.0505 0.050351.3 1.26 6.94 0.037 987 0.0748 0.1214 891 0.0758 0.081 65.3 1.37 7.93 0.047 1342 0.0865 0.1261 905 0.093 0.0993 811 0.0927 0.0798 793 0.075 0.0654 560 0.0842 0.057179.2 1.41 8.99 0.050 872 0.1371 0.1286 905 0.0974 0.1134 1164 0.0751 0.0808 1027 0.066 0.062 1022 0.0586 0.058193.2 1.51 9.73 0.055 1163 0.0947 0.1016 1142 0.0771 0.0851
107.2 2.41 13.29 0.085 952 0.1281 0.1427 792 0.1247 0.1037 761 0.0868 0.0816 997 0.0616 0.0603 744 0.0673 0.061115.4 12.38 23.94 0.126 836 0.0864 0.1198 987 0.0618 0.0934 970 0.062 0.0891 951 0.0571 0.0625 948 0.0518 0.0541
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 115.4 kNm. Date beam tested: 9 July 2001 Age of beam at testing: 90 days
Table E.12. Optimal Peak Ratio Damping Tabulations for Beam BII-12
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 1126 0.0578 0.0949 1037 0.0329 0.0954 1025 0.0338 0.0566 1018 0.0322 0.0434 1015 0.0292 0.037337.2 0.91 4.18 0.018 1103 0.0431 0.0904 1002 0.042 0.0686 56.0 1.11 5.29 0.024 1257 0.0551 0.1336 1011 0.0537 0.0746 999 0.0374 0.0628 687 0.0468 0.051 1044 0.0326 0.046569.9 1.20 6.13 0.025 1008 0.0455 0.0824 1265 0.0302 0.0635 83.9 1.26 6.44 0.028 1237 0.0238 0.1376 1004 0.0448 0.0722 1061 0.0404 0.0646 1035 0.0384 0.0434 1148 0.0297 0.04197.9 1.31 7.00 0.031 1090 0.0449 0.1111 940 0.0455 0.0879 1027 0.0341 0.0747 1031 0.0366 0.0538 1018 0.0351 0.0489
111.8 1.35 7.52 0.032 1107 0.096 0.1101 935 0.0473 0.0775 901 0.0454 0.0759 901 0.038 0.0557 887 0.0355 0.0491125.8 1.38 6.94 0.035 971 0.0613 0.1467 935 0.0554 0.0703 901 0.0419 0.0732 901 0.0386 0.0561 767 0.0383 0.0494139.8 1.64 7.46 0.038 970 0.0378 0.1689 904 0.0414 0.0881 869 0.0362 0.0773 901 0.0339 0.0581 767 0.0361 0.0565177.7 - 12.35 0.175 937 0.038 0.1786 841 0.0409 0.0828 743 0.0661 0.0804 817 0.0584 0.0676 814 0.0525 0.064
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 177.7 kNm. Date beam tested: 9 July 2001 Age of beam at testing: 90 days
Appendix E: Dam
ping Tabulations7
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.13. Optimal Peak Ratio Damping Tabulations for Beam CS1
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 819 0.0765 0.0979 759 0.1136 0.0931 740 0.1068 0.0815 730 0.0846 0.0700 724 0.0734 0.068075.9 -% 4.87 0.038 720 0.2233 0.1985 660 0.1554 0.1896 640 0.1091 0.1283 630 0.0876 0.0915 644 0.0828 0.0910
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 75.9 kNm. % No strain gauges attached. Date beam tested: 18 September 2000 Age of beam at testing: 35 days
Table E.14. Optimal Peak Ratio Damping Tabulations for Beam CS2
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 822 0.1548 0.2122 811 0.0829 0.1234 774 0.0890 0.0848 780 0.0651 0.0693 784 0.0678 0.061172.2 -% 5.23 0.042 1329 0.0960 0.1590 608 0.1317 0.1282 605 0.1241 0.1226 604 0.1169 0.0820 583 0.0909 0.0718
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 72.2 kNm. % No strain gauges attached. Date beam tested: 18 September 2000 Age of beam at testing: 35 days
Table E.15. Optimal Peak Ratio Damping Tabulations for Beam CS3
BM(kNm)
εres (mV) ∆res
(mm) Wres
(mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 823 0.1115 0.0981 811 0.0551 0.0727 808 0.0360 0.0590 755 0.0248 0.0478 744 0.0204 0.036759.0 -% 3.89 0.028 633 0.2858 0.1212 616 0.1131 0.1168 588 0.1192 0.0976 583 0.0982 0.0883 587 0.0842 0.0753
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 59.0 kNm. % No strain gauges attached. Date beam tested: 18 September 2000 Age of beam at testing: 35 days
Appendix E: Dam
ping Tabulations8
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.16. Optimal Peak Ratio Damping Tabulations for Beam CS4
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 819 0.1202 0.1716 810 0.0864 0.1160 806 0.0720 0.0875 780 0.0790 0.0495 784 0.0724 0.047325.8 -% 0.63 0 818 0.1563 0.1506 817 0.1292 0.1343 812 0.0863 0.1018 779 0.0698 0.0691 783 0.0668 0.065936.1 -% 0.86 0 820 0.0994 0.1828 817 0.1565 0.1449 778 0.1095 0.1165 780 0.0593 0.0769 744 0.0515 0.065143.8 -% 1.18 0.01 866 0.1002 0.2267 780 0.1805 0.1702 781 0.1307 0.1263 791 0.0704 0.0880 753 0.0506 0.074352.0 -% 1.89 - 922 0.2093 0.1675 870 0.1722 0.2279 846 0.1352 0.1537 830 0.0968 0.1007 -& -& -&
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 52.0 kNm. % No shear reinforcement (NSR) in beam. Because beam was designed to fail in shear, therefore there were no bending cracks or strain gauges attached. & Logdec unable to be calculated because of increased decay due to damage. Date beam tested: 1 October 2001 Age of beam at testing: 28 days
Table E.17. Optimal Peak Ratio Damping Tabulations for Beam CS5
BM(kNm)
εres (mV) ∆res
(mm) Wres
(mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 923 0.1253 0.1891 861 0.0582 0.1009 841 0.0349 0.0701 830 0.0491 0.0501 824 0.0457 0.049724.2 -% 0.87 0 943 0.1083 0.1920 815 0.0837 0.1308 810 0.0567 0.0813 811 0.0478 0.0639 808 0.0475 0.069735.5 -% 1.03 0 943 0.1199 0.1851 815 0.0940 0.1459 810 0.0544 0.0878 810 0.0460 0.0702 808 0.0485 0.0733
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 37.4 kNm. % No shear reinforcement (NSR) in beam Because beam was designed to fail in shear, therefore there were no bending cracks or strain gauges attached. Date beam tested: 1 October 2001 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations9
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.18. Optimal Peak Ratio Damping Tabulations for Beam CS6
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
-% No 0 934 0.1034 0.1630 866 0.1109 0.1139 844 0.0726 0.0863 833 0.0653 0.0502 804 0.0653 0.0468Pre-Test -% With 0 930 0.0976 0.1528 867 0.1096 0.1398 845 0.0780 0.0977 832 0.0690 0.0519 821 0.0628 0.051017.0 -% 0.52 0 968 0.1254 0.1716 822 0.1358 0.1654 815 0.1169 0.1142 817 0.0603 0.0735 813 0.0492 0.068625.6 -% 0.76 0 843 0.1125 0.1675 815 0.1499 0.1627 802 0.1209 0.1225 785 0.0695 0.0811 809 0.0515 0.067329.5 -% 1.28 0.01 1757 0.1231 0.1606 1408 0.0925 0.1619 677 0.1448 0.1303 661 0.1048 0.0875 669 0.0808 0.0665
