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
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B
0.5×YAL×(ΔL)2
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C
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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×volumeSolids 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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