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Numerical simulation of dynamic shear wall tests: A benchmark study
Computers and Structures 84 (2006) 549–562 www.elsevier.com/locate/compstruc
Numerical simulation of dynamic shear wall tests: A benchmark study _ Ilker Kazaz, Ahmet Yakut *, Polat Gu¨lkan Earthquake Engineering Research Center, Department of Civil Engineering, Middle East Technical University, Ankara 06531, Turkey Received 1 June 2005; accepted 10 November 2005 Available online 19 January 2006
Abstract This article presents the numerical simulation of a 1/3-scale, 5-story reinforced concrete load bearing structural wall model subjected to seismic excitations in the context of IAEA benchmark shaking table experiment conducted in laboratories of CEA in Saclay, France. A series of non-linear time history analyses were performed to simulate the damage experienced and response quantities measured for the specimen tested on a shaking table. The mock-up was subjected to a series of artificial and natural earthquake records. The entire model (concrete, table, masses) was discretized with 3D non-linear finite elements. An elaborate and comprehensive computer simulation process was conducted. A number of modeling iterations were performed for refinement purposes to include those details that were found to be significant in reproducing the measured behavior. The response of the structure was computed both at macro and microlevels and the results were compared with the measured quantities in order to validate the correctness of the analytical model. The comparison of analytical and experimental results yielded extremely good agreement because a numerical model that incorporates the test conditions adequately had been developed. 2005 Elsevier Ltd. All rights reserved. Keywords: Reinforced concrete wall; Dynamic test; Seismic behavior; Non-linear modeling; Finite element method; ANSYS
1. Introduction and description of the test specimen The rapid advancement of analytical and numerical capabilities must constantly be checked against empirical evidence for assessment of its accuracy. Reinforced concrete load bearing walls are commonly used in structures in seismic areas to resist laterally applied earthquake loads. A structural wall is the preferred way of ensuring lateral strength and stiffness in medium to high-rise reinforced concrete buildings. Walls are commonly encountered also in nuclear structures for similar reasons. An item of substantial research and application relevance is the computation of the dynamic response of structural systems with walls in the near-field zones of strong earthquakes where drift becomes the paramount behavior index. In order to
*
Corresponding author. Tel.: +90 312 210 5406; fax: +90 312 210 1193. E-mail address: [email protected] (A. Yakut).
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.11.002
show their capacity to sustain seismic loading and their behavior in case of near-field earthquakes, a series of seismic tests were performed on a model representative of reinforced concrete structural wall system. Understanding the structural behavior of the buildings with reinforced concrete walls not only requires experimental testing program but also many parametric studies using numerical modeling. The purpose will be to concur on the main features of an appropriate methodology to realistically account for the effects of near-field earthquakes and their safety significance on the structural shear walls. This exercise tests also the capability of current analysis tools in modeling against the irrefutable evidence of experiments in a dynamic environment. The 3D behavior of the specimen required the adoption of a non-linear finite element 3D representation. The analytical simulation of the model and verification of the experimental results by using the same experimental history were carried out.
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_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
The experimental program consisted of testing a model with scale 1/3 representative of a 5-story reinforced concrete building on the major Azalee shaking table of Commissariat a l’Energie Atomique (CEA) in the Saclay Nuclear Center, France. The specimen, named CAMUS1, had a total mass of 36 tons with the additional masses attached to it. Walls have no openings and are linked by square slabs measuring 1.7 m · 1.7 m. A heavily reinforced concrete footing allows the anchorage to the shaking table. The total height of the model is 5.10 m. Walls have thickness of 6 cm. The dimensions of the different parts and the mass distribution are shown in the sketches given in Fig. 1. The walls were loaded in their own plane. Adding lateral triangular bracing system increased the stiffness and the strength in the direction perpendicular to the table motion. This system has reduced the risk of failure that might be induced by parasitic transversal motion or a non-symmetric failure of the structural walls. Lateral bracing was designed such that it did not have any contribution to vertical load carrying capacity. Each part of the structure, the wall, floor, and basement was cast separately and assembled on the shaking table. The walls were cast in two parts in order to reproduce the construction joint at the level of each floor (few centimeters just above the middle of each floor). The specimen has reinforcement following the French PS92 seismic design code [1] with some adaptation in order to increase the shear safety factor. The distribution and the detailing of the reinforcement are quite different from those in conventional design practice where generally mesh rein-
forcement is used in the web along with the main longitudinal reinforcement at wall ends. The longitudinally reinforced regions through the 1.7 m wide walls are two edge regions in 10 cm width and the central region with 30 cm width. The amount of reinforcement decreases gradually as it goes to upper stories. The addition of central reinforcement is to limit the risk of sliding shear failure. For transverse reinforcement, bars with 3 mm diameter were used at a spacing of 6 cm along the height of the wall where longitudinal reinforcement is used. The amount of steel at each level and details of reinforcement in elevation are shown in Fig. 2. C20 microconcrete with a compressive strength of 25 MPa and a Young modulus of 28 000 MPa was chosen. These characteristics were checked by the usual compressive and splitting tests on 160 mm diameter cylinders prior to the testing program. They exhibited an unexpected compressive over strength. The compressive strength, tensile strength and modulus of elasticity were found from the average of test results to be 35 MPa, 3.8 MPa and 30 000 MPa, respectively. These values were reported in [1] and used in the analytical computations. The objective of this study is to obtain an analytical model of the tested mock-up such that the experimentally observed and measured responses could be matched by a reasonable accuracy. The challenge of capturing not only global behavior but the measured response at specific local areas has led to a detailed, sophisticated and comprehensive model that represents the features of the test specimen. During the course of the study, important and crucial parameters needed to create an adequate and reliable
Fig. 1. View of the CAMUS specimen and sketch of the walls and masses [1] (units in cm).
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Level 6
Level 5
10 cm
60 cm
30 cm
551 60 cm
10 cm
φ3 / 6 mm as stirrup
Configuration of edge and central reinforcement
Level 4
Longitudinal Reinforcement amount at each level Level 1 2 3 4 5 6
Level 3
Level 2
Level 1
Edge of the wall (mm2) 289.4 188.9 94.4 28.3 15.9 15.9
Center of the wall (mm2) 138 138 110.2 78.4 78.4 Nothing
Fig. 2. The amount and detailing of reinforcement in elevation of single wall.
Ground motions used in the loading program
1
Run5 (PGA = 0.72g)
0.5
0 -0.5
Acceleration (g)
0.5 0 -0.5 Run3 (PGA = 1.11g)
0.5
Run4 (PGA = 0.41g)
0 -0.5 Run1 (PGA = 0.24g)
Run2 (PGA = 0.13g)
0.5 0 0 -0.5 0
2
4
6
8 Time (secs)
10
12
14
Fig. 3. Nice and San Francisco signals used in the experiments.
model representing the physical test specimen were identified. We utilized ANSYS V7.0, [2], a software package capable of performing non-linear static and dynamic analyses and incorporating elements with properties that represent adequately the actual behavior of concrete. The experimental data were provided in the form of drawings, reports and photographs that are mostly contained in Combescure [1]. In the section that follows, a brief summary of the experimental background is provided in order to familiarize the reader with the tests conducted.
2. Experimental program and results 2.1. Loading program Two types of input motion were used in the experimental program. The Nice input (an artificial record) representative of far-field earthquake (FFE) ground motions, and the San Francisco input motion (a naturally recorded seismogram) that resembles the characteristics of near-field earthquake (NFE) ground motions. This benchmark study
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4
T = 0.137 sec
3.5
Acceleration (g)
Table 1 Maximum value of top displacements and internal forces at the base of wall
Acceleration Response Spectra for all Tests (ξ = 5 %)
4.5
Run1 Run2
3
Run3 2.5
Run4
2
Run5
1.5 1
Test
Run 1 Run 2
Top displacement (mm) Bending moment (kN m) Shear force (kN) Axial force Traction (kN) Compression (kN) (static loading: 163 kN)
7.0 1.54 211 75.5 65.9 23.5
Run 3 Run 4 Run 5 13.2 13.4 280 276 106 86.6
44.3 No vertical 102 36.5 Excitation 105
43.3 345 111
50 137 51.9 146
0.5 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Period (sec)
Fig. 4. Five percent damped acceleration response spectra of given ground motions.
