WOODS HOLE OCEANOGRAPHIC INSTITUTION Woods Hole, Massachusetts 02543

TECHNICAL MEMORANDUM 3-81

DESIGN PRESS

Prepared by

Arnold G. Sharp

August 1981

CURVES FOR OCEANOGRAPHIC URE-RESISTANT HOUSINGS

Off ice o f Naval Research Contract No. NO001 4-73-C-0097; NR 265-1 07

Abstract

Curves have been prepared to serve as a guide in

designing externally pressurized cylindrical and

spherical housings and flat end closures. The

p lo t s show wall thickness-to-diameter ratio as a

function of collapse depth over the range 0-10,000

meters (0-32,800 feet). Curves are plotted for the

commonly used stainless steels and aluminum and

titanium alloys. Also, there are three dimension-

less curves that can be used for any structural

material. Included is a brief review of the

equations used in plotting the curves.

Table of Contents

I, I n t r o d u c t i o n

11, R u a t i o n s

111, Design Curves for C y l i n d r i c a l Housings

fV, Design Curves for F l a t End Caps

V, Design Curves for Spherical Housings

VI, Three Dimensionless Curve S h e e t s

VII. Discussion

VIIL References

Page

1

4

i1

12

List of Figures

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Collapse Depths of Cylinders; Aluminum Alloy 6061-T6; Yield Strength 241 MPa (35.000 psi)

Collapse Depths of Cylinders; Aluminum ~lloy 7075-T6: Yield Strength 455 MPa (66,000 psi)

Collapse Depths of Cylinders; Stainless Steel Types 302, 304, 316: Yield Strength 241 MPa (35,000 p s i )

Collapse Depths of Cylinders; Stainless Steel Type 17-4 PH; Yield Strength 862 MPa (125,000 p s i )

Collapse Depths of Cylinders; Unalloyed Titanium ASTM Grade 2; Yield Strength 276 MPa (40.000 psi)

Collapse Depths of Cylinders; Titanium Alloy 6A1-4V; Yield Strength 828 MPa (120,000 psi)

Collapse Depths of Flat End Caps; Aluminum Alloy 6061-T6; Yield Strength 241 MPa (35,000 psi

Collapse Depths of Fla t End Caps; Aluminum Alloy 7O75-T6; Yield Strength 455 MPa (66,000 ' psi)

Collapse Depths of Flat End Caps; Stainless Steel Types 302, 304, 316; Yield Strength 241 MPa (35,000 p s i )

Collapse Depths of Flat End Caps; Stainless Steel Type 17-4 PH; Yield Strength 862 MPa (l25.OOO psi)

Collapse Depths of Fla t End Caps; Unalloyed Titanium ASTM Grade 2; Yield Strength 276 MPa (4O,OOO p s i )

F i g u r e 1 2

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figu re 21

Collapse Depths of Flat End Caps; Titanium Alloy 6 A l - 4 V ; Yield Strength 828 MPa (120,000 psi)

Collapse Depths of Spheres; Aluminum Alloy 6O6bT6 : Yield Strength 241 MPa (35,000 psi)

Collapse Depths of Spheres; A l u m i n u m Alloy 7075-T6; Yield Strength 455 MPa (66,000 psi)

Collapse Depths of Spheres; Stainless Steel Types 302, 304, 316; Yield Strength 241 MPa (35,000 psi)

Collapse Depths of Spheres; Stainless Steel Type 17-4 PH: Yield S t r e n g t h 862 MPa (125,000 psi)

Collapse Depths of Spheres; Unalloyed Titanium ASTM Grade 2; Yield Strength 276 MPa (40,000 p s i )

Collapse Depths of Spheres; Ti tan ium Alloy 6A1-4V; Y i e l d S t r e n g t h 828 MPa (120,000 psi)

Collapse Pressure Ratios for Cylinders ; Dimensionless Curves

Collapse Pressure Ratios for Flat End Caps; Dimensionless Curve

Collapse Pressure Ratios for Spheres; Dimensionless Curves

I. Introduction

The following pages present curves for use in the

design of externally pressurized cylindrical and spherical

oceanographic housings. Design curves also are included for

f l a t circular plates, which commonly are used as end closures

for cylindrical vessels.

