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Bulletin of Earthquake Engineering

, Volume 17, Issue 12, pp 6463–6489 | Cite as

The fixed-strut-angle finite element (FSAFE) model for reinforced concrete structural walls

  • M. Fethi Gullu
  • Kutay OrakcalEmail author
  • Kristijan Kolozvari
S.I. : Nonlinear Modelling of Reinforced Concrete Structural Walls

Abstract

This paper evaluates the capabilities and limitations of a relatively simple mesoscopic finite element modeling approach, referred to as the Fixed Strut Angle Finite Element (FSAFE) Model, in simulating the hysteretic lateral load behavior of reinforced concrete structural walls designed to yield in flexure. The behavioral characteristics of the constitutive panel (membrane) elements incorporated in the model are based on a simple fixed-crack-angle formulation, where normal stresses in concrete are calculated along fixed crack directions using a uniaxial stress–strain relationship, with modifications to represent biaxial softening effects. The constitutive panel model formulation also incorporates behavioral models for the shear-aggregate-interlock effects in concrete and dowel action on reinforcing bars, constituting the shear stress transfer mechanisms across cracks. Model predictions are compared with experimentally-measured responses of benchmark wall specimens with a variety of configurations and response characteristics. Accurate predictions are obtained for important global response attributes of the walls prior to failure, including their lateral load capacity, stiffness, ductility, and hysteretic response characteristics; although instability failures related to buckling of reinforcement and/or out-of-plane instability of the wall boundary region are not captured. The model provides accurate estimates of the relative contribution of nonlinear flexural and shear deformations to wall lateral displacements, as well as local response characteristics including the distribution of curvatures, strains, and shear deformations on the walls, prior to failure. Based on the response comparisons presented, model capabilities are assessed and possible model improvements are identified. Overall, the FSAFE model is shown to be a practical yet reliable modeling approach for simulating nonlinear wall behavior, which can be used within the framework of performance-based seismic design and assessment of building structures.

Keywords

Reinforced concrete Structural wall Analytical model Finite element Flexure Shear 

1 Introduction

Reinforced concrete (RC) structural walls are commonly used in design due to their significant contribution to resistance against lateral actions, including wind loads and earthquake effects, imposed on building structures. Presence of structural walls has considerable impact on the lateral stiffness and strength characteristics of a building, reducing both the elastic displacement demands on the structural system under wind loads or service-level earthquakes, as well as the inelastic displacement or ductility demands under severe earthquakes. In order to provide the necessary level of ductility supply required to attain the targeted seismic performance of a building under design or maximum considered earthquake levels, seismic design codes and recommendations enforce slender walls to exhibit ductile flexural behavior, by incorporating specially-detailed boundary regions and providing sufficient shear capacity for preventing brittle failures, whereas shear-controlled walls or wall segments such as wall piers and spandrels are designed to possess the necessary shear strength to experience limited levels of ductility demand under severe earthquake actions.

Considering the significant contribution of walls on the seismic performance of buildings, and especially with recent implementation of performance-based analysis and design approaches in modern design codes assessment guidelines, robust modeling of the nonlinear seismic response of walls has recently gained much importance among both researchers and engineers. Analytical modeling of the nonlinear response of structural walls can be conducted by using either microscopic (e.g., finite element) or macroscopic (e.g., fiber or plastic hinge) modeling approaches. Microscopic modeling approaches are typically not used in performance-based design or assessment procedures due to increased computational demands and complexities associated with their implementation, calibration, and interpretation of results. Macroscopic modeling approaches available in the literature, with the so-called fiber-based models being more common (e.g., Vulcano et al. 1988; Fischinger et al. 1990; Taucer et al. 1991; Orakcal et al. 2004; Perform 3D CSI 2005; Vásquez et al. 2016), are generally deemed sufficient for modeling of uncoupled shear and flexural responses of slender walls. However, typical fiber-based modeling approaches commonly-used in performance-based design applications fail to capture shear-flexure interaction effects that have been experimentally-observed for both medium-rise walls (Tran and Wallace 2015) and slender walls (Massone and Wallace 2004). Fiber-based models also cannot simulate other important response attributes of walls, including plane-sections not remaining plane, which results in amplification of compressive strains in concrete, as well as salient characteristics of nonlinear shear behavior, including effective shear stiffness, influence of axial load on shear strength, shear ductility, and shear failure type depending on whether or not horizontal web reinforcement yields. There is still a need for simple and computationally-manageable yet robust modeling approaches that can incorporate such behavioral characteristics of walls in the analysis, for more reliable performance-based seismic design applications.

Recent advances in research, as well as improvements in computational capabilities, have made finite element modeling of walls a potentially-feasible approach to be used in real-life applications of performance-based design and assessment. Various hysteretic constitutive models have been developed (e.g., Ohmori et al. 1989; Stevens et al. 1991; Vecchio 1999; Palermo and Vecchio 2003; Mansour and Hsu 2005; Gérin and Adebar 2009; Orakcal et al. 2012) for describing the nonlinear behavior of constitutive RC panel (membrane) elements to be used in finite element model formulations for walls. Numerous research efforts on finite element modeling of walls are also available in the literature (e.g., Vecchio 1989; Palermo and Vecchio 2007; Mo et al. 2008; Rojas et al. 2016; Dashti et al. 2017a; Lu and Henry 2017; Luu et al. 2017; Gullu and Orakcal 2019). However, studies on modeling of walls with both rectangular and non-rectangular cross-sections where the model is comprehensively validated against experimental results for walls with various response features (flexure-dominated, shear-dominated, shear-flexure interaction) at both global and local (deformation, strain) response levels are sparse. Relatively few studies (e.g., Rojas et al. 2016; Dashti et al. 2017a; Gullu and Orakcal 2019) have focused on a comprehensive assessment of an individual model formulation, evaluating its ability to capture various important global and local response characteristics of walls observed experimentally during different test programs.

As an international collaborative effort, comparative model validation studies were recently conducted by a group of researchers (Kolozvari et al. 2018b, 2019), considering five different macroscopic and five different finite element modeling approaches for walls, validated against experimental results for five benchmark wall specimens with rectangular cross-sections, tested as part of five different experimental programs. Although all of the benchmark specimens were designed to yield in flexure, they incorporated a range of configurations and exhibited various response characteristics. As per the objective of this Special Issue, this paper aims to expand upon one of the finite element modeling approaches in the paper by Kolozvari et al. (2019), which is the Fixed Strut Angle Finite Element (FSAFE) model developed by Gullu and Orakcal (2019). A comprehensive summary of the FSAFE model formulation is provided, followed by a detailed evaluation of model accuracy in simulating the experimentally-measured behavioral characteristics of the five benchmark wall specimens, at not only global (lateral load vs. displacement) but also various local (deformation and strain) response levels. Based on the response comparisons presented, significant attributes, strengths, and limitations of the FSAFE model are evaluated, and possible improvements to the model formulation for enhancement of its capabilities are identified.

2 Model description

The Fixed Strut Angle Finite Element (FSAFE) model is an assembly of membrane elements (with zero out-of-plane stiffness), with a smeared stress–strain formulation used to describe the plane-stress behavior of RC. The constitutive behavior of a single model element, which relates an average strain field (εx, εy, γxy) to a smeared stress field (σx, σy, τxy), follows the Fixed Strut Angle Model (FSAM) formulation developed by Orakcal et al. (2012). Working principles of the model and the material constitutive relationships implemented in its formulation are summarized in this section.

