2nd International Engineering Mechanics and Materials Specialty Conference le 2è Congrès international de mécanique et des matériaux

Ottawa, Ontario

June 14-17, 2011 / 14 au 17 juin 2011

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Nonlinear Analysis of Shear-Critical Reinforced Concrete Frames under Impact, Blast and Seismic Loads Serhan Guner1 and Frank J. Vecchio2

1Structural Engineer, Morrison Hershfield Limited, Toronto, Canada 2Professor of Civil Engineering, University of Toronto, Canada

Abstract: Due to elevated terror threats and the frequent occurrences of destructive earthquakes, nonlinear dynamic analysis methods have been increasingly used in recent years for the performance assessment, upgrading and design verification of frame structures. Currently available analytical procedures, however, are typically overly simplistic, reducing each structural component to a single-degree-of freedom system for the impact and blast analyses, and employing a lumped-plasticity approach for the seismic and progressive collapse analyses. Micro finite element methods, on the other hand, are typically overly complex and time consuming, thus having limited applicability to large frames found in practice. In both types of methods, modeling shear mechanisms remains a significant challenge, typically handled by neglecting shear effects altogether. A research initiative is undertaken to develop a nonlinear analysis procedure that accurately considers shear effects within a multi-degree-of-freedom distributed-plasticity formulation – but one that does not require extensive pre-selection of analysis options and input of hysteresis model parameters, and one that is suitable for large-scale applications. Applicable to reinforced concrete plane frames subjected to impact, blast and seismic loads, the procedure uses an explicit three-parameter time-step integration method and directly considers the effects of the loading rate on the material behaviours. Structural damping is intrinsically incorporated through the nonlinear concrete and reinforcement hysteresis models implemented. Significant second-order mechanisms, such as membrane action, concrete out-of-plane confinement effects, and reinforcement dowel action, are inherently accounted for. This paper provides an overview of the procedure developed and discusses its application to eleven previously tested specimens to examine its accuracy, reliability and practicality. 1. Background

In recent years, the need for advanced analysis procedures for dynamic loads, including impact, blast and earthquake, has increased considerably. Due to the heightened threat of terrorism, impact- and blast-resistant design is now required for strategic structures such as governmental and commercial buildings. Many such buildings, however, were designed and constructed without adequate consideration for these extreme events. A similar situation exists for earthquake loads. Many concrete buildings located in seismic areas, such as British Columbia and Québec, were built before the introduction of modern design provisions in the 1970s. Recent catastrophic earthquakes have confirmed the high vulnerability of such structures to collapse. Consequently, there is a current need to identify and upgrade such structures. For impact and blast loads, currently available analytical procedures, such as the ones contained in UFC 3-340-02 (2008), are typically overly simplified, reducing each structural component into a single-degree-of-freedom system. Unreliable and inaccurate response predictions obtained from such methods have been demonstrated in various studies, including El-Dakhakhni (2010). On the other hand, micro finite element methods, such as LS-DYNA (2010) and ABAQUS (2010), are typically overly complex and time consuming, thereby having limited applicability to large frames found in practice. Moreover, the