Curves have been plotted for t h e commonly used stain-

less s t e e l types 302, 304, 316 and 17-14 PH; the aluminum

alloys 6061-T6 and 7075-T6; and for unalloyed titanium ASTM

Grade 2, and titanium alloy 6A1-4V. In each case the thick-

ness/diameter ratio is shown as a funct ion of collapse depth

t o a maximum depth of 10,000 meters (32,800 feet) . Far thin-walled vessels, collapse can occur at wall

stress levels below the elastic limit. This is called

e l a s t i c buckling or instability failure, For relatively

thicker shells that fail a t stress levels above the elastic

limit, the failure is due to plastic yielding of the material.

Most of the present curves clearly show these two types of

failure. In p l o t t i n g the sphere curves for aluminum alloy

6061-T6, the stainless steel 300 series, and unalloyed

t i tan ium, it was found that the governing elastic portion

covers a very small range of depths (to less than 500 meters)

and so it was omitted, However, in t w o of t h e cylinder

curves (those for stainless steel 17-4 PH and titanium

alloy 6A1-4V) the elastic failure mode governs over prac-

tically the entire range of ocean depths considered.

The elastic curves are based on standard equations

found in elastic buckling theory, For spheres, the equat ion

u s e d is a modification of the classical buckling e q u a t i o n

derived for the ideal case. The inelastic portions of the

curves are based on the thick-wall theory of Lam& Speci-

fically, the am& equations used w e r e those for maximum

normal stress. All of the equations are discussed in another

section of this report.

For materials not covered by the main body of curves,

three additional non-dimensional plots have been included.

Since the parameters plotted are dimensionless ratios, any

system of units may be used, Also, because the material

yield strength sy and the modulus of elasticity E are con-

tained in the ratios, pressure vessels of any engineering

material can be designed. A minor disadvantage of these

curves is that t w o separate determinat-ions of the required

t / ~ ratio must be made, and the larger of the two must be

used. In order to allow the use of dimensionless ratios the

collapse pressure was substituted f o r col lapse depth. T h e

conversion to depth can r ead i l y be made by the u s e r .

Cyl inders

T h e curves f o r cy l inde r s are based on t h e fol lowing

equa t ions :

A, E l a s t i c Buckling Collapse Pressure

where p, = e l a s t i c buckling p r e s s u r e

E = modu1.u~ of e l a s t i c i t y

11.- = Poi s son ' s r a t i o 4*-

t = w a l l thickness of t u b e

Do = o u t s i d e diameter of t u b e

B, Tube Length ( ~ l a s t i c ~ a i l u r e )

The e l a s t i c curves f o r c y l i n d r i c a l v e s s e l s are

plotted f o r so-cal led "long tubes" where the

l eng th must be g r e a t e r than

( R e f . 1)

where r = tube r ad ius

t = w a l l t h i ckness of tube

F o r s h o r t e r t ubes the c o l l a p s e pressure would be some-

what g r e a t e r than that pred ic t ed by t h e equa t ion of

part A. above. T h i s would be the result of r e in -

forcement provided by t h e end c l o s u r e s .

