Bull Earthquake Eng DOI 10.1007/s10518-015-9739-9 ORIGINAL RESEARCH PAPER
Numerical simulation of RC infill walls under cyclic loading and calibration with widely used hysteretic models and experiments Orkun Go¨rgu¨lu¨ • Beyza Taskin
Received: 5 June 2014 / Accepted: 20 February 2015 Ó Springer Science+Business Media Dordrecht 2015
Abstract Addition of reinforced concrete (RC) infill walls into the structural system has been a commonly preferred strengthening technique within the last decades for seismic rehabilitation of RC frames. As a consequence, generating a representative numerical model of an RC infill wall has become an important issue. As the initial step of this study, measured structural responses of two selected well-known large-scale RC infill wall experiments subjected to displacement controlled cyclic loading are taken into account. Later, by calibrating the numerical model prepared in Perform-3D computer program utilizing fiber cross-sections, a practical and a compatible analytical model is obtained and proposed herein. Structural systems of the experiments are mathematically modeled by elements consisting of vertical and horizontal fiber layers to represent the bending/axial behavior and to control the out of plane displacements, respectively. Nonlinear behavior of the reinforcing steel is represented by a tri-linear backbone curve without strength degradation, while a multi-linear hysteretic behavior considering the strength loss is utilized for the structural concrete. Furthermore, those recently conducted experiments are simulated by a couple of widely used hysteretic models for comparative purposes, which are preferred in most cases by the researchers during the analytical investigation of RC structures, so that their adequacy for reflecting the nonlinear behavior of infill walls are also studied. It is shown with comparisons for the experimentally measured and the analytically derived results that the calibrated mathematical model proposed herein is more compatible with the measured values than the widely used hysteretic rules for capturing the behavior of these types of frames retrofitted by RC infill walls under reversed cyclic loading. Although numerical simulations are carried out for a limited number of tests and it is assumed that sufficient amount of anchoring dowels is provided at the interface of the existing frame and the RC infill, the proposed calibrated model conforms to both
O. Go¨rgu¨lu¨ Graduate School of Science Engineering and Technology, ˙Istanbul Technical University, Maslak, Istanbul, Turkey B. Taskin (&) Department of Civil Engineering, ˙Istanbul Technical University, Maslak, Istanbul, Turkey e-mail:
[email protected]
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experiments’ measured responses by means of seismic behavior for not only the undamaged single bay frames converted to the RC infill wall, but also pre-damaged multi-bay strengthened structures, which include structural deficiencies like low concrete strength, inadequate stiffness and insufficient confinement. Keywords Shear wall Reinforced concrete infill wall Fiber element based non-linear modeling Seismic behavior Hysteretic constitutive models
1 Introduction Most of the existing reinforced concrete (RC) structures, especially in Europe, do not comply with recent earthquake codes, since most of them were designed and built in the 1970s (Thermou et al. 2011). Due to their insufficient strength, ductility and stiffness, these buildings should be rehabilitated or replaced by means of reducing the seismic risk to acceptable levels (Varum et al. 2013). Many retrofitting techniques such as steel bracing, jacketing of columns, addition of RC infill walls and FRP wrapping are applicable for the seismic rehabilitation of existing RC structures. To decrease the vulnerability of existing frame systems, introducing the RC infill walls is one of the most preferred retrofitting techniques especially in Turkey (Canbay et al. 2002). In this technique, some of the existing RC frames are converted into the shear walls, therefore seismic capacity of the existing structure is improved significantly (Fardis et al. 2013). During the recent years, many experimental and analytical researches have been conducted to investigate the seismic behavior of RC infill walls. Higashi et al. (1980) performed several experiments on RC infill wall specimens and verified the test results by using nonlinear force controlled analysis. (Canbay 2001) and Anil and Altin (2006) investigated the cyclic behavior of RC infill walls experimentally and carried out static pushover analyses to validate the accuracy of the test results. Darwish (2006) worked through the numerical modeling of RC infill walls in terms of strengthening the bridge frame and conducted parametric analyses under cyclic loading. Sivri (2011) examined the calibration of the RC infill wall experiments by using the static pushover analyses. Strepelias et al. (2012) tested the frame strengthened by an RC infill wall subjected to displacement controlled cyclic loading and performed nonlinear dynamic simulation to verify the pseudo-dynamic test results. As a general approach, most of the researchers preferred pushover analysis method to simulate the nonlinear behavior of RC infill walls. However, this method is based on static loading and not able to capture the cyclic behavior of RC infill wall under earthquake loading. On the other hand, a few researchers performed nonlinear dynamic analyses using finite element model which is neither feasible nor practical to investigate the seismic behavior of RC infill walls due to its complexity and excessive computer time requirements during the analysis. This paper involves in the generation of a representative and a practical numerical model for RC infill walls by calibrating two different well-known experiments under cyclic loading. One of the adopted experiments for the analytical calibration was performed by Strepelias et al. (2014) to investigate RC frames rehabilitated with RC infill walls. During the research, 3:4 scaled, one-bay, four-story frames infilled with RC walls was subjected to quasi-static load cycles and then the damage was repaired before the pseudo-dynamic test had been applied. The other experimental investigation taken for consideration was
