Shear–flexureinteraction models for planar and flanged reinforced concrete walls
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Abstract
This paper presents information on development, calibration, and validation of three companion macroscopic modeling approaches for nonlinear analysis of reinforced concrete (RC) structural walls, including: (1) the baseline twodimensional twonode formulation of the Shear–FlexureInteraction MultipleVerticalLineElementModel (SFIMVLEM), (2) extension of the baseline SFIMVLEM for modeling of squat walls (SFIMVLEMSQ), and (3) extension of the baseline SFIMVLEM to a threedimensional fournode element for simulation of nonplanar RC walls under multidirectional loading (SFIMVLEM3D). The models are implemented in the computational platform OpenSees. Models presented are calibrated and validated against experimental results obtained from tests on RC wall specimens that cover a wide range of physical and behavioral characteristics, including: (1) one slender, two moderatelyslender and one mediumrise planar RC walls tested under inplane loading used for validation of the baseline SFIMVLEM, (2) three squat planar RC wall specimens tested under inplane loading used for validation of the SFIMVLEMSQ, and (3) one Tshaped and one Ushaped RC wall specimen tested under unidirectional and multidirectional loading, respectively, used for validation of SFIMVLEM3D. The comprehensive comparisons between analyticallyobtained and experimentallymeasured wall responses suggest that the analytical models proposed can accurately simulate both global and local wall responses, within their range of applicability. The models capture load–displacement response features of the walls, including lateral load capacity, lateral stiffness, cyclic stiffness degradation, and pinching characteristics, as well as nonlinear shear deformations and their coupling with flexural deformations during cyclic loading. Comparisons between experimental and analytical results at local response levels suggest that the models can also replicate distributions and magnitudes of wall vertical strains and curvatures at various locations. Overall, the analytical models presented provide robust and reliable tools for nonlinear analysis of RC walls, with a wide range of applicability.
Keywords
Finite element modeling Nonlinear analysis Performancebased design Nonplanar walls Multidirectional loading1 Introduction
Use of reinforced concrete (RC) structural walls is very popular worldwide in seismic design of new buildings and retrofit of existing buildings. Their role is to provide sufficient lateral strength and stiffness to minimize nonlinear behavior and limit lateral displacements during servicelevel earthquakes, and to provide sufficient nonlinear deformation capacity (ductility) during severe earthquakes. Application of the PerformanceBased Seismic Design (PBSD) approach (e.g., Miranda and Aslani 2003), in which nonlinear response history analysis is used to predict the key engineering demand parameters of the structural system and its components (e.g., story drifts, plastic rotations in hinge regions, strains in concrete and reinforcing steel, wall shear forces), has become common in regions where moderatetostrong earthquake shaking is anticipated. Given the crucial role of RC walls in seismic performance of buildings, and given the increasing trend in application of nonlinear analysis for design and evaluation of new and existing structures, it is essential that analytical models capable of capturing important nonlinear response characteristics of RC walls are available.
Over the past twenty years, structural and earthquake engineering fields have witnessed major advances in the availability of diverse modeling and analysis tools for RC walls. Contributions have been made from all over the globe, with significant studies reported in the U.S., Chile, New Zealand, Europe, and Japan, as well as other countries. As a result of an organized international effort, comprehensive and comparative model validation studies were recently conducted considering five different macroscopic (phenomenological) and five different microscopic (finiteelement) model formulations validated against five test specimens that reflected a broad range of wall configurations and response characteristics. Summary of results obtained through this international effort are reported in the papers by Kolozvari et al. (2018a, b). However, given the large number of models and specimens considered in these studies, some of the detailed responses representative of each of the ten models considered are not presented in the two mentioned summary papers. Therefore, one of the objectives of this paper, which is also in line with the objectives of this Special Issue, is to provide additional details to complement the validation studies reported in the papers by Kolozvari et al. (2018a, b) for the Shear–FlexureInteraction MultipleVerticalLineElementModel (SFIMVLEM; Kolozvari et al. 2015a).
