A solid structure of rectangular cross-section is simply supported, as shown in Figure Q2. The structure is made from steel with Youngs Modulus of 200GPa and is loaded by a centrally applied force, F. Using Castigliano's theorem, for the system: a) Identify and label all reaction loads, b) State the expressions for the bending moments for each section with respect to the integration length coordinate, c) Calculate the vertical deflection at Point B. d) Briefly describe how you would solve the problem, if the structure was built-in at the left hand side and simply supported on the right hand side.

Please use sigularity function to estimate the maximum deflection at the point of applying load P

The channel shape cross-section and the rectangular cross-section shown in the Figure Q4 are made of materials with elastic-perfectly plastic behaviour. The yield stress of the material used for the channel shape cross-section is oy = 473 MPa, whereas that used for the rectangular cross- section is 0.850y. Compute the thickness of the web, t, of the channel shape section if the two cross sections have identical plastic bending moment about the z-axis. In Figure 4, h= 77 mm, b=46 mm, a=11 mm, H=77 mm, d= 46 mm N b h a + N a | d H

The beam shown in the figure has a roller support at A, a fixed support at C and an internal hinge at B. The lengths of segments AB and BC are a and b respectively. A uniformly distributed load, q, is applied between points B and C, and a concentrated load P acts at a distance a/2 from the support A. EI is constant. a/2 P por quity A B Figure 2: Beam with internal hinge Determine the expression for the elastic curve (v(x)) for segment BC using the method of superposition. Use a = 2 m, b = 3 m, E = 70 GPa, I = 198 × 106 mm4, P = 10 kN and q = 2 kN/m.