Concentric Punching Shear Strength of Reinforced Concrete Flat … · 2018-12-18 · formula, as well as the punching shear formulae in some of the internationally recognised standards

Concentric Punching Shear Strength of

Reinforced Concrete Flat Plates

Fariborz Moeinaddini

Submitted in total fulfilment of the requirement of the degree of

Master of Engineering

June 2012

Centre for Sustainable Infrastructure, Faculty of Engineering and

Industrial Science

Swinburne University of Technology, Melbourne, Australia

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Abstract

Flat slabs are very popular and economical floor systems in the construction industry. These

floor systems, supported directly on columns, are known to be susceptible to punching shear in

the vicinity of the slab-column connection. The punching shear provisions of AS 3600-2009,

the current Australian Concrete Structures Standard, for the case of concentric loading are based

on empirical formulae developed in the early 1960s and have not improved significantly since

then. These provisions do not consider some of the important parameters affecting the capacity

of a slab such as flexural reinforcement ratio and slab thickness size effect. AS 3600-2009 only

recognises shearheads as an effective shear reinforcement to increase the concentric punching

shear strength of slabs, and it does not cover more practical types of reinforcement such as shear

studs and stirrups unlike most of European and North American codes of practice.

In this thesis, the available methods for calculating concentric punching shear strength of slabs

are reviewed. The analytical basis of previous work by other researchers was used to propose a

formula to calculate the punching shear strength of flat plates with good accuracy for a wide

range of slab thicknesses, tensile reinforcement ratios, and concrete compressive strengths. In

this method, it is assumed that punching shear failure occurs due to the crushing of the critical

concrete strut adjacent to the column. A large number of experimental results of slab test

specimen, reported in the literature were gathered to evaluate the accuracy of the proposed

formula, as well as the punching shear formulae in some of the internationally recognised

standards such as AS 3600-2009, ACI 318-05, CSA A23.3-04, DIN 1045-1:2001, Eurocode2,

and NZS 3101:2006.

The proposed formula was also extended to cover the case of prestressed flat plates with the use

of the decompression method. Recent experimental results of prestressed slab test specimens,

published in journal papers, were collected to assess the accuracy of the proposed formula and

provisions of aforementioned standards in the prediction of the ultimate strength of prestressed

flat plates.

Furthermore, detailing considerations for the design of shear reinforcements such as shear studs

and stirrups, which are not recognised by AS 3600-2009, were discussed. Different failure

modes of flat plates with shear reinforcement were presented. A method to calculate the

strength of the slab assuming a critical crack developing inside the shear reinforced region was

proposed. This method considers the contribution of shear reinforcement intersecting with the

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critical crack and the uncracked concrete zone adjacent to the column. In addition, a control

perimeter outside the shear reinforced zone was suggested to be used with the one-way shear

formula of AS 3600-2009 to calculate the punching shear strength of flat plates outside their

shear reinforced zone. The proposed method and provisions of ACI 318-05, CSA A23.3-04,

and Eurocode2 were evaluated against some of the reported experimental results on the flat

plates with shear reinforcement.

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Acknowledgement

This research was conducted at the Centre of Sustainable Infrastructure, Swinburne

University of Technology. The SUPRA scholarship provided by Swinburne University

of Technology is gratefully acknowledged.

I would like to sincerely thank my principal coordinating supervisor Dr. Kamiran

Abdouka for his invaluable guidance and constant support throughout this research. I

am also greatly indebted to my coordinating supervisor Prof. Emad Gad for his wise

suggestions and continuous help during my postgraduate studies.

I wish to express my deep gratitude to Emma Wenczel, Alireza Mohyeddin-Kermani

whom I lived with during my studies in Australia, for their encouragement,

understanding and support.

I owe special thanks to my valued friends and colleagues Anne Belski, Ianina Belski,

Bara Baraneedaran, Saleh Hassanzade, Hessam Mohseni, Siva Sivagnanasundram and

Stephan Zieger for their assistance and companionship during this research.

Finally, my foremost thanks and greatest gratitude goes to my beloved family Fahime,

Firoozeh, Farnaz and Faramarz for their moral support and unconditional help.

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Preface

So far, a part of this research has been presented in the following conference papers:

• Moeinaddini, F & Abdouka, K 2011, ‘Punching shear capacity of concrete slabs with no unbalanced moment’, Proceedings of Concrete 2011, Concrete Institute of Australia, Perth, Australia.

• Moeinaddini, F, Abdouka, K & Gad, EF 2010, ‘Punching shear capacity of concrete slabs: a comparative study of various standards and recent analytical methods’, Post-

graduate Research, Swinburne University of Technology, Melbourne, Australia.

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Declaration

This is to certify:

• This thesis contains no material which has been accepted for the award to the

candidate of any other degree or diploma, except where due reference is made in the

text.

• To the best of the candidate’s knowledge contains no material previously published or

written by another person except where due reference is made in the text of the

examinable outcome.

Fariborz Moeinaddini

June 2012

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Table of Content

1 INTRODUCTION ................................................................................................................ 1

1.1 Background ................................................................................................................... 1

1.2 Aim and Objectives ....................................................................................................... 5

1.3 Thesis Organisation ...................................................................................................... 5

2 LITERATURE REVIEW ..................................................................................................... 7

2.1 Introduction ................................................................................................................... 7

2.2 Reported Observations from Concentric Punching Shear Failure of Test Specimens .. 7

2.3 Mechanical Models for Punching Shear – Balanced Condition ................................... 9

2.3.1 Kinnuen and Nylander Approach ......................................................................... 9

2.3.2 Truss Model by Alexander and Simmonds ......................................................... 15

2.3.3 Bond Model by Alexander and Simmonds ......................................................... 17

2.3.4 Models Based on the Failure of Concrete in Tension ......................................... 19

2.3.5 Plasticity Approach ............................................................................................. 24

2.3.6 Flexural Approach............................................................................................... 25

2.3.7 Critical Shear Crack Theory ............................................................................... 26

2.4 Punching Shear of Prestressed Flat Plates .................................................................. 27

2.4.1 Principal Tensile Stress Approach ...................................................................... 28

2.4.2 Equivalent Reinforcement Ratio Approach ........................................................ 28

2.4.3 Decompression Approach ................................................................................... 29

2.5 Methods to Increase Punching Shear Strength of Concrete Slabs .............................. 30

2.6 Shear Reinforcement for Flat Plates ........................................................................... 31

2.6.1 Shear Reinforcement for Construction of New Slabs ......................................... 31

2.6.2 Shear Reinforcement for Retrofit of Slabs .......................................................... 35

2.7 Control Perimeter Approach and Building Code Provisions ...................................... 37

2.7.1 Australian Standard AS 3600-2009 .................................................................... 37

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2.7.2 American Code ACI 318-05 ................................................................................ 39

2.7.3 New Zealand Standard NZS 3101:2006 .............................................................. 41

2.7.4 Canadian Standard CSA A23.3-04 ...................................................................... 41

2.7.5 Eurocode2 (2004) ................................................................................................ 43

2.7.6 British Standard BS 8110-97 ............................................................................... 44

2.7.7 German Standard DIN 1045-1:2001 .................................................................... 45

2.8 Summary ...................................................................................................................... 46

3 CONCENTRIC PUNCHING SHEAR OF FLAT PLATES ............................................... 47

3.1 Introduction ................................................................................................................. 47

3.2 Strut-and-Tie Model for Punching Shear Phenomenon ............................................... 48

3.3 Proposed Formula for the Ultimate Punching Shear Strength of Flat Plates ............... 50

3.3.1 Depth of Neutral Axis ......................................................................................... .52

3.3.2 Inclination of the Critical Strut and Critical crack .............................................. .55

3.3.3 Compressive Strength of the Concrete Strut ...................................................... .58

3.3.4 Slab Size Factor ................................................................................................... 59

3.3.5 Determination of the Parameters ......................................................................... 60

3.3.6 Example ............................................................................................................... 67

3.4 Comparison of Experimental Results with Design Standards ..................................... 68

3.5 Summary ...................................................................................................................... 75

4 CONCENTRIC PUNCHING SHEAR OF PRESTRESSED FLAT PLATES ................... 77

4.1 Introduction ................................................................................................................. 77

4.2 Background .................................................................................................................. 77

4.2.1 Effect of In-plane Stresses on the Punching Shear Strength of Flat Plates ......... 78

4.2.2 Effect of Eccentricity of Prestressing Tendon on the Punching Shear Strength of

Flat Plates ............................................................................................................................ 81

4.2.3 Effect of the Vertical Component of Prestressing Tendons Passing over the Slab-

Column Connection on the Punching Shear Strength of Flat Plates ................................... 82

4.3 Ultimate Punching Shear Strength of Prestressed Flat Plates Using the Decompression

Method ..................................................................................................................................... 84

4.3.1 Available Decompression Methods ..................................................................... 86

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4.3.2 Proposed Decompression Method....................................................................... 88

4.3.3 Example .............................................................................................................. 91

4.4 Comparison of Design Standards ................................................................................ 94

4.4.1 Comparison with Experimental Results .............................................................. 94

4.5 Summary ..................................................................................................................... 99

5 CONCENTRIC PUNCHING SHEAR OF FLAT PLATES WITH SHEAR

REINFORCEMENT ................................................................................................................. 101

5.1 Introduction ............................................................................................................... 101

5.2 Detailing of Shear Reinforcement ............................................................................. 102

5.3 Ultimate Strength of Flat Plates with Shear Reinforcement ..................................... 104

5.3.1 Failure Inside the Shear Reinforced Region ..................................................... 105

5.3.2 Failure Outside the Shear Reinforced Region ................................................... 109

5.3.3 Summary of the suggested method ................................................................... 111

5.3.4 Example ............................................................................................................ 112

5.4 Comparison of Experimental Results with Design Standards .................................. 114

5.5 Summary ................................................................................................................... 114

6 SUMMARY AND CONCLUSIONS ............................................................................... 117

6.1 Summary and Findings of Literature Review ........................................................... 117

6.2 Concentric Punching Shear Strength of Flat Plates .................................................. 117

6.3 Concentric Punching Shear Strength of Prestressed Flat Plates ............................... 119

6.4 Concentric Punching Shear Strength of Flat Plates with Shear Reinforcement.........120

References…………… ...... ……………………………………………...………………….... 123

Appendix A…………………….……… .......... ……………………...………………………..125

Appendix B… ..... …….…………………………..…………………………...……………….139

Appendix C… ..... .…………………………………………………...……………..………….143

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List of Figures

Figure 1.1 Schematic view of different types of two-way concrete slabs (Wight & MacGregor

2009) ............................................................................................................................................. 1

Figure 1.2 Punching shear localised failure with pyramid-shaped failure surface (Egberts 2009 ;

Wight & MacGregor 2009) ........................................................................................................... 2

Figure 2.1 Tangential and radial cracks observed in typical punching shear test specimen (Sherif

1996) ............................................................................................................................................. 8

Figure 2.2 Comparison of deflection-load graph for slab test specimens failed by punching

shear to slab test specimens failed in flexure (Menétrey 1998) .................................................... 8

Figure 2.3 Mechanical model of Kinnunen and Nylander as shown in fib (2001) ....................... 9

Figure 2.4 Punching shear failure model proposed by Shehata and Regan (Shehata 1990) ....... 11

Figure 2.5 Radial compression stress failure proposed by Broms (1990) as shown in fib (2001)

.................................................................................................................................................... 12

Figure 2.6 Radial compression stress failure mechanism as shown in Marzouk, Rizk and Tiller

(2010) .......................................................................................................................................... 15

Figure 2.7 Truss model proposed by Alexander and Simmonds (1987) as shown in Megally

(1998) .......................................................................................................................................... 16

Figure 2.8 Curved compression strut (Alexander & Simmonds 1992) ....................................... 17

Figure 2.9 Plan view of slab and the components of Bond model proposed by Alexander and

Simmonds (1992) ........................................................................................................................ 18

Figure 2.10 Free body diagram of radial strip (Alexander & Simmonds 1992) ......................... 19

Figure 2.11 Punching shear model by Georgopoulos as shown in fib (2001) ............................ 20

Figure 2.12 Distribution of concrete tensile stresses in Georgopoulos as shown in fib (2001) .. 20

Figure 2.13 Schematic view of components of proposed method by Menetrey (2002) ............. 21

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Figure 2.14 Schematic view of model by Theodorakopoulos and Swamy (2002) ..................... 23

Figure 2.15 Plasticity model proposed by Braestrup et al. (1976) .............................................. 24

Figure 2.16 Failure pattern and parameters of the proposed method by Rankin and Long (1987)

..................................................................................................................................................... 26

Figure 2.17 Procedure to specify punching shear strength of slab according to Critical Shear

Crack Theory (Muttoni 2008)...................................................................................................... 27

Figure 2.18 Load-deflection curves of slabs strengthened by different methods (Megally &

Ghali 2000) .................................................................................................................................. 30

Figure 2.19 Shearhead reinforcement (Corley & Hawkins 1968) ............................................... 32

Figure 2.20 (a) Bent bar, (b) Single-leg stirrup , (c) Multiple-leg stirrup (d) Closed-stirrup or

Closed-tie (ACI 318-05 2005 ; Broms 2007) .............................................................................. 33

Figure 2.21 Headed shear studs (Bu 2008) .................................................................................. 33

Figure 2.22 (a) Plan view of a shearband (b) Shearbands placed in slab (Pilakoutas & Li 2003)

..................................................................................................................................................... 34

Figure 2.23 UFO shear reinforcement (Alander 2004) ............................................................... 34

Figure 2.24 Lattice shear reinforcement (Park et al. 2007) ......................................................... 35

Figure 2.25 Test specimen strengthened by steel plates (Ebead & Marzouk 2002) .................... 36

Figure 2.26 (a) Shear bolt, (b) concrete slab strengthened with shear bolts (Bu 2008)............... 36

Figure 2.27 Critical perimeter around the column as shown in AS 3600- 2009 ......................... 38

Figure 2.28 Shear reinforcement layout suggested by ACI 318-05 as shown in Kamara and

Rabbat (2005) .............................................................................................................................. 40

Figure 2.29 Critical perimeter as shown in Eurocode2 (2004) .................................................... 43

Figure 2.30 Shear reinforcement arrangement and critical perimeter outside the shear reinforced

region as shown in Eurocode2 (2004) ......................................................................................... 44

Figure 2.31 Critical perimeter as given in DIN 1045-1 (2001) ................................................... 45

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Figure 3.1 Schematic view of B-regions and D-regions in a simple structure............................ 47

Figure 3.2 Early strut-and-tie model for slab-column connection .............................................. 48

Figure 3.3 Refined Strut-and-tie model including concrete ties ................................................. 49

Figure 3.4 Punching shear by failure of concrete ties ................................................................. 49

Figure 3.5 Punching shear by crushing of concrete struts .......................................................... 50

Figure 3.6 View and cross section of the critical concrete strut around the column .................. 51

Figure 3.7 Distribution of strains, stresses and forces in elastic condition (Warner et al. 1998) 53

Figure 3.8 Strains and stresses distribution in the ultimate stage (Warner et al. 1998) .............. 53

Figure 3.9 Rectangular stress block in the ultimate stage (Warner et al. 1998) ......................... 54

Figure 3.10 Schematic view of the flexural neutral axis and the shear neutral axis

(Theodorakopoulos & Swamy 2002) .......................................................................................... 55

Figure 3.11 Observed critical crack angle versus thickness of slab ............................................ 57

Figure 3.12 Predicted angle of the critical crack using Equation 3-10 ....................................... 58

Figure 3.13 Vtest/Vuo versus effective depth of slab, tensile reinforcement ratio and compressive

strength of concrete for T-P-M-0.5 ............................................................................................. 64


strength of concrete for S-P-B-0.33 ............................................................................................ 65


strength of concrete for S-P-A-0.5 .............................................................................................. 66

Figure 3.16 Plan and elevation view of test specimen 16/1 reported in (2005) ......................... 67


strength of concrete for AS 3600-2009 and ACI 318-05 ............................................................ 70


strength of concrete for NZS3101:2006 ...................................................................................... 71

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strength of concrete for CSA A23.3-04 ....................................................................................... 72


strength of concrete for Eurocode2 and Model Code 90 ............................................................. 73


strength of concrete for DIN 1045-1 ........................................................................................... 74

Figure 4.1 Prestressing actions adjacent to the slab-column connection ..................................... 78

Figure 4.2 Geometery of BD test series (Ramos, Lúcio & Regan 2011) .................................... 79

Figure 4.3 Geometry of test specimens LP1, LP2 and LP3 as shown in Silva, Regan and Melo

(2005) .......................................................................................................................................... 80

Figure 4.4 Geometry of test specimens V5 and V6 reported in Kordina and Nolting (1984) as

shown in Silva, Regan and Melo (2005) ..................................................................................... 80

Figure 4.5 Elevation view of test setup of PC test series and the bending moment diagram which

was applied to the slab without presence of in-plane forces (Clement & Muttoni 2010) ........... 81

Figure 4.6 (a) Plan view of test specimens AR8-AR16 (b) Profile of prestressing tendons

(Ramos & Lucio 2006) ................................................................................................................ 83

Figure 4.7 Position of prestressing tendons in test specimens AR8-AR16 (Ramos & Lucio 2006)

..................................................................................................................................................... 83

Figure 4.8 Schematic view of deformation of slab after prestressing forces are applied ............ 85

Figure 4.9 (a) Prestressed slab (b) Prestressed slab at decompression stage (c) Punching shear

failure of prestressed slab ............................................................................................................ 86

Figure 4.10 Vtest/Vup versus σcp for three different methods of calculating Vup ............................ 90

Figure 4.11 (a) Plan view (b) Elevation view of test setup of specimen D2 as reported in Silva,

Regan and Melo (2005) ............................................................................................................... 92

Figure 4.12 Vtest/Vup versus σcp for AS3600-2009 ........................................................................ 96

Figure 4.13 Vtest/Vup versus σcp for AS3600-2009 when Vp is included ....................................... 96

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Figure 4.14 Vtest/Vup versus σcp for ACI 318-05 .......................................................................... 97

Figure 4.15 Vtest/Vup versus σcp for ACI 318-05 ignoring the limit on f’c .................................... 97

Figure 4.16 Vtest/Vup versus σcp for CSA A23.3-04 ..................................................................... 97

Figure 4.17 Vtest/Vup versus σcp for CSA A23.3-04 ignoring the limit on f’c ............................... 98

Figure 4.18 Vtest/Vup versus σcp for Eurocode2 ............................................................................ 98

Figure 4.19 Vtest/Vup versus σcp for DIN 1045-1 .......................................................................... 98

Figure 5.1 (a) Orthogonal type arrangement (b) Radial type arrangement (c) square type

arrangement of shear reinforcement for punching shear ........................................................... 102

Figure 5.2 Radial and tangential spacing between shear rows reinforcement in flat plates...... 103

Figure 5.3 Different types of punching shear failure in flat plates with shear reinforcement .. 104

Figure 5.4 (a) Critical tie in flat plates with shear reinforcement (b) Failure of the critical tie due

to the development of shear crack inside the shear reinforced region ...................................... 105

Figure 5.5 Vertical components of the critical tie which resist punching shear ....................... 106

Figure 5.6 Eurocode2 and Model Code 90 control perimeter outside the orthogonal shear

reinforced zone.......................................................................................................................... 110

Figure 5.7 (a) Top view of test specimen 12 (b) Arrangement of shear reinforcements in the test

specimen 12 (Birkle & Dilger 2008) ......................................................................................... 112

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List of Tables

Table 3.1 Main properties of test specimens and angle of the critical crack reported in (Pisanty

2005) ........................................................................................................................................... 57

Table 3.2 Average, SD and CV of Vtest/Vuo for different combination of parameters using the

method in Broms (1990) to calculate the depth of the neutral axis............................................. 62

Table 3.3Average, SD and CV of Vtest/Vuo for different combination of parameters using the

method in Theodorakopoulos and Swamy (2002) to calculate the depth of the neutral axis...... 62


method in Shehata (1990) to calaculate the depth of the neutral axis ......................................... 63

Table 3.5 Average, SD and CV of Vtest/Vuo for AS 3600-2009, ACI 318-05, NZ 3101:2006,

CSA A23.3-04, Eurocode2 and DIN 1045-1 .............................................................................. 69

Table 4.1 Failure load and details of BD test specimens (Ramos, Lúcio & Regan 2011) .......... 79

Table 4.2 Failure load and detail of test specimens LP1, LP2 and LP3 (Silva, Regan & Melo

2005) ........................................................................................................................................... 80

Table 4.3 Failure load and details of test specimens V5 and V6 (Silva, Regan & Melo 2005) .. 81

Table 4.4 Failure load and details of test specimens reported in Clement and Muttoni (2010) .. 82

Table 4.5 Failure load and details of test specimen AR8-AR16 (Ramos & Lucio 2006) ........... 84

Table 4.6 Average, SD and CV of Vtest/Vup for three different methods of calculating Vup ......... 89

Table 4.7 Average, SD and CV of Vtest/Vup for AS 3600-2009, ACI 318-04, CSA A23.3-04,

Eurocode2, and DIN 1045-1:2001 .............................................................................................. 95

Table 5.1 Vtest/Vuin for test specimens in which failure occurred inside the shear reinforced zone

.................................................................................................................................................. 109

Table 5.2 Vtest/Vuout for test specimens in which failure occurred outside the shear reinforced

zone ........................................................................................................................................... 111

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Table 5.3 Average, SD and CV of Vtest/Vus for ACI 318-05, CSA A23.3, Eurocode2, and the

proposed method ....................................................................................................................... 114

Table A.1 Details of collected slab test specimens.................................................................... 130

Table A.2 Predicted punching shear strength of collected test specimens ................................ 134

Table B. 1 Details of collected prestressed slab test specimens ................................................ 140

Table B. 2 Predicted punching shear strength of collected test specimens using the suggested

method ....................................................................................................................................... 141

Table B. 3 Predicted punching shear strength of collected test specimens using formulae of

design standards......................................................................................................................... 142

Table C.1 Details of collected slab test specimens with shear reinforcement ........................... 144

Table C.2 Predicted punching shear strength of slab test specimens with shear reinforcement

using the suggested method ....................................................................................................... 145


using ACI 318-05 ...................................................................................................................... 146


using Eurocode2 ........................................................................................................................ 147


using CSA A23.3-04 ................................................................................................................. 148

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Chapter One

1 INTRODUCTION

1.1 Background

Two-way concrete slabs are widely used in many types of strucutres. They can be categorised

into slabs that are supported on beams, and slabs that are supported on columns without any

beam. The beamless slabs can be further subdivided into two categories: flat slabs, which are

supported on columns through a drop panel or column capital, and flat plates, which are

supported directly on the columns. Different types of two-way concrete slabs are shown in

Figure 1.1. The early beamless slabs were flat slabs, constructed in the early 20th century. With

the devlopment of construction technology, flat plates were developed from the concept of flat

slabs and were increasingly built after World War II.

Figure 1.1 Schematic view of different types of two-way concrete slabs (Wight & MacGregor 2009)

Flat plate construction is very common in parking, office, and apartment buildings. Exclusion

of the beams, drop panels, or column capitals in the structural system optimises the storey

height, formwork, labour, construction time, and the interior space of the building. This makes

flat plate construction a very desirable structural system in view of economy, construction, and

architectural desires. However, from structural point of view, supporting a relatively thin plate

directly on a column is significantly problematic due to the structural discontinuity.

a) Concrete slab, supported on

beams

b) Flat slab concrete slab c) Flat plate concrete slab

2

Considering the flow of forces in the structure, significant biaxial bending moment and shear

force should transfer through the slab-column connection. In the absence of beams, drop

panels, or column capital, this region is considered as one of the most critical D-regions, in

which stresses are disturbed and strains are irregular, in concrete structures (fib 2001).

If the shear stresses are minor, two-way concrete slabs show significant ductility, and

redistribution of moment before the strength of the slab is reached. Where two-way slabs are

supported on beams, shear force is distributed along the beams and shear stresses are not

considerable, so a very thin slab satisfies the flexural strength criterion of the design. Generally,

in this type of concrete slab, the deflection limitations determine the thickness of the slab.

In flat plates, however, there is a considerable amount of shear to be transferred through the

slab-column connection. Typically, slab thickness would be determined either by a shear

strength criterion or deflection limitations. With the increasing use of prestressing in floor

construction, designers are capable of eliminating the excessive deflection of two-way slabs by

defining the prestressing tendon profile, and generally the critical problem which governs the

design is the so called “punching shear” (Dilger & Ghali 1981).

The punching shear or two-way shear phenomenon is a localised failure. It occurs when the

column, punches through the slab, and it can be characterised by the truncated or pyramid

failure surface. Schematic view and a saw-cut test specimen, failed by punching shear, are

shown in Figure 1.2.

Figure 1.2 Punching shear localised failure with pyramid-shaped failure surface (Egberts 2009 ;

Wight & MacGregor 2009)

This type of failure is extremely dangerous and should be prevented, since it may lead to brittle,

with little or no warning, and progressive collapse of floors. One of the most notorious

examples of the devastating punching shear failure is: the collapse of Sampoong department

store in South Korea in 1995 where more than 500 people were killed and nearly 1000 were

3

injured (Gardner, Huh & Chung 2002). Another example is the collapse of the Skyline Plaza in

Virginia in 1973 which killed more than 14 workers (Bu 2008).

Designers can increase the punching strength of beamless slabs by increasing the slab thickness,

introducing drop panels or column capitals, adding shear reinforcement adjacent to the column,

or even specifying concrete with higher strength. In some standards such as Eurocode2 (2004),

BS 8110 (1997), and DIN 1045-1 (2001) increase of flexural reinforcement also allows

designers to consider higher shear strength for the slabs.

Due to the importance of the punching shear phenomenon, an enormous volume of research has

been conducted on this topic. There have been significant attempts to propose a rational model

that can explain the flow of forces in the vicinity of the slab-column connection. However,

there is still no consensus in the literature on how to calculate the punching shear strength of

concrete slabs. Even internationally recognised concrete structure standards are significantly

different in their approach towards this problem.

Most of the international concrete structure standards have enhanced their formulae as insight

into this type of failure has improved in recent decades. Mostly, they adopt empirical or semi-

empirical formulae in their provisions for the punching shear phenomenon. Typically, they

distinguish between two conditions for punching shear. Firstly, where slab-column connections

are under no unbalanced moment and the loading of the slab produces symmetrical shear.

Secondly, where slab-column connections undergo unbalanced moment and shear forces

simultaneously. An example for the first case is where the columns are equally spaced and the

lateral loads on the structure are carried by other structural systems such as shear walls or

bracings. An example for the second case is where the slab-column structural system resists the

lateral forces in addition to the gravity loads, or at exterior slab-column connections.

Generally, the most common solution for designers to increase the punching strength of the slab

is to use different types of shear reinforcement. Some of the most common types of shear

reinforcement for punching shear are closed ties, shearheads, bent-up bars, single leg ties, and

more recently shear studs or stud rails. The slab-column connection region is highly congested

with tensile and compressive reinforcement from the column and slab. This would be worse in

the presence of post-tensioning cables. Shear reinforcement such as shearheads, which are

bulky, are not favourable in this region. Moreover, from the economical perspective, shear

reinforcement such as closed ties are time consuming and labour intensive to install in position.

Recently, more efficient shear reinforcement such as shear studs and stud rails were developed

and became very popular and common due to their easy installation and practicality. The latter

types of shear reinforcement are recognised by most European and North American standards.

4

In Australia, the Australian Standard for Concrete Structures AS 3600-2009, is still behind

many other standards in punching shear provisions. In the case of symmetric punching, the

provision is based on research work in the early 1960s. Its formula does not account for some

important parameters such as the size effect or tensile reinforcement ratio. Moreover, AS 3600-

2009 only recognises shearhead reinforcement as the allowable shear reinforcement to increase

the symmetric punching shear strength of concrete slabs, but provides no guidelines on how to

design this type of shear reinforcement. In Clause 9.2.2 of AS 3600-2009, there is a note which

states for shear reinforcement other than shearheads, strength may be determined by tests. This

has left users of AS 3600-2009 with uneconomical and architecturally unpleasant solutions such

as increasing the thickness of the whole slab or locally increasing the thickness of the slab-

column connection by introducing a drop-panel, or column capital. The European and North

American designers, however, have the option of using practical types of shear reinforcement

such as stud-rails or shear studs.

In most design standards, formulae for predicting punching shear strength of slabs with

unbalanced moment are extensions of the formulae of symmetric punching shear. Therefore,

any deficiency in the calculation of symmetric punching shear strength of slabs would be

reflected in the provisions of those for the punching shear with unbalanced moment.

In the case of punching shear with unbalanced moment, AS 3600-2009 has a totally different

approach compared to the other international standards. The provision is based on work by

Rangan and Hall (1983), and assumes that a significant amount of the unbalanced moment from

the slab is transferred by torsion to the side faces of the column. On the basis of this

assumption, AS 3600-2009 only recognises closed ties as shear reinforcement to enhance the

punching shear strength of slab-column connection in the presence of unbalanced moment. The

problem with closed ties is that they are labour intensive and cumbersome to install on site, as

compared to shear studs. Many other international standards allow designers to use more

convenient shear reinforcement such as shear studs, or single leg ties. This is based on a

considerable volume of research conducted in the last three decades (Polak, El-Salakawy &

Hammill 2005).