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conducted by Canbay et al. (2003) to observe the behavior of RC infill walls. In this second research, test specimen was constructed by two-story, three-bay and 1:3 scale frames but unlike the first experiment, these frames included the common negative features encountered in many existing RC buildings such as low concrete strength, inadequate stiffness and insufficient confinement. The cyclic behavior of the RC infill wall in each conducted experiment is numerically simulated and calibrated by using fiber element model approach utilizing the nonlinear analysis program Perform-3D. Other than the proposed mathematical model, two different commonly used hysteretic material models which are mostly preferred and employed by researchers are also considered and computations are repeated. Although these models are mostly satisfactory for reflecting the response of RC frames and shear walls under cyclic loading, their compatibility for the nonlinear behavior of RC infill walls should be investigated. Finally, numerical results obtained by the proposed model, commonly used analytical models and experimental findings are compared with each other and the capability of the proposed model to reflect the actual behavior of RC infill walls is illustrated in the paper.
2 Overview of the common constitutive hysteretic models employed in analytical investigations Numerous constitutive hysteretic models for the cyclic behavior of concrete are proposed by the analysts in recent years. Two of the earliest and widely used hysteretic models are referred by the researchers’ names: Clough model by Clough and Johnston. (1966) and Takeda model by Takeda et al. (1970). According to the first, the unloading stiffness is assumed to be equal to the initial elastic stiffness of concrete section. For the reloading cases, the concrete hysteretic model was modified by Otani (1981) providing more accuracy in the structural behavior. In this modified model, the initial stiffness (Ky) depends on the unloading stiffness (Kr) with the parameters of the unloading stiffness degradation (a) and the displacements (D) as shown in Fig. 1a. The latter, which is also modified by Otani (1981) and Kabeyasawa et al. (1983), is another preferred model, based on experimental investigations. This modified version includes the parameters for strain hardening with stiffness changes at flexural cracking and yielding point. The unloading stiffness (kr) is defined by the force (Q) and the corresponding displacement (d) at cracking
Fig. 1 Concrete constitutive models: Modified Clough Model (a) and Takeda Model (b)
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(dc; Qc), yielding (dy; Qy) and the previous maximum displacement (du), respectively as given in Fig. 1b (Esquivel 1992; Lizuka et al. 2011). Another widely used model for defining the hysteretic behavior of concrete sections is known as Modified Kent-Park model, which is introduced by Kent and Park (1971) and improved by Scott et al. (1982). This model is also employed by many researchers as in Xiaolei et al. (2008), Martinelli and Filippou (2009) and Zhou et al. (2011) for calibration of the mathematical model of RC structures under cyclic loading. In this model, the stress– strain (rc - ec) relationship of concrete is defined in three regions depending on the value of strain at maximum stress (e0) and strain in %20 of the maximum compressive stress (e20), as shown in Fig. 2. According to the model, the compressive stress (rc) in concrete is defined by the coefficients K and Z, which represent the strength increase factor due to confinement and the slope of the strain softening, respectively. Alternatively, a stress–strain relationship for concrete members subjected to cyclic loading is proposed by Mander et al. (1988) and used by researchers and engineering professionals during numerous analyses (Jalali and Dashti 2010, Xuewei et al. 2011, Jiang and Liu 2011, Yin et al. 2012). In this constitutive model, the parameter of the unloading curve is defined based on experimental investigations for not only confined but also unconfined concrete. For the reloading curve, a linear stress–strain relation followed by a parabolic transition curve between the degraded point (enew, fnew) and the returning point (ere, fre) is assumed as given in Fig. 3. Beside the concrete material model diversity, proposed constitutive hysteretic models for steel are very limited in the literature. The simplest model for simulating the hysteretic behavior of steel is the bilinear model with strain hardening used by many researchers. The only constant required to specify the bilinear model is the strain hardening parameter, which