investigates the results of five tests, in which the table was driven three times by the scaled Nice input motion (Runs 1, 4 and 5) and twice by the scaled San Francisco input motion (Runs 2 and 3). The applied ground motions were p scaled on the time axis by a factor of 1/ 3 to permit observations on the model to be extrapolated to the prototype. The acceleration waveform and corresponding response spectra for these ground motions are given in Figs. 3 and 4, respectively. It is worth noting that these ground motions were applied sequentially to the same specimen so any accumulation of inelasticity from previous runs was retained in the subsequent runs. These acceleration records were measured on the shaking table and were found to be different from the intended waveform. This is mainly due to driving mechanism of the shaking table, which altered the applied ground motions. Inspection of Fig. 4 suggests that scaling the amplitude of the table motion distorted its frequency content as well. For purposes of the study reported here, this is not a crucial point because the complete record of the actual input motion is used in our calculations. 2.2. Initial dynamic response and overall model behavior The natural frequencies of the test specimen were measured by applying low-level random excitations before each test. The first natural frequency measured in the direction of excitation (in plane bending) before the first test (Run 1) was found to be 7.24 Hz, the second was nearly 33 Hz, and the first vertical natural frequency was around 20 Hz [1]. In the direction perpendicular to the excitation, the first natural frequency was measured as 13 Hz due to the stiffening effect of the steel bracing system. The observed behavior during the tests revealed that the flexibility of the shaking table support mechanism had a significant influence on the dynamic properties of the mock-up. The vertical motion of the shaking table was measured by two vertical accelerometers at two ends of the table revealing significant vertical motion during the
tests although no vertical motion was applied. If the flexibility of the supporting mechanism of the shaking table and the wall-table connection region are not considered (the modeling is performed with fixed table supports and rigid connection region), the first vertical frequency is calculated as 43 Hz. The test results revealed that although the input signal was only applied in the horizontal direction, the extension/compression mode shape was excited during the tests as well. 2.3. Analyses of the internal forces The displacement and acceleration time histories of each floor, shear and axial forces, moments and strains throughout the specimen were obtained by means of instruments located at different parts of the mock-up. Since the exact values of masses at each floor were precisely known, internal forces could be computed from the accelerations recorded by the accelerometers. The peak values of maximal experimental top displacements; and moments, axial and shear forces at the base of the wall are given in Table 1. Strains computed in the static and dynamic analyses are presented later in this article in comparison with the experimental ones. 3. Modeling of the structure The target in developing the numerical model has been to represent the mock-up tested with the available analytical tools. A reasonably accurate model must reflect material non-linearity, dynamic properties of the system, the test boundary conditions and the loading applied. Therefore, simple models with elastic properties or frame models with approximate non-linearity, which can be analyzed by many commercially available software packages, were not considered. Instead, a finite element model that can incorporate cracking of concrete and recognize the steel reinforcement was developed. Towards this end we decided to use ANSYS as our analytical tool. ANSYS was used to model both the wall and the shaking table. The purpose of this selection was to benefit from the features of a special reinforced concrete element called SOLID65. As mentioned previously, the flexibility of the shaking table had a significant influence on the dynamic properties of the test specimen, which made it a vital
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
component of the model. For the lateral stiffening of steel bracings, the three-dimensional spar element, LINK8, a uniaxial tension–compression element with three translational degrees of freedom at each node was used. The steel rods that connect the mock-up to the shaking table were modeled with LINK10 which is a three-dimensional spar element having the unique feature of a bilinear stiffness matrix resulting in uniaxial tension-only (or compressiononly) property [2]. The tension-only option was activated in order not to allow any stiffness contribution when the element was in compression. In the numerical model vertical rods supporting the shaking table were included and assigned a stiffness to capture the measured vertical frequencies. For these rods, a spring element, COMBIN14, was used. COMBIN14 has longitudinal or torsional capability in one-, two-, or three-dimensional applications. The longitudinal spring-damper option is a uniaxial tension– compression element with up to three degrees of freedom at each node: translations in the nodal x, y, and z directions. The elastic constant of each spring element was taken as K = 400 MN/m (in accordance with the experimentally measured response) in the numerical computations. The table and walls were modeled with element SOLID 65 that has eight nodes with three translational degrees of freedom at each node. The solid element (SOLID 65) is capable of cracking (in three orthogonal directions) in tension and crushing in compression. If cracking occurs at an integration point, the cracking is modeled through an adjustment of material properties that effectively treats the cracking as a ‘‘smeared band’’ of cracks, rather than discrete cracks. The concrete material was assumed to be initially isotropic, [2]. Up to three different rebar specifications may be defined. Ties and stirrups were also modeled by making use of this property. The rebars can carry tension and compression, but not shear. They are also capable of plastic deformation and creep. The reinforcing bars were assumed to be smeared in the element volume. The reinforcement was entered as volumetric ratio of that element, defined as the rebar volume divided by the total element volume. In the formulation of stress–strain matrix, D, the relation given in Eq. (1) is used [2]. The amount of reinforcement is used as a modification factor that calibrates the concrete’s strain–stress matrix. Once a crack occurs at an integration point, a plane of weakness is introduced in
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the direction normal to the crack face to modify the stress–strain relation of concrete. ! NR NR X X R R ½D ¼ 1 V i ½DC þ VR ð1Þ i ½D i i¼1
i¼1
In this equation, NR denotes the number of different reinforcing materials, DC is the stress–strain matrix of the concrete and DR is the stress–strain matrix of reinforcement material, V R i is the ratio of the volume of reinforcing material ‘‘i’’ to the total volume of the element. The stress–strain relation of concrete in tension and the strength of cracked condition are explained in Fig. 5a. In this figure ft is the uniaxial tensile cracking strength and E is the modulus of elasticity of concrete. After cracking, a certain amount of stress relaxation can be included in the element stress formulation with the constant Tc that is taken as 0.6, the default value in our case. Rt is the secant slope defined as shown. It diminishes to zero as the solution converges. Additional concrete material data, such as the shear transfer coefficients, tensile stresses, and compressive stresses were among the parameters of the material model used. 3.1. Steel and concrete material models In the light of the experimental results, the compressive and tensile strength and the modulus of elasticity of the concrete were taken as 35 MPa, 3.8 MPa and 30 000 MPa, respectively. The stress–strain curve of the concrete in compression was represented with three segments as a multilinear isotropic hardening material, and the same model with kinematic hardening property was used in the dynamic analysis (Fig. 5b). In non-linear analysis of reinforced concrete, the shear transfer coefficient must be assumed. For closed cracks (bc), the coefficient is assumed to be 1.0, while for open cracks (bt) it should be in the suggested range of 0.05–0.5, rather than 0.0, to prevent numerical difficulties [3]. In this study, a value of 1.0 was used, which resulted in acceptably accurate predictions. The values less than 1.0 were tested, but they caused convergence problems even at very low loading levels. Only one bilinear curve was used to represent the material property of the four different reinforcing bars. The yield stress was taken as 500 MPa (an average value) at 0.002-strain value and the
σc
σt ft
fc
Tc.ft Ec 1
0.6fc
Rt
ε εck
6ε
(a) Tensile behavior of concrete
Ec 1
ε c=0.002
ck
εcu=0.0035
(b) Compressive behavior of concrete
Fig. 5. Material behavior of concrete.
ε
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_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
Fig. 6. The models created in ANSYS.
stress at the failure is assumed to be 525 MPa with 0.34% strain hardening. Initially a three-dimensional model incorporating all of the details explained above was generated. Due to lengthy computer solution time and enormous storage requirements for saving the analysis results, a 2D model was developed beside the 3D model using the perfect symmetry of the structural system (Fig. 6), i.e. plane stress assumptions appeared to be applicable since no excitation was applied in the direction perpendicular to plane of the wall. In the former representation, out-of-plane response was restrained. The masses of each story were lumped uniformly at the level of each floor by using special mass elements, MASS 21. The solid beam below the wall in Fig. 6 that models the shaking table was found to be adequate. A number of trial models indicated that the flexibility of the shaking table and especially definition of boundary conditions at wall-table connection region had substantial effect on the dynamic behavior of the specimen. Its inclusion in the model was inevitable.
its base (Fig. 7); the anchorage was provided in the middle of basement for a partial length only, approximately 1/3 of the wall length. At the two ends, the gap between the wall and table can be described as a contact surface problem. This region was filled with elastic mortar with unknown properties. Four inclined steel bars were used at the ends to anchor the wall to the shaking table. Although the apparent boundary conditions at the bottom of wall can be assumed as fully fixed, doing so led to deceptive results (the computed response in fact was different). The system was found to be more flexible than assumed. The analysis results proving this statement are presented in the following sections of this article. Once the wall is pushed in one direction one pair of steel bars becomes ineffective and the other pair acts only in tension. So the tension-only spar element, LINK10, is considered to be appropriate to model the steel bars. The diameter of the steel anchor bars was taken as 36 mm (as used in the tests). The contact surface between the wall and table was discretized with spring elements, COMBIN14, having elastic stiffness of 20 000 N/ mm. At the free ends of the wall these springs were placed between the wall and table. Influence of these springs on the first natural frequency and on the first 10 natural frequencies is shown in Fig. 8. While assigning zero stiffness to these springs leads to a first natural frequency of 7.1 Hz, assuming very rigid stiffness, i.e. fixed wall-table connection leads system dynamic response to be around 8.1 Hz. These contact spring elements are also crucial for other purposes because without them or when their stiffness is below a certain limit, sliding shear failure initiates on the plane of wall just vertically above the rigidly connected region under high lateral loads (Fig. 9). 4. Results of numerical analyses
3.2. Modeling of boundary conditions
4.1. Modal and static analyses
One of the key points that is crucial to the modeling strategy is the modeling of the boundary conditions. We modeled the table and wall connection region as they were. The wall foundation was not fully anchored to the table at
Close and reasonable results were obtained for the natural frequencies of the mock-up when the connection between the table and walls were modeled realistically and the effect of the shaking table flexibility was taken into
Fig. 7. Modeling of boundary conditions.
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Influence of the Spring Constant used between wall and table on the natural frequencies of the system
8.3
80
8.1
Frequency (Hz)
70 60 50
K=0 K=5000 N/mm K=10000 N/mm K=15000 N/mm K=18000 N/mm K=20000 N/mm K=25000 N/mm K=30000 N/mm K=35000 N/mm K=40000 N/mm K=4E7 N/mm
40 30 20 10 0 1
2
3
4
5 6 Mode Rank
7
8
9
10
Frequency (Hz)
90
555
Change in the first natural vi bration frequency due to spring stiffness
7.9 7.7 7.5 7.3 7.1 0 m m m m m m m m m m K= /m m m m m m m m m /m N/ N N/ N/ N/ N/ N/ N N/ N/ 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E 4 50 00 50 80 00 00 50 00 50 K= K= K=1 K=1 K=1 K=2 K=2 K=3 K=3 K=4
Fig. 8. Influence of the spring constant on the natural frequencies of the wall-table system.
Fig. 9. Sliding shear failure plane due to lack of vertical stiffness at the base of wall.
consideration. From the response spectrum characteristics of the ground motions applied and the expected dominant first mode response of the model, it appears that the first natural frequency has a significant role in the dynamic forces transmitted to the structure thus influencing its response (Fig. 4). So, different boundary conditions were applied to the model and the first bending and vertical modes were computed as tabulated in Table 2. As evidenced from the results contained in Table 2, both boundary conditions exert considerable influence on the modal response of the model. The flexibility of the connection between the table and the wall affects the first mode significantly but has minor influence on the vertical mode. In the light of all these preliminary analyses we realized that creating a model incorporating both boundary conditions in the table base and table-wall connection is vital for reproducing the experimental results as close as possible. Surprisingly, the static analysis results indicated that the table supporting system flexibility has negligible effect on the load deformation pattern. On the other hand, flexibility
of the table-wall connection region appears to be a significant factor influencing the initial stiffness and post elastic behavior of the load–deformation relation. It is obvious from all these findings that we face the challenging issue of modeling the mock-up by taking into account the non-linearity in the material and geometry, the dimensions of the specimen and the test boundary conditions, and load application that must be handled carefully. Therefore, the actual boundary conditions (flexible support and flexible connection) that were adopted in this study yield an initial system stiffness of 20.1 kN/mm. The computed model natural frequencies are given in Table 3 for both 2D and 3D model for comparison. The model was first subjected to statically applied inverted triangular lateral load at the level of each floor to simulate first mode response. Pushover curve that presents base shear force versus top displacement (roof) for one of the two walls (left wall) is given in Fig. 10 with other two load–deformation patterns that will be described next. The limiting point where the structure reached the state of instability due to excessive damage corresponds to the limit that indicates the capacity of the model. The computed and experimentally measured maximum response quantities for each dynamic run are also plotted on Fig. 10. The pushover curve is a powerful tool to visualize the global non-linear behavior of the structures, and it provides very useful hints about the global behavior of the test specimen in the absence of more elaborate analyses such as non-linear time history analyses. Examination of Fig. 10 reveals good agreement between the capacity curve of structure and the peak global response obtained from the dynamic tests. The information presented in Fig. 10 reveals that the structure remained in the elastic range under the ground motion applied in Run 2 whereas all other cases resulted in inelastic behavior of varying degrees. Needless to say, the largest deformation was measured in Run 5 not Run 3, the strongest shaking intensity, due to sequential application of ground motions. This figure displays also the results of the time history analyses, which are discussed next.
