Minimization of stress concentration factor in cylindrical pressure vessels

with ellipsoidal heads

K. Magnuckia,b,*, W. Szyca, J. Lewinskia

aInstitute of Applied Mechanics, Poznan University of Technology, ul. Piotrowo 3, 60-965 Poznan, PolandbInstitute of Rail Vehicles ‘TABOR’, ul. Warszawska 181, 61-055 Poznan, Poland

Received 17 April 2002; revised 11 September 2002; accepted 11 September 2002

Abstract

The paper presents the problem of stress concentration in a cylindrical pressure vessel with ellipsoidal heads subject to internal pressure.

At the line, where the ellipsoidal head is adjacent to the circular cylindrical shell, a shear force and bending moment occur, disturbing the

membrane stress state in the vessel. The degree of stress concentration depends on the ratio of thicknesses of both the adjacent parts of the

shells and on the relative convexity of the ellipsoidal head, with the range for radius-to-thickness ratio between 75 and 125. The stress

concentration was analytically described and, afterwards, the effect of these values on the stress concentration ratio was numerically

examined. Results of the analysis are shown on charts.

q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Ellipsoidal head; Pressure vessel; Stress concentration factor; Head convexity; Internal pressure

1. Introduction

Circular cylindrical vessels are usually closed with

convex heads of torispherical or ellipsoidal type. The

heads disturb the membrane stress state occurring in the

cylindrical vessels loaded with uniform pressure. Błachut

[1] numerically and experimentally investigated the stress

and strain distribution in the elastic–plastic state of

torispherical heads loaded by internal pressure. Soric and

Zahlten [2] carried out similar numerical studies of

torispherical shells. Błachut and Ramachandra [3] and

Błachut et al. [4] extended the studies to shakedown

loads. Magnucki and Szyc [5] numerically investigated

the stress and strain distributions in elastic and plastic

states of ellipsoidal heads for internal and external

pressure. They also determined internal and external

critical pressure values of these heads. Błachut and

Galletly [6] presented the results of experimental and

numerical studies of torispherical and hemispherical

shells made from fibre-reinforced plastic and loaded

with external pressure. Błachut and Galletly [7]

investigated the influence of local shape imperfections

on the elastic buckling of torispheres and hemispheres.

Błachut and Jaiswal [8] widened the studies to ellipsoidal

and toroidal shells. Ross [9] investigated vibration and

elastic buckling of thin-walled hemi-ellipsoidal domes

under uniform external pressure. Błachut [10 – 12]

determined optimal profiles of torispherical heads under

buckling constraints. Then the studies were extended to

parabolic and cubic splines together with arcs to

approximate the meridional shapes of multisegmental,

axisymmetric, externally pressurized domes. Further

generalization consists in description of meridional

shapes by a set of functions. Magnucki and Lewinski

[13,14] defined optimal shapes of heads of cylindrical

pressure vessels under strength and geometric constraints.

Wilczynski [15] numerically determined by means of

FEM an optimal shape of thin elastic shells of revolution

loaded by internal pressure. Magnucki [16] optimized

cylindrical vessels with ellipsoidal and special shapes of

heads and discussed [17] the problems of strength,

stability, and optimal shaping of thin-walled vessels. He

paid attention to stress concentration in heads and

cylindrical shells of pressure vessels. Kedziora and

Kubiak [18], using FEM, numerically calculated the

stress distribution in pressure tanks. Magnucki and

Monczak [19] provided an analytical concentration of

0308-0161/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 30 8 -0 16 1 (0 2) 00 1 01 -1

International Journal of Pressure Vessels and Piping 79 (2002) 841–846

www.elsevier.com/locate/ijpvp

* Corresponding author. Address: Institute of Applied Mechanics,

Poznan University of Technology, ul. Piotrowo 3, 60-965 Poznan,

Poland. Tel.: þ48-61-6652301; fax: þ48-61-6652307.

E-mail address: jerzy.lewinski@put.poznan.pl (K. Magnucki).

stresses in cylindrical tanks, and defined minimal

thickness values of the ellipsoidal head and the

cylindrical shell. Ziołko and Mikulski [20] investigated

the effect of head shape on stress state in a cylindrical

pressure tank.

