Investigation of Tie Bars Axial Force Demands in Composite Plate Shear Walls—Concrete Filled

Abstract

Tie bars axial force demands due to concrete dilation and prying action were investigated through numerical studies. In the first part of this study, the Karagozian and Case Concrete model, which proved to provide reasonable in-plane flexural cyclic inelastic wall response while accounting for concrete dilation effect, was used to investigate the variation of confinement inside the infill concrete, the distribution of passive confining pressures at the steel–concrete interface, and the resulting tie bar axial force demands. Finite element analyses involving C-PSW/CF having different tie spacings, wall depths, and wall thicknesses were performed. In the second part of this study, the influence of plate local buckling on tie bar axial force demand was investigated and explained by prying action. A separate finite element study was performed to investigate the significance of prying action and equations were developed from free-body diagrams. The results showed the significance of the passive confining pressures due to concrete dilation, and prying action due to local plate buckling, on imparting axial forces in tie bars. Neither of these behavior are currently considered as design parameters for tie bars. The numerical analyses and results presented here are intended to provide useful insights and direction for the design and instrumentation of future C-PSW/CF experiments by the broader research community.

Appendices

Appendix 1: Comparative Study using KCC and CSCM Concrete Models

1.1 Selection of Concrete Models

This Appendix presents details of the analyses performed to identify a simple concrete model that can reasonably predict wall strength and account for confinement effect. Note that in the aforementioned study by Polat and Bruneau (2017), the objective was to replicate the in-plane flexural wall response experimentally obtained by Alzeni and Bruneau (2017), focusing on global response and in particular on capturing the pinching effect observed in the hysteretic response, which was believed to be related to the opening and closing of the concrete cracks, and which was modeled successfully using the Winfrith model. Fig. 

Fig. 18

Comparison of experimental and numerical wall hysteresis using Winfrith model

18 shows the resulting comparison between the experimental and numerically obtained hysteretic curves using that model. Fig. 

Fig. 19

Comparison of axial strain distribution in steel plate for experimental and numerical resulting using Winfrith model

19 shows the corresponding resulting axial strain distribution in the steel plate. While the steel plate axial strains were predicted well, the wall compression depth was slightly over-predicted. Although good results were obtained with the Winfrith model, this may not be the best model when the objective is to predict the tie bar axial force demand. This is because, in C-PSW/CF, when the concrete sandwiched between steel web plates is under vertical axial compression due to combined bending and axial forces, shear dilation of the concrete can apply a horizontal pressure to the steel plate, which in turn introduces an axial force on the tie bars. This mechanism makes it essential to replicate passive confinement effects due to concrete shear dilation—which, unfortunately, is not a behavior accounted for by the Winfrith model (Wu et al., 2012).

A large number of concrete constitutive models exist that have the ability to model dilation. The KCC and CSCM models were considered here because they offer the option to generate most of the needed model parameters with little user input (although the KCC and CSCM models offer both complex material definition or parameter generation options); as such, structural engineers can easily define the nonlinear behavior of concrete per these models with knowledge of the unconfined uniaxial compressive strength and a few basic concrete properties, namely, mass density for the KCC models, and aggregate size for the CSCM models.

These models have also been investigated by various researchers. In particular, Wu et al. (2012) investigated these concrete models for their effectiveness in capturing key aspects of concrete behavior, namely, post-peak softening, shear dilation, and confinement effects for plain concrete cylinder tests, using LS-Dyna. Schwer and Malvar (2005) compared the results obtained using the KCC model when specifying only the unconfined compression strength of a well characterized 45.6 MPa concrete, based on original well-characterized concrete, with those from various material characterization tests. They reported that the complex behavior of concrete can be modeled successfully with the default parameter generation data of the KCC model. Terranova et al. (2018) used the KCC and CSCM models to generate benchmark stress–strain data using the Smooth Particle Hydrodynamics (SPH) formulation in LS-Dyna. In light of the lack of experimental data on tie bar axial forces in C-PSW/CF, the use of more complex model was also justified at this time.

1.2 Description of the Finite Element Modeling

Aspects of the finite element modeling and analysis procedures developed for the reference wall studied by Polat and Bruneau (2017) are described below, as many of these parameters are re-used here. The concrete was modeled using an eight-node solid elements (Solid 1) with reduced integration and hourglass control; the steel web plates and the boundary elements were modeled using four-node fully integrated shell elements (Shell 16), and; tie bars were modeled using two node beam elements with the Hughes-Liu beam formulation (Beam 1) with two integration points. For the steel–concrete interface, the automatic_surface_to_surface_mortar contact with an interface friction coefficient of 0.3 was used. The dimensions of the shell and solid elements were determined based on results from a convergence study. The shell elements were 25.4 mm × 25.4 mm (1 in × 1 in) and the solid elements were 25.4 mm × 25.4 mm × 25.4 mm (1 in × 1 in × 1 in) in size. Note that a total of four layers of concrete in the transverse direction were used in the numerical model of the reference wall.

