Theoretical formulation for calculating elastic lateral stiffness in a simple steel frame equipped with elliptic brace

Elliptic-braced simple resisting frame as a new lateral bracing system installed in the middle bay of frame in building facades has been recently introduced. This system not only creates a problem for opening space from the architectural viewpoint but also improves the structural behavior. Despite the researches on the seismic performance of lateral bracing systems, there are few studies performed on the effect of the stiffness parameters on the elastic story drift and calculation of period in simple braced steel frames. To overcome this shortcoming, in this paper, for the first time, an analytical solution is presented for calculating elastic lateral stiffness in a simple steel frame equipped with elliptic brace subjected to lateral load. In addition, for the first time, in this study, a precise formulation has been developed to evaluate the elastic stiffness variation in a steel frame equipped with a two-dimensional single-story single-span elliptic brace using strain energy and Castigliano's theorem. Thus, all the effective factors, including axial and shear loads as well as bending moments of elliptic brace could be considered. At the end of the analysis, the lateral stiffness can be calculated by an improved and innovative relation through the energy method based on the geometrical properties of the employed sections and specification of the used material. Also, an equivalent element of an elliptic brace was presented for the ease of modeling and use in linear designs. Application of the proposed relation have been verified through a variety of examples in OpenSees software. Based on the results, the error percentage between the elastic stiffness derived from the developed equations and the numerical analyses of finite element models was very low and negligible.