The present paper describes the effect of wall

thickness ratio of the ellipsoidal head and cylindrical

shell, as well as the relative convexity of the ellipsoidal

head on the stress concentration factor in circular

cylindrical pressure vessels. Maxima of equivalent

stresses occur in the ellipsoidal head and in the

cylindrical shell. The maximal values are different and

depend on the head and cylindrical shell thicknesses and

on convexity of the head. The effective cylindrical

pressure vessel might be defined as the one, in which

maximal values of the equivalent stresses are equal.

2. Stress state in the cylindrical pressure vessel

with ellipsoidal heads

The subject of the analysis is a pressurized cylindrical

vessel with ellipsoidal heads. The radius of the cylindrical

shell is a, thickness of its wall t2, convexity of the ellipsoidal

head is b and its thickness t1. The vessel is loaded by

uniform internal pressure p0. At the boundary between the

ellipsoidal head and the cylindrical shell, a longitudinal

force Nz ¼ ap0=2 occurs, and unknown shear force Q0 and

the bending moment M0 (Fig. 1). The quantities Q0 and M0

reflect an interaction between the connected parts of the

pressure vessel and may be determined from the theory of

boundary disturbance, the use of which in the field of

strength analysis of thin-walled vessels was discussed by

Magnucki [17].

The radial displacement u0 and the rotation angle qð1Þ0 of

the boundary ðu ¼ p=2Þ of the ellipsoidal shell-head mat be

written

u0 ¼ up0 þ a

ð1Þ11 Q0 þ a

ð1Þ12 M0;

qð1Þ0 ¼ qp

0 þ að1Þ12 Q0 þ a

ð1Þ22 M0;

ð1Þ

where

up0 ¼ 2 2 n2

a

b

� �2" #

p0a2

2Et1; qp

0 ¼ 0;

cn ¼ ½3ð1 2 n2Þ�1=4;

að1Þ11 ¼

2

Ecn

a

t1

� �3=2

; að1Þ12 ¼ 2

2

Et1

c2n

a

t1

;

að1Þ22 ¼

4

Et21

c3n

a

t1

� �1=2

;

with E being Young’s modulus and n being Poisson’s ratio.

The deflection w0 and the rotation angle qð2Þ0 of the

boundary ðz ¼ 0Þ of the circular cylindrical shell are

w0 ¼ wp0 þ a

ð2Þ11 Q0 þ a

ð2Þ12 M0;

qð2Þ0 ¼ qp

0 þ að2Þ12 Q0 þ a

ð2Þ22 M0;

ð2Þ

where

wp0 ¼ ð2 2 nÞ

p0a2

2Et2

; qp0 ¼ 0;

að2Þ11 ¼ 2

2

Ecn

a

t2

� �3=2

; að2Þ12 ¼ 2

2

Et2c2n

a

t2

;

að2Þ22 ¼ 2

4

Et21

c3n

a

t2

� �1=2

:

At the boundary connecting both shells the condition of

consistency of displacements should be met:

u0 ¼ w0; qð1Þ0 ¼ q

ð2Þ0 : ð3Þ

The above makes a system of two algebraic equations,

enabling evaluation of the shear force Q0 and the bending

moment M0. Then the stress state in the pressure vessel may

be determined.

The dimensionless meridional sð1Þu and circumferential

sð1Þw stresses in the ellipsoidal shell-head are

sð1Þu ¼

x1x2

2b

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x sin2u

p ^ f ð1ÞuBD expð2c1Þ;

sð1Þw ¼

x1x2

2b

1 2 x sin2uffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x sin2u

p 2 f ð1ÞwBD expð2c1Þ;

ð4Þ

where

f ð1ÞuBD ¼ sð1ÞM ðcos c1 þ sin c1Þ2

3

c2n

sð1ÞQ sinc1;

Fig. 1. Connection of ellipsoidal and cylindrical shells.