2.1 The constitutive fixed-strut-angle model (FSAM)

The fixed-strut-angle model (FSAM, Orakcal et al. 2012) represents the constitutive smeared stress versus average strain behavior of the RC membrane elements in the FSAFE model assembly. The formulation of the FSAM follows a fixed-crack-angle modeling approach, and its principles are based on interpretation and simplification (wherever possible) of previous hysteretic panel model formulations presented in the literature (e.g., Ohmori et al. 1989; Stevens et al. 1991; Vecchio 1999; Palermo and Vecchio 2003; Mansour and Hsu 2005; Gérin and Adebar 2009). In the FSAM, normal stresses in cracked concrete are calculated along fixed crack (strut) directions. Shear stresses developing along crack surfaces, which are calculated using a simple friction-based constitutive relationship, are superimposed with the concrete stresses along the struts for obtaining the total stress field in concrete. The behavior of reinforcing steel is described using a uniaxial stress–strain relationship applied along the orthogonal reinforcement directions. A simple constitutive model with a smeared stress approach is also implemented in the FSAM formulation for representing dowel action on reinforcing steel bars. Superposition of normal stresses (along strut directions) and shear stresses (along crack surfaces) developing in concrete, with smeared axial stresses and dowel-induced shear stresses developing in reinforcing steel bars, provides the resultant average stresses on the panel element (Fig. 1). A version of the FSAM has been implemented in the open-source computational platform OpenSees (McKenna et al. 2000) as a plane-stress constitutive nDMaterial model for RC (Kolozvari et al. 2018a).
Fig. 1

Constitutive Fixed-Strut-Angle Model (FSAM): a strain field on panel element, b concrete stresses along fixed struts, c shear aggregate interlock stresses along cracks, d uniaxial stresses in reinforcement, e dowel stresses in reinforcement

Before cracking, the plane-stress behavior of concrete in the FSAM is simulated using a rotating-principal-stress-direction approach (e.g., Vecchio and Collins 1986; Pang and Hsu 1995). The applied strain field on concrete is transformed into principal strain directions (θ in Fig. 2a), and a uniaxial monotonic stress–strain relationship for concrete is applied along the principal strain directions for obtaining the concrete principal stresses (Fig. 2a). When the principal tensile strain first exceeds the monotonic cracking strain of concrete, the first crack develops, and the principal strain direction at first cracking is assigned as the first “fixed strut” (first crack) direction in the panel (θcrA in Fig. 2b). After the first crack forms, while principal strain directions continue to rotate based on the applied strain field, the directions along which the uniaxial stress–strain relationship for concrete are applied are assumed to be fixed, in parallel and perpendicular directions to the first fixed strut (Fig. 2b). Upon load reversal, when the normal strain along the first strut (which was initially in compression) first exceeds the cracking strain of concrete upon unloading from compression (also considering the plastic compressive strain), the second crack forms, at which stage the second “fixed strut” will be activated in parallel direction to the second crack (θcrB in Fig. 2c). It is assumed that the second crack develops in perpendicular direction to the first, according to a principal-stress-based cracking criterion, which is based on the fundamental assumption that before formation of the second crack, shear stress transfer across the first crack is of negligible intensity. Therefore, principal stresses in concrete are assumed to develop approximately in parallel direction to the first crack, enforcing the second crack to form perpendicular to the first. The orthogonal crack assumption has been shown to provide accurate response predictions for the RC panel specimens upon which the FSAFE model was experimentally validated (Orakcal et al. 2012). During further loading, normal stresses in concrete are calculated along the two independent struts working under interchanging compression and tension (Fig. 2c). Although the concrete stress–strain relationship used to calculate normal stresses along the two struts is fundamentally uniaxial, it also incorporates parameters to consider biaxial softening effects under plane-stress loading.
Fig. 2

Concrete behavior in FSAM: a uncracked, b after formation of first crack, c after formation of second crack

2.2 Mechanisms for shear stress transfer across cracks

The above-described baseline formulation of the FSAM allows implementing a constitutive (shear stress vs. shear strain) relationship to represent shear stresses developing on crack surfaces, since unlike typical rotating-crack models, shear strains along crack directions can be explicitly calculated. Therefore, on top of its baseline formulation, a simple friction-based constitutive relationship for shear aggregate interlock is adopted in the FSAM, which relates the maximum shear stress developing along a crack surface to the compressive stress in concrete in perpendicular direction to the crack σc (which is equal to σcx’ or σcy’ in Fig. 1b), as well as the smeared “clamping stress” created by the perpendicular component of the yield strength of the reinforcement straddling the crack, (ρfy) (Fig. 3a). In this model, the shear stress capacity along a crack is bounded by the product of the friction coefficient (η) and the total normal stress perpendicular to the crack (σ), where σ is the sum of the compressive stress in concrete perpendicular to the crack σc (which changes during loading) and the clamping stress capacity of the reinforcement (ρfy) (which is assumed constant), similarly to the shear friction capacity approach used in ACI 318 (2014). The contribution of normal stress in concrete perpendicular to a crack (σc) to the shear stress developing along the crack surface is considered only when that normal stress is compressive; that is, only when the crack is closed. In order to describe the hysteretic characteristics of this friction-based constitutive relationship, peak-oriented hysteresis rules are used in the present model formulation, where the unloading from the shear stress versus strain envelope follows the initial elastic stiffness (a large value of 0.4Ec representing the elastic shear modulus of concrete was adopted for the initial stiffness), zero stress is maintained until the origin, and reloading to the envelope in the opposite direction is oriented towards the previous peak (Fig. 3a). These peak-oriented hysteresis rules are adopted as a simplification of more detailed hysteresis models available in the literature (e.g., Vassilopoulou and Tassios 2003) for rough crack surfaces. In the present model formulation, a friction coefficient value of η = 0.50 is adopted for the friction coefficient, similarly to the value of η = 0.44 recommended by Tassios and Vintzēleou (1987).
Fig. 3

Shear stress transfer mechanisms across cracks: a shear aggregate interlock on crack surfaces, b dowel action on reinforcement

For consideration of dowel action on reinforcing steel bars, the constitutive model by He and Kwan (2001), which uses a smeared stress versus average strain approach, is implemented in the FSAM formulation (Fig. 3b). In the He and Kwan (2001) model, shear and tensile strains in a RC panel element parallel and perpendicular to a crack are first transformed into dowel deformations (Δdow) acting on the horizontal and vertical reinforcing steel bars, using strain transformation equations and an “effective dowel length” parameter, which semi-empirically depends, based on a beam on elastic foundation analogy, on the elastic modulus of reinforcing steel, the compressive strength of concrete, rebar diameter, and the moment of inertia of the rebar cross-section. The elasto-plastic envelope of the constitutive model, which relates the dowel (shear) force on a single rebar with the dowel deformation, starts with a linear elastic region, the slope of which also depends on the same parameters. The plastic region of the model represents the dowel (shear force) capacity of a single rebar, which depends on the bar diameter, the compressive strength of concrete, and the yield strength of reinforcement. Dowel forces calculated on the horizontal and vertical rebars are then converted into smeared dowel stresses, considering the reinforcement ratios along the two orthogonal rebar directions. Finally, these dowel stresses are back-transformed, using stress transformation equations, into shear and tensile stresses developing in the panel element, in parallel and perpendicular directions to a crack. Details of the model are available in the paper by He and Kwan (2001). In the present FSAM formulation, the origin-oriented hysteresis rules shown in Fig. 3b (upon unloading from and reloading to the monotonic envelope by He and Kwan 2001) are implemented as a simplification of more detailed hysteretic models (e.g., Vintzēleou and Tassios 1986) presented in the literature.

2.3 Material constitutive models

The state-of-the-art constitutive model by Chang and Mander (1994) is adopted in the FSAM for representing the uniaxial stress–strain behavior of concrete (Fig. 4a). The model formulation reflects important behavioral characteristics of concrete such as the hysteretic transition from compression to tension and tension to compression, the progressive degradation of the tangent stiffness of the unloading and reloading curves, and gradual crack closure. The model formulation is implemented also with modifications to reflect the behavioral characteristics of concrete associated with biaxial softening behavior under plane stress loading, as well as smeared post-crack behavior in tension, by incorporating compression softening (Model B by Vecchio and Collins 1993), hysteretic biaxial damage (Mansour et al. 2002), and tension stiffening (Belarbi and Hsu 1994) relationships into its formulation.
Fig. 4

Material constitutive models: a concrete (Chang and Mander 1994), b reinforcing steel (Menegotto and Pinto 1973)

The uniaxial stress–strain relationship implemented in the FSAM for reinforcing steel bars (Fig. 4b) is the well-known hysteretic model by Menegotto and Pinto (1973), extended by Filippou et al. (1983) for representing isotropic strain hardening behavior, and further extended by Kolozvari et al. (2018a) to overcome stress overshooting upon small-magnitude strain reversals. In implementation of the Menegotto and Pinto (1973) model in the FSAM, the tensile yield strength and strain-hardening parameters of the model were calibrated also considering the empirical relationships proposed by Belarbi and Hsu (1994) to include the effect of tension stiffening on the smeared stress–strain behavior of rebars embedded in concrete. However, the present model formulation does not consider rebar buckling or fracture behavior.

