Calculating stress and strain on Loading Systems.

MODULE TITLE : MECHANICAL PRINCIPLES

TOPIC TITLE : LOADED BEAMS AND CYLINDERS

LESSON 3 : THICK-WALLED PRESSURE VESSELS

MP - 2 - 3

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________________________________________________________________________________________

INTRODUCTION ________________________________________________________________________________________

Thick-walled pressure vessels have less engineering applications than thin-

walled cases. However, we can still find examples around, e.g. hydraulic

cylinders, extrusion dies and gun barrels. It is important to understand the

stress states in the wall and their distribution throughout the wall of thick-

walled pressure vessels.

When we theoretically analysed the state of stresses in thin-walled cylindrical

and spherical pressure vessels, we made an assumption that the stress

distribution is uniform or constant throughout the wall. As we saw, if the ratio

of the inner radius to the wall thickness is greater than 10, this assumption is

reasonably correct. When the ratio is less than 10 , the wall is

considered to be thick. In this case, a different analysis technique called

elasticity method is required. The theory of elasticity methods is beyond the

scope of our course because of its complexity, although elasticity solutions are

mathematically exact for the specified boundary conditions in some particular

problems. Thus, the detailed derivation of the thick-walled formulae will not

be given. In this lesson, we will start our study with Lamé's theory and find

the equations used for the calculation of thick-walled cylindrical vessels

subject to internal pressure.

r

t i <

⎛ ⎝⎜

⎞ ⎠⎟

10

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________________________________________________________________________________________

YOUR AIMS ________________________________________________________________________________________

After studying this lesson, you should be able to:

• determine the distribution of stress in a thick-walled cylinder due to

internal pressure

• apply the formulae for computing the maximum values of the

stresses in a thick-walled cylinder.

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________________________________________________________________________________________

THICK-WALLED CYLINDRICAL VESSELS ________________________________________________________________________________________

Consider a thick-walled cylinder carrying internal pressure, the external

pressure being zero. The notations used in our analysis are shown in

FIGURE 1, in which ri and ro are inner and outer radius, respectively, and L is

the length of the cylinder.

FIG. 1 Thick-walled cylinder carrying internal pressure

With a thick-walled cylindrical pressure vessel, there will be hoop stress σ1, longitudinal stress σ2 and radial stress σ3. These have the same meanings as they did for thin-walled pressure vessels, except now they will have varying

magnitudes at different positions in the wall. Assume that the cylinder is long

in comparison to its diameter, therefore, the longitudinal stress σ2 can be treated as uniform across the thickness of the cylinder wall.

L

σ 1

σ 2

r

r o

r i

r

r o

r i

σ 1

σ 3

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HOOP STRESS σσ1 AND RADIAL STRESS σσ3

Using Lamé's theory, we can find the variations of σ1 and σ3 throughout the

wall if thick-walled vessels are subjected to internal pressure p:

The above equations, known as Lamé's equations, may be used to determine

the hoop stress σ1 and radial stress σ3 at any radius r in terms of constants a and

b, which can be obtained from the boundary conditions, such as the

dimensions of the cylinder and the loading conditions.

As the name implies, the radial stress σ3 acts along a radius of a cylinder. It is

a compressive stress and varies from a magnitude at the outer surface, which is

equal to the external pressure, to the value of the internal pressure at the inner

surface. Since hoop stress σ1 and longitudinal stress σ2 are tensile stresses,

they make vessels bigger. So we define them as having a positive value.

Conversely, σ3 is defined as having a negative value (compressive stress) when

the pressures acting on both sides of the walls of a cylinder are gauge pressure.

When cylindrical vessels are carrying internal pressure only or internal

pressure is greater than external pressure (note that they are both gauge

pressure), radial stress σ3 will have the maximum magnitude at the inner

surface and a minimum magnitude, in most cases zero (when pressure vessels

are exposed in atmosphere), at the outer surface. We will quantify the

variations of σ3 and σ1 along a radius in the following discussions.

σ1 2= a b

r – .............................................

.............

1

3 2

( )

= +and σ a b

r ................................ 2( )

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Example 1

A steel pipe of internal diameter 50 mm and external diameter 100 mm is

subject to an internal pressure of 12 MPa and an external pressure of 5 MPa.

What are the radial and circumferential stresses at the inner and outer surfaces?

