The fixed-strut-angle finite element (FSAFE) model for reinforced concrete structural walls
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Abstract
This paper evaluates the capabilities and limitations of a relatively simple mesoscopic finite element modeling approach, referred to as the Fixed Strut Angle Finite Element (FSAFE) Model, in simulating the hysteretic lateral load behavior of reinforced concrete structural walls designed to yield in flexure. The behavioral characteristics of the constitutive panel (membrane) elements incorporated in the model are based on a simple fixed-crack-angle formulation, where normal stresses in concrete are calculated along fixed crack directions using a uniaxial stress–strain relationship, with modifications to represent biaxial softening effects. The constitutive panel model formulation also incorporates behavioral models for the shear-aggregate-interlock effects in concrete and dowel action on reinforcing bars, constituting the shear stress transfer mechanisms across cracks. Model predictions are compared with experimentally-measured responses of benchmark wall specimens with a variety of configurations and response characteristics. Accurate predictions are obtained for important global response attributes of the walls prior to failure, including their lateral load capacity, stiffness, ductility, and hysteretic response characteristics; although instability failures related to buckling of reinforcement and/or out-of-plane instability of the wall boundary region are not captured. The model provides accurate estimates of the relative contribution of nonlinear flexural and shear deformations to wall lateral displacements, as well as local response characteristics including the distribution of curvatures, strains, and shear deformations on the walls, prior to failure. Based on the response comparisons presented, model capabilities are assessed and possible model improvements are identified. Overall, the FSAFE model is shown to be a practical yet reliable modeling approach for simulating nonlinear wall behavior, which can be used within the framework of performance-based seismic design and assessment of building structures.
Keywords
Reinforced concrete Structural wall Analytical model Finite element Flexure Shear1 Introduction
Reinforced concrete (RC) structural walls are commonly used in design due to their significant contribution to resistance against lateral actions, including wind loads and earthquake effects, imposed on building structures. Presence of structural walls has considerable impact on the lateral stiffness and strength characteristics of a building, reducing both the elastic displacement demands on the structural system under wind loads or service-level earthquakes, as well as the inelastic displacement or ductility demands under severe earthquakes. In order to provide the necessary level of ductility supply required to attain the targeted seismic performance of a building under design or maximum considered earthquake levels, seismic design codes and recommendations enforce slender walls to exhibit ductile flexural behavior, by incorporating specially-detailed boundary regions and providing sufficient shear capacity for preventing brittle failures, whereas shear-controlled walls or wall segments such as wall piers and spandrels are designed to possess the necessary shear strength to experience limited levels of ductility demand under severe earthquake actions.
Considering the significant contribution of walls on the seismic performance of buildings, and especially with recent implementation of performance-based analysis and design approaches in modern design codes assessment guidelines, robust modeling of the nonlinear seismic response of walls has recently gained much importance among both researchers and engineers. Analytical modeling of the nonlinear response of structural walls can be conducted by using either microscopic (e.g., finite element) or macroscopic (e.g., fiber or plastic hinge) modeling approaches. Microscopic modeling approaches are typically not used in performance-based design or assessment procedures due to increased computational demands and complexities associated with their implementation, calibration, and interpretation of results. Macroscopic modeling approaches available in the literature, with the so-called fiber-based models being more common (e.g., Vulcano et al. 1988; Fischinger et al. 1990; Taucer et al. 1991; Orakcal et al. 2004; Perform 3D CSI 2005; Vásquez et al. 2016), are generally deemed sufficient for modeling of uncoupled shear and flexural responses of slender walls. However, typical fiber-based modeling approaches commonly-used in performance-based design applications fail to capture shear-flexure interaction effects that have been experimentally-observed for both medium-rise walls (Tran and Wallace 2015) and slender walls (Massone and Wallace 2004). Fiber-based models also cannot simulate other important response attributes of walls, including plane-sections not remaining plane, which results in amplification of compressive strains in concrete, as well as salient characteristics of nonlinear shear behavior, including effective shear stiffness, influence of axial load on shear strength, shear ductility, and shear failure type depending on whether or not horizontal web reinforcement yields. There is still a need for simple and computationally-manageable yet robust modeling approaches that can incorporate such behavioral characteristics of walls in the analysis, for more reliable performance-based seismic design applications.