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consideration of shear effects remains a major weakness even for the micro finite element methods despite the fact that impact and blast loads tend to result in significant shear damage, as was observed by Saatci and Vecchio (2009a). Consequently, there remains a significant need for analysis methods which lie between those two extremes while being able to accurately model shear effects. For earthquake loads, a critical requirement for a realistic performance assessment is to accurately model shear-related effects. It is widely accepted that seismic loads typically place significant shear demands on frame members. Post-earthquake reconnaissance of concrete buildings has demonstrated the prevalence of compression and shear-dominated failures in heavily shaken buildings, as summarized by Ghannoum et al. (2008). However, currently available tools used in practice, such as IDARC 2D (Kunnath et al., 1990), RUAUMOKO (Carr, 2005), Perform3D (CSI, 2006), SAP2000 (CSI, 2009), ZEUS (Elnashai et al., 2010) and SeismoStruct (SeismoSoft, 2010), typically neglect shear-related mechanisms by default. This omission can lead to grossly inaccurate and unsafe predictions of structural performance. 2. Objectives of Current Study In addition to presenting a critical review of the current-state-of-the-art, the objective of this study is to complement the literature with a new nonlinear dynamic analysis procedure for the analysis of frames. The primary contribution of this procedure is a capability to accurately represent the coupled interaction between shear, flexural and axial effects through a modeling approach suitable for large-scale and practical applications. It is sought to eliminate the need for the complex pre-analysis calculations such as hinge calibration analyses, effective stiffness calculations, and analysis option selections. A second focus is to explicitly consider the beneficial effects of the rate of loading, which becomes particularly significant under impact and blast loads. Another focus is to present an approach through which an existing static analysis method can be modified for the consideration of dynamic loads in a total-load secant-stiffness formulation. It is sought to manipulate the dynamic equation of motion so that it can be solved by a standard static analysis algorithm. A final objective is to present the application of the procedure to previously-tested specimens to demonstrate the modeling process and to verify its accuracy. 3. Overview of Static Analysis Procedure Based on a total-load, iterative, secant-stiffness formulation, the static analysis procedure, which will be modified in this study, consists of two inter-related analyses. First, a linear-elastic global frame analysis, using a classical stiffness-based Euler-Bernoulli beam element with three degrees of freedom at each node, is performed to obtain member deformations and end actions. Using the calculated deformations, nonlinear sectional analyses are performed to determine the member sectional forces based on a distributed-nonlinearity layer approach. The differences between the global and sectional forces are termed the ‘unbalanced forces’, which are added to the ‘compatibility restoring forces’ (i.e., virtual static loads) to force member deformations in the global frame analysis to match those in the nonlinear sectional analyses. The compatibility restoring forces are applied to each member in a self-equilibrating manner. The global frame and sectional analyses are performed iteratively until unbalanced forces converge to zero. A layered analysis technique is employed for the sectional analyses, in which the cross section is divided into a number of concrete, longitudinal reinforcing bar and longitudinal prestressing steel layers, as shown in Figure (1-a). Transverse reinforcement is smeared within the concrete layers. Each layer is then analyzed for two-dimensional in-plane strain conditions according to the equilibrium, compatibility and constitutive requirements of the Disturbed Stress Field Model (Vecchio, 2000). The main sectional compatibility requirement enforced is that ‘plane sections remain plane’, while the sectional equilibrium requirements include balancing the axial force, shear force and bending moment calculated by the global frame analysis. For the consideration of shear, a parabolic shear strain distribution through the section depth is assumed. This shear-strain-based approach enables an analysis to continue into the post-peak regime and provides a fast and numerically stable execution. While the clamping stresses in the transverse direction are assumed to be zero, a ‘shear protection’ algorithm is employed to prevent premature failures of D-regions. As is typical with the sectional analysis methods using the beam theory, the procedure is only suitable for the analysis of B-regions (i.e., slender members with shear span-to-depth ratios greater than 2.0). At the conclusion of an analysis, the procedure, which is implemented into the computer code VecTor5 (Guner and Vecchio, 2008), provides sufficient output to fully describe the behaviour of the structure, including the load-deflection response, member deformations, concrete crack

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widths, reinforcement stresses and strains, deficient members, and the failure mode. The post-peak response of the structure is also provided. A more detailed description of the static analysis procedure is provided by Guner and Vecchio (2010 and 2011a).

sjy

Member CrossSection

Concrete Layers Reinforcing orPrestressingSteel Layers

layer i layer j

ciy

LongitudinalStrain

Distribution

xi

ParabolicShear StrainDistribution

xysj

(a) (b)

Figure 1: (a) Layered section analysis technique proposed; and (b) frame member proposed. 4. Modifications for Dynamic Loading Conditions The analysis procedure being modified in this study was developed to perform static nonlinear analyses through the solution of the static equilibrium equation of Eq. 1. For a dynamic analysis capability, a major requirement is the solution of the dynamic equation of motion of Eq. 2. In this study, Eq. 2 is condensed to the form of Eq. 1 so that it can still be solved by an analysis method developed for static loads. Presented below is a brief summary of the new formulations implemented into the static analysis procedure; more details can be found in Guner and Vecchio (2011b) and Guner (2008).

