Discrete Model for Membrane Analysis

Abstract

This chapter and the next chapter (Chapter 9) are intended as an intermediate step between the classical approach with differential equations and the current computational Finite Element analysis. In pre-FE days, differential equations were solved approximately by Finite Difference (FD) analysis. In that method a grid is chosen over the area of the plate, and the differential equation at each grid point (node) is displaced by an algebraic equation. Solving the set of linear equations leads to an approximate solution of the problem, a solution which becomes more accurate as the mesh is chosen finer. The Finite Element Method (FEM) is the successor of the Finite Difference Method (FDM), in a way which makes it much easier to model plates of any shape and to satisfy boundary conditions. The model for membrane states to be discussed in the present chapter serves two educational goals. First, the discussion is a simple preparation to the stiffness method. The concept of stiffness matrix is introduced, boundary effects are easily accounted for, and often the same solution is obtained as in a classic FD-analysis. Second, the structural engineer sees that different transfer mechanisms are present in a plane stress state. Some members carry horizontal axial forces in the horizontal direction, other members vertical axial forces and special members shear forces. In Chapter 9 comparable members will occur for bending in two directions, and a special member for torsion.