Bulletin of Earthquake Engineering

, Volume 17, Issue 12, pp 6391–6417 | Cite as

Shear–flexure-interaction models for planar and flanged reinforced concrete walls

  • Kristijan KolozvariEmail author
  • Kamiar Kalbasi
  • Kutay Orakcal
  • Leonardo M. Massone
  • John Wallace
S.I.: Nonlinear Modelling of Reinforced Concrete Structural Walls


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 two-dimensional two-node formulation of the Shear–Flexure-Interaction Multiple-Vertical-Line-Element-Model (SFI-MVLEM), (2) extension of the baseline SFI-MVLEM for modeling of squat walls (SFI-MVLEM-SQ), and (3) extension of the baseline SFI-MVLEM to a three-dimensional four-node element for simulation of non-planar RC walls under multi-directional loading (SFI-MVLEM-3D). 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 moderately-slender and one medium-rise planar RC walls tested under in-plane loading used for validation of the baseline SFI-MVLEM, (2) three squat planar RC wall specimens tested under in-plane loading used for validation of the SFI-MVLEM-SQ, and (3) one T-shaped and one U-shaped RC wall specimen tested under unidirectional and multi-directional loading, respectively, used for validation of SFI-MVLEM-3D. The comprehensive comparisons between analytically-obtained and experimentally-measured 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.


Finite element modeling Nonlinear analysis Performance-based design Nonplanar walls Multidirectional loading 

1 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 service-level earthquakes, and to provide sufficient nonlinear deformation capacity (ductility) during severe earthquakes. Application of the Performance-Based 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 moderate-to-strong 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 (finite-element) 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–Flexure-Interaction Multiple-Vertical-Line-Element-Model (SFI-MVLEM; 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 fiber-based models with uncoupled shear and flexural behavior (e.g., Taucer et al. 1991; Orakcal and Wallace 2006; Perform 3D 2005), the so-called uncoupled models, are generally accurate in predicting global and local responses of slender (flexure-controlled) 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 moderately-slender walls (Kolozvari 2013), and underestimate shear deformations (Massone et al. 2009) in medium-rise 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 multi-directional 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 U-shaped wall tested under multi-directional 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 (shear-controlled) 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, cross-section) and loading conditions (e.g., unidirectional vs. multi-directional), 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 widely-used uncoupled models, which is that they fail to capture the experimentally-observed 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 SFI-MVLEM developed by Kolozvari et al. (2015a, 2018c) can successfully predict the overall load-deformation 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 SFI-MVLEM 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 two-dimensional two-node beam-column element formulation and cannot be applied to non-planar walls subjected to multi-directional loading. These model shortcomings are addressed here by extending the applicability of the original SFI-MVLEM formulation (Kolozvari et al. 2015a) to squat walls and to walls with non-planar cross-sections subjected to multi-directional loading.

2 Objectives and scope

Given the aforementioned shortcomings of current macroscopic modeling approaches for RC walls, particular goals of the research results presented are to:
  1. 1.

    Provide additional information on validation of the baseline SFI-MVLEM 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. 2.
    Overcome limitations of the baseline SFI-MVLEM by:
    1. a.

      Extending the applicability of the model to squat walls, by proposing the so-called SFI-MVLEM-SQ 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.

    2. b.

      Extending the existing baseline two-node, two-dimensional model formulation to a four-node three-dimensional model formulation, referred to as the SFI-MVLEM-3D, and validating the three-dimensional model against experimental results obtained from tests on non-planar RC wall specimens subjected to uni-directional and multidirectional loading.

  3. 3.

    Evaluate capabilities of the three model formulations presented and to identify directions for future research.


In this paper, formulations of the three models (SFI-MVLEM, SFI-MVLEM-SQ, and SFI-MVLEM-3D) 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 quasi-static 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 two-dimensional two-node SFI-MVLEM is described first, followed by its extension for simulation of the response of squat walls (SFI-MVLEM-SQ), and its extension to a three-dimensional four-node model (SFI-MVLEM-3D) formulation.

3.1 Baseline two-dimensional two-node SFI-MVLEM formulation

The Shear–Flexure-Interaction Multiple-Vertical-Line-Element-Model (SFI-MVLEM, Kolozvari et al. 2015a, c) captures interaction between axial/flexural and shear responses in RC structural walls under cyclic loading. The SFI-MVLEM element incorporates a two-dimensional constitutive RC panel behavior described by the Fixed-Strut-Angle-Model (FSAM; Orakcal et al. 2018) implemented at each macro-fiber of the Multiple-Vertical-Line-Element-Model (MVLEM; Orakcal et al. 2004) as illustrated in Fig. 1. The FSAM represents a two-dimensional (plane-stress) constitutive relationship that relates the strain field imposed on a RC panel (εx, εy, and γxy) to the resultant of the stresses developing in concrete and reinforcing steel, converted into smeared stresses in concrete (σx, σy, and τxy). Therefore, coupling between axial and shear behavior is captured at each RC panel (macro-fiber), which further allows capturing the interaction between flexural and shear responses at the SFI-MVLEM element level.
Fig. 1

