Department of Chemical Engineering

Strength of Materials for Chemical Engineers (0935381)

Chapter 8

Thick Cylinders and Spheres 1) Thick Cylinders Difference in Treatment between Thin and Thick Cylinders

• In thin cylinders the hope stress is assumed to be constant across the thickness of the cylinder wall.

• In thin cylinders there is no pressure gradient across the wall. • In thick cylinders neither of these assumptions can be used and the variation of

hop and radial stress will be as shown

For any element in the wall of a thick cylinder the stresses will be radial stress, hoop stress tangential, longitudinal axial stresses.

Longitudinal stress

21

22

222

211

RRRPRP

L −−

=σ

Lame’s Theory

For cylindrical thick vessels the following differential equation is used to determine the hoop and the radial stresses at any point through the thickness of the shell.

drd

r rrH

σσσ =−

Hoop Stress For the hoop and radial stresses the following equations are used:

2

2

rBA

rBA

H

r

+=

−=

σ

σ

Where A and B are constants which can be determined using the pressure conditions inside and outside the cylinder. Case 1: Internal Pressure Only

0 ,Rrat and ,R rAt r21 ==−== σσ Pr Using these two conditions the following results will be obtained:

21

22

22

21

21

22

21 and

RRRPRB

RRPRA

−=

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−⎟⎠⎞

⎜⎝⎛

−=⎥⎦

⎤⎢⎣

⎡ −−

−=

1

1

2

22

2

222

21

22

21

Kr

R

Pr

rRRR

PRrσ

Where 1

2

1

2

RR

ddK ==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+⎟⎠⎞

⎜⎝⎛

=⎥⎦

⎤⎢⎣

⎡+

−=

1

11 2

22

2

22

21

22

21

Kr

R

PrR

RRPR

Hσ

♣ The hoop (σH) and radial (σr) stresses are maximum at r=R1

♣ The maximum shear stress occurs at the inside radius (at r=R1) and 2max rB

=τ

Case 2: Internal and external pressures:

,Rrat and ,R rAt 2r211 PPr −==−== σσ Using these two conditions the following results will be obtained:

( )21

22

22

2121

21

22

222

21 and

RRRRPPB

RRRPPRA

−−

=−−

=

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧+−

⎭⎬⎫

⎩⎨⎧+

−=

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−+⎭⎬⎫

⎩⎨⎧−

−=

2

212

222

222

1121

22

2

212

222

222

1121

22

111

111

rRRP

rRRP

RR

rRRP

rRRP

RR

H

r

σ

σ

Change of Diameter:

( )

( )LrH

LrHHD

ErD

E

νσνσσ

νσνσσξξ

−−=∆

−−==

2

tensile.are stresses all 1

Change of Length:

( )

( )HrL

HrLL

ELL

E

νσνσσ

νσνσσξ

−−=∆

−−= tensile.are stresses all 1

Comparison with Thin Cylinder σmax is the limiting factor.

For thin cylinders 2

, where 22

max KPt

dKKPt

Pd HH =∴===

σσ

For thick cylinders ( )121

2max

++=

KK

PHσ

Maximum Shear Stress The stresses on an element at any point in the cylinder wall are principal stresses.

221

maxσσ

τ−

=

Half the difference between the greatest and least principle stresses.

shape lcylindricafor 22

21max

rH σσσστ

−=

−=

σH is normally tensile, σr is compressive.

2max

2222max 21

21

rB

rBA

rBA

rBA

rBA

=

⎥⎦⎤

⎢⎣⎡ −+=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−+=

τ

τ

Greatest τmax occurs at the inside radius where r = R1. 2) Thick Spheres For the hoop and radial stresses the following equations are used:

3

3 2

rBA

rBA

H

r

+=

−=

σ

σ

Case 1: Internal Pressure Only

0 ,Rrat and ,R rAt r21 ==−== σσ Pr

31

32

32

31

31

32

31 2and

RRRPRB

RRPRA

−=

−=

( )( )

( ) ( ) 133

231

32

3

31

133

231

32

3

31

at max 2 2

at max

RrRRRr

PR

RrRRRr

PR

H

r

+−

=

−−

−=

σ

σ

Case 2: Internal and external pressures:

,Rrat and ,R rAt 2r211 PPr −==−== σσ Using these two conditions the following results will be obtained:

( )31

32

32

3121

31

32

322

311 2and

RRRRPPB

RRRPRPA

−−

=−−

=

( ) { } { }[ ]31

3322

32

33113

132

3

3

313

223

323

1131

32

222

1

111

RrRPRrRPRRr

rRRP

rRRP

RR

H

r

+−+−

=

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧−−

⎭⎬⎫

⎩⎨⎧−

−=

σ

σ