Hybrid Finite Element Method in Nonlinear Dynamic Analysis of Trusses

This paper presents a dynamic analysis of trusses with an initial length imperfection of the elements, considering geometrical nonlinearity. In the nonlinear analysis of trusses, the hybrid finite-element formulation considers the initial length imperfection of the elements as a dependent boundary constraint in the master equation of stiffness. Moreover, it was incorporated into the establishment of a modified system of equations. To overcome the mathematical complexity of dealing with initial length imperfections, this study proposes a novel approach for solving nonlinear dynamic problems based on a hybrid finite-element formulation. In this study, the unknowns of the dynamic equilibrium equations were displacements and forces, which were obtained using virtual work. The hybrid matrix of elements of the truss is established based on the hybrid variation formulation with length imperfections of elements, considering large displacements. The authors applied Newmark integration and Newton-Raphson iteration methods to solve the dynamic equations with geometrical nonlinearity. An incremental iterative algorithm and calculation programming routine were developed to illustrate the dynamic responses of trusses with initial-length imperfections. The results verified the accuracy and effectiveness of the proposed approach. The uniqueness of the proposed method is that the length imperfection of the truss element is included in the stiffness matrix and is considered a parameter that affects the dynamic response of the system. This helps to solve the problem of the dynamic response of trusses with length imperfections becoming simpler. The numerical results show that the effect of length imperfection on the dynamic response of the trusses is significant, particularly on the dynamic limit load. In addition, to completely evaluate the behavior of the trusses, this study also developed formulas and analyses to consider the inelastic and local buckling of the truss structures, named 'Inelastic post-buckling analysis (IPB).'