8-10-1 Finite Differences Examples

Let us consider a simply supported beam of constant stiffness EI carrying a uniform load q = constant. (See [21] for a different way of solving this problem.) The differential equation for the deflection of this beam [21, p. 273]

is replaced by a difference equation (see Chapter 6),

where q _i = q ₀ = constant. Because of the symmetry of the problem we put the origin of the coordinate system at the midspan (Fig. 8-86a) and consider only the portion of the beam shown in Fig. 8-86b. The symmetry implies that

Figure 8-86

In addition to the boundary conditions (8-306) we also have the boundary conditions at x = l/2,

Conditions (8-306) when applied at 0 as the pivotal point result in

Therefore,

Boundary conditions (8-307) imply that at the pivotal point x _n = l/2 we have

Together we have n + 5 equations: n + 2 Eq. (8-305), four Eqs. (8-308), and Eqs. (8-309). The number of unknowns also equals n + 5. The smallest index of the unknown in Eq. (8-305) is ?2 for i = 0 whereas the largest occurs for i = n and equals n + 2. Ultimately our equations become

where

The solution to this system gives the deflection Y _i at all points of division. Specifically, we find that

We now easily...