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 33.7 kNm. # No shear reinforcement (NSR) in beam. * Residual measurements not taken due to catastrophic shear failure. % Because beam was designed to fail in shear there were no bending cracks or strain gauges attached. Date beam tested: 1 October 2001 Age of beam at testing: 28 days
Table E.19. Optimal Peak Ratio Damping Tabulations for Beam CS7
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
0 No 0 1581 0.1147 0.1715 1557 0.0880 0.1103 1722 0.0455 0.0930 1541 0.0491 0.0620 -& -& -&Pre-T
0 With 0 1745 0.0681 0.2533 1722 0.0649 0.1518 1716 0.0466 0.0940 1696 0.0320 0.0631 -& -& -&
30.7 0.12 0.23 0.005 923 0.1860 0.2472 819 0.1665 0.2189 807 0.1148 0.1548 805 0.0789 0.0839 804 0.0628 0.077850.5 0.24 0.58 0.023 892 0.2435 0.2161 834 0.1459 0.2219 817 0.1247 0.1597 810 0.0789 0.0782 808 0.0644 0.087162.2 0.35 1.06 0.03 1635 0.0985 0.1503 1222 0.0984 0.1477 1198 0.0848 0.1007 1533 0.0404 0.0786 1546 0.0330 0.070188.0 0.47 3.33 0.04 667 0.2021 0.1784 1124 0.1563 0.1798 1110 0.1127 0.1150 604 0.1002 0.0886 -& -& 0.0730
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 88.0 kNm. & Logdec unable to be calculated because of increased decay due to damage (See ……..). Date beam tested: 8 October 2001 Age of beam at testing: 35 days
Appendix E: Dam
ping Tabulations10
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table D.20. Optimal Peak Ratio Damping Tabulations for Beam CS8
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
0 No 0 819 0.1638 0.1490 759 0.0902 0.1480 806 0.0715 0.0889 805 0.0387 0.0550 804 0.0419 0.0564Pre-T 0 With 0 818 0.2127 0.1964 763 0.0856 0.1643 809 0.0663 0.0884 804 0.0378 0.0591 804 0.0405 0.0571
23.9 0.39 0.56 0.005 819 0.1468 0.2063 809 0.1091 0.1367 781 0.0742 0.1068 780 0.0790 0.0814 784 0.0555 0.072639.2 0.50 0.95 0.015 819 0.1477 0.1807 810 0.1398 0.1696 773 0.0941 0.1131 780 0.0700 0.0854 764 0.0603 0.075951.4 0.60 1.60 0.02 1143 0.1124 0.1502 715 0.1350 0.1265 685 0.1317 0.1255 685 0.1093 0.0998 678 0.1089 0.099878.1 0.90 3.87 0.03 641 0.1937 0.2110 564 0.1340 0.1958 543 0.1049 0.1567 535 0.0979 0.1251 538 0.0908 0.1144
Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete) at a Bending Moment (BM) of 78.1 kNm. Date beam tested: 8 October 2001 Age of beam at testing: 28 days
Table E.21. Optimal Peak Ratio Damping Tabulations for Beam CS9
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T 0 0 0 842 0.2242 0.1982 814 0.0980 0.1108 809 0.0707 0.0865 785 0.0519 0.0789 788 0.0424 0.0627 22.2 - 0.74 0 842 0.2111 0.2400 821 0.0847 0.1421 780 0.0678 0.1337 760 0.0577 0.0782 769 0.0620 0.0824 34.5 0.56 1.16 0.015 843 0.2068 0.2117 765 0.0867 0.1502 743 0.0808 0.1178 761 0.0627 0.0729 749 0.0594 0.0886 48.1 0.70 1.83 0.027 720 0.1713 0.2185 665 0.1317 0.1841 676 0.0898 0.1348 655 0.0893 0.0948 644 0.0864 0.0921 61.9 0.71 2.41 0.04 720 0.2372 0.2714 659 0.1699 0.2415 606 0.1691 0.1692 605 0.1207 0.1147 604 0.0902 0.0861 73.9 0.85 10.38 0.07 617 0.3201 0.3637 507 0.1562 0.2108 505 0.1498 0.1716 479 0.1137 0.1121 483 0.0951 0.0954
Failure Mode: Beam failed in tension and shear (yielding of steel with minor shear cracking) at a Bending Moment (BM) of 73.9 kNm. Date beam tested: 8 October 2001 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations11
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.22. Optimal Peak Ratio Damping Tabulations for Beam PS3
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 1276 0.0496 0.0983 1120 0.0305 0.0612 1114 0.0404 0.0519 1085 0.0385 0.0420 1088 0.0325 0.039882.9 -% 8.87 0.042 1127 0.0963 0.1049 1063 0.0606 0.0815 1075 0.0500 0.0641 1056 0.0455 0.0587 1052 0.0429 0.0518
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 82.9 kNm. Date beam tested: 19 September 2000 Age of beam at testing: 28 days % No strain gauges attached. 1st Crack = 56.1 kNm = 62.9% Table E.23. Optimal Peak Ratio Damping Tabulations for Beam PS4
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 1276 0.0536 0.0877 1121 0.0390 0.0761 1214 0.0365 0.0540 1085 0.0371 0.0478 1088 0.0348 0.042638.8 -% 0.92 0 1124 0.0441 0.0964 1112 0.0177 0.0571 1074 0.0348 0.0522 1081 0.0390 0.0500 1085 0.0399 0.0413
129.4 -% 6 0.031 1500 0.0838 0.1453 1450 0.0657 0.1139 885 0.0812 0.0842 814 0.0732 0.0663 724 0.0730 0.0606Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 129.4 kNm. Date beam tested: 19 September 2000 Age of beam at testing: 28 days 1st Crack = 78.3 kNm = 60.5% Table E.24. Optimal Peak Ratio Damping Tabulations for Beam PS5
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 1375 0.0574 0.0935 1304 0.0367 0.0667 1046 0.0374 0.0518 1035 0.0331 0.0430 1028 0.0270 0.0359135.0 -% 11.1 0.049 1025 0.1228 0.1329 962 0.0678 0.1222 941 0.0660 0.0841 931 0.0549 0.0627 925 0.0517 0.0571
Failure Mode: Beam failed in compression (under load support) at a Bending Moment (BM) of 135.0 kNm. Date beam tested: 21 September 2000 Age of beam at testing: 35 days 1st Crack = 83.9 kNm = 62.2%
Appendix E: Dam
ping Tabulations12
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.25. Optimal Peak Ratio Damping Tabulations for Beam PS6
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 1125 0.0612 0.0764 1297 0.0348 0.0635 1041 0.0392 0.0467 1030 0.0291 0.0321 1024 0.0215 0.028247.7 -% 0.76 0 1616 0.0789 0.1371 1775 0.0310 0.0710 1046 0.0533 0.0599 1035 0.0485 0.0527 1028 0.0385 0.044979.2 -% 2.54 0.01 1849 0.0839 0.1017 1774 0.0508 0.0872 1749 0.0419 0.0845 1737 0.0352 0.0505 1727 0.0281 0.044895.1 -% 12.9 0.048 1039 0.0970 0.1558 969 0.0802 0.0935 979 0.0624 0.0839 960 0.0585 0.0600 947 0.0416 0.0527
