Risk-Optimal Arrangement of Stiffeners in Steel Plate Shear Walls


Placement of ferritic stainless steel plate shear walls in the building cores around the elevators and stairs necessitates door-type openings in these systems. Because of large dimensions of door openings, the energy dissipation capacity drops significantly and thus, the probability of out-of-plane buckling under lateral load increases. Accordingly, introducing stiffeners around the opening increases the amount of dissipated energy and improves the performance of the SPSW system. This paper evaluates the seismic risk of SPSW systems with different arrangements of stiffeners around the door opening. Risk, in this context, denotes the probability of failure times the cost of failure of a given SPSW. The probability of failure is computed through a finite element reliability analysis in which material properties, element geometries, and the lateral force are random variables. The failure event is described by a limit-state function as the exceedance of the drift ratio of the SPSW from a prescribed threshold. The drift ratio is computed by subjecting the finite element model to non-linear static analysis in ABAQUS. The reliability analysis is conducted for a variety of single-story SPSW models having door opening with different arrangements of stiffeners and also for a typical SPSW model without opening as a base model. Next, decision analysis is employed to identify the optimal arrangement, i.e., the one that is associated with the minimum risk. Finally, the effect of risk aversion on the optimal decision is studied by introducing risk-averse utility functions with different degrees of risk aversion.






Introduction


A typical 304 stainless steel plate shear wall (SPSW) consists of an unstiffened thin infill plate connected to vertical and horizontal boundary frame members, i.e., columns and beams, respectively. The lateral load is transferred through the infill plate by the principal tension stresses, as shown in Figure 1A. The infill plate is allowed to buckle in shear and consequently forms a diagonal tension field during an earthquake. Previous studies, both experimentally and numerically, have shown that this system exhibits a high ductility and hysteretic energy dissipation capacity compared with conventional braced frames and concrete shear walls (Caccese et al., 1993; Elgaaly et al., 1993; Berman and Bruneau, 2003). Another advantage of SPSWs is the ability to provide openings in the infill plate, which may be required for architectural purposes. Roberts and Sabouri-Ghomi (1992) conducted the first study on SPSW systems with opening. They performed a series of cyclic quasi-static testing on unstiffened SPSWs with a circular opening located at the center of the plate. All the SPSWs tested exhibited stable S-shaped hysteresis loops and adequate ductility. They showed that the strength and stiffness of a perforated SPSW can be approximated conservatively by applying a linear reduction factor to the strength and stiffness of a similar unperforated SPSW. Daftari and Deylami (2000) studied the effect of plate thickness, opening height to width ratio, and the areal percentage of the opening for more than 50 different SPSWs with a central rectangular opening. They determined the optimum aspect ratio for the opening. Paik (2008) obtained a closed-form empirical formula for predicting the ultimate shear strength of 316 stainless steel ship plate with central circular opening under shear loading by the regression analysis. Pellegrino et al. (2009) investigated the influence of the dimension, position, shape (circular or rectangular), and orientation of a hole with respect to the panel slenderness and aspect ratio in steel plates with one perforation subjected to shear loading. Valizadeh et al. (2012) experimentally evaluated the effects of opening dimensions and slenderness factors of plates on the seismic behavior of SPSWs with a circular opening at the center of the panel. Sabouri-Ghomi et al. (2012) studied the behavior of both stiffened and unstiffened SPSWs with a single rectangular opening with different sizes and locations through a non-linear finite element analysis. Hosseinzadeh and Tehranizadeh (2012) studied the non-linear behavior of SPSWs with fully-stiffened large rectangular openings used as windows or doors. Alavi and Nateghi (2013) experimentally investigated the seismic behavior of SPSWs with a central perforation along with diagonal stiffeners. Bhowmick (2014) developed a shear strength equation for unstiffened perforated SPSWs with a circular perforation at the center. They assessed the proposed equation by analyzing a series of single-story perforated SPSWs with different aspect ratios and different perforation diameters. Sabouri-Ghomi et al. (2015) experimentally studied the structural behavior of stiffened SPSWs with two rectangular openings and with different separations subjected to cyclic loads. Also, they determined the shear stiffness and ultimate shear strength of the SPSWs theoretically through the plate-frame interaction model.