With the significant increase in the use of post-tensioning in the construction of concrete floors

in Australia, it has become crucial to better understand the effect of prestressing on the punching

shear strength of slabs. Currently, AS 3600-2009 recognises the contribution of post-tensioning

in increasing the punching shear strength of floors by adding thirty percent of the average pre-

compression stress in the floor to the concrete component of punching shear equation. Issues

such as the effect of the post-tensioning tendon profile in the vicinity of the column on the

punching shear resistance of concrete floors, and effects of upward force resultant from

5

inclination of tendons are neglected by AS 3600-2009. More recently, some promising

mechanical methods such as decompression methods have become available in the literature to

calculate the strength of prestressed flat plates with better accuracy as compared to the current

standards’ approaches.

Considering the gap between the Australian Standard and other international standards, and the

difficulties facing AS 3600-2009 users, there is an urgent need to review and improve the

provisions of the Australian Standard for punching shear.

1.2 Aim and Objectives

The main aim of this research project is to propose a method to calculate the concentric

punching shear strength of flat plates with more accuracy as compared to the provisions of AS

3600-2009. This method should be based on a mechanical model, valid for a wide range of flat

plates and simple to use. The following objectives are covered in this project:

1. Review available mechanical methods and semi-empirical methods for concentric

punching shear strength of flat plates.

2. Propose a formula to calculate the punching shear strength of reinforced concrete flat

plates for the case of concentric punching.

3. Extend the proposed method for the case of prestressed slabs.

4. Review guidelines for detailing of shear reinforcements, and provide a method to

calculate the ultimate strength of flat plates strengthened with shear reinforcements such

as shear studs, stud rails and stirrups.

1.3 Thesis Organisation

Chapter One provides a brief background to the punching shear phenomenon and the problem

with the current Australian Standard, followed by objectives and the thesis layout.

Chapter Two is a review of the literature. Some of the influential and illustrative methods are

discussed. Different approaches by internationally recognised standards are presented.

In Chapter Three, the basis of a model developed previously by other researchers, was used to

propose a formula to calculate the punching shear strength of flat plates. Further, the accuracy

of some of the internationally recognised standards in predicting punching shear strength of flat

plates was evaluated against reported experimental results in the literature.

In Chapter Four, the proposed formula for non prestressed flat plates extended for the case of

prestressed flat plates, and provisions of various standards were assessed by some of the

available test results in the literature.

6

In Chapter Five, guidelines are provided for detailing and strength considerations of flat plates

with shear reinforcements.

Chapter Six presents the conclusions from the current research project.

7

Chapter Two

2 LITERATURE REVIEW

2.1 Introduction

In the last five decades a significant amount of research has been conducted on the topic of

punching shear in concrete floors. Many analytical and empirical methods have been proposed

based on the observations and results gathered during experimental tests. It is not possible to

cover all of the previous work on punching shear of concrete slabs herein. Therefore, in this

chapter, some of the methods which may be considered as main contributors to the current state

of knowledge on the punching shear phenomenon are presented. Other aspects of this type of

failure such as punching shear in prestressed slabs, and slabs strengthened by shear

reinforcement are reviewed briefly. Finally, the provisions of the current Australian Standard

for Concrete Structures (AS 3600 2009) and some of the internationally recognised standards

such as American code (ACI 318-05 2005), New Zealand standard (New Zealand Standard NZS

3101:Part 1 2006), European code (Eurocode 2 2004), British standard (BS 8110-97 1997), and

German standard (DIN 1045-1 2001) are presented.

2.2 Reported Observations from Concentric Punching Shear Failure of

Test Specimens

Punching shear failures, as explained in the literature, are local failures around the column or

the stub of test specimens. As reported in Kinnunen and Nylander (1960), the tangential and

radial strains of slab test specimens were measured in their test series, and it was observed that

the strains in the tangential direction are higher than the strains in the radial direction which

resulted in the formation of radial cracks prior to tangential or circumferential cracks. These

two types of cracks are shown in Figure 2.1 for clarity.

8

Figure 2.1 Tangential and radial cracks observed in typical punching shear test specimen (Sherif

1996)

As stated in (Regan 1981), generally the inclined radial cracks initiate at 1/2 to 2/3 of the

ultimate load which causes the punching failure. After the formation of inclined radial cracks,

the condition of the slab is entirely stable and it can undergo loading and reloading. As the load

increases some tangential cracks appear around the column. One of the tangential cracks will

eventually become the cone shaped surface of failure (Sherif 1996).

Figure 2.2 shows the applied load versus the deflection of test specimens reported in (Menétrey

1998). It illustrates the difference between the ductility of slabs that failed by punching

phenomenon and slabs that failed in flexure. From the sudden drop in the load-deflection graph,

it can be depicted that punching failure is a sudden failure with little warning, whereas the

specimens that failed by flexure behaved in a ductile manner before their failure.

Figure 2.2 Comparison of deflection-load graph for slab test specimens failed by punching shear to

slab test specimens failed in flexure (Menétrey 1998)

Flexural failure

Punching failure

9

2.3 Mechanical Models for Punching Shear – Balanced Condition

2.3.1 Kinnuen and Nylander Approach

Based on observations of 61 circular slab specimens, Kinnuen and Nylander (1960) proposed a

mechanical model for the punching shear of slabs with circular -ring shaped- reinforcement.

They presented a structural system for the slab-column connection as shown in Figure 2.3. In

their model, the slab is divided into a compressed conical shell and rigid elements. The

compressed conical shell part is surrounded by the shear crack, and the rigid elements are

confined at the front by a tangential crack and at the sides by the radial cracks as seen in Figure

2.3(b). The rigid elements are supported by conical compressive struts around the column as

shown in Figure 2.3(c). Under load action and after the formation of tangential and radial

cracks, the rigid segments of the slab turn around their centre of rotation at the root of the shear

crack. The failure is assumed to occur when the compressive stress in the strut and the

tangential strains at the point located under the centre of rotation reach their critical values.

Assuming that the two failure criteria coincide, the depth of the neutral axis was calculated by

iteration (Sherif 1996). The critical values for the failure criteria were calibrated based on

results of experimental tests reported by (Elstner & Hognestand 1956) and (Kinnunen &

Nylander 1960). These values were different to the well known values of strain and stress for

concrete at the ultimate stage. A major drawback of this method is the complexity and iterative

procedure of calculating punching shear strength as compared to the other methods (Megally

1998).

Figure 2.3 Mechanical model of Kinnunen and Nylander as shown in fib (2001)

Compressed conical shell

Rigid element Shear crack

10

Kinnunen (1963) further developed the previous model to include slabs with orthogonal

reinforcement. Three equations were derived from the equilibrium condition for the rigid

sector. Equation 2-1 was the result of moment equilibrium. Equation 2-2 was gained from the

equilibrium of forces in radial direction, and Equation 2-3 was derived from the equilibrium of

forces in the vertical direction.

� � �� sin � �� cos � �� 2� �� 0 (2-1)

� cos � � 2� �� 2� � � 2� �� !" � 0 (2-2)

� �1 � �� sin � (2-3)

Where P is the force causing failure, c is the diameter of test specimen, h is the effective depth

of slab, T is the compressive force in the strut around the column, κR1, R2 are the forces in

reinforcement crossing the shear crack in the tangential, and radial directions respectively, R4 is

the force resultant from the concrete compression zone as shown in Figure 2.3(b), ∆$ is the

angle of the rigid segment slice as shown in Figure 2.3(b), α is the angle between the

compressive strut and slab, y is the height of the compressive strut, λy is the distance of R4 to the

bottom of slab, B is the diameter of the stub, z1 as shown in the Figure 2.3(c), and γ is equal to

(M+D)/P, in which M is the vertical resultant of the membrane force in the reinforcement, D is

the force from the dowel-effect of reinforcement crossing the crack.

This model involves an iterative procedure to predict the punching load. First a value for (y/h)

should be assumed. Having (y/h), α can be calculated from geometry, and substituted in

Equation 2-1, 2-2, and 2-3. Punching load is the convergent value of P from above equations.

2.3.1.1 Shehata and Regan’s model

Shehata and Regan (1989) proposed a mechanical model in which the slab is divided into rigid

segments, surrounded by radial cracks on the sides and tangential cracks at the front and the

back, as shown in Figure 2.4 (b). The reinforcement crossing the circumferential crack was

assumed to reach yield prior to the failure of slab. After yield, the rigid segments are detached

from the central conical part of the slab and turn around the centre of rotation (CR), shown in

the Figure 2.4(a). Three criteria are defined for the failure:

• Inclination of the compressive force reaching 20° from the plane of the slab.

• Radial compressive strains at the face of column reaching 0.0035.

• Tangential compressive strains at a distance equal to the depth of neutral axis from the

face of column reaching 0.0035.

11

To simplify the above approach, Shehata (1990) derived a simplified formula to calculate the

punching strength of concrete slabs as expressed in Equation 2-4.

%&' � 2 � (& ) *� +�, -.*10° �500/2�/3 (2-4)

Where ro, x, and d are shown in Figure 2.4, and nc=1.4(2d/r0)0.5 is the stress concentration factor

which takes into account the effect of the multi-axial stress condition on the concrete strength.

Shehata suggested a simplified formula to calculate the depth of the neutral axis -x- which will

be presented in detail in Chapter Three (Equation 3-8).

Figure 2.4 Punching shear failure model proposed by Shehata and Regan (Shehata 1990)

2.3.1.2 Broms’ model

Broms (1990) used a similar approach as Kinnuen and Nylander (1960) in which he assumed

that the punching failure occurs when the tangential strain, or the compressive stress in the

radial direction reaches its critical values. Unlike Kinnunen and Nylander (1960) who

calibrated the aforementioned critical values by using experimental results, Broms suggested

Rigid segment

12

limitations for the strains and stresses using generally recognised properties of concrete.

Another significant difference of Broms’ method as compared to Kinnuen and Nylander (1960)

is that two types of compression zones were considered, namely the tangential compression

zone and the radial compression zone.

The limitation for high tangential compression strain is expressed in Equation 2-5.

4�5' � 0.0008�150/�)5'�8.33�25/+�,�8.33 f’c in (MPa), and xpu in (mm) (2-5)

Where xpu (mm) is the depth of the compression zone in the tangential direction, εcpu is the

tangential strain in the outermost fibre of concrete at the edge of the column and αxpu is the

height of the equivalent rectangular stress block with the stress equal to fc’. The punching force

Vε for this criterion can be obtained by the use of classical bending theory assuming εcpu as the

critical strain in the concrete. This is the punching shear load calculated using equilibrium and

Bernoulli’s compatibility conditions.

The other criterion for punching shear failure is the radial compression failure. Broms (1990)

assumed the formation of an imaginary strut around the column to transfer the applied load to

the column as shown in Figure 2.5. Broms assumed the inclination of the shear crack as 30°,

the inclination of the concrete strut as 15° and the compressive strength of the strut as 1.1 fc’ to

account for the effect of the multi-axial state of stress on the strut. Equation 2-6 was proposed

by Broms to calculate the punching load for this criterion.

Figure 2.5 Radial compression stress failure proposed by Broms (1990) as shown in fib (2001)

%σ � ��9 � 2�/-.*30°� ��;<*15°/;<*30°� 1.1+�,�150/0.5��8.333;<*15° (2-6)

Where D is the diameter of column, y is the depth of the neutral axis in the radial direction. For

the case of slabs supported on square columns with column side dimension a, D is equal to

4a/p.

Vσ

13

Equation 2-7 is suggested by Broms to calculate the depth of the compression zone in the radial

direction.

� � =>*?@A1 � 2/=>*? � 1B2 (2-7)

Where n is the ratio of elastic modulus of steel to elastic modulus of concrete n=Es/Ec, ρ is the

ratio of tensile reinforcement, d is the effective depth of section, and

kρ=(0.5D+d/tan30°)/(0.5D+y/tan30

°).

The lesser of punching shear capacities obtained from the above criteria (Vε and Vσ) is the

ultimate capacity of the slab.

Recently, Broms (2009) improved the latter model by modifying the critical tangential strain

(Equation 2.5) to the following expression.

4�5' � 0.001�150/)5'�/3�25/+�,�8. f’c in (MPa), and xpu in (mm) (2-8)

He also proposed the depth of compression zone to be calculated in the elastic condition as

shown in Equation 2-9.

)5' � *?@A1 � 2/*? � 1B2 (2-9)

Where n is the ratio of modulus of elasticity of steel to Ec10 the secant modulus elasticity of

concrete for the strain of 0.001.

Broms (2005) suggested Equation 2-10 to calculate Ec10.

C�8 � �1 � 0.6�1 � +�′/150��C�8 f’c in (MPa) (2-10)

Where Ec0 is the modulus of elasticity for concrete at zero strain which can be calculated by

Equation 2-11 as given in Model Code 90 (Model Code 90 1993).

C�8 � 21500�+�,/10�/3 f’c in (MPa) (2-11)

The punching shear strength based on the strain criterion, Vε, can be calculated from Equation 2-

12.

%E � FE GH� IJ�K/L�MN�L/K�O (2-12)

Where l is the diameter of the test specimen or the distance between points of contra-flexure in

the slab, D is the diameter of the column, and mε is the bending moment at the edge of slab-

column connection which can be calculated as following.

14

FE � ? CP 4P 2� =' �1 � )5'/32� (2-13)

In Equation 2-13, ku=(fsy/εsEs)0.2

<1.0, fsy is the yield stress of the flexural reinforcement, and εs is

the strain in the tensile reinforcement assuming elastic condition and can be calculated by

Equation 2-14.

4P � 4�5'�2 � )5'�/)5' (2-14)

Where εcpu can be calculated from Equation 2-8.

Broms also suggested an upper bound for the strength of the slab by considering the flexural

strength of the slab. This can be calculated from yield line theory as given in Equation 2-15.

%Q� � FQ �HN�L/K� (2-15)

Where my=ρ fsy d2 (1-0.59 ρ fsy/fc

’)

In the case of slabs with square columns, the column was replaced by a fictitious circular

column which gives a similar bending moment at the edge of slab-column connection D=3ap/8,

where a is the side dimension of the column.

A different εcpu was used in Broms (2005) and Broms (2009) compared to εcpu in Broms (1991) -

Equation 2-8 and 2-5 - which resulted in Vσ being less likely to govern the design. Broms

(2005) states that Vσ governing only when the thickness of the slab is large in relation to the

column dimension. This is less likely in design of flat slabs and more of the case for design of

footings.

Broms (2009) adopted the lesser of Vε from Equation 2-12 and Vy2 as the punching shear

strength of the slab.

2.3.1.3 Strut-and-tie model by Marzouk and Tiller

Tiller (1995) proposed a method in which only the radial compressive stress failure mechanism

is taken into account. The hypothetical critical concrete strut is shown in Figure 2.6. Tiller

suggested Equation 2-16 to calculate the ultimate punching shear strength of slabs.

%σ � � �9 � �QRSTU VQPWT�XOPWTU Y +��ZS[;< * �U� \ ;<�]+.�-^( (2-16)

Tiller simplified the depth of neutral axis to y=ρfsy/0.6f’c and used the formula given in

Canadian Standard CSA A23.3 for the strength of the concrete strut as expressed in Equation 2-

17. As a slab size factor, Tiller used (500/h)0.35 for concrete strength less than 40MPa and

15

(250/h)0.35 for concrete strength more than 40MPa. The angle between the crack and the plane of

the slab was assumed to be equal to 30°.

+��ZS[ � +�,/�0.8 � 1704� ` 0.85+�, (2-17)

Where fc2max is the compressive strength of the concrete strut, ε1 is the principal tensile strain in

the cracked concrete. Tiller (1995) did not specify how to calculate ε1.

Figure 2.6 Radial compression stress failure mechanism as shown in Marzouk, Rizk and Tiller

(2010)

Marzouk, Rizk and Tiller (2010) improved the latter method by using Equation 2-18 to calculate

the depth of the compression zone.

� � 0.67�*?a�8.b�35/+�,�8.b2 f’c in (MPa) (2-18)

Where n is the ratio of modulus of elasticity of steel to modulus of elasticity of concrete, ρe is

the ratio of reinforcement for a basic yield strength (500MPa) and can be calculated as ρe=

ρ(fsy/500)§0.02 where ρ is the ratio of reinforcement and fsy is the yield strength of the tensile

reinforcement. Also they suggested a range for the angle of the critical crack (θ) depending on

the thickness of the slab i.e. 25°-35° for slabs less than 250mm thick, 35°-45° for slabs 250mm-

500mm thick and 45°-60° for slabs thicker than 500mm.

2.3.2 Truss Model by Alexander and Simmonds

Alexander and Simmonds (1987) approached the punching shear phenomenon by proposing

formation of a three dimensional truss around the column. The components of the truss are

shown in Figure 2.7. The truss is broken down into the flexural tensile reinforcement acting as

ties, and the compression concrete zones acting as struts. As shown in Figure 2.7, two types of

struts are assumed, shear struts and anchoring struts. The shear struts are assumed to have an

16

angle of α to the plane of slab, and transfer shear forces from the slab to the column. The

anchoring struts are parallel to the plane of the slab and provide anchorage for the adjacent

reinforcement outside the column to transfer bending moment to the column as shown in Figure

2.7. The tensile reinforcements passing through the column plus a fraction of the tensile

reinforcement passing through a distance less than the effective depth of the slab from the side

faces of the column is considered to act in transferring shear forces to the column. It was

assumed that the reinforcement passing through the face of the column is fully effective (ζ=1)

and the reinforcement bar at the distance d from the face of column is not effective (ζ=0). The

effectiveness (ζ) of any reinforcement in between these two points is determined by linear

interpolation.

Figure 2.7 Truss model proposed by Alexander and Simmonds (1987) as shown in Megally (1998)

α the angle between the shear struts and the plane of slab was calibrated using the experimental

results available in the literature. The following expressions were proposed to calculate α.

tan � � 1 � ]N�.�be (2-19)

Where, f � �gahh2,A+�,�/�ijSk+PQ��/2�8.�b� f’c in (MPa), and d in (mm),

Seff= effective tributary width of the reinforcing bar which is equal to the spacing of

reinforcement and less than 6d ’,

17

d ‘ = cover of tensile reinforcing bar,

d= effective depth of slab,

c= dimension of column face,

Abar= area of single reinforcing bar,

fc'=compressive cylinder strength of concrete,

fsy=yield strength of tensile reinforcement steel.

Having α, the punching strength of the slab for concentric load can be calculated from Equation

2-20.

%'& � ∑ ζijSk +PQ -.*� (2-20)

Where ζ is the effectiveness of the tensile reinforcement as explained earlier.

2.3.3 Bond Model by Alexander and Simmonds

Alexander and Simmonds modified and developed their “Truss model” to the so called “Bond

model”. By monitoring the strains of the test specimens reported in (Alexander 1990),

Alexander and Simmonds (1992) suggested the shear struts are arch shaped as shown in Figure

2.8, and the geometry of the shear arch cannot be obtained by the amount of tensile

reinforcement. This is in contrast with the assumptions of the shear struts in the Truss model.

Figure 2.8 Curved compression strut (Alexander & Simmonds 1992)

18

Instead, they proposed a Bond model in which the slab is composed of four radial strips and

four quadrant slabs as shown in Figure 2.9. The assumptions of this model are:

• All the loads are transferred to the column through the radial strips, and the quadrants

components of the slab transfer the loads to the side faces of the radial strips.

• The total load on each strip is 2w and w is the ultimate internal shear that can be

resisted by the slab on each side face of the strip.

• The strength of the radial strips is limited by the flexural strength of the strip Ms.

Ms is the sum of the flexural strengths of the slab at the ends of the strip- Mneg and Mpos.

According to (Alexander 1999) Ms can be approximated by Equation (2.21).

mP � mTan � m5&P o 0.9.2��?Tan � ?5&P�+PQ (2.21)

Where a is the width of the strip -side dimension of column-, ρneg is the ratio of top

reinforcement at the column end of the strip and ρpos is the ratio of bottom reinforcement at

the shear zero end of the strip.

Figure 2.9 Plan view of slab and the components of Bond model proposed by Alexander and

Simmonds (1992)

A free body diagram of the radial strip is shown in Figure 2.10. If l is the length of applied

uniform distributed load then from equilibrium, Ms=wl2 and the maximum load Ps carried by a

strip is given by Equation 2-22.

19

�P � 2qr � 2AmPq (2-22)

Where w is the one-way shear strength of concrete from ACI 318 as expressed in Equation 2-23.

q � 0.166A+�, 2 f’c in (MPa),d in (mm), and w in N/mm (2-23)

Finally, the punching shear strength of the slab can be gained from the following Equation 2-24.

%'& � 4�P � 8tmP�0.166A+�, 2� (2-24)

Figure 2.10 Free body diagram of radial strip (Alexander & Simmonds 1992)

2.3.4 Models Based on the Failure of Concrete in Tension

Some researchers explained the punching shear phenomenon by the failure of concrete ties in

the vicinity of the column. Models by Georgopolous and Menetrey are among the models

which consider the tensile strength of concrete ties to govern the punching shear capacity of the

slab as cited in fib (2001).

2.3.4.1 Georgopoulos approach

The review of this method is based on fib (2001) as the original paper is not in English.

Georgopoulos assumed the transfer of shear from the slab to the column relies on the principal

tensile stresses in the concrete and the compression in the concrete strut around the column. He

suggested that 75 percent of the shear force transfers through the tensile strength of concrete and

the remaining 25 percent through the compressive strut. Details of the proposed model are


20

Figure 2.11 Punching shear model by Georgopoulos as shown in fib (2001)

The depth of compression zone was assumed to be 0.2 of the effective depth of the slab. The

stress distribution in the expected punching failure surface was assumed to be a polynomial of

third order as shown in Figure 2.12.

Figure 2.12 Distribution of concrete tensile stresses in Georgopoulos as shown in fib (2001)

As shown in Figure 2.11, Zb is the resultant tensile force in the cracked section. Georgopoulos

estimated Zb by integration of the stresses along the surface of failure. Consequently, he

proposed the following equation to calculate the punching strength of slabs.

%'& � uj cos � /0.75 � [email protected]�+�'ja��/3B2� cot � ��/2 � 0.2 � 0.35 cot α� (2-25)

Where α is the inclination of the failure surface, λ is the ratio of the diameter of the column to

the effective depth of the slab, fcube is the compressive strength of concrete of a cube test

specimen in MPa.

Georgopoulos suggested the following equation to predict the inclination of the critical crack

causing punching failure.

tan � � 0.56+�'ja/?+PQ � 0.3 (2-26)

Where ρ is the tensile reinforcement ratio.

21

2.3.4.2 Model by Menétrey

Menétrey (1996, 2002) assumed a strut-and-tie pattern which transfers the load from its point of

application to the column. He considered the failure to occur when the strength of the tie,

adjacent to the column, reaches the failure limit. The contributors to tensile strength of the tie

are shown in Figure 2.13.

Figure 2.13 Schematic view of components of proposed method by Menetrey (2002)

In this method, Menetrey included the tensile capacity of the concrete, the effect of dowel action

of the flexural reinforcement, the strength of the shear reinforcement and the vertical component

of the prestressing force. Equation 2-27 is suggested to calculate the ultimate punching shear

strength of a given slab.

%'& � w�R � wx&y � wPy � w5 (2-27)

Where, Fct is the vertical component of the concrete tensile strength of the hypothetical tie

shown in Figure 2.13, Fdow is the dowel-effect contribution from the flexural reinforcement

crossing the punching crack, Fsw is the contribution from shear reinforcement if there is any,

and Fp is the contribution of vertical component of forces of prestressing tendons crossing the

punching crack.

22

Fct can be calculated by Equation 2-28,

w�R � ��( � (��;+R�/3z{| (2-28)

Where rs is the radius of the column, r1=rs+d/10tan30°, r2=rs+d/tan30

°, s is the length of the

punching shear crack and is equal to √ ((r2-r1)2+(0.9d)

2), ft is the uniaxial tensile strength of the

concrete, ξ is a factor to take into account the influence of the flexural reinforcement ratio -ρ-

and can be calculated by the following expression.

ξ=min(0.87, -0.1ρ2+0.46ρ+0.35)

η and µ take into account the size effect on the tensile strength of the concrete and are

expressed as followings.

η=min(0.625, 0.1(h/rs)2+0.5(h/rs)+1.25)

µ=1.6(1+d/da)-0.5

Where h is the thickness of slab, and da is the maximum aggregate size in concrete.

The contribution of the dowel-effect Fdow is the summation of dowel-effect of each reinforcing

bar crossing the failure surface and can be calculated by the following expression.

wx&y � 1/2 ∑ }P�t+�+PQ�1 � ζ�� ;<*30° (2-29)

Where Øs is the diameter of the flexural reinforcement crossing the punching shear critical

crack, fc is the uniaxial compressive strength of the concrete, fsy is the yield stress of the

reinforcing bars, ζ=σs/fsy, and σs is the stress in the tensile reinforcement at punching which can

be quantified by the following equation.

~P � %'&/�-.* 30° ∑ iP� (2-30)

Where ∑ iP is the area of reinforcing bars crossing the punching shear failure surface.

It should be noted for calculating σs that the punching strength of the slab is needed, so the

calculation of punching shear strength is an iterative procedure in this method.

If adequate anchorage is provided, Fsw can be calculated by Equation 2-31.

wPy � iPy+Py sin �Py (2-31)

Where Asw is the area of the shear reinforcement intersecting with the punching shear crack, fsw

is the yield strength of the shear reinforcement steel, and βsw is the angle between the shear

reinforcement and the plane of the slab.

23

The contribution of prestressing Fp is given as following expression.

w5 � i5~5 sin �5 (2-32)

Where Ap is the area of prestressing steel crossing the failure surface, σp is the stress in the

tendons, and βp is the inclination of the tendon with the plane of slab as shown in Figure 2.13.

2.3.4.3 Theodorakopoulos and Swamy approach

Theodorakopoulos and Swamy (2002) proposed a method for representing the punching shear

phenomenon by considering a criterion for the tensile strength of the compression zone in the

vicinity of the column. The punching shear strength was related to the tensile strength of the

compressed concrete around the column. It was assumed that there are two types of neutral

axes adjacent to the column, namely flexural and shear. The location of the flexural neutral axis

was calculated assuming the ultimate stage in flexure and the location of the shear neutral axis

was assumed to be 0.25 of the effective depth of the slab. Equation 2-33 was suggested to

calculate the mean of the depth of the neutral axes. This will be explained further in Chapter

Three.

2T � 2�h�P/��P � �h� (2-33)

In Equation 2-33, Xf is the depth of the flexural neutral axis and Xs is the depth of the shear

neutral axis.

As show in Figure 2.14, the ultimate punching strength of slab -Vu- consists of the contribution

of the tensile strength of the compression zone, Vc , and the contribution of the dowel-effect of

flexural reinforcement.

Figure 2.14 Schematic view of model by Theodorakopoulos and Swamy (2002)

dn

Vu

24

For simplicity, Theodorakopoulos and Swamy incorporated a larger control perimeter as

compared to the perimeter of the compression zone around the column to account for the dowel-

effect action. A control perimeter, similar to BS 8110-97 (1997), was adopted in this method as

expressed in Equation 2-34.

�5 � 4. � 122 (2-34)

Where a is the side dimension of the column and d is the effective depth of the slab.

The ultimate punching shear strength of the slab was expressed as the following equation.

%'& � 2T �5 cot � +�R (2-35)

Where fct is the splitting strength of concrete, equal to 0.27(fcube)2/3

, and q was taken as 30°, dn is

calculated by Equation 2-33, and bp is calculated by Equation 2-34.

2.3.5 Plasticity Approach

Braestrup et al. (1976) proposed an upper bound model on the basis of the theory of plasticity

for punching shear phenomenon. Geometrical parameters of the model are shown in Figure

2.15. In this model, it was assumed that the vertical load V was applied to the slab by the

column with the diameter of d. The maximum diameter of punching shear failure surface is d1.

The punching failure surface was assumed to shape as curve A-B-E, shown in Figure 2.15. The

curve of the failure surface is expressed as r=r(x), and the angle of displacement vector is

expressed as α=α(x).

The work done by the punching force (Wv) should be equal to the dissipated energy (We) at the

punching shear crack surface. Equation 2-36 was suggested to express the dissipated energy

and Equation 2-37 was suggested to express the work done by the applied load.

Figure 2.15 Plasticity model proposed by Braestrup et al. (1976)

25

�a � � 0.5 � +�, �� | ;<* � �2� ( x[�&P ��8 (2-36)

�� %� (2-37)

Where δ is the displacement, λ=1-fct/fc’(k-1), µ=1-fct/fc

’(k+1), k=(1+sinφ)/(1-sinφ), fct is the

tensile strength of concrete, and φ is the friction angle of concrete as shown in Figure 2.15.

The above equations will give an upper bound punching shear strength of the slab. By

optimisation, Braestrup et al. (1976) suggested the failure surface consists of a linear conical

part (A-B) and a curved part (B-E). Thus the ultimate punching strength is the sum of P1 which

takes into account the straight line part (A-B) as expressed in Equation 2-39 and P2 which takes

into account the curved part as expressed in Equation 2-40.

%'& � � � �� (2-38)

� � � +�, �� x �� "M�� J "��N��J "��&PO " (2-39)

�� 0.5 � +�, �� 8� � � Vx�� t�x�� .�Y � | ��x�� .�� (2-40)

Where h is the thickness of the slab, h0 is the depth of inclined straight line, a=d/2+h0 tanφ,

b=c tanφ, and c=√(a2-b

2).