Recent advances in research, as well as improvements in computational capabilities, have made finite element modeling of walls a potentially-feasible approach to be used in real-life applications of performance-based design and assessment. Various hysteretic constitutive models have been developed (e.g., Ohmori et al. 1989; Stevens et al. 1991; Vecchio 1999; Palermo and Vecchio 2003; Mansour and Hsu 2005; Gérin and Adebar 2009; Orakcal et al. 2012) for describing the nonlinear behavior of constitutive RC panel (membrane) elements to be used in finite element model formulations for walls. Numerous research efforts on finite element modeling of walls are also available in the literature (e.g., Vecchio 1989; Palermo and Vecchio 2007; Mo et al. 2008; Rojas et al. 2016; Dashti et al. 2017a; Lu and Henry 2017; Luu et al. 2017; Gullu and Orakcal 2019). However, studies on modeling of walls with both rectangular and non-rectangular cross-sections where the model is comprehensively validated against experimental results for walls with various response features (flexure-dominated, shear-dominated, shear-flexure interaction) at both global and local (deformation, strain) response levels are sparse. Relatively few studies (e.g., Rojas et al. 2016; Dashti et al. 2017a; Gullu and Orakcal 2019) have focused on a comprehensive assessment of an individual model formulation, evaluating its ability to capture various important global and local response characteristics of walls observed experimentally during different test programs.
As an international collaborative effort, comparative model validation studies were recently conducted by a group of researchers (Kolozvari et al. 2018b, 2019), considering five different macroscopic and five different finite element modeling approaches for walls, validated against experimental results for five benchmark wall specimens with rectangular cross-sections, tested as part of five different experimental programs. Although all of the benchmark specimens were designed to yield in flexure, they incorporated a range of configurations and exhibited various response characteristics. As per the objective of this Special Issue, this paper aims to expand upon one of the finite element modeling approaches in the paper by Kolozvari et al. (2019), which is the Fixed Strut Angle Finite Element (FSAFE) model developed by Gullu and Orakcal (2019). A comprehensive summary of the FSAFE model formulation is provided, followed by a detailed evaluation of model accuracy in simulating the experimentally-measured behavioral characteristics of the five benchmark wall specimens, at not only global (lateral load vs. displacement) but also various local (deformation and strain) response levels. Based on the response comparisons presented, significant attributes, strengths, and limitations of the FSAFE model are evaluated, and possible improvements to the model formulation for enhancement of its capabilities are identified.
2 Model description
The Fixed Strut Angle Finite Element (FSAFE) model is an assembly of membrane elements (with zero out-of-plane stiffness), with a smeared stress–strain formulation used to describe the plane-stress behavior of RC. The constitutive behavior of a single model element, which relates an average strain field (ε_{x}, ε_{y}, γ_{xy}) to a smeared stress field (σ_{x}, σ_{y}, τ_{xy}), follows the Fixed Strut Angle Model (FSAM) formulation developed by Orakcal et al. (2012). Working principles of the model and the material constitutive relationships implemented in its formulation are summarized in this section.
