Calculation methods for plate and beam elements of box-type structure of building

The article is devoted to the dynamic calculation of the box-type structure of buildings for seismic resistance, taking into account the spatial work of the box elements under the influence of dynamic impact. The development of spatial calculations and the study of vibrations of elements of box-type structures, considering various factors, is an urgent problem of structural mechanics. In this article, a mathematical model and a numerical-analytical method were developed to solve the problem of dynamics using the finite difference method and expand the solution in terms of natural vibration modes in the spatial setting for the elements of box-type structures under the kinematic impact. The steady-state forced vibrations of box-type structures under stationary harmonic influences applied to the structure foundation are investigated, and the areas are identified where the greatest values of shearing forces and bending moments occur under harmonic influences.


Introduction
Methods for calculating the strength of a structure and its elements are based on the theory of beams, plates and shells, which is one of the most important branches of mechanics. The achievements of the theory of bending, stability and vibrations of beams, plates and shells are widely used when solving the problem of seismic resistance of buildings and structures.
The study in [1] is devoted to static accounting for higher vibration modes in the problems of dynamics of building structures under external harmonic load. With a computational software package, the displacements of nodes and internal forces in the elements of the structures under consideration were determined.
The effect of displacements, fractures of the wall panel axes in the process of their installation on the operation of large-panel structures was considered in [2]. The analysis of design schemes was conducted with an account for different types of installation errors. Forces in structural elements exceeding the permissible ones were determined with an account for the error in the installation of parts.
During earthquakes, the degree of damage to buildings and structures depends on the characteristics of seismic impact (intensity, spectral composition, etc.). The reliable design and construction of the structures depend on the soil properties of the construction site foundation [3]- [7].
The studies in [8]- [10] are devoted to improving the box-type model of the building structure, taking into account the contact conditions between the elements of the panels and beams. Equations of motion of box-type elements and graphs of displacements of plates and beams are constructed. The article deals with the problem of forced vibrations of a building of spatial box type, which consists of rectangular panels and interacting beams under a dynamic action given by the foundation displacement according to the sinusoidal law. The method of finite differences was used to solve the problem.
The studies in [11], [12] develop the methods for dynamic spatial calculation of the structure based on the finite difference method in the framework of the theory of bimoments, taking into account the spatial stress-strain state.
[13], [14] are devoted to the numerical solution of the problem of transverse and longitudinal vibrations of buildings and structures based on a plate model developed within the framework of the bimoment theory of plates. The problems of transverse and longitudinal vibrations of buildings and structures are solved using the developed model within the bimoment theory of plates [15], [16].
The data on the mechanism of tsunami wave formation and destruction were analyzed in [17], the recommendations for tsunami-resistant construction were generalized. A solution was proposed to mitigate damage from strong earthquakes and high tsunami waves.
[18] discusses the main issues of determining the reinforcement parameters of reinforced concrete structures during their inspection. The basic ways to solve these problems are analyzed. The most reliable and accurate methods for determining the parameters of reinforcement are shown.
In [19]- [24], the dynamic characteristics and vibrations of various axisymmetric and plane structures are considered, taking into account various geometries, spatial factors and inelastic properties of materials. The solution to the problem is performed by the finite element method and by expanding the solution in terms of natural vibration modes. Various mechanical effects associated with the geometry of the structure and inelastic properties of the material are revealed.
For more precise stress and strain state definition for structures are vital put in calculation scheme the reliable constitutive relations. The physical reliability and new method for the solution of plasticity problems using Il'yushin's approximating relation are described here [25]. A model of plasticity for a transversely isotropic material with allowance for complex loading is developed based on the results of experiments [26]. Some plasticity problems regarding complex loading are obtained [27] also. This is an overview of only several publications devoted to the methods for solving dynamic problems and the study of vibrations of the elements of box-type structures and other structures under various impacts. The above review shows the incompleteness of research in this direction; therefore, this article devoted to developing calculation methods and the study of vibrations of various elements of box-type structures presents a relevant problem of structural mechanics.

