hes5320 solid mechanics lab (thick-walled cylinder) by stephen bong

SWINBURNE UNIVERSITY OF TECHNOLOGY (SARAWAK CAMPUS)

FACULTY OF ENGINEERING AND INDUSTRIAL SCIENCE

HES5320 Solid Mechanics Semester 2, 2011

Lecturer: Dr. Saad A. Mutasher

Lab Supervisor: Tay Chen Chiang

Laboratory Report: Thick-walled Pressure Vessels

By

Stephen Bong (4209168)

Ngui Yong Zit (4201205)

Ling Wang Soon (4203364)

Date Performed Experiment: 28th

October 2011

Thick-walled Pressure Vessels 4209168; 4201205; 4203364

HES5320 Solid Mechanics, Semester 2, 2011 of 17

1. INTRODUCTION

Thick-walled cylinders are now extensively applicable in wide range of industries such as the pressure

vessels utilized in nuclear and steam power plants for fluids transmission. Such applications may touch

upon the existence of high pressures and temperatures exerted by the working fluids which might leads to

stress corrosion cracking (A. B. Ayob, et al., 2009; J. M Kihiu, et al., n. d. pp. 370). Hence, in order to

diminish the probability of disruptive failures, the development of rightful comprehending on the analysis

of stresses and strains distributed in the thick-walled cylinders from solid mechanics is significant.

The conversion of strains in the strain gauges obtained from this experiment to experimental stresses by

utilizing the theoretical equations and compare with theoretical stresses are the primary objectives of this

experiment. The equipment used in this experiment is the LS-22014 THICK CYLINDER APPARATUS

which comprises of a thick wall cylinder, pressure gauge, relief valve, ON/OFF switch, digital strain

meter, strain reading selector switch, hydraulic pump, and an oil refill port as shown in Fig. 1 and Fig. 2

below:

Fig. 1: LS-22014 THICK CYLINDER APPARATUS

Parts Name

A Thick wall cylinder

B Pressure gauge

C Relief valve

D ON/OFF switch

E Digital strain meter

F Strain reading selector switch

G Oil refill port (smaller screw)

Table 1: Parts labeled in Fig. 1 with their names



Fig. 2: Strain gauges in LS-22014 THICK CYLINDER APPARATUS

Gauge No. Radius (mm) Strain

1 29.50 Hoop

2 29.50 Radial

3 38.00 Hoop

4 38.00 Radial

5 47.50 Hoop

6 47.50 Radial

7 62.50 Hoop

8 62.50 Radial

9 19.00 Longitudinal

10 19.00 Circumferential

11 74.50 Circumferential

12 74.50 Longitudinal

Table 2: Description of strains contained in the strain gauges in Fig. 2

2. THEORY AND ANALYSIS

In the case of thick cylinder, the stresses distributed along the longitudinal direction, σz can be neglected

and only a biaxial stresses need to be taken into account. Fig. 3 below shows an element of a material at

some radius, r and contained within the elemental cylinder. The cylinder is subjected to an internal

pressure of pi. Due to the biaxial stress distribution based on the assumption made above, the principal

stresses σθ and σr are acting on this element where the principal strains (εr, εθ, and εz) set up by these

stresses can be determined by using the following consecutive equations:

( )

( )

( )zr

r

zr

r

zr

r

EE

EE

EE

σσυσ

ε

σσυσ

ε

σσυσ

ε

θ

θ

θ

+−=

+−=

+−=



Fig. 3: Distribution of biaxial stresses in the cylinder (P. P. Benham, et al., 1996, pp. 383)

The equations of equilibrium for the element is given by

0=−

+rdr

d rr θσσσ

which leads to the general solutions:

2r

BAr −=σ and

2r

BA +=θσ Eq. [1]

where A and B are constants which can be determined by utilizing the boundary conditions.

Consider the cylinder with piston as shown in Fig. 4 below:

Fig. 4: Cylinder with piston (P. P. Benham, et al., 1996, pp. 383)

The boundary conditions for the cylinder with piston as shown in Fig. 4 above are:

At r = ri, σr(r = ri) = – pi and At r = ro, σr(r = ro) = – po

Thus, 2

i

ir

BAp −=− and

2

o

or

BAp −=− .