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 95.1 kNm. Date beam tested: 21 September 2000 Age of beam at testing: 35 days 1st Crack =71.6 kNm = 75.3%
Table E.26. Optimal Peak Ratio Damping Tabulations for Beam PS7
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 806 0.0886 0.1326 874 0.0719 0.0982 841 0.0801 0.0743 739 0.0663 0.0565 746 0.0570 0.051637.2 -% 1.0 0 820 0.0664 0.1518 713 0.1014 0.0993 675 0.0979 0.0835 772 0.0627 0.0609 771 0.0501 0.055246.6 -% 1.36 0 733 0.0864 0.1737 822 0.0764 0.0971 675 0.0858 0.0864 658 0.0709 0.0660 580 0.0703 0.061256.0 -% 1.78 0 734 0.0938 0.1721 715 0.1039 0.2926 676 0.0841 0.0913 658 0.0644 0.0769 493 0.0757 0.062465.3 -% 2.74 0.01 820 0.1228 0.1165 772 0.0970 0.1131 672 0.0785 0.0924 754 0.0530 0.0689 744 0.0496 0.059073.1 -% 3.69 0.013 734 0.2126 0.1518 665 0.1182 0.1036 676 0.0782 0.0962 658 0.0655 0.0687 647 0.0602 0.0602
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 73.1 kNm. Date beam tested: 17 August 2001 Age of beam at testing: 43 days 1st Crack = 46.6 kNm = 63.7%
Appendix E: Dam
ping Tabulations13
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.27. Optimal Peak Ratio Damping Tabulations for Beam PS8
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 0 897 0.1087 0.0829 1167 0.0429 0.0791 1161 0.0454 0.0536 1226 0.0325 0.0412 1216 0.0279 0.039027.9 -% 0.63 0 1152 0.0648 0.0960 1214 0.0437 0.1005 1200 0.0362 0.0569 956 0.0420 0.0421 1131 0.0349 0.039337.2 -% 1.5 0 1023 0.0706 0.1190 1327 0.0493 0.0925 1142 0.0460 0.0646 1156 0.0376 0.0480 1158 0.0362 0.041055.9 -% 2.65 0.01 1022 0.0857 0.1205 779 0.1175 0.1057 772 0.0719 0.0728 704 0.0632 0.0585 864 0.0504 0.050765.3 -% 2.86 0.013 1357 0.0935 0.1491 1015 0.0742 0.0699 836 0.0553 0.0624 712 0.0647 0.0517 710 0.0519 0.042583.9 -% 7.72 0.059 1057 0.1019 0.1395 746 0.0811 0.1024 781 0.0698 0.0740 947 0.0498 0.0610 944 0.0446 0.051085.8 -% 18.10 0.120 717 0.0796 0.1396 659 0.0945 0.1053 740 0.0757 0.0778 704 0.0571 0.0674 697 0.0584 0.0568
Failure Mode: Beam failed in compression (under load support) at a Bending Moment (BM) of 85.8 kNm. Date beam tested: 17 August 2001 (Tension crack in left front face of beam prior to testing). Age of beam at testing: 47 days 1st Crack = 37.3 kNm = 43.5%
Table E.28. Optimal Peak Ratio Damping Tabulations for Beam PS9
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 No 1391 0.0807 0.0989 1289 0.0609 0.0828 1452 0.0306 0.0605 1260 0.0291 0.0414 - - -Pre-T -% 0 With 1200 0.1071 0.0964 910 0.0802 0.0947 1359 0.0358 0.0612 1187 0.0448 0.0424 1146 0.0360 0.042118.6 -% 0.52 0 1105 0.0931 0.0938 928 0.0648 0.0904 1020 0.0439 0.0668 909 0.0512 0.0523 914 0.0451 0.050537.2 -% 0.82 0 1023 0.0681 0.1354 898 0.0640 0.0943 899 0.0508 0.0693 874 0.0543 0.0626 874 0.0493 0.056055.4 -% 1.52 0 1022 0.0381 0.1103 898 0.0540 0.0901 899 0.0456 0.0720 874 0.0455 0.0612 874 0.0466 0.056769.9 -% 2.52 0.012 921 0.0750 0.1285 901 0.0341 0.0946 893 0.0392 0.0718 780 0.0424 0.0634 874 0.0353 0.051779.2 -% 5.65 0.028 933 0.0594 0.1394 901 0.0467 0.0884 867 0.0397 0.0697 871 0.0363 0.0641 856 0.0325 0.048285.0 -% 54.35 0.048 804 0.1188 0.1809 790 0.0921 0.1298 760 0.0744 0.0997 751 0.0572 0.0824 826 0.0464 0.0687
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 85.0 kNm. Date beam tested: 10 October 2001 Age of beam at testing: 60 days 1st Crack = 46.6 kNm = 54.8%
Appendix E: Dam
ping Tabulations14
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.29. Optimal Peak Ratio Damping Tabulations for Beam PS10
BM(kNm)
εres (mV)
∆res (mm)
Wres (mm)
Freq. DCM (Hz) 50#
Meas. Logdec (DCM)
50#
Calc. Logdec (TLT)
50#
Freq. DCM (Hz) 100#
Meas. Logdec (DCM)
100#
Calc. Logdec (TLT) 100#
Freq. DCM (Hz) 150*
Meas. Logdec (DCM)
150*
Calc. Logdec (TLT) 150*
Freq. DCM (Hz) 200*
Meas. Logdec (DCM)
200*
Calc. Logdec (TLT) 200*
Freq. DCM (Hz) 250*
Meas. Logdec (DCM)
250*
Calc. Logdec (TLT) 250*
Pre-T -% 0 No 914 0.0803 0.1250 983 0.0452 0.0873 984 0.0381 0.0623 844 0.0357 0.0471 971 0.0251 0.0370Pre-T -% 0 With 825 0.0827 0.1177 939 0.0515 0.0934 901 0.0480 0.0685 844 0.0399 0.0547 829 0.0426 0.046037.3 -% 1.35 0 908 0.1281 0.1857 825 0.0815 0.1132 754 0.0528 0.0608 833 0.0484 0.0564 831 0.0407 0.049555.9 -% 2.01 0 823 0.1306 0.1694 760 0.0950 0.0982 740 0.0469 0.0711 1082 0.0412 0.0662 744 0.0465 0.044665.3 -% 2.33 0 890 0.1424 0.1667 939 0.0654 0.1038 909 0.0404 0.0689 1013 0.0370 0.0605 750 0.0503 0.049174.6 -% 2.74 0 907 0.1250 0.1396 894 0.0726 0.1112 867 0.0448 0.0773 839 0.0525 0.0625 831 0.0508 0.057583.9 -% 3.21 0 907 0.1285 0.1287 900 0.0833 0.1003 829 0.0431 0.0723 908 0.0436 0.0653 826 0.0480 0.0537
102.5 -% 4.39 0.012 919 0.1192 0.1582 778 0.1130 0.1058 794 0.0597 0.0809 748 0.0582 0.0667 914 0.0458 0.0600121.1 -% 8.61 0.016 897 0.1252 0.1576 762 0.0966 0.1175 833 0.0652 0.0843 817 0.0569 0.0765 809 0.0562 0.0691125.8 -% 20.01 0.066 820 0.1250 0.1427 667 0.1108 0.0878 706 0.0672 0.0855 612 0.0681 0.0784 597 0.0682 0.0596
Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 125.8 kNm. Date beam tested: 9 October 2000 (Tension crack in centre upper top of beam prior to testing). Age of beam at testing: 59 days 1st Crack = 55.9 kNm = 44.4%
Appendix E: Dam
ping Tabulations15
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.30. Optimal Peak Ratio Damping Tabulations for Beam F1 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 37 0.1435 18 0.0820 15 0.0634 10 0.0571 8 0.0552 5 0.0604Standard Deviation 4.1 0.016 1.3 0.004 1.2 0.003 1.5 0.003 0.5 0.002 1.1 0.006
3 0 0 34 0.1557 19 0.0852 15 0.0738 13 0.0588 12 0.0510 8 0.0515Standard Deviation 8.8 0.049 5.2 0.014 2.9 0.012 3.3 0.004 1.7 0.007 0.5 0.002