[1] u k p and [2] ( ) ( ) ( ) ( )m u t c u t k u t p t

Similar to the static case, three dynamic degrees-of-freedom are considered at each node: two translational and one rotational, as shown in Figure (1-b). A lumped-mass approach is adopted due to its mathematical simplicity and diagonal characteristics. The analysis procedure considers the majority of the energy dissipation through the nonlinear concrete and reinforcement hysteresis models previously implemented by Guner and Vecchio (2011a). Under certain situations, however, it may be desirable to include some additional viscous damping into an analysis. Such a situation may arise for frames with various non-structural elements or when using Newmark’s methods. Consequently, two damping formulations are implemented. The first one is the Rayleigh damping, after Lord Rayleigh (1878), which requires the assignment of two damping ratios to two vibration modes. A second type of damping, known as ‘alternative damping’, is also implemented to enable exact specification of the damping ratios in the selected modes. Note that the procedure developed does not use a modal solution approach; the modal damping matrices are implemented for optional use. Four different types of dynamic loads are considered: base accelerations, impact or blast forces, initial velocities, and constant accelerations. An explicit time-step numerical integration method is adopted for the solution of the dynamic equation of motion. The first scheme implemented is Newmark’s Method (1959), with two well-known special cases: average and linear acceleration. Although more accurate, the Linear Acceleration Method is a conditionally stable procedure requiring a time-step length less than 0.5513 x TN, where TN is the smallest modal period of the structure. Also, both the Average and Linear Acceleration Methods may require the use of additional viscous damping for numerical stability. For these reasons, a third procedure known as Wilson’s Theta Method (Wilson et al., 1973) is implemented; it is an unconditionally stable version of the Linear Acceleration Method. The accuracy and stability of the method depends on the assumed value of the parameter θ, which is taken 1.42 as recommended by Chopra (2007). When, θ = 1.00, the method reduces to Newmark’s Linear Acceleration Method. The analysis procedure developed employs a total-load secant-stiffness formulation; therefore, the original tangent-stiffness-based formulations of Newmark’s method are modified to obtain a total-load secant-stiffness formulation, after Saatci and Vecchio (2009b). In this modification, Wilson’s Theta Method is also incorporated by the development of a three-parameter formulation, the details of which can be found in Guner (2008).

1

2

Member k

node l

Node m

Lk

4

56

3

XMYM

ZM

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0.5

1

1.5

2

2.5

3

3.5

4

Strain Rate (s-1)

Liu and Owen (1986)Model Code (1990)Lu and Xu (2004)Model Code (2010)UFC 3-340-02 (2008)Williams (1994)Tedesco et al. (1997)Grote et al. (2001)

for f'c=30 MPa

DIF

fo

r C

om

p.

Str

eng

th (

f'c)

EQ Impact Blast0.91.01.11.21.31.41.51.61.71.8

DIF

fo

r Y

ield

Str

ess

(fy)

Strain Rate (s-1)

Malvar (1998)UFC 3-340-02 (2008)CEB (1988)Filiatrault and Holleran (2002)Liu and Owen (1986)Asprone et al.(2009)

EQ Impact Blast

for fy=400 MPa

(a) (b) Figure 2: Strain rate - DIF relationships for: (a) concrete in compression; and (b) reinforcement in tension. When subjected to high rates of loading, concrete and reinforcing steel materials exhibit increased strength. In the procedure developed, this strength gain is considered through a dynamic increase factor (DIF) approach. In this formulation, the strain rate values for each concrete and steel layer are calculated from the slope of the strain-time response. The corresponding DIFs are then calculated to modify the static material properties. For concrete, the fib Model Code (2010) formulations were implemented; these may be considered lower bound values as seen in Figure (2-a). For both the longitudinal and transverse reinforcement, two different formulations were implemented: CEB (1988) and Malvar (1998). Although both models propose similar increases for the ultimate strength, there is a considerable difference in their yield stress increases, as seen in Figure (2-b). In the developed method, the Malvar (1998) formulations were selected as default to provide conservative response simulations for concrete- or shear-dominated behaviours. The lower bound CEB (1988) formulations can also be used if deemed appropriate. Other more conclusive formulations can be implemented as they are developed in future studies. 5. Verification and Application of Proposed Procedure The proposed analysis procedure is verified, using only the default models and options, with eleven previously-tested specimens, including eight simply-supported beams, two cantilever columns and one shear-critical frame, subjected to impact and earthquake loading. Multiple tests performed on the specimens result in 38 simulations in total. As the main focus is to accurately simulate shear-related mechanisms, half of the simulations performed exhibited shear-dominated behaviours in the experiments. Herein, a summary of the verification and application studies is presented. More details of the analytical modeling and response comparisons can be found in Guner and Vecchio (2011c) and Guner (2008). 5.1 Application to Beams under Impact Loads The beam specimens examined were those tested by Saatci and Vecchio (2009a), involving four pairs of beams tested under free-falling weights, dropped from a clear height of 3.26 m above the specimens, resulting in an impact velocity of 8.0 m/s. The beams were subjected to multiple testing, providing a total number of 20 impact tests. All beams had identical geometry, test setup and longitudinal reinforcement details, as shown in Figure (3-a). The main variable was the amount of the transverse reinforcement. The concrete strengths also varied slightly, ranging from 44.7 MPa to 50.1 MPa. The experimental program was comprised of Beams SS0 to SS3, where the numbers from 0 to 3 denote the transverse reinforcement ratios from 0.0% to 0.3%. The a-series and b-series beams were identical in all aspects except the loading protocol employed, as summarized in Figure (3-d). A frame model was created for one-half of each beam with member lengths in the range of one-half the cross section depth, as shown in Figure (3-b). To eliminate the need to estimate the impact force history, the impact load was simulated through a special modeling technique similar to that used by Saatci and Vecchio (2009b). For this, an artificial segment (Member 11 in Figure (3-b)) was added to the model. This

