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Question

Strain energy of a solid body due to a gradually applied load is equal to:
(Consider the cross-sectional area of the body to be uniform throughout)

A
0.5×load×extension
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B
0.5×YAL×(ΔL)2
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C
Work done by load
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D
0.5×stress×strain×volume
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Solution

The correct options are
A 0.5×load×extension
B 0.5×YAL×(ΔL)2
C Work done by load
D 0.5×stress×strain×volume
Solids are analogical to spring for storing energy. When solids are deformed by applying load, they store energy (similar to the case of extension/compression of spring)

The amount of energy stored is equal to work done by the external agent i.e load.
For spring,
Stretching force F=Kx ...(1)
where x is the extension in spring.
For elastic solids,
Load=Tension Force
F=YALΔL ...(2)
where ΔL=extension in solid
From analogy, comparing Eq. (1) and (2),
K=YAL ...(3)
& x=ΔL ...(4)
Elastic potential energy stored in spring is given as,
U=12Kx2=0.5Kx2
So, elastic potential energy for solids from Eq.(3) and (4),
Strain energy(U)=0.5×YAL×(ΔL)2
U=0.5×YAΔLL×ΔL
U=0.5×Load×extension

Or U=0.5×LoadArea×extensionLength×Area×Length
Substituting for stress and volume,
U=0.5×(stress)×(strain)×(volume)
Therefore, all four options are correct.

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