Pre-T
OVERALL AVERAGE LOGDEC 36 0.1496 18 0.0836 15 0.0686 12 0.0581 10 0.0531 7 0.0549
2.7 0.39 0 17 0.0601 12 0.06403.8 & 0.46 0 14 0.0664 10 0.06975.7 0.53 0 13 0.0652 8 0.06857.9 0.55 0 15 0.0615 10 0.062911.8 0.72 0 12 0.0816 8 0.084114.8 0.88 0 15 0.0716 10 0.077922.5 1.01 0 14 0.0671 6 0.073625.0
-
1.10 0
-
15 0.0661 7 0.0700
-
% The load levels are described in Figure E.1. * The determination of the ‘Optimal Peak Ratio’ A1/An is described in Section 5.4.2. # The average logdec is calculated at various NDP (see Section 5.4). & This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 28.8 kNm. Date beam tested: 10 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations16
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.31. Optimal Peak Ratio Damping Tabulations for Beam F2 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 35 0.1695 20 0.1119 17 0.0634 14 0.0557 11 0.0519 9 0.0451Standard Deviation 4.2 0.030 3.3 0.014 2.6 0.002 1.0 0.002 2.6 0.004 0.8 0.001
2 0 0 30 0.1533 23 0.0981 14 0.0768 14 0.0531 9 0.0556 9 0.0557Standard Deviation 2.0 0.043 2.5 0.015 2.5 0.008 1.2 0.005 2.1 0.010 - -
3 0 0 28 0.1780 23 0.0907 16 0.0729 13 0.0648 10 0.0564 9 0.0515Standard Deviation 1.5 0.019 4.5 0.010 2.2 0.003 4.2 0.008 1.7 0.004 2.6 0.006
Pre-T
OVERALL AVERAGE LOGDEG 31 0.1632 22 0.1004 16 0.0705 14 0.0583 10 0.0545 9 0.0491
1.9 0.15 - 16 0.0654 8 0.06544.1& 0.28 - 14 0.0668 9 0.06196.3 0.39 - 12 0.0706 9 0.06188.2 0.55 - 12 0.0765 9 0.063111.2 0.81 - 14 0.0706 10 0.061613.3 0.95 - 15 0.0764 10 0.071215.3 1.08 - 14 0.0721 10 0.068718.3 1.21 - 14 0.0777 9 0.073221.4 1.36 - 14 0.0721 9 0.073724.8 1.52 - 18 0.0692 8 0.072528.0 1.69 - 14 0.0805 9 0.074931.9 1.69 - 14 0.0745 10 0.074534.7 2.03 - 12 0.0706 10 0.062237.5 3.41 - 12 0.0859 9 0.080142.4
-
20.98 -
-
10 0.0770 8 0.0707
-
& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 42.4 kNm. Date beam tested: 10 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations17
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.32. Optimal Peak Ratio Damping Tabulations for Beam F3 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 33 0.1142 24 0.0676 22 0.0494 18 0.0446 14 0.0463 12 0.0377Standard Deviation 5.0 0.0 1.7 0.006 1.5 0.003 2.0 0.002 5.7 - - 0.000
2 0 0 38 0.1281 25 0.1045 16 0.0837 15 0.0586 12 0.0539 12 0.0455Standard Deviation 12.3 0.034 4.0 0.017 1.5 0.003 1.0 0.005 0.0 0.001 1.5 0.001
3 0 0 27 0.1450 16 0.0942 15 0.0719 14 0.0558 12 0.0513 10 0.0499Standard Deviation 2.5 0.021 2.0 0.005 4.0 0.010 4.9 0.009 0.7 0.001 2.1 0.005
Pre-T
OVERALL AVERAGE LOGDEG 33 0.1291 22 0.0887 18 0.0683 16 0.0530 11 0.0505 12 0.0444
2.9 0.31 - 10 0.0734 12 0.05644.9 & 0.42 - 13 0.0716 9 0.066310.2 0.78 - 18 0.0648 14 0.065619.4 1.16 - 16 0.0806 11 0.079324.0 1.28 - 10 0.0961 10 0.085626.5 1.29 - 15 0.0694 10 0.072633.1 1.41 - - - 8 0.075340.4 1.73 - - - 8 0.075843.0 1.87 - 10 0.0835 9 0.073249.1 2.03 - 10 0.0794 9 0.072655.0 2.32 - - - 12 0.063757.0 4.49 - - - 11 0.066460.4
-
14.12 -
-
- - 12 0.0650
-
& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 8.7 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 42.4 kNm. Date beam tested: 11 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations18
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.33. Optimal Peak Ratio Damping Tabulations for Beam F4 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 34 0.0936 28 0.0735 17 0.0522 15 0.0447 12 0.0437 10 0.0411Standard Deviation 5.0 0.0 7.2 0.042 2.8 0.005 2.4 0.004 1.5 0.001 0.6 0.001
2 0 0 33 0.1188 25 0.0666 17 0.0620 14 0.0551 11 0.0495 11 0.0448Standard Deviation 12.5 0.028 4.4 0.003 2.5 0.003 - - 2.1 0.000 1.5 0.004
3 0 0 33 0.1545 30 0.0639 21 0.0592 17 0.0508 12 0.0491 11 0.0446Standard Deviation 4.7 0.023 8.7 0.018 4.2 0.007 4.1 0.006 2.1 0.003 1.7 0.001
Pre-T
OVERALL AVERAGE LOGDEG 33 0.1223 28 0.0680 18 0.0578 15 0.0502 11 0.0474 11 0.0435
5.6 0.11 - 14 0.0540 10 0.04887.7 0.31 - 11 0.0572 10 0.0487
9.3 & 0.42 - 9 0.0712 7 0.064911.8 0.55 - 11 0.0634 7 0.069914.5 0.71 - 11 0.0587 7 0.063017.7 0.86 - 10 0.0676 7 0.063721.2 0.96 - 11 0.0745 9 0.073724.6 1.11 - 10 0.0872 7 0.076227.3 1.16 - 11 0.0883 11 0.072630.8 1.29 - 12 0.0843 9 0.070434.6 1.38 - 12 0.0768 9 0.070337.9
-
1.60 -
-
15 0.0755 13 0.0678
-
& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 49.4 kNm. Date beam tested: 11 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations19
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.34. Optimal Peak Ratio Damping Tabulations for Beam F5 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 37 0.1100 18 0.0749 12 0.0624 12 0.0512 9 0.0495 7 0.0478Standard Deviation 9.4 0.028 4.9 0.015 3.6 0.009 1.5 0.001 0.6 0.002 1.1 0.004
2 0 0 34 0.1373 18 0.0974 15 0.0799 12 0.0654 11 0.0539 9 0.0533Standard Deviation 4.4 0.017 5.4 0.007 3.7 0.014 - 0.015 1.0 0.004 1.9 0.006
3 0 0 38 0.0921 19 0.0678 13 0.0609 11 0.0574 9 0.0549 8 0.0466Standard Deviation 2.9 0.009 3.2 0.006 2.1 0.006 0.6 0.002 1.9 0.007 0.8 0.004
Pre-T
OVERALL AVERAGE LOGDEG 36 0.1131 18 0.0800 13 0.0677 12 0.0580 10 0.0528 8 0.0492
2.6 - - 11 0.0689 8 0.06993.8 0.13 - 11 0.0698 10 0.06745.7 0.35 - 11 0.0656 6 0.069613.3 0.58 - 10 0.0695 8 0.0637
15.3 & 0.68 - 10 0.0681 7 0.065118.2 0.83 - 11 0.0738 8 0.072520.9 0.92 - 11 0.0772 8 0.069022.9 0.96 - 11 0.0754 8 0.061327.4 1.06 - 11 0.0856 8 0.064931.7 1.17 - 12 0.0910 10 0.077838.0
-
1.32 -
-
13 0.0878 9 0.0745& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 43.4 kNm. Date beam tested: 17 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations20