2-node 2D baseline formulation of the SFI-MVLEM

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 εy on each RC panel (macro-fiber) 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 γxy on each RC panel, assuming uniform shear strain distribution across the wall cross-section. 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 SFI-MVLEM, as opposed to other commonly-used wall models with uncoupled axial/flexural and shear behavior, such as the fiber-based beam-column 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 (εx) of each RC panel element are obtained assuming that the sum of horizontal stresses associated with steel and concrete (i.e., resultant stress σx) is equal to zero. This assumption is incorporated in the SFI-MVLEM by introducing additional DOFs defined in the horizontal direction for each macro-fiber of the model element (δ1,…,δm, 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 SFI-MVLEM element is increased to 6 + m (where m is the number of macro-fibers in the model element). Previous studies (Massone et al. 2006) have shown that the assumption of zero resultant horizontal stress implemented in the SFI-MVLEM formulation is reasonable for cantilever walls with aspect ratios greater than 1.0 (slender and moderate-aspect-ratio 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 SFI-MVLEM for application to squat walls

A modified formulation of the baseline SFI-MVLEM is proposed in this study, called the SFI-MVLEM-SQ, in order to extend its applicability to walls with aspect ratios equal to or smaller than 1.0 (i.e., squat walls). In the SFI-MVLEM-SQ, the assumption of zero horizontal resultant stress at each RC panel (σx = 0), used in the baseline SFI-MVLEM to calculate panel axial strains in horizontal direction (εx), is replaced with the assumption that εx 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.

The maximum value of horizontal axial strains in the wall (εx,max) is first obtained based on the lateral drift at the top of the wall (Δ), which is calculated as wall top displacement (δ) divided by the height of the wall (hw), the horizontal reinforcement ratio (ρh), the wall length (lw), and the normalized wall axial stress \( \left( {\frac{N}{{f_{c} A_{g} }}} \right) \), according to following expressions:
  • For cantilever walls (single-curvature)

$$ \varepsilon_{x,max} = 0.0055\left( {100\rho_{h} + 0.25} \right)^{ - 0.44} \cdot \left( {100\Delta } \right)^{1.4} $$
  • For zero-rotation conditions at the bottom and the top of the wall (double-curvature)

$$ \varepsilon_{x,max} = 0.0033\left( {100\rho_{h} + 0.25} \right)^{ - 0.53} \cdot \left( {\frac{{h_{w} }}{{l_{w} }} + 0.5} \right)^{0.47} \cdot \left( {\frac{100N}{{f_{c} A_{g} }} + 5} \right)^{0.25} \cdot \left( {100\Delta } \right)^{1.4} $$
Subsequently, the horizontal strains are distributed over the wall height based on εx,max and the vertical location of the model element mid-height (y) measured from the wall base, using the following expressions:
  • For cantilever walls (single-curvature)

$$ \frac{{\varepsilon_{x,i} \left( y \right)}}{{\varepsilon_{x,max} }} = \left\{ {\begin{array}{*{20}l} {sin^{0.75} \left( {\frac{y}{{0.76h_{w} }}\pi } \right) } \hfill & {0 \le y \le 0.38h_{w} } \hfill \\ {sin^{0.75} \left( {\frac{{\left( {y + 0.24h_{w} } \right)}}{{1.24h_{w} }}\pi } \right)} \hfill & {0.38h_{w} < y \le h_{w} } \hfill \\ \end{array} } \right. $$
  • For zero-rotation conditions at the bottom and the top of the wall (double-curvature)

$$ \frac{{\varepsilon_{x,i} \left( y \right)}}{{\varepsilon_{x,max} }} = sin^{0.75} \left( {\frac{y}{{h_{w} }}\pi } \right) $$
This procedure is illustrated in Fig. 2a. Finally, assuming that calculated horizontal axial strain for each model element (εx,i) is constant over the length of the wall, the same value of εx,i is assigned to all RC macro-fibers within one element, as shown in Fig. 2b.
Fig. 2

Formulation of SFI-MVLEM-SQ: a wall model, b model element, c strain interpolation history

The expressions proposed by Massone (2010) presented above are derived based on envelopes of wall hysteretic responses and do not consider reversed-cyclic wall behavior. Therefore, simple loading and unloading cyclic rules are proposed and implemented in the formulation of SFI-MVLEM-SQ, for describing the hysteretic evolution of εx,max. These hysteretic rules are derived based on experimentally-measured axial horizontal strains in shear-controlled wall specimens tested under reversed-cyclic 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 SFI-MVLEM for εx 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 SFI-MVLEM to three-dimensional four-node model

Another modification to the original formulation of the baseline SFI-MVLEM involves its extension to a three-dimensional four-node element with a total of twenty-four DOFs (Fig. 3a), named as the SFI-MVLEM-3D, which enables simulating the response of non-planar RC walls subjected to multidirectional loading. The proposed extension of the original model formulation includes: (1) extrapolation of the six in-plane element DOFs defined at the two element centerline nodes (Fig. 1) to twelve in-plane element DOFs defined at four element nodes located at each corner of the element (Fig. 3b), and (2) implementation of the element out-of-plane stiffness. Figure 3 illustrates the two layers (and corresponding DOFs) of the SFI-MVLEM-3D model including the SFI-MVLEM layer used to represent the in-plane behavior of the model element (Fig. 3b), and a linear-elastic plate element layer used for simulating the out-of-plane response (Fig. 3c). The in-plane and the out-of-plane element behaviors are uncoupled in the current model formulation.
Fig. 3