is defined as the ratio of the post-yield and initial elastic stiffnesses. Unlike the bilinear model, the well-known Menegotto-Pinto hysteretic model (Menegotto and Pinto 1973) with many parameters to represent the behavior of steel more realistic under cyclic loading is shown in Fig. 4. This model was modified by Filippou et al. (1983) and preferred by researchers for the analytic investigation of nonlinear behavior of RC structures in most cases, as in Orakcal et al. (2006), Martinelli and Filippou (2009), Jalali and Dashti (2010), Spyrakos et al. (2012), Yang et al. (2013). According to this model, the shape of the hysteretic curve is defined by the curvature parameter (R), which depends on the initial shape parameter (R0), coefficients (a1 and a2) and absolute strain difference (n). a1 and a2 coefficients dominate the degradation of curvature, while the parameter R0 the shape of the transition curve between initial and post-yield stiffness to reflect the Baushinger & Pinching effect. Filippou et al. (1983) also proposed a modification to account for isotropic Fig. 2 Modified Kent-Park constitutive model for concrete
⎡ ⎛ε Kf c' ⎢2⎜⎜ c ε ⎣⎢ ⎝ 0
2 ⎞ ⎤ ⎟⎟ ⎥ ε c ≤ ε 0 ⎠ ⎥⎦ Kf c' [1 − Z (ε c − ε 0 ) ] ε 0 ≤ ε c ≤ ε 20
0.2 Kf c'
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⎞ ⎛ εc ⎟⎟ − ⎜⎜ ⎠ ⎝ε0
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Fig. 4 a Menegotto-Pinto constitutive model for steel and b its modified version including the degradation of the cyclic curve
strain hardening effect by introducing a stress shift to the yield asymptote by introducing the coefficients a3 and a4 (Orakcal et al. 2006).
3 Simulation of RC infill walls by using the fiber element model approach Several mathematical models used for the modeling of the inelastic response of frames strengthened with RC infill walls can be found in the literature. Higashi et al. (1980) idealized the RC infill walls as bracing elements. Canbay (2001) and Anil and Altin (2006) used equivalent beam model, while Darwish (2006) and Sivri (2011) preferred finite element model for numerical simulations of RC infill wall. Generally, these mathematical models can be classified into two groups: Microscopic and macroscopic approaches. Microscopic models such as finite element and multi-layer shell element provide detailed information of the local response but are not feasible in terms of computational efficiency and reliability (Orakcal et al. 2004). On the contrary, macroscopic models such as equivalent beam model; equivalent truss model; strut-and-tie model; multiple-vertical-lineelement model; fiber type model are computationally efficient and are considered being
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simple models compared to the microscopic procedures (Bhaumik and Raychowdhury 2012). Therefore, to obtain a practical and applicable mathematical model of RC infill walls for an extensive use, a macroscopic model approach is preferred during the numerical calibration of the selected two experiments in this study. Since one of the most preferred macroscopic model is the fiber type model (Guedes et al. 1994), the numerical simulation of the test specimens are also evaluated by introducing fiber elements in the analytical model. In this model, the discretization of RC elements is performed at the cross-section by dividing the section in an adequate number of elements, namely fibers, for not only confined and unconfined concrete but also for reinforcing steel (Varum et al. 2013). Concordantly, hysteretic behavior of each material is assigned to these fibers and the section’s force– deformation relation is integrated from each fiber by considering the material’s stress–strain response. The internal force distribution within the element (Dx) is represented by Eq. (1), in the flexibility based fiber element model. In this expression, Q is defined as the vector of element forces, while b(x) is the force interpolation functions which is determined considering the longitudinal axis of the element (x) and the entire element length (L), as given in Eq. (2). Consequently, the sectional forces are calculated from the element force which is followed by the determination of the corresponding fiber stresses, while the fiber strains and the section deformations are calculated by using the fiber stress–strain relations and the virtual force principle, respectively (Taucher et al. 1991). Dx ¼ bðxÞQ " # x 0 0x 1 bx ¼ 1 0 L L
ð1Þ ð2Þ
One of the main advantages of using fiber based element model is to allow the neutral axis to shift along the RC wall section during the analyses. Thus, calculation of the cracked moment of inertia of the cross section is not required. Moreover, by defining the material properties of the reinforcement, the confined and the unconfined concrete in the cross section, the complex nonlinear stress–strain behavior is taken into account. The fiber element model is also compatible with the complex wall structures with irregular openings (Wallace 2007). On the other hand, effect of the compression stress in the shear strength due to the internal friction, is not considered in fiber element modeling approach. Another limitation of the fiber model is that the plane cross sections remain plane which may not be true at all times, especially when significant amount of shear deformations and shear cracks are encountered. In order to handle the effect of these deficiencies in the mathematical model, the two large-scale experiments are selected such that the flexural