Placement of SPSWs in the building cores around the elevators and stairs necessitates door-type openings. Furthermore, because of large dimensions of door openings, the energy dissipation capacity drops significantly and thus, the probability of out-of-plane buckling under lateral load increases. Accordingly, introducing stiffeners around the opening increases the amount of dissipated energy and improves the performance of the SPSW system, as recommended by AISC Design Guide 20 (AISC, 2007).






However, construction of those SPSWs in which the vertical and horizontal stiffeners continue to boundary elements, as shown in Figure 1B, is significantly costly. The underlying reasons are the need for more nickel alloy steel and significantly more cutting and welding for connections, which requires further material, labor, and quality control. As shown later in the paper, based on Iran's Cost Catalog (Planning Budget Organization, 2016), the cost will increase by 15%. The extra stiffeners also elongate the construction process and entails workmanship difficulties and defects. The present paper is the first to evaluate the seismic risk of SPSW systems with different arrangements of stiffeners around the door opening. Risk, in this context, denotes the probability of failure times the cost of failure of a given SPSW. The probability of failure is computed through a finite element reliability analysis (Ghanem and Spanos, 1991; Der Kiureghian and Zhang, 1999; Haldar and Mahadevan, 2000; Imai and Frangopol, 2000; Sudret and Der Kiureghian, 2002; Haukaas and Der Kiureghian, 2007) in which material properties, element geometries, and the lateral force are random variables. The failure event is described by a limit-state function as the exceedance of the drift ratio of the SPSW from a prescribed threshold. The drift ratio is computed through a finite element model under non-linear static analysis in ABAQUS (Karlsson and Sorensen, 2013). The reliability analysis is carried out for a variety of single-story SPSW models that include door openings with different arrangements of stiffeners and also for a typical SPSW model that lacks the door opening as a base model. Then, the risk-optimal arrangement is identified through decision analysis. Finally, risk-averse utility functions are introduced to study the effect of risk aversion on the optimal decision.






The single-story SPSW considered in this research, is part of a symmetrical office building located in Tehran, Iran. The floor plan of the building and the considered SPSW is shown in Figure 2A. The roof dead and live loads are assumed 0.5 and 0.15 ton/m2, respectively. Also, Figure 2B illustrates the base SPSW model, i.e., the one without opening. The height and bay width of this model are assumed 4 and 6 m, respectively. Frames other than those containing SPSWs are gravity frames and therefore, SPSWs carry the entire lateral load. Also, gravity loads are not carried by beams of the considered SPSW and transmitted by transverse beams to beam-column connections. The structure is designed for a very high seismic zone with a site-specific earthquake acceleration of 0.35 g according to the Iranian Seismic Code (BHRC, 2014).






The SPSW is designed based on the recommendations of AISC Seismic Provisions (ANSI/AISC 341-16, 2016) and AISC Design Guide 20 (AISC, 2007), which presents a capacity design method for SPSWs with solid infill plates. The section for boundary beams is selected to carry the forces due to the yielding of the infill high-strength 2507 stainless steel plate, and the section for the columns is selected to carry the forces developed in the yielded infill plate and the plastic hinges at the ends of the top beam. Also, in order to ensure inelastic beam action at the anticipated points and to reduce the bending moment demand to columns, the beam-column connection details include reduced beam sections (RBS) at both ends. Thus, the “weak beam-strong column” criterion is guaranteed. The RBS dimensions are designed in accordance with AISC 358-16 (ANSI/AISC 358-16, 2016). The sections of beams, columns, and stiffeners for the SPSWs are W360X287, W310X202, and W310X28.3, respectively.






The door-type opening is conventionally introduced at the mid-span, as shown in Figure 3. The horizontal and vertical dimensions of the door opening are assumed 1.5 and 2.5 m, respectively. Figure 3 shows the six considered arrangements of stiffeners around the opening. Arrangement 1 is the recommendation of AISC Design Guide 20 (AISC, 2007). In the other arrangements, all different combinations of the proposed vertical and horizontal stiffeners are considered. It will be shown later in the paper that the code arrangement is the most conservative one amongst all, and will only be optimal if the designer is extremely risk averse. For this reason, arrangements that are more conservative than this arrangement, i.e., the ones which have more stiffeners, are not included in the analysis.