Nonlinear modeling of RC walls is commonly performed using macroscopic (phenomenological) models because they are fairly easy to implement and computationally efficient, and many of these models have been shown to be reasonably accurate in predicting important hysteretic response characteristics of RC walls (e.g., Taucer et al. 1991; Orakcal et al. 2004; Perform 3D 2005; Jiang and Kurama 2010; Fischinger et al. 2014; Kolozvari et al. 2015b). However, the applicability of macroscopic models is typically limited to cases for which the assumptions implemented in the model formulations are valid, which could have significant implications for their application in engineering practice. For example, the fiberbased models with uncoupled shear and flexural behavior (e.g., Taucer et al. 1991; Orakcal and Wallace 2006; Perform 3D 2005), the socalled uncoupled models, are generally accurate in predicting global and local responses of slender (flexurecontrolled) RC walls; however, they tend to underestimate flexural compressive strains in even slender walls (e.g., Orakcal and Wallace 2006), overestimate the lateral load capacity of moderatelyslender walls (Kolozvari 2013), and underestimate shear deformations (Massone et al. 2009) in mediumrise and squat walls. In addition, the majority of analytical models for RC walls have been validated for planar walls subjected to unidirectional loading only, primarily because experimental data on flanged walls subjected to multidirectional loading are sparse, and previous model validation studies (e.g., Kolozvari et al. 2017) indicate that uncoupled models such as the shear wall element in Perform 3D (CSI) can overestimate the lateral load capacity of a Ushaped wall tested under multidirectional loading by 20% to 50%, depending on the loading direction. Finally, very limited number of analytical models for RC wall can be successfully applied to squat (shearcontrolled) walls, where most of the models available are either empirical (e.g., Weng et al. 2017), or capable of capturing monotonic responses only (e.g., Massone et al. 2009), or are based on finite element model formulations (e.g., Gullu et al. 2018). Therefore, significant gaps still exist in the capability of macroscopic models to effectively simulate the seismic behavior of RC walls with various geometric/reinforcement characteristics (e.g., aspect ratio, crosssection) and loading conditions (e.g., unidirectional vs. multidirectional), which significantly hinders their applicability to a wide range of structural engineering problems.
A number of modeling approaches available in the literature are proposed to address the major shortcoming of widelyused uncoupled models, which is that they fail to capture the experimentallyobserved shear–flexural interaction behavior (Massone and Wallace 2004) in RC walls (e.g., Petrangeli et al. 1999; Massone et al. 2006, 2009; Jiang and Kurama 2010; Panagiotou et al. 2012; Fischinger et al. 2012; Kolozvari et al. 2015a). Previous studies have shown that the SFIMVLEM developed by Kolozvari et al. (2015a, 2018c) can successfully predict the overall loaddeformation behavior, nonlinear shear behavior, coupling of nonlinear flexural and shear deformations, as well as local responses (e.g., strains and rotations) for RC walls that experience significant shear–flexure interaction behavior under reversed cyclic loading conditions (Kolozvari et al. 2015b). However, the SFIMVLEM has two major shortcomings: (1) the model is not applicable to walls with aspect ratios less than about 1.0 (i.e., squat walls) due to the assumptions implemented in the model formulation, and (2) the model is based on a twodimensional twonode beamcolumn element formulation and cannot be applied to nonplanar walls subjected to multidirectional loading. These model shortcomings are addressed here by extending the applicability of the original SFIMVLEM formulation (Kolozvari et al. 2015a) to squat walls and to walls with nonplanar crosssections subjected to multidirectional loading.
2 Objectives and scope
 1.
Provide additional information on validation of the baseline SFIMVLEM against RC wall specimens identified in the paper by Kolozvari et al. (2018a) by reporting global and local response predictions that complement information provided in the prior paper.
 2.Overcome limitations of the baseline SFIMVLEM by:
 a.
Extending the applicability of the model to squat walls, by proposing the socalled SFIMVLEMSQ model formulation, and validating the model against available experimental data obtained from tests on walls with aspect ratios equal to or less than 1.0.
 b.