2.1 The constitutive fixed-strut-angle model (FSAM)
2.2 Mechanisms for shear stress transfer across cracks
For consideration of dowel action on reinforcing steel bars, the constitutive model by He and Kwan (2001), which uses a smeared stress versus average strain approach, is implemented in the FSAM formulation (Fig. 3b). In the He and Kwan (2001) model, shear and tensile strains in a RC panel element parallel and perpendicular to a crack are first transformed into dowel deformations (Δ_{dow}) acting on the horizontal and vertical reinforcing steel bars, using strain transformation equations and an “effective dowel length” parameter, which semi-empirically depends, based on a beam on elastic foundation analogy, on the elastic modulus of reinforcing steel, the compressive strength of concrete, rebar diameter, and the moment of inertia of the rebar cross-section. The elasto-plastic envelope of the constitutive model, which relates the dowel (shear) force on a single rebar with the dowel deformation, starts with a linear elastic region, the slope of which also depends on the same parameters. The plastic region of the model represents the dowel (shear force) capacity of a single rebar, which depends on the bar diameter, the compressive strength of concrete, and the yield strength of reinforcement. Dowel forces calculated on the horizontal and vertical rebars are then converted into smeared dowel stresses, considering the reinforcement ratios along the two orthogonal rebar directions. Finally, these dowel stresses are back-transformed, using stress transformation equations, into shear and tensile stresses developing in the panel element, in parallel and perpendicular directions to a crack. Details of the model are available in the paper by He and Kwan (2001). In the present FSAM formulation, the origin-oriented hysteresis rules shown in Fig. 3b (upon unloading from and reloading to the monotonic envelope by He and Kwan 2001) are implemented as a simplification of more detailed hysteretic models (e.g., Vintzēleou and Tassios 1986) presented in the literature.
2.3 Material constitutive models
The uniaxial stress–strain relationship implemented in the FSAM for reinforcing steel bars (Fig. 4b) is the well-known hysteretic model by Menegotto and Pinto (1973), extended by Filippou et al. (1983) for representing isotropic strain hardening behavior, and further extended by Kolozvari et al. (2018a) to overcome stress overshooting upon small-magnitude strain reversals. In implementation of the Menegotto and Pinto (1973) model in the FSAM, the tensile yield strength and strain-hardening parameters of the model were calibrated also considering the empirical relationships proposed by Belarbi and Hsu (1994) to include the effect of tension stiffening on the smeared stress–strain behavior of rebars embedded in concrete. However, the present model formulation does not consider rebar buckling or fracture behavior.
Similarly to the nDMaterial model for the FSAM, the above-described uniaxial constitutive models for concrete and reinforcing steel have been implemented in the open-source computational platform OpenSees (McKenna et al. 2000) as uniaxialMaterial models ConcreteCM and SteelMPF, respectively (Kolozvari et al. 2018a).
2.4 FSAFE model assembly
3 Experimental validation of the model
In line with the objective of this Special Issue, this paper aims to provide detailed information on experimental validation of the FSAFE model against test results presented in the literature for the five densely-instrumented benchmark wall specimens identified in the papers by Kolozvari et al. (2018b, 2019), which were tested as part of five different experimental programs. Information on the characteristics of the wall specimens, calibration of the FSAFE model parameters for the wall specimens, and comparison of model results with test data at both global (force and displacement) and local (deformation and strain) response levels are presented in this section.
