Modal P-delta – simplified geometric nonlinear method by using modal and buckling analysis

Abstract

Recent works show that in structures, there is a relationship between the natural period of vibration and global second-order effects. This relationship occurs because both depend on the stiffness matrix and the mass matrix of the structure. In this work, a non-linear geometric method - modal P-delta – is proposed that takes advantage of the relationship between the dynamic parameters and the non-linear effects. The methodology establishes an association between the buckling instability modes and the structural vibration modes. An interpolation of the vibration modes without axial loading and vibration modes with critical axial loading is proposed to approximate the vibration modes and frequencies of the loaded structure. In this way, through a simple formulation, the vibration modes and the natural frequencies of the loaded structure can be used to evaluate the displacements of the structures including the non-linear effects. Several numerical examples were simulated in regular structures in the plane, such as a free-fixed column and a plane frame with two different loading configurations. The results generated with the modal P-delta method provide information about the nonlinear behavior of the pre-buckling equilibrium path. These results are different from the findings in the literature, where the relationship between dynamic parameters and non-linear effects is used as a simple indicator or amplification factors to determine non-linear effects. Furthermore, our results indicate that the modal P-delta reduces the computational time spent considerably compared to the traditional P-delta iterative method.