15. A beam problem under the pure bending moment is solved using five rectangular finite ele¬

ments, as shown in the figure. Assume E = 200GPaand v = 0.3 are used. The thickness of the beam is 0.01 m. To simulate the pure bending moment, two opposite forces F = ±100,000 N are applied at the end of the beam. Using a commercial FE program, calculate strains in the beam along the bottom surface. Draw graphs of £0 and yXy, with x-axis being the beam length.

Compare the numerical results with the elementary theory of beam. Provide an explanation for the differences, if any. Is the rectangular element stiff or soft compared to the CST element?

Normally, a commercial finite element program provides stress and strain at the comers of the element by averaging with stresses at the adjacent elements. Thus, you may use nodal dis¬

placement data from FE code to calculate strains along the bottom surface of the element. Cal¬

culate the strains at about 10 points in each element for plotting purpose. Make sure that the commercial program uses the standard Lagrange shape function.

1 2 3 4 5

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1 m 5 m Repeat the above procedure when an upward vertical force of 200,000 N is applied at the tip of the beam. Use boundary conditions similar to the clamped boundary conditions of a cantilev¬

ered beam.