Question 1. Figure shows a concrete reinforced slab undergoing deflection under its own weight.

Using the FEM principles (stiffness method), calculate

a. Deflection and reactions at the nodes (define the nodes as required)

b. Draw the shear force and bending moment diagrams (using FEM)

c. Compare the maximum deflection using FEM with analytical results.

Modulus of elasticity = 5.0 × 10⁶ psi

Weight per cubic feet of slab = 150 lb

Assume width in the perpendicular direction (z-dir) = 1 in

Question 2. Many flow problems can be simplified and modeled as flow between two infinite parallel plates e.g. lubrication leakages in hydrostatic bearings. The flow between the plates is given by the following equation:

μd²u/dx² = dp/dx

u is the fluid velocity between the plates, µ is the dynamic fluid viscosity, and dp/dx is the constant pressure drop.

a. Using Galerkin's method, obtain the equation for velocity profile. Assume displacement function to be

u‾ = C₁[sin(Πy/H)]

b. Find the exact solution for the velocity profile.

c. Solve the velocity profile using finite difference approach.

d. Compare the results of all of the above approaches using the following values µ = 0.05 N.s/m, H = 0.02 mm, dp/dx = -200 MPa/m

Use an excel sheet to plot the velocity profiles.

e. How do the maximum velocities compare for each case?

Question 3. A variable cross-section plate is shown in the figure. The plate supports a load of 2500 lb.

a. Using direct stiffness method (one dimensional method), determine the deflection of the plate at nodes 2, 3 and 4. The modulus of elasticity of the material is 20e3 ksi.

b. Also calculate the stress in all the three elements.

Question 4. You have already submitted the MATLAB code to calculate the reactions and deflections in 3D truss. Calculate strains and stresses too. Submit the updated code.