Solution

From the question:

This is a thick-walled cylinder. We can therefore use Lamé's equations

σ1 2= a b

r – .............................................

.............

1

3 2

( )

= +and σ a b

r ................................ 2( )

r

r

p

i

o

i

mm 0.025 m

mm 0.05 m

MP

= = =

= = =

=

50 2

25

100 2

50

12 aa 12 10 Pa

MPa 5 10 Pa

The wall thicknes

6

o 6

= ×

= = ×p 5

ss mm

so

o i

i

t r r

r

t

= =

=

– .0 025

1

5

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When

then

This equation gives:

When

then

we have:

Subtracting the above equations gives:

Substituting b into one of the equations gives a = –2.67 × 106 Pa.

– .

– .

–

7 10 0 025 0 05

5833

6 2 2

× =

∴ =

b b

b

σ 3 = × = +– . 5 10

0 05 6

2 a

b

r r

p

= =

= = ×

o

3 o

m

Pa

0 05

5 106

.

– –σ

σ 3 = × = +– . 12 10

0 025 6

2 a

b

r r

p

= =

= = ×

i

3 i

m

Pa

0 025

12 106

.

– –σ

6

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Using equation (1)

At the outer surface where r = ro = 0.05 m

At the inner surface where r = ri = 0.025 m

Internal Pressure Only

Now we consider a thick-walled cylinder subject to just an internal pressure p.

Then, we have

With these boundary conditions, equation (2) becomes

– p a b

r

a b

r

= +

= +

i 2

o 2

and 0

σ σ3 3 0= = = =– p r r r r at and at i o

σ1 6 2

6

2 67 10 5833

0 025

6 66 10

= × +

= ×

– . .

. Pa

σ1 6 22 67 10 5833 0 05

336 800

= × +

=

– . .

– Pa

σ1 2= a b

r –

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Thus, the constants a and b can be determined as

Then, substituting a and b into equations (1) and (2) gives

Note that both hoop stress σ1 and radial stress σ3 are functions of the radius r. The maximum value and minimum value of σ3 are –p at r = ri and 0 at r = ro, respectively. The maximum value and minimum value of σ1 are

when r = ri, and

when r = ro, respectively.

σ1 2( ) =

min i 2

o 2

i 2

r

r r p

– ............................................. ...... 6( )

σ1( ) = +

max o 2

i 2

o 2

i 2

r r

r r p

– ................................................... 5( )

σ 3 21= ⎛ ⎝⎜

⎞ ⎠⎟

– –

................ r

r

r

r r po

2 i 2

o 2

i 2

..................... 3

11

( )

= +and o 2

σ r

rr

r

r r p

2

⎛ ⎝⎜

⎞ ⎠⎟

i 2

o 2

i 2–

..................................... 4( )

b r r

r r p

a b

r

r

r r

=

= =

i 2

o 2

i 2

o 2

o 2

i 2

o 2

and

–

– – ii

2 p

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FIGURE 2 shows the distribution of hoop stress σ1 and radial stress σ3 across the section of a cylindrical vessel carrying internal pressure p.

FIG. 2 Stress distribution on the cross-section of a thick-walled cylinder

subject to internal pressure

0

σ 1

σ 3

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LONGITUDINAL STRESS σσ2

The longitudinal stress σ2 for a closed thick-walled cylindrical pressure vessel can be obtained from a consideration of the equilibrium of a transverse section.

FIGURE 3 shows the arrangement for the simple case where the cylinder is

subject to internal pressure p.

FIG. 3 Longitudinal equilibrium at the end of cylinder

So assuming equilibrium conditions and resolving force balance in the

horizontal direction, we have

From the above equation, it can be seen that longitudinal stress σ2 is constant for a certain cylindrical pressure vessel, as it only depends on the internal

pressure and the geometry of the cylinder. The distribution of σ2 is uniform throughout the wall.

p r r r

r

r r p

π πi 2

o 2

i 2

i 2

o 2

i 2

and

= ( )

=

σ

σ

2

2

–

– .............................................. 7( )

p r

o r

i

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Using a similar method, we can obtain the longitudinal stress σ2 for a thick- walled cylinder subject to both internal pressure pi and external pressure po,

given by

The hoop stress σ1, longitudinal stress σ2, and radial stress σ3 are all principal stresses for cylindrical pressure vessels. Comparing equations (3), (4) and (7),

we can find that at any radius r the magnitude of theses stresses has the

following relation

σ3 < σ2 < σ1

Example 2

A cylindrical vessel has an outside diameter of 400 mm and an inside diameter

of 300 mm. For an internal pressure of 20.1 MPa, calculate the hoop stress σ1 and radial stress σ3 at the inner and outer surfaces and at points within the wall at intervals of 10 mm. Plot the graphs of σ1 versus r and σ3 versus r.