SFI-MVLEM-3D formulation: a model element, b in-plane SFI-MVLEM behavior, c linear elastic plate behavior out-of-plane

3.3.1 In-plane behavior

The in-plane DOFs defined at the four corner nodes of the SFI-MVLEM-3D element (Fig. 3b) include two translational (horizontal and vertical) and one rotational DOF at each element node, resulting in total of twelve in-plane element DOFs. These twelve in-plane DOFs at the four element nodes (δj, Fig. 4) are obtained from the six in-plane DOFs defined at the two centerline nodes of the original SFI-MVLEM model (Δj, Fig. 4), via a relatively simple geometric transformation based on kinematic principles by assuming rigid beams located at the top and the bottom of each model element.
Fig. 4

Conversion of DOFs from a 2-node to a 4-node element

In the model formulation, the conversion from the six Δi to the twelve δj DOFs takes the following form:
$$ \delta = \left[ C \right] \times \Delta $$
where matrix [C] is the geometric transformation matrix defined as:
$$ \left[ C \right]_{12 \times 6} = \left[ {\begin{array}{*{20}l} {\left[ {c_{1} } \right]_{3 \times 3} } \hfill & {\left[ 0 \right]_{3 \times 3} } \hfill \\ {\left[ {c_{2} } \right]_{3 \times 3} } \hfill & {\left[ 0 \right]_{3 \times 3} } \hfill \\ {\left[ 0 \right]_{3 \times 3} } \hfill & {\left[ {c_{3} } \right]_{3 \times 3} } \hfill \\ {\left[ 0 \right]_{3 \times 3} } \hfill & {\left[ {c_{4} } \right]_{3 \times 3} } \hfill \\ \end{array} } \right]_{12 \times 6} $$
The submatrices [c1], [c2], [c3], and [c4] in Eq. 6 correspond to each of the four nodes of the model element and represent the rigid offset matrix expressed as:
$$ \left[ {c_{i} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & x \\ 0 & 0 & 1 \\ \end{array} } \right] $$
where x is the location of each of the four nodes of the SFI-MVLEM-3D according to the coordinate system shown in Fig. 4.
The in-plane stiffness matrix of the SFI-MVLEM-3D (KSFI-MVLEM-3D) is obtained using the following procedure. First, the element stiffness matrix corresponding to the original two-node formulation of the SFI-MVLEM with six in-plane DOFs (KSFI-MVLEM) is transformed into the element stiffness matrix corresponding to the four-node model element with twelve in-plane DOFs using the geometric transformation matrix [C] as:
$$ \left[ {\widehat{K}_{{{\text{SFI}} - {\text{MVLEM}} - 3{\text{D}}}} } \right]_{12 \times 12} = \left[ C \right]^{ - 1T}_{12 \times 6} \times \left[ {K_{{{\text{SFI}} - {\text{MVLEM}}}} } \right]_{6 \times 6} \times \left[ C \right]^{ - 1}_{6 \times 12} $$
where the inverse of a non-square matrix [C] is calculated using the Moore–Penrose pseudoinverse method (Penrose 1955) as:
$$ \left[ C \right]^{ - 1}_{6 \times 12} = (\left[ C \right]^{T}_{6 \times 12} \times \left[ C \right]_{12 \times 6} )^{ - 1}_{6 \times 6} \times \left[ C \right]^{T}_{6 \times 12} $$
However, although the size of the element in-plane stiffness matrix increases from 6 × 6 to 12 × 12, the rank of the matrix does not change, which can potentially lead to singularity issues. Therefore, the rank of the obtained stiffness matrix \( \widehat{K}_{{{\text{SFI}} - {\text{MVLEM}} - 3{\text{D}}}} \) is increased, by assigning the imaginary beams (IB) to the SFI-MVLEM-3D element between each pair of top and bottom nodes (Fig. 4), allowing the final stiffness matrix of the SFI-MVLEM-3D element to be calculated as:
$$ \left[ {K_{SFI - MVLEM - 3D} } \right]_{12 \times 12} = \left[ {\widehat{K}_{SFI - MVLEM - 3D} } \right]_{12 \times 12} + \left[ {K_{IB} } \right]_{12 \times 12} $$
where \( \left[ {K_{IB} } \right] \) is expressed as:
$$ \left[ {K_{IB} } \right]_{12 \times 12} = \left[ {\begin{array}{*{20}c} {\left[ {K_{Beam} } \right]_{6 \times 6} } & {\left[ 0 \right]_{6 \times 6} } \\ {\left[ 0 \right]_{6 \times 6} } & {\left[ {K_{Beam} } \right]_{6 \times 6} } \\ \end{array} } \right]_{12 \times 12} $$

Matrix \( [K_{\text{Beam}} ] \) refers to the well-known stiffness matrix of a two-dimensional 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 in-plane stiffness of the wall element.

