Study of longitudinal oscillations of a five-storey building on the basis of plate continuum model

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
Structures such as water retaining dams, dykes, water reservoirs, etc., built and operated in seismically active regions of the Republic of Uzbekistan, are subjected to loads of both static (gravitational forces, natural external loads, etc.) and dynamic (seismic) nature. In dynamic calculation of this type of structures according to the "standard procedure, designers are limited by the possibility of obtaining only the most approximate reliability estimates.The ways to the main reserves of increasing the efficiency of the structureconstructions are unavailable, since these reserves are found only when considering the wave nature of dynamic loads (seismic) and are connected with the necessity of assuming irreversible strains in a structure ". To determine the reserve capabilities of water retaining structures in calculations, various linear and nonlinear models are used in a planeor spatial form [1][2][3][4][5][6][7].
This article proposes a method for calculating the structures for seismic resistance on the basis of a continuum plate model developed in the framework of the bimoment theory [8][9][10][11][12][13][14][15][16], taking into account the spatial stress-strain state.If to consider the law of nonlinearity of displacements distribution in the cross-sections of the plate, then in addition to tensile and shear forces, bending and torsional moments, there appear the additional force factors, called the bimoments.Among numerous objects of study in mechanics of a deformed rigid body, multi-storey buildings and structures occupy a special place. Development of dynamic models of buildings and structures with consideration of strain of spatial nature is one of the most difficult and urgent problems in mechanics.
A universal model of the building has not been developed yet. This is due to the complex structure, diversity and numerous elements of the building, especially tall largesized buildings. Therefore, one of the important tasks of modern mechanics and seismic stability of structures is the development of universal models of a building that adequately describe its spatial behavior.
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 [1][2][3][4][5][6][7]. 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. This need arises due to the fact that the existing methodology for calculating and designing buildings does not solve the problems of the optimal ratio of the sizes of a building box, the diaphragm and floor elements rigidity and so on.
2 Method In this paper, a technique for calculating the continuum plate model of buildings, developed in [8], is proposed. The motion of a building under seismic effect is represented as longitudinal and transverse oscillations of some thick cantilever plate, strained as a threedimensional body made of relatively soft, low-strength material.
In [8], formulas for the reduced density and module 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: Then, to determine the mass of the boxes, the following formula is used here 0 V -is the sum of the volumes of bearing and room partitions and floors (4) Remaining reduced elastic characteristics and densities of the building are determined by the formulas: . , , , for each building cell Here 1 ais a distance between two transverse room partitions plates; * s -is a sum of the voids areas in the cross-section of the floor.
Note that the above volumes and areas are determined, depending on the dimensions of plates, rooms and the building itself, as follows: To represent the values of dynamic characteristics of the plate model of a building, the following formula obtained in [8] is used; it connects the first natural frequency of the plate with mechanical characteristics of material: 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. The moduli of elasticity of continuum plate model of a building are determined by formulas (11).
The equations of longitudinal oscillations of a thick plate [9][10][11], 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 [9][10][11][12] in the form: Longitudinal, tangential forces and bimoments are determined with respect to the following nine unknown kinematic functions: Expressions of longitudinal and tangential bimoments have the form: Intensities of transverse and normal bimoments The system of differential equations of motion (12)  ) ( 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: From kinematic conditions it follows that in the foundation of the building the following boundary conditions must be fulfilled: On the free side faces of the building the conditions of zero force factors are fulfilled:

Results
Mechanical characteristics and geometry of the rooms of panels are accepted as follows: When comparing results based on plotted graphs for the box-like model with FEM and the plate continuum model, it can be seen that the values of displacements do not differ significantly.
The values of the generalized displacements obtained from the calculations of five-story buildings for seismic effects at different points of floor differ within the range of 10%.

Conclusion
In conclusion, it should be noted that on the basis of the bimoment theory of thick plates a dynamic plate-like model of the building has been developed, reflecting its spatial strains. Using the geometry of the building and its rooms, the reduced density, moduli of elasticity and shear of the plate model have been determined. The plate model of the building due to the choice of coefficients 23  13  12  33  22 , , , ,      , adequately reflects the laws of displacements of the points of the building. So, the plate model is acceptable for determining the displacement fields of a tall building during an earthquake and the dynamic characteristics of a building.