σ 2 = p r p r

r r i i

2 o o

2

o 2

i 2

–

– .......................................... 8( )

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Solution

From the question:

r

r

p

o

i

mm 0.2 m

mm 0.15 m

= = =

= = =

=

400 2

200

300 2

150

20 1. MMPa 20.1 10 Pa

0.15 m

0.16 m

6

i

i

= ×

= =

= + =

r r

r r

r

1

2 0 01.

33

4

5

0 01

0 01

0 01

= + =

= + =

= + =

r

r r

r r

2

3

4

0.17 m

0.18 m

.

.

. 00.19 m

0.2 m

wall thickness

o

o i

r r

t r r

6

0 2 0

= =

= =– . – .. .

.

.

15 0 05

0 15 0 05

3

=

= = ∴

m

the vessi r

t eel is thick-walled.

12

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Using equation (4), the hoop stress σ1 at different radii can be obtained as shown in the table below.

FIGURE 4 shows how the hoop stress σ1 varies with the radius r.

Using equation (3), the radial stress σ3 at different radii can be determined, as shown in the table below.

The distribution of the radial stress σ3 across the wall is shown in FIGURE 5.

radius (m) 0.15 0.16 0.17 0.1r 88 0.19 0.20

radial stress (MPa) 20.1 14.53σ – – –99.93 6.06– – .2 79 0

σ 3 21= ⎛ ⎝⎜

⎞ ⎠⎟

⎛ ⎝⎜

⎞ ⎠⎟

– –

r

r

r

r r po

2 i 2

o 2

i 2

radius (m) 0.15 0.16 0.17 0.18r 00.19 0.20

hoop stress (MPa) 71.8 66.2 61.6 571σ ..7 54 5 51 7. .

σ1 21= + ⎛ ⎝⎜

⎞ ⎠⎟

⎛ ⎝⎜

⎞ ⎠⎟

r

r

r

r r po

2 i 2

o 2

i 2–

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FIG. 4 Graph of hoop stress σ1 versus radius r

Graph of radial stress σe versus radius r

Example 3

The cylinder of a hydraulic ram has a internal diameter of 100 mm. What wall

thickness will be required to withstand an internal guage pressure of 30 MPa if

the maximum permissible tensile stresss is 60 MPa.

r o

= 200 mm

–20.1 MPa

r i = 150 mm

51.7 MPa

r i = 150 mm

r o

= 200 mm

71.8 MPa

Hoop stress in cylinder wall

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Solution

From the question:

Since σ3 < σ2 < σ1, the maximum tensile stress will be given by equation (5) as

which gives

Thus

The required wall thickness is therefore

0 087 0 05 0 037. – . .= =m 37 mm

r

r

o 2

o m 87 mm

= ×

= =

3 0 05

0 087

2.

.

σ1 6

6

60 10

60 10 0

( ) = × = +

× = +

max o 2

i 2

o 2

i 2

o 2

r r

r r p

r

–

.005

0 05 30 10

2 0 05

0 05

2

2

2 6

2

2

r

r

r

r

o 2

o 2

o 2

o

– .

.

– .

× ×

= +

22 o 2– . .2 0 05 0 052 2× = +r

r

p

i

6

mm 0.05 m

MPa 30 10 Pa

= = =

= = ×

100 2

50

30

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________________________________________________________________________________________

SELF-ASSESSMENT QUESTIONS ________________________________________________________________________________________

1. Calculate the magnitude of the maximum longitudinal, hoop and radial

stresses in a cylinder carrying helium at a steady pressure of 70 MPa. The

outside diameter of the cylinder is 200 mm and the inside diameter is

160 mm.

2. The barrel of a large field artillery piece has a bore of 220 mm and an

outside diameter of 300 mm.

(a) Calculate the magnitude of the hoop stress in the barrel at points

10 mm apart from inside to the outside surface. The internal pressure

is 50 MPa.

(b) Draw the hoop stress distribution across the wall according to the

calculation results.