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Table 2 Modal frequencies (Hz) due to different boundary conditions Boundary conditions
1st Bending mode
2nd Bending mode
3rd Bending mode
1st Vertical mode
Support and connection flexible Fixed support and flexible connection Flexible support and rigid connection Fixed support and rigid connection Fixed based wall
7.275 7.858 8.080 8.868 9.190
32.664 36.483 33.095 38.609 39.991
54.856 – 57.405 – –
22.685 42.270 22.872 43.397 44.704
Table 3 Natural frequencies (Hz) Modes
Model 1 (3D) Model 2 (2D)
1
2
3
4
5
6
7
8
9
10
7.274 7.275
9.316 –
13.698 –
22.078 22.685
31.061 –
33.438 32.664
50.340 –
50.706 –
52.975 –
57.306 54.856
160 140
Base shear (kN)
120 100
Linear Static Analyses Triangular loading Triangular loading with 1.6g gravity Rectangular loading Run1 Experimental Run2 Experimental Run3 Experimantal Run4 Experimental Run5 Experimental Run1 Computed Run2 Computed Run3 Computed Run4 Computed Run5 Computed
80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
Top displacement (mm)
Fig. 10. Pushover curve of a single wall.
The comparison of measured and computed responses superimposed on the pushover curve shows that for Runs 3 and 5, while the calculated peak displacements are in good agreement with the measured quantities, the calculated base shear is larger, especially for Run 3, than the measured ones. As discussed previously, the measured base shears are in some sense calculated, i.e. they are computed from the measured accelerations by multiplying them with the floor masses above certain wall section. Besides, the effect of additional vertical force generated in the dynamic test as a result of opening and closing of cracks was not included in the calculation of measured base shear. However, it is fundamentally known that as the axial load increases on a section, the moment and shear capacity also changes (increasing when the axial load level is high). It is known that in Runs 3 and 5, the vertical force on the base section of the wall nearly doubles both in traction and compression (Table 1) due to opening and closing of the cracks on the wall. This miscalculation of the measured base shear force is considered to be the primary reason of the discrepancy between the calculated and measured global response
for Runs 3 and 5 (Fig. 10). So we decided to perform two more pushover cases; one is for the triangular load case in which the effective mass of the system was increased 1.6 times (geffective = 1.6 g) due to vertical excitation (to show the effect of the axial force on the response) and the other one is representative of the uniform (rectangular) loading that is considered as an upper bound case for the load path. The increased effective mass is calculated as the total maximum vertical force at the base (the measured axial compression at Level 1 in Run 3 (Table 1) plus the weight of the structure (165 kN)) divided by the weight of the structure (270 kN/165 kN). The resulting load–deformation curves are presented in Fig. 10. The results support our expectations; the effect of both vertical force and load pattern is very significant on the load–deformation curve. This situation again proves the complexity of the problem and the necessity of comprehensive analysis. 4.2. Time history analyses The reference acceleration histories measured on the shaking table were applied to the model at the level of the shaking table. Time history analyses were carried out only for strong motion duration of the given ground motion records. The analyses for Runs 1–5 were carried out in a sequential order to represent the actual loading history. Top displacement was measured from the node at the top corner of the model. The shear forces at different levels of the structure were calculated by taking the sum of horizontal forces (y-component) of elements at a section. The bending moments were calculated by taking moments of vertical force components (z-component) of elements at a section about the center of section. In time history analyses, a constant damping ratio of 2% was assumed for each mode. This damping was input by means of Rayleigh damping constants ([C] = a[M] + b[K]); i.e. it was assumed that both mass- and stiffness-proportional damping was present in the system [4]. The first two modes each with 2% damping were used to define a and b.
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
In the non-linear transient dynamic analysis solution phase we encountered some problems related to ANSYS software. Direct application of ground motion is not possible within the program. The only form of acceleration input is to create acceleration field acting on all the nodes of model. Thus, the structural response of the model to the base excitation is calculated using the concept of effective earthquake forces [4]. Lumping masses at the floor levels for the wall, the absolute floor displacement vector, ut, under the ground displacement ug is computed from Eq. (2) (Fig. 11a). ut ðtÞ ¼ uðtÞ þ ug ðtÞ 1
ð2Þ
where 1 is a vector of order N with each element equal to unity and in general terms called influence vector (i) that represents the displacements of the masses resulting from the static application of a unit ground displacement. For external dynamic forces Fi(t) the general form of the equation of dynamic equilibrium can be written in the form: _ þ ½K fug ¼ fF ðtÞg ½M f€ ug þ ½C fug
ð3Þ
In the earthquake (base) excitation case, F(t) = 0 since no external dynamic force is applied and the equation of dynamic equilibrium becomes _ þ ½K fug ¼ ½M fig € ½M f€ ug þ ½C fug ug ðtÞ
ð4Þ
Comparing Eq. (3) with Eq. (4) shows that the equations of motion for the structure subjected to ground acceleration, € ug ðtÞ in Fig. 11b and externally applied dynamic load, mi €ug ðtÞ in Fig. 11c at the level of each mass are one and the same. The accelerations calculated with Eq. (4) are relative accelerations; so for computing the total acceleration related to any mass, Eq. (2) must be utilized. In view of the discussions presented above, acceleration data are entered in the form of an array to the program before the start of solution, and in the solution phase for each time integration step (Dt), corresponding acceleration is called in from the array within a small loop covering the whole time history data points. Due to application of this
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acceleration field at any particular instant, all the masses are multiplied with the ground acceleration value and the resulting force is applied to the structure as an external dynamic loading, which is the right hand side term in Eq. (4). For the given instant governing equations of motion are solved statically including the time integration effects in the calculations. Newmark time integration method is used to solve the equations of motion that are in the same form as Eq. (4). In non-linear analysis the stiffness matrix [K] is a function of unknown displacements, so Newton– Raphson procedure, which is an iterative method to solve non-linear equations, was also used. Time histories for top horizontal displacement, shear force and bending moment at level 1 (base of the shear wall), top floor absolute horizontal and vertical acceleration and corresponding 5% damped top floor response spectra, moment–curvature relationship at the base of wall, strains in the external rebars for Runs 1–5 were computed and are summarized in Figs. 12–19. These figures do not contain the results for Run 2 because it is linear, so only the maximum values are given in Table 4. The corresponding test measurements are also superimposed on the numerical plots. The maximal time history results for Runs 1–5 are summarized in Table 4 with the experimental values given in italicized parentheses. There is a fairly good match between the experimental (measured) and computed (calculated) results. These comparisons revealed the adequacy of the model for the purpose of simulating an experiment realistically by means of analytical tools. The high degree of correlation is a clear indication of the influence of all subtleties in modeling the physical test conditions as well as adequacy of the material models incorporated in the software used. As elaborated previously, the measured time history base shear values were calculated ignoring the effect of dynamic axial forces, thus producing smaller values for Runs 3 and 5 than the computed shear forces. This becomes more evident when the computed base shear using the same analogy (ignoring dynamic axial load) is
uit N=5
ui
.. − m N * u g (t )
mN All the levels have the same mass
mi
mi = m i = 1,2,..5 1
.. − mi * u g (t )
=
.. − m1 * u g (t )
m1
ug
.. u g (t )
Stationary base
(a)
(b)
(c)
Fig. 11. (a) Lumped mass system, (b) ground excitation, (c) effective earthquake forces.
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Fig. 12. Comparison of experimentally measured and numerically calculated global response parameters, such as displacement, base shear and bending moment.
compared to the measured value in Fig. 13. Note that the new maximum computed base value in Run 3 is 108 kN instead of 120.1 kN which is nearly the same as the measured base shear, 107 kN.
4.3. Local results Time history results of several parameters were derived from the computational phase including accelerations and
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559
Table 4 Results of time history analyses Run 1
Run 2
Run 3
Run 4
Run 5
Top relative displacement (mm)
6.58 (7.0)
1.49 (1.54)
13.0 (13.2)
13.2 (13.4)
35.94 (43.3)
Top abs. horizontal acceleration (g)
0.72 (0.66)
0.27 (0.28)
1.36 (1.11)
1.03 (0.93)
1.48 (1.31)
Level 1 bending moment (kN m)
215.9 (211)
75.6 (75.5)
261.3 (280)
274.9 (276)
346.8 (345)
Level 1 shear force (kN)
66.6 (65.9)
23.2 (23.5)
120.1 (106)
88.2 (86.6)
125.1 (111)
Level 1 axial traction (kN)
44 (44.3)
4.2 (0)
125 (102)
70 (50)
150.9 (137)
Level 1 axial compression (kN)
34 (36.5)
3.8 (0)
165 (105)
70 (52)
179.4 (146)
Strain in the external R-bar Level 4 (·103)
0.112 0.124
0.0275 0.0328
6.91 2.31
3.82 2.69
22.3 6.46
Strain in the external R-bar Level 3 (·103)
0.194 0.209
0.0411 0.0482
2.97 1.22
3.05 1.45
12.8 7.6
Strain in the external R-bar Level 2 (·103)
1.08 1.08
0.102 0.0765
2.08 1.13
2.05 1.67
11.68 9.82
Strain in the external R-bar Level 1 (·103)
0.611 0.385
0.267 0.281
1.75 1.27
1.85 1.50
2.23 2.22
Measured vs. Calculated Base Shear from Accelerations
125
Base Shear (kN)
100 75
Measured
50
Calculated
25 0 -25 -50 -75
-100 -125 5
6
7
8 Time (sec)
9
10
11
Fig. 13. Comparison of measured and calculated base shear force in Run 3.