Similarly to the nDMaterial model for the FSAM, the above-described uniaxial constitutive models for concrete and reinforcing steel have been implemented in the open-source computational platform OpenSees (McKenna et al. 2000) as uniaxialMaterial models ConcreteCM and SteelMPF, respectively (Kolozvari et al. 2018a).

2.4 FSAFE model assembly

The FSAFE model of a RC wall is simply generated using a direct stiffness assembly of rectangular four-node membrane elements with only in-plane translational degrees of freedom defined at the nodes. The average strain field on each model element is first calculated using the nodal displacements along the elements degrees of freedom (Fig. 5a) using the following kinematic transformation:
$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\gamma_{xy} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2b}}} \right. \kern-0pt} {2b}}} & 0 & {{1 \mathord{\left/ {\vphantom {1 {2b}}} \right. \kern-0pt} {2b}}} & 0 & {{1 \mathord{\left/ {\vphantom {1 {2b}}} \right. \kern-0pt} {2b}}} & 0 & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2b}}} \right. \kern-0pt} {2b}}} & 0 \\ 0 & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2h}}} \right. \kern-0pt} {2h}}} & 0 & { - {1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} & 0 & {{1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} & 0 & {{1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} \\ {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2h}}} \right. \kern-0pt} {2h}}} & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2b}}} \right. \kern-0pt} {2b}}} & { - {1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} & {{1 \mathord{\left/ {\vphantom {1 {2b}}} \right. \kern-0pt} {2b}}} & {{1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} & {{1 \mathord{\left/ {\vphantom {1 {2b}}} \right. \kern-0pt} {2b}}} & {{1 \mathord{\left/ {\vphantom {1 {2h}}} \right. \kern-0pt} {2h}}} & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2b}}} \right. \kern-0pt} {2b}}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {D_{1} } \\ {D_{2} } \\ \vdots \\ {D_{8} } \\ \end{array} } \right\} $$
(1)
where b and h are the width and height of model element, respectively. The average strain field is then used to obtain the average stress field acting on the model element, as well as the tangent stiffness (partial derivative) matrix of the model element at the stress–strain level, using the constitutive behavior described by the FSAM:
$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\gamma_{xy} } \\ \end{array} } \right\}\mathop{\longrightarrow}\limits{\text{FSAM}}\left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\tau_{xy} } \\ \end{array} } \right\} \, , \, \left[ {\begin{array}{*{20}c} {\frac{{\partial \sigma_{x} }}{{\partial \varepsilon_{x} }}} & {\frac{{\partial \sigma_{x} }}{{\partial \varepsilon_{y} }}} & {\frac{{\partial \sigma_{x} }}{{\partial \gamma_{xy} }}} \\ {\frac{{\partial \sigma_{y} }}{{\partial \varepsilon_{x} }}} & {\frac{{\partial \sigma_{y} }}{{\partial \varepsilon_{y} }}} & {\frac{{\partial \sigma_{y} }}{{\partial \gamma_{xy} }}} \\ {\frac{{\partial \tau_{xy} }}{{\partial \varepsilon_{x} }}} & {\frac{{\partial \tau_{xy} }}{{\partial \varepsilon_{y} }}} & {\frac{{\partial \tau_{xy} }}{{\partial \gamma_{xy} }}} \\ \end{array} } \right] $$
(2)
Components of the 8 × 1 internal (restoring) force vector of a single model element (Fig. 5b) can be calculated using equilibrium of forces along the nodal degrees of freedom with the average stresses in the model element (Fig. 5c), as:
$$ \left\{ F \right\} = \left\{ {\begin{array}{*{20}c} {F_{1} } \\ {F_{2} } \\ \vdots \\ {F_{8} } \\ \end{array} } \right\} = \begin{array}{*{20}c} {\,\left\{ { - \,\sigma_{x} \frac{ht}{2} - \tau_{xy} \frac{bt}{2}} \right.} & { - \,\sigma_{y} \frac{bt}{2} - \tau_{xy} \frac{ht}{2}} & {\,\,\sigma_{x} \frac{ht}{2} - \tau_{xy} \frac{bt}{2}} & { - \,\sigma_{y} \frac{bt}{2} + \tau_{xy} \frac{ht}{2} \cdots } \\ {\quad \cdots \sigma_{x} \frac{ht}{2} + \tau_{xy} \frac{bt}{2}} & {\,\,\,\sigma_{y} \frac{bt}{2} + \tau_{xy} \frac{ht}{2}} & { - \,\sigma_{x} \frac{ht}{2} + \tau_{xy} \frac{bt}{2}} & {\left. {\,\,\,\,\sigma_{y} \frac{bt}{2} - \tau_{xy} \frac{ht}{2}} \right\}^{T} } \\ \end{array} $$
(3)
where t is the thickness of the model element (wall thickness). Terms in the 8 × 8 tangent stiffness matrix of a model element can then be computed using partial derivatives of the restoring forces with respect to the displacements along the element nodal degrees of freedom, as:
$$ \left[ K \right] = \left[ {\begin{array}{*{20}c} {K_{11} } & \cdots & {K_{18} } \\ \vdots & \ddots & \vdots \\ {K_{81} } & \cdots & {K_{88} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial F_{1} }}{{\partial D_{1} }}} & \cdots & {\frac{{\partial F_{1} }}{{\partial D_{8} }}} \\ \vdots & \ddots & \vdots \\ {\frac{{\partial F_{8} }}{{\partial D_{1} }}} & \cdots & {\frac{{\partial F_{8} }}{{\partial D_{8} }}} \\ \end{array} } \right] $$
(4)
where the chain rule can be applied in calculation of the partial derivatives. As an example, the first term K11 in the element tangent stiffness matrix is obtained as:Finally, the 8 × 1 restoring force vectors and the 8 × 8 tangent stiffness matrices of the model elements are assembled, using a standard direct stiffness assembly, for obtaining the global restoring force vector and the global tangent stiffness matrix of the FSAFE model generated for a wall. The formulation described above becomes equivalent to a standard finite element model formulation, in which linear strain interpolation functions and only a single Gauss integration point (located at the element centroid) are used for each model element. The model formulation is implemented in MatLab (2012) together with a displacement-controlled nonlinear analysis solution strategy (Clarke and Hancock 1990), which allows direct comparison of model response predictions with test results available in the literature on RC wall specimens subjected to drift-controlled loading protocols.
Fig. 5

FSAFE model element: a Displacements along nodal degrees of freedom, b forces along nodal degree of freedom, c equilibrium of nodal forces and average internal stresses

3 Experimental validation of the model

In line with the objective of this Special Issue, this paper aims to provide detailed information on experimental validation of the FSAFE model against test results presented in the literature for the five densely-instrumented benchmark wall specimens identified in the papers by Kolozvari et al. (2018b, 2019), which were tested as part of five different experimental programs. Information on the characteristics of the wall specimens, calibration of the FSAFE model parameters for the wall specimens, and comparison of model results with test data at both global (force and displacement) and local (deformation and strain) response levels are presented in this section.

