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.
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)
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.
Manoj prasad said:
1 decade ago
fl = Longitudinal stress.
fc = Volumetric stress.
Longitudinal strain = el = fl/E-(1/m)*((fc)/E).
Volumetric strain = ec = fc/E - (1/m)*((fl)/E).
Now,
Longitudinal strain/volumetric strain.
Solve it & you will get the answer as (D).
fc = Volumetric stress.
Longitudinal strain = el = fl/E-(1/m)*((fc)/E).
Volumetric strain = ec = fc/E - (1/m)*((fl)/E).
Now,
Longitudinal strain/volumetric strain.
Solve it & you will get the answer as (D).
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)
Siva katuri said:
10 years ago
Longitudinal strain : pr/[2tE (1-2/m)].
Volumetric strain : pr/[2tE (5-4/m)].
pr/[2tE (1-2/m)] : pr/[2tE (5-4/m)].
pr/pr : 2tE (1-2/m) / 2tE (5-4/m).
: 2tE[ (m-2) /m]/2tE[(5m-4)/m].
: [(m-2)/m]/[(5m-4)/m].
: (m-2)/(5m-4).
Volumetric strain : pr/[2tE (5-4/m)].
pr/[2tE (1-2/m)] : pr/[2tE (5-4/m)].
pr/pr : 2tE (1-2/m) / 2tE (5-4/m).
: 2tE[ (m-2) /m]/2tE[(5m-4)/m].
: [(m-2)/m]/[(5m-4)/m].
: (m-2)/(5m-4).
Kiran said:
10 years ago
Longitudinal strain : Volumetric strain.
[(pd/4Et)(1-2v)] : [(pd/4Et)(5-4v)].
(1-2v) : (5-4v).
V[(1/v)-2] : v[(5/v)-4].
Where, (1/v) = m.
(m-2):(5m-4).
[(pd/4Et)(1-2v)] : [(pd/4Et)(5-4v)].
(1-2v) : (5-4v).
V[(1/v)-2] : v[(5/v)-4].
Where, (1/v) = m.
(m-2):(5m-4).
Sachin said:
1 decade ago
@Abhishek.
No answer is correct!
m is not Poisson's ratio. Its reciprocal is
m = (1/u).
Long strain = (PR/2tE)(1-2/m).
Vol strain = (PR/2tE)(5-4/m).
No answer is correct!
m is not Poisson's ratio. Its reciprocal is
m = (1/u).
Long strain = (PR/2tE)(1-2/m).
Vol strain = (PR/2tE)(5-4/m).
Abhishek said:
1 decade ago
All options are wrong.
Longitudinal strain = pr/2tE(1-2m).
Volumetric strain = pr/2tE(5-4m).
Longitudinal strain = pr/2tE(1-2m).
Volumetric strain = pr/2tE(5-4m).
Vikas said:
1 decade ago
Net longitudinal strain = pd/4tE(1-2/m).
Net volumetric strain = pdV/2tE(5/2-2/m).
Net volumetric strain = pdV/2tE(5/2-2/m).
Emlin said:
1 decade ago
1/m is poisson's ratio where m is a constant. Values ranging between 3 and 4.
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