Using a similar procedure as for the stiffness matrix, the in-plane resisting force vector of the SFI-MVLEM-3D (RSFI-MVLEM-3D) is obtained using the in-plane force vector of the original SFI-MVLEM (RSFI-MVLEM) and the geometric transformation matrix [C] described previously as:
$$ \left\{ {\widehat{R}_{{{\text{SFI}} - {\text{MVLEM}} - 3{\text{D}}}} } \right\}_{12 \times 1} = \left[ C \right]_{12 \times 6}^{T} \times \left\{ {R_{SFI - MVLEM} } \right\}_{6 \times 1} $$
In this case, the forces in the imaginary beams added in the SFI-MVLEM-3D formulation are subtracted from the resisting force vector as:
$$ \left\{ {R_{{{\text{SFI}} - {\text{MVLEM}} - 3{\text{D}}}} } \right\}_{12 \times 1} = \left\{ {\widehat{R}_{{{\text{SFI}} - {\text{MVLEM}} - 3{\text{D}}}} } \right\}_{12 \times 1} - \left\{ {R_{IB} } \right\}_{12 \times 1} $$
where \( \left[ {R_{IB} } \right] \) is the resisting force vector of a two-dimensional elastic beam (e.g., Kassimali 2010).

3.3.2 Out-of-plane behavior

The out-of-plane behavior of the SFI-MVLEM-3D is simulated using the elastic four-node Kirchhoff plate element formulation (Fig. 3c; e.g., Durán and Ghioldi 1990; Brank et al. 2015). This well-known finite element formulation is implemented in the SFI-MVLEM-3D with the following assumptions: (1) wall thickness is uniform and small compared to the remaining wall dimensions, (2) out-of-plane deflections of the wall are small compared to its in-plane 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 mid-surface develops, and (5) shear strain energy is ignored. The stiffness and force terms corresponding to the out-of-plane DOFs of the SFI-MVLEM-3D element (δ3, δ4, δ5, δ9, δ10, δ11, δ15, δ16, δ17, δ21, δ22 and δ23 in Fig. 3c) are assembled together with the in-plane element DOFs (δ1, δ2, δ6, δ7, δ8, δ12, δ13, δ14, δ18, δ19, δ20 and δ24 on Fig. 3b) to form the full (24 × 24) stiffness matrix and force vector representing three-dimensional behavior. However, as mentioned earlier, the wall out-of-plane behavior is of the wall model is uncoupled from its in-plane behavior in the present model formulation.

4 Description of test specimens and testing procedures

The three versions of the SFI-MVLEM model presented are validated using experimental results obtained for nine RC wall specimens that were selected to span a range of salient wall response parameters and characteristics, such as wall aspect ratio (slender, medium-rise, squat), axial load (between 0 and 10% of wall axial load capacity), wall cross-section (planar, non-planar), average shear stress at nominal flexural capacity, and loading regime (in-plane only, multidirectional). All specimens were tested under constant axial load and reversed-cyclic lateral loads applied at the top of the walls. Each of the three model formulations described (SFI-MVLEM, SFI-MVLEM-SQ, and SFI-MVLEM-3D) is validated using experimental data obtained for specific test specimens that are within the range of applicability of each particular model. Therefore, the baseline formulation of the SFI-MVLEM is used to simulate the behavior of the four medium-rise or relatively slender planar RC walls (aspect ratios ranging from 1.5 to 3.0) subjected to in-plane cyclic lateral loading (RW2, Thomsen and Wallace 1995; R2, Oesterle et al. 1976; WSH6, Dazio et al. 2009; RW-A15-P10-S78, Tran and Wallace 2015). The SFI-MVLEM-SQ is validated against test results obtained for three squat walls (aspect ratios equal to 0.33, 0.50, and 1.0), also tested under in-plane cyclic lateral loading (T5-S1, T2-S3, and T4-S1; Terzioglu et al. 2018). The SFI-MVLEM-3D is verified against experimental data for two flanged (T-shaped, U-shaped) slender walls subjected to both uni- and multi-directional lateral loading (TW2, Thomsen and Wallace 1995; TUB, Beyer et al. 2008). Main characteristics of the wall specimens are summarized in Table 1.
Table 1

Main characteristics of selected RC wall specimens used for validation of the models


Specimen ID


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 modea



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In-plane (web)














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aCB concrete crushing/rebar buckling, LI lateral instability, DC diagonal compression

bTop row: web or loading direction parallal to the web; Bottom row: flange or loading direction parallel to the flange

f′c concrete compressive strength, fy,BE yield strength of boundary reinforcement, ρv,BE reinforcing ratio of longitudinal boundary element reinforcement, ρh,web reinforcing ratio of horizontal web reinforcement, ρv,web reinforcing ratio of vertical web reinforcement, Ag gross cross-sectional area, Vn nominal shear capacity, Vmax maximum shear force measured, Mn nominal moment capacity, Mmax maximum moment measured

5 Model generation and calibration

5.1 Geometric discretization

The analytical models created for all considered wall specimens were discretized along the wall length using a number of RC panel elements along wall length (m) such that at least two outer panel elements (macro fibers) represented the confined wall boundaries and the remaining inner elements represented the wall web. Model discretization of the wall specimens along wall height was performed using several model elements (n) considering the locations of the instrumentation used to measure deformations on the wall specimens (e.g., LVDTs), in order to allow consistent deformation comparisons between model and experimental results, as well as the height over which nonlinear deformations are anticipated in the wall specimens (i.e., plastic hinge length). To illustrate a representative geometric discretization scheme employed in this study, Fig. 5 presents element sizes and macro-fibers used in modeling of the planar wall specimen RW2 (modeled with baseline SFI-MVLEM) and U-shaped wall specimen TUB (modeled with SFI-MVLEM-3D).
Fig. 5

