Vibration of Engineering Structures under Seismic Forcing and Related Inverse Problems: SH-Waves

The paper is concerned with oscillations of elastic structures subjected to seismic vibration, in frames of anti-plane problem of dynamic elasticity. For two models of the structure we formulate the direct and inverse problems. The direct problem is to calculate the amplitude of the vibration if a known incident seismic wave falls to the free boundary surface of the ground. The inverse problem is to optimize a position and a size of a certain enforcing laminate, so that to minimize the amplitude of the vibration. For this aim we first model the construction by a vertical rectangular elastic body. A simpler model is considered as an elastic beam clamped to the foundation. The first problem both for direct and inverse problems is reduced to an integral equation with respect to the tangential stress over the basis of the construction. For the simpler construction model in the form of elastic beam the problem is reduced to the 4x4 linear algebraic system. Some examples on solution of the inverse problem are presented for the simpler beam model. Suppression of the seismic influence to engineering constructions is one of most important problems in modern civil engineering science-both from theoretical and practical points of view. Various approaches have been proposed to reduce the amplitudes of seismic vibrations, and a good survey of nowadays seismic protection methods can be found in. Briefly speaking, the problem can be formulated in the following way. Assume an elastic structure to be joined with the ground foundation which, as a first approximation, can be modelled by a homogeneous isotropic elastic half-space. Assume an incident wave to be coming from below which causes the vibration of the upper boundary of the half-space. Being joined with the structure, this boundary surface produces oscillations of the structure itself. If the mass of the structure is significant then this problem cannot be studied in an uncoupled formulation, when one aims firstly to study the dynamics of the half-space with the boundary surface free of load and then to calculate the dynamic behaviour of the structure under known oscillations of its end points contacting to the foundation.