A Hysteretic Constitutive Model for Reinforced Concrete Panel Elements
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Abstract
A simple yet effective constitutive modelreferred to as the “Fixed Strut Angle Model” (FSAM)is presented in this paper for simulating the nonlinear axial/shear behavior of reinforced concrete membrane (panel) elements subjected to generalized and reversed cyclic loading conditions. In the formulation of 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 frictionbased constitutive relationship, are superimposed with the concrete stresses along the struts, for obtaining the total stress field in concrete. Model predictions were compared with panel tests results available in the literature, at various global and local response levels. The model was demonstrated to reasonably capture the overall response characteristics of reinforced concrete panels, including hysteretic shear stress vs. strain behavior, shear stress capacity, hysteretic shear stiffness attributes, ductility, pinching behavior, governing failure mode, principal strain and stress directions, and local deformations.
Keywords
panel membrane hysteretic constitutive model concrete crack shear wall1 Introduction
With adaptation of performancebased seismic design and assessment methodologies for reinforced concrete (RC) structures, analytical modeling of the behavior of RC members under generalized loading conditions induced by earthquake actions has recently gained substantial importance among engineers and researchers. A reliable prediction of the nonlinear earthquake response of structural systems inherently requires the use of analytical models that can accurately capture the hysteretic behavior of individual structural members, as well as their interaction within a structural system.
In seismic design of RC buildings, use of structural walls is effective for resisting earthquake actions. To counteract earthquake demands in the nonlinear response range, slender walls are designed and detailed to yield in flexure, and to undergo inelastic flexural deformations without loss of lateral load capacity. Therefore, a modeling methodology that appropriately accounts for nonlinear flexural behavior becomes sufficient for design and evaluation purposes. However, shearcontrolled squat walls (with aspect ratios typically less than 1.5) are also common in lowrise construction and at lower levels of tall buildings (for example, parkinglevel walls or basement walls), as well as in perimeter walls with perforations due to window and door openings. For low aspectratio walls or wall segments, behavior is often dominated by nonlinear shear responses, and the modeling parameters selected for shear stiffness and strength can have a significant impact on the predicted distribution of member forces and on building lateral drift. As well, obtaining reliable predictions for local deformations (e.g., amplified compressive strains at wall boundary regions associated with plane sections not remaining plane due to shear deformations) and capturing of nonlinear shear and shearflexure interaction responses (as well as a realistic value for the effective shear stiffness) in also slender and mediumrise walls are still topics of utmost interest. For performancebased design and evaluation of RC systems with structural walls, there is still a need for simple yet robust modeling approaches that capture coupled axial, shear, and flexural responses of walls with various aspect ratios and response characteristics.
Simulation of the nonlinear response of walls can be accomplished by using finite element (microscopic) or phenomenological (macroscopic) modeling approaches. For use in finite element models, various constitutive model formulations for monotonic (e.g., Modified Compression Field Theory (Vecchio and Collins 1986), RotatingAngle Softened Truss Model (Pang and Hsu 1995), FixedAngle Softened Truss Model (Pang and Hsu 1996), Disturbed Stress Field Model (Vecchio 2000), Softened Membrane Model (Hsu and Zhu 2002)) and cyclic (e.g., Ohmori et al. 1989; Stevens et al. 1991; Vecchio 1999; Palermo and Vecchio 2003; Mansour and Hsu 2005; Gérin and Adebar 2009) loading have been proposed for simulating the nonlinear response of RC panels. However, most of these models are not implemented in commonlyused opensource computational platforms for analysis of structural walls or wall systems. Although not opensource, finite element analysis software incorporating hysteretic formulations of the Modified Compression Field Theory (Vecchio and Collins 1986) and the Disturbed Stress Field Model (Vecchio 2000) is available online (http://vectoranalysisgroup.com). The Cyclic Softened Membrane Model (Mansour and Hsu 2005) has been implemented in the opensource platform OpenSees (http://opensees.berkeley.edu); however, studies on simulating the nonlinear response of structural walls using the Cyclic Softened Membrane Model are limited. It is believed that there is still a need for development, experimental validation, and opensource implementation of simple yet sufficiently accurate constitutive models to be used for nonlinear response analysis of structural walls or structural systems incorporating walls.
Therefore, an original constitutive model formulation has been developed by the authors for simulating the hysteretic response of RC panel (membrane) elements under generalized inplane loading conditions. The model formulation is based on interpretation and simplification of previous modeling approaches. The constitutive model was implemented (Kolozvari et al. 2015a, 2018c) in the opensource computational platform OpenSees (http://opensees.berkeley.edu) as a planestress constitutive relationship for reinforced concrete panel elements. As well, the FSAM has already been used as a constitutive model element in both macroscopic (fiberbased) (Kolozvari et al. 2015b, c, 2018a, 2019b) and finite element (Gullu et al. 2018, 2019; Gullu and Orakcal 2019) model formulations developed for RC walls, and has been shown to provide accurate response predictions for walls with various aspect ratios and response characteristics. This paper provides a detailed description of the mechanical constitution of the FSAM, and provides results of detailed correlation studies conducted between FSAM predictions and RC panel test results obtained from two different experimental programs reported in the literature, for experimental validation of its fundamental working principles at the stress–strain level.
2 Model Background and Description
During the past three decades, significant efforts on understanding and simulating the inelastic shear behavior of RC members have focused on developing constitutive models capable of predicting the inplane behavior of RC panel elements subjected to membrane actions. The pioneering modeling approaches, known as the first rotatingangle approaches (Vecchio and Collins 1986; Pang and Hsu 1995), were developed for monotonic loading conditions, and they simulated RC panel response using uniaxial constitutive relationships for concrete in tension and compression applied along the principal strain directions of the panel, together with uniaxial constitutive laws for reinforcing steel bars applied along rebar directions, with perfect bond assumed between concrete and reinforcing steel. This modeling approach treated cracked concrete as a new material and incorporated the smearedcrack approach, which considers average stresses and strains in concrete throughout the panel element, for satisfying the equilibrium conditions. Although the simplicity of the rotatingangle approach lies in the assumption that the principal stress direction in concrete coincides with the principal strain direction, interpretation of the rotatingangle modeling approach on how damage progresses in a RC panel element might be ambiguous, since once a crack develops at a particular location on a RC panel or wall, the crack direction remains the same and the crack progressively opens, closes, and reopens with successive load cycles. Other modeling approaches for RC panels have also been proposed, based on either a fixed crack angle that follows the principal direction of the applied stresses (Pang and Hsu 1996; Hsu and Zhu 2002; Mansour and Hsu 2005), or an angle “lag” that deviates the principal strain and stress directions (Vecchio 2000; Stevens et al. 1991). The fixedcrackangle approach used in the original FixedAngle Softened Truss Model (Pang and Hsu 1996) or its subsequent versions (Hsu and Zhu 2002; Mansour and Hsu 2005) is different from the approach used in this study in the sense that it assumes that the fixed crack directions coincide with the principal directions of the “applied” stress field on the panel element (necessitating proportional loading conditions applied on the RC panel), whereas the present model formulation does not require proportional loading.