3.1 Description of the wall specimens

The benchmark wall specimens used for experimental validation of the FSAFE model include Specimen RW2 (Thomsen and Wallace 1995); Specimen RW-A15-P10-S78 (Tran and Wallace 2015); Specimen WSH6 (Dazio et al. 2009); Specimen S6 (Vallenas et al. 1979); and Specimen R2 (Oesterle et al. 1976). One additional wall specimen, Specimen RW-A20-P10-S63 (Tran and Wallace 2015), is also included in the response comparisons presented in this paper. These wall specimens were selected to encompass a broad range of test parameters and various important response characteristics for planar walls designed to yield in flexure; including wall aspect ratio (or shear span-to-depth ratio), wall length-to-thickness ratio in plan, axial load level, average shear stress demand at nominal flexural capacity, and failure mode. The aspect ratio of the walls ranged from 1.26 to 3.00, while their shear span-to-depth ratios varied between 1.50 and 3.13, due to the different loading configurations applied during testing. For all wall specimens, the ratio of shear force at wall nominal flexural capacity to wall nominal shear strength was less than 1.0; however, the walls subjected to varying levels of average shear stress demand (ranging from \( 0.16\sqrt {f^{\prime}_{c} } \) to \( 0.61\sqrt {f^{\prime}_{c} } \)) at the measured lateral load capacity. Three specimens (RW2, WSH6, R2) showed predominantly flexural responses, whereas the remaining three (RW-A20-P10-S63, RW-A15-P10-S78, S6) experienced significant shear-flexure interaction behavior during testing. Levels of constant axial load applied on the wall specimens during the tests corresponded approximately to 0% (R2), 5% (S6), and 10% (for the remaining four specimens) of their axial load capacities. Experimentally-observed failure modes of the walls included flexure-governed compression and tension failures (concrete crushing, rebar buckling, and rebar fracture) observed for specimens RW2, WSH6, and R2; and diagonal-compression-related failures (crushing of concrete under combined shear-flexural effects) observed for specimens RW-A20-P10-S63, RW-A15-P10-S78, S6. Lateral (out-of-plane) instability of the wall boundary region was observed on four specimens (RW-A20-P10-S63, RW-A15-P10-S78, and S6) at the ultimate drift levels applied during testing, whereas specimen R2 started experiencing significant lateral instability at earlier drift levels. Important characteristics of the wall specimens are listed in Table 1, whereas specimen cross-sections at wall base are shown in Fig. 6.
Table 1

Characteristics of wall specimens

Specimen ID

lw (mm)

h (mm)

t (mm)

\( \frac{{l_{w} }}{t} \)

\( \frac{h}{{l_{w} }} \)

\( \frac{M}{{Vl_{w} }} \)

\( f'_{c} \) (MPa)

\( f_{y,BE} \) (MPa)

\( \rho_{v,BE} \) (%)

\( \rho_{h, web} \) (%)

\( \rho_{v, web} \) (%)

\( \frac{P}{{A_{g} f'_{c} }} \)

\( \frac{{V@M_{n} }}{{V_{n} }} \)

\( \frac{{V_{max} }}{{A_{CV} \sqrt {f'_{c} } }} \)

Behavior mode

Failure mode

RW-A20-P10-S63

1219

2440

152

8.0

2.00

2.00

48.6

477

7.11

0.61

0.61

0.1

0.91

0.57

SFI/DC

CB/LI

RW2

1219

3660

102

12.0

3.00

3.13

42.8

434

2.93

0.33

0.33

0.09

0.53

0.20

FL

CB

RW-A15-P10-S78

1219

1830

152

8.0

1.50

1.50

55.8

477

6.06

0.73

0.73

0.1

0.85

0.61

SFI/DC

CB/LI/SS

WSH6

2000

4030

150

13.0

2.02

2.26

45.6

576

1.54

0.25

0.46

0.11

0.75

0.29

FL

CB

S6

2412

3048

114

21.0

1.26

1.60

34.7

482

5.60

0.55

0.55

0.05

0.75

0.53

SFI/DC

CB/LI

R2

1905

4470

102

19.0

2.34

2.40

46.4

450

4.00

0.31

0.25

0.00

0.43

0.16

FL

CB/BR/LI

SFI shear/flexure interaction, DC diagonal compression, FL flexure, CB concrete crushing/bar buckling, BR bar rupture, LI lateral instability of wall boundary, SS sliding shear

Fig. 6

Wall cross-sections and reinforcement details for specimens: a RW-A20-P10-S63, b RW2, c RW-A15-P10-S78, d WSH6, e S6, f R2

3.2 Calibration of the FSAFE model for the wall specimens

FSAFE models generated for all wall specimens were geometrically-discretized using a standardized approach. On a wall cross-section, two elements side-by-side were used to represent each confined boundary region. The entire wall geometry was then discretized, using square-shaped model elements with approximately equal width and height. For example, the model for specimen RW-A15-P10-S78 was discretized using 12 model elements in the horizontal direction and 18 elements in the vertical direction, resulting in a total of 216 approximately square-shaped model elements. This method was developed based on the analytical observation that using a finer mesh of model elements did not significantly influence the model response predictions, other than the rate of degradation in lateral load after the lateral load capacity is reached. Accurate simulation of the rate of degradation in lateral load, which is associated with not only the lateral instability failures observed during the tests, but also strain localization effects in the analyses, is beyond the scope of the present study. The material constitutive relationships were therefore not regularized based on the model element size to consider strain localization effects on model results, in order to eliminate the uncertainties associated with material stress–strain regularization in interpretation of the results. All wall models were fixed-supported at the bottom wall-pedestal interfaces, via restraining the translational degrees of freedom on the nodes along the bottom interface. Rigid body constraints were defined at the top wall-loading-beam interfaces, for uniform distribution of the constant axial and cyclic lateral loads applied at the top of the wall models. Differently from other specimens, the model of specimen S6 was analyzed under a proportional lateral load pattern consisting of three horizontal forces of increasing magnitude applied at the three representative story levels along the height of the wall, together with a bending moment applied at the top of the wall, replicating the actual loading configuration applied during testing (Vallenas et al. 1979), which is also the reason why the aspect ratio (1.26) and the shear span-to-depth ratio (1.60) of specimen S6 differ significantly.

The constitutive material parameters used in the FSAFE model formulation were calibrated to match the as-tested properties of the materials used in the construction of the specimens (whenever detailed material test results were available), or based on well-established empirical relationships presented in the literature; including empirical equations by Chang and Mander (1994) for generating the ascending region of the compressive stress–stain envelope for unconfined concrete, the confinement model by Mander et al. (1988) for generating the ascending region of the compressive stress–strain envelope for confined concrete, the Saatcioglu and Razvi (1992) model for defining the descending (post-peak) slope of the compressive stress–strain envelopes for both confined and unconfined concrete, as well as the empirical relationships by Belarbi and Hsu (1994) for the tensile strength of concrete (\( f_{t} = 0.31\sqrt {f^{\prime}_{c} } \)), the cracking strain of concrete (\( \varepsilon_{t} \) = 0.00008), and the shape of the smeared stress–strain envelope for concrete in tension (aimed to represent tension stiffening behavior). The yield strength and strain hardening ratio parameters of the Menegotto and Pinto (1973) model were calibrated to represent the stress–strain curve obtained from samples of reinforcing steel bars used in the construction of the wall specimens, with the modifications by Belarbi and Hsu (1994) for consideration of tension stiffening effects. The parameters controlling the cyclic stiffness degradation characteristics of the model were calibrated as R0 = 20, a1 = 18.5, and a2 = 0.15 (Fig. 4b), as proposed originally by Menegotto and Pinto (1973), with the exception of specimen RW2 for which a value of a2 = 0.0015 was used, based on findings of previous analytical studies (e.g., Orakcal and Wallace 2006) conducted on this specimen. Details on calibration of the material model parameters are presented in the paper by Orakcal and Wallace (2006).