effects govern the behavior of the slender RC walls and no shear failure is observed during the cyclic loading of the experiments. After all, the fiber element modelling is not a flawless simulation technique for the modeling of shear walls but it offers a reasonable approach for reliable analysis and engineering practice. During the numerical analysis of a structural model, concrete infill walls can either be modeled by the shear wall element, which consists of a vertical fiber and a concrete shear layer, or the general wall element that is composed of all components including the diagonal compression layers as illustrated in Fig. 5. Vertical fiber layers are used to model the bending/axial behavior, while concrete shear layers are employed for defining the contribution of concrete to the sectional shear strength (conventional shear behavior). Diagonal layers in general wall element, transmit shear force by considering the contribution of reinforcing steel to the shear strength through the interaction with the axial-
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Fig. 5 Parallel layers to be employed for analytical modelling (Jiang and Liu 2011)
Fig. 6 Main aspect of the inelastic behavior (taken from Perform-3D user manual)
bending layers, however it may overestimate the shear strength of a wall. Connecting by the nodes, all layers are interacting with each other and the neutral axis shift can be simulated since the cross-section is based on fiber element layer (Perform 3D Components & Elements 2006). Therefore, the mathematical model of RC infill walls and frame elements are represented by using the general wall element to consider both the vertical and horizontal fiber layers. However, in order to avoid the above mentioned disadvantage of diagonal layers, only the concrete shear layer (conventional shear layer) is used for the shear behavior of elements. The skeleton curve of the material constitutive model shown in Fig. 6 is defined by the points Y, U, L and R, representing the initial stiffness, strain hardening, ultimate strength and strength loss, respectively.
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Fig. 7 Energy degradation factor: a EDF = 1.00 and b EDF \ 1.0 (taken from Perform-3D user manual)
Fig. 8 Unloading stiffness factor (taken from Perform-3D user manual)
For the implementation of hysteretic rules, the main aim is to obtain the dissipated energy, which is affected by the energy degradation factor (EDF) and the unloading stiffness factor (USF). The method for establishing a relation between the hysteretic loop and the EDF is based on the max–min deformation of the component and the corresponding EDF. In this sense, EDF for the hysteretic loop as a whole is defined by Eq. (3), where w is the weighting factor, emin and emax are the positive and the negative energy degradation at extreme deformation of the component. Thus, EDF is defined as the areal ratio of the degraded and the non-degraded hysteresis loop within a range from 0.00 to ?1.00, while the USF controls the unloading behavior of the hysteretic curve within a range from ?1.00 (maximum stiffness-minimum elastic range) to -1.00 (minimum stiffness - maximum elastic range) as shown in Figs. 7 and 8. e ¼ wemin þ ð1 wÞemax
ð3Þ
Precision of the mathematical model gets better by increasing the number of fiber and the elements along the RC wall. (Orakcal et al. 2004) have conducted analytical research to investigate the effect of the number of fiber and the elements in modeling of RC shear walls by using the multiple-vertical-line-elements (MVLEM), which has similar concept with the fiber modeling approach in the software used here (Wallace 2007). According to this research, top displacement-base shear cyclic curve response does not significantly change under different number of fibers and the elements. On the other hand, using coarse mesh in mathematical model adversely affects the strain response. Therefore, fine mesh with an adequate number of the fiber element is used for the numerical analyses of RC infill wall. For this purpose, mathematical model of the RC infill walls are implemented into the software by using at least 16 elements with 18 fibers per story, which is nearly same with
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the proposed number of fibers by Orakcal et al. (2004). Thus, the sensitivity of analytic response to section discretization is eliminated in terms of the EDF and the USF during the nonlinear constitutive numerical analyses. Connection between the existing frame and the RC infill was achieved by using epoxy grouted steel dowels to anchor the inner face of the beam and the columns in each of the numerically calibrated experiments. However analytical modelling of the interaction between the frame and the RC infill wall with dowels is a complex task and should be investigated in details. Therefore, experiments, which have negligible deformations at the interface of frame and the RC infill wall due to the conservative number of dowels, are preferably selected herein and numerically calibrated. In this regard, the steel dowels and their effect on the seismic response are not considered during the numerical calibration of the experiments. If the applicable number of anchored dowels is limited, then a comparison of the results of such experiments, the fiber modelling and one of the microscopic modeling approaches, such as finite element model, is recommended for investigating the transfer of shear stresses throughout the interface.
