Thermal Stress Analysis of Free-Free Trusses under Temperature Change Using a Self-Regularized Approach

In this study, a self-regularized approach was employed to determine the thermal stress in a free-free truss, which results in a singular stiffness matrix. Physically, the stiffness and flexibility matrices of structure with proper constraints are mutually inverse to each other. However, they are not invertible for a free-free truss from the mathematical point of view if the generalized inverse is not considered. To deal with the problem, the self-regularized approach is utilized to find the generalized flexibility matrix, which are the submatrices of the inverse of the bordered matrix for the singular stiffness matrix. The validity of this approach was confirmed through different methods such as the generalized inverse and Moore–Penrose inverse. Two analytical examples of continuous system, a rod and a beam, were given to demonstrate that the obtained inverse satisfy the Moore–Penrose inverse. Numerically speaking, two illustrative examples of a discrete system, a three-bar simple truss and a six-bar complex truss subject to uniform and nonuniform temperature change were provided. Three-bar simple truss yielded zero thermal stress under all conditions, whereas the six-bar complex truss produced thermal stress under nonuniform temperature change. The study also employed ABAQUS software for numerical analysis to validate previous results.