3.3 Comparison of model response predictions with test results

3.3.1 Specimen RW-A20-P10-S63

Figure 7 presents comparison of model predictions and test measurements for selected global and local response characteristics of Specimen RW-A20-P10-S63, including the lateral load versus top displacement response of the wall specimen (Fig. 7a), the vertical (longitudinal) strain profiles measured along the base of the wall (Fig. 7b), the lateral load versus flexural and shear deformation components of top displacement (Fig. 7c, d), and the lateral flexural and shear displacement profiles measured along wall height at various drift levels (Fig. 7e, f). The FSAFE model replicates the experimentally-measured lateral load–displacement response of the wall with reasonable accuracy (Fig. 7a). The model prediction matches the measured response in terms of lateral stiffness, lateral load capacity, and drift capacity of the wall, as well as other important cyclic response characteristics including hysteretic stiffness degradation, plastic (residual) displacements upon unloading to zero load, and pinching behavior. As also shown in the embedded plot in Fig. 7a, the model also replicates the significant contribution of shear deformations to top displacements of the wall at all drift levels, capturing the experimentally-observed shear-flexure interaction behavior. The incapability of the model in predicting the abrupt degradation in lateral load during the last loading cycle to 3% drift in the negative loading direction is because lateral instability failures not considered in the present model formulation, including rebar buckling and out-of-plane instability of the compression boundary. During testing, strength loss in the response of this specimen was caused by crushing of concrete at the wall boundary regions, along with crushing along diagonal compression struts near the wall base, leading to out-of-plane buckling of longitudinal boundary reinforcement and lateral instability of the boundary region. Although the model predicts crushing of confined concrete at the wall boundaries and provides a reasonable estimate of the drift capacity of the wall (approximately 2%), it predicts a more gradual degradation in lateral load after 2% drift, and does not capture the sudden and instability-related strength loss observed during the test at negative 3% drift.
Fig. 7

Specimen RW-A20-P10-S63: a lateral load versus top displacement, b vertical normal strain profile at wall base, c lateral load versus top flexural displacement, d lateral load versus top shear displacement, e lateral flexural displacement profile along wall height, f lateral shear displacement profile along wall height

As illustrated in Fig. 7b, the model is accurate in predicting both compressive and tensile strains measured along the base of the wall, as well as the depth of the neutral axis on the wall cross-section, at various drift levels. Unlike in fiber models, plane sections do not remain plane in the model results, which is consistent with the experimentally-measured strain profiles and provides improved predictions of longitudinal strains, particularly the compressive strains in concrete.

As shown in Fig. 7c, d, model predictions for the lateral load versus top flexural and top shear displacement responses of the wall are again, accurate. Shear deformations in the model results are calculated by multiplying the average of the shear strain values obtained for all of the individual model elements that are lined up at the same elevation, with the model element height. These shear deformations are summed up over the model height for obtaining the cumulative shear displacement at a particular elevation of the wall. Flexural displacement predictions of the model are calculated by subtracting shear displacements from total lateral displacement at the same wall elevation. Comparisons shown in Fig. 7c, d show that the model successfully replicates the load versus flexural displacement and the more-pinched load versus shear displacement response characteristics of the wall, as well as the magnitudes of the flexural- and shear-deformation-related components of top displacement.

Comparison of model results and test measurements for the lateral flexural displacement and lateral shear displacement profiles along the height of the wall is presented in Fig. 7e, f. The lateral flexural displacement profile of the wall is well-estimated by the model (Fig. 7e), with nonlinear flexural deformations concentrated along the bottom 610 mm of the wall height, corresponding to a plastic hinge length of half the wall horizontal length (lw/2). The shapes of the measured and predicted shear displacement profiles are also in agreement (Fig. 7f), with decreasing shear distortions developing towards the top of the wall, demonstrating that the model captures coupling between nonlinear shear and flexural deformations, since shear deformations are amplified in regions where nonlinear flexural deformations are concentrated. The model also replicates the distribution of nonlinear flexural and shear deformations along the height of the wall, and therefore captures the so-called “spread of plasticity” along wall height. Discrepancies between test results and model predictions at the drift level of 3.0% can be attributed to degradation in lateral load at 3.0% drift, observed in both experimental and analytical responses.

The experimentally-observed crack pattern on Specimen RW-A20-P10-S63, recorded at a drift level of 2.0%, is compared with the crack directions predicted by the model in Fig. 8a. Cracks in the model results are inherently discontinuous and crack directions in each model element are different, whereas the experimentally-observed cracks propagate in a more continuous manner. However, the model provides reasonable estimations of crack orientations in different regions of the wall, indicating that the cracking criteria and the orthogonal crack assumption in the model formulation are both reasonable. The experimentally-observed cracks are naturally not perpendicular to each other; however, they are plausibly close to being perpendicular. In both test observations and model predictions, the cracks are more horizontal at the wall boundaries, where flexural strains predominate over shear strains, and become more inclined within wall web, where shear strains are more pronounced. For the same specimen, Fig. 8b shows contours of the analytically-predicted principal compressive stresses developing in concrete, at a drift ratio of 2.0%, which approximately corresponds to the experimentally-measured lateral load capacity of the wall. As shown in the figure, the model predicts formation of a diagonal compression strut at the wall base, and concentration of compressive stresses within the wall boundary region. The analytically-predicted diagonal compression strut is consistent with the experimentally-observed behavior of the wall, in which strength degradation was initiated by crushing of concrete at the wall compression boundary, due to stresses transferred along a diagonal compression strut.
Fig. 8

Specimen RW-A20-P10-S63 at 2.0% drift: a Experimental and analytical crack patterns, b analytical stress contours for the principal compressive stresses in concrete

3.3.2 Specimen RW2

Comparisons of model results with test measurements for the global response and selected local response attributes of Specimen RW2, which is the first and the most slender of the five benchmark specimens investigated, is presented in Fig. 9. The model accurately predicts the measured lateral load–displacement response of the wall (Fig. 9a), although it cannot capture the experimentally-observed degradation in lateral load during the second loading cycle to 2.5% drift in the positive loading direction, which occurred due to initiation of buckling of longitudinal reinforcement in the wall boundary region. As shown in Fig. 9b, model prediction of the rotation time-history measured at the first story-level of the wall, due to cumulative flexural deformations developing along the bottom quarter of wall height, is also accurate. The lateral flexural displacement profile of the wall is well-predicted (Fig. 9c), with nonlinear flexural deformations concentrated along the bottom quarter (915 mm) of wall height, where flexural yielding was observed. The shape of the shear displacement profile predicted by the model is also representative of test results (Fig. 9d), although the model slightly underestimates the shear displacements measured along the height of the wall, which is acceptable considering their very small magnitude. It must be mentioned that during testing of this wall, shear deformations were measured along the bottom two quarters of wall height (first two stories) only, and test results shown in Fig. 9d assume zero shear deformations developing along the top two stories of the wall, whereas model results shown in Fig. 9d consider shear deformations developing along entire wall height. In both model results and test measurements, shear deformations are amplified along the first story height of the wall, where nonlinear flexural deformations are also concentrated. The model clearly captures the nonlinear shear-flexure interaction behavior observed experimentally even for this slender wall specimen with flexure-dominated behavior, where the contribution of shear deformations to the top displacement of the wall was measured to be as low as 10%.
Fig. 9

Specimen RW2: a lateral load versus top displacement, b first-story rotation time history, c lateral flexural displacement profile along wall height, d lateral shear displacement profile along wall height

3.3.3 Specimen RW-A15-P10-S78

Global and selected local response comparisons between analytical results and test measurements for Specimen RW-A15-P10-S78 are presented in Fig. 10. The lateral load–displacement response characteristics of the wall, including lateral stiffness, lateral load capacity, drift capacity, and pinching behavior, are again well-predicted by the model (Fig. 10a). Similarly to specimen RW-A20-P10-S63, the model does not capture the abrupt degradation in lateral load during the second loading cycle to 3.0% drift in the positive loading direction, since the model cannot simulate the out-of-plane buckling of wall boundary reinforcement and the lateral instability of wall the boundary region, which was one of the two experimentally-observed failure modes of this wall, the other being sliding shear adjacent to the wall-pedestal interface. However, the wall drift capacity estimation of the model is again reasonable, at approximately 2.0% drift, since the model captures the crushing of confined concrete in the wall boundary regions, which initiated the experimentally-observed strength degradation. As illustrated in Fig. 10b, model predictions for the contribution of shear deformation to the top displacement of the wall are also reasonably accurate at all drift levels, demonstrating the capability of the model in representing the coupled shear-flexural response characteristics of the wall. Figure 10c shows that the measured and predicted wall rotations measured at an elevation of 610 mm from base of the wall, due to flexural deformations accumulating along the wall plastic hinge length (lw/2), are also in agreement. Figure 10d compares model results and test measurements for the wall vertical growth (defined as the upward vertical displacement at the geometric centroid of the wall cross-section) versus the lateral top displacement applied on the wall. Vertical growth of a wall is associated with migration of the neutral axis on the wall cross-section, associated with cracking of concrete and yielding of longitudinal reinforcement. As shown in Fig. 10d, the model accurately predicts the vertical growth of the wall not only at the maximum lateral displacements levels applied on the wall, but also the residual (plastic) vertical growth at zero lateral displacement after unloading from the maximum displacements. Similarly to the previous two wall specimens, the lateral flexural and shear displacement profiles measured along the height of the wall are well-estimated by the model (Fig. 10e, f), with coupled nonlinear flexural and shear deformations concentrated along the bottom plastic hinge region of the wall. Differences between model results and test data at the drift level of 3.0% can be associated with degradation in lateral load at this drift level.
Fig. 10