One of the common criticisms of this method is that it ignores the effect of tensile reinforcement

on the punching shear strength of slabs.

2.3.6 Flexural Approach

A considerable number of slab test specimens, reported in the literature, have a failure load not

significantly different to their flexural capacity. As a result, some researchers such as Gesund

and Goli (1980), Gesund (1981), and Rankin and Long (1987) assumed the punching shear as a

secondary failure phenomenon and attempted to propose a method which relates the punching

shear strength of slabs to the flexural capacity of the slabs.

In this section, the flexural method proposed in Rankin and Long (1987), is reviewed. Rankin

and Long (1987) suggested that the flexural punching strength of a prototype test specimen can

be calculated from Equation 2-41.

%hKa[ � �=Q � �=Q � =j/(h�mj/mjSK�mj � =j/(h mjSK (2-41)

Where, ky1 is moment factor for overall yielding of tensile reinforcement, and for square slabs

supported on a square column is equal to 8(s/(a-c)-0.172) where a, c, s are shown in Figure

2.16.

26

kb is the ratio of the applied load to the internal bending moment at the column periphery which

is equal to (25/(ln(2.5a/c)1.5

).

rf is a factor to allow for the shape of column which is equal to 1.0 for circular columns and 1.15

for square column.

Mb is the bending moment resistance, and can be calculated by ρfsyd2(1-0.59(ρfsy/fc

’)).

Mbal is the balanced moment of resistance which was suggested to be calculated by 0.333fc’d

2.

Figure 2.16 Failure pattern and parameters of the proposed method by Rankin and Long (1987)

Rankin and Long (1987) also specified a criterion for failure caused by “internal diagonal

tension cracking”. They suggested Equation 2-42 to calculate the latter strength of slabs.

%P�aSk � 1.66A+�, �� 2�2 �100?�8.�b f’c in (MPa),Vshear in (N), c and d in (mm) (2-42)

The lesser of Vflex and Vshear is the punching shear strength of the slab.

2.3.7 Critical Shear Crack Theory

Muttoni (2008) presented a different failure criterion for punching shear based on the opening of

a critical shear crack in the vicinity of the column. According to Muttoni and Schwarts (1991),

the width of the critical shear crack (wc) is proportional to the product of the rotation of the slab

times the effective depth of slab (yd). Another relevant parameter in view of critical crack

theory is the roughness of the critical shear crack which is related to the size of the aggregates in

the concrete. With the mentioned assumptions and available experimental results, Equation 2-

43 was proposed to calculate the punching strength of concrete slabs.

S a

27

%'& � �82A+�, 8.�bM�b�yx�/�x��Mx�� f’c in (MPa),Vshear in (N),b0, d, dg, and dg0 in (mm)(2-43)

Where b0 is the control perimeter at the distance equal to d/2 from the face of column, dg0 is the

reference aggregate size and considered to be 16mm, and dg is the maximum aggregate size in

the concrete.

Rotation of slab (y) is related to the applied load V as given in Equation 2-44.

y � 1.5 k��x h�� .b (2-44)

Where rs is plastic radius around the column which can be taken as the distance between the

centre of column to the point of contraflexure, and Vflex can be calculated from yield-line theory.

To calculate the punching strength of a given slab, an iterative procedure is required.

Alternatively, the load-rotation curve can be drawn using Equation 2-44 and the failure criterion

can be drawn using Equation 2-43. The intersection of these curves determines the failure load

of the slab (Vuo). The latter procedure is shown in Figure 2.17.

Figure 2.17 Procedure to specify punching shear strength of slab according to Critical Shear Crack

Theory (Muttoni 2008)

2.4 Punching Shear of Prestressed Flat Plates

The present section reviews some of the theoretical approaches to include the effect of

prestressing forces in the calculation of punching shear strength of flat plates. According to

Regan and Braestrup (1985), the available models for punching of prestressed slabs can be

categorised to the following approaches.

Vuo

28

2.4.1 Principal Tensile Stress Approach

In this approach, the effect of prestressing was taken into account by approximation of principal

tensile stresses on the control perimeter, and consideration of the vertical component of the

tendon forces crossing the control perimeter. An example of this approach is Equation 2-45,

suggested by ACI-ASCE Committee 423 (1974), and adopted in ACI 318-05 (2005) code.

%'&/��2� � 0.29A+�, � 0.3 ~�5 � %5/��2� f’c in (MPa) (2-45)

Where u is the length of control perimeter at a distance of d/2 from the face of column, σcp is the

mean effective prestressing stress in the concrete, and Vp is the vertical component of

prestressing tendons crossing the control perimeter.

2.4.2 Equivalent Reinforcement Ratio Approach

In this approach, the effect of prestressing is considered by adding the equivalent reinforcement

ratio to the actual reinforcement ratio of the slab. The sum of the ordinary reinforcement and

the equivalent reinforcement is used in the formula, which predicts the punching strength of the

slab. There are various proposed methods to convert the prestressing stress to the equivalent

reinforcement ratio.

As cited in Sundquist (2005), FIP recommendations (1980) specifies the equivalent

reinforcement ratio by Equation 2-46.

?a� � ~�5/+PQ (2-46)

Another method for calculating equivalent reinforcement ratio proposed by Nylander,

Kinnunen, and Ingvarsson, which is cited in Regan and Braestrup (1985), is given in Equation

2-47.

?a� � ?5 +8.�/�+8.� � ~5a� (2-47)

Where ρp is the prestressing steel ratio, f0.2 is the 0.2% proof stress of the tendons, and σpe is the

effective prestress of the tendons.

Clearly, this approach is not suitable for methods which do not include the effect of the tensile

reinforcement on the punching shear strength of slabs.

29

2.4.3 Decompression Approach

Regan (1985) proposed a decompression method for the punching shear phenomenon. The state

of decompression occurs when compression stress, resulting from prestressing forces, is

cancelled out by the effect of transverse loading at a specific region (Silva, Regan & Melo

2005). In a decompression method for punching shear of slabs, it was assumed the punching

strength after the decompression stage is equal to the strength of a geometrically similar

concrete slab with the same number of reinforcement and no prestressing forces. Thus it is

possible to determine the punching resistance of prestressed slabs by adding the decompression

load to the punching strength of the ordinary concrete slab with the same amount of

reinforcement. The required bending moment for decompression of a given section can be

calculated from Equation 2-48.

m& � ~�5� ��/6 (2-48)

Where σcp* is the compressive stress in the outermost compressive fibre of the section due to

prestressing after losses.

According to Regan and Braestrup (1985) the decompression load can be taken as following.

%& � 2� m& for circular slabs

%& � 4�/r m& for rectangular slabs with breadth b, and main span l.

In Regan and Braestrup (1985), the punching shear strength of concrete slabs with no

prestressing was suggested to be calculated from the draft of British code CP 110 as following.

%'& � 0.27A500/2 A100?+�'ja¡ fcube in (MPa), d in (mm) (2-49)

Where ρ in Equation 2-49 is the sum of ordinary reinforcement area (Asr) and bonded

prestressing steel area (Asp).

? � �iPk � iP5�/�2 (2-50)

Where b is the breadth of the section and d is the equivalent effective depth of the steel and can

be calculated as expressed in Equation 2-51.

2 � �iPk +PQ 2k � iP5 +8.� 25�/�iPk +PQ � iP5 +8.�� (2-51)

Where f0.2 is the 0.2% proof stress of the prestressing steel, fsy is the yield strength of ordinary

reinforcement, dp is the effective depth of prestressing steel, and dr is the effective depth of

ordinary reinforcement.

30

2.5 Methods to Increase Punching Shear Strength of Concrete Slabs

In Polak, El-Salakawy and Hammill (2005), three common methods to increase the punching

shear strength of concrete slabs are categorised as followings:

• Expanding the area which transfers shear stresses from slab to column. In this method

designers normally increase the thickness of the slab in the vicinity of column by

introducing drop panels or column capitals. Other possibility is to increase the

dimensions of the column which results in a larger area resisting shear stresses.

• Using concrete with higher compressive strength which results in a higher punching

shear strength.

• Providing different types of shear reinforcement such as shearheads, stirrups, bent-up

bars, or shear studs in the area adjacent to the column.

In a study by Megally and Ghali (2000), four different methods were used to strengthen 150mm

thick slabs. Drop panel, column capital, stirrups (closed-ties) and shear stud rails (SSR). Then

a comparison was made between the performance and amount of increase in punching shear

capacity of slabs. The slabs were loaded to the point of failure, and the load-deflection curve

for each slab was plotted as shown in Figure 2.18. Drop panel and column capital resulted in an

increase of the punching shear strength of the slab but not the ductility of the slab. As shown,

shear studs increased both the strength and ductility of the slab. Further, it was observed in this

case that stirrups only slightly increased the punching shear strength of the test specimen due to

lack of proper anchorage (Megally & Ghali 2000).

Figure 2.18 Load-deflection curves of slabs strengthened by different methods (Megally & Ghali

2000)

31

Although all the aforementioned methods increased the punching shear strength of the tested

slabs, the issue of ductility, which is a desirable behaviour of structures in seismic regions, was

not improved by most of the provided strengthening techniques except for the slab strengthened

with shear studs. Other important considerations to decide the best strengthening method can be

economy, and practicality of the method. Designers prefer the use of shear reinforcement to

increase the punching strength of concrete slabs due to its advantages over the other methods.

In the 70s and 80s a significant amount of research was conducted on the performance of slabs

with shear reinforcement and consequently design provisions were introduced into design

codes.

2.6 Shear Reinforcement for Flat Plates

As mentioned, different types of shear reinforcement were proposed by structural engineers to

increase strength and ductility of concrete slabs. The role of shear reinforcement in the slab is

mainly to arrest the opening of the critical shear crack, increase the compression zone and

aggregate interlock which result in increase of shear strength.

In design, the radial spacing and placement of shear reinforcement is very important, and

designers should detail the position of shear reinforcement in a way that they intersect with the

inclined shear cracks. In addition, desirable types of shear reinforcement should have a good

tensile capacity, adequate ductility and enough anchorage (Polak, El-Salakawy & Hammill

2005). Providing that the shear reinforcements are placed and designed properly, it can increase

the punching shear and rotation capacity of the slab significantly. Preferably, punching shear

strength of slabs should be increased to the extent that the flexural failure occurs prior to the

punching shear failure. Generally, there are two categories of shear reinforcement for punching

shear, namely shear reinforcement for construction of new slabs and shear reinforcement for

retrofit of existing slabs.

2.6.1 Shear Reinforcement for Construction of New Slabs

Shear reinforcement for a new slab can be classified as follow (Polak, El-Salakawy & Hammill

2005).

• Shearheads, made of different types of structural steel sections as shown in Figure 2.19.

• Stirrups, single or double leg bar, bent bars, and closed-ties. This type of shear

reinforcement is made from the normal reinforcing bars as shown in Figure 2.20.

• Stud rails, shear studs, and shear bolts which are called headed shear reinforcements as


32

• Other new shear reinforcements such as shear bands, and UFO as shown in Figure 2.22

and Figure 2.23.

Figure 2.19 shows two types of fabricated shearheads which are made of channel or I

sections welded in a shape which can be fitted orthogonally at the slab-column connection.

The shear head is one of the earliest types of shear reinforcement which was used to

increase the punching shear strength of slabs. It acts as a steel frame which is hidden inside

the concrete slab. Shearheads increases ductility, shear strength and flexural strength of the

connection.

Figure 2.19 Shearhead reinforcement (Corley & Hawkins 1968)

There are several disadvantages with this type of shear reinforcement which makes it very

undesirable in industry such as the labourer intensive fabrication procedure, bulky dimensions

and interference with the longitudinal reinforcement of the slab.

Closed-ties and stirrups are common in beam sections and they are proven to increase the

punching shear capacity of slabs providing that the vertical legs of stirrups have a good

anchorage. These types of shear reinforcement are shown in Figure 2.20. As shown, the shear

reinforcement should engage the flexural bars at top and bottom to achieve a proper anchorage

(Polak, El-Salakawy & Hammill 2005). In some experimental tests, it was observed that some

of closed-ties did not reach their full yield capacity due to slip and lack of anchorage. Slabs

with smaller thicknesses are more prone to this phenomenon (Polak, El-Salakawy & Hammill

2005).

Bent bars are normal longitudinal bars which are bent and placed to intersect with the critical

shear crack as shown in Figure 2.20(a). The performance of this type of shear reinforcement

relies on its horizontal anchorages, so the horizontal part of bent bars should be long enough to

resist the pull out effect for adequate anchorage.

33

(a) (b)

(c) (d)

Figure 2.20 (a) Bent bar, (b) Single-leg stirrup , (c) Multiple-leg stirrup (d) Closed-stirrup or

Closed-tie (ACI 318-05 2005 ; Broms 2007)

The headed studs were presented in Dilger and Ghali (1981) for the first time. Since it is a very

convenient and practical type of shear reinforcement, extensive research has been conducted on

the performance of slabs strengthened with headed shear studs. In this type of shear

reinforcement, the problem of anchorage has been solved by providing large flat heads at the

both ends with the area of 10 times the stem cross-sectional area. This shear reinforcement is

available in the form of shear stud rails (SSR) in the market as shown in Figure 2.21. SSR are

easy to install, and adequate anchorage is achievable in relatively thin slabs. Most of the tests

on slabs strengthened with headed shear studs, show a ductile and satisfactory performance

(Polak, El-Salakawy & Hammill 2005), and consequently, this type of reinforcement has been

adopted by most of internationally recognised standards as an effective shear reinforcement for

slabs.

Figure 2.21 Headed shear studs (Bu 2008)

34

In recent years, other types of shear reinforcement for punching shear have been made available

in the marketplace such as Shearbands, UFOs, and lattice.

Shearbands were tested in the University of Sheffield and reported in Pilakoutas and Li (2003).

These are high ductile thin steel strips with punched holes as shown in Figure 2.22(a). The

holes are provided to increase the anchorage of strips as experimentally proven. These strips are

easily bent and shaped to place in a way to cross the shear cracks as shown in Figure 2.22(b). A

significant improvement in the ductility and strength of slabs was observed in the test specimens

reinforced with this type of shear reinforcement (Pilakoutas & Li 2003).

(a)

(b)

Figure 2.22 (a) Plan view of a shearband (b) Shearbands placed in slab (Pilakoutas & Li 2003)

UFOs are steel plates which are shaped like a cone and placed at the slab-column connection to

intersect with the critical shear crack. There are some perforated holes to allow for the

continuation of column reinforcements. This shear reinforcement is shown in Figure 2.23.

Figure 2.23 UFO shear reinforcement (Alander 2004)

35

A lattice is made of top, bottom, and web bars which are welded and prefabricated in the factory

as shown in Figure 2.24. Lattice performance as a punching shear reinforcement was first

reported in Park et al. (2007). According to the experimental observations, the strength and

ductility of the test specimens reinforced with these were increased up to 1.4, and 9.2 times

respectively as compared to the specimen with no shear reinforcement (Park et al. 2007).

Another advantage of this system is that even after failure, due to truss action of lattice system,

it can avoid sudden failure of the slab.

Figure 2.24 Lattice shear reinforcement (Park et al. 2007)

2.6.2 Shear Reinforcement for Retrofit of Slabs

Punching shear strength of existing concrete slabs may need to be increased due to the corrosion

of rebars, change in the amount of imposed load, or errors in the structural design. There are

different methods to increase the punching capacity of an existing concrete slab such as

providing external shearheads around the column, using steel plates around the column, and

providing shear bolts in the vicinity of a column (Polak, El-Salakawy & Hammill 2005). Use of

external I sections, bonded with epoxy, to increase punching shear strength of slab-column

connection was reported satisfactory in terms of strength and ductility (Polak, El-Salakawy &

Hammill 2005), but from aesthetic point of view it is not desirable. Ebead and Marzouk (2002)

used steel plates fitted around the column which are bonded to the slab with epoxy and steel

bolts as shown in Figure 2.25. An increase in the strength and ductility of test specimens

strengthen with this technique was reported in Ebead and Marzouk (2002).

36

Figure 2.25 Test specimen strengthened by steel plates (Ebead & Marzouk 2002)

Strengthening technique with shear bolts were studied by Adetifa and Polak (2005), Bu and

Polak (2009), El-Salakawy, Polak and Soudki (2003). Shear bolts are normal strength steel with

a smooth stem, forged head, large washers and threaded end as shown in Figure 2.26 (a). A

concrete slab strengthened by shear bolts is shown in Figure 2.26 (b).

(a)

(b)

Figure 2.26 (a) Shear bolt, (b) concrete slab strengthened with shear bolts (Bu 2008)

37

2.7 Control Perimeter Approach and Building Code Provisions

Traditionally, structural members such as beams or columns are checked for shear strength by

the concept of nominal shear stress on the section area at a specific distance from the support or

the point where the load is applied. This approach was first proposed by Talbot (1913) for the

case of concrete slabs as cited in (Bu 2008), and Equation 2.52 was suggested to calculate the

applied shear stress on the critical section.

¢� � %/�4�. � 22�£2� (2.52)

Where vc is the applied stress on the concrete which should be less than the nominal shear

stress. a is side dimension of the square column, d is the effective depth of slab, and jd is the

lever arm between compression and tension force which approximated to 0.9d.

Moe (1961) gathered some of available results on the slab specimen tests at the time to propose

an empirical formula for the ultimate allowable stress on the face of column. He proposed the

following formula as cited in (fib 2001).

¢'& � %'&/��&K2 � �1.246�1 � 0.059./2��&K2A+�,�/�1 � 0.436��&K2A+�,/%hKa[� (2-53)

Where vuo is the ultimate shear stress on the concrete around the column face in (MPa), a is the

side dimension of the column in (mm), ucol is the column perimeter in (mm), and Vflex is the

flexural capacity of the slab which can be calculated by yield-line theory in (N).

Based on Moe’s work, ACI-ASCE Committee 326 (1962) proposed Equation 2-54 to calculate

the ultimate shear stress on the critical perimeter u, located at a distance of d/2 from the face of

column.

¢'& � %'&/�2 � 0.33A+�, f’c in (MPa) (2-54)

Since then Equation 2-54 has been used as the basis for many internationally recognised

standards such as ACI-318, CSA 24, NZS 3101, and AS 3600.

2.7.1 Australian Standard AS 3600-2009

According to Clause 9.2.3 of AS 3600 (2009), the ultimate punching shear strength of concrete

slabs Vuo can be calculated using Equation 2-55.

%'& � �2&Z�+�¤ � 0.3~�5� (2-55)

Where dom, is the effective depth of slab, u is the perimeter around the column at a distance

equal to the half of effective depth of slab from the face of column as shown in Figure 2.27, σcp

38

is the average intensity of effective prestress in the vicinity of support in MPa, and fcv is given in

Equation 2-56.

Figure 2.27 Critical perimeter around the column as shown in AS 3600-2009

+�¤ � 0.17�1 � 2/��A+�′ ` 0.34A+�′ f’c in (MPa) (2-56)

In Equation 2-56, βh is the ratio of larger to shorter column sides.

When applied shear force on the critical perimeter is higher than the computed capacity,

calculated by Equation 2-55, AS3600-2009 permits the use of shearheads by which the fcv can

be increased using Equation 2-57.

%'& � �2&Z�0.5A+�′ � 0.3~�5� ` 0.2�2&Z+�′ f’c in (MPa) (2-57)

Unlike other international standards, AS3600-2009 does not provide any guidelines to the

design and detailing of shearhead reinforcement.

According to AS 3600-2009, the design shear strength is calculated as following.

%x � ¥%'& (2-58)

Where f is called capacity factor, and for the case of shear strength should be taken equal to 0.7.

To ensure adequate shear strength of the slab, Clause 9.1.2 of AS 3600-2009 requires 25% of

the negative bending moment in the column strip and half of the middle strip to be resisted by

the reinforcement and prestressing tendons that cross over the column and the distance of 2d

from the faces of the column.

39

2.7.2 American Code ACI 318-05

ACI- 318-05 (2005) specifies similar control perimeter around the column as AS 3600-2009.

The ultimate strength of concrete slab is the lesser of following expressions.

%'& � F<* ¦ 0.083��2 � 4/�� A+�, �20.083��P2/� � 2� A+�, �2 0.33�A+�, �2 § f’c in (MPa) (2-59)

Where λ= is a factor to account for the density of concrete and is equal to 1.0 for normal

concrete and 0.8 for low density concrete.

βc= is the ratio of the larger column side to the shorter column side.

αs= is equal to 40, 30 and 20 for interior, edge and corner columns respectively.

fc'= is the compressive strength of concrete in MPa.

For the case of prestressed slabs, Equation 2-60 was adopted by ACI 318-05 to calculate the

ultimate punching shear strength.

%'& � @0.083�5A+�, � 0.3~�5B�2 � %5 (2-60)

Where βp= lesser of (αsd/u+1.5) and 3.5

σcp= is the average intensity of effective prestress on control perimeter in (MPa)

fc'= is the compressive strength of concrete in MPa and should not be taken greater than 35 MPa

Vp is the vertical component of prestressing forces on the critical perimeter.

The design strength is calculated similar to Equation 2-58, where f is considered to be equal to

0.75 according to ACI 318-05.

ACI 318-05 recognises several types of shear reinforcement for strengthening of concrete slabs

such as headed shear studs, single-leg stirrups, double-leg stirrups and closed-ties. If shear

reinforcement is provided, the design punching shear strength is calculated for two regions,

shear strength inside the shear reinforced zone, and shear strength outside the shear reinforced

zone. The arrangement of shear reinforcement is shown in Figure 2.28.

40

Figure 2.28 Shear reinforcement layout suggested by ACI 318-05 as shown in Kamara and Rabbat

(2005)

To calculate the design punching shear strength inside the shear reinforced zone Equation 2-61

is given.

%Px � ¥%� � ¥%P§¥%ZS[ (2-61)

Where, f= 0.75,

%� � 0.17A+�,�2 f’c in (MPa)

%P � iP¤+PQ¤2/;

Asv= is the section area of one row of shear reinforcement around the column

fsyv=is the yield strength of shear reinforcement less than 414 MPa

s= is the spacing of shear between rows of reinforcement as shown in Figure 2.28

%ZS[ � 0.5A+�,�2 f’c in (MPa)

To calculate punching shear strength outside the shear reinforcement zone, Equation 2-62 is

given.

%x&'R � ¥0.17A+�,�&'R2 f’c in (MPa) (2-62)

Where uout is the critical perimeter outside the shear reinforcement zone as shown with the

broken line in Figure 2.28.

The lesser of Equation 2-61 and 2-62 governs the design.

41

2.7.3 New Zealand Standard NZS 3101:2006

The formulae of NZS 3101:2006 for punching shear are the same as formulae of ACI 318-05

except for the slab size effect factor. According to NZS 3101:2006 the contribution of concrete

shear resistance should be reduced by the slab size factor which is given in Equation 2-63. This

factor is effective to reduce the ultimate punching shear strength of slabs thicker than 200mm.

z � 0.5 ` A200/2 ` 1.0 d in (mm) (2.63)

2.7.4 Canadian Standard CSA A23.3-04

The Canadian concrete structure standard (CSA A23.3-04 2004) specifies the critical perimeter

at distance of d/2 similar to ACI 318-05 and AS 3600-2009. The ultimate punching shear

strength is given in Equation 2-64.

%'& � F<* ¦ 0.19��1 � 2/�� A+�, �2��P2/� � 0.19� A+�, �2 0.38�A+�, �2 § (2-64)

Where λ= is a factor to account for density of concrete and is equal to 1.0 for normal concrete.

βc= is the ratio of larger to shorter column sides.

αs= is equal to 4, 3 and 2 for interior, edge and corner columns respectively.

fc'= is the compressive strength of concrete in MPa.

The ultimate design strength is given as follow.

%�x � ¥�%'& (2-65)

Where fc is the partial concrete safety factor and is equal to 0.65.

A notable difference between CSA 23.3-04 and ACI 318-05 is that Canadian Standard considers

a reduction factor for slabs with effective depth more than 300mm. The reduction factor is given

in Equation 2-66 and should be multiplied by Vcd.

z � 1300/�1000 � 2� ` 1.0 (2-66)

When prestressing forces exist, the design punching shear strength of prestressed slab -Vpd- is

calculated as expressed in Equation 2-67.

%5x � V�5�¥�A+�, t1 � ¥5~�5/�0.33�¥�A+�, � Y �2 � ¥5%5 (2-67)

Where βp= lesser of (αsd/u+0.15) and 0.33

42

σcp= is the average intensity of effective prestress on control perimeter in MPa

fc'= is the compressive strength of concrete in MPa and should not be taken greater than 35 MPa

λ= is a factor to account for density of concrete and is equal to 1.0 for normal concrete

fc= is the concrete partial safety factor equal to 0.65

fp= is the prestressing steel partial safety factor equal to 0.9.

CSA 23.3-04 allows the use of stirrups and headed shear studs for strengthening of concrete

slabs. The following equation calculates the punching shear resistance of slabs strengthened

with shear reinforcements inside the shear reinforced zone.

%Px � ¥�%� � ¥P%P ` ¥�% ZS[ (2-68)

fc= Concrete partial safety factor equal to 0.65

fs= Steel partial safety factor equal to 0.85

Where headed shear studs are provided:

%� � 0.28�A+�, �2 f’c in (MPa)

%P � iP¤+PQ¤2/; f’c in (MPa)

Asv= is the section area of one row of shear reinforcement around the column

fsyv=is the yield strength of shear reinforcement less than 414 MPa

%ZS[ � 0.75 A+�, �2 f’c in (MPa)

Where stirrups are provided:

Vc and V max change to the followings.

%� � 0.19�A+�, �2 f’c in (MPa)

%ZS[ � 0.55 A+�, �2 f’c in (MPa)

Outside the shear reinforced zone punching shear strength can be calculated as:

%x&'R � ¥�0.19A+�,�&'R2

Where uout is similar to the ACI 318-05 (Figure 2.28).

43

2.7.5 Eurocode2 (2004)

As Eurocode2 (2004) and Model Code 90 (1993) are very similar in their provisions for

punching shear strength, herein only Eurocode2 provisions are presented. Eurocode2 specifies

the critical perimeter at a distance equal to 2d from the face of column which is shown in Figure

2.29. It requires designers to use rounded edges for the critical perimeter.

The concrete ultimate shear strength is calculated by Equation 2-69.

¢� � 0.18z�100?S¤a+�¨�/3 � 0.1~�5 � ¢�ZWT (2-69)

Where z � 1 � �200/2�8.b ` 2.0 d in (mm)

?S¤a � �?[?Q�8.b ` 0.02

ρx, and ρy are the tensile reinforcement ratio in two orthogonal directions.

fck=is the characteristic concrete strength in MPa which approximated to fck=fc’-1.60MPa

(Gardner 2005)

¢�ZWT � 0.035 �z��/3+�¨8.b

Figure 2.29 Critical perimeter as shown in Eurocode2 (2004)

The ultimate design punching shear strength of slab can be calculated from the following

Equation.

%�x � �2¢�/�� (2-70)

Where u1 is the critical perimeter as shown in Figure 2.29 and γc is the concrete resistance factor

equal to 1.5.

44

If headed shear studs are provided, the punching shear strength is calculated as follow.

%Px � 0.75%�x � 1.5�2/;�iP¤+PQ¤�;<*� ` %ZS[/�� (2-71)

Where α is the angle between the shear reinforcement and the plane of the slab, and

fsvvE= 250+0.25d<fsv

Vmax=0.3(1-fck/250) fck u1d

The shear strength outside the shear reinforcement zone -Vcd out- can be calculated by Equation

2-72.

%�x &'R � �&'R2¢�/�� (2-72)

In Equation 2-69, uout is the outer critical perimeter shown in Figure 2.30 with the broken lines.

In Figure 2.30, k is equal to 1.5 according to Eurocode2, whereas, k is equal to 2.0 in Model

Code 90.

Figure 2.30 Shear reinforcement arrangement and critical perimeter outside the shear reinforced

region as shown in Eurocode2 (2004)

2.7.6 British Standard BS 8110-97

In BS 8110-97 (1997), the critical perimeter is located at 1.5d from the loaded area, and the

ultimate allowable shear stress on the critical perimeter can be calculated as given in Equation

2-73.

%�x � 0.79 �100?�/3 �400/2�8�b �+�'/25�/3 �2/γZ f’cu in (MPa, and d (mm ) (2-73)

Where γm= is the material partial factor is equal to 1.25,

fcu= is the characteristic cube concrete compressive strength not less than 25 MPa and greater

than 40 MPa,

45

ρ= (ρx+ ρx)/2 <0.03, in which ρx, and ρy are the flexural reinforcement ratio in two orthogonal

directions,

(400/d)0.25 is the size factor and should be equal or less than one.

The maximum shear stress at the column face should not be greater than 5MPa, or 0.8(fcu)0.5.

There are no specific provisions for the punching shear of prestressed slabs in BS 8110-97.

The punching shear strength of slabs with shear reinforcement is calculated by the following

equation.

%Px � %�x � 0.87iP¤+PQ¤;<*� (2-74)

Where Asv is the area of one row of shear reinforcement around the column which is provided in

successive bands with spacing of 0.75d and fsyv is the yield strength of shear reinforcement.