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segment was assigned a very high stiffness to create a hard impact, and a linear-elastic compression-only behaviour to permit the separation of the drop-weight from the beam after the impact. The impact load was simulated by assigning an initial velocity of 8.0 m/s to the mass defined at Node 12. The sectional models were created using 32 concrete and two steel layers as shown in Figure (3-c). Out-of-plane reinforcement ρz was smeared within a tributary area of 5 times the bar diameter dbz at each side of the bar. Transverse reinforcement ratios were assigned to all layers except the clear cover layers, with the ratios shown in Figure (3-e). Wilson’s Theta Method, with no additional damping and a time step length of 0.00001 s, was used in all analyses. The time step length was selected in the order of the smallest natural vibration period of the beams (about 0.00003 s at the 21st mode). It was confirmed that the use of a smaller time step length did not change the analytical responses significantly.

940 1500 940

Free-FallingDrop-Weight

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1 2 3 4 5 6 7 8 9 10

235 mm x 41 2 3 4 5 6 7 8 9 10

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5353

304

250

LC

(c)

or 300.0 kgv=8.0 m/sLumped Masses (kg)

due to Self-Weight

28.9 57.8 61.50.0 61.530.8

Assigned MassAssignedVelocity

61.5 61.5 61.557.8 57.8

m = 105.5

(b)

2

LC

t=0%

z

As=1400mm2

Spacingρt (%) ρz (%) s (mm)

SS0 0.0 0.0 n/aSS1 0.1 0.19 300SS2 0.2 0.38 150SS3 0.3 0.57 100

(e)

Reinf. Ratios

Figure 3: SS Beams: (a) dimensions and test setup; (b) frame model used; (c) sectional model used; (d) loading protocol; and (e) smeared reinforcement ratios.

The peak displacements of the beams were calculated with good accuracy. Considering the 17 tests for which experimental peak displacement values were reported, a mean value of 0.99 and a coefficient of variation (COV) of 9.5% were achieved for the calculated-to-observed ratios. Furthermore, the peak displacements of previously damaged specimens were calculated accurately. For the second and third analyses of the damaged beams (10 tests), the mean ratio and COV were 0.98 and 7.1%. Selected comparisons for the experimental and computed responses are presented in Figure (4); complete comparisons for all beams can be found in Guner (2008).

-64-56-48-40-32-24-16

-80

0 0.05 0.1 0.15 0.2 0.25

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)

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ExperimentAnalysis

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SS2b-3

Figure 4: SS Beams: comparisons of mid-span displacement and reaction responses.