Extending the existing baseline twonode, twodimensional model formulation to a fournode threedimensional model formulation, referred to as the SFIMVLEM3D, and validating the threedimensional model against experimental results obtained from tests on nonplanar RC wall specimens subjected to unidirectional and multidirectional loading.
 a.
 3.
Evaluate capabilities of the three model formulations presented and to identify directions for future research.
In this paper, formulations of the three models (SFIMVLEM, SFIMVLEMSQ, and SFIMVLEM3D) are first described, followed by discussion of results of validation studies conducted using the three modeling approaches. The validation study involves comparing model results with experimental data obtained from quasistatic tests on nine RC wall specimens that cover a wide range of configurational and behavioral characteristics.
3 Model descriptions
This section provides descriptions of the analytical model formulations considered in this study. The baseline twodimensional twonode SFIMVLEM is described first, followed by its extension for simulation of the response of squat walls (SFIMVLEMSQ), and its extension to a threedimensional fournode model (SFIMVLEM3D) formulation.
3.1 Baseline twodimensional twonode SFIMVLEM formulation
Similarly to the original formulation of the MVLEM, based on the deformations at six degrees of freedom (DOFs) located at the top and bottom of the model element (Fig. 1), axial strains ε_{y} on each RC panel (macrofiber) are calculated using the plane sections assumption and assuming that the relative rotations between the top and bottom boundaries of the model element are concentrated around a point on the central axis of the element at a relative height ch. Shear deformations within the model element are also calculated from the deformations at the element DOFs, and converted to shear strains γ_{xy} on each RC panel, assuming uniform shear strain distribution across the wall crosssection. Since wall shear stiffness and strength of the element evolve according to computed RC panel responses and assumed material behavior, explicit definition of shear modeling parameters is not necessary in the SFIMVLEM, as opposed to other commonlyused wall models with uncoupled axial/flexural and shear behavior, such as the fiberbased beamcolumn element (e.g., Taucer et al. 1991) in OpenSees (McKenna et al. 2000), or the shear wall element in Perform 3D (2005).
Axial strains in horizontal direction (ε_{x}) of each RC panel element are obtained assuming that the sum of horizontal stresses associated with steel and concrete (i.e., resultant stress σ_{x}) is equal to zero. This assumption is incorporated in the SFIMVLEM by introducing additional DOFs defined in the horizontal direction for each macrofiber of the model element (δ_{1},…,δ_{m}, Fig. 1), which are kinematically independent from six DOFs at the top and bottom boundaries of the element. Therefore, the total number of DOFs of each SFIMVLEM element is increased to 6 + m (where m is the number of macrofibers in the model element). Previous studies (Massone et al. 2006) have shown that the assumption of zero resultant horizontal stress implemented in the SFIMVLEM formulation is reasonable for cantilever walls with aspect ratios greater than 1.0 (slender and moderateaspectratio walls), as well as, for columns or piles (Lemnitzer et al. 2016), while for squat walls (i.e., aspect ratio equal or less than 1.0) this assumption leads to significant underestimation of wall strength and stiffness.
3.2 Extension of baseline SFIMVLEM for application to squat walls
A modified formulation of the baseline SFIMVLEM is proposed in this study, called the SFIMVLEMSQ, in order to extend its applicability to walls with aspect ratios equal to or smaller than 1.0 (i.e., squat walls). In the SFIMVLEMSQ, the assumption of zero horizontal resultant stress at each RC panel (σ_{x} = 0), used in the baseline SFIMVLEM to calculate panel axial strains in horizontal direction (ε_{x}), is replaced with the assumption that ε_{x} is constant along the wall length and is calculated according to the empirical expressions proposed by Massone (2010), which were derived based on extensive finite element modeling studies on squat RC walls.
