Nonlinear analysis of flexible plates lying on elastic foundation

This article describes numerical procedures for analysis of flexible rectangular plates lying on elastic foundation. Computing models are based on the theory of plates with account of transverse shear deformations. The finite difference energy method of discretization is used for reducing the initial continuum problem to finite dimensional problem. Solution procedures for nonlinear problem are based on NewtonRaphson method. This theory of plates and numerical methods have been used for investigation of nonlinear behavior of flexible plates on elastic foundation with different properties.


Introduction
The design and analysis of flexible plates lying on elastic foundation constitute a complex and important domain of structural mechanics. In the analysis of plates on elastic foundation Winkler model with combination of Kirchhoff model is usually used [1][2][3][4][5][6][7][8][9]. More affective numerical algorithm can be created on the theory of flexible plates with account of transverse shear deformations [10][11][12]. This model allows to analysis an average thickness plates and plates with a low shear stiffness. In addition, the moments and transverse forces are expressed only in terms of the first derivatives of the displacements while the energy functional of the system contains no derivatives higher than the first order as in the theory of elasticity.
The main aim of this work was to develop an efficient numerical algorithm for the analysis of flexible plates on elastic foundation with account of transverse shear deformations by using a finite difference energy method and the continuation method, and investigation of their strain state.

Methods
A rectangular plate is considered in the Cartesian coordinate system (Fig.1 ;  ;  2   1  ;  2   1   2  23  1  13  1  2  12  2  22  1  11   2  1  12  2  2  22  2  1  is associated with the components of strains vector ε known relation: where D is an elasticity matrix. Lagrange functional of the theory of flexible plates on elastic foundations based on geometry (1) and physical (2) dependences is constructed. The expression for the functional is: ³³ ³³ : : : where k z is a compression ratio; , .
In this case, Euler equation being the equation of equilibrium regarding the function w in linear formulation has the form [13]: where Emodulus of elasticity of the material of the plate; ha thickness; ν -Poisson's For discretization of the problem we use the finite difference energy method [10,15,16]. In this method a grid is applied on the region Ω of the changing of independent variables. The desired function u, delivers the stationary value to the functional (3), approximately determined by its values at the nodes, and the derivatives of u are replaced by finite differences. The first order derivatives of the grid functions and values of the function at the grid cell are calculated by the formulas: where i, jnode numbers of the grid; h i , h jsteps of the grid in the directions of the x and y axes, respectively. Using formulas (5) and replace integration by summation over the area Ω occupied by the plate, the functional (3) reduces to its discrete analog, which is a scalar function of a vector argument. The solution of the nonlinear problem is performed by the Newton-Raphson method with a step change of the load parameter [14,15] where u and ε are respectively the grid vector functions of the nodal displacements and deformations.
In formulas (7) the first and second derivatives of the vector ε with respect to the nodal displacements are determined by substituting the unit vectors e i and e j in the linear Δε L and nonlinear part Δε NL of the increment of deformation: The differential formulas approximations for calculations of the first and second derivatives of ) (u W were also used in numerical analysis: Formulae (9) permit organization of a flexible computational process which can be easily reorganized depending on the specific type of variational functional. However, they lead to computation of the function ) (u W several times, which involves considerable computer time. A more economic and accurate approach is that based on computation of the first and second derivatives of ) (u W in terms of their own obvious expressions (7), (8). For solving the system of linear algebraic equations of the finite difference energy method LDL Tfactorization is applied. The compact form of storage tapes matrix by columns in one-dimensional array is used.

Results
Consider the solution of several problems about the behavior of a flexible plate on elastic foundation at various values of the compression ratio (Fig.1)  . Boundary conditions correspond to hinged bearing at the contour. For the solution of nonlinear problems step-by-step loading and the Newton-Raphson procedure (6)-(9) for each step are used. The value of load increment at each step was taken equal to Δq=4.33.
Equilibrium curves of the flexible plate (w/hdeflection in the center) lying on elastic foundations with different compression ratios which conform to the coefficients of a real soils (sand, clay etc.) are shown in Fig. 2. Values of deflections, bending moments and normal forces in the center of the plate for 50 / 'q q are given in Table 1.  In Table 2 the deflections and in Table 3 bending moments and shear forces in the plate along the line 2 / b y are shown.  SPbWOSCE-2016 4007 in the equation of equilibrium and created a certain kind of shell effect. Epures of bending moments and shear forces based on the data of Table 3, have a view typical for shallow shells on a rectangular plan with an action of a lateral uniformly distributed load.

Conclusions
For the calculation scheme considered above and parameters characterizing elastic foundation, it is possible to use a geometrically linear theory in those cases where the deflections of the plate are not small values and comparable with a plate thickness.
The presented numerical method for solving of nonlinear problems of the theory of flexible plates on elastic foundation is sufficiently universal and has a good convergence. The developed technical theory of plates is well suited not only for an analysis of thinwalled structures made of composite anisotropic materials (where the effect of shear changes the results both quantitatively and qualitatively), but also for an analysis of structures made of isotropic materials mainly with a view to simplifying the algorithms for numerical calculations. The presented algorithm can be used in particular in the analysis of the interaction of metal tanks with the ground.