By Walter Wessel
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Example text
To know what value of N to take can only be determined by solving the problem with one higher integer and determining that the previous approximation is sufficient. One thing to remember is that the result of this method is the determination of an approximate deflection w(x). To determine stresses requires solving for w" since the bending stresses are proportional to w". Taking derivatives of an approximative function causes an increase in the error through differentiation. Hence, for stress critical structural members, N must be determined to suitably approximate the maximum stress.
No attempt is made here to provide a detailed comprehensive introduction to Galerkin's method which is well treated in numerous other texts. However, consider an ordinary differential equation (although the method is equally useful for partial differential equations as well as nonlinear equations as shown below). 69) where L is any differential operator, and q(x) is a forcing function. Boundary conditions must be homogeneous; if not, a transformation of variables must be made to attain homogeneous boundary conditions.
39) = d 2 wiL) = O. 23), where the second subscript is used to denote Section 1 or Section 2. The equations to solve involve solving an 8 x 8 set of algebraic equations and lengthy manipulations. The final constants in this example are: fJ C II -- EI qo [~2L C = ~(-4L~2 _ ~4 + 31 24EI L C __ qo~2 2EIL 12 - ~3) c = -~[4L~2 + ~4J 32 24EI L C _ qo~4 24EI . 45) The location and magnitude of the maximum deflection will occur in either Section 1 or 2 depending upon the extent of the load, where ~ is located.