Various cyclic constitutive panel models (Ohmori et al. 1989; Stevens et al. 1991; Vecchio 1999; Palermo and Vecchio 2003; Mansour and Hsu 2005; Gérin and Adebar 2009), incorporating rotating or fixedangle approaches, have also been developed based on results of cyclic test programs conducted on RC panel specimens (Ohmori et al. 1989; Stevens 1987; Mansour 2001). Test observations reported in the literature indicate that after formation of cracks in a RC panel, the direction of the principal stresses in concrete does not change significantly with loading, whereas the principal strain direction in the panel may undergo significant variation (Stevens et al. 1991). The principal stress directions in concrete being insensitive to loading implies that after formation of cracks, the principal stress directions in concrete follow approximately the fixed crack directions. This observation contradicts the assumption that the principal strain and concrete principal stress directions coincide, and also implies that shear stresses along the crack have marginal influence on the concrete principal stress directions. To consider this phenomenon in various formulations of the Disturbed Stress Field Model (Vecchio 2000; Stevens et al. 1991), which uses a rotatingcrack assumption, an “angle lag” is empirically defined for the purpose of deviating the principal strain and stress directions in concrete. On the other hand, the Cyclic Softened Membrane Model (Mansour and Hsu 2005), although follows a fixedcrackangle formulation, retains the assumption that the principal strain and stress directions in concrete coincide under cyclic loading.
A more simplistic approach is used in this study. The constitutive model described herein, which is named the “FixedStrutAngle Model” (FSAM), incorporates a rotatingangle approach only until the first crack forms on the RC panel, the direction of which coincides with the direction of the principal strains applied on the panel element direction at the instant of cracking. Upon formation of the crack, the “baseline” formulation of the model (which neglects the shear stress along the crack surface) converts into a fixedangle approach that inherently fixes the angle of the crack as the principal stress direction in concrete, meaning that stresses in concrete consist of normal stresses calculated parallel and perpendicular to the crack direction. As assumed by most other RC panel models available in the literature, identical strain fields are assumed to act on concrete and reinforcing steel components of a RC panel, based on the perfect bond assumption between concrete and reinforcing steel bars. It is also assumed that reinforcing bars develop zero shear stresses perpendicular to their longitudinal direction, which implies no dowel action on reinforcement. As well, behavioral features of concrete associated with the Poisson’s ratio are neglected in the model formulation, for simplicity. While only uniaxial stresses develop in the longitudinal direction of reinforcing bars, the behavior of concrete is characterized by stress–strain relationships applied along biaxial directions, the orientation of which are fixed after cracking. On top of this baseline formulation of the FSAM, where the shear stress transfer mechanism across cracks is neglected, a simple and frictionbased constitutive relationship is adopted, and the shear aggregate interlock stresses developing along crack surfaces are superimposed with the concrete stresses developing along the crack directions. This superposition creates a marginal deviation between the crack directions and the new principal stress directions in concrete, and provides a reasonably accurate prediction of the overall behavior of the panel element, as shown later in this paper. Working principles of the FSAM are described in the following sections.
2.1 Uncracked Panel Response
The stress vs. strain behavior of uncracked concrete is simulated using a rotatingstrut (rotatingcrack) approach [similarly to the “Modified Compression Field Theory” (Vecchio and Collins 1986) and the “RotatingAngle Softened Truss Model” (Pang and Hsu 1995)]. The strain field applied on concrete is transformed into principal strain directions and a uniaxial stress–strain relationship for concrete is applied along the principal strain directions, for obtaining the principal stresses in concrete. The principal strain directions imposed on the panel element are therefore assumed to coincide the with principal stress directions in concrete. A monotonic stress–strain relationship is adopted for concrete in the uncracked state, since the monotonic and hysteretic stress–strain behavior of concrete in a panel element subjected hysteretic loading can be assumed to not differ significantly prior to first cracking. This assumption was also made to overcome the difficulties to track and store history variables in the hysteretic stress–strain relationship for concrete, along rotating stress and strain directions in a panel.
2.2 Panel Response after Formation of First Crack
At the instant when the principal tensile strain in concrete first exceeds the concrete cracking strain (ε_{t}), the first crack develops, and the principal strain direction corresponding to first cracking is assigned as the first “fixed strut” (first crack) direction in the panel (θ_{crA}). After the first crack forms, while principal strain directions continue to rotate based on the strain field applied, the directions of the principal stresses in concrete are assumed to be fixed, as parallel and perpendicular to the first fixed strut direction. This implies that the first crack (or strut) direction coincides with the principal stress directions in concrete, under the condition that zero shear stress develops along the crack.
2.3 Panel Response After Formation of Second Crack
At the instant of the second crack formation, the second “fixed strut” will develop in parallel direction to the second crack (in perpendicular direction to the first strut). During further loading, the concrete stress field comprises two independent struts, working under interchanging compression and tension (Fig. 1c), based on the applied strain field. While the principal direction of the applied strain field continues to rotate during subsequent loading, the principal stress directions in concrete are assumed to be fixed along the two strut directions, again when zero shear stresses (zero shear aggregate interlock) are assumed to develop along the two crack surfaces. Since both strut directions are fixed, the hysteretic uniaxial stress vs. strain relationship adopted for concrete can be applied along the first and second strut directions. Using the uniaxial constitutive material model for concrete, the unsoftened principal stresses in concrete are first calculated, leading to the softened principal stresses (σ_{cx’} and σ_{cy’}) after applying the softening due to history damage of concrete in compression and actual tensile strain in the orthogonal direction of each compression strut. For calculation of concrete stresses along the struts, the applied strain field should be transformed into normal strains parallel to the first and the second strut directions, instead of principal strain directions.
