Longitudinal and transverse bending by a cylindrical shape of the sandwich plate stiffened in the end sections by rigid bodies

We study the problems of deformation mechanics of sandwich constructions with taking into account the interaction with the contour reinforcing rods. To derive the basic equations of equilibrium, static boundary conditions for the shell and reinforcing rods, as well as conditions of the kinematic conjugation the carrier layers with a core, the carrier layers and a core with reinforcing rods we use a generalized variational Lagrange principle. We reduce the boundary value problem on the to the integral-algebraic system of Volterra equations of the second kind. To approximate the obtained integral equations of Volterra type a collocation method with Gaussian nodes and a method for constructing the integrating matrices are proposed. For the numerical realization of the proposed methods we have developed a software package. Numerical calculations were performed. Analyze the results of numerical experiments is carried out.


Introduction. Statement of the problem
Sandwich plates and shells with external (carrier) layers of rigid materials and cores relating to a transversely soft class (cores of cellular or folded structure, reinforced with polystyrene etc.) are one of the most popular structural elements for various purposes. The external layers of these elements in the contour are always reinforcing by the rods to ensure the transfer of the load on the carrier layers of interaction with other construction elements ( Figure 1).

a)
      1 11 13 11 0, 0, 0, 0  To derive the basic equations of equilibrium, static boundary conditions for the shell and reinforcing rods, as well as conditions of the kinematic conjugation the carrier layers with a core, the carrier layers and a core with reinforcing rods, as in [14], we use the previously proposed a generalized variational Lagrange principle [16]. With regard to the sandwich plate experiencing a cylindrical bending, derived in this way the equilibrium equations will have the form (hereinafter 1,2 k  , is the layer number,     12 1, In these equations, introduced in consideration of the tangential forces    )  1  11  11  11  11  2 ,, are the tension and compression stiffness and bending stiffness of k -th layer having a thickness , , , , where     11 13 , kk QQ ,    are the surface contact stresses that are formed at the points of coupling the reinforcing rod with a core.
We assume that the loading of plate forms in it the stress-strain state, which is symmetrical with respect to the cross section  0, 11  11  11  2  13  13  1  12   2 0, where , z TT   are the components of the external loop load нагрузки Emerging in the reinforcing rods the displacements , UW and rotation  must satisfy the kinematic conditions of the coupling the rods with carrier layers to which it is necessary to add the kinematic conditions 11th International Conference on "Mesh methods for boundary-value problems and applications" IOP Publishing in [8] a numerical method based on the method of summation identities is proposed. In this case, however, for the boundary value problem in the form of five differential equations (1) -(3) with the boundary conditions (5), (6) and ten algebraic equations (7) - (13) it is required to determine the vector function of unknowns

Reduction of the boundary value problem to the integro-algebraic form
We reduce the formulated boundary value problem (1) -(13) on the definition of the function 2 () Xx to the integral-algebraic system of Volterra equations of the second kind by using additional relationships to determine the unknown integration constants. Note that, in the original differential problem includes derivatives of order 2n of the unknown function, while the integral equation will contain only n -th derivatives. This reduction is carried out by integrating the equations (1)-(3), satisfying the conditions (5), (6) and using the following equations . As a result, relatively the vector function , we obtain the following system of integro-algebraic equations:  (18) to which must be added the algebraic equilibrium equations (7)-(9) for the rod. If the point of force application T   has a displacement A W in the direction of the axis z equal to zero, and the displacement in the direction of the axis  equal is A U , then by virtue of (1) the coupling conditions (10) -(13) take the form Thus, to determine the introduced into consideration sixteen unknowns 5 11 X H R   the system of sixteen resolution integro-algebraic equations (14) - (21) and the rod of equilibrium equations (7), (9) are composed, where 2 (0, ),

H L a 
and a is the half-length of the plate.

Approximation of integral equations by the collocation method with Gaussian nodes
To approximate the obtained integral equations of Volterra type in [17] a collocation method with Gaussian nodes and a method for constructing the integrating matrices are proposed. We introduce the integral operators  JJ. Let us consider the finitedimensional operators (1) (2) (1) (2) (1) 5 11 is a non-linear operator; in general, the right side is   Thus for the functions included in the system of differential equations (1)-(3) with the highest derivative of order 2n , the finite-dimensional scheme (22) is constructed relatively the n -th derivative of the solution of a boundary value problem. After solving the finite-dimensional problem (22) solution of the original boundary value problem is reduced by numerical integration using the previously obtained integrating matrices.

Iterative method and numerical experiments
To solve the projection scheme (22), we use the following two-layer iterative process with the lowering of non-linearity in the lower layer [8,[18][19][20][21][22][23][24] with a preconditioner, which is a linear part of the operator of the difference scheme: where (0) X is the given initial approximation, 0  (Fig. 1) (1) w (2) Figure 4 shows the axial displacements function of points of the medial surfaces of carrier layers. Note that at the lower layer they are substantially equal to zero, and on the upper layer vary along the along the length of the plate almost linearly. From Figure 5 it follows that the plate is in the moment state due to the action on the plate eccentrically compressive load, wherein the torsional load formed in the first layer is practically zero, and the force (2)

11
T along the length of the plate is not changed substantially. Figure 6 shows the variation of the tangential stresses in the core along the length of the plate. It is seen that by virtue of the acting of the eccentrically compressive force on the rod they are not equal to zero over the entire length of the plate (at the connection points of a core with the reinforcing rod, they should take the zero value in the absence of the adhesive layer) and reaches the maximum value in the neighborhood of the end cross section. Figures 7 and 8 show the functions of the generalized shear forces and shear forces in the carrier layers. By the nature of these curves we can be judged on the inclusion in the work the core and its contribution to the perception of the transverse tangential stresses.