external rebar strains on each level. The damage progress, in other words the crack development started from the base of the structure as hairline cracks and moved to the upper levels. As it was expected from the ground motions and evident from the experimental and computed results, non-linearity becomes pronounced significantly first in Run 3. The moment–curvature relations plotted for the four levels in Fig. 14 show that the second, third and fourth floor levels deformed into non-linear range after Run 3. In Fig. 15, the strain concentrations in different regions of the wall computed during Run 3 are shown. Due to progressive cracking of concrete and yielding of reinforcement under sequentially applied increasing seismic excitations, the system deformed further. This preliminary damage accumulation is the reason why the test specimen has exhibited such large deformations in Run 4 opposing the expectations. Run 5 is the ultimate loading applied to the specimen that led to excessive damage and failure of reinforcement at different sections. The special design of steel allowing the
damage to spread among different stories rather than localizing it to a particular section of the wall helped the structure survive such an intense sequential loading history. Yielding of the reinforcement occurred in the stories 1–4, but it was excessive in the upper two stories. It is evident that in the regions where the longitudinal steel is reduced, larger deformations were observed. The number of rebars changes at 10 cm below each story level along the height of the wall. For this reason, the cracks on the wall initiate from these interruption regions and progress diagonally between the stories. Smeared crack model that is used to model and detect the damage, the location of the damage and to some degree the level of damage experienced by the structure gave reasonable results when handled carefully. The main crack pattern shown in Fig. 17 at the end of testing program supports this statement. Since strains at a section or a point are the indicators of damage experienced, examining the strains measured on the external rebars from the experimental phase and
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560
Moment - Curvature _ Level1 _ Run3 300
Moment - Curvature _ Level3 _ Run3 150 100
100 0 -8E-04
-6E-04
-4E-04
-2E-04
0E+00 -100
2E-04
4E-04
6E-04
moment (kN-m)
moment (kN-m)
200
50 0 -2E-03
-2E-03
-1E-03
-5E-04
0E+00 -50
5E-04
1E-03
2E-03
-100
-200
-150 -200
-300
curvature (rad/m )
curvature (rad/m) Moment - Curvature _ Level4 _ Run3 100
Moment - Curvature _ Level2 _ Run3 200
50
100 50 0 -2E-03
-1E-03
-5E-04
0E+00 -50
5E-04
1E-03
-100 -150
m oment (kN-m )
moment (k N-m)
150
0 -2E-03
-2E-03
-1E-03
-5E-04
0E+00
5E-04
1E-03
2E-03
2E-03
-50
-100
-200 -250
curvature (rad/m)
-150
curvature (rad/m)
Fig. 14. Moment–curvature relationship of levels 1–2–3–4 after Run 3.
Fig. 15. Strain distribution and crack pattern developed in the model in Run 3.
comparing them with the calculated ones reveals the damage pattern and location in the structure. Level 3 seemed to be the most critical section as evidenced by the experimental results. In the computations very high strains, denoting excessive damage, were obtained at levels 2–4. The comparison of strains measured on the external rebars for Runs 3 and 5 is given in Figs. 16 and 17. During the ultimate Run 5, the steel yielded at all sections, and this was accompanied by the failure of the longitudinal reinforcement at level 3. The strains obtained from the simulations showed the same deformation tendency with an exception that the computed strains could not reach the failure limit (2.5%) at level 3 but at level 4. So, in experimental stage while the failure initiated at level 3, in the model the first failure indication was observed at level 4. But this is within our expectations since we know that the measured strains at upper levels (3 and 4) is very high and there is a slight difference between these strains showing that initiation of failure at a level is a matter of instant. Another difference in our computations from the measured ones is that, the com-
puted steel strains at level 2 are much greater than the experimental ones in Run 5. This can be explained within the experimental procedure itself and the instrumentation of the wall, as all the experimental measurements were made on the left wall, while it is known that the main crack pattern that was observed on the right wall at the end of the testing program was different than that on the left (Fig. 17). It is known, at least by visual inspection that, there was an excessive damage on right wall at level 2 indicating high strains, which agrees with our computations. The local results (strains) that were computed agree well with the crack pattern on the right wall. After the failure of reinforcing bars at level 3 in Run 5, the response of the system changed drastically. Natural period of the system increased to more than double (from 0.137 s to approximately 0.3 s) due to stiffness degradation in the system. This is a big challenge from the modeling point of view. The analytical model and the material laws we employed in the analysis is not equipped to retrieve a failure situation in the reinforcing bars. This is probably
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Strain Level 2 _Run3
8
2 Measured 1.5
strain (x10E-3)
strain (x10E-3)
Strain Level 4 _ Run3
10
2.5
561
Calculated
1 0.5
Measured
6
Calculated 4 2 0
0
-2
-0.5 6
7
8
6
10
9
7
8
10
9
time (sec)
time (sec)
Fig. 16. Strain time histories compared with experimental ones for Run 3 at levels 2 and 4.
Fig. 17. Maximum strains computed at each level compared with experimental ones for Run 5 and main crack patterns on right and left wall.
carried out very elaborate and intense parametric studies with different modeling techniques and tools considering the parameters such as tensile strength of concrete, crack closure stress limits, material laws and equipping the construction joints with special elements with very low tensile strength, but their results are not superior to those presented in this paper. In fact the findings presented herein produce better simulations in some aspects. The computed top floor absolute horizontal accelerations showed very good match with measured ones (Fig. 18). In order to observe the effect of the concrete cracking on the loading history, Run 4 was performed on the virgin,
the reason why the maximum experimental top displacement (43.3 mm) could not be captured in computations (36 mm) of Run 5 as good as the previous cases (Fig. 12d). The FE model and the assigned non-linear material properties can be improved to retrieve such a failure situation by adopting bond slip occurring between the concrete and rebars and spalling of concrete due to degradation in quality. We believe that better results would have been obtained if more refined material properties and laws had been defined. Although the multilinear kinematic hardening material model used is not recommended for largestrain analyses [2], it gave satisfactory results in the computations. Mazars [5] and Combescure et al. [6] have also
Top Horizontal Absolute Acceleration _ Run1
0.8
Top Horizontal Absolute Acceleration _ Run4
1.2
Measured 0.8
Calculated
Acceleration (g)
Acceleration (g)
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6
Measured Calculated
0.4 0.0 -0.4 -0.8 -1.2
-0.8 10
11
12
13
14
15
16
17
18
4
6
8
10
Time (sec)
Time (sec)
(a)
(b)
12
Fig. 18. Top floor acceleration time history comparison for (a) Run 1 and (b) Run 4.
14
16
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562
Displacement (mm)
15
Experimental Calculated Calculated_uncracked
10 5 0 -5 -10 -15 5
6
7
8
9
10
11
12
13
14
15
Time (sec)
Fig. 19. Effect of cracking of concrete and yielding of steel on the top displacement.
uncracked structure. The results were as shown in Fig. 19. While the cracked model reached a top displacement of 13.2 mm, the virgin model could only reach a translation of 6.35 mm at the top. This exercise not only proved the significance of concrete cracking and yielding of reinforcement on the behavior, but also verified the reliability of the numerical model and the software used. 5. Conclusions and comments Studies performed on the IAEA Camus benchmark project revealed many useful observations not only about the behavior of reinforced concrete load bearing walls but also about the acceptance and accuracy of the analytical modeling approach. The modeling step has proved to be the most crucial aspect of this study. Unless the boundary conditions, material properties, local non-linearity sources, structural integrity and application of the ground motion are described and modeled properly, the results tend to be deceptive. Flexibility of support mechanism needed to be considered and modeled since it is one of the dominating aspects in dynamic behavior of the structure. The physical analogue of this is of course the effect of foundation compliance on dynamic response. Non-linear dynamic analyses proved to be very powerful technique for animating and reviving the experimental studies, even though a complicated material, concrete, is handled. The numerical results indicated that, global structural behavior (displacements, forces, and moments) can be calculated rather accurately, but interpretation of local behavior requires more elaborate and refined FE modeling. While the location of the damage can be estimated quite accurately with the smeared crack model, some enhancement is required in the detection of the level of damage experienced by the structure. The numerical tool used was judged to be effective for the non-linear static analyses. The state-of-the-art in the computational power of current generation analytical tools is fully capable of meeting professional needs, and pass benchmark tests with quite adequate performance. In
non-linear dynamic analyses, convergence problems may be encountered. This study revealed that the simulation of test conditions such as load application, boundary conditions, member sizes and material properties are the key elements of the modeling. The constitutive models used for concrete and steel have been found to be satisfactory for lightly reinforced walls. The accurate prediction of the experimental measurements by the numerical models depends also on the value assigned to shear transfer coefficients for open and closed cracks. We consider that the aim of re-creating accurately the numerical model of an engineering benchmark with the results obtained from the dynamic tests instills confidence in our stress analysis capabilities. The models we developed during the course of the study could replicate both static and dynamic measurements of the Camus experiment. Acknowledgments This paper presents data and describes part of the research work performed and of the results obtained by the authors within the scope of the Co-ordinated Research Project (CRP) of the International Atomic Energy Agency (IAEA) on ‘‘Safety Significance of Near-Field Earthquakes’’, funded and organized by IAEA and the Institute for the Protection and Security of the Citizen of the Joint Research Centre (JRC) of the European Community. This CRP was based on experimental data of the Camus Project funded by the following French organisations: Fe´deration Franc¸aise du Baˆtiment, Plan Ge´nie Civil (Research Program from both Ministe`re de l’Equipement and Ministe`re de la Recherche), CEA and EdF; as well as on seismic input data provided by Japan Nuclear Energy Safety Organization (JNES). The authors would like to express their gratitude to IAEA and JRC for the opportunity to participate in this CRP. References [1] Combescure D. IAEA CRP-NFE Camus Benchmark: experimental results and specifications to the participants. Rapport DM2S. SEMT/ EMSI/RT/02-047/A; 2002. [2] ANSYS Engineering Analysis System. User and theoretical manual. ANSYS, Inc, South Pointe, Canonsburg (PA), Release 7.0 UP20021010; 2002. [3] Hemmaty Y. Modeling of the shear force transferred between cracks in reinforced and fiber reinforced concrete structures. In: ANSYS conference, Pittsburgh (PA); 1998. [4] Chopra A. Dynamics of structures: theory and application to earthquake engineering. 2nd ed. Prentice-Hall; 2000. [5] Mazars J. French advanced research on structural walls: an overview on recent seismic programs. Proceedings 11th ECEE – Paris. Rotterdam: Balkema; 1998. [6] Combescure D, Queval JC, Sollogoub P. Effect of near-field earthquake on a R/C wall structure: experimental and numerical studies. In: 11th ECEE – Paris; 1998.
