To the solution of the problem of longitudinal vibrations of multi-storey buildings on the basis of the plate model

. The article is devoted to the development of a continual spatial plate model of a multi-story building, developed in the framework of the bimoment theory of thick plates. A technique for dynamic spatial calculation for the seismic resistance of buildings under longitudinal seismic impacts is proposed. Formulas are given for determining the reduced moduli of elasticity. Numerical results of eigenfrequencies and displacements are obtained.


Introduction
Among the numerous objects of study in the mechanics of a deformable rigid body, a special place is occupied by multi-story buildings and structures. The development of dynamic models of buildings and structures, and the spatial nature of their deformation are complex topical problems of mechanics. To date, a universal model of a multi-story building has not been yet developed. This is due to the complex structure, diversity, and multiplicity of elements of the building. There are many articles and monographs devoted to the development of the theory of seismic resistance of a building and methods for calculating buildings and structures for seismic effects, taking into account various important factors. Multi-story buildings erected in seismic areas must meet the requirements of seismic resistance.
One of the important tasks of the modern theory of seismic resistance of structures is the development of calculation models of buildings that adequately describe their vibrations during earthquakes.
Buildings are complex objects of study in structural mechanics. To date, there are no universal methods for the dynamic calculation of their stress-strain state (SSS), due to the large number of elements and the complex structure of high-rise buildings. The theory of seismic resistance of buildings and structures is developing as one of the topical areas in structural mechanics. There are numerous studies devoted to the development of the theory determining the stress-strain state of earthworks, in particular earth dams. Using the finite difference method, calculation formulas and an algorithm for solving problems were developed.
In [10], the behavior and stress-strain state of structures and soils were studied, taking into account the nonlinear deformation of soil surrounding the structure. The existence of a near-contact soil layer which can play the role of seismic protection for structures, was shown.
There are numerous studies devoted to the development of the theory of seismic stability. Various methods were developed for calculating buildings and structures for seismic effects, taking into account such important factors as seismic load, soil conditions of the area and design features of building structures. These studies include the publications of the authors of this article [12][13][14].
Articles [15][16][17][18] are devoted to the dynamic calculation of the box-shaped structure of buildings for seismic resistance, taking into account the spatial work of box-shaped elements under the action of dynamic impact. A mathematical model and a numericalanalytical method for solving the problem of dynamics by the method of finite differences and expanding the solution by the modes of natural vibrations in the spatial statement of elements of box-shaped structures under kinematic action were developed. The forced oscillations of box structures under harmonic influences applied to the base of the structure were studied. The areas where the highest values of shear forces and bending moments occur under harmonic influences were determined.
Article [19] is devoted to solving the problem of transverse vibrations of a multi-story building in the framework of a spatial model of multi-story buildings under seismic action using an explicit scheme of the finite difference method. As a dynamic model of a multistory building, a continuum model in the form of an orthotropic plate is proposed, the theory of which is developed in the framework of the three-dimensional theory of elasticity and takes into account not only traditional forces and moments, but also bimoments [20]. This paper proposes a spatial continuum lamellar dynamic model, methods and programs for calculating multi-story buildings for seismic resistance, which make it possible to determine dangerous sections and butt joints of its elements at different intensities of seismic loads. Recommendations for the application of the bimoment theory of flexural and longitudinal vibrations of plate models of buildings were developed.
A multi-story building is modeled as a continuous thick plate. Seismic vibrations of a building are modeled by the movement of a thick anisotropic cantilevered plate, the deformation of which is described on the basis of the bimoment theory of thick plates. This model is considered the most suitable model for a multi-story building to conduct a dynamic spatial analysis for seismic resistance of buildings under seismic impacts.
Definitions of the reduced density and modulus of elasticity of the plate model are given in [19]; the reduced density of the building is determined by the following formula: Here 1 V is the volume of the plate forming one floor of the building. 0 V is the volume of one floor of the building.
Taking into account the geometric parameters of the building under consideration, we obtain the following formulas to calculate these volumes: h is the thickness of internal walls; пер h is the floor thickness.
In the general case, the reduced elastic characteristics and building density are determined by the following formulas: . , , ,  Depending on the dimensions of the plates, rooms and the building itself, the above areas are determined using the methodology presented in [19] in the following form: Here пер G is the shear modulus of the building floor; 2 G is the shear module of internal walls; ) 2 ( b E is the modulus of elasticity of internal walls; пер E is the modulus of elasticity of the floor plate. When determining the reduced moduli of elasticity and shear of the external walls, taking into account window openings, we apply the technique given in [21] in the form of approximate formulas: Formulas (1) -(7) determine the reduced moduli of elasticity of the discrete part of the plate model of the building. According to these formulas, the reduced moduli of elasticity are 8-30 times less than the elastic modulus of the panels, and the reduced density of the plate model of the discrete part of the building is 7-20 times less than the density of the panel material. Such a discrepancy between the modules is explained by the presence of a large number of voids in the cellular structure of the building.