Geometric discretization and load application for: a planar wall specimen RW2 (Thomsen and Wallace 1995), and b U-shaped wall specimen TUB (Beyer et al. 2008)

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 experimentally-measured properties of the reinforcing bars used in the tests. The tensile yield strength and strain-hardening 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 R0 = 20, a1 = 18.5, and a2 = 0.15, as proposed originally by Menegotto and Pinto (1973), with the exception of specimens RW2 and TW2 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 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 (Ec) 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 post-peak 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 (εt), as suggested by Belarbi and Hsu (1994). The post-peak slope of the linear tension envelope of Concrete02 was adopted as 0.05Ec, 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 plane-stress behavior of RC panel elements, incorporates two models to describe the transfer of shear stresses along concrete cracks: (1) friction-based shear aggregate interlock model, and (2) linear-elastic 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 linear-elastic relationship with a stiffness of α·Es, where Es 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 medium-rise walls using baseline SFI-MVLEM and SFI-MVLEM-3D formulations, while η = 0.35 and α = 0.001 were used for simulation of squat walls using SFI-MVLEM-SQ.

5.3 Nonlinear analysis solution strategy

The Newton-Rapson 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. Displacement-controlled analysis was used for all specimens except for specimen TUB, which was analyzed using force-controlled analysis due to complex loading protocol and specimen geometry as explained in Sect. 6.3.

6 Comparisons of analytical and experimental results

Comparison of experimentally-measured and analytically-predicted responses for the three SFI-MVLEM 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 SFI-MVLEM

The baseline formulation of the SFI-MVLEM is validated against test results for the four planar RC wall specimens selected, including: (1) RW2 (Thomsen and Wallace 1995), (2) WSH6 (Dazio et al. 2009), (3) RW-A15-P10-S78 (Tran and Wallace 2015), and (4) R2 (Oesterle et al. 1976); see Table 1 for specimen properties. Comparisons between the experimentally-measured and analytically-predicted lateral load versus top displacement responses for the four specimens are presented in Fig. 6. It can be observed from the figure that the analytical model captures well the lateral load capacity and stiffness of all specimens, for most of the applied drift levels. Analytically-predicted lateral load capacities are within the ± 5% range of the experimentally-measured capacities for all specimens considered, where lateral loads are overestimated only for specimen RW-A15-P10-S78 at drift levels lower than 0.5%. In addition, cyclic characteristics of the response, including hysteretic stiffness degradation, plastic (residual) displacements, and pinching behavior are well-represented in the model results for all wall specimens except for specimen R2, where the pinching effect is considerably underestimated compared to test results, which is associated with the zero axial load applied on the model. Significant strength degradation was observed during testing of these specimens due to compression-controlled crushing/buckling failure (RW2, WSH6), crushing/buckling failure leading to lateral instability (R2), and diagonal compression failure accompanied by sliding shear (RW-A15-P10-S78). Detailed investigation of the material behavior within the plastic hinge region in the analytical models suggests that the model captures the initiation of strength degradation associated with crushing of concrete within the boundary elements; however, the abrupt strength loss observed during the tests is not represented in the analysis results because of the inability of the model to simulate failure mechanisms associated with buckling and/or fracture of reinforcing bars or sliding shear failure near the base of the walls. Therefore, for the loading cycles applied on the specimens at their ultimate lateral drift levels, the analytical model tends to generally overestimate their lateral load/drift capacity.
Fig. 6

Load-deformation responses for specimens: a RW2, b WSH6, c RW-A15-P10-S78, d R2

To evaluate model accuracy in predicting wall behavior in the region where nonlinear deformations are expected, the force versus shear distortion and the moment versus rotation responses over the bottom 1830 mm (6.0 ft.) of specimen R2 are presented in Fig. 7. It can be observed from Fig. 7a that shear distortion predicted by the SFI-MVLEM agrees with the test measurements in both positive and negative loading directions, where minor underestimation of shear deformations is noted only at the last loading cycle where significant strength loss occurred during the test. As well, the hysteretic load-deformation response predicted by the model is characterized with highly-pinched hysteretic loops, which is also in good agreement with experimental results. The moment versus rotation comparison shown in Fig. 7b reveals that the SFI-MVLEM also predicts nonlinear flexural deformations (rotations) developing within the wall plastic hinge region with good accuracy at all drift levels except after lateral strength loss. However, due to absence of axial load, flexural cracks on the wall did not fully close upon load reversal during the test, which created significant dowel demands on longitudinal bars and caused sliding along horizontal cracks creating the pinching effects on the experimental flexural load–displacement response (Oesterle et al. 1976). The associated distortion of the longitudinal bars may have impaired uniform crack closure and led to buckling of longitudinal boundary reinforcement and 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). This complex behavioral mode of specimen R2 is naturally not represented in the flexural model response prediction where the area under the hysteretic loops in the predicted moment-rotation response is generally overestimated compared to the experimentally-measured response. Overall, results presented in Fig. 7 indicate that the analytical model successfully captures both nonlinear flexural and shear deformations on the wall, throughout most of the cyclic loading history.
Fig. 7

Shear and flexural responses along the bottom 1830 mm of specimen R2: a base shear versus shear distortion, b base moment versus cumulative rotation