Numerical simulation of dynamic shear wall tests: A benchmark study _ Ilker Kazaz, Ahmet Yakut *, Polat Gu¨lkan Earthquake Engineering Research Center, Department of Civil Engineering, Middle East Technical University, Ankara 06531, Turkey Received 1 June 2005; accepted 10 November 2005 Available online 19 January 2006
Abstract This article presents the numerical simulation of a 1/3-scale, 5-story reinforced concrete load bearing structural wall model subjected to seismic excitations in the context of IAEA benchmark shaking table experiment conducted in laboratories of CEA in Saclay, France. A series of non-linear time history analyses were performed to simulate the damage experienced and response quantities measured for the specimen tested on a shaking table. The mock-up was subjected to a series of artificial and natural earthquake records. The entire model (concrete, table, masses) was discretized with 3D non-linear finite elements. An elaborate and comprehensive computer simulation process was conducted. A number of modeling iterations were performed for refinement purposes to include those details that were found to be significant in reproducing the measured behavior. The response of the structure was computed both at macro and microlevels and the results were compared with the measured quantities in order to validate the correctness of the analytical model. The comparison of analytical and experimental results yielded extremely good agreement because a numerical model that incorporates the test conditions adequately had been developed. 2005 Elsevier Ltd. All rights reserved. Keywords: Reinforced concrete wall; Dynamic test; Seismic behavior; Non-linear modeling; Finite element method; ANSYS
1. Introduction and description of the test specimen The rapid advancement of analytical and numerical capabilities must constantly be checked against empirical evidence for assessment of its accuracy. Reinforced concrete load bearing walls are commonly used in structures in seismic areas to resist laterally applied earthquake loads. A structural wall is the preferred way of ensuring lateral strength and stiffness in medium to high-rise reinforced concrete buildings. Walls are commonly encountered also in nuclear structures for similar reasons. An item of substantial research and application relevance is the computation of the dynamic response of structural systems with walls in the near-field zones of strong earthquakes where drift becomes the paramount behavior index. In order to
*
Corresponding author. Tel.: +90 312 210 5406; fax: +90 312 210 1193. E-mail address: [email protected] (A. Yakut).
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.11.002
show their capacity to sustain seismic loading and their behavior in case of near-field earthquakes, a series of seismic tests were performed on a model representative of reinforced concrete structural wall system. Understanding the structural behavior of the buildings with reinforced concrete walls not only requires experimental testing program but also many parametric studies using numerical modeling. The purpose will be to concur on the main features of an appropriate methodology to realistically account for the effects of near-field earthquakes and their safety significance on the structural shear walls. This exercise tests also the capability of current analysis tools in modeling against the irrefutable evidence of experiments in a dynamic environment. The 3D behavior of the specimen required the adoption of a non-linear finite element 3D representation. The analytical simulation of the model and verification of the experimental results by using the same experimental history were carried out.
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The experimental program consisted of testing a model with scale 1/3 representative of a 5-story reinforced concrete building on the major Azalee shaking table of Commissariat a l’Energie Atomique (CEA) in the Saclay Nuclear Center, France. The specimen, named CAMUS1, had a total mass of 36 tons with the additional masses attached to it. Walls have no openings and are linked by square slabs measuring 1.7 m · 1.7 m. A heavily reinforced concrete footing allows the anchorage to the shaking table. The total height of the model is 5.10 m. Walls have thickness of 6 cm. The dimensions of the different parts and the mass distribution are shown in the sketches given in Fig. 1. The walls were loaded in their own plane. Adding lateral triangular bracing system increased the stiffness and the strength in the direction perpendicular to the table motion. This system has reduced the risk of failure that might be induced by parasitic transversal motion or a non-symmetric failure of the structural walls. Lateral bracing was designed such that it did not have any contribution to vertical load carrying capacity. Each part of the structure, the wall, floor, and basement was cast separately and assembled on the shaking table. The walls were cast in two parts in order to reproduce the construction joint at the level of each floor (few centimeters just above the middle of each floor). The specimen has reinforcement following the French PS92 seismic design code [1] with some adaptation in order to increase the shear safety factor. The distribution and the detailing of the reinforcement are quite different from those in conventional design practice where generally mesh rein-
forcement is used in the web along with the main longitudinal reinforcement at wall ends. The longitudinally reinforced regions through the 1.7 m wide walls are two edge regions in 10 cm width and the central region with 30 cm width. The amount of reinforcement decreases gradually as it goes to upper stories. The addition of central reinforcement is to limit the risk of sliding shear failure. For transverse reinforcement, bars with 3 mm diameter were used at a spacing of 6 cm along the height of the wall where longitudinal reinforcement is used. The amount of steel at each level and details of reinforcement in elevation are shown in Fig. 2. C20 microconcrete with a compressive strength of 25 MPa and a Young modulus of 28 000 MPa was chosen. These characteristics were checked by the usual compressive and splitting tests on 160 mm diameter cylinders prior to the testing program. They exhibited an unexpected compressive over strength. The compressive strength, tensile strength and modulus of elasticity were found from the average of test results to be 35 MPa, 3.8 MPa and 30 000 MPa, respectively. These values were reported in [1] and used in the analytical computations. The objective of this study is to obtain an analytical model of the tested mock-up such that the experimentally observed and measured responses could be matched by a reasonable accuracy. The challenge of capturing not only global behavior but the measured response at specific local areas has led to a detailed, sophisticated and comprehensive model that represents the features of the test specimen. During the course of the study, important and crucial parameters needed to create an adequate and reliable
Fig. 1. View of the CAMUS specimen and sketch of the walls and masses [1] (units in cm).
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Level 6
Level 5
10 cm
60 cm
30 cm
551 60 cm
10 cm
φ3 / 6 mm as stirrup
Configuration of edge and central reinforcement
Level 4
Longitudinal Reinforcement amount at each level Level 1 2 3 4 5 6
Level 3
Level 2
Level 1
Edge of the wall (mm2) 289.4 188.9 94.4 28.3 15.9 15.9
Center of the wall (mm2) 138 138 110.2 78.4 78.4 Nothing
Fig. 2. The amount and detailing of reinforcement in elevation of single wall.
Ground motions used in the loading program
1
Run5 (PGA = 0.72g)
0.5
0 -0.5
Acceleration (g)
0.5 0 -0.5 Run3 (PGA = 1.11g)
0.5
Run4 (PGA = 0.41g)
0 -0.5 Run1 (PGA = 0.24g)
Run2 (PGA = 0.13g)
0.5 0 0 -0.5 0
2
4
6
8 Time (secs)
10
12
14
Fig. 3. Nice and San Francisco signals used in the experiments.
model representing the physical test specimen were identified. We utilized ANSYS V7.0, [2], a software package capable of performing non-linear static and dynamic analyses and incorporating elements with properties that represent adequately the actual behavior of concrete. The experimental data were provided in the form of drawings, reports and photographs that are mostly contained in Combescure [1]. In the section that follows, a brief summary of the experimental background is provided in order to familiarize the reader with the tests conducted.
2. Experimental program and results 2.1. Loading program Two types of input motion were used in the experimental program. The Nice input (an artificial record) representative of far-field earthquake (FFE) ground motions, and the San Francisco input motion (a naturally recorded seismogram) that resembles the characteristics of near-field earthquake (NFE) ground motions. This benchmark study
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4
T = 0.137 sec
3.5
Acceleration (g)
Table 1 Maximum value of top displacements and internal forces at the base of wall
Acceleration Response Spectra for all Tests (ξ = 5 %)
4.5
Run1 Run2
3
Run3 2.5
Run4
2
Run5
1.5 1
Test
Run 1 Run 2
Top displacement (mm) Bending moment (kN m) Shear force (kN) Axial force Traction (kN) Compression (kN) (static loading: 163 kN)
7.0 1.54 211 75.5 65.9 23.5
Run 3 Run 4 Run 5 13.2 13.4 280 276 106 86.6
44.3 No vertical 102 36.5 Excitation 105
43.3 345 111
50 137 51.9 146
0.5 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Period (sec)
Fig. 4. Five percent damped acceleration response spectra of given ground motions.
