Elastic Stiffness of Semi-supported Steel Shear Walls

Semi-supported steel shear walls (SSSW) are a new lateral resisting system whose plates do not have any direct connection to the main columns of structure. Instead, they are connected to secondary columns which do not carry the gravity loads. The applied lateral loads may create overturning moment on the middle storeys. The ultimate shear capacity of the SSSWs in presence of the overturning moment has been reasonably determined with an analytical procedure. It was finalized with some applicable interaction curves between the ultimate shear capacity and the overturning moment which can be used for analysis and design of this system. In addition, some experimental studies have been conducted to find an insight for the cyclic behavior of this system. As the elastic buckling of wall plate always occurs at the low levels of lateral loads, the system stays in a relatively large region of elastic post-buckling. In this region, the geometrical nonlinearity with linear material behavior appear in the wall plate. Thus, the storey shear force has a linear variation versus the lateral displacement until the first point of wall plate is yielded. Perhaps solution of the Von-karman plate equations is the best approach to find an analytical vision for the elastic stiffness of the SSSWs. These equations are described with two coupled nonlinear fourth order differential equations. The mentioned equations have been widely solved for many plates which are under combinations of different in-plane and out of plane loads and various boundary conditions and imperfections. In this study, the Galerkin method was employed in a semi analytical procedure to solve the Von-karman plate equations for the wall plate of SSSW system in a middle storey. This solution leads to achieve the displacement field of the SSSWs at the different levels of lateral loads until the first point of the wall plate is yielded. Thus, the linear variations of the in-plane displacement versus the lateral load will be obtained. Since the ultimate capacity has been previously measured, then an ideal elasto-plastic curve can be obtained for this system. The wall plate is supposed as a thin plate whose parallel edges have two different boundary conditions: two simply supported and two stiffened free edges where the wall plate is connected to the storeys beams and the secondary columns respectively. A sine monomial is considered as the deflection function which is satisfied the boundary conditions. Then, an algorithm is analytically developed to find the out of plane deflection of plate and the two-dimensional elasticity is used to determine the in-plane displacement of plate. The obtained results are compared with those of FE analysis and the suggested algorithm can be programmed in usual computers. The results show that some parameters such as the wall plate dimensions, the geometric properties of secondary columns (i.e. cross sectional area, moments of inertia), the storey shear force and yield stress of wall plate effect on the end point of elastic post-buckling. But, the slope of this region is independent from the variation of overturning moment and section of secondary columns.