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.35. Optimal Peak Ratio Damping Tabulations for Beam F6 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 39 0.1349 26 0.0834 16 0.0708 11 0.0664 8 0.0593 8 0.0511Standard Deviation 6.6 0.030 4.8 0.008 2.8 0.009 2.5 0.003 0.6 0.003 0.6 0.003
2 0 0 34 0.1452 27 0.0849 19 0.0726 16 0.0561 11 0.0558 10 0.0475Standard Deviation 11.3 0.032 2.8 0.004 0.7 0.007 1.4 0.003 0.7 0.002 0.7 0.000
3 0 0 36 0.1354 23 0.1078 17 0.0856 15 0.0667 10 0.0675 8 0.0606Standard Deviation 3.6 0.008 2.1 0.006 2.6 0.007 - - 1.0 0.005 1.4 0.009
Pre-T
OVERALL AVERAGE LOGDEG 36 0.1385 25 0.0920 17 0.0763 14 0.0630 10 0.0609 8 0.0531
5.9 0.29 - 9 0.0787 6 0.07049.9 0.33 - 10 0.0873 7 0.0751
16.3 & 0.48 - 10 0.0811 7 0.075128.6 0.72 - 10 0.0767 8 0.064041.2
-
0.95 -
-
10 0.0905 8 0.0763& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 45.2 kNm. Date beam tested: 17 April 2002 Age of beam at testing: 28 days
Appendix E: Dam
ping Tabulations21
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.36. Optimal Peak Ratio Damping Tabulations for Beam F7 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 37 0.1216 28 0.0827 18 0.0738 13 0.0676 11 0.0512 9 0.0479Standard Deviation 11.6 0.045 6.9 0.014 3.1 0.001 6.1 0.002 0.6 0.004 1.2 0.001
2 0 0 34 0.1118 28 0.0799 19 0.0684 16 0.0550 12 0.0526 11 0.0464Standard Deviation 5.9 0.029 5.7 0.013 7.1 0.016 2.8 0.007 1.5 0.006 3.9 0.008
3 0 0 30 0.1238 25 0.0793 20 0.0679 16 0.0577 12 0.0503 10 0.0461Standard Deviation 2.4 0.015 4.5 0.010 2.7 0.003 2.1 0.003 0.6 0.002 0.5 0.003
Pre-T
OVERALL AVERAGE LOGDEG 34 0.1191 27 0.0806 19 0.0701 15 0.0601 12 0.0513 10 0.0468
11.9 0.48 - 11 0.0612 9 0.056117.0 & 0.59 - 10 0.0692 8 0.056422.6 0.68 - 11 0.0680 8 0.057229.7 0.83 - 11 0.0569 10 0.051334.9 0.91 - 9 0.0651 7 0.056242.7
-
1.01 -
-
11 0.0612 9 0.0541& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 46.5 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days
Appendix E: Dam
ping Tabulations22
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.37. Optimal Peak Ratio Damping Tabulations for Beam F8 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 42 0.0966 28 0.0793 18 0.0686 16 0.0567 10 0.0518 8 0.0496Standard Deviation 8.4 0.022 5.0 0.007 2.1 0.006 1.0 0.002 2.1 0.006 0.6 0.003
2 0 0 44 0.0932 27 0.0815 17 0.0692 14 0.0561 11 0.0529 8 0.0491Standard Deviation 0.6 0.001 2.1 0.005 2.4 0.005 2.6 0.004 2.1 0.003 1.0 0.002
3 0 0 43 0.0875 30 0.0752 23 0.0609 15 0.0461 11 0.0420 8 0.0431Standard Deviation 2.0 0.009 1.2 0.003 3.6 0.005 2.2 0.002 0.6 0.001 0.6 0.001
Pre-T
OVERALL AVERAGE LOGDEG 43 0.0925 28 0.0787 19 0.0662 15 0.0529 11 0.0489 8 0.0473
11.8 0.35 0 11 0.0504 9 0.047120.0& 0.59 0 9 0.0538 8 0.049430.2 0.77 0.02 10 0.0537 9 0.049340.9 0.95 0.04 10 0.0542 8 0.051952.9 1.15 0.05 9 0.0547 9 0.051165.3 1.35 0.06 10 0.0553 8 0.052077.5 1.63 0.07 9 0.0585 7 0.054494.0
-
2.39 0.09
-
8 0.0645 7 0.0603& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 18.1 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 94.0 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days
Appendix E: Dam
ping Tabulations23
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.38. Optimal Peak Ratio Damping Tabulations for Beam F9 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 44 0.0995 30 0.0699 21 0.0631 14 0.0558 10 0.0522 6 0.0529Standard Deviation 5.7 0.032 5.3 0.012 6.4 0.015 3.0 0.006 1.2 0.001 0.8 0.002
2 0 0 55 0.0991 35 0.0815 22 0.0718 16 0.0579 12 0.0561 7 0.0528Standard Deviation 9.1 0.041 5.1 0.030 3.0 0.017 2.6 0.003 - - 0.7 0.001
3 0 0 47 0.0974 37 0.0581 24 0.0575 17 0.0489 10 0.0476 8 0.0474Standard Deviation 9.1 0.009 5.3 0.008 3.9 0.008 3.1 0.002 1.0 0.001 0.8 0.002
Pre-T
OVERALL AVERAGE LOGDEG 49 0.0986 34 0.0698 22 0.0641 15 0.0542 11 0.0520 7 0.0510
9.6 & 0.49 0 12 0.0550 9 0.050519.2 0.86 0.01 10 0.0549 9 0.052229.3 1.14 0.03 10 0.0591 10 0.048242.5 1.39 0.04 10 0.0600 9 0.052655.7 1.68 0.08 10 0.0584 10 0.050168.5 2.94 0.12 12 0.0751 8 0.073571.9
-
17.77 -
-
13 0.0659 12 0.0546& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 13.2 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 71.9 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days
Appendix E: Dam
ping Tabulations24
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
Table E.39. Optimal Peak Ratio Damping Tabulations for Beam F10 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM
(kNm) Load
Level %∆res
(mm) Wres
(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ
1 0 0 59 0.0832 37 0.0663 28 0.0514 20 0.0425 12 0.0400 10 0.0378Standard Deviation 2.0 0.022 - - - - 4.5 0.007 1.7 0.006 0.7 0.004
2 0 0 47 0.0875 44 0.0468 29 0.0400 21 0.0410 13 0.0411 12 0.0408Standard Deviation 10.4 0.028 5.0 0.002 6.2 0.006 1.5 0.004 2.0 0.006 0.7 0.003
3 0 0 51 0.1003 33 0.0698 25 0.0603 13 0.0559 12 0.0465 10 0.0443Standard Deviation 8.1 0.021 2.1 0.008 3.1 0.006 1.2 0.004 0.6 0.002 0.6 0.003
Pre-T
OVERALL AVERAGE LOGDEG 52 0.0903 38 0.0610 27 0.0506 18 0.0465 12 0.0425 10 0.0410
11.8 & 0.45 0 14 0.0498 11 0.045522.1 0.81 0.03 15 0.0538 12 0.048033.8 1.06 0.06 13 0.0560 8 0.051545.4 1.32 0.07 13 0.0550 11 0.052456.3 1.68 0.09 10 0.0765 9 0.062067.3
-
2.51 0.12
-
12 0.0829 10 0.0681& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 12.8 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure and shear at a mid-span Bending Moment (BM) of 73.6 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days
Appendix E: Dam
ping Tabulations25
Dam
ping Charateristics of Reinforced and Prestressed N
ormal- and H
igh-Strength Concrete Beam
s
E.1 F-Series Beams Load Levels Load Level 1 – Totally Unloaded
This phase induced free-vibration to the beam whilst it had no loading whatsoever on it. This provided
the ‘benchmark’ for comparison with the following stages.