investigates the results of five tests, in which the table was driven three times by the scaled Nice input motion (Runs 1, 4 and 5) and twice by the scaled San Francisco input motion (Runs 2 and 3). The applied ground motions were p scaled on the time axis by a factor of 1/ 3 to permit observations on the model to be extrapolated to the prototype. The acceleration waveform and corresponding response spectra for these ground motions are given in Figs. 3 and 4, respectively. It is worth noting that these ground motions were applied sequentially to the same specimen so any accumulation of inelasticity from previous runs was retained in the subsequent runs. These acceleration records were measured on the shaking table and were found to be different from the intended waveform. This is mainly due to driving mechanism of the shaking table, which altered the applied ground motions. Inspection of Fig. 4 suggests that scaling the amplitude of the table motion distorted its frequency content as well. For purposes of the study reported here, this is not a crucial point because the complete record of the actual input motion is used in our calculations. 2.2. Initial dynamic response and overall model behavior The natural frequencies of the test specimen were measured by applying low-level random excitations before each test. The first natural frequency measured in the direction of excitation (in plane bending) before the first test (Run 1) was found to be 7.24 Hz, the second was nearly 33 Hz, and the first vertical natural frequency was around 20 Hz [1]. In the direction perpendicular to the excitation, the first natural frequency was measured as 13 Hz due to the stiffening effect of the steel bracing system. The observed behavior during the tests revealed that the flexibility of the shaking table support mechanism had a significant influence on the dynamic properties of the mock-up. The vertical motion of the shaking table was measured by two vertical accelerometers at two ends of the table revealing significant vertical motion during the
tests although no vertical motion was applied. If the flexibility of the supporting mechanism of the shaking table and the wall-table connection region are not considered (the modeling is performed with fixed table supports and rigid connection region), the first vertical frequency is calculated as 43 Hz. The test results revealed that although the input signal was only applied in the horizontal direction, the extension/compression mode shape was excited during the tests as well. 2.3. Analyses of the internal forces The displacement and acceleration time histories of each floor, shear and axial forces, moments and strains throughout the specimen were obtained by means of instruments located at different parts of the mock-up. Since the exact values of masses at each floor were precisely known, internal forces could be computed from the accelerations recorded by the accelerometers. The peak values of maximal experimental top displacements; and moments, axial and shear forces at the base of the wall are given in Table 1. Strains computed in the static and dynamic analyses are presented later in this article in comparison with the experimental ones. 3. Modeling of the structure The target in developing the numerical model has been to represent the mock-up tested with the available analytical tools. A reasonably accurate model must reflect material non-linearity, dynamic properties of the system, the test boundary conditions and the loading applied. Therefore, simple models with elastic properties or frame models with approximate non-linearity, which can be analyzed by many commercially available software packages, were not considered. Instead, a finite element model that can incorporate cracking of concrete and recognize the steel reinforcement was developed. Towards this end we decided to use ANSYS as our analytical tool. ANSYS was used to model both the wall and the shaking table. The purpose of this selection was to benefit from the features of a special reinforced concrete element called SOLID65. As mentioned previously, the flexibility of the shaking table had a significant influence on the dynamic properties of the test specimen, which made it a vital
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
component of the model. For the lateral stiffening of steel bracings, the three-dimensional spar element, LINK8, a uniaxial tension–compression element with three translational degrees of freedom at each node was used. The steel rods that connect the mock-up to the shaking table were modeled with LINK10 which is a three-dimensional spar element having the unique feature of a bilinear stiffness matrix resulting in uniaxial tension-only (or compressiononly) property [2]. The tension-only option was activated in order not to allow any stiffness contribution when the element was in compression. In the numerical model vertical rods supporting the shaking table were included and assigned a stiffness to capture the measured vertical frequencies. For these rods, a spring element, COMBIN14, was used. COMBIN14 has longitudinal or torsional capability in one-, two-, or three-dimensional applications. The longitudinal spring-damper option is a uniaxial tension– compression element with up to three degrees of freedom at each node: translations in the nodal x, y, and z directions. The elastic constant of each spring element was taken as K = 400 MN/m (in accordance with the experimentally measured response) in the numerical computations. The table and walls were modeled with element SOLID 65 that has eight nodes with three translational degrees of freedom at each node. The solid element (SOLID 65) is capable of cracking (in three orthogonal directions) in tension and crushing in compression. If cracking occurs at an integration point, the cracking is modeled through an adjustment of material properties that effectively treats the cracking as a ‘‘smeared band’’ of cracks, rather than discrete cracks. The concrete material was assumed to be initially isotropic, [2]. Up to three different rebar specifications may be defined. Ties and stirrups were also modeled by making use of this property. The rebars can carry tension and compression, but not shear. They are also capable of plastic deformation and creep. The reinforcing bars were assumed to be smeared in the element volume. The reinforcement was entered as volumetric ratio of that element, defined as the rebar volume divided by the total element volume. In the formulation of stress–strain matrix, D, the relation given in Eq. (1) is used [2]. The amount of reinforcement is used as a modification factor that calibrates the concrete’s strain–stress matrix. Once a crack occurs at an integration point, a plane of weakness is introduced in
553
the direction normal to the crack face to modify the stress–strain relation of concrete. ! NR NR X X R R ½D ¼ 1 V i ½DC þ VR ð1Þ i ½D i i¼1
i¼1
In this equation, NR denotes the number of different reinforcing materials, DC is the stress–strain matrix of the concrete and DR is the stress–strain matrix of reinforcement material, V R i is the ratio of the volume of reinforcing material ‘‘i’’ to the total volume of the element. The stress–strain relation of concrete in tension and the strength of cracked condition are explained in Fig. 5a. In this figure ft is the uniaxial tensile cracking strength and E is the modulus of elasticity of concrete. After cracking, a certain amount of stress relaxation can be included in the element stress formulation with the constant Tc that is taken as 0.6, the default value in our case. Rt is the secant slope defined as shown. It diminishes to zero as the solution converges. Additional concrete material data, such as the shear transfer coefficients, tensile stresses, and compressive stresses were among the parameters of the material model used. 3.1. Steel and concrete material models In the light of the experimental results, the compressive and tensile strength and the modulus of elasticity of the concrete were taken as 35 MPa, 3.8 MPa and 30 000 MPa, respectively. The stress–strain curve of the concrete in compression was represented with three segments as a multilinear isotropic hardening material, and the same model with kinematic hardening property was used in the dynamic analysis (Fig. 5b). In non-linear analysis of reinforced concrete, the shear transfer coefficient must be assumed. For closed cracks (bc), the coefficient is assumed to be 1.0, while for open cracks (bt) it should be in the suggested range of 0.05–0.5, rather than 0.0, to prevent numerical difficulties [3]. In this study, a value of 1.0 was used, which resulted in acceptably accurate predictions. The values less than 1.0 were tested, but they caused convergence problems even at very low loading levels. Only one bilinear curve was used to represent the material property of the four different reinforcing bars. The yield stress was taken as 500 MPa (an average value) at 0.002-strain value and the
σc
σt ft
fc
Tc.ft Ec 1
0.6fc
Rt
ε εck
6ε
(a) Tensile behavior of concrete
Ec 1
ε c=0.002
ck
εcu=0.0035
(b) Compressive behavior of concrete
Fig. 5. Material behavior of concrete.
ε
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_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
Fig. 6. The models created in ANSYS.
stress at the failure is assumed to be 525 MPa with 0.34% strain hardening. Initially a three-dimensional model incorporating all of the details explained above was generated. Due to lengthy computer solution time and enormous storage requirements for saving the analysis results, a 2D model was developed beside the 3D model using the perfect symmetry of the structural system (Fig. 6), i.e. plane stress assumptions appeared to be applicable since no excitation was applied in the direction perpendicular to plane of the wall. In the former representation, out-of-plane response was restrained. The masses of each story were lumped uniformly at the level of each floor by using special mass elements, MASS 21. The solid beam below the wall in Fig. 6 that models the shaking table was found to be adequate. A number of trial models indicated that the flexibility of the shaking table and especially definition of boundary conditions at wall-table connection region had substantial effect on the dynamic behavior of the specimen. Its inclusion in the model was inevitable.
its base (Fig. 7); the anchorage was provided in the middle of basement for a partial length only, approximately 1/3 of the wall length. At the two ends, the gap between the wall and table can be described as a contact surface problem. This region was filled with elastic mortar with unknown properties. Four inclined steel bars were used at the ends to anchor the wall to the shaking table. Although the apparent boundary conditions at the bottom of wall can be assumed as fully fixed, doing so led to deceptive results (the computed response in fact was different). The system was found to be more flexible than assumed. The analysis results proving this statement are presented in the following sections of this article. Once the wall is pushed in one direction one pair of steel bars becomes ineffective and the other pair acts only in tension. So the tension-only spar element, LINK10, is considered to be appropriate to model the steel bars. The diameter of the steel anchor bars was taken as 36 mm (as used in the tests). The contact surface between the wall and table was discretized with spring elements, COMBIN14, having elastic stiffness of 20 000 N/ mm. At the free ends of the wall these springs were placed between the wall and table. Influence of these springs on the first natural frequency and on the first 10 natural frequencies is shown in Fig. 8. While assigning zero stiffness to these springs leads to a first natural frequency of 7.1 Hz, assuming very rigid stiffness, i.e. fixed wall-table connection leads system dynamic response to be around 8.1 Hz. These contact spring elements are also crucial for other purposes because without them or when their stiffness is below a certain limit, sliding shear failure initiates on the plane of wall just vertically above the rigidly connected region under high lateral loads (Fig. 9). 4. Results of numerical analyses
3.2. Modeling of boundary conditions
4.1. Modal and static analyses
One of the key points that is crucial to the modeling strategy is the modeling of the boundary conditions. We modeled the table and wall connection region as they were. The wall foundation was not fully anchored to the table at
Close and reasonable results were obtained for the natural frequencies of the mock-up when the connection between the table and walls were modeled realistically and the effect of the shaking table flexibility was taken into
Fig. 7. Modeling of boundary conditions.
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Influence of the Spring Constant used between wall and table on the natural frequencies of the system
8.3
80
8.1
Frequency (Hz)
70 60 50
K=0 K=5000 N/mm K=10000 N/mm K=15000 N/mm K=18000 N/mm K=20000 N/mm K=25000 N/mm K=30000 N/mm K=35000 N/mm K=40000 N/mm K=4E7 N/mm
40 30 20 10 0 1
2
3
4
5 6 Mode Rank
7
8
9
10
Frequency (Hz)
90
555
Change in the first natural vi bration frequency due to spring stiffness
7.9 7.7 7.5 7.3 7.1 0 m m m m m m m m m m K= /m m m m m m m m m /m N/ N N/ N/ N/ N/ N/ N N/ N/ 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E 4 50 00 50 80 00 00 50 00 50 K= K= K=1 K=1 K=1 K=2 K=2 K=3 K=3 K=4
Fig. 8. Influence of the spring constant on the natural frequencies of the wall-table system.
Fig. 9. Sliding shear failure plane due to lack of vertical stiffness at the base of wall.
consideration. From the response spectrum characteristics of the ground motions applied and the expected dominant first mode response of the model, it appears that the first natural frequency has a significant role in the dynamic forces transmitted to the structure thus influencing its response (Fig. 4). So, different boundary conditions were applied to the model and the first bending and vertical modes were computed as tabulated in Table 2. As evidenced from the results contained in Table 2, both boundary conditions exert considerable influence on the modal response of the model. The flexibility of the connection between the table and the wall affects the first mode significantly but has minor influence on the vertical mode. In the light of all these preliminary analyses we realized that creating a model incorporating both boundary conditions in the table base and table-wall connection is vital for reproducing the experimental results as close as possible. Surprisingly, the static analysis results indicated that the table supporting system flexibility has negligible effect on the load deformation pattern. On the other hand, flexibility
of the table-wall connection region appears to be a significant factor influencing the initial stiffness and post elastic behavior of the load–deformation relation. It is obvious from all these findings that we face the challenging issue of modeling the mock-up by taking into account the non-linearity in the material and geometry, the dimensions of the specimen and the test boundary conditions, and load application that must be handled carefully. Therefore, the actual boundary conditions (flexible support and flexible connection) that were adopted in this study yield an initial system stiffness of 20.1 kN/mm. The computed model natural frequencies are given in Table 3 for both 2D and 3D model for comparison. The model was first subjected to statically applied inverted triangular lateral load at the level of each floor to simulate first mode response. Pushover curve that presents base shear force versus top displacement (roof) for one of the two walls (left wall) is given in Fig. 10 with other two load–deformation patterns that will be described next. The limiting point where the structure reached the state of instability due to excessive damage corresponds to the limit that indicates the capacity of the model. The computed and experimentally measured maximum response quantities for each dynamic run are also plotted on Fig. 10. The pushover curve is a powerful tool to visualize the global non-linear behavior of the structures, and it provides very useful hints about the global behavior of the test specimen in the absence of more elaborate analyses such as non-linear time history analyses. Examination of Fig. 10 reveals good agreement between the capacity curve of structure and the peak global response obtained from the dynamic tests. The information presented in Fig. 10 reveals that the structure remained in the elastic range under the ground motion applied in Run 2 whereas all other cases resulted in inelastic behavior of varying degrees. Needless to say, the largest deformation was measured in Run 5 not Run 3, the strongest shaking intensity, due to sequential application of ground motions. This figure displays also the results of the time history analyses, which are discussed next.