2.4 Shear Stress Transfer Across Cracks
As described above, the baseline formulation of the FSAM considers that when shear stresses transferred across cracks are neglected (i.e., when the crack surfaces are assumed frictionless), the concrete principal stress directions and crack directions coincide. However, this baseline formulation allows the flexibility to separately implement a constitutive relationship in the FSAM for representing shear stress transfer across cracks (e.g., a shear stress vs. shear strain relationship along a crack), since the model formulation allows calculating shear strains along the cracks. In RC members, sliding along cracks is known to develop shear aggregate interlock action, resulting in shear stress along crack surfaces. The zeroaggregateinterlock assumption incorporated in the baseline formulation of the FSAM may result in overestimation of sliding deformations along cracks, depending on the loading conditions (stress state) applied on the panel element.
2.5 Material Constitutive Models
2.5.1 Concrete
The hysteretic uniaxial constitutive model by Chang and Mander (1994) was implemented in the formulation of the FSAM for representing the stress vs. strain behavior of concrete. The Chang and Mander (1994) model is a refined nondimensional model that can reproduce the generalized hysteretic behavior of ordinary or highstrength, confined or unconfined concrete under continuous reversed cyclic compression and tension. The model formulation reflects important behavioral characteristics such as the hysteretic transition from compression to tension and vice versa, the progressive degradation of stiffness of the unloading and reloading curves for increasing values of strain, and the effects of and gradual crack closure and tension stiffening on the behavior.
In the Chang and Mander (1994) model, the monotonic curve forms the envelope for the hysteretic stress–strain relationship. Concrete in tension is modeled with a cyclic behavior similar to that in compression. The model envelopes for compression and tension can be calibrated for the slope of the stress vs. strain relationship at the origin, and the shape of both the prepeak and postpeak branches of the stress–strain behavior. The shape of the envelopes can be feasibly altered while keeping the values of the peak stress and the strain at peak stress constant. In order to define the compression and tension envelopes, Chang and Mander (1994) model uses the Tsai’s equation (1988), which is based on the equation by Popovics (1973), an equation that has proven to be very useful in describing the monotonic compressive stress–strain curve for concrete. In order to define the cyclic properties of concrete in compression, statistical regression analyses were performed by Chang and Mander (1994) on an extensive experimental database. Based on the regression analyses, empirical relations were developed for key hysteretic parameters such as those for secant stiffness and plastic stiffness upon unloading from the envelope, and stress and strain offsets upon return to the compression envelope (Fig. 2a). Further details of the model can be found in the report by Chang and Mander (1994).
2.5.2 Reinforcing Steel
The uniaxial constitutive stress–strain relationship implemented in the FSAM for reinforcing steel is the wellknown nonlinear hysteretic model of Menegotto and Pinto (1973), extended by Filippou et al. (1983) for incorporating isotropic strain hardening effects, as a relatively simple yet effective model that can accurately simulate the hysteretic stress–strain behavior of reinforcing steel bars.
In the Menegotto and Pinto model, the stress–strain (σ–ε) relationship is in the form of curved transitions (Fig. 2b), each from a straightline asymptote with slope E_{0} (modulus of elasticity) to another straightline asymptote with slope E_{1} = b·E_{0} (yield modulus) where the parameter b is the strain hardening ratio. The curvature of the transition curve between the two asymptotes is governed by a cyclic curvature parameter (R), which permits the Bauschinger’s effect to be represented. It must be emphasized that more robust stress–strain models for reinforcing steel, which also incorporate more sophisticated behavioral characteristics relevant to the seismic behavior and performance of reinforced concrete structures, including local or global inelastic bar buckling [e.g., (Massone and Moroder 2009; Massone and López 2014)], lowcycle fatigue degradation, and corrosion effects [e.g., (Tripathi et al. 2018; Kashani et al. 2015)], can easily be implemented in the FSAM formulation. The simple stress–strain model adopted in this study does not incorporate such features, since the panel test programs used for validating the working principles of the FSAM did not present any of these particular failure or degradation mechanisms.
2.5.3 Compression Softening in Concrete
An important consideration in modeling the behavior of a RC panel element under membrane actions is incorporating the compression softening effect. The softening effect for the behavior of concrete under biaxial stress state has been experimentally observed by many researchers, [e.g., Vecchio and Collins (1986)], and has been represented by analytical models mainly in the form of reduction (softening) in the compressive stresses in concrete along the principal compression direction of RC panels, due to presence of tensile strains in the perpendicular principal direction. Some compression softening models have also included softening in the compressive strain [e.g., Belarbi and Hsu (1995); Vecchio and Collins (1993)], reducing the strain at the peak compressive stress for concrete. Although all of these compression softening models were formulated for the case of monotonic loading, many of them were implemented into cyclic analysis methods (Belarbi and Hsu 1995; Vecchio and Collins 1993).
2.5.4 Tension Stiffening Effect on Concrete and Steel
The contribution of cracked concrete to the tensile resistance of RC members is known as the effect of tension stiffening. The concrete between the cracks, which is still bonded to the reinforcing steel bars, contributes to the tensile resistance of the member. The tension stiffening phenomenon plays a significant role in reducing the postcracking deformations of reinforced concrete structures, and has been proven by researchers (Pang and Hsu 1995; Stevens 1987; Belarbi and Hsu 1994; Bentz 2005; Mansour et al. 2002) to influence considerably the postcracking stiffness, yield capacity and shear behavior of reinforced concrete members.
The reason that two different constitutive models for tension stiffening were used in the FSAM formulation for comparison of model results with test results is that the tension stiffening model is not a unique attribute of the FSAM. The researchers (Mansour 2001; Stevens 1987) who conducted the panel tests considered in this study for experimental validation of the FSAM have stated that the two tension stiffening models (Belarbi and Hsu 1994; Stevens 1987) best represent the experimentallyobserved behavior of their test specimens. Therefore, the FSAM was experimentallyvalidated for the two respective test programs, with the two respective tension stiffening models in its formulation, for a more consistent evaluation of its performance. As discussed in detail by Bentz (2005), the difference between these two tension stiffening models can be attributed to differences in the reinforcement configuration of the panel specimens tested by Stevens (1987) and Mansour (2001) with regards to the bond characteristics of the reinforcing steel bars, which are influenced by bar diameter and spacing.