For cantilever walls (singlecurvature)

For zerorotation conditions at the bottom and the top of the wall (doublecurvature)
For cantilever walls (singlecurvature)

For zerorotation conditions at the bottom and the top of the wall (doublecurvature)
The expressions proposed by Massone (2010) presented above are derived based on envelopes of wall hysteretic responses and do not consider reversedcyclic wall behavior. Therefore, simple loading and unloading cyclic rules are proposed and implemented in the formulation of SFIMVLEMSQ, for describing the hysteretic evolution of ε_{x,max}. These hysteretic rules are derived based on experimentallymeasured axial horizontal strains in shearcontrolled wall specimens tested under reversedcyclic loading (Massone 2006; Orakçal et al. 2009), as illustrated in Fig. 2c. As observed from the figure, one full loading cycle, which starts from zero top lateral displacement in the positive direction, consists of the six linear loading/unloading paths presented in Fig. 2c. Although relatively simple, the strain interpolation functions implemented in the SFIMVLEM for ε_{x} provided improved model predictions for the lateral stiffness and lateral load capacity of squat walls, as demonstrated later in this paper.
3.3 Extension of baseline SFIMVLEM to threedimensional fournode model
3.3.1 Inplane behavior
Matrix \( [K_{\text{Beam}} ] \) refers to the wellknown stiffness matrix of a twodimensional elastic beam element (e.g., Kassimali 2010). Note that the properties of each imaginary beam are set such that its stiffness is significantly larger than the inplane stiffness of the wall element.
3.3.2 Outofplane behavior
The outofplane behavior of the SFIMVLEM3D is simulated using the elastic fournode Kirchhoff plate element formulation (Fig. 3c; e.g., Durán and Ghioldi 1990; Brank et al. 2015). This wellknown finite element formulation is implemented in the SFIMVLEM3D with the following assumptions: (1) wall thickness is uniform and small compared to the remaining wall dimensions, (2) outofplane deflections of the wall are small compared to its inplane dimensions, (3) wall material is homogeneous and follows linear elastic behavior, (4) the support conditions of the wall are such that no significant extension of the wall midsurface develops, and (5) shear strain energy is ignored. The stiffness and force terms corresponding to the outofplane DOFs of the SFIMVLEM3D element (δ_{3}, δ_{4}, δ_{5}, δ_{9}, δ_{10}, δ_{11}, δ_{15}, δ_{16}, δ_{17}, δ_{21}, δ_{22} and δ_{23} in Fig. 3c) are assembled together with the inplane element DOFs (δ_{1}, δ_{2}, δ_{6}, δ_{7}, δ_{8}, δ_{12}, δ_{13}, δ_{14}, δ_{18}, δ_{19}, δ_{20} and δ_{24} on Fig. 3b) to form the full (24 × 24) stiffness matrix and force vector representing threedimensional behavior. However, as mentioned earlier, the wall outofplane behavior is of the wall model is uncoupled from its inplane behavior in the present model formulation.
4 Description of test specimens and testing procedures
Main characteristics of selected RC wall specimens used for validation of the models
Model  Specimen ID  Crosssection  Lateral loading  \( \frac{{l_{w} }}{t} \)  \( \frac{{h_{w} }}{{l_{w} }} \)  \( f'_{c} \) (MPa)  \( f_{y,BE} \) (MPa)  \( \rho_{v,BE} \) (%)  \( \rho_{h, web} \) (%)  \( \rho_{v, web} \) (%)  \( \frac{N}{{A_{g} f'_{c} }} \)  \( \frac{{V_{max} }}{{A_{g} \sqrt {f'_{c} } }} \)  \( \frac{{V_{max} }}{{V_{n} }} \)  \( \frac{{M_{max} }}{{M_{n} }} \)  Failure mode^{a} 

SFIMVLEM  RW2  Inplane  12.0  3.00  42.8  434  2.93  0.33  0.33  0.09  0.20  0.52  1.16  CB  
R2  19.0  2.34  46.4  450  4.00  0.31  0.25  0.00  0.16  0.42  1.23  CB/LI  
