Numerical-analytical method to determine the stress state of an elastic rectangular plate

Abstract

A numerical-analytical method for solving the plane problem of elasticity is proposed. Systems of nonorthogonal functions are used. The method involves the minimization of a quadratic form that is equal to the integral of the sum of squared residuals of the solution and given forces. An explicit expression for stresses is derived. Bessel's inequality and the convergence of the solution are proved. The accuracy of the boundary conditions is estimated. The stress and strain distribution in the plate depending on the maximum magnitude of distributed forces and the size of their localization area is analyzed numerically. New quantitative and qualitative features of the stress distribution in the plate are established Keywords: boundary-value problem, numerical-analytical method, plane problem of elasticity, stress and strain distribution, new quantitative and qualitative features Introduction. Various analytic and numerical approaches are used [3,4,12,[19][20][21] to determine the stress state of structural members (plates and shells of different shapes). Love [22], Fadle [15], and Papkovich [8] proposed the method of homogeneous solutions (eigenfunctions) to satisfy boundary conditions, but did not follow it up with numerical calculations. Gaydon and Shepherd [17] expanded the first ten eigenfunctions in a special orthogonal system and calculated the stresses. The superposition method is widely used to solve a plane boundary-value problem [3,5]. The method of homogeneous solutions was used in [1] to analyze the stress distribution in a plate under a normal parabolic load. The analytic approach employing eigenfunctions was further developed in the papers [8,9], which give a theory of a numerical-analytic method for the determination of the coefficients of two boundary forces expanded into series in a complete system of nonorthogonal functions. This method was used in [10] to determine the stress-strain state (SSS) of a half-strip. Note that Ostrogradsky was the first to use the eigenfunction expansion to solve boundary-value problems in mechanics [7]. The relevant literature is reviewed in [2, 5, 23]. We will use the theoretical method from [8,9] for the numerical determination of the SSS of a rectangular plate. We will also prove that the solution converges and analyze the accuracy of satisfying the boundary conditions.

Solving the three-dimensional equations of the linear theory of elasticity

“…This approach is also called the method of homogeneous solutions [7]. Ostrogradsky was the first [24] to propose, in 1826, this approach to solve boundary-value problems of mechanics. In mathematics, this approach is called eigenfunction series expansion.…”

“…The function F z ( ) is the same as in the plane problem [5,24]. It has a countable number of complex zeros z k , Re( ) z k > 0, which can be numbered in ascending order of their real parts.…”

“…(5.19) tends to zero as N increases. The theoretical aspects of using this method in the case of a nonorthogonal system of functions are discussed in [24]. The deviation of the approximate solution from the exact solution of the system of equations (5.19) determines the minimum of the functional …”