2.5.5 Biaxial Damage on Concrete
An important consideration in modeling the behavior of a RC panel element under membrane actions is incorporating the cyclic damage effects on concrete subjected to biaxial loading. This cyclic damage on concrete is represented via a damage coefficient. Unlike compression softening and tension stiffening parameters, the damage coefficient is a cyclicstrainhistorydependent parameter, and is not considered in analysis of concrete under monotonic loading.
The damage coefficient is a parameter that considers the effect of the history of compressive strains, which are in perpendicular direction to a specific compressive stress direction considered (e.g., along a compression strut) for concrete. The damage coefficient is therefore defined for biaxial loading, and does not apply for uniaxial concrete stress–strain behavior. The damage coefficient, similar to the compression softening coefficient, is applied as a multiplier to the concrete compressive stress, softening the stress–strain behavior of concrete in compression.
The final softening parameter used in this study considering both compression softening and biaxial damage parameters takes the form of \(\beta = \beta_{m} \cdot \beta_{damage}\). It should be mentioned that using the two different biaxial damage coefficients in the formulation of the FSAM did not result in significant differences in model results obtained for all panel specimens investigated in this study.
2.6 Overview of the Working Principles of the FSAM
As described in the previous sections, the FSAM is merely a twodimensional (planestress) constitutive model that relates the resultant stress state on reinforced concrete to the applied strain field, where an equivalent uniaxial stress–strain relationship for concrete (considering biaxial softening and hysteretic damage effects) is applied along fixed strut (crack) directions, a frictionbased shear stress vs. sliding shear strain relationship is applied along crack surfaces, and a uniaxial stress–strain relationship is applied along reinforcement directions. Uniaxial stresses developing in concrete along two strut directions are superimposed with the shear stresses developing along crack surfaces and the uniaxial stresses developing in the reinforcing steel bars (smeared over concrete based on reinforcement ratios), in order to obtain the resultant stress field at a point. Instances of crack formation and crack directions are automatically calculated by the FSAM, based on the history of the applied strain field. The first crack is assumed to develop at the instant when the principal tensile strain first exceeds the cracking strain of concrete, in perpendicular direction to that principal tensile strain. Upon load reversal, the second crack is assumed to develop in perpendicular direction to the first, at the instant when the tensile strain parallel to the first crack exceeds the concrete cracking strain. The FSAM does not require dimensions, definition of crack angles, definition of strut widths, or any other parameter that is not related to material stress vs. strain behavior (except a friction coefficient for the crack surfaces), and it can be implemented in any model formulation as a twodimensional constitutive relationship representing the smeared stress vs. strain behavior of RC.
3 Comparison of Model Results with Experimental Data
With the mechanical principles and material constitutive relationships described in the previous section, the formulation of the FSAM was implemented in Matlab, together with a displacementcontrolled iterative nonlinear analysis solution strategy (Clarke and Hancock 1990), in order to obtain the analysis results used for the experimental validation studies presented in this section.
Very few cyclic panel tests are available in the literature (Ohmori et al. 1989; Stevens 1987; Mansour 2001). Two different test programs were considered within the scope of this study. The first of these two test programs was conducted by Stevens (1987) using the “Shell Element Tester” facility at the University of Toronto and the other by Mansour (2001) using the “Universal Element Tester” facility at the University of Houston. Findings of these experimental programs have been used for development of the hysteretic panel model formulations proposed by Stevens et al. (1991) and Mansour and Hsu (2005). A more limited suite of comparisons between FSAM results and data obtained from these experiments were presented in the conference paper by Orakcal et al. (2012). Details of these experimental programs and results of comprehensive correlation studies conducted between model predictions and test measurements are presented in the following two sections.
For all panel specimens investigated, the constitutive material parameters used in the FSAM formulation were assigned values that match the astested properties of the materials used in the construction of the panel specimens, whenever material test results relevant to a specific parameter were reported. Whenever not, the material parameters were defined based on wellestablished empirical relationships provided in the literature. Thereby, a consistent methodology was used in selecting the material parameters for all panel specimens, as opposed to adjusting the material parameters (excluding the shear aggregate interlock friction coefficient; refer to Sect. 2.4) for optimizing the accuracy of the model in predicting the test results for an individual specimen. Accordingly, compressive strength of concrete in the FSAM was defined based on compression test results on samples of concrete used in the construction of the panel specimens. Empirical relationships by Chang and Mander (1994) and Saatcioglu and Razvi (1992) were used for generating the ascending and descending (postpeak) regions of the concrete compressive stress–stain envelope, respectively. The yield strength and strain hardening ratio parameters of the Menegotto and Pinto (1973) model were assigned values that represent the stress–strain curve obtained from tests on reinforcing steel bars used in the construction of the specimens. For all panels, the parameters controlling the cyclic stiffness degradation characteristics of the model (Fig. 2b) were defined as R_{0} = 20, a_{1} = 18.5, and a_{2} = 0.15, as proposed originally by Menegotto and Pinto (1973). The behavioral parameters of the FSAM related to tension stiffening and concrete biaxial damage were defined as reported by the researchers who have conducted the tests, to best represent the experimentallyobserved tension stiffening and biaxial damage characteristics of the test specimens, as also discussed in Sects. 2.5.4 and 2.5.5 of this paper.
3.1 Tests by Stevens (1987)
Panel test parameters, Stevens (1987).
