3

3. The potential energy for a simply supported beam under uniform distributed load (see the figure) is H— rEI fdy) 2 (147x(H — x) 2 W.7c) 2 y)1 dx 
in which: y is the transverse deflection of the beam W= transverse distributed load ef, 447 3 ) a) Use the Euler equation to solve for the deflection equation y(x) of the beam C -r_, , b) If we were to set 811 = 0, the result would be the Euler equation plus this boundary terril: 4_ a41 C i?) = 0 
El 

E, I, and H are constants independent of x 
and the boundary conditions are y(0) = 0 and 
AM= 0 

X 

What does this boundary term tell us about the boundary conditions that must exist at the ends of the beam? Dt Zkc 
4. Concerning FEA models with 3-node triangular and 4-node rectangularelements: a) Using words and/or equations, explain why these elements perform poorly in bending. b) Does "refining the mesh" (adding more of these elements) improve the performance of models with these elements? Explain. c) What other approach can be taken to improve the performance of these models? Explain.