2.7.7 German Standard DIN 1045-1:2001

DIN 1045-1 (2001), similar to BS 8110-97, specifies the critical perimeter to be located at a

distance equal to 1.5d from the face of column as shown in Figure 2.31. The ultimate punching

shear strength of slabs is calculated by Equation 2-75.

%�x � 0.21z�100?S¤a+�¨�/3/�� 0.12~�5 fck in (MPa) (2-75)

Where z � 1 � �200/2�8.b ` 2.0

?S¤a � �?[ � ?Q�/2 ` 0.02 .*2 ` 0.23+�¨/+PQ

γc=is the material partial safety factor equal to 1.5

Figure 2.31 Critical perimeter as given in DIN 1045-1 (2001)

If shear reinforcement is provided, the punching shear strength of slabs can be increased to the

maximum of 1.9Vcd for slabs reinforced with double headed shear studs and 1.5Vcd for other

types of shear reinforcement.

46

The first row of shear reinforcement, placed at the distance of d/2 from the face of column,

should be capable of resisting the punching shear force, so Equation 2-76 is suggested by DIN

1045-01.

%Px � %�x � =P0.87iP¤+PQ¤ (2-76)

For the strength of remaining rows can be calculated by Equation 2-77.

%Px � %�x � =P0.87iP¤+PQ¤2/; (2-77)

Where s is the spacing of shear reinforcement, fsyv is the yield strength of shear reinforcement

not more than 500 MPa, and ks is a parameter to take into account the effect of slab thickness in

anchorage and efficiency of shear reinforcement. ks can be calculate as following.

0.7 ` =P � 0.7 � 0.3�2 � 400�/400 ` 1.0 d in (mm) (2-78)

2.8 Summary

There has been an extensive research on the topic of punching shear of flat slabs. Major

previous analytical methods were briefly presented. There are various available approaches to

the punching shear phenomenon and there are significant differences between many of them.

Solutions to include the effect of prestressing forces on punching shear strength of flat plates

were discussed. Further, different types of strengthening technique and shear reinforcement for

punching shear were reviewed. Finally, the provisions of several internationally recognised

standards for punching shear strength of concrete slabs, prestressed slabs and concrete slabs

with shear reinforcement were presented. Despite the large volume of research conducted on

punching shear capacity and the large number of proposed mechanical models, none of the

internationally recognised standards has yet to adopt any of these mechanical models for its

design equations of punching shear capacity. It is clear that most of the standards still use the

empirical formulae originally proposed by Moe (1961) with minor modifications for different

factors such as slab thickness and concrete compressive strength.

47

Chapter Three

3 CONCENTRIC PUNCHING SHEAR OF FLAT PLATES

3.1 Introduction

From the structural point of view, concrete structures consist of two types of regions, main

regions, and local regions. The main regions -sometime referred to as the B-regions- are where

the distribution of stresses and strains are regular and this distribution can be presented by

mathematical expressions. In B-regions, force equilibrium and compatibility conditions

determine the state of stresses and strains (Hsu 1993). On the other hand, in the local or

disturbed regions -sometime referred as D-regions-, stresses are disturbed and strains are

irregular. Figure 3.1 shows main regions and local regions in a simple structure. In local

regions, it is very difficult to provide a mathematical solution for the flow of forces. Especially,

the compatibility conditions are not applicable, which leads to the use of equilibrium conditions

alone as the solution to the design of local regions. Prior to cracking, the stress pattern and

stress values can be quantified by the use of elastic finite-element analysis. After cracking, the

stress field will be disrupted and reoriented.

D-region

B-region

Figure 3.1 Schematic view of B-regions and D-regions in a simple structure

Historically, engineers designed local regions by “good practice”, by rule of thumb, or more

recently by empirical methods (Wight & MacGregor 2009). However, in the last three decades,

structural engineers have had a giving renewed interest in the strut-and-tie method as an

48

alternative solution for the design of D-regions. Basically, a strut-and-tie model consists of

concrete struts acting in compression and steel ties acting in tension, which form a truss to

transfer the internal forces.

As stated in fib (2001), one of the most critical D-regions in structures is where the slab meets a

supporting column. The statistical discontinuity and existence of significant bending moment

and shear force result in a very complicated three dimensional state of stress. To deal with this

D-region, there have been valuable efforts by researchers to introduce empirical or semi-

empirical methods, which some of them were reviewed in the previous chapter. The truss

analogy or strut-and-tie method has been used by various researchers to model the transfer of

internal load in the slab-column connection. In this chapter models that explain the transfer of

force from slab to column are presented. Then a formula is proposed to calculate the ultimate

punching shear strength of flat plates and its accuracy is assessed against a large number of

reported experimental results in the literature. Further, punching shear formulae of AS 3600-

2009, ACI 318-05, NZS 3101:2006, Eurocode2, and DIN 1045-1 are used to predict the

punching shear strength of the same test specimens to evaluate and compare their accuracy with

the proposed formula.

3.2 Strut-and-Tie Model for Punching Shear Phenomenon

As mentioned earlier, the strut-and-tie method can be considered as a very powerful analytical

tool to predict the ultimate capacity of D-regions. This method is a lower bound method on the

strength of a portion of a structure. Conventionally, an idealised truss model, transferring the

load from its point of application to the support, consists of concrete compression struts and

reinforcement ties. Applying this approach to model the slab-column connection of a prototype

test specimen, a compression strut should be drawn from the column to the point where the load

is applied. This compression strut is tied by tensile flexural reinforcements as shown in Figure

3.2. Although this model was used in the early days of design of flat slabs, it is considered to be

an unsafe and implausible load path. This model may result in overestimation of punching shear

capacity of slabs (fib 2001).

Top reinforcement ties

Concrete struts

Figure 3.2 Early strut-and-tie model for slab-column connection

49

Although concrete has some tensile strength, it is conservatively neglected in strut-and-tie

modelling. To achieve a more accurate and plausible mechanical strut-and-tie layout, it is

necessary to consider the tensile capacity of concrete. An alternative arrangement of struts and

ties can be envisaged if the tensile strength of concrete is taken into account. A very straight

forward model is shown in Figure 3.3 where solid lines represent ties and broken lines represent

compressive struts.

Top reinforcement ties

Concrete strutsConcrete ties

Figure 3.3 Refined Strut-and-tie model including concrete ties

Considering this model in 3D, the critical tie is the closest concrete tie to the column, which has

the least concrete area to transfer tension. Some researchers such as Menetry, and Georgopoulos

(fib 2001) as presented in 2.3.4.2 and 2.3.4.1, quantify the ultimate punching shear strength of

slabs by calculating the strength of the concrete tie shown in Figure 3.4.

Concrete tie failure

Figure 3.4 Punching shear by failure of concrete ties

According to Regan and Braestrup (1985) the crack which causes the punching shear

phenomenon, initiates approximately at 70 percent of the ultimate punching load. Even after the

development of this crack, test specimens were able to resist unloading and reloading. Broms

(1990) concluded that punching shear is not a “pure shear” problem and the resistance

mechanism of slab against punching shear relies on the compression zone where there is no

crack.

Consequently, an alternative failure criterion to the one shown in Figure 3.4, can be envisaged

in which the compressive strength of the critical strut, adjacent to the column, governs the

50

ultimate strength of the slab-column connection as shown in Figure 3.5. This has been the basis

for a number of proposed mechanical methods for the punching shear phenomenon such as

Kinnunen and Nylander (1960), Shehata (1990), Broms (1990), Hallgren (1996), Tiller(1995),

and Marzouk, Rizk and Tiller (2010) as reviewed in 2.3.1, 2.3.1.1, 2.3.1.2 and 2.3.1.3.

Furthermore, it has been observed by Kinnunen and Nylander (1960), Hallgren (1996) that the

radial compressive strains in a slab-column connection increase as the load increases, but just

before the punching shear failure occurs, strains start to decrease to zero at the soffit of the

connection. Broms (2005) considered this phenomenon as an evidence that the failure was

triggered by a crack in the compression zone at the soffit of the slab. Based on this observation,

Muttoni (2008) suggested an “elbow-shaped” compressive strut and horizontal tie develop in

the vicinity of column just before the punching shear failure.

There is an agreement between researchers on the formation of the critical strut beneath the

critical shear crack in the vicinity of the column, which transfers the load from slab to the

column (Broms 1990), (Shehata 1990), (Tiller 1995), (Muttoni 2008), and (Marzouk, Rizk &

Tiller 2010).

Critical shear crack

Critical concrete strut

Figure 3.5 Punching shear by crushing of concrete struts

3.3 Proposed Formula for the Ultimate Punching Shear Strength of Flat

Plates

Considering the mentioned observations in experimental tests, it is more rational to quantify the

punching shear capacity of slabs using the criterion for crushing of the critical compressive

strut. A schematic view of the critical compressive strut is shown in Figure 3.6.

As suggested by Broms (1990), Tiller(1995), and Marzouk, Rizk and Tiller(2010), Equation 3-1

can be used to calculate the ultimate punching shear strength of slabs (Vuo).

51

%'& � ��9 � 2©� \ - \ +� \ z \ sin ��/2� (3-1)

Where D, B, t and q are shown in Figure 3.6. fc and z are the compressive strength of the

concrete strut and a slab size factor respectively. For the case of square columns an equivalent

circular column with a similar perimeter has been considered.

Figure 3.6 View and cross section of the critical concrete strut around the column

A number of parameters such as dimensions of the critical strut, compressive strength of the

critical strut, slab size factor, and inclination of the critical strut should be quantified before

trying to calculate the punching shear strength of a slab using Equation 3-1. Herein, a prismatic

strut was chosen to simplify the model similar to Broms (1990), Tiller (1995), and Marzouk,

Rizk and Tiller (2010). Also it was assumed that the inclination of the critical strut is half of the

inclination of the critical shear crack similar to previous researchers such as Shehata (1990),

Broms (1990), Tiller (1995) and Marzouk, Rizk and Tiller (2010). Thickness of the idealised

prismatic strut can be quantified by determining boundary conditions for the geometry of the

compressive strut. If the top of the concrete strut is assumed to be fixed at the level of the

neutral axis, and the bottom of the strut is assumed to be fixed at the soffit of the slab-column

θ/2

B

tC

θ

h

Column CL

D/2

Ν.Α.

Critical shear crack

Critical strut

52

connection, it is possible to determine the thickness of the strut. Referring to Figure 3.6, B and t

can be quantified by Equation 3-2 and 3-3 respectively.

© � 2T/-.*� (3-2)

- � 2T sin��/2� /;<*� ª 2T/2 (3-3)

In following sections, methods to calculate parameters such as depth of neutral axis dn in the

vicinity of column, inclination of crack q, strength of concrete strut fc, and slab size factor will

be discussed.

3.3.1 Depth of Neutral Axis

As reviewed in the previous chapter, significant research has been carried out on the punching

shear of concrete slabs, but there is no agreement on how to calculate the depth of the

compression zone in the vicinity of column. This can be attributed to the existence of shear

forces adjacent to the column in addition to a complex triaxial state of stress, which results from

the bending moment around the column. Different approaches to calculate the depth of the

neutral axis adjacent to column are presented as following.

3.3.1.1 Depth of neutral axis in the elastic condition

Figure 3.7 shows the distribution of strains and the forces in the elastic condition of a section

subject to bending moment in B-region of structure, where the Bernoulli compatibility condition

is valid. The depth of the neutral axis in the elastic condition can be quantified by using

Hooke’s uniaxial constitutive law (4P � ~P/CP , 4� � ~�/C�), the strain distribution considering

Bernoulli compatibility condition (4P � 4&�2 � 2T�/2T� , and equating the tensile force in

reinforcing bars to the compressive force in the concrete (C=T). The depth of the neutral axis

can be expressed as Equation 3-4.

2T � *?@A1 � 2/*? � 1B2 (3-4)

Where n is the ratio of elastic modulus of steel to elastic modulus of concrete n=Es/Ec, ρ is the

ratio of tensile reinforcement and d is the effective depth of the section.

Broms (1990) used the basis of this method to calculate the location of the neutral axis in the

vicinity of the column. He included a modification factor kρ to reflect the inclined crack effects

as expressed in Equation 3-5. It should be noted that this effect is included because the region

adjacent to the column is a D-region, where both the bending moment and the shear forces are

significant.

53

2T � =>*?@A1 � 2/=>*? � 1B2 (3-5)

Where, => � �0.59 � 2/-.*30°�/�0.59 � 2T/-.*30°� . Hence, calculating the depth of the

neutral axis, based on this method, requires an iterative procedure.

Figure 3.7 Distribution of strains, stresses and forces in elastic condition (Warner et al. 1998)

3.3.1.2 Depth of neutral axis in the ultimate stage

The depth of the neutral axis can be calculated in the ultimate stage i.e. the maximum bending

moment resistance of section is reached. Assuming the provided tensile reinforcement ratio of

the section is less than the balanced reinforcement ratio, failure occurs when the outermost

compressive fibre of the section has reached its maximum strain εu. Figure 3.8 shows the strain

and stress distribution in a section at the ultimate stage for bending only.

To simplify calculation of the bending moment strength and the depth of the compression zone

at the ultimate stage, most of design standards allow using the equivalent rectangular stress

block as shown in Figure 3.9. Where ku=dn/d, γ is a parameter to convert the depth of the

neutral axis to the length of the equivalent rectangular stress block.

Figure 3.8 Strains and stresses distribution in the ultimate stage (Warner et al. 1998)

54

Figure 3.9 Rectangular stress block in the ultimate stage (Warner et al. 1998)

AS 3600-2009 specifies the ultimate strain for concrete as εu=0.003, and γ=1.05-0.007f’c. The

magnitude of the equivalent uniform stress is given as α2=1.0-0.003f’c. In this method, the

depth of the compression zone can be calculated by equating the compressive force to the

tensile force which will result in Equation 3-6.

2T � �?+PQ�2/��+�′� (3-6)

Theodorakopoulos and Swamy (2002) proposed two types of neutral axes in the region adjacent

to the column, namely the flexural neutral axis Xf and the so called “shear neutral axis” Xs.

Theodorakopoulos and Swamy (2002) pointed out that the ratio of fcube / (ρfsy) for test specimens

which yielded prior to punching shear had a value between 5 to 9. They assumed that the shear

neutral axis is equal to the flexural neutral axis i.e. Xf=Xs in test specimens which yielded before

punching shear occurs. In their model, the flexural neutral axis was calculated for the ultimate

stage and considering fcube / (ρfsy) is equal to average value of 7 then Xf=Xs=0.25d. In the

opinion of Theodorakopoulos and Swamy (2002), Xf is influenced by the amount of flexural

reinforcement and the compressive strength of concrete whereas Xs is unaffected. A schematic

view of the flexural neutral axis and the shear neutral axis is shown in Figure 3.10. In this

figure, point A is the intersection of the column and the slab. If two lines are drawn from the tip

of shear crack and the tip of flexural crack to the point A, the angle between the lines is ϕ as

shown in Figure 3.10. For cases where fcube / (ρfsy)≠7, Theodorakopoulos and Swamy (2002)

argued that the harmonic mean of Xf and Xs gives a more realistic approximation of the depth of

the compression zone. This is due to the characteristic of the harmonic value which tends to

mitigate the impact of the larger of Xf or Xs and aggravate the impact of the smaller one. ϕ tends

to zero as the flexural and shear cracks are very close or coincide (Theodorakopoulos & Swamy

2002). Consequently, it was suggested that the depth of the neutral axis could be calculated

using Equation 3-7.

55

Figure 3.10 Schematic view of the flexural neutral axis and the shear neutral axis

(Theodorakopoulos & Swamy 2002)

2T � 2��h � �P�/��h�P� (3-7)

In Equation 3-7, Xf=(ρfs-ρ’f’s)d/(k1fcu) where k1 is the concrete stress block parameter, fs is the

stress in the tensile reinforcement, f’s is the stress in the compressive reinforcement and

Xs=0.25d as suggested by Theodorakopoulos and Swamy (2002).

3.3.1.3 Simplified formula for depth of neutral axis

Shehata (1990) suggested a simplified formula to calculate the depth of the neutral axis for a

given slab in the elasto-plastic condition as given in Equation 3-8. In this formula, the shear

was accounted for by assuming punching shear occurs prior to concrete reaching the fully

plastic range.

2T � 0.8A*?aA35/+�, 2 f’c in MPa (3-8)

In Equation 3-8, n is the ratio of modulus of elasticity of steel to modulus of elasticity of

concrete, ρe is the ratio of reinforcement for a basic yield strength (500MPa) and can be

calculated as ρe= ρ(fsy/500)§0.02 where ρ is the ratio of reinforcement and fsy is the yield

strength of the tensile reinforcement.

3.3.2 Inclination of the Critical Strut and Critical crack

As shown in Figure 3.6, the angle of the critical strut was assumed to be half of the critical crack

angle. Shehata (1990), based on his experimental observations, suggested the inclination of the

critical crack to be 20°. While, Broms (1990), Tiller (1995) used 30

± as a typical critical crack

angle in their method which agrees with Regan and Braestrup (1985) experimental observations.

The assumption of treating the inclination of critical crack as a single value seems to be

inaccurate, as it has been observed in more recent experiments such as Hegger, Sherif and

Ricker (2006), and Guandalini, Burdet and Muttoni (2009). In these experiments, some test

dn

56

specimens failed with a 45° critical crack angle. Generalising the critical crack angle to a

specific value such as 20° or 30° may result in inaccuracy in the prediction of the punching

shear capacity of a slab.

As discussed in Chapter2, section 2.3.4.1, an attempt by Georgopoulos to approximately

quantify the angle of critical crack is cited in fib (2001). Georgopoulos suggested a formula to

predict the inclination of the critical crack by correlating the tangent of the crack angle to the

ratio of flexural reinforcement and compressive strength of concrete as given in Equation 3-9.

tan�� 0.056/« � 0.3 ` 1.0 (3-9)

Where ω=ρfy/fcube, and fcube can be approximated to1.25f’c.

Recently Marzouk, Rizk and Tiller (2010) suggested a range for the angle of the critical crack

depending on the thickness of the slab. They proposed a crack angle of 25°-35° for slabs less

than 250mm thick, 35°-45° for slabs 250mm-500mm thick and 45°-60° for slabs thicker than

500mm. Herein, the variation in the crack angle is investigated by test specimens reported by

Pisanty (2005). The test specimens had a relatively similar ratio of reinforcement and concrete

compressive strength. The main variable between them was their thickness h. The effective

depths d, the average compressive strength of concrete fcm, the ratio of reinforcement ρ, the yield

strength of tensile reinforcement fsy, the side dimension of the square column a, and the side

dimension of square slab l are provided for each test specimen in Table 3.1. In this experiment,

each test specimen was saw-cut and the angle of the critical crack q was reported as given in

Table 3.1. As suggested by Marzouk, Rizk and Tiller (2010), an increase was observed in the

angle of critical cracks as the thickness of the test specimens increased. This is shown in Figure

3.11 where tan(q) is plotted against the thickness of test specimens. Considering the suggested

values by Marzouk, Rizk and Tiller (2010) and the reported crack angles in Table 3.1, a linear

relation between the thickness of slab and the tangent of the angle of critical crack was

suggested by the author of this report as expressed in Equation 3-10.

The suggested values by Marzouk, Rizk and Tiller (2010) were used as the upper and lower

limits for Equation 3-10. This is shown in Figure 3.12 in which tan(q) is plotted against the

thickness of slab for Equation 3-10 along side the upper and lower limits by Marzouk, Rizk and

Tiller (2010) and the observed angle of the critical crack in Pisanty (2005). As can be seen in

Figure 3.12, the range of experimental results of 140mm to 200mm has been extended slightly

in both directions to cover the range of 90mm-300mm. This is justified because it only involves

a minor extrapolation in both upper and lower limits.

57

Table 3.1 Main properties of test specimens and angle of the critical crack reported in (Pisanty

2005)

Test Specimen 14/1 14/2 16/1 16/2 18/1 18/2 20/1 20/2

h(mm) 140 140 160 160 180 180 200 200

d(mm) 112 112 133 133 151 151 171 171

fcm(MPa) 26.4 22.8 25 19 23.3 25.5 24.1 21.8

ρ 0.013 0.013 0.009 0.009 0.012 0.012 0.01 0.01

fsy(MPa) 500 500 500 500 500 500 500 500

a(mm) 200 200 200 200 250 250 300 300

l(mm) 1700 1700 1700 1700 1700 1700 1700 1700

Angle of crack q 30° 33° 32° 35° 35° 31° 37° 40°

Angle of crack

Equation 3-9 30° 29° 34° 30° 30° 31° 32° 31°

Angle of crack

Marzouk, Rizk and Tiller (2010) 25°<q<35°

Figure 3.11 Observed critical crack angle versus thickness of slab

0.45 ` -.*�� 0.0027�� 0.2 ` 1.0 (3-10)

Where h is the thickness of slab in (mm) and q in degree.

tan(θ)= 0.0027h + 0.2

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

130 140 150 160 170 180 190 200 210

tan(θ)

Thickness of slab (mm)

Observed critical shear crack in test specimens

58

Figure 3.12 Predicted angle of critical crack using Equation 3-10

3.3.3 Compressive Strength of the Concrete Strut

The idealised prismatic strut in the slab is subjected to lateral compressive stress. The

compressive strength of a prismatic strut can be influenced by the state of lateral stress around

it. Mehta and Monteiro (2006) suggested that the strength of concrete specimens under biaxial

state of stress can be 27 percent more than the similar specimen under uniaxial compression

stress. Schlaich simplified the compressive strength of the concrete strut for the following cases

(Warner et al. 1998).

Where concrete is uncracked and there is uniaxial stress:

+� � 0.85+�, (3-11)

Where lateral compressive stress exists:

+� � 1.1 \ 0.85+�, (3-12)

Broms (1990) suggested Equation 3-13 as the compressive strength of the concrete strut in his

method to account for the lateral compressive stress on the strut.

+� � 1.1+�, (3-13)

Marzouk, Rizk and Tiller(2010) adopted the suggested compressive strength for a prismatic

strut in the Canadian Standard (CSA A23.3-04 2004) as expressed in Equation 3-14.

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250 300 350 400 450 500

tan (θ)

Thickness of slab (mm)

Equation 3-10

Marzouk, Rizk and

Tiller (2010) h<250mm

Marzouk, Rizk and

Tiller (2010) 250mm<h<500mm

Observed critical

crack angle

59

+� � +�,/�0.8 � 1704� ` 0.85+�, (3-14)

Where 4 is the principal tensile strain in the cracked concrete. Marzouk, Rizk and Tiller (2010)

used 0.001 for 4 in their method.

The recent AS 3600-2009 gives Equation 3-15 as the capacity of prismatic concrete struts.

+� � 0.9+�, (3-15)

Muttoni, Schwarts and Thurlimam (2003) suggested the following equations to quantify the

compressive strength of the concrete strut.

Where there is lateral confining compressive stress (σ1) the compressive capacity of strut is:

+� � 20�+�,/20��/3� � 4~ for +¬� � 20m�. (3-16)

Where there is no lateral stress (σ1 ) the compressive capacity of strut is:

+� � 20�+�,/20��/3� for +¬� � 20m�. (3-17)

As expressed in Equation 3-16, an increase in the lateral confining stress σ1 will result in

increase of the compressive strength of the concrete strut. For the strut, σ1 is not uniform and it

changes depending on the distance to the neutral axis. Therefore, it is not possible to specify a

value for σ1 in Equation 3-16.

In this report, Equation 3-13, Equation 3-15, and Equation 3-17 are used as the compressive

strength of the concrete strut in Equation 3-1.

3.3.4 Slab Size Factor

Marzouk, Rizk and Tiller (2010) based on fracture mechanics, suggested a slab size factor for

their proposed strut-and-tie model as expressed in Equation 3-18.

g<�] +.�-^( � �r��/�� (3-18)

lch is called characteristic length. This parameter is not a physical property and reflects the

fracture characteristic of the concrete. Marzouk, Rizk and Tiller (2010) suggested to use α=0.33

for various concrete compressive strengths. lch can be calculated by Equation 3-19.

r�� C�h/+�R� (3-19)

Where Ec is the concrete modulus of elasticity, fct is the tensile strength of concrete, and Gf is

called fracture energy. Gf represents the amount of energy which causes a unit area of crack.

This parameter can be quantified by calculating the area under the curve of load-crack width

60

graph. Marzouk and Chen (1995) suggested that the average value of lch for normal strength

concrete is 500mm and for high strength concrete is 250mm. Marzouk, Rizk and Tiller (2010)

in the outline of the design procedure for their strut-and-tie model, suggested to obtain the

characteristic length either from a simple fracture mechanics test or the latter approximate

values.

Broms (2005) proposed to use the compression zone dimensions, instead of the thickness of the

slab, as the reference dimension to consider the slab size effect. Justification for this assumption

relies on the hypothesis of the compressive failure in the soffit of the slab-column connection.

Broms used (150/dn)0.33 in his method to consider the size effect on the strain capacity of slabs,

and (150/t)0.33 to consider the slab size effect on the compressive strength of concrete struts in

which t is expressed in Equation 3-3 and shown in Figure 3.6. 150mm is the reference value,

chosen based on the diameter of the standard test cylinder specimen. If the failure occurs before

the concrete goes into the non-linear mode, 0.5 is a suitable exponent to reflect the size effect,

but for cases in which concrete goes into the plastic range and performs non-linearly then 0.25

is a suitable exponent. Therefore, Broms (2005), pointed out that the 0.5 exponent, suggested

by Hallgren (1996), exaggerates the size effect on the failure capacity of slabs. Instead he

suggested 0.33 as a more realistic exponent for the case of punching shear failure. In this report,

the author used (150/t)a as the slab size factor. Considering Equation 3-3, t can approximate to

(@ dn/2). Different values were used as the exponent for this ratio to determine the most suitable

exponent.

3.3.5 Determination of the Parameters

As it was discussed, for a given slab, there is no agreement in the literature on how to quantify

some of the aforementioned parameters such as the depth of the compression zone, the size

effect, the inclination of the critical crack and the strength of the critical strut. Therefore, a large

number of reported experimental tests were gathered from fib (2001) and some of other recent

papers, which are not included in fib (2001), such as Birkle and Dilger (2008), Li (2000),

Marzouk and Hussein (1991), Guandalini, Burdet and Muttoni (2009), and Pisanty (2005).

Slabs which reportedly failed in flexure were excluded from the database, and only slabs which

reportedly failed by punching shear were considered. Details of these test specimens are

provided in Appendix A.

An Excel spreadsheet was written to predict the capacity of each slab based on Equation 3-1.

The depth of the neutral axis was quantified, using Equation 3-5, Equation 3-7 and Equation 3-

8. To account for the slab size effect, the ratio of (300/dn) with four different exponents, 0 -no

size effect-, 0.25, 0.33, and 0.5, were considered. Further, the proposed expression by

Georopoulos in Equation 3-9 and that proposed by the author in Equation 3-10 were used to

61

predict the inclination of the critical crack. Moreover, Equation 3-13, Equation 3-15, and

Equation 3-17 were used to calculate the compressive strength of the critical strut. In total 72

different combinations of parameters were considered using Equation 3-1.

The ratio of the predicted capacity, over the reported failure load (Vtest/Vuo) was calculated for

each test specimen of the database. Consequently, average, standard deviation -SD-, and

coefficient of variation -CV- of these ratios were calculated to compare the capability of each

combination of parameters.

In Table 3.2 to Table 3.4, the column of parameters indicates the parameters, which were used

in Equation 3-1 to calculate the punching strength of slabs. The first letter expresses the method

which was used to calculate the depth of the neutral axis. So, B represents the depth of the

compression zone based on the method suggested in Broms (1990), T represents the method

suggested in Theodorakopoulos and Swamy (2002), and S represents the method suggested in

Shehata (1990). The second letter stands for the method which was used to quantify the

inclination of shear crack. Letter G standing for the method, suggested by Georgopoulos

(Equation 3-9), and P standing for the proposed formula by the author of this thesis (Equation 3-

10). The third letter represents the method which was used to calculate the compressive

strength of the critical strut. Here A standing for the suggested method in AS 3600-2009, B

standing for the suggested method in Broms (1990), and M represents the suggested method in

Muttoni, Schwarts and Thurlimam (2003). Finally, the last figure represents the exponent of the

slab size factor (300/dn). This is, 0 (no size effect is considered), 0.25 represents (300/dn)0.25,

0.33 represents (300/dn)0.33 and 0.5 represents (300/dn)

0.5.

As an example, S-G-B-0.25 indicates that Shehata’s method (Equation 3-7) was used to

calculate the depth of the compression zone, Georgopoulos’s method (Equation 3-9) was used

to calculate the inclination of the critical crack, Broms’s method (Equation 3-13) was used to

calculate the strength of the concrete strut, and finally the ratio of (300/dn)0.25 was used as the

slab size effect. Therefore, Equation 3-1 for the case of S-G-B-0.25 is shown as below.

Vuo(S-G-B-0.25)� � �9 � �x®¯°J�U� \ x®±WT�XO±WT�U� \ 1.1+�, \ g<* �U� \ �388x® �8.�b

Table 3.2 gives the average, SD, and CV of Vtest/Vuo for different combination of parameters

where Equation 3-5 was used to calculate the depth of the neutral axis. Similarly, Table 3.3,

and Table 3.4 show the average, SD, and CV of Vtest/Vuo for different combination of parameters

where Equation 3-7 and Equation 3-8 were used respectively to calculate the depth of the

neutral axis. The desired method would have the lowest CV and an average value close to

unity.

