Solution Procedure

As before, an approximate solution is assumed as a series of N+ 1 terms :

(4.1)

However, here, the unknown coefficients are denoted by rather than as (as in equation 3.2). Note that, the Dirichlet boundary condition (equation 2.1b) implies

at x = 0

(4.2)

However, all except are zero at x = 0 (Fig. 4.3). Therefore, to satisfy this condition, the first coefficient must be zero :

(4.3)

Then, the series (4.2) starts from i = 1 rather than from i = 0. Therefore,

(4.4)

When the expression (4.4) is substituted in the integral form (equation 3.52), we get

,   for i = 1,2,., N

(4.5)

where the expressions for the stiffness matrix and the force vector are given by equations (3.13) and (3.53). In matrix form, equation (4.5) can be written as

(4.6)