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Strength of Materials - Engineering

Q1:

A thin cylindrical shell of diameter (d) and thickness (t) is subjected to an internal pressure (p). The ratio of longitudinal strain to volumetric strain is

A

B

C

D

ANS:D -

el = (Sl / E ) - v/E * Sh.

Where Sl = longitudnal stress.
Sh = Hoop stress.

eh = (Sh / E ) - v/E * Sl
v = Poisson ratio

Substitute Sl = Pd/4t & Sh = Pd / 2t gives,

el = (Pd/4Et) (1- 2v) & eh = (Pd/4Et) (2 - v) --------------- (1).

Now volume of cylinder V = (Pi/4) d^2 * L
Differentiating gives

dV = (Pi/4) (2d d + d^2 L) ==> vol sttrain = (dV/V) = 2 d /d + L / L
ie, vol strain ev = 2 eh + el -------------- (2)
now el/ev = el /( 2 eh + el ) substituting relations from (1) & (2) gives
= (1 - 2v) / (2(2 - v) + (1 - 2v)) = (1 - 2v) / (4 - 2v + 1 - 2v) = (1 - 2v) /(5 -4 v) , put v = 1/m
==> el/ev = (m - 2) / (5m - 4)