Panel specimen:  SE8  SE9  SE10 

Loading type:  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy}\! :Reversed Cyclic\)  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy}\! :Reversed Cyclic\)  \(\sigma_{x} =  \left {{{\tau_{xy} } \mathord{\left/ {\vphantom {{\tau_{xy} } 3}} \right. \kern0pt} 3}} \right\) \(\sigma_{y} =  \left {{{\tau_{xy} } \mathord{\left/ {\vphantom {{\tau_{xy} } 3}} \right. \kern0pt} 3}} \right\) \(\tau_{xy}\! :Reversed Cyclic\) 
ρ _{ x}  0.03  0.03  0.03 
ρ _{ y}  0.01  0.03  0.01 
f_{y,x} [MPa]  492  422  422 
f_{y,y} [MPa]  479  422  479 
f’_{c} [MPa]  37  44  34 
ε _{ co}  0.0026  0.0026  0.0023 
f_{ct} [MPa]  2.0  2.2  2.0 
ε _{ t}  0.0001  0.0001  0.00013 
In this experimental program, there were two different parameters investigated; the loading type and reinforcement ratio. While specimens SE8 and SE10 were used to examine the effect of loading type on panel response (pure shear stress applied on SE8 and SE9; shear stress with proportionallyapplied normal stresses σ_{x} = σ_{y} = –τ_{xy}/3 applied on SE10), specimens SE8 and SE9 were utilized to investigate the effect of reinforcement ratio on the response (ρ_{x} = 0.03, ρ_{y} = 0.01 for SE8 and SE10; ρ_{x} = ρ_{y} = 0.03 for SE9).
3.1.1 Global Response
In this section, the overall shear stress τ_{xy} vs. shear strain γ_{xy} behavior of the test specimens are compared with predictions of the FSAM.
Specimen SE9 General behavior of Specimen SE9 is captured by the model with reasonable accuracy (Fig. 4b). Although the shear stress capacity appears to be slightly overestimated by the model (by 11% of the test measurement); pinching characteristics, cracking stresses, and cyclic stiffness properties of the behavior are all wellpredicted (Fig. 4b). The widths of the loading and unloading loops in the stress–strain behavior are predicted to be only slightly narrower than the test results, with the cumulative area under the loops estimated at 88% of the experimentallymeasured response. The main reason of this discrepancy may be attributed with variation in the cyclic behavior of concrete, which governs the overall response of this specimen. As a measure of ductility, the shear strain capacity of the specimen (at which degradation in shear stress initiates), is wellestimated by the model, at approximately 95% of the test result. Since the specimen has 3% reinforcement in both directions, in the model for this specimen, the concrete compression struts are able to reach their maximum “softened and damaged” compressive stress capacity, prior to yielding of steel. However, since the tests were stresscontrolled, none of these tests were continued until the failure mode of the specimens was clearly indentified.
Specimen SE10 Specimen SE10 was the one of the most important specimens of this test program, with its unequal reinforcement ratio in x and y directions, and with applied σ_{x} and σ_{y} compressive stresses which are both proportional to applied shear stress. Specimen SE10 is a replication of specimen SE8, with 1% reinforcement ratio in y direction and 3% reinforcement in x direction; the only difference being the compressive normal stresses (σ_{x} and σ_{y}) applied during testing. The shear stress capacity of this specimen is overestimated by only 5% of the test result (Fig. 4c). Other behavioral features of the response including cracking stress, stiffness of the loading and unloading curves, and pinching characteristics are predicted accurately. The model slightly overestimates the cumulative area under the shear stress vs. strain loops by 14% of the test result, and moderately underestimates the shear strain capacity of the specimen (at initiation of degradation in shear stress), at 80% of the test measurement in the negative loading direction.
Sensitivity to friction coefficient All model results presented in this paper for the specimens tested by Stevens (1987) are obtained using an optimal aggregate interlock friction coefficient value of η = 0.1. Modification of the interlock coefficient to a value of η = 0.2 [as used for the specimens by Mansour (2001)], results in approximately 30% increase in the shear stress capacity of Specimen SE8. For Specimen SE9, which is symmetricallyreinforced and subjected to a pure shear stress state, model results are not sensitive to the value of the friction coefficient, since shear strains do not develop along the cracks. In the case of Specimen SE10, although it is nonsymmetrically reinforced (similarly to SE8), the normal stresses applied on the specimen reduce the impact of the friction coefficient on the model response, yielding only 1% increase in the predicted shear stress capacity when the friction coefficient is increased from 0.1 to 0.2.
3.1.2 Local Responses
Specimen SE8 When the normal strains in x direction of Specimen SE8 are compared, it is observed that the strain history is predicted accurately by the model (Fig. 5a). Model predictions for the normal strains in y direction are also accurate. An increasing trend in strains in the y direction that starts at approximately the 900th step (corresponding to the first yield of reinforcement in y direction) can be clearly identified in both model and test results (Fig. 5b). When the principal strain directions are compared for Specimen SE8, as illustrated in Fig. 6a, the model accurately captures the measured principal strain direction history of the specimen, which changes noticeably throughout loading. Only a slight underestimation at the beginning of the analysis (for small shear strain values) is observed in the model results.
When the principal stress direction in concrete vs. panel shear stress behaviors are compared for Specimen SE8, it is observed that the model predictions for the principal stress direction in concrete do not significantly deviate from the two crack directions. As can be observed in Fig. 7a, the concrete principal stress direction in the model results undergoes marginal variation with the magnitude of the shear stress applied on the panel, due shear aggregate interlock stresses developing along crack surfaces, which deviate the principal stress directions from the two fixed crack directions. However, the variation is not significant, because the compressive stresses developing along the struts dominate over the shear stresses developing along the crack surfaces. There exists larger variation in the test measurements, probably due to a more complicated shear stress transfer mechanism across cracks; although these deviations are typically associated with unloading and reloading branches of the behavior, and the upper and lower bounds for the principal stress directions do not vary significantly with the magnitude of the shear stress (Fig. 7a). Thus, the model reasonably captures the experimental behavior with a frictionbased model for shear aggregate interlock, maintaining its simplicity. It must be mentioned that the test result given in the figure is the envelope of the test measurements for the specimen, as the original graphic (Stevens 1987) was too congested for digitizing purposes.
The test results presented in Figs. 6a, 7a clearly demonstrate that for this specimen, although the principal strain direction noticeably changes with loading (under increasing strains), the direction of principal stresses in concrete do not deviate significantly from the two crack directions. This behavior is clearly captured in the model results, since the model does not follow the assumption that principal strain and concrete principal stress directions coincide.
