Technical Memorandum WHOI 3 81
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Transcript of Technical Memorandum WHOI 3 81
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.