(15 Points) Prandtl's Soap Film Analogy. Aside from a few simple geometries, it is difficult to find analytical solutions to Saint-Venant's equations for torsion of prismatic beams. However, in 1903 Ludwig Prandtl recognized that the partial differential equation for the torsional stress function was analogous to the equation describing the shape of a soap bubble under constant pressure. If we consider the base of the soap bubble to be the shape of the cross section of our bar in the xy plane, then the height, z, of the bubble satisfies the equation 2z=x22z+y22z=Sp with the boundary condition z=0 on the boundary A.p is the pressure within the bubble and S is the surface tension in the soap film. By comparing this equation to the equation for the stress function of a bar under torsion, we immediately see the analogy: z,Sp2G1. While solving this equation is just as difficult as solving the original equation, the soap bubble analogy does provide a useful visualization for the stress function. In this problem, you will use the intuition provided by imagining a soap bubble with a base of a given shape to derive approximate expressions for the shear stress in bars with thin-walled cross sections under torsion. Figure 2 shows a sketch of the contour lines for a soap bubble obeying equation (2) over a thin rectangular surface of breadth b and thickness t. (a) Use the observation that the gradient of z in the x direction is much smaller than the gradient in the y direction to show that the expression for the height z in the sectionAA is given by z=2Sp(4t2y2) (b) Using the analogy between the soap film and the torsional stress function show that the maximal shear stress is max=G1t. Recall that the maximum shear stress in the torsion problem is located where the slope (gradient) of the stress function is maximal. (c) By neglecting edge effect near x=b/2, show that the volume of the soap film is approximately V=12Spt3b (d) Using the soap film analogy, show that the applied torque, T, is related to the rate of twist for a thin rectangular cross section by T=G13bt3. (e) Lastly, combine equations (4) and (6) to get an expression for the maximal shear stress as a function of the applied torque and the geometric parameters b and t. (f) Based on the contours of constant z shown in Figure 2, where would you expect the shear stress to be greatest? Where will it be minimal? Figure 2. Contour lines of a soap film over a thin rectangular surface