K. Magnucki et al. / International Journal of Pressure Vessels and Piping 79 (2002) 841–846842

f ð1ÞwBD ¼ sð1ÞM

c2n

37 n

!cos c1 2

c2n

3^ n

!sin c1

" #

2 sð1ÞQ cos c1 7

3n

c2n

sin c1

� �;

sð1ÞM ¼

6M0

t21

; sð1ÞQ ¼ 2cn

ffiffiffiffiffiffix1x2

p Q0

t1

;

c1 ¼ cn

ffiffiffiffiffiffiffix1x2

b

r ðp=2

u

du

ð1 þ x sin2uÞ5=4;

the upper sign corresponding to the internal surface, the

lower sign to the external one.

The dimensionless longitudinal sð2Þz and circumferential

sð2Þw stresses in the cylindrical shell follow as

sð2Þz ¼ 1

2x2 ^ f ð2ÞzBD expð2c2Þ;

sð2Þw ¼ x2 2 f ð2ÞwBD expð2c2Þ;

ð5Þ

where

f ð2ÞzBD ¼ sð2ÞM ðcos c2 þ sin c2Þ þ

3

c2n

sð2ÞQ sin c2;

f ð2ÞwBD ¼ sð2ÞM

c2n

37 n

!cos c2 2

c2n

3^ n

!sin c2

" #

þ sð2ÞQ cos c2 7

3n

c2n

sin c2

� �;

sð2ÞM ¼

6M0

t22

; sð2ÞQ ¼ 2cn

ffiffiffix2

p Q0

t2; c2 ¼

cnffiffiffix2

pz

t2

:

The following dimensionless parameters of the structure are

used here:

x1 ¼ t2=t1—the ratio of wall thicknesses,

x2 ¼ a=t2—the characteristic ratio of the thin-walled

cylindrical shell,

b ¼ b=a—the relative convexity of the ellipsoidal head

of the pressure vessel.

The meridional or longitudinal and circumferential

stresses are the products of respective dimensionless

stresses and the uniform pressure p0. Then

sðiÞj ¼ s

ðiÞj p0;

for i ¼ 1; j ¼ u;w; and i ¼ 2; j ¼ z;w:

ð6Þ

The equivalent stress

sðiÞeq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½s

ðiÞj �2 2 s

ðiÞj s

ðiÞw þ ½s

ðiÞw �

2q

: ð7Þ

The stress concentration factor is defined in the following

form

aSC ¼seq;max

sð1Þeq;0

; ð8Þ

where seq;max ¼ max½sð1Þeq;max; s

ð2Þeq;max� : maximal equivalent

stress in the pressure vessel, sð2Þeq;0 ¼ ð

ffiffi3

p=2Þx2 : the

equivalent stress of the cylindrical shell in the membrane

state.

The value of the stress concentration factor (8) may be

adjusted by means of the dimensionless parameters x1, x2,

and b. The pressure vessel is effectively designed, when the

maximal equivalent stresses in the ellipsoidal head and the

cylindrical shell are equal:

sð1Þeq;max ¼ sð2Þ

eq;max: ð9Þ

This condition will serve for determining effective values of

the dimensionless structural parameters (x1ef, bef) of

pressure vessels.

3. Numerical analysis

Numerical calculation enables tracing the stresses in the

ellipsoidal and cylindrical shells of the vessel. Example

results of such calculations are shown in Fig. 2 in the form

of charts presenting the values of dimensionless equivalent

stresses of Eq. (7) in both shells of the vessel, at their

external and internal surfaces, versus u for the ellipse and

z=t2 for the cylinder. The results shown here are related to a

cylindrical vessel with ellipsoidal head of relative convexity

b ¼ 0:5; with equal thicknesses of both shells of the vessel

x1 ¼ 1; with the parameter x2 ¼ 100: One can remark that

the maximal stress value occurs at the external surface of the

ellipsoidal head, near the boundaries of both shells and

clearly exceeds the values existing in the cylindrical part of

the vessel. Such a solution is ineffective; therefore, an

ellipsoidal head of greater thickness ðx1 , 1Þ or bigger

relative convexity ðb . 0:5Þ should be applied. Any of the

approaches may lead to equalization of maximal values of

the equivalent stress in both parts of the vessel. Adjusting

the thickness ratio, e.g. to x1 ¼ 0:595 shall lead to the

desired effect, that is shown in Fig. 3. However, such a

solution is disadvantageous in practice, as in order to meet

the condition of Eq. (9) the cylindrical part of the vessel is to

be connected with a head of thickness nearly 1.7 greater.