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Table 2 Modal frequencies (Hz) due to different boundary conditions Boundary conditions
1st Bending mode
2nd Bending mode
3rd Bending mode
1st Vertical mode
Support and connection flexible Fixed support and flexible connection Flexible support and rigid connection Fixed support and rigid connection Fixed based wall
7.275 7.858 8.080 8.868 9.190
32.664 36.483 33.095 38.609 39.991
54.856 – 57.405 – –
22.685 42.270 22.872 43.397 44.704
Table 3 Natural frequencies (Hz) Modes
Model 1 (3D) Model 2 (2D)
1
2
3
4
5
6
7
8
9
10
7.274 7.275
9.316 –
13.698 –
22.078 22.685
31.061 –
33.438 32.664
50.340 –
50.706 –
52.975 –
57.306 54.856
160 140
Base shear (kN)
120 100
Linear Static Analyses Triangular loading Triangular loading with 1.6g gravity Rectangular loading Run1 Experimental Run2 Experimental Run3 Experimantal Run4 Experimental Run5 Experimental Run1 Computed Run2 Computed Run3 Computed Run4 Computed Run5 Computed
80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
Top displacement (mm)
Fig. 10. Pushover curve of a single wall.
The comparison of measured and computed responses superimposed on the pushover curve shows that for Runs 3 and 5, while the calculated peak displacements are in good agreement with the measured quantities, the calculated base shear is larger, especially for Run 3, than the measured ones. As discussed previously, the measured base shears are in some sense calculated, i.e. they are computed from the measured accelerations by multiplying them with the floor masses above certain wall section. Besides, the effect of additional vertical force generated in the dynamic test as a result of opening and closing of cracks was not included in the calculation of measured base shear. However, it is fundamentally known that as the axial load increases on a section, the moment and shear capacity also changes (increasing when the axial load level is high). It is known that in Runs 3 and 5, the vertical force on the base section of the wall nearly doubles both in traction and compression (Table 1) due to opening and closing of the cracks on the wall. This miscalculation of the measured base shear force is considered to be the primary reason of the discrepancy between the calculated and measured global response
for Runs 3 and 5 (Fig. 10). So we decided to perform two more pushover cases; one is for the triangular load case in which the effective mass of the system was increased 1.6 times (geffective = 1.6 g) due to vertical excitation (to show the effect of the axial force on the response) and the other one is representative of the uniform (rectangular) loading that is considered as an upper bound case for the load path. The increased effective mass is calculated as the total maximum vertical force at the base (the measured axial compression at Level 1 in Run 3 (Table 1) plus the weight of the structure (165 kN)) divided by the weight of the structure (270 kN/165 kN). The resulting load–deformation curves are presented in Fig. 10. The results support our expectations; the effect of both vertical force and load pattern is very significant on the load–deformation curve. This situation again proves the complexity of the problem and the necessity of comprehensive analysis. 4.2. Time history analyses The reference acceleration histories measured on the shaking table were applied to the model at the level of the shaking table. Time history analyses were carried out only for strong motion duration of the given ground motion records. The analyses for Runs 1–5 were carried out in a sequential order to represent the actual loading history. Top displacement was measured from the node at the top corner of the model. The shear forces at different levels of the structure were calculated by taking the sum of horizontal forces (y-component) of elements at a section. The bending moments were calculated by taking moments of vertical force components (z-component) of elements at a section about the center of section. In time history analyses, a constant damping ratio of 2% was assumed for each mode. This damping was input by means of Rayleigh damping constants ([C] = a[M] + b[K]); i.e. it was assumed that both mass- and stiffness-proportional damping was present in the system [4]. The first two modes each with 2% damping were used to define a and b.
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
In the non-linear transient dynamic analysis solution phase we encountered some problems related to ANSYS software. Direct application of ground motion is not possible within the program. The only form of acceleration input is to create acceleration field acting on all the nodes of model. Thus, the structural response of the model to the base excitation is calculated using the concept of effective earthquake forces [4]. Lumping masses at the floor levels for the wall, the absolute floor displacement vector, ut, under the ground displacement ug is computed from Eq. (2) (Fig. 11a). ut ðtÞ ¼ uðtÞ þ ug ðtÞ 1
ð2Þ
where 1 is a vector of order N with each element equal to unity and in general terms called influence vector (i) that represents the displacements of the masses resulting from the static application of a unit ground displacement. For external dynamic forces Fi(t) the general form of the equation of dynamic equilibrium can be written in the form: _ þ ½K fug ¼ fF ðtÞg ½M f€ ug þ ½C fug
ð3Þ
In the earthquake (base) excitation case, F(t) = 0 since no external dynamic force is applied and the equation of dynamic equilibrium becomes _ þ ½K fug ¼ ½M fig € ½M f€ ug þ ½C fug ug ðtÞ
ð4Þ
Comparing Eq. (3) with Eq. (4) shows that the equations of motion for the structure subjected to ground acceleration, € ug ðtÞ in Fig. 11b and externally applied dynamic load, mi €ug ðtÞ in Fig. 11c at the level of each mass are one and the same. The accelerations calculated with Eq. (4) are relative accelerations; so for computing the total acceleration related to any mass, Eq. (2) must be utilized. In view of the discussions presented above, acceleration data are entered in the form of an array to the program before the start of solution, and in the solution phase for each time integration step (Dt), corresponding acceleration is called in from the array within a small loop covering the whole time history data points. Due to application of this
557
acceleration field at any particular instant, all the masses are multiplied with the ground acceleration value and the resulting force is applied to the structure as an external dynamic loading, which is the right hand side term in Eq. (4). For the given instant governing equations of motion are solved statically including the time integration effects in the calculations. Newmark time integration method is used to solve the equations of motion that are in the same form as Eq. (4). In non-linear analysis the stiffness matrix [K] is a function of unknown displacements, so Newton– Raphson procedure, which is an iterative method to solve non-linear equations, was also used. Time histories for top horizontal displacement, shear force and bending moment at level 1 (base of the shear wall), top floor absolute horizontal and vertical acceleration and corresponding 5% damped top floor response spectra, moment–curvature relationship at the base of wall, strains in the external rebars for Runs 1–5 were computed and are summarized in Figs. 12–19. These figures do not contain the results for Run 2 because it is linear, so only the maximum values are given in Table 4. The corresponding test measurements are also superimposed on the numerical plots. The maximal time history results for Runs 1–5 are summarized in Table 4 with the experimental values given in italicized parentheses. There is a fairly good match between the experimental (measured) and computed (calculated) results. These comparisons revealed the adequacy of the model for the purpose of simulating an experiment realistically by means of analytical tools. The high degree of correlation is a clear indication of the influence of all subtleties in modeling the physical test conditions as well as adequacy of the material models incorporated in the software used. As elaborated previously, the measured time history base shear values were calculated ignoring the effect of dynamic axial forces, thus producing smaller values for Runs 3 and 5 than the computed shear forces. This becomes more evident when the computed base shear using the same analogy (ignoring dynamic axial load) is
uit N=5
ui
.. − m N * u g (t )
mN All the levels have the same mass
mi
mi = m i = 1,2,..5 1
.. − mi * u g (t )
=
.. − m1 * u g (t )
m1
ug
.. u g (t )
Stationary base
(a)
(b)
(c)
Fig. 11. (a) Lumped mass system, (b) ground excitation, (c) effective earthquake forces.
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_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
Fig. 12. Comparison of experimentally measured and numerically calculated global response parameters, such as displacement, base shear and bending moment.
compared to the measured value in Fig. 13. Note that the new maximum computed base value in Run 3 is 108 kN instead of 120.1 kN which is nearly the same as the measured base shear, 107 kN.
4.3. Local results Time history results of several parameters were derived from the computational phase including accelerations and
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Table 4 Results of time history analyses Run 1
Run 2
Run 3
Run 4
Run 5
Top relative displacement (mm)
6.58 (7.0)
1.49 (1.54)
13.0 (13.2)
13.2 (13.4)
35.94 (43.3)
Top abs. horizontal acceleration (g)
0.72 (0.66)
0.27 (0.28)
1.36 (1.11)
1.03 (0.93)
1.48 (1.31)
Level 1 bending moment (kN m)
215.9 (211)
75.6 (75.5)
261.3 (280)
274.9 (276)
346.8 (345)
Level 1 shear force (kN)
66.6 (65.9)
23.2 (23.5)
120.1 (106)
88.2 (86.6)
125.1 (111)
Level 1 axial traction (kN)
44 (44.3)
4.2 (0)
125 (102)
70 (50)
150.9 (137)
Level 1 axial compression (kN)
34 (36.5)
3.8 (0)
165 (105)
70 (52)
179.4 (146)
Strain in the external R-bar Level 4 (·103)
0.112 0.124
0.0275 0.0328
6.91 2.31
3.82 2.69
22.3 6.46
Strain in the external R-bar Level 3 (·103)
0.194 0.209
0.0411 0.0482
2.97 1.22
3.05 1.45
12.8 7.6
Strain in the external R-bar Level 2 (·103)
1.08 1.08
0.102 0.0765
2.08 1.13
2.05 1.67
11.68 9.82
Strain in the external R-bar Level 1 (·103)
0.611 0.385
0.267 0.281
1.75 1.27
1.85 1.50
2.23 2.22
Measured vs. Calculated Base Shear from Accelerations
125
Base Shear (kN)
100 75
Measured
50
Calculated
25 0 -25 -50 -75
-100 -125 5
6
7
8 Time (sec)
9
10
11
Fig. 13. Comparison of measured and calculated base shear force in Run 3.