4 Experimental studies and their analytical modeling It is aimed to propose an analytical model for RC infill walls subjected to cyclic loads, which will reflect the real behavior better and also can be handily used during the numerical analysis of strengthened systems by the addition of shear walls. Henceforth, two well-known experimental studies are considered for the calibration purpose of the analytical model to be proposed in this paper. 4.1 Analytical investigation of the first experiment The first of the two experiments involving the wall specimens that is selected for numerical investigation was conducted by Strepelias et al. (2012) to examine the cyclic behavior of frames strengthened with RC infill walls. In this experiment, single bay, four story test specimen, called SW1, were prepared with the scale of 3:4 and then converted to the RC infill wall with a thickness equal to the frame members. The steel dowels, which were used for connecting the frame and the infill RC wall, were anchored into frame members by using epoxy grout. The construction phase of test specimen SW1 is illustrated in Fig. 9. The height of the test specimen was 9000 mm with a 190 mm thickness and 2200 mm width. The concrete strength of the test specimen was approximately 27 MPa. During the construction, plain bars denoted by ST1 were used for the stirrups, while deformed bars denoted by B500c were preferred for the longitudinal and web reinforcement, as indicated in Fig. 10. Initially, the test specimen was subjected to the cyclic loading and after the damage had been repaired, pseudo-dynamic loading was applied. Displacement controlled cyclic loading was subjected to the test specimen with an inverted triangular distributed lateral load and displacement of the top floor was adjusted to ±5, ±60 and ±70 mm, respectively as exhibited in Fig. 11. An axial force of 465 kN, was applied to the test specimen constantly during the test. Mathematical model calibration is performed for only the displacement controlled loading step of the experiment and pseudo-dynamic loading phase is not considered within the contents of this study to create a more simplistic loading for consecutive analyses.
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Fig. 9 Construction phase of test specimen SW1 (Strepelias 2012)
Fig. 10 Reinforcement pattern of test specimen SW1: a infill wall; b beam and c column (Strepelias 2012)
(b)
(c)
80
Top Displacement (mm)
(a)
60 40 20 0 - 20 - 40 - 60 - 80
Fig. 11 Geometry (a); experiment setup (b) and displacement controlled cyclic loading of test specimen SW1 (c) (Strepelias 2012)
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During the numerical simulation of the test specimen, columns, beams and RC infill walls are modeled separately by using general wall element in the software Perform-3D but as mentioned in the previous chapter, the diagonal layers are ignored in order to consider the conventional shear behavior only. Within the model, stirrup reinforcement of the frame and horizontal bars of the RC infill wall are defined into the fiber section in terms of the area ratio which is described as the rebar area divided by the concrete area. The longitudinal bars of the frame and RC infill wall element are implemented in their exact locations for each fiber in the program. Rotation gage elements are used to compute the RC infill wall’s sectional rotations in the analytic model. Since the gage elements introduced in the mathematical model can only be defined between the nodes, the rotation gage element is placed 630 mm above the base of the test specimen, which is nearly the midpoint of the two experimentally measured locations, namely 450 and 900 mm. The non-buckling inelastic steel material and inelastic 1D concrete material component is used for the definition of reinforcement and concrete material, respectively. The skeleton of the stress– strain curve for concrete in compression was represented by the curve that includes strength loss and tensional strength, while the reinforcement constitutive model was implemented into the program as a tri-linear curve without strength degradation as shown in Fig. 12. The relationship between shear stress and strain for the RC infill wall is defined with bilinear model by using Eq. (1) defined in Requirements for Design and Construction of RC Structures (TS500, 2000) for cracking shear strength (Vcr). Vcr ¼ 0:65fctk Ach
ð4Þ