Load Level 2 – Loading Beam Only
At this stage, the loading beam and loading cell were placed into position, at which point more free-
vibration damping testes were conducted. It should be noted here, that the ‘untested’ damping
experiments for the B-, CS- and PS-Series beams were conducted at this Load Level.
Load Level 3 – Additional Weights Hung
Two large concrete blocks were hung from the pin and roller supports of the loading beam. These third
points were selected so that the suspension slings did not come into contact with the test beam, and to
ensure minimal interference with the free-vibration damping tests. Each concrete block weighed 136.5
kg. The tests were limited to two of these blocks due to the size restrictions of the blocks. Nevertheless,
they were sufficiently large enough to induce a small observable level of stress in the beam, which was
monitored via the deflection gauges.
100 tonne capacitycylinder
100 tonne capacitycylinder
Loading CellWeight of loading beam (221 kg)+ Weight of cylinder loading discs
(40 kg)
2000mm 2000mm2000mm
100 tonne capacitycylinder
Loading CellWeight of loading beam (221 kg)+ Weight of cylinder loading discs
(40 kg)
2000mm 2000mm2000mm
136.5 kg 136.5 kg
LoadLevel
1
LoadLevel
2
LoadLevel
3
Figure E.1: Load Levels for F-Series Beam Tests
Appendix E: Damping Tabulations26
Damping Charateristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-1
APPENDIX F
Serviceability Curves
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-2
F.1 Bending Moment versus Instantaneous Deflection
a) AA
AAA
AA
AA
AA
AA
AA
BBB
BBB
BB
BB
BB
BBB
CCC
C
C
C
CC
CC
DD
D
D
D
D
DD
DD
E
EE
E
E
E
EE
E
FF
FF
F
F
F
FF
F
F
GGGGGG
GG
GG G
G GGGGG G
HHHHH H HH H H H
HH
H HH HHHH H H
II I
I
I
I
I
III I I I
JJ
JJ
JJ
JJ
JJ
JJJ J J
KKK
KK
K
K
K
K
K
KK K
LLL
LL
L
L
L
L
L
L
L
L
LL
LL
LL L
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
0 25 50 75 100 1250
25
50
75
100
125
150
175
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b) 1
11
11
11
11
11
11
11
11
11
11
11
1
22
2222 2
22 2 22
22
22
22
22
22
22
33
3333
33
33 3
33
33
33
33
33
3 3
444
444
44
444
44
44
44
44
4
555
555
555555
5555555 5
66666
66666
66666
66
77777
7777
777
777
77
77 7
77
77
77
77
77 7
77
7
888
88888
888888888
8888 88 8 8 88 8 88
8 88 8 8 8 8 8 8 8 8
9999999999999999999 9 99 9 99 99 9 9
999 9 9 9 9 9 9 9 9 9
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
0 2 4 6 8 10 12 14 16 18 20 220
10
20
30
40
50
60
70
80
90
CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c) a a
aaaaaa
a a a a a a a a a a a a
bbbbbbbbbbb
bb
bb
bb
bbb
ccccccccccc
cc
cc
cc
c
dddddddddddddddddddd
dd
dd
dd d d
dd
eeeee
eeeeeeeeeeeee
ee
e
ee
ee
e
fff
f
f
f
fff
ff
ff
ff f f
g
g
g
g
g
g
gg g
h
h
h
h
h
hh
hh
hh
h h
i
i
i
i
i
ii
j
j
j
j
j
j
j
j
j
j
j
jj
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
-20 0 20 40 60 80 100 120 140 1600
10
20
30
40
50
60
70
80
90
100
110
120
130
PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
abcdefghij
Figure F.1: BM (kNm) versus Instantaneous Deflection for: a) BI-1 to BII-12; b) CS1 to
CS9; c) PS1 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-3
a) AA
AA
A
A
A
A
A
A
A
A
A
A
A
BB
BB
B
B
B
B
B
B
B
B
BBB
C
CC
C
C
C
C
C
C
C
D
DD
D
D
D
D
DDD
E
EE
E
E
E
E
EE
F
FF
F
F
F
F
F
FF
F
GGGGGG
G
G
G
G
G
G
GGGGG G
HH
HH
HH
HH
HH
H
H
H
H
HH HH
HH H H
I
II
I
I
I
I
I
I
I II
I
J
J
J
J
J
J
J
J
J
J
JJ J
J J
KKK
K
K
K
K
K
K
K
KK
K
LLL
LL
L
L
L
L
L
L
L
L
LLL L
LL L
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 50 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
11
11
11
11
11
1
11
11
11
11
11
11
2
222
22
22
22
22
22
22
22
22
22
22
3
33
33
33
33
33
33
33
33
33
33
33
444444
44
444
4
44
44
44
44
5
55
555
555555
555555
55
666
66
666
66
66
666
6
6
77777
777
77
77
777
77
77
77
77
7 77 7
77
77
77
7
88888
888
888
888
88888
888
8 88 8
8 8 88
8 888
8 8 88
88 8
999
9999
999
9999999
999 99
9 9999 9 9
9 999 9
9 99 9 9 9
9 9
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 2 4 6 8 10 12 14 16 18 20 220
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c) a
a
a
aa
a
a
a
aa
aa
aa
aa
aa
aa
bbbbbbbbbbb
bb
bb
bb
bbb
c
c
ccccc
c
c
ccc
cc
cc
cc
dddddddddddddddddddd
dd
dd
dd d d
dd
eeeee
eeeeeeeeeeeee
ee
e
ee
ee
e
f
ff
f
f
f
ff
f
f
f f
f f
f f f
g
g
g
g
g
g
g
gg
h
h
h
h
h
hh
hh
hh
h h
i
i
i
i
i
i
i
j
j
j
j
j
j
j
j
j
j
j
jj
Instantaneous Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
-20 0 20 40 60 80 100 120 140 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
abcdefghij
Figure F.2: BM (normalised) versus Instantaneous Deflection for: a) BI-1 to BII-12; b)
CS1 to CS9; c) PS1 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-4
F.2 Bending Moment versus Residual Deflection
a) A
AAAAAAA
AA
AA
B
BB
BBBBB
BB
B
C
C
C
C
CC
CC
D
D
D
D
DD
D D
E
E
E
E
EE
E
F
F
F
F
FF
F
F
G
GGGG
GG G
H
H HHHHHHH
HH
I
I
I
I
II
J
JJJJ
JJJ
JJ
J J
K
K
K
K
K
K
K
KK
L
L
L
L
L
L
L
L
L
LL
Residual Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
0 20 40 600
20
40
60
80
100
120
140
160
180
BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
4
4
4
4
5
5
5
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
9
Residual Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
90
CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c) a
a
a
a
b
b
b
b
c
c
c
c
d
d
d
e
e
e
e
e
e
eee
f
f
f
f
f
ffff
ff
f f
g
g
g
g
g
gg
h
h
h
h
h
h
h
h
h
h