62

In Table 3.2, Table 3.3, and Table 3.4, three formulae show a reasonable accuracy, and they are

T-P-M-0.5, S-P-A-0.5, and S-P-B-0.33.


method in Broms (1990) to calculate the depth of the neutral axis

Parameters Average SD CV Parameters Average SD CV

B-G-A-0 1.24 0.55 0.44 B-P-A-0 1.32 0.53 0.40

B-G-A-0.25 0.82 0.33 0.41 B-P-A-0.25 0.86 0.31 0.36

B-G-A-0.33 0.71 0.28 0.40 B-P-A-0.33 0.75 0.26 0.35

B-G-A-0.5 0.54 0.21 0.39 B-P-A-0.5 0.57 0.19 0.33

B-G-B-0 1.02 0.45 0.44 B-P-B-0 1.08 0.43 0.40

B-G-B-0.25 0.67 0.27 0.41 B-P-B-0.25 0.70 0.36 0.36

B-G-B-0.33 0.58 0.23 0.40 B-P-B-0.33 0.61 0.21 0.35

B-G-B-0.5 0.44 0.17 0.39 B-P-B-0.5 0.46 0.15 0.33

B-G-M-0 1.28 0.32 0.25 B-P-M-0 1.37 0.32 0.23

B-G-M-0.25 0.85 0.19 0.22 B-P-M-0.25 0.90 0.17 0.18

B-G-M-0.33 0.74 0.16 0.22 B-P-M-0.33 0.79 0.14 0.17

B-G-M-0.5 0.56 0.13 0.23 B-P-M-0.5 0.60 0.10 0.17

Table 3.3 Average, SD and CV of Vtest/Vuo for different combination of parameters using the method

in Theodorakopoulos and Swamy (2002) to calculate the depth of the neutral axis


T-G-A-0 2.88 0.99 0.34 T-P-A-0 3.14 0.97 0.31

T-G-A-0.25 1.62 0.55 0.34 T-P-A-0.25 1.75 0.49 0.28

T-G-A-0.33 1.34 0.46 0.34 T-P-A-0.33 1.44 0.40 0.28

T-G-A-0.5 0.92 0.32 0.35 T-P-A-0.5 0.98 0.27 0.27

T-G-B-0 2.36 0.81 0.34 T-P-B-0 2.57 0.79 0.31

T-G-B-0.25 1.33 0.45 0.34 T-P-B-0.25 1.43 0.28 0.28

T-G-B-0.33 1.10 0.37 0.34 T-P-B-0.33 1.18 0.33 0.28

T-G-B-0.5 0.75 0.27 0.35 T-P-B-0.5 0.80 0.22 0.27

T-G-M-0 3.03 0.54 0.18 T-P-M-0 3.35 0.69 0.21

T-G-M-0.25 1.71 0.32 0.19 T-P-M-0.25 1.87 0.30 0.16

T-G-M-0.33 1.41 0.28 0.20 T-P-M-0.33 1.54 0.24 0.15

T-G-M-0.5 0.97 0.22 0.23 T-P-M-0.5 1.05 0.17 0.16

63

Table 3.4 Average, SD and CV of Vtest/Vuo for different combination of parameters using the method

in Shehata (1990) to calaculate the depth of the neutral axis


S-G-A-0 3.03 0.48 0.16 S-P-A-0 3.38 0.71 0.21

S-G-A-0.25 1.68 0.31 0.18 S-P-A-0.25 1.84 0.30 0.16

S-G-A-0.33 1.38 0.28 0.20 S-P-A-0.33 1.51 0.24 0.15

S-G-A-0.5 0.94 0.23 0.25 S-P-A-0.5 1.02 0.17 0.17

S-G-B-0 2.48 0.40 0.16 S-P-B-0 2.76 0.58 0.21

S-G-B-0.25 1.37 0.25 0.18 S-P-B-0.25 1.51 0.24 0.16

S-G-B-0.33 1.13 0.23 0.20 S-P-B-0.33 1.23 0.19 0.15

S-G-B-0.5 0.77 0.19 0.25 S-P-B-0.5 0.83 0.14 0.17

S-G-M-0 3.37 0.89 0.26 S-P-M-0 3.80 1.23 0.32

S-G-M-0.25 1.85 0.47 0.25 S-P-M-0.25 2.06 0.54 0.26

S-G-M-0.33 1.52 0.39 0.26 S-P-M-0.33 1.68 0.42 0.25

S-G-M-0.5 1.03 0.29 0.28 S-P-M-0.5 1.12 0.27 0.24

To compare these three methods, Vtest/Vuo is plotted against effective depth of slab (d), tensile

reinforcement ratio (ρ), and compressive strength of concrete (f’c) in Figure 3.13, Figure 3.14,

and Figure 3.15. The linear trendline is shown for the ratio of Vtest/Vuo for each of the latter

methods. Similar to fib (2001), the linear trendline is used to approximately evaluate the

capability of the predicting method. The trendline indicates if a method can keep its accuracy as

a variable such as effective depth of slab, compressive strength of concrete or tensile

reinforcement ratio changes. A horizontal trendline demonstrates that the model is capable of

maintaining its accuracy for a wider range of test specimens. Conversely, an inclined line

indicates that the model is not capable of keeping its accuracy for a broad range of test

specimens. As given in Table 3.4, S-P-B-0.33 has the lowest CV compared to the other

methods. Also as shown in Figure 3.13, the trendlines are horizontal and the method is

consistent for a wide range of test specimens. The minimum value of Vtest/Vuo for S-P-B-0.33 is

0.78 as compared to 0.69 and 0.63 for T-P-M-0.5 and S-P-A-0.5 respectively. In this study, the

author decided to adopt S-P-B-0.33 Equation 3-20 to calculate the punching strength of slabs.

%'& � ��9 � 22T/-.*�� \ 2T/2 \ 1.1+�, \ �300/2T�8.33 \ sin��/2� (3-20)

Where, the depth of the neutral axis was expressed in Equation 3-8 (dn=0.8√(nρe)√(35/f’c)d) ,

the inclination of the critical crack is quantified by the proposed formula Equation 3-10

(tanθ=0.0027h+0.2), 1.1f’c is used as the concrete strut strength, and (300/dn)0.33 is the slab size

effect parameter. The predicted punching shear strength of test specimens using this method is

provided in Appendix A.

64


strength of concrete for T-P-M-0.5

65


strength of concrete for S-P-B-0.33

66


strength of concrete for S-P-A-0.5

67

3.3.6 Example

Herein, the ultimate punching shear strength of test specimen 16/1 reported in Pisanty (2005) is

calculated as an example to illustrate the procedure of calculating the punching shear strength of

a slab using the suggested formula.

The geometry and arrangement of tensile reinforcement of the test specimen 16/1 are shown in

Figure 3.16. The compressive strength of concrete was 25 MPa, the ratio of tensile

reinforcement was 0.95%, the yield strength of normal reinforcements was 500 MPa, the mean

of effective depths of tensile reinforcements was 133 mm, and the thickness of the test specimen

was 160 mm.

Figure 3.16 Plan and elevation view of test specimen 16/1 reported in Pisanty (2005)

Having the above information and given dimensions in Figure 3.16, the ultimate punching shear

strength of the test specimen can be predicted as following.

Vuo according to Equation 3.20:

%'& � ��9 � 22T/-.*�� \ 2T/2 \ 1.1+�, \ �300/2T�8.33 \ sin��/2�

In which the equivalent diameter:

D=4a/p=4µ200mm/3.14=254.8mm

φ12@85mm

φ12@95mm

1700m

m

1700mm

200mm

V

13

3m

m

68

The depth of neutral axis using Equation 3-8:

dn=0.8√(nρe)√(35/f’c)d

n=Es/Ec

According to AS 3600-2009, Ec for the concrete with the density of 2400 kg/m3 and f’c§40MPa

can be calculated as:

Ec=24001.5µ0.043√f’c=2400

1.5µ0.043µ√25=25.3µ103 MPa

n=200µ103/25.3µ10

3=7.91

ρe=ρ(fsy/500)=0.0095(500/500)=0.0095

dn=0.8√(7.91µ0.0095)µ√(35/25)µ133=35.5mm

The angle of the critical crack based on Equation 3-10:

tan(q)=0.0027h+0.2=0.0027µ160+0.2=0.632

The predicted angle of the critical crack q=32°.

The observed angle of the critical crack, reported in (Pisanty 2005), q=32°.

Using Equation 3-20, the predicted ultimate punching shear strength of the test specimen is:

Vuo=p(254.8+2µ35.5/tan32°)µ35.5/2µ1.1µ25µ(300/35.5)0.33µsin(32°/2)=315kN

Observed punching shear strength of test specimen was reported as Vtest=376kN

Vtest/Vuo=1.19

3.4 Comparison of Experimental Results with Design Standards

In this section, the collected experimental results were used to assess formulae of AS 3600-

2009, ACI 318-05, NZS 3101:2006, CSA A23.3-04, Eurocode2, and DIN 1045-1 for concentric

punching shear. It should be noted that the formulae of AS 3600-2009 and ACI 318-05 are

similar. NZS 3101:2006 is also similar to formulae of ACI 318-08 and AS3600-2009 except

that it includes a size effect factor (Equation 2-63). In Appendix A, the predicted punching

shear strength of the gathered test specimens using aforementioned standards are provided.

Table 3.5 provides the average, SD, and CV of the failure load to the predicted capacity

(Vtest/Vuo) for AS 3600-2009, ACI 318-05, NZS 3101:2006, CSA A23.3-04, Eurocode2, and DIN

69

1045-1. In Table 3.5, AS 3600-2009, ACI 318-05, and NZS 3101:2006 have a significant

higher average and CV compared to Eurocode2 and DIN 1045-1.

Table 3.5 Average, SD and CV of Vtest/Vuo for AS 3600-2009, ACI 318-05, NZ 3101:2006, CSA

A23.3-04, Eurocode2 and DIN 1045-1

Predicting method Average

Vtest/Vuo SD

Vtest/Vuo CV

Vtest/Vuo

AS 3600-2009 & ACI 318-05 1.39 0.28 0.20

NZS 3101:2006 1.45 0.28 0.19

CSA A23.3 1.24 0.25 0.20

Eurocode2 1.20 0.20 0.17

DIN 1045-1 1.24 0.20 0.16

Further, Vtest/Vuo is plotted against the effective depth of slabs, the ratio of tensile reinforcement,

and concrete compressive strengths for AS 3600-2009, ACI 318-05, NZS 3101:2006, CSA

A23.3-04, Eurocode2, and DIN 1045-1 in Figure 3.17 to Figure 3.21. The linear trendline is

drawn similar to the previous section to approximately evaluate the capability of the mentioned

standards in predicting the punching shear strength of flat plates. As shown in Figure 3.17, the

ratio of Vtest/Vuo decreases as the effective depth of the slabs increases for AS3600-2009. AS

3600-2009 seems to overestimate the capacity of thick slabs due to neglecting of the slab

thickness size effect. As shown in Figure 3.18, NZS 3101:2006 does not overestimate the

punching shear strength of thick slabs because of considering the slab thickness size factor. In

Figure 3.17 to Figure 3.19, due to neglect of tensile reinforcement ratio in the punching shear

formula of AS 3600-2009, NZS 3101:2006 and CSA A23.3-04, the punching shear strength of

heavily reinforced slabs is underestimated. According to Eurocode2 and DIN 1045-1, punching

shear capacity of a slab is proportional to the third root of the tensile reinforcement ratio of the

slab. As shown in Figure 3.20 and Figure 3.21, horizontal trendlines demonstrate Eurocdoe2

and DIN 1045-1 are very good in the estimation of the effect of tensile reinforcement ratio.

However, it seems latter standards cannot keep their accuracy for a wide range of slab

thicknesses.

70


strength of concrete for AS 3600-2009 and ACI 318-05

71


strength of concrete for NZS3101:2006

72


strength of concrete for CSA A23.3-04

73


strength of concrete for Eurocode2 and Model Code 90

74


strength of concrete for DIN 1045-1

75

3.5 Summary

The proposed formula to predict the concentric punching shear strength of flat plates, considers

the compressive strength of the critical strut adjacent to the column to govern the punching

shear strength of slabs. It was discussed that there is no agreement between researchers on the

method to specify the depth of the neutral axis in the vicinity of slab-column connections, the

inclination of the critical crack, the slab size effect and the compressive strength of the critical

strut. Therefore, a large number of experimental test specimens were gathered and the best

combination of the mentioned parameters was chosen to achieve a relatively accurate formula to

predict the punching shear strength of slabs. This method has a low coefficient of variation and

its accuracy is consistent for a wide range of slab thicknesses, tensile reinforcement ratios, and

concrete compressive strengths. AS 3600-2009 formula for punching shear with no unbalanced

moment does not consider two important parameters, namely the slab size effect and tensile

reinforcement effect. Comparing experimental test results, reported in the literature, to the

predicted strength of slab by AS 3600-2009 formula, it was revealed that due to neglecting of

slab size effect, the capacity of thick slabs is overestimated, and due to neglect of tensile

reinforcement ratio, the capacity of heavily reinforced slabs is underestimated. Further, AS

3600-2009, ACI 318-05, and CSA A23.3-04 had the worst coefficient of variation as compared

to the other mentioned standards. NZS 3101:2006 shows a better estimation of thick concrete

slab capacity as the slab thickness size effect is included in the formula. Eurocode2 and DIN

1045-1 give a good prediction of the failure load, and have a lower coefficient of variation as

compared to AS 3600-2009, ACI 318-05, NZS 3101:2006 and CSA A23.3-04.

76

77

Chapter Four

4 CONCENTRIC PUNCHING SHEAR OF PRESTRESSED

FLAT PLATES

4.1 Introduction

The use of prestressing technique in construction of concrete slabs has been increasing recently.

It solves serviceability issues such as excessive deflection and cracking, and allows designers to

achieve relatively thin slabs for large spans. This reduces the self-weight and overall height of

the building which is desirable in seismic regions and results in more economical structures. As

explained in Chapter Three, the state of stress is complex in the vicinity of column and the

presence of in-plane forces makes it even more difficult to determine the stresses adjacent to the

column.

In the following sections, the effect of prestressing tendons on the punching shear strength of

slabs is reviewed, and based on the available experimental results, the proposed method in

Chapter Three is extended to calculate the punching shear strength of prestressed slabs. Further,

the provisions of standards, presented in Chapter Two, are used to predict the punching shear

resistance of the same test results, and comparisons are made between them to determine their

accuracy in the prediction of punching shear strength of test specimens.

4.2 Background

In Chapter Two, three different approaches were reviewed for calculating the punching shear

strength of prestressed slabs namely: the principle tensile stress method, the equivalent

reinforcement ratio method, and the decompression method. These approaches are empirical or

semi-empirical, and a fully satisfactory mechanical method is yet to be developed to explain the

effect of prestressing on the punching shear phenomenon. The topic of punching of prestressed

slabs has been reviewed and presented by several researchers such as Scordelis, Pister and Lin

(1958), Regan (1985), Shehata (1990), Silva, Regan and Melo (2005), Clement and Muttoni

(2010), and Ramos, Lucio and Regan (2011). Inclusion of prestressing tendons in slabs imposes

three main actions in the analysis of stresses. Two of them are the resultant of the compressive

force in the tendons which can be divided into horizontal (Np) and vertical (Vp) components.

78

The third action is the bending moment (Mp) which is the resultant of the eccentricity

prestressing force from the neutral axis. These actions are shown in Figure 4.1.

P P

Prestressing tendon

Np

Vp

MpNp

Vp

Mp

Figure 4.1 Prestressing actions adjacent to the slab-column connection

As discussed in Silva, Regan and Melo (2007), the majority of prestressed slab specimens,

tested before the mid 1980s, cannot be used to draw a general conclusion on how prestressing

effects punching shear strength of slabs. This is due to the individual features of the test series

such as small slab thicknesses or lack of bonded reinforcement, and also in some instances some

of the important information about the test specimens such as the depth and profile of the

prestressing tendons were not clearly documented (Silva, Regan & Melo 2007). However, in

recent years, there has been valuable experimental work which sheds light on the effects of

prestressing on the punching shear strength of prestressed slabs.

Effects of prestressing have been investigated globally in most of the experimental test series

(Clement & Muttoni 2010). The test specimens were prestressed in the way that Np, Mp, and Vp

were applied to the slab simultaneously. As a result, it is not possible to investigate the effect of

each individual parameter on the punching shear strength of flat plates.

There are very limited experimental results available in which the effect of one of the

aforementioned parameters can be observed. Herein, the author selected the test specimens

reported in the literature from which the effect of one of the parameters -Np, Mp, Vp- can be

investigated on the overall strength of prestressed slabs.

4.2.1 Effect of In-plane Stresses on the Punching Shear Strength of Flat Plates

In this section, the effect of in-plane compressive stress is presented as deduced from some of

the reported experimental results on prestressed slab specimens. Two criteria were considered

to select the following presented test specimens. First, prestressed slab test specimens should be

similar to their reference test specimens -the specimen with no prestressing- in specifications

such as concrete compressive strength, ratio of reinforcement and dimensions. Second, the only

difference between the reference slab and prestressed slab should be the presence of in-plane

forces.

79

Regan (1983) reported experimental results which investigated the effect of in-plane force only

in one direction of the slab. The dimensions and the loading configuration of the reference test

specimen BD2 and the prestressed test specimen BD4 are shown in Figure 4.2 (a). Test

specimens were 1500mm by 1500mm square slabs which were supported on two edges and

loaded by 100mm by 100mm steel plates at the centre until failure. In addition, test specimens

BD4 and BD8 were tested in which the slab was supported on four edges as shown in Figure 4.2

(b). Prestressing forces were applied by unbonded prestressing tendons at the centre of the slab

thickness. The thickness of all test specimens were 125mm and the effective depth of tensile

reinforcement was 101mm. Table 4.1 gives the failure load and details of the test specimens.

(a) (b)

Figure 4.2 Geometery of BD test series (Ramos, Lúcio & Regan 2011)

Table 4.1 Failure load and details of BD test specimens (Ramos, Lúcio & Regan 2011)

Slab fcube(MPa) ρ(%) σcpx(MPa) σcpy(MPa) Vtest(kN)

BD2 49.0 1.28 0 0 268

BD1 52.8 1.28 7.65 0 293

BD8 44.1 1.28 0 0 251

BD4 46.0 1.28 7.65 0 293

In Table 4.1, fcube is the concrete cube compressive strength, ρ is the ratio of flexural

reinforcement, σcpx is the average in-plane compressive stress in the slab in the x direction, σcpy

is the average in-plane compressive stress in the slab in the y direction, and Vtest is the reported

punching shear failure load of the test specimen.

Silva, Regan and Melo (2005) cited experimental results of Correa (2001) in which unbonded

prestressing tendons in two of test specimens LP2 and LP3 were positioned horizontally in two

perpendicular directions at the mid thickness of slabs. As a result, the only difference between

LP2 and LP3 to the reference specimen was the presence of in-plane compressive stress. Test

specimens were 135mm thick square slabs supported on a 150mm by 150mm square columns,

80

and were loaded on 8 points as shown in Figure 4.3. The geometry and details of test specimens

LP1, LP2 and LP3 are given in Table 4.2.

Figure 4.3 Geometry of test specimens LP1, LP2 and LP3 as shown in Silva, Regan and Melo (2005)

Table 4.2 Failure load and detail of test specimens LP1, LP2 and LP3 (Silva, Regan & Melo 2005)

Slab fc’ (MPa) ρ(%) σcpx(MPa) σcpy(MPa) Vtest(kN)

LP1 50.7 1.17 0 0 327 LP2 52.4 1.17 2.19 2.19 355

LP3 52.4 1.17 4.28 4.28 415

Silva, Regan and Melo (2005) also cited test results by Kordina and Nolting (1984) in which

test specimen V6 was prestressed by a horizontal unbonded tendon at mid thickness of the slab.

Slabs were 150mm thick supported on 200mm diameter circular columns. The test setup is

shown in Figure 4.4 in which dimensions are in metres.

Figure 4.4 Geometry of test specimens V5 and V6 reported in Kordina and Nolting (1984) as shown

in Silva, Regan and Melo (2005)

560

mm

16

00 m

m

2000

mm

150

mm

81

Table 4.3 Failure load and details of test specimens V5 and V6 (Silva, Regan & Melo 2005)

Slab fc’ (MPa) ρ(%) σcpx(MPa) σcpy(MPa) Vtest(kN)

V5 36.8 0.9 0 0 349.5

V6 30.4 0.62 2.19 1.77 375

From the presented experimental specimens, it can be concluded that an increase in the in-plane

compressive stresses results in increase of the punching shear strength of slabs. This effect has

been included in most of the available methods for calculating punching shear strength of

prestressed slabs.

4.2.2 Effect of Eccentricity of Prestressing Tendon on the Punching Shear Strength of

Flat Plates

Most of reported experimental results simultaneously investigated the effect of eccentricity of

the prestressing tendons from the neutral axis with the effect of in-plane compressive stresses.

Recently, a very illustrative test series reported in Clement and Muttoni (2010) demonstrated

the effect of the eccentricity of prestressing simulated by applying bending moment mp on the

test specimens without the presence of any in-plane compressive stresses. Test specimens were

a 3000mm by 3000mm square slab with 250mm thickness and 210mm effective depth

supported on 260mm by 260mm square column. Shear forces were applied at 8 points, and the

slab was subjected to a bending moment mp at the region around the column. This was made

possible by two diagonal steel frames which were used to introduce two equal couples of forces

(Fh, -Fh, Fv, and -Fv) as shown in Figure 4.5.

Figure 4.5 Elevation view of test setup of PC test series and the bending moment diagram which

was applied to the slab without presence of in-plane forces (Clement & Muttoni 2010)

82

The bending moment is constant at the centre of the slab as shown in Figure 4.5. Two sets of

slab specimens with 0.77%, and 1.5% ratio of flexural reinforcement were tested. Each set

included one reference test specimen with no applied bending moment, one test specimen with

75 kNm/m bending moment and one specimen with 150 kNm/m bending moment. Details and

failure loads of the experiment are provided in Table 4.4.

Table 4.4 Failure load and details of test specimens reported in Clement and Muttoni (2010)

Slab fc’ (MPa) ρ(%) mp(kNm/m) Vtest (kN)

PG19 46.2 0.77 0 860

PC1 44.0 0.77 75 1201

PC3 43.8 0.77 150 1338

PG20 51.7 1.50 0 1014

PC2 45.3 1.50 75 1397

PC4 44.4 1.50 150 1433

From Table 4.4, it is clear the applied bending moment resulted in the increase of punching

shear resistance of the test specimens PC1, PC2, PC3, and PC4. It can be concluded the

eccentricity of tendons in the vicinity of the column, which creates a similar bending moment,

can play an important role in the punching shear capacity of slabs. Unfortunately, most of

current standards, reviewed in Chapter Two, do not take into account this parameter in their

punching shear formula. The only available method, which include the effect of eccentricity of

the prestressing tendon, is the decompression method which will be discussed later in this

chapter.

4.2.3 Effect of the Vertical Component of Prestressing Tendons Passing over the Slab-

Column Connection on the Punching Shear Strength of Flat Plates

The other effective parameter on the punching shear strength of prestressed slabs is the vertical

component of the prestressing tendon crossing the punching shear failure zone. This vertical

load acts against the shear force around the column and is a resultant of the deviation of

prestressing tendons. Test specimens AR8 to AR16, reported in Ramos and Lucio (2006), were

tested to investigate the latter parameter. In this test series, slab AR 9 was the reference slab.

The position and profile of the prestressing tendons of test specimens are shown in Figure 4.6

and Figure 4.7. Test specimens were prestressed by four prestressing tendons with 12.7 mm

diameter in each direction, and the position of the tendons was varied as shown in Figure 4.7.

In Table 4.5, the vertical deviation of prestressing tendons -a- and the prestressing force in

tendons -P- are given. A steel frame was used to avoid transfer of any in-plane force to the slab.

The failure load and detail of test specimens AR8 to AR16 are provided in Table 4.5.

83

(a)

(b)

Figure 4.6 (a) Plan view of test specimens AR8-AR16 (b) Profile of prestressing tendons (Ramos &

Lucio 2006)

Figure 4.7 Position of prestressing tendons in test specimens AR8-AR16 (Ramos & Lucio 2006)

84

Table 4.5 Failure load and details of test specimen AR8-AR16 (Ramos & Lucio 2006)

Slab fc’ (MPa) ρ(%) a(mm) P(kN) Vtest (kN)

AR9 41.6 1.68 0 0 251

AR8 37.1 1.68 40.3 448 380

AR10 41.4 1.68 40.5 348 371

AR11 38.0 1.68 41.9 239 342

AR12 31.3 1.68 36.8 448 280

AR13 32.5 1.68 38.3 446 261

AR14 28.2 1.68 35.2 431 208

AR15 31.7 1.68 36.9 445 262

AR16 30.6 1.68 41.5 442 351

As shown in Figure 4.7, prestressing tendons in AR8, AR10, AR11, and AR16 are concentrated

around the column, and prestressing tendons are positioned outside the column band in test

specimens AR12, AR13, AR14 and AR15. Considering the failure loads, presented in Table

4.5, the test specimens in which the tendons are passing over the column show higher punching

shear resistance as compared to the slabs with prestressing tendons outside the column band. It

can be concluded the vertical component of the prestressing force crossing the failure surface

increases the strength of prestressed slabs. Ramos and Lucio (2006) suggested that the tendons

passing within the distance of d/2 from the faces of the column are effective in increasing the

punching shear strength of prestressed concrete slabs.

4.3 Ultimate Punching Shear Strength of Prestressed Flat Plates Using the

Decompression Method

Most internationally recognised standards use the “principal tensile stress” approach to include

the effects of prestressing. Generally, the allowable shear stress on the control perimeter is

increased by adding a percentage of the horizontal prestressing stress. Also the majority of

standards include the vertical force, resulting from the deviation of the tendons passing the

critical perimeter, in punching shear formulae. A shortcoming in the “principal tensile stress”

approach is that the effect of eccentricity of prestressing tendons has been neglected.

In the “equivalent reinforcement ratio” approach, the prestressed reinforcement, or the

prestressing stress is converted to the equivalent normal reinforcement. Then the equivalent

reinforcement ratio is added to the actual ratio of normal reinforcement to be used in the

punching shear formula. Similar to “principal tensile stress” approach, this method does not

85

take into account the effect of eccentricity of tendons on the punching shear strength of

prestressed slabs.

Decompression approaches are more mechanically acceptable and promising as they take into

account all of the actions imposed on the slab by prestressing tendons in calculating the

punching shear strength of slabs. As discussed in section 4.2, it has been observed that the

compressive in-plane stress, the eccentricity of prestressing tendons from the neutral axis, and

the vertical component of prestressing tendons can influence the punching shear strength of

slabs.

The schematic deformation of prestressed slab after applying the prestressing forces is shown in

Figure 4.8. Vdec is the shear force at a section which corresponds to the decompression moment

being reached at that section. The amount of compressive stress in the extreme fiber depends on

the intensity of force in the tendons and also the eccentricity of tendons from the neutral axis of

the section. Therefore, the decompression action is divided into two components, Vo which is

the force needed to cancel out the compressive stress of the in-plane force of prestressing -Np- at

the outermost fibre, and Ve which is the force needed to cancel out the compressive stress from

the imposed bending moment of prestressing Mp in the outermost fibre. After the

decompression stage the remaining punching shear strength of prestressed slab is assumed to be

equal to the similar slab without the presence of prestressing actions. Figure 4.9 schematically

shows the component of decompression method and the punching shear strength of prestressed

slabs.

P P

Prestressing tendon

Deformed slab after application of prestressing forces

Figure 4.8 Schematic view of deformation of slab after prestressing forces are applied

86

P P

(a)

P P

(b)

Vdec=Vo+Ve

P P

(c)

Vup=Vuo+Vdec

Figure 4.9 (a) Prestressed slab (b) Prestressed slab at decompression stage (c) Punching shear

failure of prestressed slab

4.3.1 Available Decompression Methods

There are three decompression methods available in the literature for predicting the punching

shear resistance of prestressed flat plates. First is the method proposed by Regan (1985) which

presented in Chapter Two. Second is a “direct decompression approach” which presented in

Silva, Regan and Melo (2005). Third is a more complex decompression method suggested in

FIP recommendations for design of post-tensioned slabs and foundations (1998).

Silva, Regan and Melo (2005) suggested the decompression force is a force which creates a

bending moment at the face of the column annulling the compressive stress in the extreme fibre.

According to Silva, Regan and Melo (2005), the decompression force can be calculated by the

87

following equation considering the eccentricity of prestressing tendons and the in-plane

compressive stress.

%'5 � %'& � %& � %a � %'& � F&�%/F� � Fa�%/F� (4-1)

Where Vup is the ultimate punching shear strength of the prestressed slab

Vuo is the ultimate punching shear strength of similar slab with no-prestressing force using

formula of Eurocode2 (Equation 2-69)

mo=σcph2/6 in which σcp is the average in-plane compressive stress in the slab due to

prestressing.

me is the average moment due to the eccentricity of the tendon at the column.

(V/m) is the ratio between shear and the average bending moment at the face of the column.

To calculate the bending moment at the face of the column simple elastic analysis is suggested.