In this expression, cracking shear depends on the cross-section area of wall (Ach) and characteristic tensile strength of concrete (fctk). Since the study for the test specimen is focused on modeling flexural failure mode without shear distress of the web, the non-linear shear strain behavior is not foreseen for the RC infill wall in the mathematical model. Therefore, elastic-perfectly-plastic (e-p-p) model is selected to represent the hysteretic behavior of the shear stress–strain relation. In this model, stiffness and strength range stay constant during cyclic loading, as shown in Fig. 13. The proposed numerical model configuration of the test specimen with an adequate number of the section discretization and component is illustrated in Fig. 14. Due to the test having been carried out in a horizontal position, self-weight of the system is ignored during analyses, however a total 450 kN vertical axial load is applied to the top nodes of each column, as similar to the test procedure. Displacement controlled cyclic loading compatible with the experiment is performed with inverted triangular distributed lateral loads, which are defined at each node corresponding to the floor level of the mathematical model. The calibration philosophy of the EDF and the USF is based on the comparison of the experimentally measured and analytically calculated base shear-top displacement curve. In (b)
Concrete Constuve Model 30
Stress (N/mm2)
Stress (N/mm2)
25 20 15 10 5 0 -0.002
-5
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Steel Constuve Model 700 600 500 400 300 200 100 0
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Shear Stress - Strain Model
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
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Shear Strain
0.003
Fig. 12 Implemented constitutive material models for: a concrete; b steel and c shear stress–strain curve for the test specimen SW1
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Fig. 13 Elastic-perfectly-plastic material model
30 25 20 15 10 5 0 -0.002
-5
0.001
0.004
0.007
0.01
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S
700 600 500 400 300 200 100 0 0
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Fig. 14 The established analytical model of the test specimen SW1
this sense, different sets of EDF and USF are implemented into the program and the obtained results are investigated for each set, respectively. According to the results, the EDF and USF are readjusted by considering the numerically obtained base shear-top displacement curve. For instance, if the curve obtained from the analysis result does not match with the measured one by means of elastic range, USF is recalibrated to reflect the experiment behavior properly. As for the dissipated energy, which is the area under the loop, EDF is recalibrated for the next step. Thus, after consecutive analyses of the numerical model under reversed cyclic loading, the EDF and the USF for hysteretic rules compatible with the experimental measurements are obtained. Since the EDF and USF controls the different response properties of the hysteretic curve, the obtained calibrated sets are unique and not possible to arrive at the same prediction with entirely different sets of the EDF and USF. Calibrated EDF corresponding to the steel and concrete material hysteretic model is given in Fig. 15. The calculated base shear-top displacement and base moment-rotation curves obtained from the analytical modelling are compared with the
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0.7
0.35
L
U
Stress
Calibrated EDF for Steel Constuve Model
(b)
Stress
(a)
1.0 Y
0.5
0.3
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U 1.0 Y
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R
Strain
Strain
Fig. 15 Calibrated EDF for: a concrete and b steel constitutive models
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300
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(b) 1500 Base Moment (kN-m)
Base Shear (kN)
200 150 100 50 0 -50 -100 -150
Proposed Model
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Measured
1000 500 0 -500 -1000 @450 mm (Measured) -1500
@630 mm (Calculated)
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@900 mm (Measured) -2000
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-0.008
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0
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Fig. 16 Comparison of experimentally measured and analytically calculated base shear-top displacement variation (a) and base moment-rotation curve for SW1 specimen (b)
measured experimental results, which indicate that the calibrated analytic model has successfully reflected the measured response of the test specimen by means of not only force–deformation but also base-rotation relationship (Fig. 16). Since the maximum obtained base shear from the calibrated mathematical model (Vmax = 236 kN) is smaller than the calculated cracking shear strength (Vcr = 350 kN) of the wall, non-linear shear deformation has not occurred similar as the test specimen’s observed behavior. 4.2 Analytical investigation of the second experiment The second experimental program, which is considered for the numerical investigation, was performed over two consecutive steps by Canbay et al. (2003). Initially, reversed cyclic loading was applied to a three-bay, two-story, 1:3 scale bare frames, which included common deficiencies like low concrete strength, inadequate stiffness and insufficient confinement in the building stock of Turkey. Later, the damaged frame was rehabilitated by adding an RC infill wall at the central span of the specimen. Without repairing the damage of the frame elements, a second reversed cyclic loading was applied to the strengthened RC frame, named R1. Detailed dimensions of the test specimen, including the frame element section properties are illustrated in Fig. 17. The specimen was constructed approximately to have a 2500 mm total height and the RC infill wall was generated with a 70 mm thickness. Connection between the frame and the RC infill was achieved by using epoxy grouted dowels to anchor the inner face of the beam and columns. The measured