WSH6  13.0  2.02  45.6  576  1.54  0.25  0.46  0.11  0.29  0.83  1.11  CB/LI  
RWA15P10S78  8.0  1.50  55.8  477  6.06  0.73  0.73  0.10  0.61  0.89  1.19  DC  
SFIMVLEMSQ  T5S1  Inplane  12.5  1.00  35.0  420  9.75  0.68  0.34  0.05  0.67  1.15  0.90  DC  
T2S3  12.5  0.50  29.0  420  5.15  0.68  0.68  0.05  0.84  1.01  0.95  DC  
T4S1  12.5  0.33  34.8  420  3.95  0.68  0.68  0.05  0.82  0.99  0.74  DC  
SFIMVLEM3D  TW2  Inplane (web)  12.0  3.00  42.8  448  1.4  0.44  0.44  0.10  0.47  1.08  1.26  LI  
TUB^{b}  Multidirectional  13.0  2.58  54.7  471  3.39  0.45  0.45  0.04  0.62  1.29  –  DC  
10.5  0.24  0.51  – 
5 Model generation and calibration
5.1 Geometric discretization
5.2 Calibration of material models
5.2.1 Steel stress–strain relationship
Behavior of reinforcing steel is simulated using the uniaxial OpenSees material SteelMPF (Kolozvari et al. 2018c), which is based on the stress–strain relationship proposed by Menegotto and Pinto (1973) and extended by Filippou et al. (1983). Parameters of the material model were calibrated to represent the experimentallymeasured properties of the reinforcing bars used in the tests. The tensile yield strength and strainhardening parameters were modified according to empirical relations proposed by Belarbi and Hsu (1994) to include the effect of tension stiffening on steel bars embedded in concrete. The parameters controlling the cyclic stiffness degradation characteristics of the model were calibrated as R_{0} = 20, a_{1} = 18.5, and a_{2} = 0.15, as proposed originally by Menegotto and Pinto (1973), with the exception of specimens RW2 and TW2 for which a value of a_{2} = 0.0015 was used, based on findings of previous analytical studies (e.g., Orakcal and Wallace 2006) conducted on these specimen.
5.2.2 Concrete stress–strain relationship
Behavior of concrete (confined and unconfined) is described using constitutive model for concrete developed by Yassin (1994), which is available in OpenSees as uniaxial material Concrete02. This constitutive model for concrete was selected because it takes into account important behavioral characteristics of the material behavior such as concrete damage, concrete tensile strength, tension stiffening, and hysteretic behavior; however, it is also relatively simple, computationally efficient, and numerically stable. The monotonic envelope of the stress–strain model for unconfined concrete in compression was calibrated to agree with material properties obtained from cylinder tests on the day of testing by matching the compressive strength (f′_{c}) and the strain at compressive strength (ε′_{c}), where the initial tangent modulus (E_{c}) is calculated automatically within the Concrete02 formulation, as per the Yassin (1994) model. The stress–strain envelope for confined concrete in compression was obtained by computing the peak stress of confined concrete (f′_{cc}) and the strain at peak stress (ε′_{cc}), based on the area, configuration, spacing, and yield stress of the transverse reinforcement, using the confinement model by Mander et al. (1988). The postpeak slope of the strain–stress envelopes for both unconfined and confined concrete were calibrated to match the monotonic curve proposed by Saatcioglu and Razvi (1992). The tensile strength of concrete was determined from the relationship 0.31√f′_{c} (MPa), and a value of 0.00008 was selected for the strain at the peak monotonic tensile stress (ε_{t}), as suggested by Belarbi and Hsu (1994). The postpeak slope of the linear tension envelope of Concrete02 was adopted as 0.05E_{c}, as suggested by Yassin (1994) to represent tension stiffening effects on concrete.
