Analysis of non-stationary reaction of elastic shell to impulse load

Abstract

An effective numerical method for studying non-stationary vibrations of thin elastic shells is proposed. The method is based on the finite element model of a thin elastic inhomogeneous shell and the reduced model created on its basis for the dynamics problems.The finite-element shell model is based on the relations of the three-dimensional theory of thermo-elasticity and is developed with the use a tensor calculus apparatus, a geometrically nonlinear formulation of the problem in increments and the application of the moment finite-element scheme. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness which is an efficient approach. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing (of the shell’s sections without stepwise-variable thickness), it should be possible to vary the thickness of the lateral edges of the finite element; the lateral edges of the neighboring finite elements should be in continuous contact; and it should be possible to model sharp bends and the multilayer structure of the shell. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The first static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The next is the nonclassical kinematic hypothesis of deformed straight line: though stretched or shortened during deformation, a straight segment along the thickness remains straight. This segment is not necessarily normal to the mid-surface of the shell.The method for studying non-stationary vibrations of the shells under the action of short-term loads is based on the application of reduced models. The use of the basic nodes method allowed us to develop a simple and effective algorithm for solving this problem. We have transformed a system of coupled differential equations describing the motion of a shell to independent ones. The solution of obtained Cauchy problems is easily found by the well-developed Runge-Kutta numerical method.The possibility of applying the developed method to assess the effect of short-term load on the behavior of a thin-walled structure is shown on the test problems. Convergence of solutions is investigated and a comparison with theoretical data and results obtained with the help of the SCAD software is made.