By solving these two equations simultaneously gives:

22

22

io

ooii

rr

rprpA

−

−= and

22

22)(

io

oioi

rr

rrppB

−

−=

Substitute A and B into Eq. [1] gives the following Lamé’s formulations:

+−

+

−=

−

−+

−

−=

−−

−

−=

−

−−

−

−=

2

2

2

222

22

22

22

2

2

2

222

22

22

22

111

1)(

111

1)(

r

rkp

r

rp

krr

rrpp

rr

rprp

r

rkp

r

rp

krr

rrpp

rr

rprp

i

o

o

i

io

oioi

io

ooii

i

o

o

i

io

oioi

io

ooii

r

θσ

σ

Eq. [2]

where σr = Radial stress (MPa)

σθ = Circumferential stress (Hoop stress) (MPa)

ri = Inner radius (m)

ro = Outer radius (m)

k = ro/ri (Radius ratio)

For the case which the cylinder only exposed to internal pressure (po = 0), the radial and hoop stresses are

given by:

+

−=

+

−=

−

−=

−

−=

2

2

2

22

2

2

2

2

22

2

11

1

11

1

r

r

k

p

r

r

rr

rp

r

r

k

p

r

r

rr

rp

oio

io

ii

oio

io

ii

r

θσ

σ

Eq. [3]

At the inner surface, r = ri,

Radial stress, ir p−=σ (radial compressive stress)

Circumferential or hoop stress, ip

k

k

1

12

2

−

+=θσ

At the outer surface, r = ro,

Radial stress, 0=rσ

Circumferential or hoop stress, 1

22

−=

k

pi

θσ



Fig. 5: Stress distribution for σr and σθ for the thick cylinder which subjected to internal pressure only

with a k ratio of 3 (P. P. Benham, et al., 1996, pp. 387)

Since the thick cylinder is only subjected to internal pressure, pi, and due to the biaxial stress distribution,

by the consecutive equations, the principal strains set up the principal stress σr and σθ are:

( )

( )

( )

+=

−=

−=

θ

θθ

θ

σσυ

ε

υσσε

υσσε

rz

r

rr

E

E

E

1

1

Eq. [4]

Substitute Eq. [3] into Eq. [4] yields

−−

+

−=

+−

−

−=

22

2

22

2

11)1(

11)1(

r

r

r

r

kE

p

r

r

r

r

kE

p

ooi

ooi

r

υε

υε

θ

Eq. [5]

where E = 6.895 × 1010

Pa (Young modulus for Aluminium)

33.0=υ (Poisson ratio)

Cylinder diameter: 150 mm (ro = 75 mm) for this experiment take (ro = 74.5 mm)

Cylinder length: 320 mm

Wall thickness: 55 mm (ri = 20 mm) for this experiment take (ri = 18.625 mm)

Radius ratio = ro/ri = 74.5/18.625 = 4



Substitute these values into Eq. [5] gives:

−−

+×=

+−

−×=

−

−

22

13

22

13

5500133.0

5500110524.9

5500133.0

5500110524.9

rrp

rrp

i

ir

θε

ε

Eq. [6]

The relationship which used for the computation of stresses based on the value of strains obtained can be

derived from Eq. [4].

From Eq. [4], the radial and hoop stresses can be expressed as:

[B] Eq. ----------

[A] Eq. ----------

r

rr

E

E

υσεσ

υσεσ

θθ

θ

+=

+=

By substituting Eq. [B] into Eq. [A] gives:

−

+=

−

+=

2

2

1

)(

1

)(

υ

υεεσ

υ

υεεσ

θ

θ

θ

r

r

r

E

E

Eq. [7]

3. PROCEDURE

(1) The main power supply (D) was switched ON and current was allowed to pass through the

gauges about five to eight minutes in order to ensure all the strain gauges at steady state

temperature condition.

(2) The relief valve (C) was relieved to ensure the pressure reading is zero.

(3) All the initial readings for 12 strain gauges with zero pressure in the system were recorded from

the strain meter (E).

(4) The relief valve was tightened.

(5) Some pressure was applied slowly into the cylinder by means of the hand pump. The pressure

was pumped up to approximately 10 bar.

(6) All the readings for 12 strain gauges were recorded from the strain meter.

(7) The experiment was repeated until 50 bar with an increment of 10 bar.