Specimen SE9 When the normal strains in the x direction are compared for Specimen SE9, it is observed that the general ascending trend is captured by the model (Fig. 5b). The general ascending trend in normal strains in y direction is again captured, with increasing discrepancies at later stages of loading (Fig. 5b). At this point, it should be clarified that the normal strains predicted by the model in x and y directions are approximately identical. Considering that the specimen had equal reinforcement ratios in both directions and pure shear loading is applied, the model predictions are mechanically consistent. Therefore, differences between normal strains measured in the x and y directions during the test may be attributed to imperfections in the test conditions. When the principal strain directions are compared for this specimen, as depicted in Fig. 6b, test results are wellpredicted by the model. Model predictions for the principal strain directions are 45° and 135°, since the pure shear stress state applied on the symmetricallyreinforced specimen creates a pure shear strain state, which is also observed in the test measurements. Furthermore, under the pure shear stress state applied on the panel specimen, because of identical reinforcing steel ratios and yield strengths along the two reinforcement directions, concrete is also subjected to a pure shear stress state. Therefore, the principal stress directions in concrete and the fixed crack directions coincide at 45° and 135° angles during the entire loading history, which is also in agreement with the test results, as shown in Fig. 7b.
Specimen SE10 For specimen SE10, when the normal strains in x direction are compared, it is observed that the general response is reasonably captured (Fig. 5c). When the normal strains in y direction are compared, it is seen that the general trend is captured for most of the loading history, with increasing discrepancies at later stages of loading (Fig. 5c), probably due to degradation in the overall shear stress vs. strain behavior predicted by the model. When the principal strain directions are compared, as shown in Fig. 6c, the test measurements are predicted accurately by the model. The measured and predicted principal stress directions in concrete (vs. shear stress on the panel) are compared in Fig. 7c. Similarly to Specimen SE8, variation in the principal stress directions predicted by the model are not significant, because of larger concrete compressive stresses developing along the struts compared to smaller shear friction stresses developing along the crack surfaces. There is slightly more variation in the experimentallymeasured principal stress directions; however, as shown in Fig. 7c, such variation is typically limited to the unloading and reloading branches of the response, at regions of small shear stress. At regions of high shear stress, the measured principal stress directions do not vary significantly, and are in good agreement with model predictions, validating the modeling approach used.
3.2 Tests by Mansour (2001)
There are two important characteristics of the experimental program by Mansour (2001). First, these tests (12 fullsize reversed cyclic panel tests) were performed under strain control, which revealed the sudden stiffness drop as an effect of first cracking on panel response, as well as degradation in shear stress during later stages of loading due to the behavior of concrete in compression. Second, the test program was aimed to investigate the effect of reinforcement ratio to the overall behavior (i.e., ρ_{x} = ρ_{y} = 0.0077 for CA2; ρ_{x} = ρ_{y} = 0.017 for CA3; ρ_{x} = ρ_{y} = 0.027 for CA4), asymmetry of reinforcement ratio in the orthogonal directions (i.e., ρ_{x} = 0.017, ρ_{y} = 0.0077 for CB3; ρ_{x} = 0.027, ρ_{y} = 0.0067 for CB4) and different loading conditions (i.e., pure shear stress on CA2, normal and shear stresses on CD4, and normal stresses on CE4).
Panel test parameters, Mansour (2001).
Panel specimen:  CA2  CA3  CA4  CB3  CB4  CD4  CE4 

Loading type:  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy} :RC\)  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy} :RC\)  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy} :RC\)  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy} :RC\)  \(\sigma_{x} = 0\) \(\sigma_{y} = 0\) \(\tau_{xy} :RC\)  \(\sigma_{x} = 1.05\tau_{xy}\) \(\sigma_{y} =  1.05\tau_{xy}\) \(\tau_{xy} :RC\)  \(\sigma_{x} = RC\) \(\sigma_{y} =  \sigma_{x}\) \(\tau_{xy} = 0\) 
ρ _{ x}  0.0077  0.017  0.027  0.017  0.027  0.02  0.019 
ρ _{ y}  0.0077  0.017  0.027  0.0077  0.0067  0.02  0.019 
f_{y,x} [MPa]  424  425  453  425  453  453  453 
f_{y,y} [MPa]  424  425  453  424  424  453  453 
f’_{c} [MPa]  45  44.5  45  48  47  43  47 
ε _{ co}  0.0025  0.0024  0.0028  0.0026  0.0024  0.0024  0.0022 
f_{ct} [MPa]  2.1  2.1  2.1  2.2  2.2  2.1  2.2 
ε _{ t}  0.00008  0.00008  0.00008  0.00008  0.00008  0.00008  0.00008 
3.2.1 Global Response
In this section, the overall shear stress τ_{xy} vs. shear strain γ_{xy} behavior of the selected test specimens are compared with model results.
Specimen CA3 Specimen CA3 has a reinforcement ratio of 1.7% in both directions. The analytical model accurately predicts the overall shear stress vs. shear strain behavior of this specimen in terms of its stiffness, cracking stress, and pinching attributes (Fig. 8b). The shear stress capacity of the specimen is underestimated by only 5% of the test result. The difference between model and test results is mostly associated with the shape of the envelope of the response, which was found to be influenced both by the shape of the concrete stress–strain behavior in compression and the effective yield strength of reinforcing steel. The cumulative area under the shear stress vs. strain loops estimated by the model corresponds to 82% of the test result, and the model predicts the shear strain capacity of the specimen (at initiation of shear stress degradation) at 88% of the measured value.
Specimen CA4 Specimen CA4 had the largest reinforcement ratio, with the value of 2.7% in both directions. The model wellpredicts the overall shear stress vs. shear strain behavior of this specimen well, with slight discrepancies in cyclic stiffness characteristics. Features including the pinching effect, cracking stress, and shear stress capacity are all captured accurately (Fig. 8c). The shear stress capacity of the specimen is captured with 100% accuracy, and its shear strain capacity (at initiation of shear stress degradation) is overestimated by only 7% of the test measurement. The model also wellpredicts the cumulative area under the measured shear stress vs. strain loops for the specimen, at 96% of the test result. The shape of the envelope of the predicted response was found to be significantly influenced by the shape of the concrete stress–strain behavior in compression.
