parametric studies of cylindrical pressure vessels with different end closures

PARAMETRIC STUDIES OF CYLINDRICAL PRESSURE VESSELS WITH DIFFERENT END CLOSURES

S. K. AGGARWAL

Mechanical Engineering Department, Punjab Engineering College, Ctlandigarh, India

&

G. C. NAYAK & SHANKAR LAL

University of Roorkee, Roorkee-247 672, India

(Received: 31 January, 1978)

ABSTRACT

Recent dez'elopments in the finite element technique using increnwntal elements permit an easier and more precise determination of stresses, strains and displacements in cylindrical pressure l'essels ha~'ing different end closures. In this paper the geometo' analysed is a cylindrieal pressure ~'essel ha~'ing hemispherical, torispherical, semiellipsoidal and torieonical heads. An axisymmetric solid finite element program employing incremental elements was used Jor a useful range of t'essel parameters.

TheJormulation oj the method used and the results of the parametric stud), obtained when internal pressure is applied to the t'essel are presented. Results are reported in theJorm of stress intensity parameters based upon the nwan circumferential stress of the eylindrical portion of the z'essel away Ji'om the ~Tlinder head junction.

INTRODUCTION

The applications of cylindrical pressure vessels with different end closures in the form of processing vessels and nuclear containment vessels demand their economic, reliable and safe design. Various theoretical ~ - s and experimental b- 9 methods and approaches are available in the literature for this purpose. The analysis of cylindrical pressure vessels with different end closures and complex geometries is difficult to obtain theoretically, especially for the cylinder head junctions. Experimentally, the solution of this type of problem is not easy either, because the reproduction of

417 Int. J. Pres. Ves. & Piping {6) (1978)--~" Applied Science Publishers Ltd, England, 1978 Printed in Great Britain

418 S . K . AGGARWAL, G. C. NAYAK, SHANKAR LAL

complicated shapes 9 of a given prototype in a model analysis requires a great deal of skill and considerable effort.

The conventional method of design and analysis of cylindrical pressure vessels with different heads is the use of Codes 1°- 12 prevalent in different countries. The chief shortcoming of the Code approach is that it introduces inconsistencies I a and thereby incorrectly predicts the stresses at junctions between mating shells, at points of support and at local sources of bending.

In addition to the Code approach, various numerical techniques are in current use whereby the analytical and computational labour can be reduced. Kraus et al . 14.15

have used a finite difference approach for the analysis of different heads using Love-Messner equations. Kraus 16 has also compared the results of hemispherical and torispherical heads obtained by CEGB17 and Bettisl 8 programs with the earlier photoelastic results of Fessler and Stanley s and Dally and Schneider. 7 The CEGB program, using a finite difference approach, is based upon Love's first approximation to the theory of thin elastic shells and the Bettis program, using a stiffness matrix method, is based upon a higher order theory of shells. Further, the CEGB program 1 o, 17.19 assumes the pressure to act on the middle of the surface--an assumption which is not valid for higher thicknesses--and the accuracy of the Bettis program l o, 18.19 is particularly poor at the junction between segments or for vessels with rapidly varying geometry giving stresses on the low side compared with the measured values. However, this program takes into account the effect of transverse shear and normal stresses while the CEGB program does not.

The complex requirements in pressure vessel structural design and the advent of high speed digital computers 2° have caused the pressure vessel industry to rely heavily on finite element analysis. The analysis of axisymmetric shells has, therefore, received considerable attention in the finite element literature 21 wherein a knowledge of variational calculus, matrix algebra, various constitutive relations and computer programming is a prerequisite for a complete understanding of the process in the finite element method. 22

SCOPE OF THE PARAMETRIC STUDY

The present investigation is concerned with the use of a finite element method (FEM) for the computation of various stress intensity parameters of cylindrical pressure vessels having hemispherical, torispherical, semiellipsoidal and toriconical head closures. The pressure vessel with a torispherical head is a composite shell consisting of cylindrical, torus and spherical segments, whereas that with a toriconical head is a composite shell consisting of cylindrical, torus and conical segments.

A parametric study to investigate the effect of the cited end closures on the different stress intensity parameters in cylindrical pressure vessels has been conducted. The vessel parameters studied for different heads are given in Table 1.

PARAMETRIC STUDIES OF C Y L I N D R I C A L PRESSURE VESSELS 419

T A B L E 1 D E T A I L S OF P A R A M E T R I C S T U D Y

Type of head Geometric ratios D/t L/t R/t a/t a /R A/B

Hemispher ica l 0 0 0 0 0 0

Tor ispher ica l ~

Semiellipsoidal

Tor iconica l a

25 12.5 12.5 50 25 25 0

100 50 50 0 200 100 100 0 300 150 150 0 400 200 200 0

25 18-75 6.25 6.25 50 37.50 12.50 12.50

100 75 25 25 200 150 50 50 300 225 75 75 400 300 100 100

25 - - - - - - 50 - - - -

100 - - - -

200 - - - - - - 300 - - - - - - 400 - - - -

25 11.25 2.5 10 50 22.50 5 20

100 45 10 40 200 90 20 80 300 135 30 120 400 180 40 160

m

4 4 4 4 4 4

2 2 2 2 2 2

" I n each case the torus subtends an angle of 60 °.

The dimensions in each case are in accordance with the nomenclature shown in Fig. 1, wherein:

D = mean diameter of the cylindrical portion of the vessel t = thickness of vessel

L = mean radius of spherical cap R = mean radius of torus a = distance of centre of torus from the axis of the vessel

A = semi-major axis of middle surface of semiellipsoidal head B = semi-minor axis of middle surface of semiellipsoidal head

Incremental ring elements of the type having cubic displacement in the ~ direction and linear displacement in the r/direction (see Fig. 2) were used in the analysis. This element has provided results which agree well with alternative solutions. A considerable reduction in the total number of degrees of freedom is possible with these elements because less elements (number of elements: 12) were required in this case for a sufficiently accurate analysis. This, incidentally, helps in reducing the data errors due to fewer elements resulting in a saving in the data preparation and computer time.

420 S. K. AGGARWAL, G. C. NAYAK, SHANKAR LAL

I

~=9o °

[-i D/2 -~i ~- (a) HEMI-SPHERICAL HEAD

~-_0 °

Z

1' J i_ , ~ D I2 - -

-.-It p- (c) SEMI-ELLIPSOIDAL HEAD

'~-

0/2 -~t "-

(b} TORI-SPHERICAL HEAD

! L¢ =9o*

~ o/2 N L

(C) IORI-CONICAL HEAD

Fig. I. Mesh configuration of cylindrical pressure vessels.

PARAMETRIC STUDIES OF C Y L I N D R I C A L PRESSURE VESSELS 421

I 3 S 7

INCREMENTAL NODE

( o} LOCAL CO-ORDINATES

lq

• - r

(b ) CARTESIAN MAP

Fig. 2. Inc rementa l cubil inear e lement .

FINITE ELEMENT F O R M U L A T I O N IN DISPLACEMENT ANALYSIS

Using a displacement finite element formulation 22 we have, for a typical element, the displacement field defined by nodal displacements {6}e as:

where the matrix [N] is composed of appropriate shape functions. The strains are, therefore, given by:


I~:I = [L][N]I61 ~' = [B] 16 '¢" (2)

where [L] and [B] are operator matrix and strain displacement matrix, respectively. Thus, the virtual work principle yields the equilibrium condition as:

[ B ] r / a l d V - . ,F, = 0 (3)

where { F} represents forces corresponding to displacements and 16 } and Icr } are the stresses at any point of the region.

The material constitutive relation defining {a} in terms of lel is given by:

~r I = [D](Ic I - *~o I) + l~ro} (4)

where I%} and {%1 are initial stresses and strains, respectively and [D] is the elasticity matrix.

INCREMENTAL ELEMENTS

These are the elements wherein the displacement variations in the ~ and r/directions are different. In the case of elements with higher aspect ratios, the presence of large off-diagonal terms associated with the adjacent nodes across the thickness creates ill- conditioning of the stiffness matrix in the usual isoparametric elements. This can be avoided by eliminating the rigid body displacements across the thickness by making suitable modifications in the shape function and the strain displacement matrices IN] and [B] locally.

The incremental cubilinear element 24 used in this analysis is shown in Fig. 2. The basic expressions required in the derivation of the element stiffness matrix are given below for axisymmetric elements. Their shape functions are written by taking the displacements and co-ordinates of the nodes on the t /= + 1 surface as the difference of displacements and co-ordinates, respectively between the q = - 1 surface and the t /= + 1 surface, i.e. for an incremental cubilinear element replace variables 62, 64, 66 and 6 8 at t/ = 1 on the top surface by A62, A64, A66 and Abs, wherein:

Ad z = 62 -- 61, A64 = 64 -- 63,

A6 o = 6 6 - 65, A68 = 6 s - 67

For a typical element (see Fig. 2) with eight nodes the displacement variations are defined by eqn. (1).

In order to redefine the shape function matrix for the incremental element, the relations for the two nodes 1 and 2 across the thickness can be written as:

{ ; : } = [11 0 ~ U 1 }, { ; :} = [ : 0~t~' 1 )z l_][Au I l ] [Av~J (5)

or~

P A R A M E T R I C STUDIES OF C Y L I N D R I C A L PRESSURE VESSELS 423

/')1 = 1 00/,~U1 ~ 0 , I

4 × 4

= [ T ] { 6 ' } e ( 6 )

Therefore:

where:

Io j{1} = . = [ N ] [ T ] { ~ 5 ' } e = [ N ' ] { 6 ' } ~ ( 7 )

N 1 0 N 2 u 2 2 x l 2 x 4

U 2

4 x l

[N,] = INI + N 2 0 N 2 0 1 0 N 1 + N 2 0 N 2

is the modified shape funct ion matrix:

N' 1 = N 1 + N z = ~6(9~ z - 1)(1 - ~)

N~ = N 2 = ~11 ~9~232 ~, - - 1)(1 - {)(1 + ~/)

The other shape funct ions can be evaluated similarly.

Therefore, the modified shape funct ion relat ionship for the whole element becomes:

{5}= ~U)[N']{6'} ~ ( V ) 2 x 16116 x 1 2 x l

(8)

The strain-displacement relat ion for the axisymmetric ring element is:

4 x

- 9 -

~ - 0 d r

0 0

c~z #r

1 0 r

4 x 2

{ ~ } = [L][N']{5'}~= [B']{6'} ~ 4 x 4 4 x l

2 x l

(9)


where [B'] is known as the modified strain displacement matrix. The element stiffness matrix and the equivalent nodal forces due to surface

pressure, p, are derived in eqns. (10) and (11) by using the virtual work principle:

[Kl"= 2~ f+l f ' l [B'l"[Dl[B'],qJ, Id4d,, (10)

and:

I It' f 1 j*+ 1 ~F,p = 2~r [N']llp',rl Jctd~. d~ l 1 1

where I J,,I is the determinant of the co-ordinate Jacobian matrix: r.

(11)

i ~ r ~ : 1

75 cc [J"] = i~r i'7_ (12)

RESULTS AND DISCUSSION

Parametric studies have been conducted to investigate the effect of different end closures on the various stress intensity parameters in cylindrical pressure vessels.

The solutions are presented in graphical form in terms of pertinent parameters (see Table 1).

In the cases under study a cylindrical pressure vessel having a mean diameter D = 250 cm was subjected to an internal pressurep. The entire vessel was assumed to be of constant thickness. The vessel material was taken as steel which is characterised by non-dimensional parameters Y/E = 0.001067 and v = 0.3 throughout. Here Y, E and v are yield point, modulus of elasticity and Poisson's ratio, respectively. The results of the parametric study have been reported as non-dimensional stress intensity parameters (ao/%, %/%) where 0 o = pD/2t is the mean circumferential stress in the cylindrical portion of the vessel far away from the cylinder head junction where the bending effects, brought about by the attachment of a head, are very slight and are disregarded.

The discretised assemblage in all the four head shapes investigated is shown in Fig. l(a) to (d). The details of element geometry are shown in the respective head shapes along with variation of position co-ordinate 4~. The circumferential stress intensity parameters, ao/% and meridional stress intensity parameters, ~%/% in the vessels are plotted for various degrees of location in the vessel heads and at various axial locations in the cylinder portion for both the inner and outer surfaces of the vessels. The curves begin at the top of each head (~b = 0 °) and continue as a function of the

PARAMETRIC STUDIES OF CYLINDRICAL PRESSURE VESSELS 425

position co-ordinate ~b to the cylinder head junction (q5 = 90 o) (see Fig. l(a) to (d)). Further, in the cylindrical portion the curves start at the cylinder head junction (Z/D = 0) and continue until the stresses attain fairly steady values at, say, Z/D = 0-5. The axial co-ordinate in the cylindrical portion, Z/D is the dimensionless ratio of axial position Z from the cylinder head junction and the mean cylinder diameter, D.

Figures 3 to 6 show the superimposed plots of~0/% and tr~/a o for inner and outer surfaces, respectively with all six D/t ratios of the hemispherical head. The D/t ratios are shown on the respective curves. The corresponding values of other parameters are given in Table 1. Similarly, Figs. 7 to 10, 11 to 14 and 15 to 18 are the corresponding plots for torispherical, semiellipsoidal and toriconical heads, respectively. These curves (Figs. 3 to 18) give a range of results of interest to pressure vessel designers. The discontinuity of stresses between the adjacent elements is so small that it is not apparent in these figures. This shows the high accuracy that can be obtained with the use of higher order elements for the discretisation of the pressure vessels of revolution. The various regions of interest have been clearly illustrated by these figures.

Because of the use of angular position as the abscissa for the data in each head there are two discontinuities in the slope of the circumferential and meridional stress intensity parameters for the torispherical and toriconical heads and one discontinuity in the slope of these data for the hemispherical and semiellipsoidal heads. These occur because the meridional radii of curvature are discontinuous at the junctions of the segments of which these vessels are comprised.

A study of the results plotted in Figs. 7, 9, 11, 13, 15 and 17 shows the existence of high compressive stresses in the circumferential direction for the torispherical, semiellipsoidal and toriconical heads for both the inner and outer surfaces in the torus portion (for q5 = 30 ° to 90°). Similarly, the existence of compressive meridional stress intensity parameters for both the inner and outer surfaces for the semiellipsoidal and toriconical head shapes is very clearly visible in Figs. 12, 14, 16 and 18. In the case of the torispherical head shape the compressive %/a o is only for D/t = 400 at the outer surface (see Fig. 10). It is seen that the compressive values of o~/a 0 are quite small except for the case of the toriconical head wherein the magnitude of %/a o in the compressive direction is quite significant compared with tensile values (see Fig. 18).

It is quite apparent from Figs. 3 to 18 that there are stress concentration regions in the cylinder portion near the cylinder head junction and the head portions of torispherical, semiellipsoidal and toriconical vessels. The designer requires peak values of these stresses along with their locations. It is therefore desirable to plot these stress concentration factors for various D/t ratios. Along with these plots information about the shifting of locations of stress concentrations with D/'t ratio is also useful. Seeing the importance of such information, the peak values oftro/a o and %/a o and their locations in the cylindrical portion of the pressure vessel for various

426 S . K . AGGARWAL, G. C. NAYAK, SHANKAR LAL

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432 S, K. AGGARWAL, G. C, NAYAK, SHANKAR LAL

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O_o l en S~I313~IV~Vd t115N31NI S$3~1$ 7VlIN3~3~I~In9~II9

PARAMETRIC STUDIES OF CYLINDRICAL PRESSURE VESSELS 4 3 3

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N

U.I

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L~


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0.8

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~ 0 . 2

z l,u

z - - 0

(/I u~ w pc ~-o.2 J

~-0-~ u./ I,L

~-0.6

- 0 . 8

-1.0 t..-.....,........,-~ 0

Z

20 40 60 80 90 0-1 0.2 0.3 O.l. ~o -~-" Z ID

SEMIELLIPSOIDAL HEAD ~ CYLINDER-

Fig. 11. Circumferential stress intensity parameters (inner surface).

N.';


1.8.

1 . 6 '

g~ 1./,.

O3

W

bJ 1.2 <

<

] , 0 =

f.-

cr~ Z

G.B Z

Ul

0.6-

0-4- Z c2_ Q tw

:~ 0.2-

t rXt

0"

-0.2 L----- 0 20 /-0 60 80 90 0.1 0.2 0.3 0.4 0-5

I . ,q l~ - - - SEMIE L LI P$OI DA L HEAD "J- CYLINDER =.~ v C , . .

Fig. 12. Meridional stress intensity parameters (inner surface).

436 s . K . AGGARWAL, G. C. NAYAK, SHANKAR LAL

I./,-

1.2-

1.0 -F===

0.8 03 r,,'

W

Z 0 .6 r,...

IX.

>_ O.L'

(/} z tu

0.2'

tO U.I

~ O'

. J

~ - 0 . 2 W

tLI t.L

~ - 0 . / ' C)

L)

-0 .6

PSOIDAL HEAD d q'

CYLINDER " J = z / O "-~

-O.B

- | .0

2O

Fig. 13.

/ '0 60 " t~ 90 0.! 0.2 0.3

Circumferential stress intensity parameters (outer surface).

0.~ 0.5


1.~I

~ t . 0 (/1

-=t u~ 0.8 ~E

°1 O.

) . I--

z 0.~

Z

(/1 Or) UJ n,, l -- (/1

0" Z o {3 E~ ~:-0.2"

/

-0.6L-- 0 L,..

- - SEMIELLIPSOIDAL H E A D ~ CYLINDER ;-~

20 ~0 60 80 90 0.1 0.2 0.3 0.1. 0.5 J - Z / D _ l

Fig. ]4. Meridiona] stress intensity parameters (outer surface).

438 S. K . A G G A R W A L , G . C. N A Y A K , S H A N K A R L A L

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~1.2

W "

W :~ 0.8- < n,,.

~ -

>. 0.4

or) z LLI

~ o

U.I

~-o.4 (./)

V--'-O .E Z Ud

UJ 14.

U

-1.6

\

... CONICAL _ . ~ . ~ . ~ TORUS -2.0., CAP

0 2'0 4'o 6"0 e'o 90 0~1 012 013 0~4 L_- ¢0 _ I_ Z / D ' - - - 7 ~

i - TORI CONICAL H E A D __L.. CYLINDER

Fig. 15. Circumferential stress intensity parameters (inner surface).

o.'5 - - i

=i

P A R A M E T R I C STUDIES OF C Y L I N D R I C A L PRESSURE VESSELS 439

3.2-

2@

rd

2.0 ttJ

r,,

a . 1.6

> -

z 1.2 w

z

O.t~ o'1, w re-

O.&

z 0

0 0

u.l

0.4

0 .8

0

t . . - i -

o o

. . . . . . CAP " - - % ~ " ~ - - - - TORUS ~ t I ,

20 40 60 80 90 0.1 0.2 0.3 ~D ° ~-14 .... Z / D

TORICONICAL H E A D = ' , = C Y L I N D E R

Fig. 16. Meridional stress intensity parameters (inner surface),

0./., 0.5

4 4 0 S. K. A G G A R W A L , G . C. N A Y A K , S H A N K A R L A L

1.5

oo 1.0 m,,

u.l

a~ 0.5

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~ - 0 . 5 LO t.~

~ -1.0.

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-2.5

"7 \ , ~ . C O N I C A L . 4 , , , ~ _. C A P -ql ~

0 20 40 I - - (~o

l-- - T O R I C O N I C A L F-

Fig. l 7.

k,._,..A TO R U S - ~

60 80 g0 04 0.2 0.3

=;-- Z /O H E A D ~-{~- - C Y L I N D E R

Circumferential stress intensity parameters (outer surface).

O.Z O3 ~ J

PARAMETRIC STUDIES OF CYLINDRICAL PRESSURE VESSELS 4 4 ]

2.0

16

. 1 . 2

p, .

UJ Go.8 <

~ 0 . ~

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7 o

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-2.0

t C3C3C~t~

° ° 7 '7 / /

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o 2'o ' ~'o 6'o

;= T O R I C O N I C A L H E A D

e'o 90 0'.~ 012 0'.3 o'.t. o.'s - ~ - Z I D - ' =-~" C Y L I N D E R " _ l w !

Meridional stress intensity parameters (outer surface). Fig. 18.

442

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1.08

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1.02

1.00

S. K. AGGARWAL, G. C. NAYAK, SHANKAR LAL

j 4

'%

0.~"

0.3

O i - -

0.2

O

0.1

Fig. 19.

\ ', 7~.'~."~,'c~ . "~

, ~ ~,'c~o~,'2°'°~.~ ~'°

1 ' 2 ' 3 F ~

o lOO ' 2 6 0 3~o ' ' ~o~ D/ t RATIO

Peak circumferential stress intensity parameters and locations in cylindrical portion (inner surface).

heads and D/t ratios are plotted for both the inner and outer surfaces (Figs. 19 to 22), the other corresponding parameters being shown in Table I. The corresponding plots of peak meridional stress intensity parameters and their locations in head portions are shown in Fig. 23.

Dilation or radial growth of cylindrical pressure vessels having different end


1-6"

I./.

:•1.2 =£

~ I.0

0.8

0.6

0.5

1 HEMISPHERICAL HEAD 2 TORISPHERICAL HEAD

3 SEMIELLIPSOIDAL HEAD

/. TORICONICAL HEAD

L

3 3 2 I

- - - . . i i

o 0.2'

o 0.I

N

0 0

Fig. 20.

i

100 200 300 /.60

D / t RATIO

Peak meridional stress intensity parameters and locations in cylindrical portion (inner surface).

closures was also obtained th roughou t the surface of the vessels corresponding to an internal pressure p = 1 kg/cm: .

Equat ion (13) gives the dilation, 6 , of a cylindrical vessel of inner diameter D~, purely f rom the membrane solution:

p D 2 ( 2 - v) 6~ - 8tE (13)

Table 2 gives a compar ison of values o f dilation of a cylindrical vessel having a hemispherical head for the inner surface at Z/D = 0-6, i.e. far away f rom the junct ion where the effect of junct ion can be disregarded. The magni tude o f the

AAA S.K. AGGARWAL, G. C. NAYAK, SHANKAR LAL

1 . ~ °

1.3

~1.2

u=l {2

1.0

0.9

0.3"

0 0.2 ,,¢ n,-

D .., 0.1 ̧ N

0 0

Fig. 21.

1 H E M I S P H E R I C A L HEAD

2 TORISPHERICAL HEAD

3 S E M I E L L I P S O I D A L HEAD

.../. /. TORICONICAL HEAD

~b0 ' ' 2b0 ' 3b0 ' ~ O / t RATIO

Peak circumferential stress intensity parameters and locations in cylindrical portion (outer surface).

co r r e spond ing values is a lmos t the same for o ther head shapes. Table 2 clear ly i l lustrates the fact that as the D/t ra t io is increased the thickness effect is min imised and the F E M solut ion gives results close to the m e m b r a n e solut ion. The discrepencies in the d isp lacements are due to higher thickness with lesser D/t ra t ios as its effect is d i s regarded in the m e m b r a n e solut ion whereas, in the F E M solut ion , its effect is taken into account .

Fu r the r , it is wor th while to c o m p a r e the results of stresses in the cyl inder at Z/D = 0-6 ob ta ined by F E M with those o f Lam6's equat ions .


1.6

1.2

:E

10

O.B

0.6

0 3

/.

3

f- f2 #'

1 H E M I S P H E R I C A L HEAD

0.2' 2 TORISPHERICAL HEAD

O 3 SEMIELLIPSOIDAL HEAD

n, 4 4 TORICONICAL HEAD 0.!

Q

N 1,2 )r ~ . ,

0 ' ' • ,o0 260 ' 3 a o ~6 0

O / t RATIO

Fig. 22. Peak meridional stress intensity parameters and locations in cylindrical portion (outer surface).

Equations 14(a) and (b) give the values of maximum and minimum circumferential stress intensity parameters in the cylinder portion at the inner and outer surfaces, respectively. These equations have been derived from the original Lamb equations:

fro, [ t~___£] - - = 1 + ( 1 4 ( a ) ) o" o

troD= 1 - (14(b)) o- o

446

3.2

2.8 ¸

2./.

X a

E 2.0

t .2

0.8

S. K. AGGARWAL, G. C. NAYAK, SHANKAR LAL

/ / ~ ~ /

MAJOR PEAK . . . . MINOR PEAK

(Z) TORISPHERiCAL HEAD ~) SEMIELLIPSOIDAL HEAD (~) TORICONICA L HEAO

I I/ ~-- . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

90'

8 0

6 0

/ .0

2 0

0

Fig, 23.

f

s'o ,bo " ,6o

. . . . . . - - " : -

260 2;~o 360 3~o ~6o D/t RATIO

Peak meridional stress intensity parameters and locations in pressure vessel heads (inner surface).

PARAMETRIC STUDIES OF CYLINDRICAL PRESSURE VESSELS

TABLE 2 DILATION OF CYLINDRICAL PRESSURE VESSEL HAVING HEMISPHERICAL HEAD (INNER

SURFACE)

447

D/t Dilation of eylindrical pressure vessel Percentage ratio at Z/D = 0.6 dijJerence

( × 10 -3) FEM solution Membrane solution

(era) (cm)

25 0.632 0.58285 7.777 50 1-26564 1.21479 4.018

100 2.5312 2-47941 2.046 200 5.06293 5-00905 1.064 300 7.58393 7.53877 0.595 400 10.1193 10-06851 0.502

Table 3 gives a comparison of these values with the FEM solution at Z/D = 0.6 for a cylindrical pressure vessel having a hemispherical head. These values are approximately of the same order for other head shapes. The slight differences can be attributed to the interaction of the head with the cylinder. With the increase in thickness the flexural disturbance at the junction travels up to 0.68D for D/t = 25. 23

In order to find the peak values according to Von Mises and Tresca yield criteria,

maximum values of the ratios of x//3 J~/2/Cro and (a~ - ~o)/ao are plotted in Fig. 24, along with their locations.

The study, therefore, gives a very clear relative and overall picture of the effect of end closures on the stress intensity parameters at the areas of interest in cylindrical pressure vessels having the range of parameters cited in Table 1. The design charts presented in this paper can go a long way towards helping the designer in the proper design of pressure vessels of revolution with different end closures.

CONCLUSIONS

(1) A finite element approach to the parametric study of cylindrical pressure vessels

TABLE 3 COMPARISON OF LAME'S AND FEM SOLUTIONS FOR CYLINDRICAL PRESSURE VESSEL HAVING HEMISPHERICAL

HEAD AT Z/D = 0"6

D/t Lamk's solution FEM solution Percentage ratio difference of

~o,/~o ~Oo/~O ~o,/~o ~Oo/~O ~o,/~o ~Oo/~O

25 1"0016 0.9216 1"03256 0"88888 2'998 3-55 50 1.0004 0.9604 1"01504 0'94528 1"442 1.574

100 1-0001 0.9801 1"00764 0.9726 0.748 0.765 200 1"000025 0"990025 1"004 0"98659 0'396 0"347 300 1"0000111 0.9933443 1-00153 0.98993 0.152 0.343 400 1.00000625 0.99500625 1'0018 0"99305 0"179 0"197

448 S. K, AGGARWAL, G. C. NAYAK, SHANKAR LAL

/..0

3,6

3.2

T 2.8

C' I 2.~

2.0

I I I

{ 1.6

~ e

0,8

/ J , TRESCA YIELD CRITERION

/ / / .__ VON MISES YIELD CRITERION

/ / / (~ TORISPHERICAL HEAD

/'/ (~ SEMIELLIPSOIDAL HEAD

(E) TORICONICAL HEAD

~ _ ~ ~ w q ~ -

f

Fig. 24.

90'

70

50

3 0 0

I ~ ) ..

do ' ,6o ,%o 26o 2~o 3oo 3~o 46o Ol t RATIO

Variation of peak effective stress ratios and locations in pressure vessel heads (inner surface).


with different end closures using an axisymmetric solid finite element program, PVSAGG, employing incremental elements has been presented.

(2) The results show that the technique is very useful since the whole vessel can be analysed in a few minutes' computer time and all mid-side nodes of higher order elements are generated within the computer program and need not be specified. Only twelve elements were needed in the present analysis for an adequate solution, resulting in a reduction in the number of degrees of freedom compared with conventional elements.

(3) Testing of incremental elements having paralinear and cubilinear displacements revealed that cubilinear elements, whose results are reported in this paper, are more accurate and economical. The discrete values of stress intensity parameters ~0/~o and ~ / a o and their locations for the six D/t ratios in the vessel heads and cylinder portions for both the inner and outer surfaces of the vessels have been presented.

(4) The discontinuity of stresses between the adjacent elements is so small that it is not apparent in the plots, showing thereby that the results of investigations in the form of design charts provide a reliable basis for the design of a cylindrical pressure vessel with different vessel parameters and head closures.

(5) The behaviour of pressure vessels depends upon the interaction of membrane and bending effects. Because of this, the effect of vessel thickness is reflected in the curves.

(6) The effect of increased thickness is reflected by an increase in bending action. (7) The peak stresses occur near the junctions. The cylindrical wall always

experiences circumferential stresses causing longitudinal cracking and the head portions experience peak meridional stresses near the junctions which are likely to cause circumferential cracking.

(8) The peak values of various stress intensity parameters and their locations in the cylindrical and head portions for various head and D/t ratios are also presented for both the inner and outer surfaces of the vessel.

(9) The peak yield stress ratios according to Von Mises and Tresca yield criteria are also presented, along with their locations.

(10) The discrepencies in the displacements and stresses are attributed to the vessel thickness whose effect has been accounted for in the FEM solution whereas the membrane solution disregards this effect.

ACKNOWLEDGEMENTS

The authors are grateful to the Department of Mechanical Engineering, University of Roorkee, for providing the necessary funds to carry out the investigation and to Delhi University for making the computer facilities available at a concessional rate. The first author is also grateful to the Ministry of Education, Government of India, for QIP sponsorship to work at the University of Roorkee.


REFERENCES

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parametric studies of cylindrical pressure vessels with different end closures

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