4. RESULTS

The experimental strains obtained from the strain meter are recorded and tabulated in Table 3 & 4 below:

Radius (mm) Hoop Strain, εθ (µm) Experimental

0 10 20 30 40 50

29.5 0 5E-06 3E-06 6E-06 5E-06 6E-06

38.0 0 3E-06 2E-06 3E-06 2E-06 4E-06

47.5 0 2E-06 2E-06 1E-06 3E-06 1E-06

62.5 0 1E-06 2E-06 1E-06 1E-06 1E-06

Radius (mm) Radial Strain, εr (µm) Experimental

0 10 20 30 40 50

29.5 0 -5E-06 -6E-06 -6E-06 -6E-06 -4E-06

38.0 0 -2E-06 -4E-06 -3E-06 -2E-06 -3E-06

47.5 0 -1E-06 -2E-06 -3E-06 -1E-06 -2E-06

62.5 0 1E-06 -0.000002 -1E-06 -0.000001 -0.000002

Table 3 & 4: Experimental strains obtained from the strain meter

The theoretical strains are calculated from Eq. [5] by using Microsoft Excel Spreadsheet and the results

are tabulated in Table 5 & 6 below:

Radius (mm) Hoop Strain, εθ (µm) Theoretical

0 10 20 30 40 50

29.5 0 8.64363E-06 1.73E-05 2.59E-05 3.46E-05 4.32E-05

38.0 0 5.46277E-06 1.09E-05 1.64E-05 2.19E-05 2.73E-05

47.5 0 3.72589E-06 7.45E-06 1.12E-05 1.49E-05 1.86E-05

62.5 0 2.42161E-06 4.84E-06 7.26E-06 9.69E-06 1.21E-05

Radius (mm) Radial Strain, εr (µm) Theoretical

0 10 20 30 40 50

29.5 0 -0.000007367 -1.47E-05 -2.21E-05 -2.95E-05 -3.68E-05

38.0 0 -4.18655E-06 -8.37E-06 -1.26E-05 -1.67E-05 -2.09E-05

47.5 0 -2.44967E-06 -4.9E-06 -7.35E-06 -9.8E-06 -1.22E-05

62.5 0 -1.14539E-06 -2.29E-06 -3.44E-06 -4.58E-06 -5.73E-06

Table 5 & 6: Theoretical strains calculated by using Eq. [5]



The graphs of hoop strain, εθ (µm) vs. radius, r (mm) and radial strain, εr (µm) vs. radius, r (mm) for

pressures of 10 bar and 30 bar are plotted by using Microsoft Excel as shown in Fig. 6, 7, 8 & 9 below

respectively.

Fig. 6: Plot of hoop strain, εθ (µm) vs. radius, r (mm) for pi = 10 bar

Fig. 7: Plot of hoop strain, εθ (µm) vs. radius, r (mm) for pi = 30 bar

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

6.00E-06

7.00E-06

8.00E-06

9.00E-06

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Ho

op

Str

ain

, εθ (µ

m)

Radius, r (mm)

Hoop Strain, εθ (µm) vs. Radius, r (mm) (pi = 10 bar)

Experimental

Theoretical

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Ho

op

Str

ain

, εθ (µ

m)

Radius, r (mm)

Hoop Strain, εθ (µm) vs. Radius, r (mm) (pi = 30 bar)

Experimental

Theoretical



Fig. 8: Plot of radial strain, εr (µm) vs. radius, r (mm) for pi = 10 bar

Fig. 9: Plot of radial strain, εr (µm) vs. radius, r (mm) for pi = 30 bar

-8.00E-06

-7.00E-06

-6.00E-06

-5.00E-06

-4.00E-06

-3.00E-06

-2.00E-06

-1.00E-06

0.00E+00

1.00E-06

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Rad

ial

Str

ain

, ε

r (µ

m)

Radius, r (mm)

Radial Strain, εr (µm) vs. Radius, r (mm) (pi = 10 bar)

Experimental

Theoretical

-2.50E-05

-2.00E-05

-1.50E-05

-1.00E-05

-5.00E-06

0.00E+00

5.00E-06

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Rad

ial

Str

ain

, ε

r (µ

m)

Radius, r (mm)

Radial Strain, εr (µm) vs. Radius, r (mm) (pi = 30 bar)

Experimental

Theoretical



The experimental and theoretical stresses are computed by using Eq. [7] and the results are tabulated in

Table 7, 8, 9 & 10 below:

Radius (mm)

