Mechanical Engineering - Strength of Materials - Discussion
Discussion Forum : Strength of Materials - Section 1 (Q.No. 20)
20.
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
Discussion:
20 comments Page 1 of 2.
Apoorva said:
5 years ago
All options are wrong.
Longitudinal strain= (longitudinal stress/ E) - (m circumferential stress/ E).
= Pd(1-2m)/4tE.
Volumetric strain= longitudinal strain+ circumferential strain.
= Pd(5-4m)/4tE.
Ratio= (2m-1)/(4m-5).
Longitudinal strain= (longitudinal stress/ E) - (m circumferential stress/ E).
= Pd(1-2m)/4tE.
Volumetric strain= longitudinal strain+ circumferential strain.
= Pd(5-4m)/4tE.
Ratio= (2m-1)/(4m-5).
(6)
Prem said:
6 years ago
We have constitutive relations :
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) , Answer is 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) , Answer is D.
(2)
Aditya said:
7 years ago
Yes, I agree with you @Koushik.
Prasenjit Deb said:
7 years ago
D is the right answer. I agree with the given answer.
Koushik said:
7 years ago
The Answer will be m/m-2, where 1/m is Poisson's ratio.
Sanny said:
7 years ago
In numerator it's (1-2m).
Neeraj said:
9 years ago
D is right, I agree.
(1)
Yallus said:
9 years ago
Longitudinal strain = pr/2tE(1-2/m).
Volumetric strain = pr/2tE(5-4/m).
Longitudinal strain/volumetric strain = (Pr/2tE(1-2/m))/(pr/2tE(5-4/m)).
Cancel pr/2tE.
Next will be getting
(1-2/m)/(5-4/m).
Take lcm and cancel m.
We get, m-2/5m-4.
Volumetric strain = pr/2tE(5-4/m).
Longitudinal strain/volumetric strain = (Pr/2tE(1-2/m))/(pr/2tE(5-4/m)).
Cancel pr/2tE.
Next will be getting
(1-2/m)/(5-4/m).
Take lcm and cancel m.
We get, m-2/5m-4.
Yashwanth said:
9 years ago
Please, explain clearly.
Pavan said:
10 years ago
Please solve it in simple manner.
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