Consider a beam with a solid rectangular section as shown below. The beam has length L, Youngs modulus E, and density r. The beam is built-in on the left end and supported by two linear springs on the right end with spring constants ky and kz for the y- and z-directions respectively. The beam is loaded as shown below by a distributed partial-span load of magnitude f0 [N/m] and a point force on the right end of magnitude F [N] at the location shown. Gravity is also acting along the negative z-direction.

a) Formulate the boundary value problem(s) present (i.e., find all of the necessary governing differential equations and boundary conditions). (b) Assuming that a>>b and recalling that the moment of inertia of a rectangular beam of height h for bending about the thickness, t, axis is given by ht3/12, evaluate all geometric quantities present in part (a). (c) Assuming that f0 = 0 and ky = kz =0, solve for the deformation of the beam.