Hoop Stress, σθ (MPa) Experimental

0 10 20 30 40 50

29.5 0 0.259211 0.078924 0.311053 0.233676 0.362121

38.0 0 0.18106 0.052616 0.155526 0.103684 0.232903

47.5 0 0.129218 0.103684 0.000774 0.206595 0.026308

62.5 0 0.10291 0.103684 0.051842 0.051842 0.026308

Radius (mm)

Hoop Stress, σθ (MPa) Theoretical

0 10 20 30 40 50

29.5 0 0.480691123 0.961382 1.442073 1.922764 2.403456

38.0 0 0.315788402 0.631577 0.947365 1.263154 1.578942

47.5 0 0.22574505 0.45149 0.677235 0.90298 1.128725

62.5 0 0.158128496 0.316257 0.474385 0.632514 0.790642

Radius (mm) Radial Stress, σr (MPa) Experimental

0 10 20 30 40 50

29.5 0 -0.259211 -0.387655 -0.311053 -0.336587 -0.1563

38.0 0 -0.07815 -0.258437 -0.155526 -0.103684 -0.129992

47.5 0 -0.026308 -0.103684 -0.206595 -0.000774 -0.129218

62.5 0 0.10291 -0.103684 -0.051842 -0.051842 -0.129218

Radius (mm) Radial Stress, σr (MPa) Theoretical

0 10 20 30 40 50

29.5 0 -0.349355163 -0.69871 -1.048065 -1.397421 -1.746776

38.0 0 -0.184452442 -0.368905 -0.553357 -0.73781 -0.922262

47.5 0 -0.09440909 -0.188818 -0.283227 -0.377636 -0.472045

62.5 0 -0.026792536 -0.053585 -0.080378 -0.10717 -0.133963

Table 7, 8, 9 & 10: Theoretical and experimental stresses computed by using Eq, [7]



The plots of hoop and radial stresses vs. radius for pressure of 10 bar and 30 bar are shown in Fig. 10, 11,

12 & 13 below:

Fig. 9: Plot of hoop stress, σθ (µm) vs. radius, r (mm) for pi = 10 bar

Fig. 10: Plot of hoop stress, σθ (µm) vs. radius, r (mm) for pi = 30 bar

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Ho

op

Str

ess

, σθ (

MP

a)

Radius, r (mm)

Hoop Stress, σθ (MPa) vs. Radius, r (mm) (pi = 10 bar)

Experimental

Theoretical

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Ho

op

Str

ess

, σθ (

MP

a)

Radius, r (mm)

Hoop Stress, σθ (MPa) vs. Radius, r (mm) (pi = 30 bar)

Experimental

Theoretical



Fig. 11: Plot of hoop stress, σr (µm) vs. radius, r (mm) for pi = 10 bar

Fig. 12: Plot of hoop stress, σr (µm) vs. radius, r (mm) for pi = 30 bar

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0R

ad

ial

Str

ess

, σr (

MP

a)

Radius, r (mm)

Radial Stress, σr (MPa) vs. Radius, r (mm) (pi = 10 bar)

Experimental

Theoretical

-1.05

-0.85

-0.65

-0.45

-0.25

-0.0525.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0

Ra

dia

l S

tre

ss, σr (

MP

a)

Radius, r (mm)

Radial Stress, σr (MPa) vs. Radius, r (mm) (pi = 30 bar)

Experimental

Theoretical



5. DISCUSSION AND CONCLUSION

The hoop strain and radial strain at various locations (strain gauges) with the increment of pressure are

measured in this experiment. The principal stresses distributed along the LS-22014 THICK CYLINDER

APPARATUS are the hoop (circumferential) stress, σθ and radial (longitudinal) stress, σr due to the

biaxial stress distribution property. The primary objectives of this experiment are to convert both the

experimental and theoretical strains to stresses and compare the deviations of the experimental and

theoretical stresses.

Based on the results obtained from numerical computations by using Microsoft Excel Spreadsheet which

are tabulated in tables above, the deviations between the values of strains obtained from experiment and

theoretical calculations are quite small. Even the distinctions between the results obtained by utilizing

both the two methodologies are small, but there is still the existence of errors. Among the reasons which

leads to the occurrence of errors are accuracy of data obtained from the experiment or rounding of

decimal places during the numerical computations by using Microsoft Excel, human error which occurs

during the experiment as a result of fluctuation occurs on the strain meter especially when the selector

switch (F) is controlled in order to obtained the appropriate internal pressure which required in this

experiment. Apart from that, machine error or calibrating error will occurs in conjunction with the

experiment as well.