Methods
a. Mathematical models of the problem A spatial box consisting of beam and plate elements is taken as a design model of a building (Fig. 1). It is assumed that the bottom part of the box is rigidly fixed and, under dynamic action, moves together with the foundation.
The fixed bottom part of the box under seismic impact in the direction of the OZ axis moves according to a given harmonic law, i.e. ) ( 0 t U [8,9]:  The displacements of the shear plate elements are indicated by The plate elements of the box are connected by beam elements. Based on this condition, it follows that the beam elements are subjected to bending and torsion. Deflections and torsion angles of the beams are indicated by where: I, II, III, IV is the number of beams. It is assumed that the floor slab (plate element 5) is also deformed. The law of motion of its points is determined following the deformation forms of the upper edges of the vertical contacting plate elements. The functions for floor slab points displacement are indicated by Further, the following designations are introduced for the plate elements of the building box: where: is the cylindrical stiffness under transverse bending.
The expressions for the longitudinal and tangential forces of plate shear elements are given in the following form: where: Bending and torsional moments of beams are: , , Where kr EI is the torsional rigidity of the beam, EJ is the bending stiffness of the beam.
The expression for the shear force of the plates bent in the zones of their connections with the beams is as follows: Longitudinal and tangential forces of plate elements in contact zones, working in shear, have the following expressions: Where bi and ci are the corresponding coordinates of beam-and-column elements. The displacement field of the beam-and-column elements is determined based on the Bernoulli hypothesis in the form The torsion of the beam-and-column element is induced by the bending moments at the edges of the plate elements 1 or 3.
b. Method for solving the problem. An analytical-numerical method is proposed for solving the problem of box-type structure vibrations, taking into account spatial strains with complete contact conditions in the zones of joints of plate and beam elements of the structure box.
Based on representation (1) The displacements of plate and beam elements are given in the following form: Consider a theoretical calculation of the building box under the dynamic impact, taking into account the spatial work of transverse and longitudinal plate elements.
Let us compose the equations of motion for each plate and beam elements of the boxtype structure of a building [8][9][10]. Plate elements are considered to be thin elastic plates obeying the Kirchhoff-Love hypothesis. Each beam is subject to bending and torsion. When constructing the equation of motion and the boundary conditions of the plates and beams, the expressions for the moments of forces and displacements (2) -(6) are used.
The system of equations for bending, torsional moments and shear forces is presented in the following form The system of equations of motion of shear plate elements is taken as: The system of equations for bending and torsional vibrations of the beams is:   (11, v) where: kr GI is the torsional rigidity of the beam.
The boundary conditions at the foundation of the building box are written as for rigid fixing. The lower part of the building moves with the foundation, and there is no torsion.
Boundary conditions (12) with (7) are rewritten as: The contact conditions in the zone of connection of beam and plate elements working in shear are written in the following form , ) , The displacements of the upper points of the beam and plate elements working in bending and shear are denoted by ). , , Based on the notation (15), the distribution law for the displacements of the floor slab points are given by the following expressions ). , The boundary conditions at the upper ends of the elements of the building box at a x  are the following contact conditions between these elements and the floor.
The contact conditions at the joints of the floor and plate bending elements have the following form . 0 , (17) where: . 0 The contact conditions at the joints of the floor and plate shear elements, relative to the contact tangential and normal stresses, are written in the following form:  , The vibration modes (8) must satisfy the equations of motion (9) -(11), boundary conditions (13), (19), and (20), and contact conditions (14), (17), and (18). The general solution to the problem of forced vibrations of bending plate elements of the box is described by a function represented as the sum of the solution to the problem of forced and natural vibrations: The solution to the problem of forced vibrations of a plate element in bending is expressed through natural vibration modes [19][20][21] , ) , are the expansion coefficients.
In the calculations, it was sufficient to restrict ourselves to the one-term approximation. The general solution to the equation of bending vibrations of the panels is taken in the following form: where: 1 p is the first eigen frequency, is the mode of forced vibrations, 1 С is a constant to be determined.
Substituting (24) into (23) and using zero initial conditions, we obtain . By virtue of this expression and taking into account (24), we obtain the general solution of the problem for bending plate elements in the following form [8][9][10]: The expression for displacement of shear plate elements has the form The kinematic functions of the beams are written as: The problem of determining the unknown coordinate functions in expressions (25) - (27) is solved by the finite difference method.
For this purpose, in this study, a method (based on the method of finite difference schemes) was developed for the dynamic calculation of the box-type structure of buildings.

Results and Discussion
In the calculations, the value of the frequency of external influence is Let us proceed to a discussion of the numerical results obtained for the bending moment on bent plate elements. Figures 3 and 4 show the graphs of the time variation of the deflections at the upper points of the plate elements working in bending, without considering the initial conditions (dashed line) and with the initial conditions (solid line).
As can be seen, the deflection values increase up to 20% (Fig. 3.) when considering the initial conditions of the problem.

Conclusions
1. A mathematical model was developed for dynamic calculations of plate and beam elements of box-type structures of buildings, taking into account the spatial nature of their work under various kinematic impacts. 2. A numerical-analytical method was developed to study the forced vibrations of elements of box-type structures of a building using the finite difference method and the expansion of the solution in terms of natural spatial modes of vibrations. 3. The steady-state forced vibrations of box-type structures under stationary harmonic kinematic effect on the foundation were investigated.