MAT003 (with kinematic hardening) was used for the steel web plates, half-HSS boundary elements, and tie bars. Note that MAT003 is a bi-linear model which requires the definition of the elastic modulus, $E_{s}$, the yield strength, $F_{y}$, the tangent modulus, $E_{T}$, and a hardening parameter for kinematic hardening or isotropic hardening. The elastic modulus used for the steel web was E_s = 205,463 MPa (29,800 ksi) and for the boundary element E_s = 189,605 MPa (27500 ksi), based on experimental results of steel coupons for the specimens. The yield strength used for the steel web was 427 MPa (62 ksi) and for the boundary element 317 MPa (46 ksi). Tangent modulus used for the web plate was 551 MPa (80 ksi) and for the boundary elements 344 MPa (50 ksi). Concrete compressive strength was set as f_c' = 47.6 MPa (6900 psi). In the LS-Dyna model, a concrete tensile strength of f_t' = 4.76 MPa (690 psi) and a Poisson’s ratio, v, of 0.2 were also specified. For the CSCM concrete model, the unconfined compression strength is 30 MPa (4.351 ksi) (Murray, 2007). Note that for the KCC and CSCM concrete models, the user has the option of directly inputting material properties or requesting default material properties based on the unconfined compressive strength of concrete—as mentioned previously. Note also that, for the default option in the CSCM concrete, the internally calculated values for a number of parameters by the model are reported to have been derived based on the properties of concrete ranging in strength from 20 to 58 MPa (2.901 to 8.412 ksi) (LSTC, 2013; Murray, 2007). In all cases, aggregate size was specified as 7.9 mm (0.31 in.).

The elastic modulus of the concrete, in the benchmarked model, was adjusted to 0.5E_c (MPa or ksi) where $E_{c} \left( {{\text{psi}}} \right) = 57000\sqrt {f_{c}^{^{\prime}} \left( {{\text{psi}}} \right)}$ to match the experimentally measured wall elastic stiffness. For the KCC model, the elastic modulus of concrete is internally calculated. The value of the internally calculated value for the KCC model can be obtained by checking the output log files from the analysis, which, in this case, confirmed that the values of the elastic modulus was equal to E_c for the wall with the KCC model.

The walls were subjected to in-plane lateral displacements at their top end, applied to all the steel and concrete nodes at that height. The experimental loading protocol provided by (Alzeni & Bruneau, 2017) was used in the numerical solution—details on this protocol are not provided here due to space concerns.

1.3 Comparison of Inelastic Cyclic Wall Response Using KCC and CSCM Concrete

Wall response using the KCC and CSCM models were compared against that of the benchmark wall using the Winfrith model, using fixed-based boundary conditions in all cases. Note that the benchmarked wall was originally developed by explicitly accounting for foundation flexibility of the specimens, but this was avoided here because it was unnecessary. Comparison of the resulting hysteretic responses in terms of base moment versus wall drift ratio is shown in Fig. 

Fig. 20

Comparison of cyclic in-elastic base moment hysteresis: a Winfrith and KCC, and; b Winfrith and CSCM

20a–b, where the results are compared for the walls with: (a) Winfrith and KCC models, and; (b) Winfrith and CSCM models. Note that the wall with the CSCM model exhibits almost no pinching, whereas the amount of pinching captured by the wall with KCC model is somewhat between what has been obtained for the walls with the Winfrith and CSCM models. The prediction of maximum wall strength by the KCC model is almost the same as that for the Winfrith model, whereas, for the CSCM model, it is 9% higher than for the Winfrith model at a 2% drift ratio. Given that confinement is related to wall strength, and that tie bar demands are affected by confinement pressures, the KCC model was deemed to be more adequate for the investigation presented in the main body of the paper, in spite of its shortcoming in capturing global hysteretic response.

1.4 Comparison of Confinement Through Concrete Axial Stress Distribution Through Wall Thickness

Fig. 

Fig. 21

Axial stress distribution in steel and concrete along half wall depth from FE models with concrete models: a Winfrith, b KCC, and; c CSCM (vertical dotted lines denote tie locations)

21a–c shows the axial stress distribution in the steel and the concrete obtained from static inelastic pushover analyses along half the wall depth under compression, but excluding the half-HSS ends of the wall. In this figure, the zero position corresponds to the center of the wall and the dotted lines in these figures represent the tie bar locations. Note that, here, the wall thickness is represented as layers equal to the concrete finite element mesh segments through the thickness. Layer 1 corresponds to the outermost layer and Layer 4 corresponds to the innermost layer at the center of the concrete wall. For each model, concrete stresses are reported for each layer to observe the variation of confinement through the thickness. The stresses are reported at the wall elevation corresponding to half the distance between the first and second row of tie bars [= 203 mm (8 in)] from the base.

