Applications of the Plate Membrane Theory

Abstract

In this chapter we will give solutions for plates, which are loaded only on their edges. This implies that no distributed forces px and py occur, and the fourth-order bi-harmonic equation (1.23) reduces to the simple form $$abla^2abla^2 u_x=0$$ (2.1) When a general solution has been found for ux, the solution for uy can be derived from the relation between ux and uy as given in Eq. (1.17). If we choose the first equation, the relation is (Px = Py = 0) $$\left(\frac{\partial^2}{\partial x^2}+\frac{1-u}{2}\frac{\partial^2}{\partial y^2}\right)u_x+\left(\frac{1+u}{2}\frac{\partial^2}{\partial x\partial y}\right)u_y=0$$ (2.2) We will demonstrate two types of solution. In the first type, solutions for the displacements ux and uy will be tried, which are polynomials in x and y. We will see that interesting problems can be solved through this ’inverse method‘. The second type of solution is found by assuming a periodic distribution (sine or cosine) in one direction. Then in the other direction an ordinary differential equation has to be solved. This approach is suitable for deep beams or walls.