Specimen CB4 Specimen CB4 was similar to CB3, but with a larger reinforcement ratio in the x direction, which made the shape of the experimentallymeasured response even more influenced by the concrete stress–strain behavior in compression (Fig. 9b). Similar discrepancies between model and test results to those observed for specimen CB3 are observed for this specimen. Although the model wellpredicts the shear stress capacity of the specimen, at 99% of the measured capacity, it does not capture the degradation in shear stress at high shear strains, and the cumulative area under the shear stress vs. strain loops predicted by the model corresponds to 64% of the test result.
Specimen CD4 CDseries specimens were subjected to the most complex loading configuration, with normal and shear stresses applied simultaneously. The model slightly underestimates the shear stress capacity of specimen CD4 by only 10% of the measured; however, similar discrepancies (to those observed for specimens CB3 and CB4) related to the hysteretic shape of the behavior are observed, although to a lesser degree (Fig. 9c). Differently for this specimen, the model captures degradation in shear stress (in the positive loading direction) associated with crushing of concrete, albeit at a much larger shear strain compared to the test result. The cumulative area under the shear stress vs. strain loops is again underestimated by the model, at 61% of measured response.
The experimentallyobserved failure (strength degradation) mode of specimens CB3, CB4, and CD4 (Fig. 9a–c) was crushing of concrete along compression struts, after yielding of reinforcement. Therefore, the most probable reason why the FSAM did not capture the degradation associated with concrete crushing (Fig. 9a, b), or predicted initiation of degradation a higher shear strain than measured (Fig. 9c), is the relatively simple “compression softening” model implemented in its formulation to represent softening in the compressive stress–strain behavior of concrete along the principal compression direction, due to presence of large tensile strains in the perpendicular direction. As described in Sect. 2.5.3 of this paper, the socalled “Model B” by Vecchio and Collins (1993), which considers reduction only in the concrete compressive stresses, was adopted in the FSAM formulation, since it is more suitable for a hysteretic panel model formulation. It is believed that implementation of a more robust compression softening model, which also considers reduction in the strain at peak compressive stress on the monotonic stress vs. strain envelope of concrete (strain softening behavior), such as “Model A” by Vecchio and Collins (1993), may improve the ductility predictions of the FSAM for these particular specimens. However, Model A is not suitable for hysteretic implementation, unless adhoc manipulations are made to convert its monotonic nature into a hysteretic one.
Specimen CE4 CEseries specimens were subjected to loading (normal stresses) parallel to the reinforcement directions. The model for specimen CE4 model recovers most features observed in the experimental response (stiffness, strength, hysteretic response), which is governed by the uniaxial stress vs. strain behavior of the reinforcing steel bars (Fig. 9d). The shear stress capacity and the cumulative area under the stress vs. strain loops predicted by the model for this specimen correspond to 100% and 99% of the experimentallymeasured values, respectively, which validates the reliability of the uniaxial stress–strain model implemented in the FSAM for reinforcing steel.
Sensitivity to friction coefficient All model results presented for the test specimens by Mansour (2001) are generated using an optimal aggregate interlock friction coefficient value of η = 0.2. Similarly to the tests by Stevens (1987), modification of the friction coefficient noticeably changes the analytical response obtained for some of these specimens. For specimen CA2, CA3, CA4, and CE4, model results are not sensitive to the friction coefficient, due to the loading conditions imposed on these specimens and symmetry of the reinforcement. Differences in model results are observed for Specimens CB3, CB4, and CD4. Reduction of the coefficient to a value of η = 0.1 [as used for the specimens by Stevens (1987)], results in approximately 20% and 25% decrease in the shear stress capacity of Specimens CB3 and CB4, respectively Interestingly for Specimen CD4, which was subjected to the complex loading condition, reducing the friction coefficient to a value of 0.1 does not significantly change the predicted shear stress capacity of the specimen, but decreases the shear strain at which shear stress degradation initiates by approximately 20%.
Similarly to the discrepancy in tension stiffening behavior observed in these two experimental programs (Bentz 2005), the necessity of using different interlock friction coefficients to obtain the best correlation between model and test results for the two programs can be attributed to different shear aggregate interlock characteristics of the specimens, such as the mean aggregate size used in the concrete mix. Recent studies on implementation of the FSAM into finite element formulations for modeling of RC walls (Gullu and Orakcal 2019; Gullu et al. 2019) have shown that using a coefficient of η = 0.5 consistently provided accurate response predictions for numerous RC wall specimens that were tested as part of multiple experimental programs, where the dimensions of the wall specimens were significantly larger than the panel specimens analyzed in the present study. This variation is most likely due to the different characteristics of concrete (particularly the mean aggregate size) used in the construction of these larger wall specimens, as well as more nonuniform internal stress distributions and crack orientations developing on the walls. For modeling of reallife walls, adopting a larger value (e.g., η = 0.5) for the friction coefficient therefore seems more reasonable.
3.2.2 Local Responses
Specimens CB4 and CD4 incorporated asymmetry either in reinforcement ratios or loading conditions, resulting in asymmetry in the normal strains observed in x and y directions (Fig. 10c, d). A similar situation is observed for the strain histories predicted by the model. In general, the progression of residual strain, as well as the maximum strain values attained during each cycle or incursions into negative (compressive) values, are well captured.
4 Overview of FSAM Implementation in Structural Models
Since the FSAM is intended to simulate the smeared stress vs. average strain behavior of reinforced concrete under generalized plane stress loading, it is suitable for implementation in model formulations for simulating the inplane nonlinear behavior of reinforced concrete walls, in which the constitutive model elements are membrane elements. The FSAM can also be used to construct a shell element formulation, using a layered membrane assembly.
Research has been conducted on implementation of the FSAM into a “macrofiber” model formulation [SFIMVLEM, Shear Flexure Interaction Multiple Vertical Line Element Model (Kolozvari et al. 2015b, c, 2018a, c, 2019b)] for simulating the coupled axialshearflexural responses of reinforced concrete walls or columns, where the uniaxial line elements in a fiber model formulation were replaced with membrane elements, the planestress vs. strain behavior of which were described with FSAM. Accordingly, axial and shear stress vs. strain responses of each fiber were coupled, which further allowed capturing of the experimentallyobserved coupling of axialshearflexural responses in reinforced concrete walls or columns, through a fiber model formulation. Detailed information on implementation of the FSAM in the computational platform OpenSees [http://opensees.berkeley.edu] as a planestress material model (nDMaterial FSAM) used within the macrofiber model formulation SFIMVLEM, is provided by Kolozvari et al. (2015a, 2018c). Results of earthquake response history analyses on wallframe assemblies and coupled wall systems (Kolozvari et al. 2018d; Kolozvari and Wallace 2016), in which the FSAM was used within the SFIMVLEM formulation adopted for the walls, have also shown that the FSAM is capable of providing computationallystable and accurate response predictions for coupled walls or wallframe systems subjected to dynamic loading.
