16. Consider a tapered bar of circular cross-section. The length of the bar is 1 m, and the radius varies as r(x) = 0.050 — 0.040x, where r and x are in meters. Assume Young's modulus = 100 MPa. Both ends of the bar are fixed, and a uniformly distributed load of 10,000 N/m is applied along the entire length of the bar. Determine the displacements, axial force dis¬

tribution, and wall reactions using

(a) three elements of equal length;

(b) four elements of equal length.

Compare your results with the exact solution by plotting w(x) and P(x) curves for each case (a)

and

(b) and the exact solution. How do the finite element results for the left and right wall reactions, and Rr, compare with the exact solution?

Chapter 3 Weighted Residual and Energy Methods for One-Dimensional Problems Hint: To approximate the area of cross-section of a bar element, use the geometric mean of the end areas of the element, i.e., AW = ^jAjAj — jr/'/r,-. The exact solution is obtained by solving the following differential equation with the boundary conditions «(0) = 0 and u(l) = 0:
=-pM --ROOD The axial force distribution is found from P(x) = A(x)Edu/dx. The wall reactions are Rl = -P(0) a.ndRR = P(l).