In Regan (1985), a linear relation between the applied force and the resultant bending moment

can be calculated. The ratio between the applied load V and the bending moment in the elastic

condition is a constant value which depends on the span of the slab, and side dimensions of the

column. For further illustration, an example is provided later in this chapter in which it is

shown how to calculate the ratio of V/m.

The other available decompression method is the formula in FIP (1998) which is more

complicated as compared to the latter method and needs iterative calculations. According to

FIP (1998), the punching strength of a prestressed slab can be calculated by the following

expression.

%'5 � %'& � %5 � %& � %a � %'& � %5 � F&, �% � %5�/�F, � Fa, � (4-2)

Where Vp is the vertical component of prestressing forcing crossing perimeter around the

column at the distance equal to the half of the thickness of slab (h/2).

V is the applied shear force.

m'o=σ’cph2/6 in which ~�5, is the average in-plane compressive stress on the critical perimeter of

slab, located at 2d from the face of the column.

F, is the average bending moment over the width of critical perimeter due V.

Fa, is the average moment due to the eccentricity of the tendon over the width of the critical

perimeter.

88

For any individual slab finite element analysis should be used to obtain m’, m’o, and m’e. This

may not be a convenient method for every day design cases.

4.3.2 Proposed Decompression Method

As discussed, all prestressing actions -Np, Vp, and Mp- are effective parameters in the punching

shear resistance of prestressed slabs. Decompression methods are the only available methods

which include Mp in the punching shear strength of slabs. Further, to take into account the

vertical component of the prestressing force, crossing within the distance of d/2 from the faces

of the column, Vp should be added to the punching shear strength of the slab as concluded in

Ramos and Lucio (2006), and Silva, Regan and Melo (2007). In the absence of a fully

satisfactory mechanical model to calculate the punching shear strength of prestressed flat plates

any proposed method should be validated by experimental results. Therefore, the author

gathered a database of 46 tested prestressed slab specimens which reported in the literatures

after mid 1980s. These tests are reported in Clement and Muttoni (2010), Ramos and Lucio

(2006), Ramos, Lucio and Regan (2011), Silva, Regan and Melo (2007) , and provided in

Appendix B.

To calculate the punching shear strength of prestressed slabs, the author suggested using a

decompression method with the proposed formula in Chapter Three. Three different scenarios

were considered to calculate the strength of prestressed slabs. In the first scenario, only the

effect of the in-plane compressive stress was considered and the punching shear strength of

prestressed slabs were calculated as Vuo+Vo in which Vuo is the punching shear strength of a

similar slab with no prestressing by Equation 3-20 and Vo is the load to cancel out the

compressive stress of the outermost compressive fibre due to the in-plane prestressing stress. In

the second scenario, strength of prestressed slabs were calculated as Vuo+Vo+Ve in which Ve is

the load to cancel out the compressive stress of the outermost compressive fibre due to

eccentricity of prestressing tendons at the face of column. For simplicity Vo, and Ve are

calculated in a manner similar to that in Equation 4-1. Finally, in the third scenario, in addition

to the previous effects, the contribution of the vertical component of the prestressing force in the

tendons was considered and the punching resistances of the slabs were calculated as

Vuo+Vo+Ve+Vp. As some details such as forces in each tendon at failure are not available for a

number of test specimens of the database, the calculated Vp in Silva, Regan and Melo (2007) for

the tendons within the distance d/2 from faces of the column were used in the latter method.

Similar to Chapter Three, the ratio of the observed failure load Vtest over the predicted punching

shear strength Vup was calculated for the three different scenarios as presented in Appendix B.

The average, standard deviation -SD-, and coefficient of variation -CV- for the ratios were

calculated and presented in Table 4.6.

89

Table 4.6 Average, SD and CV of Vtest/Vup for three different methods of calculating Vup

Method Average SD CV

Vup=Vuo+Vo 1.43 0.25 0.18

Vup=Vuo+Vo+Ve 1.16 0.19 0.16

Vup=Vuo+Vo+Ve+Vp 1.10 0.15 0.13

Figure 4.10 shows Vtest/Vup versus σcp for these methods. The third method, in which Vo, Ve, and

Vp were added to the punching shear resistance of the similar non prestressed slab, is a more

accurate method as it has an average closer to one and has a lower CV in comparison to the

other two methods.

The test specimens which isolated the effect of eccentricity of the prestressing tendons or the

effect of Vp are the ones with σcp=0 and positioned on the vertical axis of Figure 4.10. As it can

be seen in Figure 4.10, the method which takes into account Vo,Ve, and Vp predict the punching

shear strength of these test specimens with a better accuracy (Vtest/Vup closer to one).

As a result, the author of this report suggests Equation 4-3 for calculating the ultimate punching

shear strength of prestressed slabs.

%'5 � %'& � %5 � %& � %a � %'& � %5 � F&�%/F� � Fa�%/F� (4-3)

Where Vuo is the punching shear strength of similar slab with no prestressing using Equation 3-

20.

Vp is the vertical component of prestressing tendon crossing within the distance of d/2 from

faces of the column.

mo=σcph2/6 in which h is the thickness of the slab and σcp is the average in-plane compressive

stress in the slab due to prestressing.

me is the average moment due to the eccentricity of tendons from the neutral axis of the section

at the column.

(V/m) is the ratio between shear and the average bending moment at the face of column.

90

Figure 4.10 Vtest/Vup versus σσσσcp for three different methods of calculating Vup

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

Vup=Vuo+Vo

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

Vup=Vuo+Vo+Ve

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

Vup=Vuo+Vo+Ve+Vp

91

It should be mentioned that all the considered prestressed test specimens had unbonded

prestressing tendons. The only available test specimens that studied the punching of prestressed

slabs with bonded tendons are the ones investigating the effect of prestressing on bridge slabs.

These tests were spanning and prestressed in only one direction (Silva, Regan & Melo 2005).

In the case of prestressed slabs with bonded prestressing tendons, the ratio of prestressing

tendon -ρp- can be added to the ratio of normal reinforcement -ρ-, and the effective depth of the

section can be calculated by Equation 2-51 (Silva, Regan & Melo 2005).

Herein an example is provided to clarify the procedure of calculating the punching shear

strength of a prestressed slab using Equation 4-3.

4.3.3 Example

The test specimen D2 reported in Silva, Regan and Melo (2005) is presented as an example to

illustrate the procedure of calculating the punching shear strength of a prestressed slab using the

suggested method.

The plan view of test specimen D2 and positions of the supports are shown in Figure 4.11 (a).

The slab was loaded by a jack below the column and supported on eight nodes. The elevation

of the test specimen and the profile of prestressing tendons are shown in Figure 4.11 (b). The

compressive strength of the concrete was 44.1 MPa, the yield strength of normal reinforcements

was 540 MPa, the effective depths of normal reinforcements was 106mm, the ratio of normal

reinforcement was 0.5%, the effective depth of prestressing tendons over the support was 90

mm, the average force of each prestressing tendon at the beginning of the test was 137 kN, and

the mean in-plane compressive stress in the concrete was 2.23 MPa.

Having the above information and given dimensions in Figure 4.11, the ultimate punching shear

strength of the test specimen can be predicted as following.

The ultimate punching shear strength of a similar slab with no prestressing Vuo according to

Equation 3-20:

%'& � ��9 � 22T/-.*�� \ 2T/2 \ 1.1+�, \ �300/2T�8.33 \ sin��/2�

Equivalent circular diameter:

D=4a/p=4µ200mm/3.14=254.8mm

The depth of neutral axis using Equation 3-8:

dn=0.8√(nρe)√(35/f’c)d

92

n=Es/Ec=200µ103/32.8µ10

3=6.09

ρe=ρ(fsy/500)=0.005(540/500)=0.0054

dn=0.8√(6.43µ0.0054)µ√(35/44.1)µ106=13.7mm

The angle of the critical crack using Equation 3-10:

tan(q)=0.0027h+0.2=0.0027µ123+0.2=0.532, so q=28°

The ultimate punching shear strength using Equation 3-20:

Vuo=p(254.8+2µ13.7/tan28°)µ13.7/2µ1.1µ44.1µ(300/13.7)0.33µsin(28°/2)=217.2kN

560m

m

1600mm

2000m

m

200mm

100mm

100mm

200mm

Prestressing tendons

(a)

100mm

200mm

100mm

123

mm

55

mm

90m

m

Prestressing tendons

V

Supports

(b)

Figure 4.11 (a) Plan view (b) Elevation view of test setup of specimen D2 as reported in Silva, Regan

and Melo (2005)

93

The decompression load to cancel out the in-plane compressive force of prestressing tendons in

the outermost fiber:

Vo=mo(V/m)

Where

mo=σcph2/6=2.23µ123

2/6=5.623kN.m/m

m=(2µV/8µ(560/2-200/2)+ (2µV/8µ(1600/2-200/2))/2000=0.11V

V/m=9.091

Vo=5.623µ9.091=51.1kN

The decompression load to cancel out the compressive stress due to the eccentricity of

prestressing tendons:

Ve=me(V/m)

In-plane force per meter= σcph

Eccentricity of tendon= (dp-0.5h)

me= σcph.(dp-0.5h)=2.23µ123µ(90-0.5µ123)=7.817kN.m/m

(V/m)=9.091

Ve=7.817µ9.091=71.1kN

The sum of vertical forces in the tendons Vp crossing the width a+d over the column:

a+d=200+106=306mm

According to Figure 4.11, there are two tendons in each direction passing the width a+d.

Considering that the profile of the tendons is “circular-arc” the vertical force in each tendon can

be calculated by the following formula:

Vp=∑ P. sin(β)

Where P is the average prestressing force in each tendon, β is the inclination of tendon from the

plane of the slab at the distance of d/2 from the face of column.

The average force in each tendon at the start of the test was 137kN. β is equal to 0.6° from

geometry of the tendon. Considering four tendons in two directions crossing a+d:

94

Vp=8µ137µsin 0.6°=11.5kN

The predicted punching shear strength of the test specimen Vup:

Vup=Vuo+Vo+Ve+Vp=217.2+51.1+71.1+11.5=350.9kN

The reported failure load:

Vtest=385kN

Vtest/Vup=385/350.9=1.09

4.4 Comparison of Design Standards

As presented in Chapter Two, AS 3600-2009, ACI 318-05, NZS 3101:2006, and CSA A23.3-04

use the same control perimeter and relatively similar formulae for punching of non-prestressed

slabs. In the case of prestressed slabs, AS 3600-2009 differ to the other standards due to

ignoring the contribution of Vp. Further, ACI 318-05, NZS 3101:2006, and CSA A23.3-04 limit

the compressive strength of the concrete f’c to 35 MPa and increasing the concrete compressive

strength more than 35 MPa does not increase the punching shear strength of prestressed slabs.

AS 3600-2009, however, allows the use of concrete compressive strength up to 100 MPa.

AS 3600-2009, ACI 318-05 and NZS 3101:2006 add 30% of the in-plane compressive stress to

the concrete shear strength to account for the prestressing contribution. CSA A.23.3-04 has a

different approach to consider the effect of prestressing as shown in Equation 2-67.

Ramos (2006) discussed two contradictory clauses 6.4.3 and 9.4.3 in Eurocode2 (2004). Clause

6.4.3 suggests the vertical component of the prestressing tendons at a distance of 2d from the

face of column should be included in punching shear strength of the slab, whereas clause 9.4.3

suggests the distance to be d/2 from the face of column. Ramos (2006) based on his

experimental results, which studied effects of the vertical component of prestressing forces and

the position of tendons on the punching shear strength of slabs, concluded that tendons

positioned within the distance of d/2 from the faces of column are effective in the punching

shear resistance of slabs.

4.4.1 Comparison with Experimental Results

AS 3600-2009 formula for punching shear -Equation 2-55- was used to predict the punching

shear strength of each test specimen of the gathered database of prestressed test series. The

ratio of the observed failure load over the predicted punching shear strength was calculated for

46 test specimens, and the average, SD, and CV of the ratios are provided in Table 4.7. As

mentioned, AS3600-2009 does not include Vp in its punching shear formula unlike other

95

standards. To investigate the effect of including Vp in the accuracy of AS3600-2009, the

vertical component of prestressing tendons, located within the distance of d/2 of the face of

column, was added to the predicted punching shear strength by AS36000-2009. The ratio of the

observed failure load to the predicted punching shear strength was calculated for each test

specimen of the database and the average, SD, and CV of these ratios is provided in Table 4.7.

As mentioned earlier according to ACI 318-05, NZS 3101:2006 and CSA A23.3-04 the

compressive strength of concrete in the punching shear formula of ACI 318-05 should not be

taken as more than 35 MPa. The ratios of the observed failure load over the predicted punching

shear resistance of test specimens are calculated using formulae of ACI 318-05, NZS 3101:2006

and CSA A23.3-04 for two scenarios namely, including the limit on the concrete strength, and

ignoring the limit on the concrete strength. The average, SD, and CV of Vtest/Vup for both

scenarios is presented in Table 4.7. Finally, Eurocode2 and DIN 1045-1 were used to predict

the strength of test specimens, and the average, SD, and CV of Vtest/Vup for these standards are

given in Table 4.7.

Table 4.7 Average, SD and CV of Vtest/Vup for AS 3600-2009, ACI 318-04, CSA A23.3-04,

Eurocode2, and DIN 1045-1:2001


AS3600-2009 1.40 0.26 0.19

AS3600-2009 (including Vp within the distance of d/2 of the face of column)

1.29 0.19 0.14

ACI 318-05 and NZS 3101:2006 1.54 0.26 0.17

ACI 318-05 and NZS 3101:2006 (ignoring the limit on f’c)

1.46 0.23 0.16

CSA A23.3 1.32 0.24 0.18

CSA A23.3 (ignoring the limit on f’c)

1.25 0.21 0.17

Eurocode2 1.35 0.17 0.13

DIN 1045-1 1.36 0.18 0.14

The only difference between the punching shear formula of NZS 3101:2006 to the formula of

ACI316-05 is inclusion of a size factor which is effective for slabs with effective depth more

than 200mm. As the majority of available prestressed test specimens have effective depth less

than 200mm, given values for ACI 318-05 in Table 4.7 are the same for NZS 3101:2006.

In Figure 4.12 and Figure 4.13, Vtest/Vup is plotted against σcp for AS3600-2009, and for the case

when Vp is added to AS3600-2009 respectively. Considering the average, SD, and CV of

AS3600-2009 in Table 4.7 and comparing Figure 4.12 to Figure 4.13, it is clear including Vp in

96

the punching shear formula of AS3600-2009 significantly increases the accuracy of the

predicted resistance of prestressed slabs. As given in Table 4.7, the current formula of AS3600-

2009 has the highest CV and relatively high average as compared to the other standards. In

Figure 4.14, Figure 4.16, Figure 4.18, and Figure 4.19, Vtest/Vup is plotted against σcp for ACI

318-05, CSA A23.3, Eurocode2, and DIN 1045-1 respectively.

As shown, ACI 318-05 and underestimate the punching shear strength of prestressed slabs, and

its accuracy can be improved if the limit on f’c is ignored (Figure 4.15). Eurocode2 and DIN

1045-1 have a lower CV and average as compared to the other standards.

Figure 4.12 Vtest/Vup versus σσσσcp for AS3600-2009

Figure 4.13 Vtest/Vup versus σσσσcp for AS3600-2009 when Vp is included

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

AS3600-2009

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

AS3600-2009+Vp

97

Figure 4.14 Vtest/Vup versus σσσσcp for ACI 318-05

Figure 4.15 Vtest/Vup versus σσσσcp for ACI 318-05 ignoring the limit on f’c

Figure 4.16 Vtest/Vup versus σσσσcp for CSA A23.3-04

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (mm)

ACI 318-05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

ACI 318-05 no limit on f'c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

CSA A23.3-04

98

Figure 4.17 Vtest/Vup versus σσσσcp for CSA A23.3-04 ignoring the limit on f’c

Figure 4.18 Vtest/Vup versus σσσσcp for Eurocode2

Figure 4.19 Vtest/Vup versus σσσσcp for DIN 1045-1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

CSA A23.3-04 no limit on f'c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

Eurocode2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vte

st/V

up

σcp (MPa)

DIN 1045-1

99

4.5 Summary

In this chapter, the proposed method in Chapter Three was extended to calculate the punching

shear strength of prestressed slabs with the use of the decompression method. The proposed

formula has the advantage of taking into account some of the effective parameters such as in-

plane compressive stresses, the eccentricity of prestressing tendons from the neutral axis, and

the vertical component of the prestressing force of tendons, crossing in the region adjacent to

the column. Then the proposed method is used to predict the strength of some of the

experimental results reported in the literature. By comparing the observed failure load to the

predicted strength, the accuracy of the model assessed. Further, formulae of the standards,

presented in Chapter Two, are used to predict the strength of the same experimental results. It is

shown the suggested formula has a better accuracy in comparison to the current design

standards. Also it is concluded, by adding the vertical component of prestressing tendons which

are located within a distance d/2 from faces of the column to the predicted punching shear

strength, the accuracy of AS 3600-2009 can be improved. ACI 318-05, NZS 3101:2006, and

CSA A23.3-04 underestimate the strength of prestressed slabs, and it is shown that by ignoring

the current limitation on f’c in these standards the accuracy of formulae in predicting strength of

prestressed slabs is improved. The ratios of the observed failure load over the strength

predicted by Eurocode2 and DIN 1045-1 show less divergence as compared to ACI 318-05,

NZS 3101:2006, and CSA A23.3-04.

100

101

Chapter Five

5 CONCENTRIC PUNCHING SHEAR OF FLAT PLATES

WITH SHEAR REINFORCEMENT

5.1 Introduction

In Chapter Two, different methods of strengthening of concrete slabs against punching shear

were presented. As discussed, the use of shear reinforcement is a more favorable solution to

increase punching shear strength of concrete floors due to its aesthetical advantages. Among

the different types of shear reinforcement available in the market, headed shear studs are the

most popular in Europe and North America because of their lower cost, easy installation

procedure, and proven adequate anchorage. Even some of the design guidelines such as ACI

Committee 421 (1999) and CSA A23.3 (2004) allow higher punching shear strength for flat

plates strengthened by shear studs compared to similar flat plates with other types of shear

reinforcement such as stirrups. Issues such as placement, anchorage, and strength of shear

reinforcement should be addressed by the designer to ensure the shear reinforcement is effective

against punching shear. Unfortunately, AS 3600-2009 does not mention shear studs as a type of

shear reinforcement that increases the punching shear strength of flat slabs unlike most

internationally recognised standards. Further, there are no guidelines or design

recommendations for the design of any type of shear reinforcement (including shearheads)

against punching shear of concrete flat plates in AS 3600-2009 unlike other internationally

recognised standards. Here in this chapter, two aspects of design of shear reinforcement are

discussed, namely detailing considerations, and ultimate strength considerations.

In the detailing section, issues such as the arrangement of shear reinforcement, the spacing

between shear reinforcement and the anchorage of shear reinforcement are discussed. In the

ultimate strength section, different types of failure in flat plates reinforced with shear

reinforcement are presented, and a method to calculate the ultimate punching shear strength is

suggested. Further, a comparison is made between the accuracy of different standards and the

proposed method in predicting the ultimate strength of flat plates with shear reinforcement.

102

5.2 Detailing of Shear Reinforcement

Although the recommendations for detailing of shear reinforcement against punching shear are

slightly different among various standards, the main objective of detailing is to provide shear

reinforcement which effectively intersects with critical shear cracks, and delays the punching

shear phenomenon.

The layout of shear reinforcement can be divided into a number of different arrangements such

as the orthogonal type arrangement recommended by standards such as ACI 318-05, CSA

A23.3-04 and NZS 3101:2006, the radial type -star-shape- arrangement recommended in

Eurocode2, DIN 1045-1, and the square arrangement recommended in BS 8110. These types of

arrangement are shown in Figure 5.1. Generally, all mentioned arrangements are capable of

increasing the punching shear strength of a slab provided they are placed symmetrically around

the column (Polak, El-Salakawy & Hammill 2005). However, the orthogonal type arrangement

is found to be more economical (Vollum et al. 2010), and the positioning of shear reinforcement

is more convenient as the flexural reinforcement is also placed orthogonally.

(a) (b) (c)

Figure 5.1 (a) Orthogonal type arrangement (b) Radial type arrangement (c) square type

arrangement of shear reinforcement for punching shear

Another important issue in detailing is the spacing between shear reinforcement. There are

restrictions on the distance between the first row of shear reinforcement and the face of the

column -so-, the radial spacing between rows of shear reinforcement -sr-, and the tangential

spacing between shear reinforcement -st- in design standards. For illustration, so, sr and st are

shown for the orthogonal and the radial arrangement of shear reinforcement in Figure 5.2.

103

st

so sr

st

srso

Figure 5.2 Radial and tangential spacing between shear rows reinforcement in flat plates

The limitation on so is to avoid a premature failure at the face of the column in which the shear

crack develops without intersecting with any of the shear reinforcement elements. This type of

failure is shown in Figure 5.3. Potential shear cracks in flat plates have an angle between 25° to

45° to the plane of the slab, so to ensure they intersect with the first row of shear reinforcement

ACI 318-05, Eurocode2 and CSA A.23.3-04 limit so to less than 0.5d. ACI-ASCE Committee

421 (1999) is more stringent and suggests to place the first row of shear reinforcement between

0.35d and 0.4d from the face of the column.

The radial spacing between two consecutive rows of shear reinforcement -sr- should be limited,

to ensure shear cracks which develop in the shear reinforced zone intersect with shear

reinforcement. ACI 318-05, CSA A23.3-04 and NZS 3101:2006 limit the radial spacing to

0.5d, whereas Eurocode2 and DIN 1045-1 allow 0.75d as the maximum radial spacing.

Further, the limitation on the tangential spacing st was introduced to provide enough

confinement for concrete during loading and reloading which is especially important for the

seismic design of flat plates (Polak, El-Salakawy & Hammill 2005). ACI 318-05, CSA A23.3-

04 and NZS 3101:2006 limit the tangential spacing to 2d, and Eurocode2 and DIN 1045-1 limit

this spacing to 1.5d.

The other important issue in the design of shear reinforcement is to ensure that shear

reinforcement can develop their tensile resistance against punching of the slab. This can be

achieved by providing enough anchorage at the ends of shear reinforcement. In Elgabry d Ghali

(1990), it was suggested shear studs with a steel strip or plate at both ends, having at least area

equal ten times of the stem, can develop a tensile stress of 414 MPa. When stirrups are

provided, they should tie to the flexural reinforcements at the top and bottom with a 135°-180°

hook to ensure the anchorage of stirrups (ACI 318-05 2005).

104

In practice, designers tend to match the spacing of punching shear reinforcements with the

spacing of top flexural reinforcement to ensure shear reinforcement will not interrupt the

flexural reinforcement, but this should not violate mentioned limitations on detailing of shear

reinforcements (Polak, El-Salakawy & Hammill 2005).

5.3 Ultimate Strength of Flat Plates with Shear Reinforcement

Different types of failure were observed in experimental tests on slabs with shear reinforcement.

The first type of failure occurs when the critical crack develops from the bottom surface of the

slab at the face of column to the top surface of the slab with a steep angle as shown in Figure

5.3. In this type of failure, the critical crack misses the first row of provided shear

reinforcement and does not intersect with any of the shear reinforcement. As mentioned in the

previous section, this failure can be avoided by limiting the maximum distance between the first

row of shear reinforcement and the face of the column. The second type of failure occurs when

the critical crack propagates in the region where shear reinforcement is provided. In this case,

shear reinforcement intersects with the surface of failure and increases the strength of slab by

arresting the critical crack from opening. The third type of failure is a phenomenon known as

“web crushing failure” in which the concrete in the region where shear reinforcement is

provided crushes prior to the latter failure. The fourth type of failure occurs when a critical

shear crack develops outside the shear reinforced region. The fifth type of failure is the flexural

failure which can precede any of the other mentioned types of failure. In the case of flexural

failure, the slab shows a ductile behaviour prior to the failure and the ultimate flexural strength

of slabs can be calculated by yield-line theory.

Figure 5.3 Different types of punching shear failure in flat plates with shear reinforcement

Column CL Shear reinforcement

Premature failure at the column face

(the critical crack does not intersect with the shear reinforcement)

Failure in the shear reinforced zone

(the critical crack intesrsect with the shear reinforcement

or web crushing occurs)

Failure outside the shear reinforced zone

105

5.3.1 Failure Inside the Shear Reinforced Region

A critical tie adjacent to the column can be envisaged in flat plates with shear reinforcement as

shown in Figure 5.4 (a). The crack needs to cross the shear reinforcement before causing failure

in this region Figure 5.4 (b). The strength of slab against the development of the critical crack

inside the shear reinforced zone can be quantified by the tensile strength of the critical tie.

(a)

Tensile failure of the critical tie

occurs by the development of cracks

inside shear reinforced region

(b)

Figure 5.4 (a) Critical tie in flat plates with shear reinforcement (b) Failure of the critical tie due to

the development of shear crack inside the shear reinforced region

The schematic view of vertical components of tensile strength of the critical tie is shown in

Figure 5.5. The tensile strength of the critical tie can be divided into the tensile strength of

shear reinforcement intersecting with the hypothetical failure surface and the tensile strength of

the un-cracked concrete section. Vts is the vertical tensile resistance of the shear reinforcement

intersecting with the critical crack and can be quantified by the use of the truss analogy as

expressed in Equation5-1.

%RP � iP¤+PQ¤ 2 �^-�/;k (5-1)

Where

Asv is the cross sectional area of shear reinforcement in one row around the column,

fsyv is the yield strength of the shear reinforcement,

Critical tie

Shear reinforcement

106

sr is the radial spacing between rows of shear reinforcement,

q is the angle between the critical crack and the plane of slab.

In designing members undergoing one-way shear with shear reinforcement, q is assumed to be

between 30° to 45° (Warner et al. 1998). This is in agreement with the reported angle of the

critical crack in two-way flat plate specimens in Pisanty (2005). ACI 318-05, NZS 3101:2006,

and CSA A23.3-04 use q=45°, whereas Eurocode2 uses q=34° angle to calculate the

contribution of shear reinforcement in shear resistance of slabs under punching shear.

Column CL

D/2

dnd

sr

Concrete compression zone

θ

VtsVtc

Critical crack

Shear reinforcements

fct,sp

Figure 5.5 Vertical components of the critical tie which resist punching shear

In addition to the tensile strength of shear reinforcement, there is a contribution from the tensile

strength of the concrete in the compression zone as shown in Figure 5.5. Warner et al. (1998)

discussed that ignoring the contribution of the concrete in the calculation of the ultimate shear

strength of members with shear reinforcement results in a very conservative prediction of the

shear capacity of the member. This is recognised by most of design standards and a proportion

of the ultimate shear strength of concrete is added to the strength of shear reinforcement to

calculate the ultimate punching shear strength of the slab. There is no rationale for the latter

method and the contribution of concrete was obtained empirically.

As shown in Figure 5.5, the author suggested the contribution of the un-cracked concrete can be

taken into account by including the vertical component of the splitting strength of the un-

107

cracked concrete zone. Considering splitting of the compressive zone in 3D, Vtc can be

expressed as following.

%R� � �4. � 42T/-.*�� 2T/-.*��+�R,P5�^;� for slabs with square column (5-2)

%R� � ��9 � 42T/-.*�� 2T/-.*��+�R,P5�^;� for slabs with circular column

Where

a is the side dimension of square column,

D is the diameter of circular column,

dn is the depth of the neutral axis,

q is the angle between the critical crack and the horizontal plane of slab.

fct,sp is the splitting tensile strength of concrete which can be calculated by Equation 5-3.

As discussed in Chapter Three, the depth of the neutral axis can be calculated using Equation 3-

8. Considering recommendations of Model Code 90 (1993) the splitting tensile strength of

concrete can be calculated by Equation 5-3.

+�R,P5 � 0.337�+�,��/3� (5-3)

In which fc’ is the concrete compressive strength in MPa.

The ultimate resistance of a slab against the development of the critical crack inside its shear

reinforced zone -Vit- can be obtained by Equation 5-4.

%WR � %RP � %R� (5-4)

Where, Vts and Vtc are calculated from Equation 5-1 and 5-2 respectively.

As mentioned, another type of failure is web-crushing failure. This failure can occur in flat

plates which are heavily reinforced with shear reinforcement. In these slabs the crushing of the

concrete in the shear reinforced zone may occur prior to the failure of the critical tie in tension.

The web-crushing capacity of two-way concrete slab can be quantified by the available

empirical formulae in standards. AS 3600-2009, ACI 318-05, and NZS 3101:2006 specify the

web crushing strength of a given slab by Equation 5-5.

108

%y�k � 0.5A+�,�&2 (5-5)

Where

Vwcr is web-crushing strength of the slab,

fc’ is concrete compressive strength in MPa,

d is the effective depth of slab,

bo is the perimeter around the column at a distance of d/2 from the face of column.

The lesser of Vit, Vwcr and Vflex should be chosen as the ultimate strength of the flat plate inside

its shear reinforced zone -Vuin- is the lesser of the.

%'WT � F<* �%WR, %y�k, %hKa[� (5-6)

A database of reported experimental test series, which investigated the punching shear strength

of flat plates with shear reinforcement, was gathered from journal articles such as Vollum et al.