5.2.3 Shear resisting mechanisms across cracks
The biaxial material model FSAM, used in the presented wall model formulations for representing the planestress behavior of RC panel elements, incorporates two models to describe the transfer of shear stresses along concrete cracks: (1) frictionbased shear aggregate interlock model, and (2) linearelastic model for reinforcement dowel action, as described by Kolozvari et al. (2015a). In the shear aggregate interlock model, the shear stress along concrete crack is restrained to zero value when the concrete normal stress perpendicular to the crack is tensile (crack open) and is bounded by the product of a shear friction coefficient η and the concrete normal stress perpendicular to the crack, when the concrete normal stress is compressive (crack closed). In the reinforcement dowel action model, the relationship between shear strain acting on a panel element in the horizontal plane of the wall and the resulting shear stress on vertical reinforcing steel bars is described by a linearelastic relationship with a stiffness of α·E_{s}, where E_{s} is the modulus of elasticity of reinforcing steel and α is the stiffness coefficient. In this study, values of η = 1.0 and α = 0.01 were used for simulation of slender and mediumrise walls using baseline SFIMVLEM and SFIMVLEM3D formulations, while η = 0.35 and α = 0.001 were used for simulation of squat walls using SFIMVLEMSQ.
5.3 Nonlinear analysis solution strategy
The NewtonRapson algorithm is used to perform nonlinear analyses for specimens considered in this study, with convergence assessed using the norm of the displacement vector and a tolerance of 10^{−5} for all specimens. Convergence is initially pursued with current (i.e., updated tangent) stiffness matrix of the model; however, if convergence is not achieved after 100 iterations, the algorithm updates to use the initial (i.e., undeformed) stiffness matrix. Convergence was achieved using the current stiffness matrix for at least 75% of analysis steps for all models. Displacementcontrolled analysis was used for all specimens except for specimen TUB, which was analyzed using forcecontrolled analysis due to complex loading protocol and specimen geometry as explained in Sect. 6.3.
6 Comparisons of analytical and experimental results
Comparison of experimentallymeasured and analyticallypredicted responses for the three SFIMVLEM formulations described previously and the RC wall specimens selected, are presented in this section. Response comparisons are presented for the global load–displacement behavior as well as various local response characteristics of the walls.
6.1 Validation of the baseline SFIMVLEM
Results presented in Fig. 7a demonstrate that the SFIMVLEM can simulate, in a mechanical manner, the evolution of nonlinear shear deformations developing on a wall throughout the loading history, which allows the model to capture the cyclic degradation of wall shear stiffness. This is an important feature of the SFIMVLEM since the majority of analytical models available in commercial (e.g., shear wall element in Perform 3D, CSI) and researchoriented (displacement/forcebased nonlinear beam column element in OpenSees) computational platforms treat wall shear behavior via userdefined adhoc force–deformation (or strain–stress) relationship that are independent of the modeling approach to simulate nonlinear flexural behavior. Previous research (e.g., Kolozvari and Wallace 2016) has shown that selection of the effective stiffness value for the linearelastic relationship for shear (most common approach) can have significant implications on predicted behavior of RC walls under dynamic loading, where a typical value of 0.2E_{c}A_{g} (e.g., LATBSDC 2017; PEERTBI 2.0 2017) leads to underestimation of story drifts and overestimation of wall shear demands. Therefore, the ability of the SFIMVLEM to capture the experimentallymeasured degradation of shear stiffness as a function of increasing lateral deformations is evaluated in the following paragraphs.
Results presented in Fig. 9 reveal that the SFIMVLEM can capture the general trend of shear stiffness degradation with increasing drift levels throughout the loading cycles. The analyticallypredicted uncracked effective shear stiffness value is approximately equal to theoretical value of 0.4E_{c}A_{g}, while the experimentallymeasured initial shear stiffness values (based on the point of first load reversal) are considerably lower, at approximately 0.24E_{c}A_{g} and 0.28E_{c}A_{g} for specimens RW2 and RWA15P10S78, respectively. However, after shear cracking occurs at a drift level of approximately 0.2%, the effective secant shear stiffness reduces significantly in both analytical and experimental results. For specimen RW2, the analytical model predicts effective shear stiffness that is approximately two times larger than the experimentallymeasured values, while for specimen RWA15P10S78, the analyticallypredicted effective shear stiffness is in excellent agreement with experimental results. The overestimation of the effective shear stiffness of specimen RW2 can be associated with very small shear deformation magnitudes measured during the test. It is also important to note that a commonly used value for the effective shear stiffness of 0.2E_{c}A_{g} (e.g., LATBSDC 2017; PEERTBI 2.0 2017; Fig. 9) is four to ten times larger than the shear stiffness values measured during the tests and captured by the SFIMVLEM at postcracking drift levels (i.e., larger than 0.2%), which varies between approximately 0.02E_{c}A_{g} and 0.05E_{c}A_{g}.