3.1 Description of the wall specimens
Characteristics of wall specimens
Specimen ID | l_{w} (mm) | h (mm) | t (mm) | \( \frac{{l_{w} }}{t} \) | \( \frac{h}{{l_{w} }} \) | \( \frac{M}{{Vl_{w} }} \) | \( f'_{c} \) (MPa) | \( f_{y,BE} \) (MPa) | \( \rho_{v,BE} \) (%) | \( \rho_{h, web} \) (%) | \( \rho_{v, web} \) (%) | \( \frac{P}{{A_{g} f'_{c} }} \) | \( \frac{{V@M_{n} }}{{V_{n} }} \) | \( \frac{{V_{max} }}{{A_{CV} \sqrt {f'_{c} } }} \) | Behavior mode | Failure mode |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RW-A20-P10-S63 | 1219 | 2440 | 152 | 8.0 | 2.00 | 2.00 | 48.6 | 477 | 7.11 | 0.61 | 0.61 | 0.1 | 0.91 | 0.57 | SFI/DC | CB/LI |
RW2 | 1219 | 3660 | 102 | 12.0 | 3.00 | 3.13 | 42.8 | 434 | 2.93 | 0.33 | 0.33 | 0.09 | 0.53 | 0.20 | FL | CB |
RW-A15-P10-S78 | 1219 | 1830 | 152 | 8.0 | 1.50 | 1.50 | 55.8 | 477 | 6.06 | 0.73 | 0.73 | 0.1 | 0.85 | 0.61 | SFI/DC | CB/LI/SS |
WSH6 | 2000 | 4030 | 150 | 13.0 | 2.02 | 2.26 | 45.6 | 576 | 1.54 | 0.25 | 0.46 | 0.11 | 0.75 | 0.29 | FL | CB |
S6 | 2412 | 3048 | 114 | 21.0 | 1.26 | 1.60 | 34.7 | 482 | 5.60 | 0.55 | 0.55 | 0.05 | 0.75 | 0.53 | SFI/DC | CB/LI |
R2 | 1905 | 4470 | 102 | 19.0 | 2.34 | 2.40 | 46.4 | 450 | 4.00 | 0.31 | 0.25 | 0.00 | 0.43 | 0.16 | FL | CB/BR/LI |
3.2 Calibration of the FSAFE model for the wall specimens
FSAFE models generated for all wall specimens were geometrically-discretized using a standardized approach. On a wall cross-section, two elements side-by-side were used to represent each confined boundary region. The entire wall geometry was then discretized, using square-shaped model elements with approximately equal width and height. For example, the model for specimen RW-A15-P10-S78 was discretized using 12 model elements in the horizontal direction and 18 elements in the vertical direction, resulting in a total of 216 approximately square-shaped model elements. This method was developed based on the analytical observation that using a finer mesh of model elements did not significantly influence the model response predictions, other than the rate of degradation in lateral load after the lateral load capacity is reached. Accurate simulation of the rate of degradation in lateral load, which is associated with not only the lateral instability failures observed during the tests, but also strain localization effects in the analyses, is beyond the scope of the present study. The material constitutive relationships were therefore not regularized based on the model element size to consider strain localization effects on model results, in order to eliminate the uncertainties associated with material stress–strain regularization in interpretation of the results. All wall models were fixed-supported at the bottom wall-pedestal interfaces, via restraining the translational degrees of freedom on the nodes along the bottom interface. Rigid body constraints were defined at the top wall-loading-beam interfaces, for uniform distribution of the constant axial and cyclic lateral loads applied at the top of the wall models. Differently from other specimens, the model of specimen S6 was analyzed under a proportional lateral load pattern consisting of three horizontal forces of increasing magnitude applied at the three representative story levels along the height of the wall, together with a bending moment applied at the top of the wall, replicating the actual loading configuration applied during testing (Vallenas et al. 1979), which is also the reason why the aspect ratio (1.26) and the shear span-to-depth ratio (1.60) of specimen S6 differ significantly.
The constitutive material parameters used in the FSAFE model formulation were calibrated to match the as-tested properties of the materials used in the construction of the specimens (whenever detailed material test results were available), or based on well-established empirical relationships presented in the literature; including empirical equations by Chang and Mander (1994) for generating the ascending region of the compressive stress–stain envelope for unconfined concrete, the confinement model by Mander et al. (1988) for generating the ascending region of the compressive stress–strain envelope for confined concrete, the Saatcioglu and Razvi (1992) model for defining the descending (post-peak) slope of the compressive stress–strain envelopes for both confined and unconfined concrete, as well as the empirical relationships by Belarbi and Hsu (1994) for the tensile strength of concrete (\( f_{t} = 0.31\sqrt {f^{\prime}_{c} } \)), the cracking strain of concrete (\( \varepsilon_{t} \) = 0.00008), and the shape of the smeared stress–strain envelope for concrete in tension (aimed to represent tension stiffening behavior). The yield strength and strain hardening ratio parameters of the Menegotto and Pinto (1973) model were calibrated to represent the stress–strain curve obtained from samples of reinforcing steel bars used in the construction of the wall specimens, with the modifications by Belarbi and Hsu (1994) for consideration of tension stiffening effects. The parameters controlling the cyclic stiffness degradation characteristics of the model were calibrated as R_{0} = 20, a_{1} = 18.5, and a_{2} = 0.15 (Fig. 4b), as proposed originally by Menegotto and Pinto (1973), with the exception of specimen RW2 for which a value of a_{2} = 0.0015 was used, based on findings of previous analytical studies (e.g., Orakcal and Wallace 2006) conducted on this specimen. Details on calibration of the material model parameters are presented in the paper by Orakcal and Wallace (2006).