According the strains and stresses versus radius plots as shown in previous section, the graphs of hoop

strain of the experimental values follows the same trend and acts consistently to the series of theoretical

values. Likewise, both the graphs of the hoop stress and the radial stress behave alike except for the plot

of radial stress vs. radius for an internal pressure of 30 bar as shown in Fig. 12 above. As the radial

distance increased, the stresses distribution obtained from experiment and theoretical calculations seems

like to overlap each other when the radial distance approach the outer diameter of the thick-walled

cylinder.



6. APPENDICES

6.1 Sample Calculations

The sample calculations shown below are the calculations for experimental and theoretical strains and

experimental and theoretical stresses with an internal pressure of pi = 10 bar and a radius of r = 29.5 mm.

Given that E = 6.985 × 1010 Pa

33.0=υ

ro = 74.5 mm

k = 4

Theoretical Hoop Strain

( )

( ) ( )( )

6108.7164

−

−

−

×=

−−

+××=

−−

+×=

−−

+

−=

22

513

22

13

2

2

2

2

2

)5.29(

5550133.0

)5.29(

55501101010524.9mm 29.5 bar, 10

5550133.0

5550110524.9

11)1(

,

θ

θ

ε

υε

rrp

r

r

r

r

kE

prp

i

ooi

i

Theoretical Radial Strain

( )

( ) ( )( )

6107.4402

−

−

−

×−=

+−

−××=

+−

−×=

+−

−

−=

22

513

22

13

2

2

2

2

2

)5.29(

5550133.0

)5.29(

55501101010524.9mm 29.5 bar, 10

5550133.0

5550110524.9

11)1(

,

r

i

ooi

ir

rrp

r

r

r

r

kE

prp

ε

υε

Experimental Hoop Stress

Based on the tables above, the experimental hoop strain and radial strain at r = 29.5 mm with an internal

pressure of pi = 10 bar are εθ = 5E-06 and εr = -5E-06.

( )

( ) ( ) ( )[ ]( )

MPa 0.259210=

=

−

×−+××=××−

−

+=

−−

−−

Pa 53.259210

33.01

10533.0105)10895.6(105,105

1

)(,

2

661066

2

θ

θθθ

σ

υ

υεεεεσ r

r

E



Experimental Radial Stress

( )

( ) ( ) ( )[ ]( )

MPa 0.259210−=

−=

−

×+×−×=××−

−

+=

−−

−−

Pa 53.259210

33.01

10533.0105)10895.6(105,105

1

)(,

2

661066

2

r

r

rr

E

σ

υ

υεεεεσ θ

θ

Theoretical Hoop Stress

Based on the table above, the theoretical hoop strain and radial strain at pi = 10 bar and r = 29.5 mm are εθ

= 8.64363E-06 and εr = -7.367E-06.

( )

( ) ( ) ( )[ ]( )

MPa 0.48070167=

=

−

×−+××=××−

−

+=

−−

−−

Pa 67.480701

33.01

10367.733.01064363.8)10895.6(1064363.8,10367.7

1

)(,

2

661066

2

θ

θθθ

σ

υ

υεεεεσ r

r

E

Theoretical Radial Stress

( )

( ) ( ) ( )[ ]( )

MPa 0.3493231−=

−=

−

×+×−×=××−

−

+=

−−

−−

Pa 10.349323

33.01

1064363.833.010367.7)10895.6(1064363.8,10367.7

1

)(,

2

661066

2

r

r

rr

E

σ

υ

υεεεεσ θ

θ



6.2 References

Ayob, A. B., Tamin, M. N. & M. Kabashi Elbasheer, ‘Pressure Limits of Thick-Walled Cylinders’,

Proceedings of the International MutiConference of Engineers and Computer Scientists 2009 Vol.

II, IMECS: 2009, March 18, Hong Kong.

J. M. Kihiu, S. M. Mutuli & G. O. Rading, n. d., Stress characterization of autofrettaged thick-walled

cylinders, pp. 370, International Journal of Mechanical Engineering Education, 31/4, Department of

Mechanical Engineering, University of Nairobi.

P. P., Benham, R. J., Crawford & C. G., Armstrong, 1996, ‘Chapter 14 – Applications of the Equilibrium

and Strain-Displacement Relationships’, in Mechanics of Engineering Materials, 2nd

edn., Pearson

Longman, China.

hes5320 solid mechanics lab (thick-walled cylinder) by stephen bong

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