Elasticity Analysis of Orthotropic Plate under Concentrated Force

The complex variable function including a material parameter is analyzed fully. Typical boundary value problem is considered to the plane orthotropic materials. By constructing new stress function, the mechanic analysis for the plate carrying a concentrated force is carried out. The boundary problems of the basic partial equation are studied and the formulae for stress fields are derived in the rectangular and polar coordinates. Introduction The complex variable theory provides a very powerful tool for the solution of many boundary value problems in the elastic body. Such theory was originally found by Russian researchers for solving general boundary problems in isotropic materials [1~3]. Furthermore, the complex variable technique has also been expanded to use for anisotropic materials. Complex variable methods prove to be very useful for the solution of many full-space and half-space problems. Fiber-reinforced polymer matrix materials are the most typical composites, which are also as anisotropic materials at the macroscopic level [4, 5]. The orthotropic plate may have been the base of composites in common engineering use. Typical half-space examples include concentrated force and moment systems applied to the free surface. The feasible method to solve stress-field problems in anisotropic composites is to use complex analytic function theory, and the results have been reported [6]. But the general solutions may have some weakness. So the purpose of this paper is to focus attention on a new solution of the boundary-value problem for the orthotropic plate. Concentrated Force on Straight Boundary Consider now the half plane carrying a general concentrated force P on a horizontal straight boundary surface of an infinitely large plate as shown in Figure 1. The direction of inclined force P is defined by the angle between P and free straight boundary. The distribution of the load along the thickness of the plate is uniform. The thickness of the plate is taken as unity, so that P is the load per unit thickness. The general inclined force P can be resolved into two components, which are cos P horizontally and sin P vertically. Figure 1. Scheme of the force acting on straight boundary. 329 The elasticity analysis of the plane stress problem is of great importance to the usual engineering application. The distribution of any stress depends on the forces acting on the complete closed boundary. It is supposed that the remote boundary is constrained (Figure 1), so that the stress is infinitely small at the place far away from the point of applied force. For a general isotropic material, the solution of stress distribution in Figure 1 can be found out from a book on the theory of elasticity. But for an anisotropic material, the solution of the typical problem is hard to find in books or articles. Therefore, the aim of this article is to give an example of showing the method to solve the boundary loading problem particularly for an orthotropic plate. Basic Equations The plane stress state of composite sheets is common and very importance for the application. It is the key point to solve stress-field problems in orthotropic materials. Suppose the principal elastic directions of the plate coincide with the coordinate directions (x, y), and let the directions 1, 2 parallel to the axes x, y, respectively. Linear elastic strain-stress relations are known generally as Hooke’s law, and the linear constitutive equations for the orthotropic materials are given as follows: 12 1 12 2 1 12 1 , , G E E E E xy xy x y y y x x (1) It is well known that the compatibility condition of strains must satisfy as follows: y x x y xy y x 2 2 2 2 2 (2) In the case of plane stress state, the equilibrium equations are as (body forces are absent): 0 , 0 y x y x y xy xy x (3) Usually, the method of solving the equations is by introducing a new function U of x and y, called the stress function. Through taking any real function U, it is easily checked that the equilibrium equations are satisfied by putting the following expressions for the stress components: y x U x U y U xy y x 2 2 2 2 2 , , (4) By means of using above relations, the governing equation of the compatibility condition can be expressed by the stress function U(x, y), which is