Abstract
The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.
Highlights
The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences
After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion
The study area is divided by spatial variables into finite elements of the first order
Summary
Для решения задачи о моделировании нестационарных упругих волн в деформируемых областях сложной формы рассмотрим некоторое тело Γ в прямоугольной декартовой системе координат XOY, которому в начальный момент времени t = 0 сообщается механическое воздействие. Предположим, что тело Γ изготовлено из однородного изотропного материала, подчиняющегося упругому закону. Точные уравнения двумерной (плоское напряженное состояние) динамической теории упругости имеют вид. Где σ x , σ у и τ ху – компоненты тензора упругих напряжений; ε x , ε у и γ ху – компоненты тензора упругих деформаций; u и v – составляющие вектора упругих перемещений вдоль осей ОХ и ОY соответственно; ρ – плотность материала; Cp =. Систему (1) в области, занимаемой телом Γ , следует интегрировать при начальных и граничных условиях. Для решения двумерной плоской динамической задачи теории упругости с начальными и граничными условиями (1) используем метод конечных элементов в перемещениях
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