1. A simply supported beam has a concentrated downward force P at a distance of a from the left support, as shown in the figure below. The flexural rigidity EI is constant. Find the equation of the Elastic Curve by successive integration.
2. Determine the rotations at A and B due to an applied moment MB on the beam, as shown in the figure below. Use the Method of Virtual Work.
3. Find the strain energy stored per unit volume for the materials listed below when they are axially stressed to their respective proportional limits.
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Material
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Proportional Limit (N/mm2)
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Modulus of Elasticity Proportional Limit (N/mm2)
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Mild Steel
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247
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2.06 x 105
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Aluminium
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412
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7.20 x 104
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Rubber
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2.06
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2.06
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4. As shown in the figure below, find the downward deflection of the end C caused by the applied force of 2 kN in the structure. Neglect deflection caused by shear. Let E = 7 x 107 kN/m2.
5. For the loaded beam, as shown in the figure below, determine the magnitude of the counter weight Q for which the maximum absolute value of the bending moment is as small as possible. If this beam section is 150 mm x 200mm, determine the maximum bending stress. Neglect the weight of the beam.
6. A wooden beam with sectional dimensions of 150 mm x 300 mm, carries the loading as shown in the figure below. Determine the maximum shearing and bending stress for the beam.
7. For the box beam shown in the figure below, determine the maximum intensity w of the distributed loading that can be safely supported if the permissible stresses in bending and shear are 10 N/mm2 and 0.75 N/mm2 respectively.
8. A beam of rectangular section 450 mm wide and 750 mm deep has a span of 6 metres. The beam is subjected to a uniformly distributed load of 20 kN per metre run (including the self-weight of the beam) over the whole span. The beam is also subjected to a longitudinal axial compressive load of 1500 kN. Find the extreme fibre stresses at the middle section span.
9. A hollow alloy tube 5 metres long with external and internal diameters equal to 40 mm and 25 mm respectively, was found to extend by 6.4 mm under a tensile load of 60 kN. Find the buckling load for the tube when it is used as a column with both ends pinned. Also find the safe compressive load for the tube with a Factor of Safety of 4.
10. A cantilever beam of length l carrying a distributed load varies uniformly from zero at the free end to w per unit run at the fixed end. Find the slope and downward deflection of the free end B.