C, Ine la s t i c F a i l u r e

(Ref. 1)

where ( ~ ' m a x . = maximum s t r e s s a t tube i n n e r s u r f a c e

p = external p r e s s u r e

b = o u t e r radius of tube

a = inner r a d i u s of tube

F a i l u r e i s cons idered t o occur a t t h a t va lue of pres-

s u r e where (s),,,. i s equa l t o t h e y i e l d s t rength of

the m a t e r i a l ,

D. F l a t End Caps

C y l i n d r i c a l p re s su re c a s e s commonly use f l a t c i r c u l a r

p la tes as end c losu res . These plates usually contain

some type of s e a l , such as an o - r ing , and some means

of attachment t o t h e cylinder, For p r a c t i c a l design

purposes t h e equat ion f o r a f l a t circular p l a t e , simply

supported around i t s edge, can be used. The diameter

used in the equation i s the diameter of the unsupported

p o r t i o n of t h e p l a t e i n t h e a c t u a l case . For g r e a t e s t

accuracy t h e p l a t e t h i c k n e s s should be no more than

about one-quarter of t h e e f f e c t i v e diameter . The flat

p l a t e equat ion i s

(Ref. 1)

where (sImax, = maximum s t r e s s at center of plate

p = e x t e r n a l pressure (p res su re uniformly d i s t r i b u t e d over p l a t e s u r f a c e )

a = r ad ius of unsupported p l a t e

t = t h i ckness of unsupported p l a t e

rn = r e c i p r o c a l of Poisson ' s r a t i o

F a i l u r e i s considered t o occur a t that v a l u e of pres-

s u r e where (s)~,,. i s equal t o t h e yield s t r e n g t h of

t h e m a t e r i a l ,

Spheres

The following equat ions served a s the bas i s for p l o t t i n g

t h e curves f o r s p h e r i c a l p ressure v e s s e l s ,

A. E l a s t i c Buckling Collapse Pressure

T h e c l a s s i c a l e l a s t i c buckling p res su re f o r an i d e a l

sphere i s g iven by

( R e f . 1)

where p, = e l a s t i c buckling p res su re

E = modulus of e l a s t i c i t y

t = spherical s h e l l t h i ckness

r = radius to shell midsurface

f X = Poisson's ratio

This reduces to

- 1 . 2 1 ~ ( t / r ) * Pe -

where Poisson's ratio is taken as 0.3,

Early experimental work showed wide disagreement

with the above equation, probably due to imper-

fections in geometry and material. Extensive re-

cent testing at the David Taylor Naval Ship Research

and Development Center (DTNSRDC) has indicated that

the coefficient 1.21 can be attained in an ideal

spherical shell, but that in most cases, even when

imperfections are almost unmeasurably small, t h e

coefficient more properly should be about 70 per-

cent of the classical value, DTNSRDC has recommended

that the classical equation be modified for near-

perfect spherical shells to be

pe = 0.84 E (t/ro) 2 (~ef. 2)

where Poisson's r a t i o is equal t o 0.3, and r, is the

radius to the shell outside surface, The e l a s t i c

portions of t h e spherical shell curves of this report

are based on the above modified equation.

B. I n e l a s t i c F a i l u r e -

(Ref, 1)

where ( s ) ~ , , ~ = maximum s t r e s s a t sphere inne r s u r f ace

p = e x t e r n a l pressure

b = o u t e r r a d i u s of sphere

a = i n n e r r a d i u s of sphere

A s i n the c a s e of t h e c y l i n d e r s , f a i l u r e i s con-

s i d e r e d t o occur when (dm,,- is equal t o the y i e l d

s t r e n g t h of the m a t e r i a l .

Thin Wall Stress Equations

Where t h e w a l l t h i c k n e s s t of a vessel i s l e s s t h a n

about one t e n t h t h e r a d i u s r t h e s t r e s s e s can be con-

s i d e r e d uniform throughout t h e th i ckness of t h e w a l l

and t h e th in -wa l l , o r membrane stress equat ions

(~)max. = EL (cylinder)

and (''max. = pr (sphere) 2t

can be used t o p r e d i c t i n e l a s t i c behavior. The presen t

curves d i d not make use of t h e s e equat ions , however. T h e

so-ca l led ~ a m e ' equa t ions , discussed above, were used

throughout the i n e l a s t i c ranFe of t/D values. The membrane

or thin-wall curves would practically coincide with

the ~ a m 6 curves at the lower end of the range- It

should be noted, however, that it is in t h a t region

of t/D values that elastic or instability failure

might govern. For the higher values of t/D t h e thin-

wall equations could be in error by as much as 20 per-

cent - Dimensionless Curves

The three dimensionless curves (E'igs. 19, 20, and 21)