4. Effective values of relative convexity of the ellipsoidalhead and the ratio of wall thickness

Suitable numerical procedures have been developed in

order to enable searching for the cases exactly satisfying

the condition (9) in the space of the x1 and b parameters

for an imposed value of x2. Results of the search are

depicted by the curve shown in Fig. 4, being a pattern of

the points satisfying the condition (9) with x2 ¼ 100:

Coordinates of the points belonging to ðb; x1Þ may be

considered as effective ðbef ; x1efÞ: Corners in lower and

upper parts of the curve are also justified. The lower

corner is a result of a change in location of the maximal

K. Magnucki et al. / International Journal of Pressure Vessels and Piping 79 (2002) 841–846 843

value of the equivalent stress in the cylindrical shell from

its border, i.e. the place of connection with the head (for

smaller b values), to the area located at a distance equal

to more than ten thicknesses t2 (for greater b values).

The reason for the occurrence of the upper corner of the

curve, in which the maximal value existing in its top part

(for smaller b values) is less than the value at the shell

edge (for greater b values), is similar.

Assuming that wall thicknesses of joined shells are to be

equal, one could easily state that the best relative head

convexity should amount to b ø 0:6: Small changes in x1

(up to ^10%) do not considerably affect the selected value

of the relative convexity.

Should the thicknesses t1 and t2 of both shells differ

considerably from each other, their connection might

give rise to technological difficulties and, therefore,

would not be suitable. Moreover, a better solution

would be characterized by a possibly smaller value of

stress concentration factor, Eq. (8). Therefore, the

changes in the value of the stress concentration factor

should be analyzed, with regard to selected effective

values of the b and x1 parameters. As x1ef is closely

related to bef (Fig. 4), it is sufficient to interrelate the asc

values of one of these parameters. Results of such an

analysis for x2 ¼ 100 are shown in Fig. 5. For bef , 0:5

the values of the stress concentration factor are

Fig. 3. Equalization of maximal equivalent stresses (b ¼ 0:5; x1 ¼ 0:595; x2 ¼ 100).

Fig. 2. Equivalent stresses in ellipsoidal and cylindrical shells (b ¼ 0:5; x1 ¼ 1; x2 ¼ 100).

K. Magnucki et al. / International Journal of Pressure Vessels and Piping 79 (2002) 841–846844

undesirably high, while for bef ¼ 0:5–0:65 the asc values

remain nearly unchanged. In spite of the above, selection

of optimal wall thickness ratio x1 is relatively difficult.

The range x1ef ¼ 0:8–1:0 seems to be the most

reasonable. However, in order to appraise a given variant

accurately another criterion of economic character should

be formed and optimization procedure undertaken, aimed

at obtaining more detailed results.

5. Conclusions

The calculations and analysis enable the following notes

and conclusions to be formulated

in the case of equal wall thicknesses of the cylindrical

part of the vessel and its ellipsoidal head ðx1 ¼ 1Þ

the most effective geometry will be a head of relative

convexity b ø 0:6;

the use of ellipsoidal heads of smaller convexities

induces an increase of the head thickness, that increases

the value of the stress concentration factor asc; in the

range 0:5 # b # 0:6 the increase remains small, but

becomes very high for b , 0:5;

head convexities in the range 0:48 , b , 0:86 produce

maximal equivalent stresses beyond the boundary

between both shells, i.e. beyond the welding zone,

a suitable variant of the pressure vessel (having the most

advantageous parameters) may be selected on the

grounds of other accepted criteria, e.g. minimal mass

of the vessel.

