Bending by explosion of a multilayered concrete beam on a visco-elastic basis

In this work, the problem of bending of a multilayered concrete beam of arbitrary cross-section by explosive loading on a visco-elastic basis is considered. It is assumed that different grades of concrete can be realized in layers in the cross-section. The property of concrete resistance to tension and compression is considered in work. It is assumed that the dynamic loading is caused by consecutive explosion of two charges over the middle of the span of beam. The distribution of bending moments and deflections of the beam at each time is determined. The time of the end of motion and the residual deflection of beam are found.


Introduction
The beginning of the research of reinforced concrete structures was laid by A. A. Gvozdev [1] who introduced a widely known model of ideal rigid-plastic material into calculation practice and carried out many calculations of the bearing capacity of various building structures on its basis. The model turned out to be quite simple and convenient in design engineering practice and was widely developed [2][3][4][5][6]. It should be noted, however, that even though in the mentioned works it is often noted that the calculations were carried out including the building structures made of reinforced concrete, the specific property of concrete and reinforced concrete structures that is associated with their significant resistance to tension and compression was not taken into account when carrying out specific calculations within the framework of that model [7][8][9].
In scientific literature, the calculation of reinforced concrete structures is often limited to the case of the simplest forms of rod cross-sections and the simplest conditions of loading and fastening. Modern technological means of creating flexible sets of hybrid laminated structures, where in cross-section different grades of concrete can be realized in layers, are not considered.
In construction practice, there are often situations when deformation is accompanied by a repulse of environment [10] and there is a problem of estimating the bearing capacity of a structure and reducing its damage level in the presence of such repulse under the influence of dynamic loads. A similar problem of dynamic loading of a multilayered pinched beam on a viscoelastic basis was considered in [11][12].
Materials included in the real construction in the field of plastic deformations can have completely different dependences. An analysis of possible general relations for the description of the stress-strain state of structures made of hybrid reinforced concrete is given in [13].

Methods
Creating of a calculation method with a search of all known dependencies is completely unreal and impractical. Therefore, we previously introduce and use further some useful concepts to develop a uniform method of calculation. First, plastically equivalent materials [14]. Suppose that considered as a part of a layered beam ( fig. 1) i-material in plastic area of deformation is described by the dependence where ** , ii   are maximum allowable strains at compression and tension respectively,

,
ii AA are experimentally determined coefficients.
Assume that the coefficients 12 , ii AA are determined from experiments performed for samples of the corresponding phase materials, for example, by the method of least squares. For this purpose, it is necessary to have real diagrams of tension-compression. As the published data of specific tests are usually not sufficiently complete, the following simplified method of calculation can be used to determine the parameters. We assume that the modulus of concrete elasticity i E under compression and tension is identical. We assume that concrete behaves like an elastic body in the segment where *i   is a limit of tensile strength.
From the ratio (1), we can obtain Thus, it is sufficient to have three traditional characteristics Substituting in expressions (7) the values of coefficients obtained in the ratios (2), (4), Substitute all layers i S in fig. 1 for plastically equivalent ideal materials. Then the expression for the force N and the bending moment M has the form   According to Kirchhoff-Lyav's hypothesis, we have in examined case Assuming 0 N  in equation (9), we obtain an expression for

Results
Consider the case when the beam is hinged. Two states rigid where 00 ( ) px is a distribution function of the blast wave, depending on the location of charge, which we take in the form [15] where m is a distributed mass of the beam, w is a deflection, s q is a distributed weight of the beam, 12 , kk are coefficients of viscous and elastic resistance of the basis.
For rigid segment the curvature , then from the condition of fixing the left end of the beam and condition at the border of two areas get expressions for the deflection and the moment in section should be fair everywhere the site In plastic area, the moment has reached its limit value then for the specified section we obtain the differential equation for determining the deflection in plastic region where 1 s q f m  . (21) Equation (20) is a second-order linear inhomogeneous differential equation with constant coefficients.
where the function 01 () fx has the form (13). The first limit load can be found from the expression 1 () xt found in (30) by putting  . From equation (26) we can find the time of stop of the structure motion