Continual model and dynamic calculation of buildings under seismic impacts

Continuum plate model in the form of a cantilever anisotropic plate developed in the framework of the bimoment theory of plates describing seismic oscillations of buildings is proposed in this paper as a dynamic model of a building. Formulas for the reduced moduli of elasticity, shear and density of the plate model of a building are given. Longitudinal oscillations of a building are studied using the continuum plate and box-like models of the building with Finite Element Model. Numerical results are obtained in the form of graphs, followed by their analysis.


Introduction
Studies in the theory of structures showed that in strength calculation, the study of dynamic behavior and assessment of the stress-strain state of various structures, taking into account physical and geometrical nonlinearity, viscoelastic and anisotropic properties of the material, as well as inhomogeneous structural features under the influence of external impacts are important.
The effect of earthquakes of various intensities and frequency on the seismic resistance of a wooden building was considered in [1]. Evaluation of seismic resistance was made on the basis of experimental and design-theoretical calculations. Based on the obtained experimental data, a design-theoretical assessment of the frame building was performed on simple and complex models under the effect of earthquakes of various intensities and frequency content.
Structurally-nonlinear problems with one-way connections are often encountered in calculation of various types of structures and buildings. Problems considering the friction on the contact and under dynamic action of load present certain difficulties. The study in [2] is devoted to the construction of computational models and methods for solving problems with non-ideal one-way connections under dynamic loading.
In [3][4][5], various dynamic problems were considered, devoted to the estimation and prediction of dynamic behavior of various structures, taking into account physical and geometrical nonlinearity, inhomogeneous structural features under multicomponent kinematic effect.
In [6] the solution to the problem of optimizing the projects of industrial buildings designed in seismically hazardous areas was considered. Economic efficiency was taken as an optimality criterion, depending on certain variable parameters adopted at the design stage.
In [7], the problem of evaluating the impact of wall panel fractures when estimating large-panel structures strength was solved. The forces exceeding the permissible values in structural elements were calculated The model of a box-like structure of a building is improved in [8,9] taking into account forces and moments in the contact zones of beam and plate elements interaction. The equations of motion of the box-like elements, the boundary conditions in the box base and the contact conditions between the box elements are given; the graphs of plates and beams displacements are constructed. The problem of forced oscillations of a building of spatial box type is considered in the paper; it composes of rectangular panels and interacting beams under dynamic effect set by base displacement according to a sinusoidal law. The finite difference method was used in the problem solution. Numerical results of stresses, displacements in the hazard areas of the box-like building are obtained.
The studies in [10,11] are devoted to the development of the methods for dynamic spatial calculation of a structure based on the finite difference method in the framework of the bimoment theory, which takes into account the spatial stress-strain state.
Solutions to the problem of transverse and longitudinal vibrations of buildings and structures were obtained using a plate model developed in the framework of the bimoment theory of plates [12,13].
There are numerous articles and monographs devoted to the development of the theory of seismic resistance of a building. Various methods for calculating buildings and structures for seismic actions have been developed, taking into account important factors, such as seismic loading, soil conditions of terrain and structural features of buildings [10]. Note that the analysis of the consequences of strong earthquakes has shown the shortcomings of existing methods of calculating buildings and structures for seismic resistance. One of the most common design schemes of the building is a multi-mass elastic cantilever rod. Oscillations of a spatial construction are reduced to the consideration of oscillations of a plane system consisting of several concentrated masses connected by certain rigidities. Many researchers, criticizing the cantilever design of buildings, recognize the need to move to improved calculation schemes that are more adequate to real structures.

Statement of the problem
To describe the motion of the plate model of a building, introduce a Cartesian coordinate system with variables 2 1 , x x and z . The origin is located in the lower left corner of the median surface. We direct the axes OX1 and OX2 along the length and height, and the axis OZ -along the thickness (the width of the building) of a plate model. Assume that seismic motion of soil occurs in the direction of the axis OX (the length of the building).
In [10], formulas for the reduced density and moduli of elasticity and shear of a plate model of the building are obtained. Mechanical and physical characteristics of the building are defined under the assumption that the building consists of numerous boxes (rooms) with volumes determined by the formula:   [10].
Note that the above volumes and areas are determined, depending on the dimensions of plates, rooms and the building itself, as follows: 01 = 1 , 02 = , 03 = 1 , 11 = 2 1 ℎ 1 + 1 ℎ 2 + ℎ fpl , The values of coefficients 23  13  12  33  22  11 , , , , ,       are determined for each cell (room) of the building. In general, these coefficients are variables and are the functions of two spatial coordinates, which should be determined for the building in question from multiple numerical theoretical experiments and existing experimental data.
Longitudinal, tangential forces and bimoments are determined with respect to the following nine unknown kinematic functions: Let's introduce the elastic constants of Hooke's law 33 12 11 ,..., , E E E , which determine the components of the stress tensor via strain tensor's components for threedimensional cases [13].
The equations of longitudinal oscillations of a thick plate [11][12][13], built with internal forces and bimoments within the framework of the spatial theory of elasticity, are taken as the equation of motion of a building under seismic action directed along the longitudinal direction, and are written in the following form The equations of motion for determining the displacements of external longitudinal walls, obtained by meeting the boundary conditions on the faces of the plate h z − = and h z + = using the method of displacements expansion into the Maclaurin infinite power series, are constructed in [12,13] in the form: Expressions of longitudinal and tangential forces are written in the form: 11  Where ̄= 420(̄+ 6̄− 15). The system of differential equations of motion (6) -(10) constitutes a joint system of seven equations with respect to nine unknown functions Let us write down the boundary conditions for the problem of bending and shear vibrations of buildings. Suppose that the foundation points move according to a given law ) ( 0 t u , and the lower part of the building moves horizontally with the foundation. From kinematic considerations it follows that the displacements will be written in the form: The problem is solved by the Finite Differences Method. To approximate the first derivatives, the central difference schemes are taken.

Method of the solution
The problem is solved by the finite difference method. To approximate the first derivatives, the central difference schemes is taken.
To substantiate the expressions obtained with continuous model of calculation, consider a specific problem with the use of necessary physical and mechanical characteristics of a box-like model. Calculation of the box-like models has been carried out by the FEM, the main point of which is the replacement of a real system (a multi-story box-like building) with a discrete model of rectangular plane elements (walls and floors) connected at the nodal points and satisfying the equilibrium conditions of the converging system of forces at each node.
Mechanical characteristics and geometry of the rooms panels are accepted as follows: bending bearing panels have a modulus of elasticity = 20000М а; density = 2700 So, it can be said that the considered effect is in a certain sense analogous in predominant period to the Gazli earthquake (1976). The scheme of the building in question is shown in Figure. 1.  , respectively.  in the external load-bearing wall of a two-story building.