Problem 1 A beam AB is clamped at one end and joined to a bar, BC, at the other end by a hinge. The bar is pinned at end C. The length of the beam is L while the length of the bar is L/2. The beam and bar have the same circular cross section but are made of different materials. The beam is subject to a vertical load that varies linearly from qo at the clamp to zero at the hinge. 90 L CROSS SECTION N A B L/2 d D Figure 1 a) Write the system of equations needed to find the expressions of the functions describing how the forces, moments, rotation, and displacement components vary along the length of the beam and the bar. Assume that and E are the Young's moduli of the beam and bar respectively, that the cross sectional area is and that the second moment of area is I. Write the system of equations in matrix form [20 pts] b) Solve the system of equations assuming that: L= 500 mm, D=25 mm, d = 22 mm, E1=68.9 GPa, E2-3 GPa, and q = 10 N/mm. Provide the expressions describing how the shear force and bending moment vary along beam AB using the appropriate units [20 pts] c) Plot the relevant force, moment, rotation, and displacement diagrams for the beam and bar. Moreover, show the reaction diagram for the entire structure (no explanation needed) [20 pts] d) Determine the maximum compressive and tensile stresses and indicate where in the structure they occur [10 pts] Problem 2 a Figure 2 a a The cross section shown in Fig. 2 consists of a solid square of size a = 50 mm attached to a rectangle of width b = 10 mm. The square has a hole of radius r = 15 mm at its center. a) Find the position of the centroid of the cross section [10 pts] b) Find the principal axes of the cross section and the corresponding second moments of area. Sketch and label the axes on the cross section. [20 pts]