3.3 Comparison of model response predictions with test results
3.3.1 Specimen RW-A20-P10-S63
As illustrated in Fig. 7b, the model is accurate in predicting both compressive and tensile strains measured along the base of the wall, as well as the depth of the neutral axis on the wall cross-section, at various drift levels. Unlike in fiber models, plane sections do not remain plane in the model results, which is consistent with the experimentally-measured strain profiles and provides improved predictions of longitudinal strains, particularly the compressive strains in concrete.
As shown in Fig. 7c, d, model predictions for the lateral load versus top flexural and top shear displacement responses of the wall are again, accurate. Shear deformations in the model results are calculated by multiplying the average of the shear strain values obtained for all of the individual model elements that are lined up at the same elevation, with the model element height. These shear deformations are summed up over the model height for obtaining the cumulative shear displacement at a particular elevation of the wall. Flexural displacement predictions of the model are calculated by subtracting shear displacements from total lateral displacement at the same wall elevation. Comparisons shown in Fig. 7c, d show that the model successfully replicates the load versus flexural displacement and the more-pinched load versus shear displacement response characteristics of the wall, as well as the magnitudes of the flexural- and shear-deformation-related components of top displacement.
Comparison of model results and test measurements for the lateral flexural displacement and lateral shear displacement profiles along the height of the wall is presented in Fig. 7e, f. The lateral flexural displacement profile of the wall is well-estimated by the model (Fig. 7e), with nonlinear flexural deformations concentrated along the bottom 610 mm of the wall height, corresponding to a plastic hinge length of half the wall horizontal length (l_{w}/2). The shapes of the measured and predicted shear displacement profiles are also in agreement (Fig. 7f), with decreasing shear distortions developing towards the top of the wall, demonstrating that the model captures coupling between nonlinear shear and flexural deformations, since shear deformations are amplified in regions where nonlinear flexural deformations are concentrated. The model also replicates the distribution of nonlinear flexural and shear deformations along the height of the wall, and therefore captures the so-called “spread of plasticity” along wall height. Discrepancies between test results and model predictions at the drift level of 3.0% can be attributed to degradation in lateral load at 3.0% drift, observed in both experimental and analytical responses.