are based on the same equations that were used f o r the

other curves , but their arrangement is somewhat d i f -

ferent. On the horizontal axes are plotted dimension-

less ratios that involve the collapse pressure, and

either the modulus of elasticity E or the yield

strength sy. The vertical scales express as before

the ratio of thickness t to outside diameter Do. Thus,

for any collapse depth and any material, t / D o values

can be found for elastic and inelastic failure modes.

Of the two t / ~ , ratios determined, t h e greater value

must be used, The left hand vertical scale and the

bottom horizontal scale are used with the inelastic

curve, while t he right hand vertical and upper hori-

zontal scales go with the elastic buckling curve.

.Pressure-Depth Conversion

In performing the calculations required for plotting

the curves the following approximate pressure-depth

relationship was used:

where p = pressure in megapascals ( M P ~ )

h = depth in meters

where p = pressure in pounds per square inch

h = depth in feet

A more exact, non-linear determination of the pressure-

depth equation would take into account the effect of

the compressibility of sea w a t e r , but t h i s added pre-

cision was not considered necessary here. The approx-

imate linear equation shown above is in error by about

two p e r c e n t a t a depth of 10,000 meters.

Poisson's Ratio

In all cases where Poisson's ratio was required in t h e

calculations, a value of 0.3 was used.

111. Design Curves for C y l i n d r i c a l Housings

F i g u r e 4 Collapse Depths of Cylinders; S t a i n l e s s S t e e l Type 17-4 PH; Y i e l d S t r e n g t h 062 MPa (125,000 p s i )

LV. Design Curves For F l a t End Caps

V. Design Curves For Spherical Housings

VI. Three Dimensionless Curve Sheets

(PC/€) 10 - EL AST/C BUCKLING

Figure 19 Collapse Pressure Rat ios fo r Cylinders; Dimensionless Curves

V I I . Discussion

It i s common t o des ign v e s s e l s t o have a f a c t o r of

safety s u i t a b l e t o t h e intended a p p l i c a t i o n , L e . , t o have

a c o l l a p s e depth g r e a t e r t han t h e o p e r a t i n g depth by some

known f a c t o r , I n some cases proof t e s t procedures are re-

qu i red t h a t dictate a larger safety f a c t o r (and g r e a t e r

weight pena l ty) t h a n o therwise might have been used. For

example, i n t h e case of ALVIN, a U S , Navy c e r t i f i e d deep-

submersible, all implodable ins t rument housings a r e proof-

t e s t e d t o a pressure that i s 1.5 t i m e s t h e ope ra t ing pres-

s u r e - T h i s means t h a t t h e des ign c o l l a p s e p res su re must

exceed t h e t e s t p re s su re by a s a f e margin.

The p resen t curves a r e in tended t o s e r v e a s a guide

i n designing o r i n p r e d i c t i n g t h e c o l l a p s e depth of cy l in -

d r i c a l o r s p h e r i c a l p re s su re cases . The actual c o l l a p s e

depth of a tube o r sphere depends upon s e v e r a l f a c t o r s not

considered h e r e , such a s out-of-roundness, v a r i a t i o n s i n w a l l

t h i ckness , flat s p o t s , and f laws o r lack of homogeniety i n

ma te r i a l . These d e f e c t s a r e u s u a l l y smal l i n magnitude and

may be expected t o dec rease c o l l a p s e s t r e n g t h s s l i g h t l y from

those values given by t h e curves . On t h e other hand, the

curves of t h i s r e p o r t are based on t h e minimum mechanical

properties of the materials in question. Average values of

yield strength, however, are often found to be 10 to 15 per- .'

cent greater than the minimum. This could result in a ten-

b dency toward conservative design when using the present

curves (see Figure 13).