The results of calculation analyses presented in the

present paper were related exclusively to the characteristic

ratio x2 ¼ 100: However, verification calculations in the

range x2 ¼ 75–125 did not show any considerable changes.

References

[1] Błachut J. Plastic loads for internally pressurised torispheres. Int J

Pressure Vessels Piping 1995;64:91–100.

[2] Soric J, Zahlten W. Elastic–plastic analysis of internally pressurized

torispherical shells. Thin-Walled Struct 1995;22:217–39.

[3] Błachut J, Ramachandra LS. Shakedown and plastic loads for

internally pressurised vessel end closures. Int Conf Carrying Capacity

Steel Shell Struct, Brno, Czech Republic 1997;130–6.

[4] Błachut J, Ramachandra LS, Krishnan PA. Plastic and shakedown

loads for internally pressurised domes made from strain hardening

material. Ninth Conf Pressure Vessel Technol, Sydney 2000;57–66.

[5] Magnucki K, Szyc W. Stability of ellipsoidal heads of cylindrical

pressure vessels. Appl Mech Engng 2000;5(2):389–404.

[6] Błachut J, Galletly GD. Externally pressurized hemispherical fibre-

reinforced plastic shells. Proc Inst Mech Engrs 1992;206:179–91.

[7] Błachut J, Galletly GD. Influence of local imperfections on the

collapse strength of domed end closures. Proc Inst Mech Engrs 1993;

207:197–207.

[8] Błachut J, Jaiswal OR. On the choice of initial geometric imperfec-

tions in externally pressurized shells. J Pressure Vessel Technol 1999;

121:71–6.

[9] Ross CTF. Vibration and elastic instability of thin-walled domes under

uniform external pressure. Thin-Walled Struct 1996;26(3):159–77.

[10] Błachut J. Search for optimal torispherical end closures under

buckling constraints. Int J Mech Sci 1989;31(8):623–33.

[11] Błachut J. Influence of meridional shaping on the collapse strength of

FRP domes. Engng Optim 1992;19:65–80.

[12] Błachut J. Externally pressurized filament wound domes—scope for

optimization. Comput Struct 1993;48(1):153–60.

[13] Magnucki K, Lewinski J. Shape optimization of heads of

cylindrical pressure vessels. In: Topping BHV, editor. Advances

Fig. 5. Stress concentration factor asc for effective parameters of the vessel

ðx2 ¼ 100Þ:

Fig. 4. Effective thickness ratio x1 versus relative convexity b ðx2 ¼ 100Þ:

K. Magnucki et al. / International Journal of Pressure Vessels and Piping 79 (2002) 841–846 845

in computational structural mechanics. Edinburgh, Scotland: Civil-

Comp Ltd; 1998. p. 421–5.

[14] Magnucki K, Lewinski J. Fully stressed head of a pressure vessel.

Thin-Walled Struct 2000;38:167–78.

[15] Wilczynski B. Shape optimization of thin elastic shell of revolution

with axisymmetric loading. 16th Conf Poliopt CAD Koszalin 1998;

396–403.

[16] Magnucki K. Optimal design of cylindrical vessels under strength and

stability constraints. Ninth Int Conf Pressure Vessel Technol, Sydney

2000;867–74.

[17] Magnucki K. Strength and optimization of thin-walled vessels.

Warszawa-Poznan: Scientific Publishers PWN; 1998. in Polish.

[18] Kedziora S, Kubiak T. Application of FEM for calculation of stresses

andstrains in pressure tanks.Dozor Techniczny1999;4:83–5. in Polish.

[19] Magnucki K, Monczak T. Determination of minimal wall thickness of

circular cylindrical tank with ellipsoidal heads. Arch Mech Engng,

XLV 1998;2:73–85.

[20] Ziołko J, Mikulski T. Effect of bottom shape upon the strain in

cylindrical pressure tank. XLIII Sci Conf, Comm Civil Engng PAN,

Krynica 1997;157–64. in Polish.

K. Magnucki et al. / International Journal of Pressure Vessels and Piping 79 (2002) 841–846846