The peak support reactions were calculated with acceptable accuracy. Considering all 20 tests, a mean value of 0.95 and a COV of 21.3% were achieved for the calculated-to-observed ratios. The post-peak

a-series b-seriesTest 1 211 600Test 2 600 600Test 3 600 211

Drop Weight (kg)

(d)

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damping characteristics and the vibrational periods of the beams were calculated reasonably well. For SS0 and SS1 beams, the analytically-determined crack widths, damage levels and failure modes, when applicable, showed good correlations with the experimental observations. The analyses found shear-related mechanisms to be the major cause of the damage, consistent with the experimental observations. Note that both SS0 and SS1 beams were shear-critical under static loads; similarly, they exhibited shear-dominated behaviours under impact loads both in the experiments and in the analyses. For SS2 and SS3 beams, the crack widths, dominant behaviour and damage levels were also calculated accurately. Analyses indicated flexural behaviours in the first impact tests, accurately calculated the shift in the behaviours towards shear cracking in the second impact tests, and captured the experimental observation of no significant change in damage levels in the third impact tests. Note that SS2 and SS3 beams were flexure-critical under static loads; when subjected to impact loads, however, they exhibited shear-dominated behaviours in the experiments and in the analyses. This demonstrates the importance of considering shear effects when assessing structural performance under impact loads. 5.2 Application to Columns under Seismic Loads The column specimens tested by Hachem et al. (2003), involving four columns subjected to lateral earthquake loading on a shake table, were examined. The columns, constructed integral with a square top slab and a footing slab, had well-confined circular cross sections with 1.20% longitudinal and 0.27% spiral reinforcement ratios, consistent with modern design guidelines. The top slab carried three concrete blocks, resulting in a total mass of 29.5 tons including the slab weight. The footing slab was connected to the shake table through steel rods. The only difference between the specimens was the loading applied. Examined herein are Specimens A1 and B1, which were subjected to a unidirectional earthquake loading. Specimen A1 was tested eight times under a scaled version of the Olive View record of the 1994 Northridge earthquake. After scaling, the peak ground acceleration (PGA) values (in g) were 0.16, 0.58, 0.90, 0.60, 1.04, 1.02, 0.57, and 1.03 respectively. Similarly, Specimen B1 was tested nine times subjected to the modified version of the Llolleo record of the 1985 Chile earthquake. The PGA values were 0.03, 0.08, 0.14, 0.19, 0.48, 0.87, 0.46, 0.89 and 0.91. A frame model of the structure was created using member lengths in the range of one-half the cross section depth. Since frame models are typically based on centreline dimensions, stiffened end zones were used to account for the overlapping portions in the joint regions, as shown with bold lines in Figure (5-a). The top slab was modeled at the elevation corresponding to the mass centre of the top slab and the weight blocks. Three member types were used to create the sectional models of the column, with 36 concrete layers as shown in the inset of Figure (5-a), and slabs; an additional member type was used for the stiffened joint members. Additional masses and gravity loads were applied to the top slab to consider the weight blocks. The acceleration history data used in all analyses were those recorded during the tests as the shake table output. Wilson’s Theta Method, with no additional damping and a time step length of 0.0005 s, was used in all analyses. The smallest natural vibration period of both specimens, at the 60th mode, was calculated to be 0.00011 s; therefore, the selected time step length corresponds to 4.5 times the smallest period. For both specimens, it was confirmed that the use of a smaller time step length did not change the analytical responses significantly. The peak base shear forces were calculated with good accuracy. Considering all 17 analyses performed on A1 and B1 columns, a mean value of 0.97 and a coefficient of variation (COV) of 17.5% were achieved for the calculated-to-observed ratios. The peak deflections were also calculated reasonably well. Considering all 17 analyses, a mean value of 0.94 and a COV of 17.8% were achieved for the calculated-to-observed ratios. Representative base shear and displacement responses are presented in Figure (5-b). The post-peak damping characteristics of the columns were also captured reasonably well. As seen from Figure (5-b), the slight tendency in the analytical responses was to dampen out more quickly than the experimental responses. Note that these analyses employ Wilson’s Theta Method and use no additional viscous damping. In the case of using Newmark’s methods, which tend to require additional damping for numerical stability, less accurate calculations would have been obtained. The vibration periods were calculated successfully with a slight underestimation. Considering all runs, a mean of 0.92 and a COV of 9.8% were obtained for the calculated-to-observed average periods. All columns exhibited flexural behaviours and damage modes in the analyses, consistent with the experimental observations.