3. The cylinder of a hydraulic actuator has a bore of 100 mm and is required

to operate up to a pressure of 12 MPa. Determine the required wall

thickness for a limiting tensile stress of 36 MPa.

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NOTES ________________________________________________________________________________________

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ANSWERS TO SELF-ASSESSMENT QUESTIONS ________________________________________________________________________________________

1. From the question:

Using equation (5), the maximum hoop stress is

σ1

2 2

2

0 1 0 08 0 1 0 08

( ) = +

= +

max o 2

i 2

o 2

i 2

r r

r r p

–

. .

. – . 22 6

6

70 10

319 10

× ×

= ×

=

Pa

319 MPa

p

r

r

= = ×

= = =

= =

70

160 2

80

200 2

1

MPa 70 10 Pa

mm 0.08 m

6

i

o 000

0 1 0 0

mm 0.1 m

The wall thickness o i

=

= =t r r– . – . 88 0 02

0 08 0 02

4

=

∴ = =

.

.

.

m

(the cylini r

t dder is thick-walled)

18

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Using equation (7), the maximum longitudinal stress

The maximum radial stress is:

Note that all these maximum stresses are at the inner surface of the

cylinder.

σ 3 max MPa( ) = =– –p 70

σ 2

2

2 2 60 08

0 1 0 08 70 10

( ) =

= × ×

=

max i 2

o 2

i 2

r

r r p

–

. . – .

1124 106×

=

Pa

124 MPa

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2. (a) From the question:

Using equation (4), the hoop stress σ1, at different radii can be calclated

The results are given in the table below.

radius (m) 0.11 0.12 0.13 0.14 0.15

h

r

ooop stress (Pa) 166 10 149 10 136 10 1251 6 6 6σ × × × ×× ×10 116 106 6

σ1 21= + ⎛ ⎝⎜

⎞ ⎠⎟

r

r

r

r r po

2 i 2

o 2

i 2–

wall thickness mo i

i

t r r

r

t

= = =

=

– . – . .

.

0 15 0 11 0 04

0 111 0 04

2 75 .

.= ∴ the cylinder is thick-wallled( )

r

r

p

i

o

mm 0.11 m

mm 0.15 m

M

= = =

= = =

=

220 2

110

300 2

150

50 PPa 50 10 Pa

m

m

6

i

1

= ×

= =

= + =

=

r r

r r

r r

1

2

3

0 11

0 01 0 12

.

. .

22

3

4

m

m

+ =

= + =

= + =

0 01 0 13

0 01 0 14

0 01 0 1

4

5

. .

. .

. .

r r

r r 55 m

20

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(b)

3. From the question:

The maximum tensile stress is 36 MPa = 36 × 106 Pa

Since the maximum tensile hoop stress occurs at the inner surface, then

Using equation (5)

we have

σ1

6 2

36 10 0 05

( ) = +

× = +

max o 2

i 2

o 2

i 2

o 2

o 2

r r

r r p

r

r

–

.

– 00 05 12 10

2 6

. × ×

σ1 636 10 0 05( ) = × =max i Pa at mr .

r

p

i

6

mm 0.05 m

MPa 12 10 Pa

= = =

= = ×

100 2

50

12

0.11

170 × 106

160 × 106

150 × 106

140 × 106

130 × 106

120 × 106

110 × 106

0.12 0.13 0.14

σ 1

(Pa)

0.15 r (m)

Hoop stress distribution across wall

21

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Therefore, the required wall thickness for a limiting tensile stress of

36 MPa is

t r ri=

=

= =

o

m 21 mm

–

. – .

.

0 071 0 05

0 021

3 0 05

0 05

3 3 0 05 0 05

2

2

2

2 2

= +

× = +

r

r

r r

o 2

o 2

o 2

o 2

.

– .

– . .

rr

r

r

o 2

o 2

o m 71 m

= ×

= ×

∴ = =

4 0 05

2 0 05

0 071

2

2

.

.

. mm

22

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________________________________________________________________________________________

SUMMARY ________________________________________________________________________________________

You should now be able to apply Lamé’s theory and boundary conditions to

calculate hoop stress, longitudinal stress and radial stress occurring in the wall

of a thick-walled cylindrical pressure vessel. Also, you should be aware to that

hoop stress and radial stress vary with radius, unlike those in thin-walled

cylinders.

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