Specimen RW-A15-P10-S78: a lateral load versus top displacement, b contribution of shear deformation to top displacement, c base rotation versus drift, d vertical growth versus top lateral displacement, e lateral flexural displacement profile along wall height, f lateral shear displacement profile along wall height

3.3.4 Specimen WSH6

Similarly to specimen RW2, specimen WSH6 showed a predominantly-flexural response during testing. The observed failure mode of the specimen was crushing of confined concrete (caused by rupture of confining hoops) within the boundary region of the wall and buckling of the longitudinal boundary reinforcement, leading to a sudden drop of the lateral load at a drift level of 2.0% in the negative loading direction. As shown in Fig. 11a, the model accurately predicts the experimentally-measured lateral load versus top displacement response characteristics of this wall, until failure at negative 2.0% drift. However, as expected, the model does not capture the experimentally-observed degradation in lateral load associated with rebar buckling, and therefore overestimates the drift capacity of the wall. Nevertheless, initiation of concrete crushing is reflected in the analysis results, in which a very gradual degradation in lateral load, starting at drift level of approximately 1.5%, is observed. Model predictions for the ratio of the contribution of shear deformations to the top displacement of the wall to that of flexural deformations, for increasing values of top displacement, are reasonably accurate, as depicted in Fig. 11b. Figure 11c, d compare model predictions of the distribution of vertical (longitudinal) strains along the height of the wall, with measurements obtained using mechanical strain gauges mounted on all four corners of the wall specimen, at various drift levels applied in positive and negative loading directions. In Fig. 11e, the distribution of average curvatures, measured along wall height using LVDT chains attached to the wall boundaries, are compared with model results. Finally, a comparison of model predictions with test measurements for the cumulative contribution of flexural deformations (relative rotations) measured along wall height to the total flexural displacement at the top of the wall is shown in Fig. 11f. Comparisons presented in Fig. 11c–f clearly demonstrate that the model provides accurate estimations of the distribution of flexural deformations (vertical strains, curvatures, rotations) over the entire wall geometry, indicating that model is reliable in simulating the spreading of flexural plasticity in a wall.
Fig. 11

Specimen WSH6: a lateral load versus top displacement, b shear deformation to flexural deformation ratio, c vertical strains at wall boundaries along wall height under positive displacements, d vertical strains at wall boundaries along wall height under negative displacements, e curvature profile along wall height, f cumulative contribution of flexural deformations along wall height to top flexural displacement

3.3.5 Specimen S6

The medium-rise wall specimen S6 exhibited a shear-flexure interaction response during testing, similarly to specimens RW-A20-P10-S63 and RW-A15-P10-S78. Failure of the specimen was due to crushing of concrete and out-of-plane buckling of the wall boundary element, creating sudden strength loss during loading to a drift level of 1.6% in the negative loading direction. As illustrated in Fig. 12a, the model accurately simulates the lateral load versus top displacement response of the wall until failure, but fails to predict the sudden strength loss related to lateral instability of the wall boundary. At the negative 1.6% drift level, mild degradation in lateral load, associated with initiation of crushing of confined concrete within the wall boundary region, is observed in the model response prediction. Model estimations for the contribution of shear deformations to the top lateral displacements of the wall (Fig. 12b), as well as the lateral load versus flexural (Fig. 12c) and shear (Fig. 12d) deformation components of top displacement are accurate, consolidating the reliability of the model in simulating coupled shear and flexural responses of walls.
Fig. 12

Specimen S6: a lateral load versus top displacement, b shear deformation contribution to top displacement, c lateral load versus top flexural displacement, d lateral load versus top shear displacement

3.3.6 Specimen R2

Response predictions of the model for Specimen R2, which is the last of the five benchmark specimens, are not accurate. As shown in Fig. 13a, the experimentally-measured lateral load versus top displacement response of this wall follows a relatively pinched behavior, the lateral stiffness of the wall undergoes significant hysteretic degradation with increasing drift, and the wall experiences abrupt strength degradation during loading to a drift ratio of 3.4%. The model, although provides a reasonable estimate of the lateral load capacity of the wall, predicts a hysteretic response that comprises wide load–displacement loops (similar to the cyclic stress–strain behavior of reinforcing steel), with minimal pinching observed at large drift levels only. As illustrated in Fig. 13b, which compares model predictions with test results for the lateral load applied on the wall versus the shear distortion measured along the bottom 40% of wall height, the model also does not capture the large the shear distortions measured during the last three drift levels applied on the wall specimen. Specimen R2 is a relatively slender wall, having the second-largest shear span-to-depth ratio among all wall specimens investigated. The nominal shear strength of the wall is more than twice its nominal flexural capacity, and the average shear stress demand on the wall at its lateral load capacity is the lowest among all specimens investigated. Also considering the zero axial load on the wall, it is believed that the flexure-dominated response prediction of the model represents the expected (conventional) behavior of the wall, with yielding of the longitudinal bars in tension, followed by their yielding in compression (upon load reversal) prior to closure of cracks, creating the wide load–displacement loops shown in Fig. 13a.
Fig. 13

Specimen R2: a lateral load versus top displacement, b lateral load versus shear distortion along the bottom 1830 mm (40%) of wall height

However, the experimentally-observed behavior and failure mode of this wall was unique. Test observations reported by Oesterle et al. (1976) indicate that due to absence of axial load, flexural cracks on the wall did not fully close upon load reversal, which created significant dowel demands on longitudinal bars and caused sliding along horizontal cracks, amplifying the shear distortion measurements shown in Fig. 13b and creating the pinching effects on the experimental load–displacement response shown in Fig. 13a. The associated distortion of the longitudinal bars may have impaired uniform crack closure and led to out-of-plane instability of the wall boundary, which initiated at drift levels as low as 1.7%, also due to the relatively low out-of-plane flexural stiffness of the boundary (small wall thickness). The complex behavioral mode and failure mechanism of specimen R2, the main source of which is believed to be the distortion of longitudinal bars as they go from tension to compression, and the related non-uniform crack closure, is naturally not represented in the model response prediction. Out-of-plane instability failures in RC walls (first discussed by Paulay and Priestley 1993) has recently gained much interest among researchers, especially with damage observations after recent earthquakes (e.g., Elwood 2013, Maffei et al. 2014), and has been the subject of both experimental (e.g., Rosso et al. 2016; Dashti et al. 2017b, 2018a) and analytical modeling (e.g., Dashti et al. 2018b) studies conducted on walls, as well as experimental studies conducted on isolated wall boundary elements (e.g., Welt et al. 2017, Rosso et al. 2018). Recent research efforts (e.g., Abdullah and Wallace 2019) also focus on revision of wall seismic design provisions to consider reduction of wall drift capacity associated in-part with out-of-plane instability of wall boundary regions. Implementation of a behavioral modeling approach in the formulation of the FSAFE model for simulating the lateral instability of wall boundaries can potentially improve the model response predictions, not only for specimen R2 but also the other wall specimens investigated in this paper. Test observations on specimen R2 also highlight the shortcoming that current seismic performance evaluation methods for RC walls do not yet consider the effect of relevant parameters (such as in-plane slenderness and axial load) on development on wall instability failures, which may result in significant differences between observed and expected failure modes.