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(a) (b)

Figure 5: A1 and B1 Columns: (a) frame model used; and (b) comparisons of responses. 5.3 Application to Frames under Seismic Loads The frame specimen examined was that tested by Elwood and Moehle (2003), involving a one-storey two-bay frame tested under horizontal earthquake loading on a shake table. The frame consisted of two well-confined circular outer columns with 2.0% longitudinal and 1.4% transverse reinforcement ratios, and one poorly-confined square centre column with 2.5% longitudinal and 0.18% transverse wire reinforcement ratios. The columns supported a 1537 mm wide and 343 mm deep transfer beam which supported piled steel weights, resulting in a total mass of 22.8 tons, including the beam weight. The frame was secured to the shake table through bolted footing beams. The frame was subjected to a modified version of the Vile del Mar record of the 1985 Chile earthquake with a PGA of 0.79g, representing a maximum level earthquake. Two identical frames were tested with the only difference being the special prestressing technique used for Specimen 2. Herein, only Specimen 1 is examined. A frame model of the structure was created using member lengths in the range of one-half the cross section depth. Stiffened end zones were used for the joint core members as shown with bold lines in Figure (6-a). Four member types were used for the sectional models of the top beam, columns and footing beams; an additional four member types were used for the stiffened end zone members. Additional masses and gravity loads were applied to the top beam to simulate the steel weights. The acceleration data used in the analysis were those recorded during the test as the table output. Wilson’s Theta Method, with no additional viscous damping and a time step length of 0.00025 s, was used in the analysis. The smallest natural vibration period of the frame, at the 114th mode, was calculated to be 0.0000534 s; therefore, the selected time step length corresponds to 4.7 times this value. Note that if the Linear Acceleration Method was used for the analysis, a time step length of less than 0.000029 s would have been required for a numerically stable execution. The requirement of such an extremely small time step length and excessive computation time demonstrates the importance of using an unconditionally stable integration method for the nonlinear seismic analysis of frames. As seen in Figure (6-b), the base shear response of the frame was captured accurately with a calculated-to-observed peak shear force ratio of 1.08. The peak deflection, calculated at Node 1 in Figure (6-a), which occurred shortly after the shear failure of the centre column, was calculated reasonably well, being 0.83 times the experimental value. In the analysis, the first significant cracking occurred at about 17.7 s at the top part of all three columns. The centre column suffered up to 0.8 mm wide shear cracks while the outer columns experienced up to 2.0 mm wide flexural cracks. This time stage marked the first peak in the displacement response, shown with ▲ in Figure (6-b). At about 25.7 s, the central column had reached

As=260mmper layer

As=130mm2

406

237x3

203x

4

MT4

1478

1593

MT

127

1 x

6

2642

Member 22

237x3

massesdue to

self-weight

305

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203x2

#4

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)

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19.6

21.2

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40.8

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additionalmasses M

T4

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305

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1821

1593

(kg)

3800

2

As=130mm2

t=0.27%z=0.27%

COLUMN (MT1)

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its shear capacity at Member 57. At this time stage, shown with ■ in Figure (6-b), shear crack widths of the central column reached 1.8 mm while the outer columns sustained 3.2 mm wide flexural cracks. At about 28.8 s, the analysis indicated the shear failure of the central column at Member 57 initiated by the fracture of its tie. This time stage, shown with ♦ in Figure (6-b), marked the beginning of significant deterioration of the frame’s lateral stiffness and significant redistribution of the axial force from the central column to the outer columns. The frame exhibited a highly similar behaviour in the experiment, including the shear failure of the central column. After this failure, significant spalling of the concrete and subsequent buckling of the longitudinal reinforcement at the top portion of the central column was observed. The analytical procedure, on the other hand, conservatively dropped the load of the member once it failed in shear and distributed its load to the adjacent columns. Also shown with × in Figure (6-b) is the first yielding of the central column longitudinal reinforcement in the positive and negative loading directions, which correspond well to the experimental observations.

MT

1

MT

3

MT5

MT7

MT2 MT4

MT2MT4

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124.6 135.2 136.0 123.3 (kN/m)

1679 3796 41181822 1832 4141 3756 1662 (kg)

masses due toself-weight

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Member 57

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symm.