Vertical profiles of the measured and predicted wall curvatures for specimen WSH6 at the applied drift levels of 0.22%, 0.85%, 1.15%, and 1.72% in positive and negative loading directions, are compared in Fig. 8. It can be observed from the figure that the SFI-MVLEM accurately captures the curvature magnitudes and distributions over the height of the wall specimen at all of the drift levels considered, where the analytically-predicted wall curvatures are generally 5% to 10% smaller than the experimentally-measured values. This suggests that hysteretic material models and the plane-sections assumption incorporated within the SFI-MVLEM formulation produce reasonable predictions of vertical (flexural) wall strains along the entire height of the wall, and the model can capture the concentration of nonlinear flexural deformations within the wall plastic hinge region.
Fig. 8

Experimentally-measured and analytically-predicted vertical profiles of wall curvature for specimen WSH6 (Dazio et al. 2009)

Results presented in Fig. 7a demonstrate that the SFI-MVLEM 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 SFI-MVLEM since the majority of analytical models available in commercial (e.g., shear wall element in Perform 3D, CSI) and research-oriented (displacement/force-based nonlinear beam column element in OpenSees) computational platforms treat wall shear behavior via user-defined ad-hoc 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 linear-elastic relationship for shear (most common approach) can have significant implications on predicted behavior of RC walls under dynamic loading, where a typical value of 0.2EcAg (e.g., LATBSDC 2017; PEER-TBI 2.0 2017) leads to underestimation of story drifts and overestimation of wall shear demands. Therefore, the ability of the SFI-MVLEM to capture the experimentally-measured degradation of shear stiffness as a function of increasing lateral deformations is evaluated in the following paragraphs.

Comparisons of the effective secant shear stiffness obtained from experimentally-measured and analytically-predicted shear force versus shear deformation responses are presented in Fig. 9 for the slender, flexure-controlled specimen RW2 and the medium-rise specimen RW-A15-P10-S78 with significant shear–flexure-interaction behavior. In both analytical and experimental results, the effective secant shear stiffness is calculated at the peaks of the cyclic loading history, based on corresponding shear deformation (Δs) and shear force (F) values, and then normalized with respect to EcAg for consistent comparison among the specimens as:
$$ \frac{{GA_{eff} }}{{E_{c} A_{g} }} = \frac{{h_{w} }}{{E_{c} A_{g} }}\left( {\frac{F}{{\Delta_{s} }}} \right) $$
where Ec is the concrete modulus of elasticity, Ag is the gross wall cross-section area, hw is the height of the wall, and GAeff is the effective shear stiffness. It should be noted that shear deformations used to obtain shear stiffness are measured during the tests over different heights for the two specimens: over the bottom 914 mm (36 in.; 1st story height) of specimen RW2 and over the entire wall height of 1829 mm (72 in.) for specimen RW-A15-P10-S78. Shear deformations in the analytical model results are obtained by summing up shear deformations from the model elements that are located within the wall regions considered in the experiments to allow consistent comparisons.
Fig. 9

Experimentally-measured and analytically-predicted degradation of the effective shear stiffness of specimen: a RW2 and b SP4

Results presented in Fig. 9 reveal that the SFI-MVLEM can capture the general trend of shear stiffness degradation with increasing drift levels throughout the loading cycles. The analytically-predicted uncracked effective shear stiffness value is approximately equal to theoretical value of 0.4EcAg, while the experimentally-measured initial shear stiffness values (based on the point of first load reversal) are considerably lower, at approximately 0.24EcAg and 0.28EcAg for specimens RW2 and RW-A15-P10-S78, 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 experimentally-measured values, while for specimen RW-A15-P10-S78, the analytically-predicted 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 0.2EcAg (e.g., LATBSDC 2017; PEER-TBI 2.0 2017; Fig. 9) is four to ten times larger than the shear stiffness values measured during the tests and captured by the SFI-MVLEM at post-cracking drift levels (i.e., larger than 0.2%), which varies between approximately 0.02EcAg and 0.05EcAg.

6.2 Validation of SFI-MVLEM-SQ

Accuracy of the SFI-MVLEM-SQ is evaluated using experimental results obtained from three planar squat specimens tested by Terzioglu et al. (2018): (1) T5-S1 with aspect ratio of 1.0, (2) T2-S3 with aspect ratio of 0.5, and (3) T4-S1 with aspect ratio of 0.33. All three specimens experienced significant shear deformations contributing up to 35%, 75%, and 80% to the top lateral displacement of the walls, respectively, and failed in diagonal compression under combined shear and flexural effects. Figure 10a–c compare the experimentally-measured and analytically-predicted lateral load versus top displacement responses of the three wall specimens considered obtained using the SFI-MVLEM-SQ model, while Fig. 10d compares the experimental load-deformation response of specimen T4-S1 with the hysteretic behavior predicted using the baseline SFI-MVLEM formulation. It can be observed from Fig. 10c–d that the baseline SFI-MVLEM, with σx = 0 assumption implemented at each wall macro panel (see model description), underestimates by 50% the lateral load capacity of the squat wall specimen T4-S1 (Fig. 10d), while incorporation of the empirical horizontal axial strain interpolation functions by Massone (2010) in the SFI-MVLEM-SQ significantly improves the prediction of the wall load capacity (Fig. 10c). Figure 10a–c demonstrate that the SFI-MVLEM-SQ captures the measured hysteretic response of the three walls considered with good accuracy. The model accurately replicates the initial stiffness, the lateral load capacity, as well as the post-peak behavior of the specimens. Note that the post-peak response of the specimens was dependent on the post-peak slope of the concrete strain stress curve, which was calibrated to match the experimentally observed behavior. Alternatively, strain localization can be studied by revising the fracture energy of concrete in order to avoid calibration, which is outside the scope of this work. Analysis results also reflect the highly-pinched shape of the hysteretic load–displacement responses observed, which is associated with shear-dominant wall behavior.
Fig. 10