6.2 Validation of SFIMVLEMSQ
6.3 Validation of SFIMVLEM3D
The capability of the proposed SFIMVLEM3D model to simulate the behavior of nonplanar wall specimens subjected to cyclic loading is evaluated using experimental results obtained for a Tshaped wall specimen TW2 (Thomsen and Wallace 1995) tested under unidirectional loading, and a Ushaped wall specimen TUB (Beyer et al. 2008) tested under complex multidirectional loading.
Figure 12 shows the comparison between load–displacement responses measured during the experiments and recorded during the analysis for loading cycles in E–W direction (Fig. 12b), cycles in N–S direction (Fig. 12c), and diagonal cycles (Fig. 12d,e), as well as the resultant squarerootofsumofsquares (SRSS) moment at the base of the wall for the diagonal loading cycles (Fig. 12f). It can be observed from Fig. 12b that for cycles in the E–W direction, the SFIMVLEM3D model accurately captures wall lateral load capacity and stiffness, where the overall hysteretic characteristics of the load–displacement response are also well represented by the model, including cyclic stiffness degradation, reloading/unloading stiffness, and pinching characteristics of the response. For loading cycles in N–S direction (Fig. 12c), the SFIMVLEM3D overestimates the lateral load capacity of the wall by less than 5% at low and intermediate levels of lateral displacements (μ = 1.0–4.0), where the overestimation in lateral load is slightly larger (approximately 10%) at large displacement ductility levels (μ = 6.0). For the N–S loading cycles, the hysteretic response characteristics are wellpredicted, where slight overestimation of pinching is observed in the analytical results. Furthermore, the response comparison shown in Fig. 12d reveals that the SFIMVLEM3D predicts reasonably well the measured forces in the actuator in the E–W direction for diagonal loading cycles, where considerable overestimation (of approximately 30%) of the actuator forces can be observed at the displacement ductility level of μ = 6.0 for the loading position where compression is imposed on the S–W corner of the flange (point E in Fig. 12a). However, prediction of the actuator forces in the N–S direction for diagonal cycles (Fig. 12e) are not as accurate, where the model overestimates the measured force for the displacement ductility levels larger than 2.0 by approximately 30% for position D (web in compression) and by 40% for position C (flanges in compression). The degree of accuracy of model predictions for the resultant moment (SRSS) at the wall base for diagonal cycles (Fig. 12e) is inbetween the accuracy of predicted forces in E–W (Fig. 12d) and N–S (Fig. 12e) directions discussed above. One possible reason for the notable discrepancy between analytical and experimental results for the diagonal cycles is related to the inability of the SFIMVLEM3D model to accurately capture nonlinear strain distributions along the wall flange/web (shearlag effect) due to the planesections assumption incorporated in the model element formulation. Finally, although the model captures initiation of strength degradation associated with concrete crushing at the boundaries of the wall web, analytical results do not reflect the significant strength loss that was observed in the experiment due to crushing of concrete along a diagonal compression strut in the unconfined part of the web, which occurred during loading H → B (sweep) at μ = 6.0 (Beyer et al. 2008).
