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1. For each of the cross-sections shown in Fig.Q1, calculate the moment of the inertia about a horizontal axis passing through the centroid. Evaluate also the corresponding minimum and maximum elastic section moduli.
2. A simply-supported beam of span 2 m is subjected to a lateral uniformly distributed load of 20 kN/m which causes a maximum vertical shear force of 20 kN at each support. The beam cross- section consists of a hollow square section of external dimensions 100 mm x 100 mm and with a wall thickness of 10 mm. Determine the maximum and average vertical shear stresses within the beam over the supports.
3. A welded plate girder (beam cross-section made up of plates welded together) has a web plate measuring 1000 mm x 12 mm and a top flange of a larger area than that of the bottom flange. The top flange measures 400 mm x 65 mm, the centroid of the section is at a distance of 464 mm from the upper surface of the top flange and Ixx = 12.5 x 109 mm4. If the vertical shear force at a section along the beam is 600 kN, calculate the shear force per unit length which is to be carried by the web/flange welds just underneath the top flange.
4. The beams A and B in Fig.Q4 are to support an applied lateral load that a sagging bending moment of 72.3 kNm is carried by the beams acting in combination. Calculate the maximum bending stress developed if
(a) the beams are free to slide at the interface, and
(b) the beams are rigidly connected at the interface.
5. A cantilever beam is of length, L, and has a depth, d, which varies linearly from a value of d = L/12 at the free end to a value of d = L/3 at the fixed end. The cross-section of the beam is rectangular and of width, b = L/12 throughout the length of the beam. Derive an expression for the maximum bending stress due to a point load, W, acting at the free end of the cantilever.
Note that the bending moment at any point along a beam is equal to the sum of the products of the loads to one side of that point multiplied by their respective lever arms from that point.
6. Consider a composite beam with the cross-sectional dimensions shown in Fig.Q6. The upper 150 mm x 250 mm part is made of wood with Ew = 10 000 N/mm2, while the bottom 150 mm x 10 mm strip is made of steel Ew = 200 000 N/mm2. If this beam is subjected to a sagging bending moment of 0.03 MNm about the horizontal neutral axis, determine the maximum bending stresses in the wood and steel.
7. Using Mohr´s circle construction, determine the position of the principal axes with respect to the original X-Y axes, and the principal moments of inertia for the asymmetrical angle section with unequal legs shown in Fig.Q8.