Part A-

1. (a) A simply-supported beam of span L is discretized into 5 nodes (Nodes 1 and 5 at the pinned supports and the other nodes spaced equally at L/4). The stiffness influence coefficient giving the force required at the j^th node to cause a unit displacement at the i^th node is defined as k_ij. Using the Principle of Superposition, obtain an expression for the deflection at the mid-span of the beam if point loads of P₁, P₂, and P₄ are applied respectively at Nodes 1, 2 and 4 along the beam.

(b) Explain briefly why, in the case of I-section beams, it is possible to assume that the bending moment is resisted by the flanges, while the shear force is resisted by the web of the beam. State whether or not this assumption is conservative (i.e. safe), providing reasons for your answer.

(c) For which type of beam cross-sections is it necessary to determine the position of the principal axes of bending? Without resorting to any mathematical derivations, explain briefly why the principal moments of inertia of a beam cross-section also happen to be the maximum and minimum moments of inertia for that cross-section.

Part B-

2. From strain gauge readings taken at a point within a structure under plane stress conditions, it was found that the stresses with respect to a set of X-Y co-ordinate axes were σ_x = 80N/mm², σ_y = 30N/mm² and τ_xy = 40N/mm². Using Mohr's Circle graphical construction, determine

(i) the direct and shear stresses acting on a plane at an angle of 35° to the Y-axis;

(ii) the magnitude and direction of the maximum and minimum principal stresses;

(iii) the magnitude and direction of the maximum shear stress.

Assuming that the material elastic constants are E = 200,000N/mm² and ν = 0.30, find also the principal strains at that point.

3. A simply-supported beam has a T-shaped cross section with an overall depth of 320mm. The top flange has a width of 120mm and a thickness of 20mm, while the web thickness is 10mm. Given that the maximum shear force at a section along the beam is 60kN, determine the maximum shear stress in the web of the beam and the shear stress at the web/top flange junction.

4. A simply-supported beam has an I-shaped cross-section with unequal flange widths. The top flange has a width of 180mm, the bottom flange has a width of 100mm and both flanges are 20mm thick. The overall depth of the beam cross-section is 450mm, while the web thickness is 10mm. Given that the maximum shear force and the maximum sagging bending moment within the beam are 75kN and 135kNm respectively, determine the maximum shear stress and the maximum tensile and compressive bending stresses within the cross-section of the beam.

5. A timber beam with a cross-section of 250mm wide x 400mm deep is strengthened by a steel plate, having cross-sectional dimensions of 250mm x 12mm, that is glued to the bottom face of the timber beam. The modulus of elasticity of steel and timber are 200kN/mm² and 10kN/mm² respectively. Furthermore, the maximum allowable bending stresses in the steel and the timber are 120N/mm2 and 10N/mm2 respectively. Determine the maximum bending moment of resistance of the composite beam cross-section.

6. A steel beam, which has a 150mm x 60mm x 6mm L-shaped angle cross-section, consists of two legs of unequal length at right angles to each other. The beam cross-section has an overall depth of 150mm, the bottom flange has an overall width of 60mm and a thickness of 6mm, while the vertical web has a thickness of 6mm. Using Mohr's Circle graphical construction, determine the position of the principal axes of bending and the magnitude of the principal moments of inertia of the cross-section of the beam. What would you expect to find if, instead of an asymmetric L-shaped cross-section, the steel beam had a bi-symmetric I-shaped cross-section?