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Fig. 17 a Dimension details, b beam and c column reinforcement pattern of R1 specimen (Canbay 2001)
(a)
(b) 60 45.4 40
Lateral Load (kN)
49.6
52.6 53.1 52.5 51.6 53.2
40 34.9 25
30
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10.7 0 0 -9.8 -20
-15.4 -20.1 -24.9 -30.2
-40
-35.2
-40.1 -45.2
-60
-42.6 -47.1
-49.7 -51.8
-52.7-53.2
Cycles
Fig. 18 General view of the test setup (a) and cyclic loading pattern for R1 specimen (b) (Canbay et al. 2002)
concrete compressive strength was 13.8 MPa for the first story; 16.7 MPa for the second story and 30.8 MPa for the infill wall. Therefore, concrete material models are implemented into the model by considering the differences in the concrete compressive strength for each floor level, respectively as in Fig. 20a. Longitudinal and transverse reinforcements of the frames were detailed by plain bars that have individual yield strengths of approximately 400 and 322 MPa, respectively. The mesh reinforcement pattern of the RC infill wall was constituted with plain bars (2U6/150) that have 378 MPa yield strength. The general view of the test setup for the frame with the RC infill wall and the reversed cyclic loading, which was subjected to the test specimen at the second story, is given in Fig. 18 (Canbay et al. 2002). Within the scope of this numerical study, only simulation of the frame rehabilitated with the RC infill wall (R1) is investigated to verify the calibrated analytic model parameters from the experiment (SW1) achieved by Strepelias et al. (2012). The mathematical model of the test specimen R1 is evaluated under the same analytic model principle of the first calibrated specimen SW1. The general wall element, which includes similar number of sectional discretization and components used in the analytical
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35 30 25 20 15 10 5 0 0
0.003
0.006
0.009
0.012
600 500 400 300 200 100 0 0
0.05
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Fig. 19 Mathematical model established for R1 test specimen
(b)
Concrete Constuve Model C16.7
25 C30.8 20 15 10 5 0 0
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0.006
0.009
0.012
500 400 300 200 100 0 0
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(c)
Steel Constuve Model
600
C13.8
30
Stress (N/mm2)
Stress (N/mm2)
35
0.05
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Shear Stress (N/mm 2)
(a)
Shear Stress - Strain Model 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.001
0.002
0.003
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Fig. 20 Implemented constitutive material models in Perform-3D: a concrete; b steel and c shear stress– strain curve for the test specimen R1
model of SW1, is adopted as the mathematical model of the R1 specimen (Fig. 19). Although the stress–strain relationship for the concrete and reinforcement is adapted by using a tri-linear curve similar to the SW1 mathematical model, the tensile strength of concrete is ignored due to pre-damage of the frame elements (Fig. 20). Calibrated hysteretic rules for the steel and concrete constitutive models from the SW1 analyses are implemented into the mathematical model of R1. Shear stress–strain curve is defined as a bilinear model employing the cracking shear strength calculated as approximately Vcr = 78.5 kN by Eq. (4). Since the test specimen R1 has also flexure-controlled failure mode, the hysteretic behavior of the shear stress–strain curve is implemented using elasticperfectly-plastic material model. Due to the bond/anchorage problem in the longitudinal reinforcement of the members, capacity reduction, which is explained in details by Canbay et al. (2003), was applied to the frame elements in the mathematical model. Other than the self-weight of the system, which is calculated automatically by the software, a constant axial force (9 kN) corresponding to the external weights shown in Fig. 18a is applied to the top nodal points of each column. Cyclic loading compatible with the experiment is realized as a lateral load, which is defined at the top node of the analytical model. The calculated base shear-top displacement variation from the calibrated mathematical model shown in Fig. 21 is compared successfully with the measured experimental results reflecting the measured behavior of the test specimen under cyclic loading. Since the computed maximum base shear for the mathematical model Vmax = 52 kN does not exceed the cracking shear strength of the wall, nonlinear behavior due to shear force is not observed with the test results.