h
hh
Residual Deflection (mm)
Ben
ding
Mom
ent(
kNm
)
0 20 40 600
20
40
60
80
100
120
140
PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)
abcdefgh
Figure F.3: BM (kNm) versus Residual Deflection for: a) BI-1 to BII-12; b) CS1 to
CS9; c) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-5
a) A
A
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
B
C
C
C
C
C
C
C
C
D
D
D
D
D
DD D
E
E
E
E
E
EE
F
F
F
F
F
FF
F
G
G
G
G
G
G
G
G
H
HH
HHH
H
H
H
H
H
I
I
I
I
I
I
J
J
J
J
J
J
J
J
J
JJ J
K
K
K
K
K
K
K
K
K
L
L
L
L
L
L
L
L
L
LL
Residual Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 20 40 600
BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
4
4
4
4
5
5
5
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
9
Residual Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 1 2 3 4 5 6 7 8 9 10 110
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS4CS5CS6CS7CS8CS9
123456789
c) c
c
c
c
d
d
d
d
e
e
e
e
f
f
f
g
g
g
g
g
g
gg g
h
h
h
h
h
hhh
hh
hh h
i
i
i
i
i
i
j
j
j
j
j
j
j
j
j
j
j
jj
Residual Deflection (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 2 4 6 8 10 12 14 16 18 20 220
20
40
60
80
100
120
140
PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
cdefghij
Figure F.4: BM (Normalised) versus Residual Deflection for: a) BI-1 to BII-12; b) CS1
to CS9; c) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-6
F.3 Bending Moment versus Average Instantaneous Crack Width
a) A
AAAAAAAA
AA
B
BBB
BBBB
BBB
B
C
C
C
C
C
CCC
C
D
D
D
D
DD
DD
E
E
E
E
EE
E
F
F
F
F
F
FF
F
G
GG
GGGGG
H
H HH HHHHH
HH
I
I
I
I
II
JJ
JJ
JJJ
JJ
JJ
J
K
K
K
K
K
K
K
K
L
L
L
L
L
L
L
L
L
L
Average Instantaneous Crack Width (mm)
Ben
ding
Mom
ent(
kNm
)
0 0.5 1 1.50
25
50
75
100
125
150
175
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b) c
c
c
d
d
d
f
f
f
g
ggg
h
h
h
h
h h
i
i
ii
j
j
jj
Average Instantaneous Crack Width (mm)
Ben
ding
Mom
ent(
kNm
)
0 0.02 0.04 0.06 0.08 0.1 0.120
20
40
60
80
100
120
140
PS3 (c)PS4 (d)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
cdfghij
Figure F.5: BM (kNm) versus Average Instantaneous Crack Width for: a) BI-1 to BII-
12; b) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-7
a) A
AA
A
A
A
A
A
A
A
A
B
BB
B
B
B
B
B
B
B
BB
C
C
C
C
C
C
C
C
C
D
D
D
D
D
DDD
E
E
E
E
E
EE
F
F
F
F
F
F
FF
G
G
G
G
G
G
G
G
H
HH
HH
H
H
H
H
H
H
I
I
I
I
I
I
J
J
J
J
J
J
J
J
J
J
JJ
K
K
K
K
K
K
K
K
L
L
L
L
L
L
L
L
L
L
Average Instantaneous Crack Width (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.25 0.5 0.750
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b) c
ccc
cc
cc
d
dd
d
dd
ddd
ddd
f
f
f
f
g
g
g
gg
h
h
h
h
h
h
h
i
i
i
i
j
j
j
j
j
j
j
j
Average Instantaneous Crack Width (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
PS3 (c)PS4 (d)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
cdfghij
Figure F.6: BM (Normalised) versus Average Instantaneous Crack Width for: a) BI-1 to
BII-12; b) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-8
F.4 Bending Moment versus Average Residual Crack Width
a) A
AAA
AA
AA
AA
B
BB
BBBB
BB
BB
C
C
C
C
CC
C
D
D
D
D
DD
D
E
E
E
E
EE
E
F
F
F
FF
F
G
GGGG
GG
H
H HHHH HHHH
H
I
I
I
I
II
J
JJJ
JJ
JJ
JJ
J
K
K
K
K
K
K
K
KK
L
L
L
L
L
L
L
L
L
L
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
kNm
)
0 0.05 0.1 0.15 0.20
25
50
75
100
125
150
175
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
kNm
)
0 0.02 0.04 0.06 0.080
25
50
75
CS1CS2CS3CS7CS8CS9
123789
c) c
c
c
d
d
d
f
f
f
g
ggg
h
h
h
h
h h
i
i
ii
j
j
jj
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
kNm
)
0 0.02 0.04 0.06 0.08 0.1 0.120
20
40
60
80
100
120
140
PS3 (c)PS4 (d)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
cdfghij
Figure F.7: BM (kNm) versus Average Residual Crack Width for: a) BI-1 to BII-12; b)
CS1 to CS3 and CS7 to CS9; c) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-9
a) A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
BB
C
C
C
C
C
C
C
D
D
D
D
D
DD
E
E
E
E
E
EE
F
F
F
F
FF
G
G
G
G
G
G
G
H
HH
HH
H
H
H
H
H
H
I
I
I
I
I
I
J
J
J
J
J
J
J
J
J
JJ
K
K
K
K
K
K
K
K
K
L
L
L
L
L
L
L
L
L
L
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.05 0.1 0.15 0.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.02 0.04 0.06 0.080
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
c) c
c
c
d
d
d
f
f
f
g
g
gg
h
h
h
h
h h
i
i
i
i
j
j
jj
Average Residual Crack Width (mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.02 0.04 0.06 0.08 0.1 0.120
0.2
0.4
0.6
0.8
1
PS3 (c)PS4 (d)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)
cdfghij
Figure F.8: BM (Normalised) versus Average Residual Crack Width for: a) BI-1 to BII-
12; b) CS1 to CS3 and CS7 to CS9; c) PS3 to PS10
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-10
F.5 Bending Moment versus Instantaneous Steel Strain
a) A
AA
A
A
A
A
A
A
A
A
BB
BB
B
B
B
B
B
C
CC
C
C
C
C
D
DD
D
D
E
E
E
E
E
F
F
F
F
F
GGGGG G G
GG
GG
HHH
H HH
HH
HH
H
II
II
I
I
J
J
J
J
J
J
K
KK
K
K
K
K
L
LL
LL
L
L
L
L
Instantaneous Steel Strain (mm/mm)
Ben
ding
Mom
ent(
kNm
)
0 0.0005 0.001 0.0015 0.0020
10
20
30
40
50
60
70
80
90
100BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
11
11
11
11
11
11
111
11
11
11
11
22
22222
222 22
22
22
22
22
22
2
33
3333
33
33 3
33
33
33
33
33
3 3
777
777
77
77
77
777
77
77 7
77
77
77
77
77 7
77
7
888
88888 8888
88 8888 88 88 888 8888 8
8 88 8 8 8 8 8 8 8
999
99999999999 999999 9999 99999
999 999 99 9999 99