As shown in Fig. 21a for the wall with the Winfrith model, the compression axial stress distribution through the layers of the infill concrete does not change significantly. Moreover, all the concrete layers exhibit a stress block with a peak strength of approximately 60 MPa (8.7 ksi). This gives the wall a minor increase in flexural strength considering the unconfined strength of 48 MPa (6.9 ksi). This indicates that some confinement is provided by the pressures developed in the boundary elements. The consequence of web plate buckling is also visible, as expressed by the progressive reduction of axial stresses resisted by the steel plate, most significantly observed at 3% and 4% drift.

As shown in Fig. 21b for the wall with the KCC model, the compression axial stress distribution varies through the thickness with higher values at the innermost layers than at the outermost ones—this is attributed to the model’s ability to simulate shear dilation and hence capture confinement effects. Note that the maximum compressive stresses reached at mid-layer is more than double the values measured for the wall with the Winfrith model. For example, the peak strength is about 150 MPa (21.8 ksi) in Layer 4 (compared to 60 MPa (8.7 ksi) in the wall with the Winfrith model). For the wall with the KCC model, the compression depth is approximately 100 mm (3.94 in) less than for the wall with the Winfrith model.

As shown in Fig. 21c for the wall with the CSCM model, the compression axial stress distribution also varies through the thickness as a consequence of concrete shear dilation. The maximum compressive stress reached at mid-thickness of the wall is about 75 MPa (10.9 ksi). Note that, the compression depth is increased by as much as 75 mm (2.95 in) compared to the wall with the Winfrith model. For this wall, the axial stress distribution of the steel plate indicates significant plate buckling.

1.5 Comparison of Tie Bars Axial Force Demand

Here, axial force response of tie bars obtained from the static inelastic pushover analyses are compared. Fig. 

Fig. 22

Axial force of tie bars of the walls with concrete material models of: a Winfrith; b KCC, and; c CSCM

22a–c shows the axial force demands for the four tie bars identified in Fig. 1b. As shown in Fig. 22a, relatively low axial forces are obtained when using the Winfrith model compared to the ones using the KCC (Fig. 22b) and the CSCM (Fig. 22c) models. Larger tie bar forces are created when using the KCC and CSCM models, evidently due to the ability of these models to account for shear dilation. For example, for Tie₂₂, axial demands obtained at 4% drift are 6 kN (1.4 kips), 120 kN (27 kips), and 59 kN (13.3 kips) for the walls with the Winfrith, KCC and CSCM models, respectively. Results indicate that, in this case, tie bars remain elastic for all cases. Assuming a 345 MPa (50 ksi) yield strength of tie bars and a 25.4 mm (1 in) bar diameter, the yield capacity of a tie bar is 175 kN (39.3 kips). Note that maximum force demand occurs in different tie bars for each model. For the wall with the Winfrith and CSCM models, maximum tie bar force occurs in Tie₁₁ (interior tie bar), whereas it occurs in Tie₁₂ (exterior tie bar) for the wall with the KCC material model. This is attributed to differences in the width of the concrete compressive stress blocks—previously shown in Fig. 21—which is closer to interior tie bars (e.g., Tie₁₁ and Tie₁₂) for the walls with the Winfrith and CSCM models, and closer to exterior tie bars (e.g., Tie₂₁ and Tie₂₂) for the wall with the KCC model.

Summary of Findings on Selection of Concrete Model

For the work presented study, various aspects of wall behavior were compared to select a concrete model that is able to provide an acceptable representation of global wall response while at the same time allowing to consider concrete confinement and associated tie bars axial force responses. For this purpose, inelastic cyclic wall responses obtained using the KCC and CSCM models were first compared with the previously benchmarked wall response (from prior research) using the Winfrith concrete. Then, concrete confinement behavior was investigated for the three concrete models by plotting and comparing the concrete axial stress distributions across the wall depth and concrete thickness. Finally, tie bars axial force responses were investigated for the three concrete models by plotting and comparing axial forces for a certain set of tie bars. On the basis of this work, it was determined, as described in Sect. 3.2, that while both the KCC and CSCM models could not simulate the pinching experimentally observed in the inelastic hysteretic flexural behavior of the walls, results from the wall using KCC model was better in agreement with the actual wall response. Likewise, with respect to axial stress distribution along the cross-section, compression depth obtained with KCC model was found to be in better agreement with experimental results. Consequently, of the three models considered, the KCC model was retained for the exploratory work presented in this paper.