Find the equation of elastic curve by successive integration

Find the equation of elastic curve by successive integration

Find the equation of elastic curve by successive integration

Civil Engineering

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.

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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.

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3. Find the strain energy stored per unit volume for the materials listed below
when they are axially stressed to their respective proportional limits.

Material

Proportional Limit (N/mm2)

Modulus of Elasticity Proportional Limit (N/mm2)

Mild Steel

247

2.06 x 105

Aluminium

412

7.20 x 104

Rubber

2.06

2.06

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.

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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.

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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.

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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.

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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.

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