3.3.2 Specimen RW2
3.3.3 Specimen RW-A15-P10-S78
3.3.4 Specimen WSH6
3.3.5 Specimen S6
3.3.6 Specimen R2
However, the experimentally-observed behavior and failure mode of this wall was unique. Test observations reported by Oesterle et al. (1976) indicate that due to absence of axial load, flexural cracks on the wall did not fully close upon load reversal, which created significant dowel demands on longitudinal bars and caused sliding along horizontal cracks, amplifying the shear distortion measurements shown in Fig. 13b and creating the pinching effects on the experimental load–displacement response shown in Fig. 13a. The associated distortion of the longitudinal bars may have impaired uniform crack closure and led to out-of-plane instability of the wall boundary, which initiated at drift levels as low as 1.7%, also due to the relatively low out-of-plane flexural stiffness of the boundary (small wall thickness). The complex behavioral mode and failure mechanism of specimen R2, the main source of which is believed to be the distortion of longitudinal bars as they go from tension to compression, and the related non-uniform crack closure, is naturally not represented in the model response prediction. Out-of-plane instability failures in RC walls (first discussed by Paulay and Priestley 1993) has recently gained much interest among researchers, especially with damage observations after recent earthquakes (e.g., Elwood 2013, Maffei et al. 2014), and has been the subject of both experimental (e.g., Rosso et al. 2016; Dashti et al. 2017b, 2018a) and analytical modeling (e.g., Dashti et al. 2018b) studies conducted on walls, as well as experimental studies conducted on isolated wall boundary elements (e.g., Welt et al. 2017, Rosso et al. 2018). Recent research efforts (e.g., Abdullah and Wallace 2019) also focus on revision of wall seismic design provisions to consider reduction of wall drift capacity associated in-part with out-of-plane instability of wall boundary regions. Implementation of a behavioral modeling approach in the formulation of the FSAFE model for simulating the lateral instability of wall boundaries can potentially improve the model response predictions, not only for specimen R2 but also the other wall specimens investigated in this paper. Test observations on specimen R2 also highlight the shortcoming that current seismic performance evaluation methods for RC walls do not yet consider the effect of relevant parameters (such as in-plane slenderness and axial load) on development on wall instability failures, which may result in significant differences between observed and expected failure modes.
4 Summary and conclusions
Although the working principles of the FSAFE model are relatively simple, the model was shown to provide accurate predictions of the experimentally-measured lateral load versus displacement response characteristics of all wall specimens except one (specimen R2), including their lateral stiffness, lateral load capacity, hysteretic stiffness degradation, and pinching behavior. The model reasonably captured the drift capacity of three of the wall specimens only (RW-A20-P10-S63, RW-A15-P10-S78, and S6), although with much more gradual lateral load degradation in the analysis results. For all specimens, the model could not simulate the experimentally-observed abrupt degradation in lateral load, since instability failures related to buckling of reinforcement or out-of-plane instability of the wall boundary region are not considered in its formulation, which is also the reason why the model was incapable of predicting the drift capacity of specimens RW2, WSH6, and R2.
Prior to their experimentally-observed instability failures, local response characteristics of all wall specimens excluding R2, including the contribution of shear and flexural deformations to lateral displacement, the experimentally-observed coupling between nonlinear flexural and shear deformations, the distribution of shear deformations, flexural deformations, and vertical normal strains along wall height, the vertical growth behavior of the walls, and the nonlinear distribution of vertical normal strains measured along the base of the walls, were all well-estimated by the model. Simulation of nonlinear shear-flexure interaction behavior and capturing of the nonlinear flexural strain gradient along wall length, which creates increased compressive strain demands on concrete and leads to reduction in wall drift capacity, are significant strengths of the FSAFE model over typical fiber-based models used for walls that are expected to yield in flexure.
The accuracy of the model in predicting wall drift capacity, as well as the rate of degradation in lateral load after the capacity is reached, is to be improved upon implementation of constitutive models in its formulation for representing buckling of reinforcement (e.g., Dhakal and Maekawa 2002), as well as incorporation of a modeling approach to simulate lateral instability of wall boundary regions. The inadequacy of the present model in capturing the experimentally-observed response of Specimen R2, which experienced initiation of lateral instability in the wall boundaries at early drift levels, due to the zero axial load level and the large in-plane slenderness of the wall, highlights the potential of such an approach towards improving the capabilities of the model.
Overall, with the features and potential improvements identified in this paper, the FSAFE model is presented as a relatively simple yet effective modeling approach for simulating the nonlinear seismic response of RC walls. Ongoing studies focus on improvement of the existing model formulation–using the simplest approach possible–for incorporating lateral instabilities in the model response predictions, and conducting seismic response simulation studies on building systems using the FSAFE model for structural walls, towards improvement of performance-based seismic design and assessment procedures.
Notes
Acknowledgements
The authors would like to thank Prof. John Wallace and Dr. Thien Tran from UCLA, and Dr. Alessandro Dazio from ROSE School, for sharing test data.
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