Rework Example 6.2 using the trigonometric Ritz approximation w j = sin jx/l. Develop a two-term approximate

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Rework Example 6.2 using the trigonometric Ritz approximation w= sin jπx/l. Develop a two-term approximate solution and compare it with the displacement solution developed in the text. Also compare each of these approximations with the exact solution (6.7.9) at midspan x = l/2.

Data from example 6.2

Consider a simply supported Euler-Bernoulli beam of length l carrying a uniform loading qo. This one-dimensional problem has displacement boundary conditions:

and tractions or moment conditions w = 0 at x = 0,1 dw dx The Ritz approximation for this problem is of the

With no nonhomogeneous boundary conditions, wo = 0. For this example, we choose a polynomial form for the trial solution. An appropriate choice that satisfies the homogeneous conditions (6.7.4) is wj = x (l – x). Note this form does not satisfy the traction conditions (6.7.5). Using the previously developed relation for the potential energy (6.6.12), we get:

2 dw II = = [[5/7 (2)  - 9ow]dx dx 2 N   L'H * ( = 0-1)-2-j0j +1)xj-] N -90cx (1-x) dx - 39/4

Equation 6.7 .4

w = 0 at x = 0,1

Equation 6.7 .5

dw  dx -= 0 at x = 0, /

Retaining only a two-term approximation (N=2), the coefficients are found to be 901 C1 = 24EI and this gives

Actually, for this special case, the exact solution can be obtained from a Ritz scheme by including polynomials of degree three.

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