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A square bar of side 40 mm and length 1 m is subjected to an axial load P. The same bar is then used as a cantilever beam and  subjected to an end load P. The ratio of the strain energies, stored in the bar in the second case to that stored in the first case, is
  • a)
    16
  • b)
    400
  • c)
    1000
  • d)
    2500
Correct answer is option 'D'. Can you explain this answer?
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A square bar of side 40 mm and length 1 m is subjected to an axial loa...
Given data:
Side of the square bar, a = 40 mm = 0.04 m
Length of the bar, L = 1 m
Axial load, P = End load, W

To find: Ratio of the strain energies stored in the bar when it is subjected to an axial load and when it is used as a cantilever beam and subjected to an end load.

Assumptions:
1. The material of the bar is homogeneous and isotropic.
2. The deformation is elastic.
3. The stress-strain relationship is linear.

Calculation:
The strain energy stored in a body is given by the formula:
U = (1/2) σεV
Where,
σ = Stress
ε = Strain
V = Volume

The stress in the bar when it is subjected to an axial load is given by the formula:
σ = P/A
Where,
A = Cross-sectional area of the bar = a^2

The strain in the bar when it is subjected to an axial load is given by the formula:
ε = σ/E
Where,
E = Young's modulus of elasticity of the material of the bar

The volume of the bar is given by the formula:
V = AL
Where,
L = Length of the bar

Substituting the values in the formula for strain energy, we get:
U1 = (1/2) (P/A) (P/E) (a^2L)

The stress in the bar when it is used as a cantilever beam and subjected to an end load is given by the formula:
σ = (Wl)/(2I)
Where,
W = End load
l = Length of the cantilever
I = Moment of inertia of the cross-section of the bar

For a square cross-section, the moment of inertia is given by the formula:
I = (a^4)/12

The strain in the bar when it is used as a cantilever beam and subjected to an end load is given by the formula:
ε = (y/L) (Wl)/(2I)
Where,
y = Distance from the neutral axis to the outermost fiber

For a square cross-section, the distance from the neutral axis to the outermost fiber is given by the formula:
y = a/2

Substituting the values in the formula for strain energy, we get:
U2 = (1/2) [(Wl)/(2I)] [(y/L) (Wl)/(2I)] (a^2L)

Dividing U2 by U1, we get:
U2/U1 = [(Wl)/(2I)] [(y/L) (Wl)/(2I)] / (P/A) (P/E)

Substituting the values of the variables, we get:
U2/U1 = [(W/2) (a^2/12) (1/L)] [(a/2)/L (W/2) (a^2/12)] / (P/a^2) (E)

Simplifying the expression, we get:
U2/U1 = (16/3) (W/P) (L/a)^3

Since the bar has the same dimensions in both cases, i.e. when it is subjected to an axial load and when it is used as a cantilever beam and subjected to an end load, the ratio of the strain energies stored in the
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A square bar of side 40 mm and length 1 m is subjected to an axial load P. The same bar is then used as a cantilever beam and subjected to an end load P. The ratio of the strain energies, stored in the bar in the second case to that stored in the first case, isa)16b)400c)1000d)2500Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A square bar of side 40 mm and length 1 m is subjected to an axial load P. The same bar is then used as a cantilever beam and subjected to an end load P. The ratio of the strain energies, stored in the bar in the second case to that stored in the first case, isa)16b)400c)1000d)2500Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A square bar of side 40 mm and length 1 m is subjected to an axial load P. The same bar is then used as a cantilever beam and subjected to an end load P. The ratio of the strain energies, stored in the bar in the second case to that stored in the first case, isa)16b)400c)1000d)2500Correct answer is option 'D'. Can you explain this answer?.
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