Overall, the results presented demonstrate that the SFIMVLEM3D is a reliable tool for predicting the hysteretic behavior of nonplanar RC walls subjected to unidirectional and multidirectional loading. The analytical predictions at both global and local responses are generally more accurate when compared to analysis results reported by Kolozvari et al. (2017) using the shear wall or general wall elements available in Perform3D (CSI), due to the capability of the SFIMVLEM3D to capture interaction between axial/flexural and shear responses at the model element level, which are not considered in current commerciallyavailable macroscopic models. The major shortcoming of the SFIMVLEM3D compared to commonlyused microscopic models (e.g., VecTor, DIANA, LSDYNA) is the use of planesections assumption, which could impair the accuracy of predicted wall local responses, particularly for nonplanar walls subjected to multidirectional loading. However, the significance of the SFIMVLEM3D lies in its macroscopic formulation that provides a good balance between the model capabilities, accuracy, and computational efficiency, which may be advantageous over detailed microscopic model formulations in many cases.
7 Summary and conclusions
This paper provides information about formulation, calibration, and validation of three accompanying analytical models for nonlinear analysis of RC structural walls, including: (1) twodimensional baseline formulation of the Shear–FlexureInteraction MultipleVerticalLineElementModel (SFIMVLEM), (2) extension of the baseline SFIMVLEM formulation to simulation of squat wall behavior (SFIMVLEMSQ), and (3) a threedimensional version of the baseline SFIMVLEM formulation for simulating the response of nonplanar RC walls under multidirectional loading (SFIMVLEM3D). The model formulations are implemented in the widely used opensource computational platform OpenSees. The models are calibrated and validated against experimental results obtained from tests on RC walls specimens that cover a wide range of wall physical and behavioral characteristics, including: (1) four relativelyslender or mediumrise planar RC walls specimens tested under inplane loading used for validation of the baseline SFIMVLEM, (2) three squat planar RC wall specimens tested under inplane loading, used to verify the accuracy of the SFIMVLEMSQ, and (3) one Tshaped and one Ushaped RC wall specimen tested under unidirectional and multidirectional loading, respectively, used to validate the SFIMVLEM3D model. The analyticallyobtained and experimentallymeasured wall responses are compared at global and local responses levels to assess the capabilities of the proposed models to simulate a wide range of RC wall behaviors.
Comparisons between experimentallymeasured and analyticallypredicted lateral load–displacement responses of the walls reveal that all of the wall models proposed are capable of accurately capturing the cyclic load–displacement response attributes of wall specimens within their range of applicability, including their lateral load capacity, lateral stiffness, cyclic stiffness degradation, and pinching characteristics. In addition, comparison of experimentallymeasured and analyticallyobtained shear deformations show that the SFIMVLEM captures the magnitude of nonlinear shear deformations and degradation of the effective secant shear stiffness under cyclic loading, whereas comparison of vertical curvature profiles revealed that the model also accurately predicts the experimentallymeasured nonlinear flexural deformations. Results presented also demonstrate that the SFIMVLEMSQ captures well the hysteretic sheardominated behavior of squat RC walls. Finally, validation of the SFIMVLEM3D shows that the model can simulate, with reasonable accuracy, the hysteretic loaddeformation response of nonplanar RC walls subjected to uniaxial and multidirectional loading, as well as vertical profiles of tensile and compressive strains measured during the experiments. A major shortcoming of the proposed model formulations, which is identified based on the response comparisons presented, is the inability of the models to capture the strength degradation observed in the experimental results due to failure modes associated with buckling (or fracture) of reinforcing steel bars and lateral instability.
Future work includes providing official OpenSees releases of the SFIMVLEMSQ and SFIMVLEM3D models (baseline SFIMVLEM is already publicly available) and publishing corresponding OpenSeesWiki pages with user manuals and examples to make the models available to the broad research and engineering community. Future studies will also focus on development and implementation of various failure mechanisms (e.g., buckling and fracture of reinforcement, lateral instability) to allow simulation of strength loss in RC walls, validation of the models against dynamic tests on RC wall structures, and their application to systemlevel studies.
Notes
Acknowledgements
This work was supported by the National Science Foundation, Award No. CMMI1563577. Any opinions, findings, and conclusions expressed herein are those of the authors and do not necessarily reflect those of the sponsors.
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