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4.3 Comparison of the proposed and widely used hysteretic models As the third step of the evaluation of the accuracy of the proposed mathematical modelling for RC infill walls, each of the experimental structures are modelled employing commonly used hysteretic models. Among the previously described models, Mander model for concrete and the modified version of Menegotto-Pinto model for steel are selected for comparison, considering their being the two most accepted and widely used constitutive models during the analytical research and investigation of RC structures and structural systems. Therefore, these two material models are selected and adapted to the configured mathematical model of test specimens. Concrete’s hysteretic behavior defined by Mander model is implemented into the numerical modelling by using the adjusted energy degradation factor defined by Xuewei et al. (2011), as given in Fig. 22a. The values in the parenthesis denote the calibrated proposed model in the same figure. For the modified Menegotto-Pinto steel constitutive model, EDF and USF are revised according to parameters (R0 = 20; a1 = 18.5; a2 = 0.15; a3 = 0.01;
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a4 = 7) given by Filippou et al. (1983). The modified hysteretic model and the proposed original steel constitutive model are compared in Fig. 22b. As it can be seen from the figures, although the adopted mathematical model has to be linear due to the limitation of the computer program, no significant difference appears between implemented hysteretic model and original proposed model. Indeed the amount of energy dissipation under cyclic loading is almost the same as the original proposed model. The other parameters, such as section discretization, assigned element fiber type, loading phase, are likewise preserved in the mathematical model. Figure 23 shows the base shear-top displacement variations for both experiments’ analytically obtained results by employing Mander and Menegotto-Pinto models for concrete and steel, respectively. The proposed model and the corresponding test measurements are also superimposed into the graphs for comparative purposes. When the plots are comprehensively inspected by means of the adequacy of encountering the cyclic response of the mathematical models and the experimental results, significant divergence for capturing the experimental pattern by utilizing the widely used analytical model is observed compared to the proposed model, particularly for R1 specimen. Moreover, this observation is also viable for the amount of the dissipated energy for the same specimen. When SW1 test specimen is considered, the numerical differences seem to decrease, however the proposed model matches the experimental measurement results much better. One of the main reasons for the diversity observed for specimen R1 with three bays seems to be the pinching in the cyclic behavior of the system due to the bond-slip effect caused by the insufficient amount of development lengths in the beam-column connections of the existing frame. On the other hand, this phenomenon is not clearly observed in SW1 since the system is consisting of a single bay with adequate detailing of reinforcing bars. Another reason depends on the differences in the variation in the unloading and reloading braches of the cyclic behavior mostly based on low concrete compressive strength, which is calibrated by introducing the energy degradation and unloading stiffness factors in the proposed model.
5 Conclusion The numerical simulation of two different well-known large-scale experiments, in which RC frames rehabilitated with RC infill walls are constructed and response measurements
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are carried out subjected to reversed cyclic loading, is carried out by using the software Perform-3D. Columns, beams and RC infill walls are modeled with general wall element based on fiber layers. The mathematical model configuration of RC infill walls in accordance with the test results is also explained in details. Consecutive analyses indicated that the developed fiber element model is successful in reflecting the nonlinear behavior of RC infill walls subjected to cyclic loading. Indeed, with an adequate number of components and section discretization, the general wall element without diagonal layers can simulate the nonlinear behavior of RC infill walls sufficiently, as shown in the present study. Thus, a realistic and a representative numerical model, which is compatible with the experimental evaluation and also practical for extensive use, is obtained for numerical simulation of RC infill walls. Another major result of the numerical simulation is that, the obtained model including the calibrated hysteretic rules is compatible with the experimentally measured results by means of seismic behavior of not only undamaged single bay RC infill walls but also predamaged multi-bay strengthened frames, which include structural deficiencies encountered in many earthquake prone countries. On the other hand, analyses presented herein indicate that the widely used hysteretic constitutive models for the numerical investigation of the nonlinear seismic behavior of RC structures might occasionally be insufficient to capture the behavior of existing poor quality frames retrofitted by RC infill wall. Consequently, while the established fiber layered analytical model with proposed calibrated hysteretic constitutive rules might need some refinements in the light of the results of new verifications; it gives a more accurate result than the widely used hysteretic material models for the specific two cases handled during this research. This is because the cyclic behavior of the entire structural system is calibrated in terms of energy degradation and unloading stiffness factors. Moreover, the usage of the macroscopic modeling based on fiber elements approach is more practical and needs less computational efforts during the applications for representing the nonlinear behavior of RC infill walls. Based on the measurements realized at the interface of the existing frame and the RC infill of the selected experiments, it is assumed during the analytical modelling that the anchoring steel dowels are sufficient to transfer the stresses and preserve the bond so that they are not included in the model. On the other hand, since the number of the investigated experiments is limited in this numerical research, the further investigation should be undertaken by the new numerical analyses to increase the credibility of the proposed model. Acknowledgments This research was financially supported by the Scientific and Research Council of Turkey (TUBITAK) with the Grant No. 2211.
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