4 Summary and conclusions

The objective of this paper was to provide a comprehensive review of the so-called Fixed-Strut-Angle Finite Element (FSAFE) model by Gullu and Orakcal (2019), and to provide detailed information on experimental validation of the model using test results for the five benchmark wall specimens identified in the collaborative research papers by Kolozvari et al. (2018b, 2019), plus one additional specimen, at global and various local response levels. The wall specimens used for validation of the model represented a broad range important response characteristics for rectangular walls designed to yield in flexure, including slenderness along height, slenderness in plan, reinforcement characteristics, axial load levels, relative flexural and shear capacities, average shear stress demands at flexural capacity, as well as experimentally-observed behavioral modes and failure mechanisms. Based on the detailed correlations studies presented in this paper, the following main conclusions on significant features of the FSAFE model are derived:
  • Although the working principles of the FSAFE model are relatively simple, the model was shown to provide accurate predictions of the experimentally-measured lateral load versus displacement response characteristics of all wall specimens except one (specimen R2), including their lateral stiffness, lateral load capacity, hysteretic stiffness degradation, and pinching behavior. The model reasonably captured the drift capacity of three of the wall specimens only (RW-A20-P10-S63, RW-A15-P10-S78, and S6), although with much more gradual lateral load degradation in the analysis results. For all specimens, the model could not simulate the experimentally-observed abrupt degradation in lateral load, since instability failures related to buckling of reinforcement or out-of-plane instability of the wall boundary region are not considered in its formulation, which is also the reason why the model was incapable of predicting the drift capacity of specimens RW2, WSH6, and R2.

  • Prior to their experimentally-observed instability failures, local response characteristics of all wall specimens excluding R2, including the contribution of shear and flexural deformations to lateral displacement, the experimentally-observed coupling between nonlinear flexural and shear deformations, the distribution of shear deformations, flexural deformations, and vertical normal strains along wall height, the vertical growth behavior of the walls, and the nonlinear distribution of vertical normal strains measured along the base of the walls, were all well-estimated by the model. Simulation of nonlinear shear-flexure interaction behavior and capturing of the nonlinear flexural strain gradient along wall length, which creates increased compressive strain demands on concrete and leads to reduction in wall drift capacity, are significant strengths of the FSAFE model over typical fiber-based models used for walls that are expected to yield in flexure.

  • The accuracy of the model in predicting wall drift capacity, as well as the rate of degradation in lateral load after the capacity is reached, is to be improved upon implementation of constitutive models in its formulation for representing buckling of reinforcement (e.g., Dhakal and Maekawa 2002), as well as incorporation of a modeling approach to simulate lateral instability of wall boundary regions. The inadequacy of the present model in capturing the experimentally-observed response of Specimen R2, which experienced initiation of lateral instability in the wall boundaries at early drift levels, due to the zero axial load level and the large in-plane slenderness of the wall, highlights the potential of such an approach towards improving the capabilities of the model.

Overall, with the features and potential improvements identified in this paper, the FSAFE model is presented as a relatively simple yet effective modeling approach for simulating the nonlinear seismic response of RC walls. Ongoing studies focus on improvement of the existing model formulation–using the simplest approach possible–for incorporating lateral instabilities in the model response predictions, and conducting seismic response simulation studies on building systems using the FSAFE model for structural walls, towards improvement of performance-based seismic design and assessment procedures.

Notes

Acknowledgements

The authors would like to thank Prof. John Wallace and Dr. Thien Tran from UCLA, and Dr. Alessandro Dazio from ROSE School, for sharing test data.

