I2

The beam < < , is built-in at the end = 0 and loaded by a uniform shear traction = on the upper edge, = , the remaining edges, = , = being tractionfree Mechanics of Materials considerations suggest that the bending moment will vary linearly with x and hence that will vary with , requiring a 4th order stress function. The most general such function is given by (,) = 14 +23 + 322 +43 +54 +63 +72 +8 2 + 9 3 + 10 2 + 11 + 12 2 . a. Determine the constants in the stress function (,) i. by imposing strong boundary conditions on the upper and lower edges = , ii. by satisfying weak boundary conditions at = and iii. by satisfying the biharmonic equation 4 = 0 b.