(2010), Birkle and Dilger (2008), Gomes and Regan (1999), Marzouk and Jiang (1997),

Mokhtar, Ghali and Dilger (1985), and Seible, Ghali and Dilger (1980). Test specimens with

shear reinforcement placed in the orthogonal type arrangement were used, and these test

specimens were reinforced with different types of shear reinforcement such as shear stud rails,

stirrups, and short cut-offs of steel I beams.

The specimens which reportedly failed in the shear reinforced region were separated to

determine a value for q, inclination of the critical crack, in Equations 5-1 and 5-2. Vit was

calculated for three different scenarios q=45°, q=34°, and q=30°. Then for each scenario, the

ultimate strength of each test specimen of the database which failed inside the shear reinforced

zone Vuin was calculated using Equation 5-6. The ratio of the observed failure load -Vtest- over

the predicted ultimate strength was calculated as provided in Table 5.1. The average, SD, and

CV of Vtest/Vuin were calculated for each scenario to enable the author to choose a value for q.

As it is shown in Table 5.1, q=30° results in an average closer to one and a lower, SD and CV.

Consequently, q=30° is suggested to be used in Equation 5-1 and 5-2.

109

Table 5.1 Vtest/Vuin for test specimens in which failure occurred inside the shear reinforced zone

Reference Specimen

Vtest/Vuin

for

q=30°

Vtest/Vuin

for

q=34°

Vtest/Vuin

for

q=45°

(Birkle & Dilger 2008)

2 1.15 1.15 1.43

8 0.95 1.15 1.28

9 1.14 1.57 1.66

11 0.93 1.08 1.22

12 1.00 1.36 1.47

(Mokhtar, Ghali & Dilger 1985)

AB3 1.02 1.02 1.06

AB4 0.90 0.90 0.90

AB5 0.96 1.02 1.16

AB6 0.90 0.90 0.90

AB8 0.92 0.92 0.92

(Vollum et al. 2010) 2 1.16 1.16 1.16

5 1.18 1.18 1.18

(Gomes & Regan 1999) S2 1.15 1.60 1.68

Average 1.03 1.15 1.23

Standard deviation 0.11 0.23 0.26

Coefficient of variation 0.11 0.20 0.21

5.3.2 Failure Outside the Shear Reinforced Region

As mentioned, in some test specimens punching shear failure occurred outside the shear

reinforced zone. The shear strength of slabs outside the shear reinforced zone can be treated

similar to the shear in beams outside the shear reinforced zone as the confining effect of the

tangential stress is significantly lower in regions away from the column in comparison with the

region adjacent to the column (Polak, El-Salakawy & Hammill 2005).

To deal with this type of failure, standards such as ACI 318-05 and Eurocode2 define a

perimeter outside the shear reinforced zone and require the shear stress on the perimeter to be

less than the allowable one-way shear stress. Unfortunately, AS 3600-2009 does not provide

any provision for designers to check the shear strength outside the shear reinforced zone of slabs

even if they are reinforced with shearheads. Considering that AS 3600-2009 has a very similar

one-way shear formula to the one used in Eurocode2, it is suggested by the author to adopt a

similar control perimeter as Eurocode2 for AS 3600-2009. Although using a similar formula,

Eurocode2 and Model Code 90 do not agree on the distance of the outer control perimeter to the

last row of shear reinforcements. Eurocode2 suggests the control perimeter at a distance equal

to 1.5d from the last row of shear reinforcements whereas Model Code 90 suggests the distance

of 2d. Figure 5.6 shows the outer perimeter for the case of orthogonal type arrangement of

shear reinforcement in Eurocode2 and Model Code 90 and it can be calculated by Equation 5-7.

110

k.d

k.d

st

X

Figure 5.6 Eurocode2 and Model Code 90 control perimeter outside the orthogonal shear

reinforced zone

�&'R � �4;R � 2=�2 � 4√2�� ` �4;R � 2=�2 � 82� (5-7)

Where, uout is the critical perimeter outside the shear reinforced zone, k, X and st are shown in

Figure 5.6.

In this research, test specimens in the gathered database which reportedly failed by punching

outside the shear reinforcement were separated and used to determine the distance of the outer

control perimeter from the last row of shear reinforcement. Three different scenarios were

considered for the outer control perimeter, namely k=1, k=1.5, and k=2. Considering the one-

way shear formula of AS3600-2009, the ultimate strength of the slab outside of the shear

reinforced zone can be calculated by Equation 5-8.

%'&'R � 1.1�1.6 � 2/1000� �?+�,��/3� �&'R 2 (5-8)

Where, uout is given in Equation 5-7 and f’c in MPa.

The punching shear strength of each test specimen, which reportedly failed in the region outside

the reinforced zone, is calculated using Equation 5-8 using three different values of k. Then the

ratio of the observed failure load over the predicted punching shear strength was calculated as

given in Table 5.2. The average, SD, and CV of Vtest/Vuout for each scenario were calculated and

presented in Table 5.2.

√2 X§2d

111

Table 5.2 Vtest/Vuout for test specimens in which failure occurred outside the shear reinforced zone

Reference Specimen

Vtest/Vuout

for

k=1

Vtest/Vuout

for

k=1.5

Vtest/Vuout

for

k=2

(Birkle & Dilger 2008) 4 1.36 1.19 1.06

(Marzouk & Jiang 1997) HS22 1.31 1.15 1.03

HS23 1.28 1.12 1.00

(Seible, Ghali & Dilger 1980)

SC7 1.46 1.29 1.16

SC11 1.39 1.23 1.11

SC12 1.39 1.23 1.11

SC13 1.36 1.20 1.08

SC9 1.39 1.23 1.10

(Gomes & Regan 1999) S4 1.59 1.36 1.18

S5 1.55 1.32 1.15

Average 1.41 1.23 1.10

Standard deviation 0.10 0.07 0.06

Coefficient of variation 0.07 0.06 0.05

In Table 5.2, k=2 results in a better prediction of ultimate punching shear strength of slabs. The

author suggests a similar outer control perimeter as the one shown in Figure 5.6 at the distance

of 2d from the last row of shear reinforcement. This control perimeter can be used with the one-

way shear formula of AS3600-2009 for calculating the punching shear strength of slabs outside

their shear reinforced zone.

5.3.3 Summary of the suggested method

To summarise, the orthogonal arrangement of the shear reinforcement is suggested for the

design proposes due to its convenient placement as compared to the other types of arrangement.

The radial spacing between the first row of shear reinforcement and the face of the column

should be limited to d/2. This is similar for the radial spacing between consecutive rows of

shear reinforcement. Further, the tangential spacing of shear reinforcement should not be more

than 2d. For shear studs, both ends should have an area of at least ten times that of the stem,

and for stirrups, they should tie to the top and bottom flexural reinforcement with a 135°-180°

hook.

For strength considerations, the punching shear strength inside the shear reinforced zone can be

calculated as the lesser of Equation 5-5, and Equation 5-4. Further, the punching shear strength

outside the shear reinforced zone can be calculated by Equation 5-8. The lesser of the

112

aforementioned strengths and the flexural strength, calculated by yield-line theory, determines

the ultimate strength of the slab.

5.3.4 Example

Herein, the ultimate punching shear strength of test specimen 12 from (Birkle & Dilger 2008) is

calculated as an example to illustrate the procedure of the suggested method. Figure 5.7 (a)

shows the top view of the test specimen 12. In Figure 5.7 (a), Bc is equal to 1900mm, and side

dimension of the square column is a=350mm. Figure 5.7 (b) shows the arrangement and the

radial spacing of the shear reinforcement. The effective depth of the test specimen is d=260mm,

the concrete compressive capacity is f’c=33.8MPa, the tensile flexural reinforcement ratio is

ρ=1.1%, and the yield strength of the tensile reinforcement is fsy=524MPa. The provided shear

reinforcement is headed shear stud with the cross sectional area equal to 127mm2, and yield

strength equal to 409MPa.

(a) (b)

Figure 5.7 (a) Top view of test specimen 12 (b) Arrangement of shear reinforcements in the test

specimen 12 (Birkle & Dilger 2008)

Ultimate strength of slab using yield-line theory

According to (Birkle & Dilger 2008), Vflex for a circular slab with a square column is:

Vflex=2p (ρ fsy d2(1-0.59ρfsy/f’c))(Bc)/(Bc-(4a/p))

Vflex=2p(0.011µ524µ2602µ(1-0.59µ0.011µ524/33.8))µ1900/(1900-4µ350/p)=2875kN

Ultimate web crushing strength

Vwcr=0.5√f’c (bo d)=0.5√33.8µ(4µ350+4µ260)µ260=1844kN

Ultimate strength of slab if the crack develops inside the shear reinforced zone

Vit=Vts+Vtc

113

Vts=Asv fsyv d cotq / sr

Eight shear studs are provided in one row of shear reinforcements and be q=30° as suggested

earlier in this chapter.

Vts=(8µ127)µ409µ260µcot30°/195=960kN

Vtc= (4a+4dn/tanq) (dn/tanq) fct,sp cosq

dn=0.8√(nρe)√(35/f’c)d

n=Es/Ec=200µ103/29.4µ10

3=6.8

ρe=ρ(fsy/500)=0.011(524/500)=0.0115

dn=0.8√(6.8µ0.0115)µ√(35/33.8)µ260=59.2mm

fct,sp=0.337(33.8)(2/3)

=3.52MPa

Vtc=(4µ350+4µ59.2/tan30°)µ(59.2/tan30°)µ3.52µcos30°=566kN

Vuit=Vts+Vtc=960+566=1523kN

Ultimate punching shear strength of the test specimen outside the shear reinforced zone

Vuout=1.1 (1.6-d/1000) (ρf’c)(1/3)

uout d

uout= lesser of (4st+2kpd+4√2X) and (4st+2kpd+8d)

uout= (4µ350+2µ2µpµ260+4√2µ(90+5µ195))=10692mm

§ (4µ350+2µ2µpµ260+8µ260)=6747mm

uout=6747mm

Vuout=1.1µ(1.6-260/1000) µ(0.011µ33.8)(1/3)µ6747µ260=1859kN

The ultimate strength of the slab is the lesser of above calculated strengths:

Vus=min(Vwcr, Vuit, Vuout, and Vflex)=1523kN

Reported failure load 1520kN, and the location of failure was reported inside the reinforced

zone as predicted above.

Vtest/Vus=1.00

114

5.4 Comparison of Experimental Results with Design Standards

ACI 318-05, CSA A23.3-04, Eurocode2, and the proposed method were used to predict the

ultimate punching shear strength of the gathered database which included 30 slab specimens

with shear reinforcement. DIN 1045-1 does not recognise the orthogonal type arrangement of

shear reinforcement (Hegger, Sherif & Beutel 2005). To compare the accuracy of mentioned

standards and the proposed formulae, the ratio of Vtest/Vus for each slab were calculated. The

average, SD, and CV of Vtest/Vus for each standard and the proposed method are also given in

Table 5.3. From Table 5.3, it can be concluded ACI 318-05 underestimates the punching shear

strength of test specimens with shear reinforcement, and the accuracy of the formula deviates

significantly. As presented in Chapter Two, the main difference between the CSA A23.3-04

and ACI 318 approaches is that the CSA A23.3-04 adds a higher proportion of the concrete

shear strength to the ultimate punching strength of slabs when shear studs are provided. As a

result CSA A23.3-04 has a closer average to unity and lower CV as compared to the ACI 318-

05. The proposed method and Eurocode2 give a more accurate prediction of the ultimate

strength of slabs with shear reinforcement in comparison with the other two standards.

Table 5.3 Average, SD and CV of Vtest/Vus for ACI 318-05, CSA A23.3, Eurocode2, and the proposed method


ACI 318-05 1.47 0.41 0.28

CSA A23.3 1.26 0.31 0.24

Eurocode2 1.11 0.10 0.09

Proposed method 1.07 0.10 0.09

5.5 Summary

In this chapter, the available recommendations in design guidelines and standards for detailing

of shear reinforcement were reviewed, and the importance of specifying proper spacing between

the shear reinforcement to avoid premature failure was discussed. Then different types of

potential failure in flat plates were explained. In this research it is assumed that the failure

inside the reinforced zone occurred either by the failure of the critical tie in the vicinity of the

column or by web crushing of the slab. A method proposed to calculate the ultimate strength of

the critical tie using a refined truss analogy. In this method, the contribution of the tensile

strength of shear reinforcement intersecting with the critical crack was added to the contribution

of the tensile strength of the un-cracked concrete zone. Further, a control perimeter outside the

115

shear reinforced zone of orthogonal type shear reinforcement arrangement was proposed. This

can be used with the current one-way shear formula of AS 3600-2009 to calculate the punching

shear strength of flat plates outside the shear reinforced zone. Finally, formulae from ACI 318-

05, CSA A23.3, Eurocode2, and the proposed method were used to predict the ultimate

punching shear strength of a number of test specimens reported in the literature, and the

accuracy of each method assessed against the experimental results. It was observed ACI 318-

05, and CSA A23.3 have lower accuracy as compared to Eurocode2 and the proposed method.

116

117

Chapter Six

6 SUMMARY AND CONCLUSIONS

Summary and conclusions of this thesis can be divided into four sections as follow.

6.1 Summary and Findings of Literature Review

• Earlier models for symmetric punching shear failure of flat plates were reviewed and

discussed briefly in Chapter Two. There are various approaches available to quantify

the ultimate punching shear strength of flat plates some of which are significantly

different to others.

• Current available methods to include effects of prestressing forces in the punching shear

strength of flat plates such as the principal tensile stress approach, the equivalent

reinforcement ratio approach, and the decompression approach were discussed.

• The strengthening techniques for increasing punching shear strength of prospective

concrete slabs and existing concrete slabs were briefly presented. This was followed by

a review of different types of shear reinforcement for strengthening of flat plates.

• Current provisions of some of the internationally recognised standards for design of

concrete structures such as ACI 318-05, AS 3600-2009, BS 8110-97, CSA A23.3-04,

DIN 1045-1:2001, Eurocode2, and NZS 3101:2006 for punching shear of flat plates

with no unbalanced moment were reviewed.

• AS 3600-2009 neglects effects of the tensile reinforcement ratio and the slab size factor

on the punching shear stress resistance of flat plates and differs from most of

aforementioned standards.

6.2 Concentric Punching Shear Strength of Flat Plates

• The strut-and-tie method was used to model the transfer of shear force from the slab to

the column. Based on experimental observations it is plausible to assume the punching

shear failure occurs as a result of crushing of the critical concrete strut adjacent to the

column.

• In this study, the basis of the critical compressive strut model, developed by previous

researchers, was used to quantify the punching shear strength of flat plates based on the

118

assumption of crushing of the critical compressive strut. In this model, there is no

consensus on the method to calculate the depth of the neutral axis in the vicinity of the

slab-column connection, compressive strength of the critical prismatic concrete strut,

and the size effect factor, and the inclination of the critical shear crack.

• Three different available methods in the literature were considered to calculate the

depth of the neutral axis, three different formulae were used to calculate the

compressive strength of the critical strut, four different conditions were considered to

calculate the size effect factor, and two different methods were considered to predict the

inclination of the critical shear crack. In total, 72 different formulae were constructed

using various combinations of the above parameters to calculate the punching shear

strength of flat plates. To evaluate the accuracy of these formulae, 152 slab test

specimens, reported in the literature, were gathered. The ratio of the observed failure

load to the predicted failure load was calculated for each of the test specimens using the

mentioned formulae. The average, standard deviation, and coefficient of variation of

these ratios were calculated for each formula. The formula which produced the lowest

coefficient of variation and an average ratio close to unity was selected to predict the

punching shear strength of flat plates.

• The selected formula produced an average of 1.23, standard deviation of 0.19, and

coefficient of variation of 0.15. Further, it was shown the predicted strengths by this

formula have a consistent accuracy for a wide range of slab thicknesses, tensile

reinforcement ratios, and concrete compressive strengths.

• Provisions of AS 3600-2009, ACI 318-05, CSA A23.3-04, DIN 1045-1:2001,

Eurocode2, and NZS 3101:2006 were used to predict the punching shear strength of the

same 152 experimental specimens. The ratio of the actual failure load to the predicted

punching shear strength for each test specimen was calculated. These ratios for AS

3600-2009 and ACI 318-05 have the average of 1.39, standard deviation of 0.28, and

coefficient of variation 0.20. It seems AS 3600-2009, and ACI 318-05 overestimate the

punching shear strength of thick slabs, and underestimate the punching shear strength of

heavily reinforced slabs due to neglect of size effects, and tensile reinforcement effect

in their punching shear formula. NZS 3101:1006 has a similar punching shear to AS

3600-2009 except for including a size effect factor which results in a better prediction

of punching shear strength of thick concrete slabs. DIN 1045-1:2001, and Eurocode2

are capable of predicting the punching shear strength of slabs with a better accuracy as

compared to the other mentioned standards which can be attributed to the inclusion of

the tensile reinforcement ratio. It seems the size factor used in formula of DIN 1045-

1:2001 and Eurocode2 is not capable of maintaining its accuracy for a wide range of

slab thicknesses.

119

6.3 Concentric Punching Shear Strength of Prestressed Flat Plates

• It was discussed that the presence of prestressing tendons can introduce three actions

adjacent to the slab-column connection, namely the in-plane compressive stress due to

prestressing force in tendons, the bending moment due to the eccentricity of tendons

from the neutral axis of the section, and the vertical component of prestressing force in

tendons due to the slope of profile of the tendons. Based on some of the reported

experimental results of prestressed slabs, it was shown these three actions affect the

punching shear strength of slabs.

• The proposed formula for the punching shear strength of concrete slabs with no

prestressing was extended by a decompression method to include the effect of

prestressing forces on the punching shear strength of flat plates. Three different cases

were investigated for the proposed method, namely a case in which only the effect of

the in-plane compressive stress is considered, a case in which the effect of in-plane

force and the effect of the eccentricity of tendons are considered, and a case in which

the effect of all three actions of prestressing forces are considered. To evaluate accuracy

of each case, 46 prestressed slab test specimens, reported in the literature and that had

failed by punching shear, were gathered. The average, standard deviation, and

coefficient of variation of the ratios of the observed failure load to the predicted strength

were calculated for the three cases. The third case had a lower coefficient of variation

of (0.13) and an average closer to one (1.10) as compared to the other two cases. This

method was suggested by the author to be used to calculate the punching shear strength

of prestressed flat plates.

• The current provisions of AS 3600-2009 were used to predict the punching shear

strength of the gathered results of prestressed test specimens. The average, standard

deviation, and coefficient of variation of ratios of the observed failure load to the

predicted failure were 1.40, 0.26, and 0.19 respectively.

• The current punching shear formula of AS 3600-2009 does not include the contribution

of the vertical component of the prestressing tendons in its punching shear formula. It

was shown by including the vertical component of the prestressing tendons, positioned

within the distance of d/2 from the face of column, the accuracy of the current

provisions of AS 3600-2009 can be improved. The average, standard deviation and

coefficient of variation of ratios of the observed failure load to the predicted failure load

are improved to 1.29, 0.19, and 0.14 respectively.

• The provisions of ACI 318-05, NZS 3101:2006, and CSA A23.3-04 were used to

predict the punching shear strength of the gathered results of test specimens. These

standards limit the concrete compressive strength in their formula to the maximum

120

value of 35MPa. It was shown if this limitation is neglected, similar to AS 3600-2009,

a better accuracy in the prediction of punching shear strength of prestressed slabs can be

obtained.

• Eurocode2 and DIN 1045-1:2001 were bench marked against the experimental results,

and both standards show a very good accuracy in prediction of the punching shear

strength of prestressed slabs as compared to ACI 318-05, AS 3600-2009, NZS

3101:2006 and CSA A23.3-04.

6.4 Concentric Punching Shear Strength of Flat Plates with Shear

Reinforcement

• Issues such as arrangement, spacing, and adequate anchorage for detailing of shear

studs and stirrups, which are not mentioned in AS 3600-2009, were discussed.

• Different modes of failure which were observed in the experimental tests by previous

researchers were reviewed.

• It was suggested that the premature failure can be prevented by limiting the radial

spacing of shear reinforcement.

• The formula in ACI 318-05 and AS 3600-2009 for calculating the web crushing

strength of slabs was suggested to be used to quantify the web crushing strength of

slabs.

• To calculate strength of flat plates for the case of failure by the critical shear crack

developing inside the shear reinforced zone, a method was proposed based on the

tensile strength of the critical tie adjacent to the column. This method calculates the

tensile strength of the critical tie by considering the tensile strength of shear

reinforcements intersecting with the critical shear crack and tensile strength of

uncracked concrete zone.

• To calculate punching shear strength outside the shear reinforced zone, it was suggested

to use the one-way shear formula. This approach is adopted by most other standards

such as ACI 318-05 and Eurocode2. Considering the failure load of test specimens

which reportedly failed outside the shear reinforced zone, a control perimeter at a

distance of 2d outside the shear reinforced zone was suggested to be used with the one-

way shear formula of AS 3600-2009 to quantify the punching shear strength of flat

plates outside the shear reinforced zone.

• The ultimate strength of flat plates reinforced with shear reinforcement can be

determined as the lesser of aforementioned strengths and its flexural strength.

• Results from 30 test specimens were gathered to evaluate the latter approach. The ratio

of the observed failure load to the predicted strength was calculated for each test

121

specimen. The average, standard deviation, and coefficient of variation for these ratios

are 1.07, 0.10, and 0.09 respectively.

• This method shows a very good accuracy in prediction of the strength of flat plates

reinforced with shear reinforcements.

• ACI 318-05, CSA A23.3-04, and Eurocode2 were used to predict the punching shear

strength of the same gathered test specimens. The ratios of the observed failure load to

the predicted strength were calculated for the test specimens. The average, standard

deviation, and coefficient of variation of these ratios for ACI 318-05 are 1.47, 0.41, and

0.28. On the other hand, CSA A23.3-04 produced an average of 1.26, standard

deviation of 0.31, and coefficient of variation of 0.24. Eurocode2 resulted in an average

of 1.11, standard deviation of 0.10, and coefficient of variation of 0.09 which shows it is

significantly more accurate as compared to ACI 318-05 and CSA A23.3-04.

122

123

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129

Appendix A

In this appendix, details of test specimens which were used to determine the proposed formula

and to evaluate the accuracy of formulae of standards are presented. Also the reported failure

load of test specimens and the predicted punching shear strength of test specimens, using

formulae in design standards, are provided.

In Table A.1, a reference number is given to each test specimen, and the source in which the test

specimen was reported is provided. The shape of the column of each test specimen is given in

Table A.1 where C stands for the circular column and S stands for the square column. Further,

the thickness of the slab (h), the effective depth of the slab (d), the average compressive strength

of concrete (fcm), the tensile reinforcement ratio of the slab (ρ), the yield strength of the tensile

reinforcement (fsy), the diameter of circular columns (D), and the side dimension of square

columns (a) are provided in Table A.1.

In Table A.2, the predicted punching shear strength of each test specimen of the databank is

provided using the proposed formula, and formulae of design standards.

It should be noted that the reference number for each test specimen is the same in Table A.2 and

Table A.1.

130

Table A.1 Details of collected slab test specimens

No. Source Specimen Column h d f cm ρ f sy D or

shape (mm) (mm) (MPa) (%) (MPa) a(mm)

1 PG1 S 250 210 27.6 1.50 573 260

2 (Guandalini, PG11 S 250 210 31.5 0.75 570 260

3 Burdet & PG3 S 500 456 32.4 0.33 520 520

4 Muttoni 2009) PG6 S 125 96 34.7 1.50 526 130

5 PG7 S 125 100 34.7 0.75 550 130

6 NS1 S 120 95 42.0 1.47 490 150

7 HS2 S 120 95 70.0 0.84 490 150

8 HS7 S 120 95 74.0 1.19 490 150

9 HS3 S 120 95 69.0 1.47 490 150

10 HS4 S 120 95 66.0 2.37 490 150

11 (Marzouk & NS2 S 150 120 30.0 0.94 490 150

12 Hussein 1991) HS5 S 150 125 68.0 0.64 490 150

13 HS6 S 150 120 70.0 0.94 490 150

14 HS8 S 150 120 69.0 1.11 490 150

15 HS9 S 150 120 74.0 1.61 490 150

16 HS10 S 150 120 80.0 2.33 490 150

17 HS12 S 90 70 75.0 1.52 490 150

18 HS13 S 90 70 68.0 1.87 490 150

19 HS14 S 120 95 72.0 1.47 490 220

20 HS15 S 120 95 71.0 1.47 490 300

21 (fib 2001) HSLW1.5 S 150 115 75.5 1.50 435 250

22 HSLW2 S 150 115 74.0 2.00 435 250

23 NSLW1 S 150 115 36.2 1.00 435 250

24 NSC1 S 200 158 35.0 2.17 400 250

25 HSC1 S 200 138 69.0 2.48 400 250

26 HSC2 S 200 128 70.0 2.68 400 250

27 HSC3 S 200 158 67.0 1.67 400 250

28 HSC4 S 200 158 61.0 1.13 400 250

29 HSC5 S 150 113 70.0 1.88 400 250

30 NS4 S 300 218 40.0 0.73 400 250

31 HS2 S 300 218 64.7 0.73 400 250

32 HS3 S 350 263 65.4 1.44 400 400

33 NS5 S 400 313 40.0 1.58 400 400

34 ND65-1-1 S 320 275 64.3 1.50 500 200

35 ND65-2-1 S 240 200 70.2 1.70 500 150

36 ND95-1-1 S 320 275 83.7 1.50 500 200

37 ND95-1-3 S 320 275 89.9 2.50 500 200

38 ND95-2-1 S 240 200 88.2 1.70 500 150

39 ND95-2-1D S 240 200 86.7 1.70 500 150

40 ND95-2-3 S 240 200 89.5 2.60 500 150

131


Column h d f cm ρ f sy D or

shape (mm) (mm) (MPa) (%) (MPa) a (mm)

41 ND95-2-3D S 240 200 80.3 2.60 500 150

42 ND95-2-3D+ S 240 200 98.0 2.60 500 150

43 ND95-3-1 S 120 88 85.1 1.80 500 100

44 ND115-1-1 S 320 275 112.0 1.50 500 200

45 ND115-2-1 S 240 200 119.0 1.70 500 150

46 ND115-2-3 S 240 200 108.1 2.60 500 150

47 P100 S 135 100 39.4 0.97 488 200

48 P150 S 190 150 39.4 0.90 488 200

49 P200 S 240 200 39.4 0.83 465 200

50 P300 S 345 300 39.4 0.76 468 200

51 P400 S 450 400 39.4 0.76 468 300

52 P500 S 550 500 39.4 0.76 433 300

53 (Birkle & 1 S 160 124 33.1 1.54 488 250

54 Dilger 7 S 230 190 33.5 1.30 531 300

55 2008) 10 S 300 260 31.0 1.10 524 350

56 HSC0 C 240 200 94.0 0.80 643 250

57 HSC1 C 245 200 91.0 0.80 627 250

58 HSC2 C 240 194 86.0 0.80 620 250

59 HSC4 C 240 200 92.0 1.20 596 250

60 HSC6 C 239 201 109.0 0.60 633 250

61 N/HSC8 C 242 198 95.0 0.80 631 250

62 12 C 125 98 60.4 1.30 550 150

63 13 C 125 98 43.6 1.30 550 150

64 14 C 125 98 60.8 1.30 550 150

65 21 C 125 98 41.9 1.30 650 150

66 22 C 125 98 84.2 1.30 650 150

67 25 C 125 98 32.9 1.20 650 150

68 26 C 125 98 37.6 1.20 650 150

69 27 C 125 98 33.7 1.00 650 150

70 S2.1 C 240 200 24.2 0.80 657 250

71 S2.2 C 240 199 22.9 0.80 670 250

72 S2.3 C 240 200 25.4 0.50 668 250

73 S2.4 C 240 197 24.2 0.50 664 250

74 S1.1 C 120 100 28.6 0.80 706 125

75 S1.2 C 120 99 22.9 0.80 701 125

76 I/1 S 100 77 25.8 1.39 500 200

77 I/2 S 100 77 23.4 1.20 500 200

78 I/3 S 100 77 27.4 0.92 500 200

79 I/4 S 100 77 32.3 1.20 500 200

80 I/5 S 100 79 28.2 0.87 480 200

No. Source Specimen

(fib 2001)

(Li 2000)

(fib 2001)

(fib 2001)

132




81 I/6 S 100 79 21.9 0.80 480 200

82 II/1 S 250 200 34.9 1.00 530 250

83 II/2 S 160 128 33.3 1.00 485 160

84 II/3 S 160 128 34.3 1.00 485 160

85 II/4 S 80 64 33.3 1.00 480 80

86 II/5 S 80 64 34.3 1.00 480 80

87 II/6 S 80 64 36.2 1.00 480 80

88 III/1 S 120 95 23.2 0.80 494 150

89 III/2 S 120 95 9.5 0.80 494 150

90 III/3 S 120 95 37.8 0.80 494 150

91 III/4 S 120 93 11.9 1.50 464 150

92 III/5 S 120 93 26.8 1.50 464 150

93 III/6 S 120 93 42.6 1.50 464 150

94 V/1 S 150 118 34.3 0.80 628 54

95 V/2 S 150 118 32.2 0.80 628 170

96 V/3 S 150 118 32.4 0.80 628 110

97 V/4 S 150 118 36.2 0.80 628 102

98 A1/M1 S 140 114 16.3 1.10 255 203

99 A1/M2 S 140 117 15.5 1.50 282 203

100 A1/M3 S 140 121 14.2 1.90 282 203

101 A1/M4 S 140 124 14.0 1.00 432 203

102 A1/M5 S 140 117 21.0 1.20 432 203

103 A2/M2 S 140 117 32.8 1.50 282 203

104 A2/M3 S 140 121 32.5 1.90 282 203

105 A2/T1 S 140 124 39.3 1.00 432 203

106 A2/T2 S 140 124 41.4 1.70 432 203

107 A3/M1 S 140 124 18.8 1.00 255 203

108 A3/M2 S 140 102 19.3 1.70 282 203

109 A3/M3 S 140 117 27.3 1.90 282 203

110 A3/T1 S 140 121 20.6 1.00 432 203

111 A3/T2 S 140 119 16.0 1.20 432 203

112 A4/M1 S 140 114 38.3 1.10 255 203

113 A4/M2 S 140 119 29.2 1.50 282 203

114 A4/M3 S 140 117 32.2 1.90 322 203

115 A4/T1 S 140 114 32.8 1.10 432 203

116 A4/T2 S 140 117 29.3 1.20 432 203

117 II-1 C 102 82 10.5 1.20 457 221

118 II-4a C 102 82 17.9 0.90 559 221

119 II-4b S 102 82 9.8 0.90 466 201

120 II-4c S 102 82 13.9 0.90 510 201

(fib 2001)

Source SpecimenNo.