Lateral load–displacement responses for squat wall specimens: a T5-S1, b T2-S3, and cd T4-S1

6.3 Validation of SFI-MVLEM-3D

The capability of the proposed SFI-MVLEM-3D model to simulate the behavior of non-planar wall specimens subjected to cyclic loading is evaluated using experimental results obtained for a T-shaped wall specimen TW2 (Thomsen and Wallace 1995) tested under uni-directional loading, and a U-shaped wall specimen TUB (Beyer et al. 2008) tested under complex multi-directional loading.

The model of specimen TW2 was loaded according to the experimental cyclic loading protocol, by applying a point load at the top of the model in the direction parallel to the web of the wall. It should be noted that despite the fact that the wall was loaded only along one loading direction, a full three dimensional model of the specimen was generated where one SFI-MVLEM-3D element was used for the web and two SFI-MVLEM-3D elements (in plane perpendicular to the web) were used for the flange (Fig. 11a). Specimen TW2 is also modeled with the baseline two-dimensional 2-node SFI-MVLEM in which wall web was discretized in the same manner as for the SFI-MVLEM-3D and wall flange was modeled using a single macro-fiber, with the fiber thickness equal to flange length to allow consistent comparison between the two models (Fig. 11b). Identical material parameters are used in both models. Figure 11 compares the experimentally-measured and analytically-predicted lateral load versus top displacement responses of specimen TW2, where results for SFI-MVLEM-3D are shown in Fig. 11a and results for SFI-MVLEM are shown in Fig. 11b. Results presented in the Fig. 11a show that SFI-MVLEM-3D model captures the lateral load capacity and lateral stiffness of the T-shaped wall specimen in both loading direcitons. In the positive loading direction, where the wall flange is in compression and the wall web is in tension, the model slightly underestimates the unloading stiffness of the wall, likely due to inability of Concrete02 material to capture gradual closure of concrete cracks, resulting in smaller plastic (residual) displacements and higher-level pinching compared to the experimentally-measured behavior. In the negative loading direction, where the flange is in tension and the web is in compression, the model overestimates the experimentally-measured unloading stiffness and residual displacements of the wall, which generally leads to moderate overestimation of the area under the hysteretic loops. Results shown in Fig. 11b reveal that the load–displacement response of specimen TW2 predicted using the baseline SFI-MVLEM is notably less accurate compared to SFI-MVLEM-3D results. The wall lateral load capacity is overestimated by the SFI-MVLEM during cycles in the negative loading direction by 13%, and the SFI-MVLEM results show a more pronounced overestimation of the unloading stiffness values when compared to SFI-MVLEM-3D, possibly due to inability of the model to capture shear-lag effect using the plane sections assumption implemented at the model element level. Similarly to planar RC walls, the SFI-MVLEM-3D and SFI-MVLEM do not capture the strength loss of specimen TW2 that occurred due to lateral instability of the web in compression.
Fig. 11

Lateral load–displacement response of specimen TW2 (Thomsen and Wallace 1995): a SFI-MVLEM-3D, b SFI-MVLEM

The last wall specimen investigated in this study is the U-shaped specimen TUB (Beyer et al. 2008) tested under a complex multi-directional loading history applied at the top of the wall. The loading pattern applied during the experiment at the wall height of 2950 mm included cycles in E–W (A–B), N–S (C–D) and diagonal (E–F) directions, as well as a complex O–A–G–D–O–C–H–B–O cycle (sweep), as illustrated in Fig. 12a. Torsional deformation (twisting) of the specimen was restrained during the experiment. Peak top displacements for each loading cycle corresponded to displacement ductility levels (μ) of 1.0, 2.0, 3.0, 4.0, and 6.0. In the model of specimen TUB, the lateral load was applied using the force load patterns that corresponded to locations and directions of the actuators, as shown in Fig. 5b. Because the cross-section of the specimen is not torsionally symmetrical for E–W, diagonal, and sweep loading directions, and the E–W force resultant acts outside the shear center of the cross-section, there is a tendency of the wall to move out of the plane of applied loading. Therefore, in order to prevent this out-of-plane movement of the model, as well as to ensure proper boundary conditions at the top of the wall (i.e., no twisting), three uniaxial springs with very high stiffness values were assigned to the wall model at these locations and in the directions of the actuators used in the experiment (Fig. 5b). The prescribed cyclic loading history is applied by imposing large forces in the direction of the actuator forces (i.e., against horizontal supporting springs), where force histories were defined for each of the actuator forces in E–W and N–S directions, such that their combination produces the wall displacement at the height of 2950 mm imposed during the experiment. Therefore, the analysis performed to obtain the model results was a quasi-static force-controlled analysis, via application of appropriate scaling factors to the prescribed force patterns to impose various peak top displacements on the model, in compliance with the experimental loading protocol.
Fig. 12

Lateral load–displacement response of specimen TUB (Beyer et al. 2008): a Top displacement history, b E–W cycles, c N–S cycles, df diagonal cycles

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 square-root-of-sum-of-squares (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 SFI-MVLEM-3D 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 SFI-MVLEM-3D 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 well-predicted, where slight overestimation of pinching is observed in the analytical results. Furthermore, the response comparison shown in Fig. 12d reveals that the SFI-MVLEM-3D 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 in-between 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 SFI-MVLEM-3D model to accurately capture nonlinear strain distributions along the wall flange/web (shear-lag effect) due to the plane-sections 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).