1270

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1270 559 1270559

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ase

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(a) (b)

Figure 6: Specimen 1: (a) frame model used; and (b) comparison of base shear responses. 6. Summary and Conclusions Although studies have demonstrated the significance of shear-dominated behaviours under impact, blast and seismic loads, most currently available frame analysis methods continue to ignore shear influences altogether. The ones that do consider shear mechanisms are overly complex, demanding the input of the sectional shear hysteresis responses, and yet are still unable to accurately represent the coupled shear and flexural responses. Moreover, most currently available dynamic frame analysis methods continue to use the lumped-plasticity formulation, an approach developed in the 1960s when computational power was far less advanced; this approach is known to be incompatible with the reinforced concrete behaviour. In this study, a nonlinear dynamic analysis procedure was developed for the impact, blast and seismic load analyses of reinforced concrete plane frames. The procedure employs multi-degree-of-freedom distributed-plasticity frame analysis algorithms in a nonlinear mode based on an unbalanced force approach. The nonlinear sectional analyses utilize a two-dimensional implementation of the Disturbed Stress Field Model. The advantage of the procedure lies in its capability to accurately model shear and significant second-order mechanisms, including membrane action, out-of-plane confinement effects, and reinforcement dowel action, through a modeling approach suitable for large-scale applications. The procedure uses default material models and analysis options, and requires the input of geometry, support conditions, basic material properties and reinforcement details, all of which can be obtained from the design drawings. The developed procedure is suitable for the analysis of frames with unusual or complex cross section details and has a broad range of applicability from seismic time-history to progressive collapse analyses. The formulations developed can be incorporated into most static frame analysis procedures to enable a dynamic load analysis capability. To investigate the accuracy, reliability and practicality of the procedure, the verification studies were also undertaken for previously tested specimens including eight simply-supported beams, two cantilever columns and one shear-critical frame, comprising 38 simulations in total. The results of the studies conducted support the following conclusions:

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1 - Currently available dynamic analysis methods typically neglect shear-related effects despite the fact that various research studies and post-earthquake reconnaissance of frame structures have demonstrated the significance of shear effects under impact, blast and seismic loads.

2 - Newmark’s and Wilson’s Theta Methods can successfully be employed within a total-load secant stiffness solution algorithm, resulting in excellent numerical stability.

3 - Newmark’s Linear Acceleration Method requires extremely small time step lengths due to its numerical stability requirement. Consequently, employing it for a seismic time-history analysis becomes computationally prohibitive, even when using a fast frame analysis tool.

4 - Wilson’s Theta Method provides unconditional numerical stability and permits the use of larger time step lengths, thereby reducing the computational demand. The numerical damping provided by this method to stabilize the analysis was found to be not a practical concern.

5 - The Disturbed Stress Field Model can successfully be employed within a layered sectional analysis algorithm for the consideration of transverse shears in nonlinear dynamic analyses.

6 - Dynamic material properties are widely considered to be strain-rate dependent. However, there are considerable discrepancies between the dynamic increase factors proposed by various researchers, in particular with the yield stress increase factor for reinforcing bars.

7 - The use of dynamic increase factors in a nonlinear frame analysis using a lumped-mass approach provides improved response simulations, in particular for impact loads.

8 - Impact analyses can be conveniently and accurately performed through a modeling approach employing the impacting mass and the contact velocity. This eliminates the complex pre-analysis calculations required to estimate the impact force history.

9 - Multiple successive dynamic analyses can be successfully undertaken for previously loaded structures, taking previous damage into account. This was achieved in this study by storing stress and strain histories in a data file and supplying it in a subsequent analysis.

10 - The strategy adopted here to modify an existing static analysis procedure for a dynamic analysis capability provides a convenient and viable platform for further code development.

11 - The analysis procedure developed accurately simulates the experimental responses of frames subjected to impact and seismic loads. Strengths, peak deflections, and damage and failure modes, including shear failures, are captured accurately. Considering all 38 simulations, mean values of 0.96 and 0.96 and COVs of 14.3% and 19.2% were obtained for the calculated-to-observed peak displacement and peak reaction ratios, respectively. Computed parameters such as crack widths, reinforcement strains and vibrational characteristics were also simulated well.

12 - The analytical procedure developed exhibits excellent convergence and numerical stability, requiring a fraction of the analysis time demanded by micro finite element methods. Each impact analysis reported herein, for example, required 15 minutes on a laptop computer.

13 - Further work is required to consider the bond slip and buckling behaviours of the reinforcing bars. In addition, the implementation of a 2D nonlinear joint model would eliminate the need, which may arise in some cases, to undertake detailed local analyses to determine the joint response. Future work will be directed towards these areas, as well as to extending the procedure to 3D frames.

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