The paper is devoted to the numerical solution of the problem of transverse oscillations of a multi-storey building within the framework of a continuous plate model under seismic effects. Cantilevers anisotropic plate is proposed as a building dynamic model, the theory of which is developed in the framework of a three-dimensional dynamic theory of elasticity and takes into account not only structural forces and moments but also the bimoments. The proposed plate model of a building allows us to take into account and study all types of different spatial oscillations of the building structure under the impacts different in direction. Formulas are given for the reduced density, elastic moduli, and shear of the plate model of the building. The base acceleration, given in time by a harmonic law, is taken as a seismic impact. The problem is solved by the finite difference method. Examples are considered and numerical results are obtained. Waveform, displacements, and accelerations distribution diagram of multi-storey high-rise buildings under transverse oscillations are plotted.

The cultivated soil environment changes its structure and is deformed; therefore, the considered model of the processes of interaction of the working body with soil remains understudied. The influence of soil criteria on the working body behavior can be taken into account through its density and tensile strength. To describe the movement of the soil near the leg during finite deformations, a plastic medium model proposed by academician Kh.A. Rakhmatulin and simplified equations obtained on the basis of the hypothesis of flat sections were used. When using the model of linearly elastic and compressible plastic medium, the resistance forces of the soil medium are determined when the legs of the subsoiler, presented in the form of a circular cone, move. It has been established that the magnitude of this force substantially depends on the type of contact conditions between the body and the soil, and its greatest value is achieved in the case of continuous motion. The dependence of the resistance force on time is obtained. According to the results of graph analytical studies, it is obvious that at the initial stage, while the contact area of the circular cone with the soil is variable, the resistance force depending on time changes according to a parabolic law, and then it remains constant. In the case of the movement of the subsoiler leg at a constant speed, it was found that, depending on the coefficient of internal friction and soil traction, a zone of increased soil density can form near the working body of the cultivator, where there is a significant increase in resistance force. With an increase in the angle of internal friction, a slight decrease in the resistance force is observed. The calculations were carried out based on the methods of mechanics of a deformable solid, soil mechanics, and were performed in the Maple-8 programming environment.

In modern engineering and construction, thin-walled plates and shells of variable thickness, subjected to various static and dynamic loads, are widely used as structural elements. Advances in the technology of manufacturing thin-walled structural elements of any shape made it possible to produce structures with predetermined patterns of thickness variation. Calculations of strength, vibration and stability of such structures play an important role in design of modern apparatuses, machines and structures. The paper considers nonlinear vibrations of viscoelastic orthotropic cylindrical panels of variable thickness under periodic loads. The equation of motion for cylindrical panels is based on the Kirchhoff-Love hypothesis in a geometrically nonlinear statement. Using the Bubnov-Galerkin method, based on a polynomial approximation of deflections, the problem is reduced to the study of a system of ordinary integro-differential partial differential equations, where time is an independent variable. The solution to the resulting system is found by a numerical method based on the feature elimination in the Koltunov-Rzhanitsyn kernel used in the calculations. The behavior of a cylindrical panel with a wide range of changes in physico-mechanical and geometrical parameters is investigated.

А ННОТА Ц И Я Введение. Рассматриваются изотропные вязкоупругие цилиндрические панели переменной толщины, находящиеся под действием равномерно распределенной вибрационной нагрузки, приложенной по одной из параллельных сторон, приводящей (при определенных сочетаниях частот собственных колебаний и возмущающей силы) к параметрическому резонансу. Материалы и методы. Считается, что под воздействием указанной нагрузки цилиндрические панели допускают перемещения (в частности, прогибы), соизмеримые с их толщиной. На основе классической гипотезы Кирхгофа-Лява построена математическая модель задачи о параметрических колебаниях вязкоупругой изотропной цилиндрической панели переменной толщины в геометрически нелинейной постановке. Выведены соответствующие нелинейные уравнения колебательного движения рассматриваемых панелей (в перемещениях). Предложена методика решения рассматриваемой нелинейной задачи на основе применения метода Бубнова-Галеркина при многочленной аппроксимации перемещений (и прогиба), а также численного метода, использующего квадратурные формулы. В качестве слабо-сингулярного ядра выбрано ядро Колтунова-Ржаницына с тремя различными реологическими параметрами. Результаты. Исследованы параметрические колебания вязкоупругих цилиндрических панелей переменной толщины под воздействием внешней нагрузки. При этом осуществлялся учет влияния на области динамической неустойчивости геометрической нелинейности, вязкоупругих свойств материала, а также других физико-механических и геометрических параметров и факторов (начальных несовершенств формы, соотношений сторон, толщины, граничных условий, коэффициента возбуждения, реологических параметров). Выводы. Разработаны математическая модель и метод для оценки параметрических колебаний вязкоупругой цилиндрической панели переменной толщины с учетом геометрической нелинейности при действии периодических нагрузок. Полученные результаты хорошо согласуются с результатами и данными других авторов. Проверена сходимость метода Бубнова-Галеркина.

The problem of parametric oscillations of an isotropic viscoelastic shallow shell of variable thickness under periodic load is considered. It is believed that under the influence of specified load, the shallow shell allows displacements (in particular, deflections), commensurate with its thickness. In a geometrically nonlinear statement, taking into account the viscoelastic properties of material, a mathematical model of the problem has been developed using the classical Kirchhoff-Love hypothesis. Using the Bubnov-Galerkin method based on the polynomial approximation of the deflections, the problem is reduced to the study of the system of integro-differential equations, where time is the independent variable. The solution of the system of integrodifferential equations is determined by the proposed numerical method. Based on this method, a numerical solution algorithm is described. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. At the same time, the effect of geometric nonlinearity, viscoelastic properties of material, as well as other physicomechanical and geometric parameters and factors (rheological parameters, thickness, initial shape imperfections, aspect ratios, boundary conditions, excitation coefficient) on the area of dynamic instability is taken into account. The results obtained in this study are in good agreement with the results and data obtained by other authors.