Recent studies (e.g., Gullu and Orakcal 2019; Gullu et al. 2018, 2019; Kolozvari et al. 2018b, 2019a) have focused on adopting the FSAM in finite element model formulations for walls, using fournode quadrilateral membrane or shell elements, the inplane stress vs. strain behavior of which are described by the FSAM. Considering contemporary needs towards improving seismic design and performancebased assessment procedures, a reliable analysis model should be able to capture, with a reasonable level of accuracy, important response characteristics of walls with various behavioral characteristics (e.g., flexurecontrolled, shear controlled, various levels of shear–flexure interaction, sliding shear, etc.), at both global (load vs. displacement) and local (e.g., strain) response levels, under generalized loading conditions. Towards this goal, current stateoftheart rapidly progresses towards nonlinear finite element modeling of walls. Finite element modeling inherently considers numerous important aspects of wall behavior pertaining to performancebased seismic design, including characterization of nonlinear shear behavior, consideration of coupled shear and flexural responses, plane sections not remaining plane due to shear deformations, definition of a cracked shear stiffness, and coupling of fluctuating axial load with wall shear capacity. All of these aspects are reallife issues that are presently being discussed among the research and engineering community, towards improvement of design applications and performancebased assessment procedures for wall buildings. Finite element modeling methodologies to be used for such practical purposes must incorporate constitutive model formulations that depend on welldefined parameters related to physical behavior, so that the model can be understood and calibrated by the engineer using it. This has been the primary motivation behind development of the FSAM. As described in this paper, the formulation of the FSAM is based on clear assumptions (crack directions are fixed, cracks are perpendicular, concrete along struts follows a uniaxial stress–strain relationship, shear stress transfer across cracks is frictionbased), with as few adhoc model parameters (e.g., friction coefficient) as possible. Other than its inherent assumptions, mechanical principles of the FSAM are based purely on physical material behavior, which can be updated as more information on material stress–strain behavior becomes available.
5 Summary and Conclusions

The FSAM has proven to be a relatively simple constitutive model that is capable of simulating the coupled axial/shear stress–strain behavior of reinforced concrete panel elements. Despite its simple formulation and welldefined assumptions, results of the correlation studies conducted in this paper indicate that the FSAM consistently provides reasonably accurate response predictions for RC panels with various reinforcement configurations, subjected to various loading conditions (stress states). The model was shown to generally capture the overall behavioral attributes of most of the RC panel specimens investigated, including the overall shear stress vs. shear strain behavior, cracking shear stress, shear stiffness, cyclic stiffness degradation, ductility, pinching behavior, and failure mode (crushing of concrete vs. yielding of reinforcement).

Comparison of the overall shear stress vs. shear strain responses reveal the FSAM provides shear stress capacity predictions with an error margin of approximately ± 10% of the test results, when a shear aggregate interlock friction coefficient of 0.1 was used for all specimens tested by Stevens (1987) and 0.2 for all specimens tested by Mansour (2001). The friction coefficient remains as the only adhoc parameter of the model, and is believed to be sensitive to parameters of test specimens associated with the shear aggregate interlock characteristics of the crack surfaces, such as the mean aggregate size used in the concrete mix.

Averaged over all panel specimens investigated in this paper, the shear stress capacity prediction of the FSAM corresponds to 98% of the experimentallymeasured capacity of the specimens, whereas the hysteretic strain energy dissipation capacity prediction of the FSAM (defined as the cumulative area under the hysteretic shear stress vs. strain loops) corresponds to 81% of the measured. The FSAM was not able to accurately capture the ductility characteristics of the measured shear stress vs. strain responses (characterized by the shear strain at initiation of degradation in shear stress) for only three (CB3, CB4, CD4) of the ten panel specimens considered, possibly due to simplifications in behavioral modeling of compression softening effects on concrete.

The FSAM was also shown to be capable of providing accurate predictions of local deformations for the specimens investigated, including normal strains in two orthogonal directions, the direction of principal strains developing in the panel element, and the direction of principal stresses developing in concrete. The experimental observation that even when the principal strain directions rotate during loading, the direction of principal stresses in concrete do not deviate significantly from the two crack directions, is captured by the FSAM.

The flexible formulation of the FSAM allows revisions including implementation of different constitutive material models and various shear stress transfer mechanisms across cracks, as well as incorporation of different compression softening, tension stiffening, and biaxial damage parameters. However, the stateoftheart constitutive material models used in this study for concrete and reinforcing steel have been shown to be satisfactory in simulating the material behavior and failure modes of the RC panels investigated.
With the features and potential improvements described in this paper, the FSAM is believed to provide a good balance between simplicity and accuracy in simulating nonlinear RC behavior under planestress loading conditions. Formulation of the FSAM has been implemented in the opensource computational platform OpenSees (http://opensees.berkeley.edu) as an efficient and flexible constitutive model that can be used for response prediction RC walls with various behavioral characteristics (flexurecontrolled, shear controlled, or shear–flexure interaction responses), under seismic actions. Current studies focus on seismic response simulation of RC walls and wall systems using either macroscopic or finite element modeling approaches, in which the FSAM is used to describe the constitutive behavior of the model elements.
Notes
Acknowledgements
Test data shared by Dr. Mohammad Mansour from Bennett Offshore (Houston, TX) is gratefully acknowledged. The authors would also like to thank Dr. M. Fethi Gullu from Harran University for his help with the analysis of the panel specimens.
Authors’ contributions
KO and LMM jointly developed and coded the FSAM formulation. DU conducted the analyses, using the FSAM, of the panel specimens investigated. All authors read and approved the final manuscript.
Funding
Financial support provided by Chile’s National Commission on Scientific and Technological Research (CONICYT) as travel Grant under the project Fondecyt 2008, Initiation into Research Funding Competition, Grant No. 11080010, is gratefully acknowledged.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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