Formulation of the problem
Longitudinal oscillations of a multi-story building within the framework of a plate model of a multi-story building are considered in the Cartesian coordinate system The origin of coordinates is located in the lower left corner of the middle surface of the continual plate model of a multi-story building. Let us direct the OX1 and OX2 axes along the length and height, and the OZ-axis -along the thickness (width of the building) of the plate model.
The problem of longitudinal vibrations of a multi-story bimoment theory of plate structures consists of two equations for longitudinal and shear forces and four additionally constructed bimoment equations for nine unknown kinematic functions: The system of equations of motion of the plate model of the building relative to the longitudinal and shear forces generated under longitudinal oscillations of the plate model of the building are built in the following form: It should be noted that the system of two equations (15) Two more equations of plate motion with respect to the intensity of transverse bimoments, which are absent in the conventional theory of plates, are constructed in the following form: Based on Hooke's law and expressions (18), we obtain the following expressions for bimoments  (19) At the base of the plate model of a multi-story building, the boundary conditions for flexural-shear vibrations have the following form: It is assumed that the seismic ground motion occurs in the direction of the OZ-axis (width or thickness of the building).
Based on the consideration, the external seismic impact is given as the acceleration of

Solution method
For the numerical solution to the problem posed, the method of finite differences was applied. To approximate the derivatives of displacements with respect to spatial coordinates, we use the formulas of central difference schemes. In this case,

Analysis of numerical results
Calculations were made for nine-and twelve-story buildings at seven-, eight-and ninepoint earthquakes, which are specified through the corresponding seismicity coefficients kс.
At that, internal and external walls, ceilings and floors are considered to consist of reinforced concrete. To obtain numerical results, the following initial data were used for the structures of the considered plate model of a multi-story building.
We consider that the external walls are made of reinforced concrete with elastic The reduced elastic characteristics of the building are determined by the following formulas: . Figure 2 shows graphs of changes in the values of longitudinal horizontal displacements u1 at the edges of a twelve-story building in time t.
From the graph ( Figure 2) it can be seen that in the middle of a twelve-story building, the maximum longitudinal displacement is.   Table 1).
The values of the natural frequency of the longitudinal vibrations of a twelve-story building are calculated depending on two values of the width of the building H=11 m and H=13m, which are p1=4.863 Hz and p1=5.178 Hz, and the periods of the fundamental tone of vibration are T1=1/p1=0.205 s and T1=1/p1=0.193 s ( Table 1).
The value of the natural frequency of the longitudinal vibrations of a sixteen-story building, depending on two values of the width of the building H=11 m and H=13m, is p1=3.086 Hz, and the period of the fundamental tone of vibrations is T1=1/p1=0.323 s ( Table 1).

1.
A spatial continuum plate model of longitudinal oscillations of multi-story buildings, developed in the framework of the bimoment theory of thick plates, was proposed. Formulas were given for determining the elastic characteristics of a plate model of multi-story buildings, taking into account design features. 2. A technique, algorithm and program for the numerical calculation of the displacements of a multi-story building within the framework of a plate model using an explicit scheme of the finite difference method were developed. 3. The numerical results of the maximum values of displacements at the upper level of nine-and twelve-story buildings were obtained, for which the first frequencies and periods of natural oscillations were determined. The laws of changes of displacements in time were determined in the form of graphs in the state of beating and in resonant mode.