Analytical and Numerical Approach to Detect Limit and Bifurcation Points of Mises Truss with Out-of-Plane Lateral Linear Spring

With the increasing use of high-strength steel, structures are becoming more resistant but also more slender. Therefore, the phenomenon of structural instability must be considered and prevented. It is imperative for the structural engineer to understand in detail, analytically and numerically, the detection and classification of critical points in the primary equilibrium path of slender structural systems. For this purpose, this work uses the total Lagrangian formulation to describe the kinematics of a three-dimensional biarticulated bar element. Through this formulation, the internal force vector and the tangent stiffness matrix, including the geometric nonlinearity effects, are obtained. An elastic linear constitutive model is assumed for the uniaxial stress-strain state. This constitutive model uses the natural logarithm strain and the Kirchhoff axial stress, which are energetically conjugate tensors. For developing the analytical approach, as a study case, the article presents a simple structural system with three degrees of freedom made up of two biarticulated prismatic three-dimensional (3D) bars and an out-of-plane lateral linear spring. Finally, for the numerical approach, an in-house program was developed to perform geometrical nonlinear analyses using the 3D truss element.