question1: Plate with uniform moments. Consider a rectangular plate of side lengths a and b with edges that are subjected to bending moments (per unit length) M1a on the sides of length a and M1b on the sides of length b, and no other loading.

(a) Find the deflections in the plate. Hint 1: consider an ansatz of the form w(x, y) = C1 + C2x + C3y + C4x2 + C5xy + C6y2. Hint 2: constrain rigid body displacements and rotations by choosing deflections w, and slopes w/x, w/y to be zero at the origin x = 0, y = 0.

(b) In the case that M1b = 0, what are the curvatures of the surface in the x and y directions?

QQuestion2: Simple plate bending. The deflection of a square plate of side length a is given by w(x, y) = Cxy.

(a) Show that this deflection satisfies the plate bending equation in the case of no externally applied dis- tributed load (i.e. p(x, y) = 0).

(b) What is the mean curvature of the deflected midsurface? What is the direction of the maximum curvature and what is the magnitude of the curvature in this direction?

(c) What is the effective shear force on the edges of the plate that lead to the given deflection field?