Problem 1- A uniform, rigid beam 8 ft long is supported by a hinge at one end (0) and 3 elastic cables DA, DD, and DC as shown, The beam weighs w = 50 pounds/ft and it sustains an end load P = 1000 pounds. The three elastic cables are have the same elastic stiffness coefficient k = EA pounds.

A) Assuming the rotation of the beam is small, write an appropriate expression for the total potential energy of the system and use the minimum potential energy theorem to determine the equation of equilibrium.

B) Assuming the rotation of the beam is not small, write an appropriate expression fur the total potential energy of the system and use the minimum potential energy theorem to determine the equation of eq uilibrium.

C) Linearize the equation of equilibrium from part b. and compare to the result of part a.

Problem 2- A cantilever beam has a tapered cross section, Cross section and the taper are such that the weight per unit length decreases linearly from the fixed and to the free end, and the cross-sectional moment of inertia varies quadratically with the distance from the end,

w = w₀(1-(x/l)) + w₁(x/l);                              I = I₀[1-(x/l)²] + I₁(x/l)²

The modulus of the beam material is a constant value E.

A) Determine the potential energy expression ror the beam deflecting under its own weight

B) Use the Ritz method to obtain an approximate solution for the deflection at the end u(x=l). Use the approximating function u~ = x²(a₁ + a₂x + a₃x²).

Problem 3- The square plate with side lengths l =1 is originally at an equilibrium temperature T0. The left and right sides of the plate are suddenly and simultaneously exposed to ambient air. The top and bottom edges are kept at T0. The ambient temperature on the left side is Ta > T0 and the ambient temperature on the right side is Tb < T0. The properties of the plate are thickness b, heat capacity C. mass density p and conductivity The convective heat transfer coefficient for both the left and right sides is h.

A) Set up the IVBP and develop a series solution for the transient temperature distribution.

B) Determine an approximate solution for the steady state distribution using Galerkin's method and a function of the form T(x, y)= sin (πy/l)(a₁ + a₂ (x/l))

Problem 4- Using the mesh below, set up a finite element model for the steady suite version of problem 3 above. The result should be a set of 9 linear algebraic equations for the no temperatures.