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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).’

A numerical method for evaluating the plastic limit load of modified 9Cr-1Mo steel pressure vessel structures containing volume defect at $650^{\circ }$C is proposed based on the plastic limit load concept under high temperature. Firstly, the creep analysis of the defected pressure vessel is conducted with the Liu–Murakami creep model to obtain the creep damage after a prescribed service time. Secondly, the obtained creep damage is introduced into Ramberg–Osgood model through the hardness ratio to characterize the material deterioration during the creep process. Thirdly, the plastic limit load of the defected pressure vessel under high temperature is obtained through the classic zero curvature criterion with the modified Ramberg–Osgood model. The numerical examples for the pressure vessels with different sizes of volume defects are performed, and the failure modes of pressure vessel structures at the limit state are revealed and the fitting formulae between the plastic limit load ratio and the dimensionless defect factor are established based on the numerical results. Results show that the plastic limit load and the service time of pressure vessel structures under high temperature are sensitive to the volume defect ratio and can be determined easily through the fitting formulae which are convenient for engineering applications.