References

  1. Abdullah SA, Wallace JW (2019) Drift capacity of RC structural walls with special boundary elements. ACI Struct J 116(1):183–194CrossRefGoogle Scholar
  2. ACI (2014) Building code requirements for reinforced concrete (ACI 318-14)Google Scholar
  3. Belarbi A, Hsu TTC (1994) Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete. ACI Struct J 91:465–474Google Scholar
  4. Chang GA, Mander JB (1994) Seismic energy based fatigue damage analysis of bridge columns : part 1—evaluation of seismic capacity. NCEER Technical Report No. NCEER-94-0006. Buffalo, USAGoogle Scholar
  5. Clarke MJ, Hancock GJ (1990) A study of incremental-iterative strategies for non-linear analyses. Int J Numer Meth Eng 29:1365–1391CrossRefGoogle Scholar
  6. Dashti F, Dhakal RP, Pampanin S (2017a) Numerical modeling of rectangular reinforced concrete structural walls. J Struct Eng 143:04017031CrossRefGoogle Scholar
  7. Dashti F, Dhakal RP, Pampanin S (2017b) Tests on slender ductile structural walls designed according to New Zealand Standard. Bull NZ Soc Earthq Eng 50:504–516Google Scholar
  8. Dashti F, Dhakal RP, Pampanin S (2018a) Evolution of out-of-plane deformation and subsequent instability in rectangular RC walls under in-plane cyclic loading: experimental observation. Earthq Eng Struct Dynam 47:2944–2964CrossRefGoogle Scholar
  9. Dashti F, Dhakal RP, Pampanin S (2018b) Validation of a numerical model for prediction of out-of-plane instability in ductile structural walls under concentric in-plane cyclic loading. J Struct Eng 144:04018039CrossRefGoogle Scholar
  10. Dazio A, Beyer K, Bachmann H (2009) Quasi-static cyclic tests and plastic hinge analysis of RC structural walls. Eng Struct 31:1556–1571CrossRefGoogle Scholar
  11. Dhakal RP, Maekawa K (2002) Modeling for postyield buckling of reinforcement. J Struct Eng 128:1139–1147CrossRefGoogle Scholar
  12. Elwood KJ (2013) Performance of concrete buildings in the 22 February 2011 Christchurch earthquake and implications for Canadian codes. Can J Civ Eng 40:759–776CrossRefGoogle Scholar
  13. Filippou FC, Bertero VV, Popov EP (1983) Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Report No. UCB/EERC-83/19. Earthquake Engineering Research Center, University of California, Berkeley, CaliforniaGoogle Scholar
  14. Fischinger M, Vidic T, Selih J, Fajfar P, Zhang HY, Damjanic FB (1990) Validation of a macroscopic model for cyclic response prediction of RC walls. Computer aided analysis and design of concrete structures, 2nd edn. Pineridge Preess, Swansea, pp 1131–1142Google Scholar
  15. Gérin M, Adebar P (2009) Simple rational model for reinforced concrete subjected to seismic shear. J Struct Eng 135:753–761CrossRefGoogle Scholar
  16. Gullu MF, Orakcal K (2019) Nonlinear finite element modeling of reinforced concrete walls with varying aspect ratios. J Earthq Eng.  https://doi.org/10.1080/13632469.2019.1614498 CrossRefGoogle Scholar
  17. He XG, Kwan AKH (2001) Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Comput Struct 79:595–604CrossRefGoogle Scholar
  18. Kolozvari K, Orakcal K, Wallace JW (2018a) New opensees models for simulating nonlinear flexural and coupled shear-flexural behavior of RC walls and columns. Comput Struct 196:246–262CrossRefGoogle Scholar
  19. Kolozvari K, Arteta C, Fischinger M, Gavridou S, Hube M, Isakovic T, Lowes L, Orakcal K, Vasquez J, Wallace JW (2018b) Comparative study on state-of-the-art macroscopic models for planar reinforced concrete walls. ACI Struct J 115(6):1637–1657CrossRefGoogle Scholar
  20. Kolozvari K, Biscombe L, Dashti F, Dhakal RP, Gogus A, Gullu MF, Henry R, Massone L, Orakcal K, Rojas F, Shegay A, Wallace J (2019) State-of-the art in nonlinear finite element modeling of isolated planar reinforced concrete walls. Eng Struct (in press)Google Scholar
  21. Lu Y, Henry RS (2017) Numerical modelling of reinforced concrete walls with minimum vertical reinforcement. Eng Struct 143:330–345CrossRefGoogle Scholar
  22. Luu CH, Mo YL, Hsu TTC (2017) Development of CSMM-based shell element for reinforced concrete structures. Eng Struct 132:778–790CrossRefGoogle Scholar
  23. Maffei J, Bonelli P, Kelly D, Lehman DE, Lowes L, Moehle J, Telleen K, Wallace J, Willford M (2014) Recommendations for seismic design of reinforced concrete wall buildings based on studies of the 2010 Maule, Chile earthquake. NIST GCR 14-917-25. National Institute of Standards and Technology, CaliforniaGoogle Scholar
  24. Mander JB, Priestley MJN, Park R (1988) Theoretical stress-strain model for confined concrete. J Struct Eng 114:1804–1826CrossRefGoogle Scholar
  25. Mansour M, Hsu TTC (2005) Behavior of reinforced concrete elements under cyclic shear. II: theoretical model. J Struct Eng 131:54–65CrossRefGoogle Scholar
  26. Mansour M, Hsu TTC, Lee JY (2002) Pinching effect in hysteretic loops of R/C shear elements. ACI Spec Publ 205:293–322Google Scholar
  27. Massone LM, Wallace JW (2004) Load-deformation responses of slender reinforced concrete walls. ACI Struct J 101:103–113Google Scholar
  28. MatLab M (2012) The language of technical computing. The MathWorks, Inc. http://www.mathworks.com
  29. McKenna F, Fenves G, Scott M, Jeremic B (2000) Open system for earthquake engineering simulation (OpenSees)Google Scholar
  30. Menegotto M, Pinto E (1973) Method of analysis for cyclically loaded reinforced concrete plane frames including changes in geometry and nonelastic behavior of elements under combined normal force and bending. In: Proceedings of IABSE symposium on resistance and ultimate deformability of structures acted on by well-defined repeated loads. Lisbon, PortugalGoogle Scholar
  31. Mo YL, Zhong J, Hsu TTC (2008) Seismic simulation of RC wall-type structures. Eng Struct 30:3167–3175CrossRefGoogle Scholar
  32. Oesterle RG, Fiorato AE, Johal LS, Carpenter JE, Russel HG, Corley WG (1976) Earthquake resistant structural walls—tests of isolated walls. Report to National Science Foundation. Grant No. GI-43880. Research and Development Construction Technology Laboratories Portland Cement Association. Skokie, IllinoisGoogle Scholar
  33. Ohmori N, Takahashi T, Inoue H, Kurihara K, Watanabe S (1989) Experimental studies on nonlinear behaviors of reinforced concrete panels subjected to cyclic in-plane shear. Proc AIJ 403:105–118Google Scholar
  34. Orakcal K, Wallace JW (2006) Flexural modeling of reinforced concrete walls-experimental verification. ACI Struct J 103:196–206Google Scholar
  35. Orakcal K, Wallace JW, Conte JP (2004) Flexural modeling of reinforced concrete walls-model attributes. ACI Struct J 101:688–698Google Scholar
  36. Orakcal K, Ulugtekin D, Massone LM (2012) Constitutive modeling of reinforced concrete panel behavior under cyclic loading. In: Proceedings of 15th World Conference on Earthquake Engineering, Lisbon, PortugalGoogle Scholar
  37. Palermo D, Vecchio FJ (2003) Compression field modeling of reinforced concrete subjected to reversed loading: formulation. Struct J 100:616–625Google Scholar
  38. Palermo D, Vecchio FJ (2007) Simulation of cyclically loaded concrete structures based on the finite-element method. J Struct Eng 133:728–738CrossRefGoogle Scholar
  39. Pang XB, Hsu TTC (1995) Behavior of reinforced concrete membrane elements in shear. ACI Struct J 92:665–679Google Scholar
  40. Paulay T, Priestley MJN (1993) Stability of ductile structural walls. ACI Struct J 90:385–392Google Scholar
  41. Perform 3D (2005) 3D performance-based design software. Computers and Structures IncGoogle Scholar
  42. Rojas F, Anderson JC, Massone LM (2016) A nonlinear quadrilateral layered membrane element with drilling degrees of freedom for the modeling of reinforced concrete walls. Eng Struct 124:521–538CrossRefGoogle Scholar
  43. Rosso A, Almeida JP, Beyer K (2016) Stability of thin reinforced walls under cyclic loads: state-of-the-art and new experimental findings. Bull Earthq Eng 14:455–484CrossRefGoogle Scholar
  44. Rosso A, Jiménez-Roa LA, Almeida JP, Zuniga APG, Blandon CA, Bonett RL, Beyer K (2018) Cyclic tensile-compressive tests on thin concrete boundary elements with a single layer of reinfocement prone to out-of-plane instability. Bull Earthq Eng 16:859–887CrossRefGoogle Scholar
  45. Saatcioglu M, Razvi SR (1992) Strength and ductility of confined concrete. J Struct Eng 118:1590–1607CrossRefGoogle Scholar
  46. Stevens NJ, Uzumeri SM, Collins MP (1991) Reinforced concrete subjected to reversed cyclic shear–experiments and constitutive model. Struct J 88:135–146Google Scholar
  47. Tassios TP, Vintzēleou EN (1987) Concrete-to-concrete friction. J Struct Eng 113:832–849CrossRefGoogle Scholar
  48. Taucer F, Spacone E, Filippou FC (1991) A fiber beam-column element for seismic response analysis of reinforced concrete structures, vol 91. Earthquake Engineering Research Center, College of Engineering, University of California Berkeley, CaliforniaGoogle Scholar
  49. Thomsen JH, Wallace JW (1995) Displacement—based design of reinforced concrete structural walls: an experimental investigation of walls with rectangular and T-shaped cross-sections. Report No. CU/CEE-95/06. Department of Civil Engineering, Clarkson University, Potsdam, New YorkGoogle Scholar
  50. Tran TA, Wallace JW (2015) Cyclic testing of moderate-aspect-ratio reinforced concrete structural walls. ACI Struct J 112:653–666CrossRefGoogle Scholar
  51. Vallenas JM, Bertero VV, Popov EP (1979) Hysteretic behavior of reinforced concrete structural walls. Report to National Science Foundation. Report No. UCB/EERC-79/20. College of Engineering, University of California, Berkeley, CaliforniaGoogle Scholar
  52. Vásquez JA, de la Llera JC, Hube MA (2016) A regularized fiber element model for reinforced concrete shear walls. Earthq Eng Struct Dyn 45:2063–2083CrossRefGoogle Scholar
  53. Vassilopoulou I, Tassios TP (2003) Shear transfer capacity along a RC crack, under cyclic sliding. In: Proceedings of fib symposium concrete structures in seismic regions (electronic source), Athens, GreeceGoogle Scholar
  54. Vecchio FJ (1989) Nonlinear finite element analysis of reinforced concrete membranes. ACI Struct J 86:26–35Google Scholar
  55. Vecchio FJ (1999) Towards cyclic load modeling of reinforced concrete. ACI Struct J 96:193–202Google Scholar
  56. Vecchio FJ, Collins MP (1986) The modified compression field theory for reinforced concrete elements subjected to shear. ACI Struct J 83:219–231Google Scholar
  57. Vecchio FJ, Collins MP (1993) Compression response of cracked reinforced concrete. J Struct Eng 119:3590–3610CrossRefGoogle Scholar
  58. Vintzēleou EN, Tassios TP (1986) Mathematical models for dowel action under monotonic and cyclic conditions. Mag Concr Res 38:13–22CrossRefGoogle Scholar
  59. Vulcano A, Bertero VV, Colotti V (1988) Analytical modeling of R/C structural walls. In: Proceedings of 9th World Conference on Earthquake Engineering 6:41–46Google Scholar
  60. Welt TS, Massone LM, LaFave JM, Lehman DE, McCabe SL, Polanco P (2017) Confinement behavior of rectangular reinforced concrete prisms simulating wall boundary elements. J Struct Eng 143:401620CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Bogazici UniversityIstanbulTurkey
  2. 2.California State University FullertonFullertonUSA

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