external rebar strains on each level. The damage progress, in other words the crack development started from the base of the structure as hairline cracks and moved to the upper levels. As it was expected from the ground motions and evident from the experimental and computed results, non-linearity becomes pronounced significantly first in Run 3. The moment–curvature relations plotted for the four levels in Fig. 14 show that the second, third and fourth floor levels deformed into non-linear range after Run 3. In Fig. 15, the strain concentrations in different regions of the wall computed during Run 3 are shown. Due to progressive cracking of concrete and yielding of reinforcement under sequentially applied increasing seismic excitations, the system deformed further. This preliminary damage accumulation is the reason why the test specimen has exhibited such large deformations in Run 4 opposing the expectations. Run 5 is the ultimate loading applied to the specimen that led to excessive damage and failure of reinforcement at different sections. The special design of steel allowing the
damage to spread among different stories rather than localizing it to a particular section of the wall helped the structure survive such an intense sequential loading history. Yielding of the reinforcement occurred in the stories 1–4, but it was excessive in the upper two stories. It is evident that in the regions where the longitudinal steel is reduced, larger deformations were observed. The number of rebars changes at 10 cm below each story level along the height of the wall. For this reason, the cracks on the wall initiate from these interruption regions and progress diagonally between the stories. Smeared crack model that is used to model and detect the damage, the location of the damage and to some degree the level of damage experienced by the structure gave reasonable results when handled carefully. The main crack pattern shown in Fig. 17 at the end of testing program supports this statement. Since strains at a section or a point are the indicators of damage experienced, examining the strains measured on the external rebars from the experimental phase and
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
560
Moment - Curvature _ Level1 _ Run3 300
Moment - Curvature _ Level3 _ Run3 150 100
100 0 -8E-04
-6E-04
-4E-04
-2E-04
0E+00 -100
2E-04
4E-04
6E-04
moment (kN-m)
moment (kN-m)
200
50 0 -2E-03
-2E-03
-1E-03
-5E-04
0E+00 -50
5E-04
1E-03
2E-03
-100
-200
-150 -200
-300
curvature (rad/m )
curvature (rad/m) Moment - Curvature _ Level4 _ Run3 100
Moment - Curvature _ Level2 _ Run3 200
50
100 50 0 -2E-03
-1E-03
-5E-04
0E+00 -50
5E-04
1E-03
-100 -150
m oment (kN-m )
moment (k N-m)
150
0 -2E-03
-2E-03
-1E-03
-5E-04
0E+00
5E-04
1E-03
2E-03
2E-03
-50
-100
-200 -250
curvature (rad/m)
-150
curvature (rad/m)
Fig. 14. Moment–curvature relationship of levels 1–2–3–4 after Run 3.
Fig. 15. Strain distribution and crack pattern developed in the model in Run 3.
comparing them with the calculated ones reveals the damage pattern and location in the structure. Level 3 seemed to be the most critical section as evidenced by the experimental results. In the computations very high strains, denoting excessive damage, were obtained at levels 2–4. The comparison of strains measured on the external rebars for Runs 3 and 5 is given in Figs. 16 and 17. During the ultimate Run 5, the steel yielded at all sections, and this was accompanied by the failure of the longitudinal reinforcement at level 3. The strains obtained from the simulations showed the same deformation tendency with an exception that the computed strains could not reach the failure limit (2.5%) at level 3 but at level 4. So, in experimental stage while the failure initiated at level 3, in the model the first failure indication was observed at level 4. But this is within our expectations since we know that the measured strains at upper levels (3 and 4) is very high and there is a slight difference between these strains showing that initiation of failure at a level is a matter of instant. Another difference in our computations from the measured ones is that, the com-
puted steel strains at level 2 are much greater than the experimental ones in Run 5. This can be explained within the experimental procedure itself and the instrumentation of the wall, as all the experimental measurements were made on the left wall, while it is known that the main crack pattern that was observed on the right wall at the end of the testing program was different than that on the left (Fig. 17). It is known, at least by visual inspection that, there was an excessive damage on right wall at level 2 indicating high strains, which agrees with our computations. The local results (strains) that were computed agree well with the crack pattern on the right wall. After the failure of reinforcing bars at level 3 in Run 5, the response of the system changed drastically. Natural period of the system increased to more than double (from 0.137 s to approximately 0.3 s) due to stiffness degradation in the system. This is a big challenge from the modeling point of view. The analytical model and the material laws we employed in the analysis is not equipped to retrieve a failure situation in the reinforcing bars. This is probably
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I. Strain Level 2 _Run3
8
2 Measured 1.5
strain (x10E-3)
strain (x10E-3)
Strain Level 4 _ Run3
10
2.5
561
Calculated
1 0.5
Measured
6
Calculated 4 2 0
0
-2
-0.5 6
7
8
6
10
9
7
8
10
9
time (sec)
time (sec)
Fig. 16. Strain time histories compared with experimental ones for Run 3 at levels 2 and 4.
Fig. 17. Maximum strains computed at each level compared with experimental ones for Run 5 and main crack patterns on right and left wall.
carried out very elaborate and intense parametric studies with different modeling techniques and tools considering the parameters such as tensile strength of concrete, crack closure stress limits, material laws and equipping the construction joints with special elements with very low tensile strength, but their results are not superior to those presented in this paper. In fact the findings presented herein produce better simulations in some aspects. The computed top floor absolute horizontal accelerations showed very good match with measured ones (Fig. 18). In order to observe the effect of the concrete cracking on the loading history, Run 4 was performed on the virgin,
the reason why the maximum experimental top displacement (43.3 mm) could not be captured in computations (36 mm) of Run 5 as good as the previous cases (Fig. 12d). The FE model and the assigned non-linear material properties can be improved to retrieve such a failure situation by adopting bond slip occurring between the concrete and rebars and spalling of concrete due to degradation in quality. We believe that better results would have been obtained if more refined material properties and laws had been defined. Although the multilinear kinematic hardening material model used is not recommended for largestrain analyses [2], it gave satisfactory results in the computations. Mazars [5] and Combescure et al. [6] have also
Top Horizontal Absolute Acceleration _ Run1
0.8
Top Horizontal Absolute Acceleration _ Run4
1.2
Measured 0.8
Calculated
Acceleration (g)
Acceleration (g)
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6
Measured Calculated
0.4 0.0 -0.4 -0.8 -1.2
-0.8 10
11
12
13
14
15
16
17
18
4
6
8
10
Time (sec)
Time (sec)
(a)
(b)
12
Fig. 18. Top floor acceleration time history comparison for (a) Run 1 and (b) Run 4.
14
16
_ Kazaz et al. / Computers and Structures 84 (2006) 549–562 I.
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Displacement (mm)
15
Experimental Calculated Calculated_uncracked
10 5 0 -5 -10 -15 5
6
7
8
9
10
11
12
13
14
15
Time (sec)
Fig. 19. Effect of cracking of concrete and yielding of steel on the top displacement.
uncracked structure. The results were as shown in Fig. 19. While the cracked model reached a top displacement of 13.2 mm, the virgin model could only reach a translation of 6.35 mm at the top. This exercise not only proved the significance of concrete cracking and yielding of reinforcement on the behavior, but also verified the reliability of the numerical model and the software used. 5. Conclusions and comments Studies performed on the IAEA Camus benchmark project revealed many useful observations not only about the behavior of reinforced concrete load bearing walls but also about the acceptance and accuracy of the analytical modeling approach. The modeling step has proved to be the most crucial aspect of this study. Unless the boundary conditions, material properties, local non-linearity sources, structural integrity and application of the ground motion are described and modeled properly, the results tend to be deceptive. Flexibility of support mechanism needed to be considered and modeled since it is one of the dominating aspects in dynamic behavior of the structure. The physical analogue of this is of course the effect of foundation compliance on dynamic response. Non-linear dynamic analyses proved to be very powerful technique for animating and reviving the experimental studies, even though a complicated material, concrete, is handled. The numerical results indicated that, global structural behavior (displacements, forces, and moments) can be calculated rather accurately, but interpretation of local behavior requires more elaborate and refined FE modeling. While the location of the damage can be estimated quite accurately with the smeared crack model, some enhancement is required in the detection of the level of damage experienced by the structure. The numerical tool used was judged to be effective for the non-linear static analyses. The state-of-the-art in the computational power of current generation analytical tools is fully capable of meeting professional needs, and pass benchmark tests with quite adequate performance. In
non-linear dynamic analyses, convergence problems may be encountered. This study revealed that the simulation of test conditions such as load application, boundary conditions, member sizes and material properties are the key elements of the modeling. The constitutive models used for concrete and steel have been found to be satisfactory for lightly reinforced walls. The accurate prediction of the experimental measurements by the numerical models depends also on the value assigned to shear transfer coefficients for open and closed cracks. We consider that the aim of re-creating accurately the numerical model of an engineering benchmark with the results obtained from the dynamic tests instills confidence in our stress analysis capabilities. The models we developed during the course of the study could replicate both static and dynamic measurements of the Camus experiment. Acknowledgments This paper presents data and describes part of the research work performed and of the results obtained by the authors within the scope of the Co-ordinated Research Project (CRP) of the International Atomic Energy Agency (IAEA) on ‘‘Safety Significance of Near-Field Earthquakes’’, funded and organized by IAEA and the Institute for the Protection and Security of the Citizen of the Joint Research Centre (JRC) of the European Community. This CRP was based on experimental data of the Camus Project funded by the following French organisations: Fe´deration Franc¸aise du Baˆtiment, Plan Ge´nie Civil (Research Program from both Ministe`re de l’Equipement and Ministe`re de la Recherche), CEA and EdF; as well as on seismic input data provided by Japan Nuclear Energy Safety Organization (JNES). The authors would like to express their gratitude to IAEA and JRC for the opportunity to participate in this CRP. References [1] Combescure D. IAEA CRP-NFE Camus Benchmark: experimental results and specifications to the participants. Rapport DM2S. SEMT/ EMSI/RT/02-047/A; 2002. [2] ANSYS Engineering Analysis System. User and theoretical manual. ANSYS, Inc, South Pointe, Canonsburg (PA), Release 7.0 UP20021010; 2002. [3] Hemmaty Y. Modeling of the shear force transferred between cracks in reinforced and fiber reinforced concrete structures. In: ANSYS conference, Pittsburgh (PA); 1998. [4] Chopra A. Dynamics of structures: theory and application to earthquake engineering. 2nd ed. Prentice-Hall; 2000. [5] Mazars J. French advanced research on structural walls: an overview on recent seismic programs. Proceedings 11th ECEE – Paris. Rotterdam: Balkema; 1998. [6] Combescure D, Queval JC, Sollogoub P. Effect of near-field earthquake on a R/C wall structure: experimental and numerical studies. In: 11th ECEE – Paris; 1998.