Instantaneous Steel Strain (mm/mm)
Ben
ding
Mom
ent(
kNm
)
0 0.001 0.002 0.003 0.004 0.0050
10
20
30
40
50
60
70
80
90
CS1CS2CS3CS7CS8CS9
123789
Figure F.9: BM (kNm) versus Instantaneous Steel Strain for: a) BI-1 to BII-12; b) CS1
to CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-11
a) A
AA
A
A
A
A
A
A
A
A
BB
BB
B
B
B
B
B
C
CC
C
C
C
C
D
DD
D
D
E
E
E
E
E
F
F
F
F
F
GGG
GG
GG
G
G
G
G
HH
HH
HH
HH
HH
H
I
II
I
I
I
J
J
J
J
J
J
KK
KK
K
K
K
LLLL L
L
L
L
L
Instantaneous Steel Strain (mm/mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.0005 0.001 0.0015 0.0020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
11
11
11
11
11
11
11
11
11
11
11
1
2
222
22
22
22
22
22
22
22
22
22
2
3
33
33
33
33
33
33
33
33
33
33
33
777
777
777
77
77
777
77
77
77
77 7
777
77
77
77
88888
888
88 8
88 8
8888 8
8 88
888 8
88 88
8 8 88
8 8 88
88
999
99 99
999
9 99 9999
999 99
99 999 99
99 9
99 999
999999
Instantaneous Steel Strain (mm/mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.001 0.002 0.003 0.0040
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
Figure F.10: BM (Normalised) versus Instantaneous Steel Strain for: a) BI-1 to BII-12;
b) CS1 to CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-12
F.6 Bending Moment versus Instantaneous Steel Stress
a) A
AA
A
A
A
A
A
A
A
A
BB
BB
B
B
B
B
B
C
CC
C
C
C
C
D
DD
D
D
E
E
E
E
E
F
F
F
F
F
GGGGG G G
GG
GG
HHH
H HH
HH
HH
H
II
II
I
I
J
J
J
J
J
J
K
KK
K
K
K
K
L
LL
LL
L
L
L
L
Instantaneous Steel Stress (MPa)
Ben
ding
Mom
ent(
kNm
)
0 100 200 300 4000
10
20
30
40
50
60
70
80
90
100BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
11
11
11
11
11
11
11
1
2
22
22
22
22
22
22
2
22
2
2
3
33
33 3
33
33
33
33
33
33
77
77
77
77
7
77
77
77
77
77
77
88
88
88
88
88
88
88
88 8
88
88
88
88
99
99
99
99 9
99
99
99
99
99
99
99
99
9
Instantaneous Steel Stress (MPa)
Ben
ding
Mom
ent(
kNm
)
0 100 200 3000
10
20
30
40
50
CS1CS2CS3CS7CS8CS9
123789
Figure F.11: BM (kNm) versus Instantaneous Steel Stress for: a) BI-1 to BII-12; b) CS1
to CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-13
a) 1
1
1
2
2
2
3
3
3
7
7
7
8
8
8
9
9
9
Instantaneous Steel Stress (MPa)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
b) 1
11
11
11
11
11
11
11
1
2
22 2
22
22
22
22
22
22
22
3
33
33
33
33
33
33
33
33
3
77
77 7
77 7
77
77
77 7
77
77
77
88
8 8 88
8 88
8 88
8 88 8 8
8 88 8
88 8
8
99 9
99 9 9
9 99
9 9 9 99 9 9
9 99 9 9
9 9 99
Instantaneous Steel Stress (MPa)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
Figure F.12: BM (Normalised) versus Instantaneous Steel Stress for: a) BI-1 to BII-12;
b) CS1 to CS3 and CS7 CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-14
F.7 Bending Moment versus Residual Steel Strain
a) A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
C
C
C
C
C
D
D
D
E
E
E
E
F
F
F
F
G
GG
GG
H
HH
HH
HH
I
I
I
J
J
J
J
J
K
K
K
K
L
L
L
L
L
Residual Steel Strain (mm/mm)
Ben
ding
Mom
ent(
kNm
)
0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.00080
10
20
30
40
50
60
70
80
90
100BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
8
8
8
9
9
9
Residual Steel Strain (mm/mm)
Ben
ding
Mom
ent(
kNm
)
0 5E-05 0.0001 0.00015 0.0002 0.000250
5
10
15
20
25
30
35
40
45
50
55
CS1CS2CS3CS7CS8CS9
123789
Figure F.13: BM (kNm) versus Residual Steel Strain for: a) BI-1 to BII-12; b) CS1 to
CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-15
a) A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
C
C
C
C
C
D
D
D
E
E
E
E
F
F
F
F
G
G
G
G
G
H
HH
HHH
H
I
I
I
J
J
J
J
J
K
K
K
K
L
L
L
L
L
Residual Steel Strain (mm/mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 0.0005 0.001 0.0015 0.0020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
8
8
8
9
9
9
Residual Steel Strain (mm/mm)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 5E-05 0.0001 0.00015 0.0002 0.000250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
Figure F.14: BM (Normalised) versus Residual Steel Strain for: a) BI-1 to BII-12; b)
CS1 to CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-16
F.8 Bending Moment versus Residual Steel Stress
a) A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
C
C
C
C
C
D
D
D
E
E
E
E
F
F
F
F
G
GG
GG
H
HH
HH
HH
I
I
I
J
J
J
J
J
K
K
K
K
L
L
L
L
L
Residual Steel Stress (MPa)
Ben
ding
Mom
ent(
kNm
)
0 40 80 120 1600
10
20
30
40
50
60
70
80
90
100BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
8
8
8
9
9
9
Residual Steel Stress (MPa)
Ben
ding
Mom
ent(
kNm
)
0 10 20 30 40 500
5
10
15
20
25
30
35
40
45
50
55
CS1CS2CS3CS7CS8CS9
123789
Figure F.15: BM (kNm) versus Residual Steel Stress for: a) BI-1 to BII-12; b) CS1 to
CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams
APPENDIX F: Serviceability Curves F-17
a) A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
C
C
C
C
C
D
D
D
E
E
E
E
F
F
F
F
G
G
G
G
G
H
HH
HH
H
H
I
I
I
J
J
J
J
J
K
K
K
K
L
L
L
L
L
Residual Steel Stress (MPa)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 40 80 120 160 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)
ABCDEFGHIJKL
b) 1
1
1
2
2
2
3
3
3
7
7
7
8
8
8
9
9
9
Instantaneous Steel Stress (MPa)
Ben
ding
Mom
ent(
Nor
mal
ised
)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CS1CS2CS3CS7CS8CS9
123789
Figure F.16: BM (Normalised) versus Residual Steel Stress for: a) BI-1 to BII-12; b)
CS1 to CS3 and CS7 to CS9
Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams