The design of ends of cylindrical pressure vessels in glass reinforced plastic

Int. J. Mech. Sci. Vol. 26, No. 3, pp. 177-199, 1984

0020-7403/84 $3.00 + .00 Pergamon Press Ltd.

Printed in Great Britain.

THE DESIGN OF ENDS OF CYLINDRICAL PRESSURE VESSELS IN GLASS REINFORCED PLASTIC D. C. BARTON,A. R. AMOS,P. D. SODEN and S. S. GILL Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, England (Received 14 December 1982; in revised form 13 March 1983)

Sununary--A design procedure for torispherical and semi-ellipsoidal ends of cylindrical chopped strand mat glass-fibre reinforced plastic pressure vessels is presented. The method is applicable where all layers of reinforcement are of the same type. The procedure takes account of the effect of the change of shape of the end due to pressure. The procedure is based on either maximum stress or maximum strain. The criteria used aim to achieve consistency with the BS.4994 method of calculating the design unit loading of laminates and allow for the biaxial stress states which occur in the vessels.

NOTATION D Di

E h hi

K

K, Ko Ks g0.002

nx

P P0.002 P a r ri

R Ri

S t U UL Us

Ux Wx

x,

)CLAM Emin Emax EL ER E ELAM

Oo O'ma~

mean diameter of cylinder inside diameter of cylinder Young's modulus of laminate height of pressure vessel end- tangent line to pole internal height of pressure vessel end overall design factor in BS.4994t strain concentration factor (Sn.C.F.) stress concentration factor (S.C.F.) stress factor for domed ends (Table 4, BS.4994) strain concentration factor at 0.2~ maximum strain number of layers of type x:~ internal pressure internal pressure to give 0.2~o maximum strain design strength of laminate unit load at point under considerationt mean radius of torus inside radius of torus mean radius of sphere inside radius of sphere ratio of circumferential to meridional stresses shell thickness, thickness of end ultimate tensile unit strengtht load-limited allowable unit loadingt strain-limited allowable unit loading'l" design unit loading for layer of type x t weight of glass per unit surface area for one layer of type x§ extensibility of layer of type x# overall extensibility of laminate under consideration (equation 11 of BS4994) permitted strain for a load-limited laminater maximum tensile strain in pressure vessel end load-limited allowable straint failure strain for the unreinforced resint strain-limited allowable straint lesser of qmn and E meridonal angle of sphere/torus intersection maximum tensile stress in pressure vessel end failure stress of G.R.P. in uniaxial tension (or compression)

S.C.F. = /co= Sn.C.F. =

K,-

2tO'max

pD~ 2EtCmax

pDI

tThese quantities are as defined in BS4994~'), Table 1, page 182; for example ultimate tensile unit strength is the strength of a constituent layer of a laminate expressed as force per unit width per unit weight of reinforcement and has units N/mm (per kg/m 2 glass). SAs defined in Section 2.2 of BS4994tl). §As defined in Section 3.4 of BS4994(°. 177

178

D.C. BARTON et

al.

INTRODUCTION

The design of ends of cylindrical pressure vessels in glass reinforced plastic (G.R.P.) is covered in BS.4994[1], A. D. Merkblatt, NI[2] and A.S.M.E. Section X.[3] and a large programme of theoretical and experimental work has been carried out[4-7] particularly related to the design methods in Ref. [1]. All the work is based on G.R.P. using chopped strand mat glass reinforced polyester resin. Making due allowance for the variability using hand lay-up manufacture, it has been concluded from the experimental work[4-5] that a computer program BOSOR 4 [8] could be usefully used to predict the behaviour of pressure vessel ends in G.R.P. The program BOSOR4 is a very comprehensive one for the analysis of stress, buckling and modal vibration of shells of revolution. For the particular work reported here, its main advantage is that it solves the axisymmetric thin shell problem for a linear elastic material but takes account of the effect of change of geometry on the equations of equilibrium. This effect is significant in the case of G.R.P. which has a relatively large linear elastic range. Based on experience from the experimental work and discussions with designers it was considered that a strain criterion might be appropriate in some cases for G.R.P. construction and the paper presents design factors for G.R.P. ends based on limiting the maximum tensile strain in the end to a nominated value. In certain cases, however, it is shown to be advisable to retain a maximum stress criterion in view of the limited data on the properties of G.R.P. under biaxial loading. The design procedure proposed considers both the maximum stress and maximum strain in the end of the pressure vessel but retains the BS4994 procedure for determining the allowable unit loads in the laminate layers, based on uniaxial data. BASIS OF DESIGN M E T H O D IN BS.4994 One very important feature of BS.4994 is that unlike the metal pressure vessel codes, the usual engineering idea of stress is not used. The design method is based on the concept[9] of unit strength, allowable unit loadings and design unit loading of the layers o f a laminate. These are defined in Table 1 of BS.4994. Further details of the background to the method are given in [10, 11]. The ultimate tensile unit strength, u, is defined as the strength of a constituent layer of a laminate expressed as force per unit width per unit weight of reinforcement. For stiffness the corresponding property is extensibility, X, defined as the ratio of the load per unit width to the corresponding direct strain in a loaded tensile specimen. The design unit loading, ux, is the permitted load for a constituent layer of a laminate and its value depends on whether the laminate is strain-limited or load-limited. The strain-limited design unit loading is given by u~ = XxE where E is the maximum permissible strain in the resin which is the lower of 0.2~ or 0.1ER. The load-limited design unit loading is uL = ( u / K ) where K is a safety factor built up from part factors depending on the method of manufacture, operating conditions, etc. In designing a laminate consisting of a number of layers, if u~ is less than uL for all the layers then the laminate is strain-limited and u, = ux is the design unit loading of each layer. If for some o f the layers uL is less than us then the allowable strain for each layer eL = uL/Xx. The lowest value of eL for all layers (E~in) is used to give the design unit loading ux = XxE~n for each layer of a load-limited laminate. From these considerations, the maximum load per unit width, P, which may be applied to a laminate, i.e. the design strength (Table 1 of BS.4994) is given by P = UlWln I + u2w2n 2 + . . . . . .

uxwxn~.

(1)

It proves convenient in the context of the present paper to formulate the BS.4994 procedure for the design of a laminate in a slightly different way which leads to exactly the same result. From uL, the load-limited allowable strain for each type o f layer is found eL = ( u i j X x ) and E~n is the lowest value of eL as in BS.4994. The strain-limited allowable strain is E as in BS.4994. Now define ELAM(not used in BS.4994) as the lesser of E and Emio.Then the maximum unit load which may be applied to the laminate, i.e. the design strength is given by: (2)

P = ELAM XLA M where

XLA M :

~ : l W i n l -q- X 2 w 2 n 2 -~- . . .

XxWxnxf r o m equation (11) o f

BS.4994.

In the design o f vessels it is necessary to ensure that P > Q for all parts of the vessel, where for internal pressure loading pD,

Q =2

for cylinders, (but note equation (9) o f BS.4994 for combined loading).

(3)

Q = pDi for spheres

(4)

Q = 0 . 5 5 p D ~ Ks for domed ends.

(5)

4

Design of ends of cylindrical pressure vessels

179

The presence of the factor 0.55 in equation (5) is considered later in the discussion. The values of the factor Ks for torispherical ends are given in Table 4 of BS.4994 for (hi/D~) = 0.2, 0.25, 0.32, 0.4, 0.5, (t/D~) = 0.01, 0.02, 0.03, 0.04, 0.05. Different sets of Ks values are given for 0.1 _<(rJD~)<_ 0.15 and for (rJD~) > 0.15. Values of Ks for semi-ellipsoidal ends are given for the same head heights and are stated to be independent of thickness. A single value of Ks (Ks = 0.65) is given for all hemispherical ends. The values of Ks for torispherical ends are taken from Crisp[12] and those for semi-ellipsoidal ends from Kraus, Bilodeau and Langer[13]. It should be noted that Ks is essentially a stress concentration factor although the design for the laminate is either load-limited or strain-limited. In fact BS.4994 avoids the use of the term "stress" by using the unit load concept, but in the discussion and proposals which follow it is convenient to use the term "stress" although the final proposal reverts to the use of the unit load concept. SOME COMMENTS ON THE BS.4994 PROCEDURE It is quite clear that the procedure for evaluating the design strength, P, for a laminate in BS4994 is based entirely on data from uniaxial tests and that the procedure for calculating the applied unit load Q for domed ends is based on the maximum stress in the end, even though the term stress is not used. The BS4994 procedure for determining the allowable design unit loads for the laminate layers, carefully ensures that neither uL nor us is exceeded in any layer when the laminate is subjected to uniaxial load but makes no allowance for the biaxial stresses which occur in the actual vessel. Problems occur because the stress system in the domed end is biaxial with different ratios of the two principal stresses at different points. Two cases arise: (a) Where the principal stresses have the same sign, the stresses associated with any given magnitude of strain are higher than would be obtained from a uniaxial stress-strain or load-extension curve. (b) when the principal stresses have opposite signs a larger strain may exist than implied by using the maximum principal stress and a uniaxial stress-strain or load-extension criterion. Fig. 1. shows two theoretical failure envelopes of circumferential stress versus meridional stress in a vessel subject to biaxial stresses. One envelope assumes that failure will occur when the maximum stress reaches a critical value (which is different in tension and compression) and the other plots the stresses when the maximum strain reaches a critical value similar to that which produces failure in a uniaxial tension or compression test. The design procedure proposed later in the paper ensures that the stresses in the vessel are always less than or equal to the smaller of the values given by the two criteria (i.e. the stresses are kept within the combined inner envelope shown in Fig. 1). Fig. 1 also includes experimental failure envelopes given by Owen[14] for C.S.M. reinforced polyester resin laminates under biaxial stresses. The experimental results indicate that where the biaxial stresses are both tensile, the experimental failure stress is less than that for a uniaxial tensile test. The maximum stress criterion of failure would thus be optimistic and the maximum strain criterion very optimistic in these cases (see Fig. 1). The maximum stress criterion is also optimistic where one of the stresses is tensile and the other compressive, but the maximum strain criterion is less so. The combined inner envelope gives a better description of the experimental results than either criterion alone. Fig. 1 compares the theoretical and experimental results at fracture, but it can be seen that similar observations apply for fibre/matrix debonding and resin cracking. These considerations lead to an alternative design procedure for torispherical ends which considers both the maximum stress and the maximum strain in the ends but retains the BS4994 procedure for determining the allowable design unit loads. Before giving the procedure it is necessary to define the geometry of a torispherical end and outline the procedure used to calculate the stress and strain concentration factors which will be used. END GEOMETRIES Ends of cylindrical vessels may be either torispherical, semi-ellipsoidal or hemispherical. The ellipsoidal end is completely specified by its major and minor axes (see Fig. 2a). For the torispherical end (Fig. 2b) of given head height, h, and cylinder diameter, D, the relationships between the sphere radius, R, and toms radius, r, are given in the Appendix which also discusses optimum torispherical ends. CALCULATION OF STRESS AND STRAIN CONCENTRATION FACTORS The computer program BOSOR 4 was used to study a wide range of torispherical and semi-ellipsoidal ends. In all cases the end was of constant thickness and joined to a cylinder of the same thickness and the loading was by internal pressure only. The cylinder was sufficiently long to avoid any interaction between stresses at the remote end of the cylinder and the torispherical or ellipsoidal end being investigated. The material was assumed to be homogeneous and isotropic. Values of Young's modulus of 7.0 GPa and Poisson's ratio of 0.34 were used based on average data found in the experimental work, but, in fact, all of the results presented are independent of Young's modulus and only slightly dependent on Poisson's ratio. For each geometry investigated, BOSOR 4 results were obtained for a very small initial pressure increment (0.1 KPa) followed by a series of larger increments chosen so that the final increment gave a strain of just over 0.2~o in the end. By interpolation the pressure to give 0.2~o was evaluated and for this pressure various quantities were calculated. These are given in Table 1. The first four columns define the geometry and column 5 gives the values of K~from BS.4994, where available. Columns 6 and 7 give the results for the stress concentration factor and strain concentration factor from BOSOR 4 at the first pressure increment which corresponds to results from analyses which do not take account of any change of shape of the vessel under pressure. Stress Concentration Factor (S.C.F.) is defined as K~ -

2tOmax pD~

(6)

i.e. the ratio of the maximum stress in the head ( a ~ ) to the hoop stress in a cylinder of the same thickness.

180

D . C . BARTON et al.

150

I00

I

~1 I~'''''~

//

//

/ / / ÷ / Lu.~Lu.~ ,: ,.o/

/6 50

Z

.,d~3,.~"

//

/

s:,o.3~

/ / / I . s:O 0 0. in

50

i ~'~" s : - 0.5

-

-

i00

/•

L¸ b

i \

s:--I'O

Theoretical Envelopes Maximum struir criterion - - - - Maximum stress criterion LUJJ / Inner er'velope Experimental Envelopes ,~frorr,(18))

150

Debonding - - 0 - Resin crocking Rupture _o" e NOTE : s - - -

L- 200

I

I

1

I

5O

I O0

150

200

Principal stress o-¢ M N / m

2

FIG. 1. Theoretical and experimental failure envelopes for thin CSM: polyester resin laminates under biaxial stress.

Strain Concentration Factor (Sn.C.F.) is defined as K~ = 2EtCma--x

(7)

pDr

where Em~ is the maximum strain in the end and E is taken as 7 GPa as in the computer analysis. The form of equation (7) has been chosen for compatibility with the BS4994 design procedures (see later discussion). Columns 8 and 9 give respectively the S.C.F. and Sn.C.F. at a pressure P0.002 which gives a maximum strain of 0.2~ in the end. It will be seen that because of the effect of change of geometry under pressure, these values are slightly lower than the corresponding values in columns 6 and 7. Column 10 gives Po.oo2/E. Column 11 gives the ratio of Ko to K, at P0.002- All the results in columns 6--11 are independent of the value of Young's modulus used in BOSOR 4 since the way in which the end deforms to reach a maximum strain of 0.2~o is dependent only on the geometry and thickness. The results will be slightly dependent on Poisson's ratio. The results apply to any isotropic linear elastic material. BS.4994 is based on inside dimensions since the manufacturing techniques for G.R.P. vessels usually use an internal mould shape. On the other hand, thin elastic shell theory is based on the deformation of the mid-surface of the shell. Hence in using BOSOR 4 to compute the values in Table 1, all dimensionless parameters are based on inside dimensions, but the shell co-ordinates used in the computer program for each case were the co-ordinates of the mid-surface, i.e. cylinder diameter = Dr + t, sphere radius = Rr + t/2 and tarns radius = r r + t/2. This introduces a difficulty in the case of semi-ellipsoidai ends of constant thickness in that either the mid-surface or inner surface, but not both, can be accurate ellipses. The results given use an ellipsoidal mid-surface which is more appropriate for thin shell theory and the major and minor axes of the mid-surface ellipses were (Dr + t)/2

181

Design of ends of cylindrical pressure vessels

Ellipsoid-~

Tan

line Cylinder

( a ) Geometrical parameters of Semi - El lipsoidal End

Pole Sphere : Torusjunclion' = ~ ] = = = = ~ Torus

Ton,he

7f .

i .

.

.

.

.

II

I Zr? 7" 7

I

hi

i, R, / Cylinder

Di

{ b ) Geometricalparametersof TorisphericalEnd FIG. 2(a). Geometrical parameters of semi-ellipsoidal ends. (b) Geometrical parameters of

torispherical ends. and hr + t/2 respectively. The results are therefore for inner surface values of hdDr although the inner surface is not exactly an ellipse. The error is greater for thicker ends. The thin shell theory used in the program is likely to be unreliable for shell thickness/radius of curvature ratios greater than about 0.2 and hence in the case of a torispherical end the ratio t/rr limits the validity of the results. For example, for an end having (hdDr) = 0.2, (RJDr) = 0.9, (rJDr) = 0.0875, a value of (t/Dr) = 0.0175 (which is quite small) gives (t/rr) = 0.2. BS.4994 gives values of K, up to (t/Di) = 0.05 for which t/ri) > 0.2 for these particular parameters. Table 1 also gives values up to (t/Dr) = 0.05 but they should be treated with caution. Figs. 3(a-c) show values of Sn.C.F. at 0.2% plotted against t/Dr for a series of torispherical ends having different head heights (hdDi), torus radii (h/Dr) and sphere radius (Rr/Dr). Figs. 4(a--c) show equivalent graphs for S.C.F. at 0.2% maximum strain. Both Figs. 3 and 4 are graphical presentations of the values in Table 1. Also shown in Figs. 3 and 4 are the curves of K, taken from Table 4 of BS.4994. PROPOSED DESIGN PROCEDURE Instead of using equation (5) for pressure vessel ends it is proposed that it could take the form

O = 0.5pDrK,

(8) equation (7).

o0

l"

"l

12e/J Using P > Q we obtain, using equation (2), ELAMXLAM ~ Et(.max~ and since XLAM= Et, the design criterion ensures that the maximum strain in the head, Em~x,is less than the permissible maximum strain in the laminate. MS Vol. 26, No. 3 ~

Ri/D,

(2)

1.00

0.90

0.80

hJDi

(1)

0.20

0.20

0.20

0.050

0.0875

0.110

(3)

ri/Di

0.005 0.010 0.015 0.022 0.030 0.040 0.050 0.005 0.010 0.015 0.0175 0.030 0.040 0.050 0.005 0.0065 0.0085 0.010 0.015 0.020 0.025

(4)

t/Di°~

K~

-2.85 2.75 2.62 2.50 2.35 2.25 ---------------

(5)

(BS4994)

2.77 2.71 2.65 2.55 2.45 2.34 2.24 3.01 2.95 2.82 2.76 2.50 2.35 2.24 4.15 3.95 3.72 3.58 3.22 2.96 2.77

(6)

S.C.F.

3.20 2.99 2.83 2.63 2.46 2.29 2.16 3.47 3.20 2.96 2.85 2.48 2.28 2.13 4.53 4.23 3.92 3.72 3.26 2.95 2.72

(7)

Sn.C.F.

Results at p = 0.1 kP a

2.51 2.59 2.57 2.50 2.41 2.31 2.22 2.81 2.85 2.76 2.70 2.47 2.33 2.22 3.99 3.83 3.64 3.50 3.17 2.93 2.74

(8)

S.C.F. t2)

2.92 2.85 2.74 2.57 2.42 2.26 2.13 3.25 3.09 2.88 2.79 2.44 2.25 2.11 4.36 4.10 3.82 3.64 3.21 2.91 2.69

(9)

Sn.C.F. ~2)

6.9 14.0 21.9 34.2 49.7 70.8 94.0 6.2 13.0 20.8 25. t 49.2 71.1 95.0 4.6 6.3 8.9 11.0 18.7 27.5 37.1

(10)

( x 106)

p/E

Results at p = 0 . 2 ~ M a x i m u m Strain

TABLE l(a). RKSULTS FROM B O S O R 4 SURVEY--TORISPHERICAL ENDS OF 0.2 HEAD HEIGHT

0.860 0.909 0.938 0.973 0.996 1.022 1.042 0.865 0.922 0.958 0.968 1.012 1.036 1.052 0.915 0.934 0.953 0.962 0.988 1.007 1.019

(11)

K,/K,

2

© Z

Ri/D ~

(2)

1.00

0.90

0.80

hJD~

(1)

0.25

0.25

0.25

0.1458

0.1719

0.187

(3)

r~/D~

0.005 0.010 0.015 0.020 0.030 0.0375 0.040 0.050 0.005 0.010 0.015 0.024 0.0343 0.040 0.050 0.005 0.010 0.015 0.020 0.0292 0.040 0.050

(4)

t/D~ t)

Ks

-1.80 1.78 1.75 1.72 1.71 1.70 1.70 -1.80 1.78 1.74 1.71 1.70 1.70 -2.25 2.18 2.10 1.90 1.85 1.75

(5)

(BS4994)

1.95 1.80 1.76 1.75 1.73 1.72 1.72 1.75 1.77 1.69 1.69 1.70 1.70 1.70 1.68 1.68 1.73 1.77 1.79 1.77 1.73 1.70

(6)

S.C.F.

2.13 1.94 1.86 1.81 1.75 1.70 1.69 1.64 1.96 1.83 1.79 1.74 1.69 1.66 1.60 1.91 1.89 1.87 1.83 1.75 1.67 1.60

(7)

Sn.C.F.

Results at p = 0.1kPa

1.72 1.69 1.68 1.68 1.69 1.69 1.69 P1.70 1.58 1.59 1.62 1.66 1.67 1.67 1.66 1.53 1.65 1.72 1.75 1.74 1.71 i .68

(8)

S.C.F. ~2)

1.87 1.80 1.76 1.74 1.70 1.67 1.65 1.61 1.75 1.72 1.72 1.70 1.66 1.63 1.58 1.74 1.80 i.81 1.79 1.72 1.64 1.58

(9)

Sn.C.F. t2)

10.7 22.2 34.1 46.0 70.7 90.1 96.7 124.5 11.4 23.2 34.9 56.6 83.0 98.3 126.5 11.5 22.2 33.1 44.7 67.8 97.3 126.6

(10)

( x 106)

p/E

Results p = 0.2~o M a x i m u m Strain

TABLE l(b). RESULTS FROM B O S O R 4 SURVEY--TORISPHERICAL ENDS OF 0.25 I-mAD HEIGtrr

0.942 0.976 1.006 1.025 1.063 0.879 0.917 0.950 0.978 1.012 1.043 1.063

0.924

0.920 0.939 0.955 0.966 0.994 1.012 1.025 1.056 0.903

(11)

K,/K,

p,

"1.

e.s

(3)

0.100

(2)

0.7083

0.6645

(1)

0.25

0.25

0.060

ri/D i

RffD,

hffDi

0.005 0.010 0.015 0.020 0.030 0.040 0.050 0.005 0.0075 0.010 0.012 0.015 0.020 0.030

(4)

t/Di (1)

-2.25 2.18 2.10 1.98 1.85 1.75 --------

(5)

Ks (BS4994)

2.18 2.22 2.15 2.07 1.94 1.84 1.76 3.14 2.92 2.75 2.63 2.49 2.31 2.07

(6)

S.C.F.

2.49 2.35 2.20 2.06 1.87 1.73 1.62 3.37 3.04 2.80 2.65 2.47 2.25 1.96

(7)

Sn.C.F.

Results at p = 0.1 kPa

2.06 2.16 2.11 2.04 1.92 1.82 1.75 3.02 2.85 2.69 2.59 2.45 2.28 2.06

(8)

S.C.F. 12)

2.34 2.28 2.15 2.03 1.84 1.71 1.61 3.24 2.96 2.74 2.60 2.43 2.22 1.94

(9)

Sn.C.F.(2)

8.6 17.5 27.9 39.4 65.1 93.5 124.1 6.2 10.2 14.6 18.5 24.7 36.1 61.9

(10)

piE ( x 106)

Results at p = 0.2~, Maximum Strain

TABLE l(c). RESULTS FROM B O S O R 4 SURVEY--TORISPHERICAL ENDS OF 0.25 EmAD HEIGHT (CONTINUED)

0.880 0.947 0.981 1.005 1.043 1.064 1.087 0.932 0.963 0.982 0.996 1.008 1.027 1.062

(ll)

Ko/K,

Z

~3

0

0.288

0.280

0.210

0.90

0.647

0.32

0.32

(3)

1.00

(2)

(1)

rj/Dj

0.32

RJD i

hJD~

0.032 0.042 0.050

0.021

0.005 0.010 0.015 0.025 0.030 0.035 0.040 0.057 0.005 0.010 0.030 0.045 0.056 0.005 0.010

(4)

t/Dia)

K~

-1.45 1.42 1.39 1.38 1.36 1.35 1.30 -1.45 1.38 1.32 1.30 -1.45 1.40 1.37 1.34 1.30

(5)

(BS4994)

1.53 1.51 1.49 1.52 1.57 1.62 1.65 1.66 1.36 1.34 1.35 1.44 1.48 1.00 1.02 1.16 1.21 1.22 1.22

(6)

S.C.F.

1.52 1.40 1.32 1.23 1.20 1.18 1.15 1.10 1.34 1.24 1.11 1.08 1.06 1.02 1.06 1.13 1.12 1.11 1.09

(7)

Sn.C.F.

Results at p = 0.1 kPa

S S S S S P P P S S S P P S

1.34 1.39 1.40 1.45 1.49 1.54 1.57 1.61 1.21 1.25 1.30 1.39 1.44 0.92NC 0.98NC 1.14 1.19 1.20 1.21

(8)

S.C.F. <2)

1.30 1.27 1.24 1.17 1.15 1.13 1.12 P 1.06 1.16 1.15 1.07 1.05 1.04 0.95 1.01 1.10 1.11 1.09 1.08

(9)

Sn.C.FP )

15.4 31.4 48.6 85.2 104.2 123.7 143.5 216.1 17.2 34.9 112.1 171.1 215.0 21.2 39.7 76.3 115.7 153.9 185.6

(10)

p/E ( x 106)

Results at p = 0.2% M a x i m u m Strain

TABLE l(d). RESULTS FROM B O S O R 4 SURVEY TORISPHFAUCALENDS OF 0.32 HEAD HEIGHT

1.031 1.094 1.129 1.239 1.296 1.363 1.402 1.519 1.043 1.087 1.215 1.324 1.385 0.968 0.970 1.036 1.072 1.101 1.120

(11)

K,/K,

OO t~

,<

?,

:::2.

o~

(2)

0.595

0.574

0.32

0.32

Ri/Di

(1)

hi/Di

0.100

0.150

(3)

ri/Di

0.005 0.010 0.020 0.030 0.040 0.050 0.005 0.010 0.015 0.020 0.030 0.040 0.050

(4)

t/Di ~xJ

Ks

-1.85 1.60 1.50 1.40 1.30 -1.85 1.72 1.60 1.50 1.40 1.30

(5)

(BS4994)

1.32 1.45 1.44 1.39 1.35 1.31 1.96 1.83 1.71 1.62 1.50 1.42 1.36

(6)

S.C.F.

1.44 1.47 1.37 1.28 1.20 1.15 2.05 1.80 1.63 1.51 1.35 1.24 1.17

(7)

Sn.C.F.

Results at p = 0.1 kPa

1.23 1.41 1.42 1.38 1.34 1.30 1.86 1.78 1.68 1.60 1.48 1.41 1.35

(8)

S.C.F. t2)

1.34 1.42 1.34 1.26 1.19 1.14 1.94 1.75 1.60 1.49 1.34 1.23 1.16

(9)

Sn.C.FJ 2)

14.9 28.2 59.6 95.4 134.4 175.7 10.4 22.9 37.6 53.8 89.9 129.7 172.2

(10)

( × 106)

p/E

Results at p = 0.2% M a x i m u m Strain

TABLE l(e). RESULTS FROM B O S O R 4 SURVEY--TOR1SPHERICAL ENDS OF 0 . 3 2 HEAD HEIGHT (CONTINUED)

0.918 0.993 1.060 1.095 1.126 1.140 0.959 1.017 1.050 1.074 1.104 1.146 1.164

(11)

K,/K,

z

©

(5) 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65

(4) 0.005 0.010 0.015 0.020 0.030 0.040 0.050 0.070 0.100

(3)

(2)

0.50

(1)

0.50

Ks

(BS4994)

t / D ] l)

rJDi

Ri/Di

hJDi

0.652 0.655 0.659 0.662 0.669 0.676 0.682 0.696 0.716

(6)

S.C.F.

0.441 0.443 0.445 0.447 0.452 0.455 0.460 0.468 0.480

(7)

Sn.C.F.

Results at p = 0.1 kPa

(10) 50.0 95.1 139.7 183.8 270.8 356.2 440.1 603.7 838.5

(9) 0.400 0.420 0.430 0.435 0.443 0.449 0.454 0.464 0.477

0.616 0.636 0.645 0.651 0.662 0.670 0.678 0.692 0.714

NC NC NC NC NC NC NC NC NC

p/E ( x 1o~)

(8) NC NC NC NC NC NC NC NC NC

Sn.C.F.I~/

Results at p = 0.2% Maximum Strain

TABLE l(f). RESULTS FROM B O S O R 4 SURVEY--HEMISPHERICAL ENDS

1.540 1.514 1.500 1.497 1.494 1.492 1.493 1.491 1.497

(11)

KJK,

<

g

1.244 1.239 1.234 1.229 1.224 1.209 1.200 0.998 0.995 0.993 0.990 0.988 0.986 0.984 0.977 0.780 0.780 0.779 0.778 0.777 0.775 0.774

0.20

0.079 0.078 0.077 0.077 0.076 0.073 0.071 0.124 0.124 0.123 0.122 0.122 0.121 0.121 0.199 0.204 0.204 0.204 0.204 0.203 0.202 0.202

(3) 0.005 0.010 0.015 0.020 0.025 0.040 0.050 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.050 0.005 0.010 0.015 0.200 0.030 0.040 0.050

(4)

tlD} I)

2.00 2.00 2.00 2.00 2.00 2.00 2.00 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.04 1.04 1,04 1.04 1.04 1.04 1.04

(5)

Ks (BS4994)

1.78 1.94 2.05 2.16 2.23 2.33 2.37 1.24 1.32 1.38 1.43 1.47 1.50 1.53 1.57 0.907 0.942 0.972 0.996 1.036 1.067 1.092

(6)

S.C.F.

2.08 2.14 2.18 2.23 2.26 2.27 2.27 1.36 1.40 1.42 1.44 1.46 1.47 1.48 1.47 0.871 0.892 0.908 0.920 0.940 0.951 0.959

(7)

Sn.C.F.

Results at p = 0.1 kPa

1.71 1.90 2.02 2.13 2.20 2.30 2.35 1.18 1.29 1.36 1.41 1.46 1.49 1.52 1.56 0.856 N C 0.916 N C 0.953 N C 0.982 N C 1.025 N C |.058 N C 1.084 N C

(8)

S.C.F32)

1.99 2.09 2.14 2.20 2.22 2.24 2.24 1.29 1.36 1.39 1.41 1.44 1.45 1.46 1,45 0.811 N C 0.860 N C 0.885 N C 0.902 N C 0.927 N C 0.940 N C 0.949 N C

(9)

Sn.C.F. ~)

10.0 19.2 28.1 36.5 45.0 71.4 89.2 15.5 29.4 43.1 56.6 69.6 82.9 96.1 137.5 24.7 46.6 67.8 88.7 129.5 170.3 210.9

(10)

p/E ( × 106)

Results at p = 0.2% M a x i m u m Strain

0.859 0.909 0.944 0.968 0.991 1.027 1.049 0.915 0.949 0.978 1.000 1.014 1.028 1.041 1.076 1.055 1.065 1.077 1.089 1.106 1.126 1.142

(11)

KolK<

Key: (1) Underlined values of t/D i are where ttr~ = 0.2; (2) Both S.C.F. and Sn.C.F. at P0002 occur in the torus unless indicated by prefix P(pole) or S(sphere). If both S.C.F. and Sn.C.F. occur in the sphere or at the pole, both have prefixes. S.C.F. and Sn.C.F. values at p = 0.1 K P a do not have prefixes. In general these are at the same position as for Pooo2. Suffix N C indicates that a higher value o f S.C.F. or Sn.C.F. than in the value shown, occurred in the cylinder. (3) A higher value of S.C.F. or Sn.C.F. t h a n the value shown occurred in the circumferential direction in the end.

0.32

0.25

(2)

RJD, rdD ~ (at pole) (at tan line)

(i)

hi/Di

Maximum Minimum

TABLE l(g). RESULTS FROM B O S O R 4 SURVEY--ELLIPSOIDAL ENDS

© Z e~

).

(3

Design of ends of cylindrical pressure vessels

189

4.5

4"0

5'5

Ri = 0 ' 8 0 ri

D~-

~'i : 0 ' 0 5

Ri

ri

o ~D

~i-i = 0 " 9 0 "~-i = 0 " 0 8 7 5

30

-

BS 4994

Ri 2"5

-

z o 0

ri =l.o~=o.~

-

I

I

I

I

I

0.01

0-O2

0-03

O.04

O'05

t D, FIG. 3(a). Variation of strain concentration factor (Sn.C.F.) at 0.2% maximum strain with thickness for torispherical ends of differing radii and (hJD~) = 0.20. Note that the maximum tensile strain occurs in the meridional direction on the inside surface of the torus.

It is suggested that K, in equation (8) should be/(0.0o2 the value of K when q~x = 0.002 (column 9 of Table 1) which makes allowance for the change of geometry under pressure. For cases where the allowable strain in the laminate is to be limited to less than 0.2%, it may be considered necessary to modify the value of K, (see later). There are, however, some parameters for which the use of this method of limiting the maximum strain would give possibly unacceptably high stresses and the design criterion should be (9)

Q = 0.5pD~K~ = 0.5pDiF~-~]

from equation (6)

= trmaxt.

Now from equation (2) in cases where the design is load-limited p = ELXLAM = EL[X1w~nl + X2w2n~ + . . . . . Xxwxnx]

(10)

190

D . C . BARTON et al. 4.0

Ri = 0.665 ~i : 0 ' 0 6 3.5

m

Ri =0.708 ~i

0.10

'0 ~ : 0.1875 Di 3.0--

ii

•9r--' =0.1719 D~

o 0

(D O9

'80 r , : 0.1458 Di

2.5--

BIS. 4994 ri -->0.15 Di

2.0--

B.S. 4994 D.l
Di

1.5, 0

1 0.01

I

1

I

1

0.02

0-03

0.04

0.05

t

Di Fig. 3(b). Variation of strain concentration factor (Sn.C.F.) at 0.2% maximum strain with thickness for torispherical ends of differing radii and (h~/D) = 0.25. Note that the maximum tensile strain occurs in the meridional direction on the inside surface of the torus.

In the case of a load-limited laminate where all the layers are o f the same type (e.g. all CSM)

" ~ XxW~x

P = KX x U

Using P > Q gives u ~( " ~, Wxnx > trmax t

which, noting that the uniaxial tensile (or compression) strength of a laminate with all layers of one type is

reduces to O'f > O'max

K

Design of ends of cylindrical pressure vessels

191

5"0

,-Ri = 0 . 5 7 4 rl :O.i Di Di

2.5

/

-

/

/

7-- BS 4994

2.0

v'

o o Lt(._)

"

/

/ / /

.-5:,o :0288 0

03

/

...

1.5

o,

1.0

o.s o

I o.o,

1 o.o2

I o.os

I o.o,;

1 o.o5

!

Di

FIG. 3(C). Variation of strain concentration factor (Sn.C.F.) at 0.2% maximum strain with thickness for torispherical ends of differing radii and (hJD~) = 0.32. Note that the maximum tensile strain occurs in the meridional direction on the inside surface of the torus.

which ensures that the maximum stress in the head is less than the permissible stress in the laminate by an overall design factor K. The value of Ko in equation (9) in some cases is at a strain of e = 0.002 (or 0.1 eR) and in other cases at a strain eL = (U/KXx) but allowance is made for the appropriate change of geometry of the end (see later). Thus, summarising, using equations (8) and (2) together 0.SpDiK~ ~ eLAMXLAM

(11)

ensures that the maximum strain in the vessel end does not exceed the maximum allowable strain for the laminate determined by the BS4994 procedure (i.e. EMAX< EL~ where ELAM is the smaller of E and eL) and also using equations (9) and (10) together O. 5pD,K, < ELXLAM

(I 2)

ensures that, for a laminate having all layers of the same type, the maximum stress in the vessel is less than the fracture stress by a factor K. It is proposed that pressure vessel ends should be designed according to equations (11) and (12) and it is necessary to establish which equation should be used in any given case. It will be seen from Table 1 that for certain parameters K, is greater than / ~ whilst in other cases K o is significantly greater than K,. (It is obvious from Hooke's law that the latter will occur when the two principal stresses have the same sign). Ideally the design procedure should ensure that the designer effectively checks both conditions (11) and (12) and uses whichever gives the largest required value of XL~. Comparing the values of

D . C . BARTON et al.

192 4'5

4.0

/

5.5

'

0 i, tj G'J

Ri

r__, = 0 . 0 8 7 5

Ri

ri

/ - ~i = 0.9 oi

//-- Di : 0 " 8 D i = 0 " 0 5 /

//

IRi

5.0

r¸

BS. 4 9 9 4 ri

0.1<~-~ <~0.15 2.5

_

2.0 0

/~'~"

1

1

1

I

I

0.01

0.02

0"03

0.04

005

t DI

Fro. 4(a). Variation of stress concentration factor (S.C.F.) at 0.2%. m a x i m u m strain with thickness for torispherical ends of differing radii and hi/D i = 0.20. Note that the m a x i m u m tensile stress occurs in the meridional direction on the inside surface of the torus.

XLAM required by equations (11) and (12) shows that equation (12) should be used if K.>

EL

g~

ELAM

(13)

and equation (1 l) should be used if g~

K,

~<

EL

. ELAM

(14)

The ratio (K,/K,) is given in Table 1 for all parameters investigated at a strain of 0.002. For a strain limited laminate (i.e. when EtAM = E, usually 0.2%) Ec will already have been calculated from the design factor K and extensibility X so equations (13) and (14) determine which of equations (11) or (12) should be used. Examination o f the ratio (KJK,) in the table will show that in the majority o f cases equation (11) will apply. Cases where (Ka/K,) is high are usually where the m a x i m u m stress occurs at a different point on the structure from the point of m a x i m u m strain (see next section). In all cases for strain-limited laminates K, and K~ are the values at ¢. If E # 0.002 it is possible to evaluate the stress and strain concentration factors at any E using an interpolation formula: K~ = S.C.F. o - ELAMIS C F - S.C.F.ooo2 ) 0.002 ~ " ' 'o

(15)

Design of ends of cylindrical pressure vessels

193

4-0

Ri=0.665r_ i =0.06 D~ )i

3.5

R

ri

IL = 0,708 _~ = 0.10 Di

I, = 1.0 r i = 0 . 1 8 7 5 3-0

h

Di

ti = 0 9 )i

D,

-

o o it-

rl = 0 1 7 1 9

(.,9

!i

-

2.5

'i

=

0 ' 8 ri ~ii = 0 " 1 4 5 8

S 4994 '>0"15 i

2.0

-S 4994 "1~0.15

1.5

0

L

I

I

I

I

0"01

0'02

0"03

0'04

0"05

t DI FIG. 4(b) Variation of stress concentration factor (S.C.E.) at 0.2~o maximum strain with thickness for torispherical ends of differing radii and (hJDi) = 0.25. Note that the maximum tensile stress occurs in the meridional direction on the inside surface of the torus. Except for (R~/D~)= 1.0; (ri/Di) = 0.1875 and (t/Di)= 0.050 where it occurs at the pole.

K, = Sn.C.F.0 - ~

(Sn.C.F.0 - Sn.C.F.0.0o2)

(16)

0.002

where S.C.F. 0, S.C.F.0.002, Sn.C.F. 0, Sn.C.F.0.002 are the values of stress and strain concentration factors from columns 6, 8, 7 and 9 o f Table 1. For a load limited laminate (i.e. when ELAM= EL) equation (13) becomes (Kcr/KE)> 1 and equation (14) becomes (Ka/KQ < 1. Strictly in these cases Ko and K, are at a strain EL and the above interpolation formula can be used. It is slightly unconservative but in general acceptance to use Ko and K, at E = 0.002 in the table and not use the interpolation formula except for special cases. If the values of Ko and K, were taken at 0 . 2 ~ strain throughout, the proposed procedure would be greatly simplified and even in the extreme worst case considered, the error would be unlikely to exceed 10~o and in most cases would be m u c h less than 10~ P O S I T I O N O F M A X I M U M STRESS A N D S T R A I N As given in Figs. 3(a)-(c) and 4(a)-(c) the m a x i m u m stress and strain occur at about the same point and are in the meridional direction on the inside surface o f the torus. For some cases (mainly (hflDi) = 0.32) the m a x i m u m stress is in the meridional direction on the outside surface of the sphere (marked S in Table 1) and sometimes the m a x i m u m stress is at the pole (marked P in Table 1). There are also cases of torispherical ends of large torus radius and small sphere radius which are more like hemispheres and of course for hemispheres themselves when the m a x i m u m stress is in the circumferential direction in the cylinder. Crisp[12] pointed out that when the m a x i m u m stress occurs in the cylinder, it is of no

194

D . C . BARTON et al. 5.0

Ri

r; 0.574 E:o.,o

BS 4994 2.5

0'1 ~ ~i ~ 0'15 =0'595

/

=0.15

B.S.4994

2.0 Ri

ri

1.0~i = 0 . 2 8 8

U,..

L~ CO

Ri

0.9

ri

= 0.280

1.5 "/._ Ri ri ~ii : 0"647 ~ii = 0'21

•

1.0

0.5 o

I

1

1

I

I

o.o~

0.o2

o.o3

0.04

o.o5

t Di FIG. 4(c). Variation of stress concentration factor (S.C.F.) at 0.2%. Maximum strains with thickness for torispherical ends of differing radii and ( h J D ) = 0.32. Note that the maximum tensile stress occurs in the meridional direction on the inside surface of the torus for: ( R J D ) = 0.574, (rJD~) = 0.10; ( R J D ) = 0.595, (ri/D~) = 0.15 and (R~/D) = 0.647, (rJD) = 0.21 for ( t / D ) >t 0.01. Otherwise the maximum tensile stress occurs on the outside surface of the sphere. The discontinuity as t / D i increases for (Rt/D) = 1.0; (rJD) = 0.288 and (R~/D) = 0.9, ( r J D ) = 0.280 occurs when the maximum stress is at the pole.

great importance because the S.C.F. is found to be only slightly greater than unity. In all cases where this has occurred the circumferential stresses in the cylinder have been ignored and the values of Sn.C.F. and S.C.F. in Table 1 refer to the maximum values in the ends. These cases are labelled "NC" in Table 1. In the case of hemispherical ends, the circumferential stresses and strains in the ends have also been ignored. Only the meridional stresses and strains are presented in Table 1 because the higher circumferential stresses and strains at the cylinder/hemisphere junction will be dealt with in practice by a suitable transition region from cylinder to hemisphere. All the maximum stresses and strains shown in Table 1 for ellipsoidal ends occur on the inside surface in the meridional direction near to the cylinder/end junction. THE EFFECTS OF CHANGE OF SHAPE DUE TO PRESSURE Fig. 5 shows the variation of S.C.F. and Sn.C.F. with pressure for three different torispherical ends all having R J D i = 1.0 and t / D i = 0.01 but different values of rJD~ (and hence hJD). These variations in S.C.F. and Sn.C.F. are due to changes in geometry of the vessels with increasing pressure. The vertical lines in Fig. 5 indicate the values of pDi/2tE at which the maximum strain is 0.2%. The curves have been discontinued at different values of p D J 2 I E to indicate that vessels with smaller torus radii will fracture at lower values of p D J 2 t E due to the higher stress and strain concentrations.

Design of ends of cylindrical pressure vessels

195

3.0 (..)

2.5 U')

2.0

1.5

I-0 Strain concentration factor Stress concentration factor

0.5

Pressure at which 0 . 2 % maximum strain predicted

I

I

I

I

I

2

4

6

8

I0

PDi

12

xlO-3

2rE FIG. 5. Variation of stress and strain concentration factors with (PD~/2tE) for torispherical ends of common thickness (t/D i = 0.010) and sphere radius ((Ri/Di) = 1.00) at different values of (ri/D~).

Fig. 5 cleady shows the effect of change of geometry in reducing the S.C.F. and Sn.C.F. with increasing pressure. Comparing column 6 and 7 of Table 1 with columns 8 and 9 show that the values of S.C.F. and Sn.C.F. at P0002 are all slightly less than the corresponding values at very low pressure, the reduction being greatest for thinner vessels [4]. With allowance for change of shape the S.C.F. and Sn.C.F. values are generally slightly lower than the values of K, given in BS.4994 but there are some cases for hi/D~ = 0.32 (see Fig. 4(c) and Table 1) where the maximum stress occurs in the sphere and the S.C.F.'s are significantly greater than the shape factors K, given in BS.4994. As the pressure is increased from P0.002 towards the pressure which produces fracture, there is a significant further reduction in S.C.F. and Sn.C.F. for all ends (Fig. 5). The criterion for design based on Sn.C.F. and S.C.F. at 0.2~ maximum strain is therefore reasonably conservative. EFFECT OF LARGE VALUES OF t/ri As stated in the section on the calculation of stress and strain concentration factors, results have been presented in Table I and Figs. 3 and 4 for values of t/r~ much larger than those for which thin shell theory is valid. In (12), Crisp studies the problem by a curved beam analysis of the torus region and gives correction factors which can be applied to the S.C.F., but these correction factors do not appear to be incorporated in BS.4994. In the present case, which is based on strain as well as stress, there are similar difficulties in that thin shell theory his not valid. It is assumed that initially plane sections of the shell remain plane and that the neutral axis in bending coincides with the mid-surface of the shell. Strictly a full three-dimensional analysis is required and the values of Sn.C.F. for high t/r~ should be treated with a little caution.

196

D . C . BARTON et al.

DESIGN OF TRANSITION C Y L I N D E R / E N D It should be noted that all calculations have been carried out for an end and cylinder of the same thickness. In a design, the thickness of the end and the cylinder will be different and a suitable tapered transition region in accordance with BS.4994 is required. Strictly speaking this will give a distribution of stress and strain in the actual vessel which differs from the distribution in a uniform thickness vessel on which the design criterion is based. This is satisfactory with an adequate design of the transition region. It appears that the Ks factors in BS.4994 for semiellipsoidal ends, which are taken from (13) make some allowance for different thicknesses in the cylinder and end since (13) presents data for such cases. H E M I S P H E R I C A L ENDS All practical design procedures will result in hemispherical ends which are thinner than the cylinders to which they are attached. The ASME X code/3] recommends a thickness for hemispherical ends equal to that given by equation (4) for a thin sphere or hemisphere which is equivalent to equation (9) with a shape factor K~ = 0.5. The German code/2] employs a value of K~ = 0.6. BS.4994 uses Ks = 0.65 which, when combined with the 0.55 factor in equation (5), gives an equivalent value of K~ = 0.715 in equation (9). In Table 1 the maximum S.C.F. based on meridional stresses in a uniform end/cylinder is 0.714, a value which is very close to the effective K~ used in BS.4994. In reality the maximum stresses in the ends will depend entirely on the form of the transition between the cylinder and hemisphere. It seems that the effective value of K~(0.715) in BS.4994, whilst possibly conservative, would be a reasonably safe assumption in the absence of data for realistic cylinder/sphere transitions. The Sn.C.F. values in Table 1 are as low as 0.4 for thin hemispheres. If these low Sn.C.F. values were used in equation (8) the resulting ends would be thinner than spheres and hemispheres designed using equation (4). This discrepancy arises, of course, because of the biaxial stresses of the same sign in the sphere. LOCAL TORUS R E I N F O R C E M E N T For most torispherical ends the strains in the spherical portion of the end are lower than the peak strain in the torus, but in certain cases, as discussed previously, the stress in the sphere can be as high as or even higher than the stress in the torus. Although not mentioned in BS.4994, it is suggested in the Explanatory Supplement/10] that K s factors for ends can be applied to the torus region and that with a suitable blending o f sphere and torus laminates the spherical portion can be designed using equation (4). Because considerable bending can occur in the sphere and even at the pole in some cases this suggestion is not conservative and indeed some recent experimental work not yet reported suggests that the sphere is under-designed on this basis. The German Code/2] allows different factors for the sphere and torus regions but for their two standard torispherical ends the sphere factor is 1.5 and 2.0 times the value for simple spherical vessels. Where such a design is used it would be advisable to extend the same wall construction used in the torus for about a decay length into the sphere and then gradually taper off to the required sphere thickness. Ideally, as in the ease of the hemispherical ends discussed previously, allowance should be made for the stresses associated with the particular shape of transition region used. C O M M E N T S ON BS.4994 A N D P R E S E N T A T I O N OF RESULTS For a given head height BS.4994 tabulates one set o f Ks factors for use in the range 0.1 < r~/Dj~<0.15 and another for r/D i > 0.15. It is quite clear from Figs. 3 and 4 and Table 1 that for a given head height there can be a considerable variation of Sn.C.F. and S.C.F. with different combinations of r~/D~ and R~/D~ to give a particular h~/D~(see Appendix). The sharp distinction at r~/D~= 0.15 in BS.4994 does not seem to represent actual results particularly well. This is very evident for h/D~ = 0.25 in Figs. 3(b) and 4(b) for r/D~ = 0.1458, 0.1719 and 0.1875. It is also noticeable from Figs. 3 and 4 that for rjD~ < 0.1, the Sn.C.F's increase significantly and hence there is good reason to require r/D~ > 0.1 in a design code, but in principle this is no reason why the data presented here could not be used to design special ends outside the code limits. It has already been mentioned under the section on "The effects of change of shape due to pressure" that for (h~/D~)= 0.32 (see Fig. 4c) the S.C.F.'s are significantly greater than the shape factors Ks given in BS4994. Inspection of Fig. 4(c) shows that for 0.1 ~<(r~/D~)<~0.15 Ks from BS4994 is similar to the curve for (RJD~)=0.574 (rjDi)=O.lO. The Ks values from BS4994 for (r/D~)>O.15 are similar to the curve for (Ri/Di) =0.595 (rJDi)= 0.15 and BS4994 factor is obviously conservative for (Ri/Di)=0.647 (r~/Di)=0.21. However for (RJD~) = 0.9 (rJD~) = 0.28 and (R,/D~) = 1.0 (rJDe) = 0.288 the BS4994 values are unconservative for higher values of (t/D~) where the maximum stress occurs in the sphere or at the pole. One minor point in the code concerns the case h~/Di = 0.2 for which a K s value is given for 0.1 ~< r/Di <~O.15 whereas it is clear from Fig. AI that rjD~ <~0.11 if R~/D~ cannot exceed unity. Note also that the apparently arbitary limit RJD~ < 1 prevents the designer from selecting the optimum geometry for the head height hJD~ = 0.2 (see Fig. A2 and comments in the Appendix). The use o f a slightly larger Re/D ~and hence a larger knuckle radius (the critical feature in this case) would give a lower value of Ks. However, the savings in thickness o f the end would not be large. USE OF 0.5 INSTEAD OF 0.55 IN EQUATIONS (8) A N D (9) Note that 0.5 is used in equations (8) and (9) instead of 0.55 as in equation (5). The value 0.55 in equation (5) is used since the work of Crisp[12] is based on mid-surface shell parameters. In the current work all calculations o f S.C.F. and Sn.C.F. have been calculated using thin shell centre-line theory, but the dimensionless parameters presented are all based on inside dimensions. It follows, therefore, that the factor 0.55 in BS.4994 calculations for Q (where Ks is based on centre-line calculations) can be replaced by the less conservative value of 0.5 for the factors presented here. It has already been noted that for hemispherical ends BS.4994 gives Q = 0.55pD~ (0.65)= 0.357pD~ whereas the present work would give the very similar Q = 0.5pD~ (0.716)= 0.358 pD~ as the highest value. Similar remarks apply to ellipsoidal ends where the S.C.F. is slightly greater than K~ in BS.4994.

Design of ends of cylindrical pressure vessels

197

FURTHER COMMENTS ON DESIGN CRITERIA FOR GRP LAMINATES An important feature of the BS4994 procedure is that it attempts to allow for the design of mixed laminates with a variety of different types of reinforcement. This invariably means that the effective laminate stiffness in bending is different from that in direct loading. Even for a laminate of one type of reinforcement only, e.g. CSM, the low modulus resin-rich gel coats on the inner and outer surfaces of the laminate mean that the effective stiffness in tension can be nearly 50~o greater than that in pure bending[15]. In the BS4994 method of calculating laminate properties, the laminate stiffness in direct loading only is derived even though the mode of deformation may be predominantly bending as in the torus of a torispherical end. Ideally, laminate theory[16] should be used in such a situation to derive a more realistic value for the effective stiffness (or extensibility) in bending. However, this would mean a further increase in the complexity of the code. Another problem in the case of laminates having layers of different properties is that stresses may vary greatly across the thickness depending on the arrangement of the layers and that the maximum bending stress is not necessarily at the surface of the laminate. Further careful consideration would be required before applying equation (12) under such circumstances. The maximum strain, on the other hand, will always be on one or other of the extreme surfaces and will depend largely on the overall laminate bending stiffness. Thus a design based on limiting the maximum strain using (say) the lesser of the direct and bending stiffnesses would seem to have a greater theoretical validity than the present BS4994 procedure. However, the reservations expressed earlier about the general applicability of a maximum strain failure criterion under biaxial loading should be remembered in this context. For all the ends considered in Table 1 where the maximum Sn.C.F. and S.C.F. occur in the torus region, the ratio of circumferential to meridional stresses (S) is in the range + 0.35 to - 0.5. Fig. 1 shows that in this region the discrepancy between maximum stress and maximum strain criteria and the experimental results is not large considering the scatter in the experimental results and the combined inner envelope fits the results very well. Where the maximum stresses and strains occur in the spherical crown the stress ratio S is about + 1; the biaxial tension condition which corresponds to the maximum discrepancy between the experimental and the two theoretical envelopes. This is an example of a situation where the adoption of a maximum strain criterion is unsafe. CONCLUSIONS Based on the experience of a large programme of experimental work on torispherical e n d s o f glass r e i n f o r c e d p l a s t i c p r e s s u r e vessels, a c o m p u t e r p r o g r a m B O S O R 4 h a s b e e n u s e d to g e n e r a t e d e s i g n d a t a f o r t o r i s p h e r i c a l a n d s e m i - e l l i p s o i d a l e n d s . T h e d e s i g n p r o c e d u r e p r o p o s e d is b a s e d o n d a t a c a l c u l a t e d f o r stress a n d s t r a i n c o n c e n t r a t i o n f a c t o r s at a v e r y l o w p r e s s u r e a n d a p r e s s u r e to g i v e a m a x i m u m s t r a i n in t h e e n d o f 0.002. T h e p r o c e d u r e is c o m p a t i b l e w i t h t h e s e c t i o n in B S . 4 9 9 4 o n t h e d e s i g n o f l a m i n a t e s b a s e d o n u n i a x i a l tensile stress a n d s t r a i n d a t a . T h e p r o c e d u r e e n s u r e s t h a t n e i t h e r t h e m a x i m u m stress n o r t h e m a x i m u m s t r a i n in the e n d e x c e e d s t h e a l l o w a b l e limits. A l t h o u g h m o r e c o m p l e x t h a n t h e e x i s t i n g B S . 4 9 9 4 p r o c e d u r e it w o u l d n o t b e difficult to f o r m u l a t e t h e s e p r o p o s a l s in c o d e f o r m a n d t h e y h a v e the a d v a n t a g e o f b e i n g a r a t i o n a l a t t e m p t to c o r r e l a t e l a m i n a t e d e s i g n a n d s t r u c t u r e d e s i g n t a k i n g i n t o c o n s i d e r a t i o n t h e b i a x i a l stress states w h i c h o c c u r in t h e vessels. Acknowledgements--We wish to thank Mr. B. D. Gray and Dr. G. C. Eckold of Plastics Design and Engineering Ltd. for their advice throughout the project and preparation of the paper, and the Polymer Engineering Directorate of the Science and Engineering Research Council for supporting the work. We are particularly grateful to Professor P. Stanley who has been closely associated with the preparation of BS4994 for his many thoughtful contributions during the development of this paper.

REFERENCES 1. BS4994, Specification for Vessels and Tanks in Reinforced Plastics, British Standards Institution (1973). 2. A. D. MERKBLATTNI, Pressure Vessels in Glass Fibre Reinforced Plastics, Vereinignng der Technischen Uberwachungs-Verein e.v. (Vd TUV), Essen, West Germany, (1969). 3. ASME Boiler and pressure vessel code, section X. Fibre Glass-Reinforced Plastic Pressure Vessels (Aug. 1971). 4. D. A. HUGHES, O. C. BARTONP, D. SODENand S. S. GILL, The effect of thickness/diameter ratio on the structural behaviour of the torispherical ends of glass reinforced plastic cylindrical pressure vessels. North Western Branch o f Int. Chem. Engng and Int. Mech. Engng Joint Syrup. on Reinforced Plastic Construction Equipment in the Chemical Process Industry, Manchester (1980). 5. D. C. BARTON, P. D. SODENand S. S. GILL, The strength and deformation of torispherical ends for glass reinforced plastic pressure vessels--1. Effect of torus radius/cylinder diameter ratio for ends with sphere radius equal to cylinder diameter. Int. J. Pres. Ves. Piping 9, 285-318 (1981). 6. D. C. BARTON, P. D. SODENand S. S. GILL, The strength and deformations of torispherical ends for glass reinforced plastic pressure vessels, part 2: effect of torisphere geometry for ends of fixed head height. Int. J. Pres. Ves. Piping 10, 31-53 (1982). 7. D. C. BARTON, The strength and deformations of torispherical ends for glass reinforced plastic pressure vessels. Ph.D. Thesis, Faculty of Technology, University of Manchester (April 1981). MS Vok 26, No. 3 D

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8. D. BUSHNELL, Stress, stability and vibration of complex branches shells of revolution, Comput. Structures 4, 399-437 (1974). 9. P. T. MOORE, Unit strength and extensibility--a new design concept for glass-fibre reinforced plastics. British Plastics Federation 9th Int. Reinforced plastics Conf., Brighton (1974). 10. PD 6480, Explanatory Supplement to BS.4994. Specification for Vessels and Tanks in Reinforced Plastics. British Standards Institution (1977). 11. P. STANLEYand P. T. MOORE, The design of vessels in reinforced plastics under buckling conditions. Int Mech. Engng Con,/. '" Vessels under Buckling Conditions", pp. 1-8. London (Dec.'1972). 12. R. K. CRISp, A computer survey of the behaviour of torispherical drumheads under internal pressure loading--The elastic analysis. CEGB Rep. RD/B/NI005 (Feb. 1968). 13. H. KRAUS, G. G. BILODEAUand B. F. LANGER, Stresses in thin-walled pressure vessels with ellipsoidal heads. Trans. ASME J. Engng Industry (Feb. 1961). 14. M. J. OWEN, Designing against fatigue in glass reinforced plastics. Int. Mech. Engng. Conf. on Designing with Reinforced Plastics (C233/77) (1977). 15. D. C. BARTONand P. D. SOt)EN, An investigation of the short term in-plane stiffness and strength properties of chopped strand mat reinforced polyester laminates. Composites 13, 66-78 (1982). 16. D. C. BARTON and P. D. SODEN, Stiffness and strength properties of asymmetric glass-reinforced polyester laminates. Composites 13, 180-201 (1982). 17. H. FESSLERand P. STANLEY,Stresses in torispherical drumheads: a photoelastic investigation. J. Strain Anal. 1. (1), 69-82 (1965). 18. E. P. ESZTERGARand H. KRAUS, Analysis and design of ellipsoidal pressure vessel heads. J. Engng Industry, ASME Paper No. 70-PVP-26, 805-817 (1970). 19. H. C. BOARDMAN, Optimum Torispherical Pressure Vessel Heads, Water Tower, Edn 8-9 (Jan. 1949). 20. G. A. THURSTON and A. A. HOLSTON, Buckling of cylindrical shell end closures by Internal Pressure. NASA CR-540 (July 1966). 21. H. FESSLERand P. STANLEY,Stresses in torispherical drumheads: a critical evaluation. J. Strain Anal. 1 89-101 (1966).

APPENDIX

Geometry of torispherical ends (including a discussion of the optimum geometry) It is evident from Fig. 2(b) that for a given head height, h, and cylinder diameter, D, there is an infinite number of combinations of r and R to give a torispherical end of spherical radius, R, and torus radius, r, with continuity of slope at the sphere/torus and torus/cylinder junction. An excellent presentation of the relationships between (h/D), (r/D), (R/D), the dimensionless geometrical parameters, was given in (17). The other dimensionless parameter is t/D. It may be seen that ( O - r)2 +_( R - h ) 2 = (R - r) 2 which gives 4-D

+

-2

+2

=0

(A1)

Part of Fig. 2 in (17) is reproduced here as Fig. A1 giving contours of (h/D ) with ordinate (r/D) and absicissae (R/D) (or (D/R) using a linear scale). The figure also shows the limitation on torisphere geometry imposed by BS.4994. Esztergar and Kraus[18] state that the structural behaviour of a torispherical end is affected by the discontinuity of meridional radius (i.e. r to R) at the sphere/torus junction and state that for this reason torispherical ends are weaker than ellipsoidal ends. Boardman [19] states that for a given hiD the best torispherical end is that which makes r/R a maximum. Thurston and Holston[20] also define an optimum torispherical end as one which approximates to an ellipsoid and give the criterion tan q~0= (2h/D). Neither paper gives a full explanation of the criterion but both lead to the equation R

A 1 , 2 =~+~#(A -A)

where

and then (r/D) is given by equation (AI) for the optimum geometry. The curve relating (R/D) and (r/D) for maximum (r/R) is shown chain dotted in Fig. A1. A discussion of optimum shapes of torispherical ends is given in (21). Fig. A2 shows plots of Sn.C.F.0.002 against (ri/Di) for (hl/D) = 0.2, 0.25, 0.32 and (t/D) = 0.05 and 0.1 as computed for i the other results presented in this paper (Figs. 3 and 4). For these parameters it is clear that a minimum Sn.C.F.0.002 exists for (hiD) = 0.2, 0.25, 0.32 which is not significantly dependent on (t/D) and occurs at a value of (rffDi) reasonably near to the optimum parameters indicated in Fig. AI. For (hiD) = 0.2 the optimum is outside the limits of BS.4994, i.e.

Design of ends of cylindrical pressure vessels

199

D R 2.0 0.5

1.8

I-6

1.4

I-2

1.0

\

0.5

0.6

i ~

,.----

0"4

I

~

/

0.45

- - - - Limitsof B S 4 9 9 4 _ , r maximum R

~

,0.4 0.35

_.---f,

-5 0-2

_

o

-- 0.25

I 0.5

0.6

0.7

0"8

0"9

1.5

R

5 FlG. AI. Parametric "carpet" plot for torispherical ends.

hi 4

5

--

/

o.25

/

D

"\//

o LL:

c~

2--

__

'r : 0.01 Di

I

I

L

I

0-I

02

L

I 0.3

ri

F[6. A2. Variation of strain concentration factor (Sn.C.F.) at 0.2% maximum strain with torus radius for torispherical ends of differing head heights. at approximately (rgDi) = 0.125 for which (R~/Di) is approximately 1.2. It is worth noting that the limit R~ < D~ in all pressure vessel codes is based on simple membrane theory for ends and cylinders of the same thickness. However, for an end thicker than the cylinder there is no a priori reason why R~ cannot be greater than D~ and for hJD~= 0.2 it leads to an optimum design. This rather brief note on optimum geometry might be useful to designers aiming to achieve a torispherical end of given (h~/Di) of minimum thickness.