THE STUDY OF THE STIFFNESS COEFFICIENT OF THE SEAMDEPENDING ON THE QUANTITY OF SYMMETRICALLY LOCATED SHIFT ONNECTION-SIN AN OVAL TWO-LAYERPLATE

The paper considers a two-layer isotropic plate on pliable connections, for which the theory of cal-culation of composite plates by A.R.Rzhanitsinwas adopted. With the help of bending moments Mx(My), the numerical stiffness coeffi cient of the seam is calculated. The numerical basic studies were carried out on an oval two-layered plate in the case of a rigid and hinged support of the plate along the contour. It proved that the stiffness coeffi cient of the seam ξ depends on the main coeffi -cient after the hard oscillation frequency ω. The authors constructed the graph of the change of the natural oscillation frequency (ω) and the graph of the change of the stiffness coeffi cientof the seam (ξ) on the number of shift connections (nss./ne).


INTRODUCTION
A large number of works are devoted to the calculation of solid and composite plates [01, 02, 03, 04]. In [05-08], composite plates of square and round shapes were studied depending on the number of symmetrically and unevenly located shift connections, and the stiffness coeffi cients of the seams were determined depending on the frequency of the natural oscillations of composite plates. The authors also studied the stiffness coeffi cients of the seam for triangular composite plates [09,10]. The present paper researched the stiffness coeffi cients of seams depending on the frequency of natural oscillations of oval composite plates with a different number of evenly and symmetrically located shift connections. Let us consider a composite plate consisting of a number of layers joined together not only by the shift connections, but also by transverse connections, which prevent the removal or rapprochement of the layers. This approach is based on A.R. Rzhanitsin'stheory of composite plates. For each layer, a hypothesis of direct normals is valid. The number of seams or gaps between plates is n, and the total number of layers is n + 1 (Figure 1). There is a linear dependence between the differences of longitudinal displacements and tangential stresses in the connections of the i-th seam shift: where, c i is the distance between the median planes of the layers lying on both sides of the i-th seam; ξ i is the stiffness coeffi cient of the connections of the i-th seam shift.

Figure 1: Accepted designations for a composite plate
Andrey Viktorovich Turkov -The study of the stiffness coeffi cient of the seamdepending on the quantity of symmetrically located shift onnection-sin an oval two-layerplate For bending and twisting moments in the i-th seam: , one can draw equations: By transforming the right-hand side of the system and expressing lateral forces through the sum of the momentary loads in adjacent layers, we have: As a result of the transformations, we get: We express the left side of equation through the biharmonic operator from the defl ection: where D 0 is the cylindrical rigidity of a composite plate devoid of shift connections, which is deter-mined by the formula: μ isPoisson's ratio; hi is the thickness of the i-th layer. Accepting we substitute A i into equation (8): and lowering the order, we get: 3 Equation (10)  In the composite plate, T j is the total shifting force in the i-th plate, which is equal to , and N i is the longitudinal forces in the i-th layer. The bending moments M X and M Y will be considered equal to the total bending moments in the composite plate devoid of shift connections: where μ усл is Poisson's ratio of a conventional solid plate. Let us consider a specifi c case of a composite plate of two layers. For this, we set n = 1 for equations (12) and (13). We have the system of equations: where D 0 is actual cylindrical stiffness equal to where is the modulus of elasticity of the layers in the composition of the composite plate, while the indices of the seams are omitted, since the seam is one. 23)

THE STUDY OF THE SEAM STIFFNESS COEFFICIENT
In the framework of the present study, the research task of studying the stiffness coeffi cient of the seam ξ depending on the the boundary conditions of the layers and the ratio of the number of elements with shift connections (n ss ) to the number of fi nite elements of one layer (n e = 288) symmetrically located along the area of the plate was being solved. In turn, we introduce symmetrically shift connections into the net of fi nite elements, according to schemes a-e in Figure 3, while the rigidity of the shift connections in all cases is constant and equal to EA SS = 10 kN. The calculation was performed in the SCAD software package by the fi nite element method [11]. As a result of the calculation, the fundamental frequency of the transverse oscillations and the value of the distributed moments were determined.
The following schemes of composite plates with symmetrically located shift connections were considered ( Figure 2). The studies of oval composite two-layer plates were carried out using the method of fi nite elements; for this purpose, both layers were conventionally divided into sectors (Figure 2), and this division of the dependence allows approximating the original plate rather precisely. Two conditions were considered for the plate support: hinged and rigid pinching. The supports along the contour of the plate were located at the nodes of the fi nite elements of the layers, while their boundary conditions were the same. The composite material on a wood base (a chipboard plate) with a thickness δ = 10 mm is used as layers. All the characteristics were taken from the product passport: the thickness is δ = 10 mm, the average density is ρ = 740 kg / m 3 , the modulus of elasticity at bending is E = 260,000 MPa. To determine the dynamic calculation of the mass in the nodes, the results were collected in ac-cordance with the construction with a volumetric weight and the loading node area. In the case of a static calculation, a uniformly distributed load with intensity of 1kN/ m 2 was applied to the top layer. The distance between the layers was assumed to be equal to the distance between the gravity centers of layers. The calculation was performed in the SCAD software package by the fi nite ele-ment method [11].
The results of numerical studies of composite plates, which are pivotally supported and rigidly clamped along the contour, are shown in Table 1. By the results of the calculation, the bending moments in the layers of the composite plate and the natural frequencies of the structure oscillations were determined, depending on the ratio nss./ne. According to the data in Table 1, graphs of the oscillation frequency change depending on the stiffness coeffi cient of the seam (Figure 4), as well as changes in oscillation frequencies and stiffness coeffi cient of the seam, depending on the ratio of the number of shift connections to the number of fi nal elements in the layer n ss ./n e (Figures 5 and 6) were constructed.