Bilinear Load–Displacement Curve of Semi-supported Steel Shear Walls

Abstract

Elastic stiffness and ultimate shear capacity are two main parameters of a structural system to obtain its ideal bilinear load–displacement. In the previous studies, the ultimate shear capacity of semi-supported steel shear walls (SSSW) which are a new lateral resisting system, has been determined. In this system, wall plates do not have any direct connection to the main columns of structure and they are connected to secondary columns which do not carry the gravity loads. The used thin plate in the SSSW elastically buckles at low levels of lateral loads and the wall plate stays on a fairly vast region with elastic post-buckling behavior (elastic stiffness). In this study, the Von-Karman plate equations are solved by the Galerkin method to find displacement field of the wall plate in the elastic post-buckling region as well as the maximum shear load after which the plasticity expand in the wall plate. Thus, the elastic stiffness of system is calculated. As the analytical procedure is complicated, the method is applied on 144 examples with different material and geometrical properties. Using linear regression technique, a concise formula is proposed to predict the elastic stiffness of system. The dimensions of wall plate are only the effective parameters in the suggested formula and the elastic stiffness is independent of the overturning moment, section of secondary columns and yield stress of material. Using the ultimate shear capacity and elastic stiffness, an ideal bilinear curve is obtained for the lateral load versus the horizontal displacement. The shear capacity at the end of elastic post-buckling region and out of plane displacement are acceptably validated with those of FE analysis for some examples.

Appendix: Influence of β on the Out of Plane Deflection of Secondary Column

The initial deflection function is defined with Eq. (7) in which a parameter, β is used to consider the out of plane deflection of secondary columns. Then, using the second part of Eqs. (4) and (5), Eq. (8) is obtained to determine $\beta = \beta \left( y \right)$. To find the maximum deflection of the secondary columns, the absolute maximum and minimum of $\beta$ must be obtained. Then in Eq. (8):

$$\frac{d\beta }{dy} = 0$$

(59)

As $\beta$ is dependent to Eq. (6) and it is a two criteria function, Eq. (60) is obtained which has two parts in the right side. Each part must be separately solved to achieve the positions of extremum points.

$$3\tan \frac{m\alpha \pi y}{h} + \frac{m\alpha \pi y}{h} = \left\{ {\begin{array}{*{20}l} {0;} \hfill & {0 \le \frac{y}{h} \le \frac{1}{2}} \hfill \\ {m\alpha \pi ; } \hfill & { \frac{1}{2} \le \frac{y}{h} \le 1} \hfill \\ \end{array} } \right.$$

(60)

These equations do not have a closed form solution and must be numerically solved. The solution leads to some acceptable answers such as $\lambda_{i} = \frac{{y_{i} }}{h}$ and the maximum and minimum of β are obtained as:

$$\beta_{i}^{max/min} = \frac{{\pi^{3} t^{3} r}}{{576I_{xx} \left( {1 - \nu^{2} } \right)}}\left[ {r^{2} \left( {1 + 3m^{2} } \right) + 2 - \nu } \right]\sin \left( {m\alpha \pi \lambda_{i} } \right) \times \left\{ {\begin{array}{*{20}l} {\lambda_{i}^{3} ;} \hfill & {0 \le \lambda_{i} \le \frac{1}{2}} \hfill \\ {\left( {1 - \lambda_{i} } \right)^{3} ; } \hfill & { \frac{1}{2} \le \lambda_{i} \le 1} \hfill \\ \end{array} } \right.$$

(61)

Supposing that $m = 3$, $\alpha = 1.3$ and $\nu = 0.3$, Table 7 shows the acceptable answers in the defined domains and the corresponding $\beta$ s for some models which is defined in Table 1. Table 7 shows that the absolute maximum and minimum values of $\beta$ are 1925.16 × 10⁻⁸ and − 3501.53 × 10⁻⁸ respectively. As seen in Eq. (62) which shows the deflection of wall plate center, the maximum/minimum value of $\beta$ has negligible effect on the $w\left( {\frac{b}{2},\frac{h}{2}} \right)$. It can be easily shown that for the other points, the values of $\beta$ are ignorable and out of plane deflection of secondary columns is approximately zero.

$$w\left( {\frac{b}{2},\frac{h}{2}} \right) = A\left[ { - \sin \left( {0.45\pi } \right) + \beta_{i}^{max/min} } \right]$$

(62)

Table 7 The maximum and minimum of $\beta$ in Eq. (7)