133




121 IIB20-2 C 128 108 15.0 0.90 500 201

122 IIB30-1 C 102 80 17.6 2.00 403 300

123 II-2 C 102 82 9.8 1.30 373 221

124 II-3 S 102 82 13.5 1.30 491 301.5

125 II-6 C 102 82 21.6 1.30 456 221

126 II-9 S 102 79 9.3 0.85 550 201

127 II-3 C 102 82 18.1 1.20 559 221

128 II-7 C 102 82 10.0 0.70 456 119

129 II-10 C 102 82 11.7 1.00 385 119

130 S1-60 S 152 114 23.3 1.10 399 254

131 S1-70 S 152 114 24.5 1.10 483 254

132 S5-60 S 152 114 22.2 1.10 399 254

133 S5-70 S 152 114 23.0 1.10 483 254

134 R1 S 152 114 26.6 1.40 328 254

135 R2 S 152 114 27.6 1.40 328 254

136 H1 S 152 114 26.1 1.10 328 254

137 M1A S 152 114 20.8 1.50 481 254

138 VIII B-9 S 152 114 35.1 2.00 341 254

139 VIII B-11 S 152 114 40.4 3.00 325 254

140 VIII-14 S 152 114 38.2 0.90 303 254

141 14/1 S 140 112 26.4 1.31 500 200

142 14/2 S 140 112 22.8 1.31 500 200

143 16/1 S 160 133 25.0 0.95 500 200

144 (Pisanty 16/2 S 160 133 19.0 0.95 500 200

145 2005) 18/1 S 180 151 23.3 1.18 500 250

146 18/2 S 180 151 25.5 1.18 500 250

147 20/1 S 200 171 24.1 1.04 500 300

148 20/2 S 200 171 21.8 1.04 500 300

149 IA15a-5 C 149 117 27.9 0.80 454 150

150 IA30a-6 C 151 118 25.8 0.80 441 150

151 IA30a-24 C 158 128 25.9 1.00 456 300

152 IA30a-25 C 154 124 24.6 1.10 451 300

No.

(fib 2001)

(fib 2001)

Source Specimen

134

Table A.2 Predicted punching shear strength of collected test specimens

Proposed AS 3600 NZS CSA EC2 DIN

No. V test formula ACI 318 3101 A23.3-04 MC90 1045-1

(kN) (kN) (kN) (kN) (kN) (kN) (kN)

1 1023 973 705 668 788 931 910

2 763 728 753 714 842 775 742

3 2153 1824 3445 2215 3438 2307 2198

4 238 224 174 169 194 219 211

5 241 169 184 179 206 187 179

6 320 243 205 199 229 239 233

7 249 233 265 257 296 237 230

8 356 278 272 264 304 271 263

9 356 296 263 255 294 284 276

10 418 335 257 250 287 310 301

11 396 242 241 234 270 272 261

12 365 286 386 374 431 341 326

13 489 330 369 358 412 365 349

14 436 353 366 355 409 384 367

15 543 430 379 368 424 445 426

16 645 490 394 383 440 491 470

17 258 212 181 176 203 179 178

18 267 223 173 168 193 186 185

19 498 411 345 335 386 333 332

20 560 531 430 417 481 382 391

21 538.5 582 496 481 554 486 483

22 613.4 649 491 477 549 531 528

23 432.1 353 343 333 384 330 328

24 678 684 516 501 577 685 667

25 788 806 602 584 673 692 680

26 801 762 548 532 612 615 607

27 802 826 714 693 799 807 785

28 811 673 682 662 762 686 668

29 480 585 464 450 519 493 491

30 882 721 875 814 977 869 829

31 1023 907 1112 1035 1243 1026 978

32 2090 2089 1913 1620 2138 1955 1897

33 2234 2000 1915 1487 2120 2198 2109

34 2050 1381 1425 1179 1592 1774 1650

35 1200 845 798 774 891 1095 1019

36 2250 1534 1625 1345 1816 1941 1805

37 2400 1798 1684 1394 1883 2189 2035

38 1100 921 894 868 999 1183 1102

39 1300 915 886 860 991 1176 1095

40 1450 999 901 874 1007 1255 1169

135





41 1250 960 853 828 953 1210 1127

42 1450 1034 942 915 1053 1294 1205

43 330 246 208 201 232 254 242

44 2450 1729 1880 1556 2101 2142 1992

45 1400 1035 1039 1008 1161 1309 1219

46 1550 1073 990 961 1106 1338 1246

47 330 267 256 249 286 246 243

48 583 428 448 435 501 470 452

49 904 576 683 663 763 752 711

50 1381 872 1280 1015 1431 1372 1271

51 2224 1537 2390 1640 2481 2343 2182

52 2681 1826 3415 2096 3308 3366 3103

53 483 479 363 352 406 416 412

54 825 909 733 711 819 849 826

55 1046 1349 1201 1022 1342 1306 1257

56 965 936 932 905 1042 997 941

57 1021 921 917 890 1025 986 931

58 889 864 853 828 954 917 867

59 1041 1055 922 895 1031 1133 1070

60 960 903 1010 979 1130 959 906

61 944 953 923 896 1032 983 929

62 319 245 202 196 225 255 244

63 297 218 171 166 192 228 218

64 341 246 202 196 226 256 244

65 286 234 168 163 188 225 215

66 405 301 238 231 266 286 273

67 244 212 149 144 166 201 192

68 294 218 159 154 178 211 201

69 227 193 151 146 168 191 182

70 603 574 473 459 529 624 589

71 600 569 456 443 510 606 572

72 489 465 484 470 541 542 512

73 444 450 462 449 517 519 490

74 216 157 128 125 144 165 156

75 194 150 113 110 127 150 142

76 194 194 147 143 165 158 159

77 176 177 140 136 157 146 147

78 194 164 152 147 170 141 142

79 194 195 165 160 184 163 164

80 165 162 159 154 178 145 146

136





81 165 144 140 136 157 129 130

82 825 643 567 551 635 764 721

83 390 291 289 281 323 328 314

84 365 293 294 285 328 331 317

85 117 72 72 70 81 82 78

86 105 73 73 71 82 83 79

87 105 64 59 57 66 79 75

88 197 131 120 116 134 147 141

89 123 129 98 95 109 113 110

90 214 177 195 189 218 188 183

91 154 180 106 103 119 148 144

92 214 207 159 154 178 199 193

93 248 236 201 195 224 234 228

94 170 142 162 157 181 214 195

95 280 276 262 254 293 267 258

96 265 207 208 202 233 238 224

97 285 202 213 206 238 243 228

98 322 184 198 193 222 233 228

99 346 229 200 195 224 264 259

100 307 263 201 195 225 293 286

101 259 240 206 200 231 245 239

102 346 273 233 226 261 274 269

103 419 282 292 283 326 346 339

104 430 324 304 295 340 395 386

105 419 318 346 336 386 355 346

106 439 414 355 344 397 431 420

107 247 199 239 232 267 273 266

108 336 225 186 180 208 239 236

109 298 298 266 258 297 351 344

110 328 257 242 235 270 271 265

111 298 261 208 202 233 255 250

112 259 243 304 295 340 316 310

113 341 276 282 273 315 342 334

114 541 333 289 280 323 372 364

115 384 291 281 273 315 299 293

116 402 299 276 268 308 309 302

117 181 138 86 83 96 112 111

118 245 147 112 109 126 125 124

119 162 128 99 96 111 106 106

120 215 145 118 114 132 121 121

137





121 307 180 138 134 154 177 171

122 239 221 136 132 152 178 181

123 152 127 83 80 93 112 111

124 244 223 157 153 176 165 170

125 240 167 123 120 138 151 150

126 157 129 92 89 102 95 96

127 201 170 113 110 126 138 137

128 117 73 56 54 62 75 71

129 98 83 60 58 67 90 85

130 389 308 275 267 308 289 288

131 393 341 282 274 316 294 293

132 343 303 269 261 300 284 283

133 378 335 274 266 306 288 287

134 312 328 294 286 329 329 327

135 394 333 300 291 335 333 332

136 372 294 291 283 326 301 300

137 433 375 260 253 291 308 307

138 505 430 338 328 378 408 406

139 578 442 363 352 405 428 427

140 334 301 352 342 394 322 321

141 390 308 244 237 273 284 278

142 355 296 227 220 254 269 264

143 376 315 301 292 337 332 322

144 445 300 263 255 293 301 292

145 581 485 398 386 444 463 452

146 606 497 416 404 465 479 467

147 835 618 538 522 601 589 577

148 822 599 511 496 572 569 557

149 255 175 176 171 197 226 214

150 275 171 171 166 192 223 211

151 430 346 298 289 333 341 335

152 408 340 279 271 312 328 323

138

139

Appendix B

In this appendix, details of prestressed slab test specimens which were used to evaluate the

suggested method are presented. Also, the reported failure load of the test specimens and the

predicted punching shear strength of the test specimens using formulae of design standards are

provided.

In Table B. 1, a reference number is given to each test specimen, and the source in which the

test specimen was reported is provided. The shape of the column of the test specimen is given

in Table B. 1 where C stands for the circular column and S stands for the square column.

Further, the thickness of slabs (h), the effective depth of tensile reinforcement (d), the average

compressive strength of concrete (fcm), the tensile reinforcement ratio of the slab (ρ), the yield

strength of the tensile reinforcement (fsy), the diameter of circular columns (D), the side

dimension of square columns (a), the ratio of the average bending moment to the shear force

(m/V), the average compressive stress in the slab (σcp), and the depth of prestressing tendons

over the column (dp) are given in Table B. 1.

In Table B. 2, the predicted punching shear strength of each test specimen of the databank is

provided using the proposed formula.

In Table B. 3, the predicted punching shear strength of each test specimen is given using AS

3600-2009, ACI 318-05, CSA A23.3-04, Eurocode2, and DIN 1045-1. Also, the predicted

punching shear of each test specimen is given using AS 3600-2009 with inclusion of the vertical

component of tendons crossing within a distance of d/2 from faces of the column. The

predicted punching shear strength of test specimens are also calculated for ACI 318-05 and

CSA A23.3-04 with ignoring the limit on f’c.

It should be mentioned that the reference number for each test specimen is the same in Table B.

1, Table B. 2, and Table B. 3.

140

Table B. 1 Details of collected prestressed slab test specimens

Column h d f'c

ρ fsy

D or €cpd

p

shape (mm) (mm) (Mpa) (%) (MPa) a (mm) (MPa) (mm)

1 A1 S 125 109 37.8 0.62 553 100 3.31 91

2 A2 S 127 113 37.8 0.47 553 100 2.14 97

3 A3 S 128 109 37.8 0.62 553 100 3.16 86

4 A4 S 129 104 37.8 0.51 553 100 1.98 86

5 B1 S 124 114 40.1 0.60 553 200 3.39 98

6 B2 S 124 110 40.1 0.48 553 200 2.23 94

7 B3 S 124 108 40.1 0.63 553 200 3.12 90

8 B4 S 124 106 40.1 0.50 553 200 2.16 89

9 C1 S 126 111 41.6 0.61 525 300 3.33 94

10 C2 S 122 105 41.6 0.50 525 300 2.26 89

11 C3 S 124 106 41.6 0.64 525 300 3.48 90

12 (Silva, C4 S 123 102 41.6 0.52 525 300 2.31 85

13 Regan & D1 S 124 100 44.1 0.68 540 200 3.34 83

14 Melo 2005) D2 S 123 106 44.1 0.50 540 200 2.23 90

15 and D3 S 125 103 44.1 0.51 540 200 2.27 90

16 (Silva, D4 S 125 111 44.1 0.48 540 300 2.22 95

17 Regan & LP2 S 130 105 52.4 1.70 500 150 2.19 65

18 Melo 2007) LP3 S 130 105 52.4 1.70 500 150 4.28 65

19 LP4 S 130 105 50.7 1.70 500 150 0.8 81

20 LP5 S 130 105 50.7 1.70 500 150 1.33 81

21 LP6 S 130 105 52.4 1.70 500 150 1.76 81

22 SP1 S 175 140 36.5 2.70 500 150 3.94 135

23 SP4 S 175 140 41.7 2.70 500 150 4.28 135

24 SP5 S 175 140 40.9 2.70 500 150 3.28 135

25 SP6 S 175 140 42.5 2.70 500 150 3.5 135

26 M4 S 160 134 51.9 0.92 500 180 1.95 120

27 V1 C 150 124 33.6 0.62 500 200 1.7 114

28 V2 C 150 123 36 0.90 500 200 1.66 114

29 V3 C 150 122 36 0.62 500 200 3.09 114

30 V6 C 150 120 30.4 0.62 500 200 1.77 75

31 V7 C 150 124 31.2 0.62 500 200 1.77 114

32 V8 C 150 124 35.2 0.62 500 200 1.77 114

33 PC1 S 250 210 44 0.77 591 260 0 0

34 (Clemente & PC3 S 250 210 43.8 0.77 591 260 0 0

35 Muttoni 2010) PC2 S 250 210 45.3 1.40 577 260 0 0

36 PC4 S 250 210 44.4 1.40 577 260 0 0

37 AR5 S 100 80 35.7 1.60 523 200 2 0

38 AR7 S 100 80 43.9 1.60 523 200 2.75 0

39 AR8 S 100 80 41.6 1.60 481 200 0 0

40 AR10 S 100 80 41.4 1.60 481 200 0 62

41 (Ramos & AR11 S 100 80 38 1.60 481 200 0 62

42 Lucio 2006) AR12 S 100 80 31.3 1.60 481 200 0 62

43 AR13 S 100 80 32.5 1.60 481 200 0 62

44 AR14 S 100 80 28.2 1.60 481 200 0 62

45 AR15 S 100 80 31.7 1.60 481 200 0 62

46 AR16 S 100 80 30.6 1.60 481 200 0 62

No. Source Specimen

141

Table B. 2 Predicted punching shear strength of collected test specimens using the suggested

method

V test V uo mo V o me V e V p V up

(kN) (kN) (kN.m)/m (kN) (kN.m)/m (kN) (kN) (kN)

1 380 142 9 70 12 96 9 3182 315 129 6 47 9 74 10 2613 353 144 9 70 9 73 0 2874 321 125 5 45 5 45 0 2145 582 247 9 79 15 138 32 4966 488 217 6 52 9 80 30 3807 520 240 8 73 11 98 13 4248 459 214 6 50 7 66 13 3439 720 341 9 90 13 133 41 60610 557 297 6 58 8 79 35 46911 637 333 9 91 12 124 18 56612 497 296 6 60 7 68 15 44013 497 240 9 78 9 79 10 40714 385 217 6 51 8 71 12 35115 395 219 6 54 8 71 0 34416 532 322 6 59 9 93 40 51317 355 322 6 53 0 0 0 37518 415 322 12 104 0 0 0 42619 390 318 2 19 2 14 8 36020 475 318 4 32 3 24 13 38721 437 322 5 43 4 32 11 40722 988 460 20 159 33 258 19 89523 884 476 22 172 36 281 21 95024 780 473 17 132 27 215 0 82025 728 479 18 141 29 229 0 84926 773 377 8 59 12 88 27 55027 450 219 6 51 10 79 66 41428 525 264 6 50 10 77 61 45129 570 220 12 93 18 143 116 57130 375 206 7 53 0 0 0 25931 475 214 7 53 10 82 68 41632 518 222 7 53 10 82 70 42733 1201 839 0 0 75 375 0 121434 1338 837 0 0 150 750 0 158735 1397 1099 0 0 75 375 0 147436 1433 1090 0 0 150 750 0 184037 251 240 3 24 0 0 0 26438 288 259 5 33 0 0 0 29139 380 244 0 0 0 0 72 31640 371 243 0 0 0 0 56 29941 342 236 0 0 0 0 40 27642 280 222 0 0 0 0 66 28843 261 225 0 0 0 0 34 25944 208 216 0 0 0 0 0 21645 262 223 0 0 0 0 0 22346 351 221 0 0 0 0 74 295

No.

142

Table B. 3 Predicted punching shear strength of collected test specimens using formulae of design

standards

AS 3600- AS 3600- ACI 318-05 ACI 318-05 CSA A23.3 CSA A23.3 DIN

No. 2009 2009+V p f' c§ 35MPa no limit on f' c f' c§ 35MPa no limit on f' c 1045-1

(kN) (kN) (kN) (kN) (kN) (kN) (kN) (kN)

1 281 290 256 262 301 309 311 2552 263 273 237 244 282 290 283 2313 277 277 243 249 288 296 274 2434 228 228 196 202 235 242 220 1935 454 486 424 441 495 517 440 3756 385 415 356 372 420 440 344 3087 411 424 366 382 431 452 364 3258 363 376 320 335 380 399 314 2779 582 623 536 565 627 662 501 44410 488 524 443 469 523 555 381 36011 557 575 493 520 578 612 440 40412 473 489 411 436 489 520 360 33113 391 401 336 362 396 428 343 30414 380 392 322 349 383 416 323 28515 367 367 300 326 358 390 297 26316 534 573 475 513 561 607 420 39217 334 334 254 296 305 354 365 35418 401 401 297 338 374 427 407 39619 285 293 217 255 256 299 350 33020 302 315 240 277 284 328 373 34621 320 331 252 293 299 348 392 35622 526 544 468 474 569 577 774 65923 565 586 471 496 587 621 809 69724 513 513 439 462 519 548 688 64025 530 530 450 478 530 566 721 65426 511 538 415 477 491 567 540 49027 313 379 342 342 397 397 424 39328 317 377 337 340 392 396 452 42629 366 482 442 445 503 506 545 48130 290 290 257 257 308 308 286 30431 307 374 339 339 393 393 422 39032 322 392 354 355 410 411 438 40433 890 890 679 761 771 864 877 83934 888 888 679 759 771 862 875 83835 903 903 679 772 771 877 1083 103736 894 894 679 764 771 868 1075 102937 236 236 208 209 249 251 226 22838 276 276 228 246 271 294 252 25439 196 268 226 240 247 263 280 28140 196 252 210 223 231 246 264 26541 188 228 194 200 215 222 242 24242 170 236 212 212 231 231 254 25543 174 208 182 182 203 203 259 22644 162 162 138 138 157 157 212 18245 172 172 147 147 166 166 189 19046 169 243 218 218 238 238 261 262

EC2

143

Appendix C

In this appendix, details of test specimens with shear reinforcement which were used to evaluate

the suggested method are presented. Also, the reported failure load of test specimens and the

predicted punching shear strength of test specimens using formulae of ACI 318-05, CSA A23.3-

04, and Eurocode2 are provided.

In Table C.1, a reference number is given to each test specimen, and the source in which the test

specimen was reported is provided. Further, the thickness of the slab (h), the effective depth of

tensile reinforcement (d), the average compressive strength of concrete (fcm), the tensile

reinforcement ratio of the slab (ρ), the yield strength of the tensile reinforcement (fsy), the side

dimension of square columns (a), the cross sectional area of shear reinforcement in one row

around the column (Asv), the yield strength of shear reinforcement (fsvy), the radial distance

between the first row of shear reinforcement and the face of column (so), the radial spacing

between rows of shear reinforcement (sr), and the tangential spacing between shear

reinforcement are given in Table C.1.

In Table C.2, the predicted punching shear strength of each test specimen of the databank is

provided using the suggested formulae.

In Table C.3, the predicted punching shear strength of each test specimen is provided using ACI

318-05.

In Table C.4, the predicted punching shear strength of each test specimen is presented using

Eurocode2.

In Table C.5, the predicted punching shear strength of each test specimen is given using DIN

1045-1.

It should be mentioned that the reference number for each test specimen is the same in Table

C.1, Table C.2, Table C.3, Table C.4, and Table C.5.

144

Table C.1 Details of collected slab test specimens with shear reinforcement

No. of side

Test h d f' c ρ f sy A sv f svy s o s r s t shear dimnesion of

specimen (mm) (mm) (MPa) (%) (MPa) (mm) 2 (MPa) (mm) (mm) (mm) reinforcment square column

rows (mm)

1 4 160 124 38 1.54 488 568 465 30 60 250 5 250

2 (Birkle & 2 160 124 29 1.54 488 568 393 45 90 250 6 250

3 Dilger 8 230 190 35 1.3 531 568 460 50 100 300 5 300

4 2008) 9 230 190 35.2 1.3 531 568 460 75 150 300 6 300

5 11 300 260 30 1.1 524 1016 409 65 130 350 5 350

6 12 300 260 33.8 1.1 524 1016 409 95 195 350 6 350

7 AB3 150 142 23 1.1 516 852 278 70 105 250 8 250

8 (Mokhtar, AB4 150 142 41 1.1 516 852 278 70 105 250 8 250

9 Ghlia & AB5 150 142 30 1.1 516 852 278 70 105 250 8 250

10 Dilger AB6 150 142 29 1.1 516 852 278 70 105 250 6 250

11 1985) AB7 150 142 35 1.1 448 852 278 70 105 250 6 250

12 AB8 150 142 30 1.1 448 852 278 70 105 250 5 250

13 (Marzouk & HS22 150 120 60 1.1 490 2120 400 60 90 250 3 250

14 Jiang 1997) HS23 150 120 60 1.1 490 942 400 60 90 250 4 250

15 SC7 150 121 33.6 1.17 450 868 350 60 120 310 2 310

16 SC11 150 121 33.6 1.17 450 992 500 60 120 310 2 310

17 (Seible, SC12 150 121 33.6 1.17 450 496 500 30 50 310 4 310

18 Ghali & SC13 150 121 33.6 1.17 450 496 500 30 50 310 4 310

19 Dilger 1980) SC8 150 121 33.6 1.17 450 900 490 60 120 310 2 310

20 SC9 150 121 33.6 1.17 450 600 490 60 60 310 3 310

21 SC10 150 121 33.6 1.17 450 500 490 60 40 310 4 310

22 (Vollum, 2 220 174 24 1.28 567 628 560 90 90 270 10 270

23 Abdel-Fattah, 3 220 174 27.2 1.28 567 628 560 90 90 270 6 270

24 Eder & 4 220 174 27.2 1.28 567 628 560 90 90 270 6 270

25 Elghazouli 5 220 174 23.3 1.28 567 628 560 90 90 270 10 270

26 2010) 6 220 174 23.3 0.64 567 628 560 90 90 270 10 270

27 S2 200 159 34.5 1.26 670 225 450 80 80 150 2 200

28 (Gomes & S3 200 159 39.2 1.26 670 300 450 80 80 150 2 200

29 Regan 1999) S4 200 159 32.1 1.26 670 400 450 80 80 150 3 200

30 S5 200 159 34.7 1.26 670 630 450 80 80 150 4 200

No. Source

145

Table C.2 Predicted punching shear strength of slab test specimens with shear reinforcement using

the suggested method

V test V flex V wcr V ts V tc V it V uout V us

(kN) (kN) (kN) (kN) (kN) (kN) (kN) (kN)

1 634 922 572 204 944 1149 598 572

2 574 922 499 217 532 749 546 499

3 1050 1845 1102 381 859 1240 1158 1102

4 1091 1845 1105 381 573 953 1160 953

5 1620 2875 1737 588 1438 2026 1787 1737

6 1520 2875 1844 566 960 1523 1859 1523

7 545 590 534 228 554 783 565 534

8 583 648 713 201 975 1176 685 648

9 583 646 610 214 554 768 617 610

10 541 615 600 216 975 1191 610 600

11 572 562 659 190 554 745 650 562

12 508 550 610 197 975 1172 617 550

13 605 623 688 157 1956 2113 590 590

14 590 623 688 157 869 1026 590 590

15 623 623 605 199 530 729 538 538

16 596 623 605 199 865 1065 538 538

17 595 623 605 199 1038 1238 538 538

18 580 623 605 199 1038 1238 538 538

19 592 623 605 199 769 969 538 538

20 594 623 605 199 1026 1225 538 538

21 537 623 605 199 1282 1482 538 538

22 876 1225 757 360 1176 1536 858 757

23 884 1225 806 347 1176 1524 894 806

24 888 1225 806 347 1176 1524 894 806

25 880 1225 746 363 1176 1539 849 746

26 748 752 746 236 1176 1412 674 674

27 693 1403 671 256 348 604 669 604

28 773 1431 715 247 464 711 698 698

29 853 1385 647 261 619 880 721 647

30 853 1404 672 255 975 1230 740 672

No.

146


ACI 318-05

V test V flex

(kN) (kN) V max (kN) V sd (kN) V uout (kN) V us (kN)

1 634 922 572 670 393 393

2 574 922 499 472 471 471

3 1050 1845 1102 806 818 806

4 1091 1845 1105 660 1227 660

5 1620 2847 1737 1404 1318 1318

6 1520 2847 1844 1163 2104 1163

7 545 590 534 497 689 497

8 583 648 713 708 921 648

9 583 646 610 522 787 522

10 541 615 600 670 620 600

11 572 562 659 538 681 538

12 508 550 610 674 552 550

13 605 623 688 1358 426 426

14 590 623 688 729 507 507

15 623 623 605 506 310 310

16 596 623 605 610 310 310

17 595 623 605 692 310 310

18 580 623 605 692 310 310

19 592 623 605 572 310 310

20 594 623 605 696 310 310

21 537 623 605 820 310 310

22 876 1225 757 748 966 748

23 884 1225 806 764 714 714

24 888 1225 806 764 714 714

25 880 1225 746 744 951 744

26 748 752 746 744 951 744

27 693 1403 671 405 342 342

28 773 1431 715 480 365 365

29 853 1385 647 539 399 399

30 853 1404 672 735 487 487

No.ACI 318-05

147


Eurocode2

V test V flex


1 634 922 1199 823 540 540

2 574 922 954 628 491 491

3 1050 1845 2059 1128 1084 1084

4 1091 1845 2069 969 1086 969

5 1620 2847 2883 1929 1640 1640

6 1520 2847 3192 1650 1711 1650

7 545 590 890 786 509 509

8 583 648 1460 868 624 624

9 583 646 1125 817 559 559

10 541 615 1092 826 553 553

11 572 562 1282 835 590 562

12 508 550 1125 830 559 550

13 605 623 1642 1513 535 535

14 590 623 1642 853 535 535

15 623 623 1309 670 488 488

16 596 623 1309 722 488 488

17 595 623 1309 806 488 488

18 580 623 1309 806 488 488

19 592 623 1309 683 488 488

20 594 623 1309 810 488 488

21 537 623 1309 938 488 488

22 876 1225 1223 1004 788 788

23 884 1225 1367 1026 824 824

24 888 1225 1367 1026 824 824

25 880 1225 1191 999 780 780

26 748 752 1191 903 619 619

27 693 1403 1135 610 634 610

28 773 1431 1261 694 663 663

29 853 1385 1068 751 690 690

30 853 1404 1140 961 709 709

Eurocode2No.

148


CSA A23.3-04

V test V flex


1 634 922 858 801 439 439

2 574 922 749 587 545 545

3 1050 1845 1652 1059 962 962

4 1091 1845 1657 914 1419 914

5 1620 2847 2606 1804 1556 1556

6 1520 2847 2766 1587 2439 1587

7 545 590 801 619 792 590

8 583 648 1069 872 1058 648

9 583 646 915 662 905 646

10 541 615 899 808 717 615

11 572 562 988 689 788 562

12 508 550 915 814 641 550

13 605 623 1032 1516 501 501

14 590 623 1032 888 591 591

15 623 623 907 645 365 365

16 596 623 907 749 365 365

17 595 623 907 831 365 365

18 580 623 907 831 365 365

19 592 623 665 602 365 365

20 594 623 665 726 365 365

21 537 623 665 850 365 365

22 876 1225 833 785 1112 785

23 884 1225 886 804 833 804

24 888 1225 886 804 833 804

25 880 1225 820 781 1096 781

26 748 752 820 781 1096 752

27 693 1403 1006 559 415 415

28 773 1431 1072 645 443 443

29 853 1385 970 688 478 478

30 853 1404 1009 890 578 578

No.CSA A23.3-04

Documents

Concentric Punching Shear Strength of Reinforced Concrete Flat … · 2018-12-18 · formula, as well as the punching shear formulae in some of the internationally recognised standards