To assess the capability of the SFI-MVLEM-3D model to capture local wall responses, analytically-obtained vertical strains over the height of the wall specimen TUB are compared with experimentally-measured strain profiles at wall boundaries. Figure 13 compares experimental and analytical strain profiles at various locations (marked with a star) corresponding to loading positions B (E–W cycles), and C/D (N–S cycles), at displacement ductility levels of μ = 1.0 (low displacements), μ = 3.0 (intermediate displacements), and μ = 6.0 (large displacements and failure). Based on comparisons of results presented in Fig. 13, it can be observed that strain profiles predicted using the SFI-MVLEM-3D adequately represent the height over which strain concentration occurs (i.e., plastic hinge length), where both experimental and analytical results suggest a plastic hinge length of approximately 900 mm. It can be further observed from Fig. 13a–c that the SFI-MVLEM-3D model predicts average tensile strains over the plastic hinge region with ± 25% accuracy for the displacement ductility levels of μ = 1.0 and 3.0, although with considerable dispersion at some locations, whereas at the ductility level of μ = 6.0 the analytically-predicted tensile strains are within ± 50% of the experimental measurements over the majority of the wall height and overestimated up to two times at the base of the wall. Analytically-predicted vertical profiles of compressive strains shown in Fig. 13d-f are generally characterized with reduced accuracy and more dispersion compared to experimentally-measured values, where the analytical model captures the general trend of the experimental results for the displacement ductility levels of μ = 1.0 and 3.0, but typically overestimates the compressive strains 2.0–3.0 times at the base of the wall at the ductility level of μ = 6.0 for positions B and D. Discrepancy between analytical and experimental results could be due to two major reasons: (1) the plane-sections assumption implemented in formulation of the SFI-MVLEM-3D cannot capture nonlinear strain distributions over the horizontal sections in the non-planar walls subjected to multidirectional loading, as discussed by previous researchers (e.g., Constantin 2016), and (2) locations from which strains are obtained in the model and experiment are not in exact agreement, where model uses points at the wall centerline and in the experiment the strains were measured at the outer face of the wall.
Fig. 13

Experimental and analytical strain profiles for specimen TUB: a position B, tension strains, b position C, tensions strains, c position D, tensions strains, d position B, compression strains, e position C, compression strains, f position D, compression strains

Overall, the results presented demonstrate that the SFI-MVLEM-3D is a reliable tool for predicting the hysteretic behavior of non-planar 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 Perform-3D (CSI), due to the capability of the SFI-MVLEM-3D to capture interaction between axial/flexural and shear responses at the model element level, which are not considered in current commercially-available macroscopic models. The major shortcoming of the SFI-MVLEM-3D compared to commonly-used microscopic models (e.g., VecTor, DIANA, LS-DYNA) is the use of plane-sections assumption, which could impair the accuracy of predicted wall local responses, particularly for non-planar walls subjected to multi-directional loading. However, the significance of the SFI-MVLEM-3D 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) two-dimensional baseline formulation of the Shear–Flexure-Interaction Multiple-Vertical-Line-Element-Model (SFI-MVLEM), (2) extension of the baseline SFI-MVLEM formulation to simulation of squat wall behavior (SFI-MVLEM-SQ), and (3) a three-dimensional version of the baseline SFI-MVLEM formulation for simulating the response of nonplanar RC walls under multi-directional loading (SFI-MVLEM-3D). The model formulations are implemented in the widely used open-source 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 relatively-slender or medium-rise planar RC walls specimens tested under in-plane loading used for validation of the baseline SFI-MVLEM, (2) three squat planar RC wall specimens tested under in-plane loading, used to verify the accuracy of the SFI-MVLEM-SQ, and (3) one T-shaped and one U-shaped RC wall specimen tested under unidirectional and multi-directional loading, respectively, used to validate the SFI-MVLEM-3D model. The analytically-obtained and experimentally-measured 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 experimentally-measured and analytically-predicted 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 experimentally-measured and analytically-obtained shear deformations show that the SFI-MVLEM 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 experimentally-measured nonlinear flexural deformations. Results presented also demonstrate that the SFI-MVLEM-SQ captures well the hysteretic shear-dominated behavior of squat RC walls. Finally, validation of the SFI-MVLEM-3D shows that the model can simulate, with reasonable accuracy, the hysteretic load-deformation 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 SFI-MVLEM-SQ and SFI-MVLEM-3D models (baseline SFI-MVLEM 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 system-level studies.



This work was supported by the National Science Foundation, Award No. CMMI-1563577. 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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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.California State University, FullertonFullertonUSA
  2. 2.Bogazici UniversityIstanbulTurkey
  3. 3.University of ChileSantiagoChile
  4. 4.University of CaliforniaLos AngelesUSA

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