The curves for flat end caps are those for the basic

sol id flat circular plate with a simply supported edge. Any

holes or penetrations in such a plate would cause stress

concentrations to be present near the hole boundaries. These

regions of higher stress are quite localized, but an increase

in plate thickness may be necessary to prevent loca l yielding

of t h e end cap material. Standard handbooks on the use of

stress concentration factors, such as that of Peterson (Ref.

6), are available. With their use the designer can determine

peak values of stress around penetrator holes, and select a

suitable thickness/diameter ratio accordingly.

On two of the curves (Figures 13 and 18) documented

results of collapse tests known to the author have been plotted

f o r comparison with the curves. These were spheres designed

for the variable ballast and fixed buoyancy systems of the deep-

submersible ALVIN. The aluminum alloy spheres were two of those

originally installed in ALVIN, while the titanium alloy samples

were i n t e n d e d f o r destructive t e s t i n g , p a r t of t h e group of

new spheres that rep laced the o l d e r aluminum ones.

For c r i t i c a l a p p l i c a t i o n s ( f o r example, i n manned

I deep submersible h u l l s ) d e v i a t i o n s from idea l geometry can

be measured and their e f f e c t s on t h e u l t i m a t e s t r e n g t h of

the vesse l determined. Researchers a t t h e David Taylor Naval

Ship Research and Development Center have developed a method

fo r s p h e r i c a l s h e l l s t h a t makes use of l o c a l geometry L e . ,

local th ickness and r a d i u s of curva tu re over a predetermined

c r i t i c a l arc length . Krenzke (Ref. 7 ) r e p o r t e d on this pro-

cedure t h a t p r e d i c t s c l o s e l y t h e collapse p r e s s u r e s of near-

p e r f e c t and i n i t i a l l y - i m p e r f e c t spheres . T h i s technique i n

m o s t c a ses would p r e d i c t c o l l a p s e a t a somewhat lower pressure

than do t h e curves of t h i s r e p o r t . However, the method re-

q u i r e s many a c c u r a t e measurements of r a d i u s and t h i c k n e s s and .

i s the re fo re expensive. I n less c r i t i c a l uses, t h e curves of

t h i s report, combined w i t h an adequate f a c t o r of s a f e t y and a

proof- tes t procedure, should provide a practical approach t o

pressure vessel design.

VIII, References

1. Roark, R. J. and Young, W. C . , Formulas f o r Stress and S t r a i n , Fifth E d i t i o n , McGraw H i l l , New York, 1975.

2 . Kiernan, T. J . , P r e d i c t i o n s of the Collapse S t r e n g t h of Three HY-100 S t e e l S p h e r i c a l Hu l l s F a b r i c a t e d f o r t h e Oceanographic Research Vehicle ALVIN, David Tay lo r Model Bas in Report N o . 1792, March 1964.

3 , Mavor, J. W., Jr., Summary Report on F a b r i c a t i o n , In- spection and T e s t of ALVIN Fixed and V a r i a b l e Buoyancy Spheres, Woods Hole Oceanographic I n s t i t u t i o n Techn ica l Memorandum N o . DS-15, January 1965.

4. Humphries, P. E., Hydros t a t i c T e s t s of a 6 A1-4V Titanium Buoyancy Sphere f o r ALVIN, David Taylor Model Basin Report No. 720-145, January 1967,

5 , Waterman, R, L. and Humphries, P. E. , H y d r o s t a t i c Creep T e s t s of a 6 A1-4V Titanium Buoyancy Sphere f o r ALVIN, Naval Ship Research and Development Cen te r , T and E Report N o . 720-6, June 1967.

6. Peterson, R. E . , Stress Concentration Design F a c t o r s , John Wiley and Sons, Inc. , New York, 1953.

7 . Krenzke, M. A . , The Elastic Buckling S t r e n g t h of Near- Pe r f ec t Deep Spherical Shells With Ideal Boundaries, David Taylor Model Basin Report 1713, Ju ly 1963.