A Review of Simple Formulae for Elastic Hoop Stress

7/26/2019 A Review of Simple Formulae for Elastic Hoop Stress

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Accepted Manuscript

A review of simple formulae for elastic hoop stresses in cylindrical and spherical

pressure vessels: what can be used when

G.B. Sinclair, J.E. Helms

PII: S0308-0161(15)00007-1

DOI: 10.1016/j.ijpvp.2015.01.006

Reference: IPVP 3430

To appear in: International Journal of Pressure Vessels and Piping

Received Date: 24 June 2014

Revised Date: 13 January 2015

Accepted Date: 16 January 2015

Please cite this article as: Sinclair GB, Helms JE, A review of simple formulae for elastic hoop stresses

in cylindrical and spherical pressure vessels: what can be used when,International Journal of Pressure

Vessels and Piping(2015), doi: 10.1016/j.ijpvp.2015.01.006.

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A review of simple formulae for elastic hoop stresses in cylindrical and spherical

pressure vessels: what can be used when

G.B. Sinclaira*

J.E. Helmsb

Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803,

U.S.A. E-mails: a [email protected], b [email protected]

Abstract

Classical simple formulae for elastic hoop stresses in cylindrical and spherical pressure

vessels continue to be used in structural analysis today because they facilitate design procedures.

Traditionally such formulae are only applied to thin-walled pressure vessels under internal

pressure. There do exist, however, some variations of these formulae that remain simple yet

permit wider use. Here, by reviewing various underlying rationales for simple hoop stress

formulae, we make a determination of when and how well different formulae apply. For the

formulae that do apply to thicker vessels than usually recognized, we give companion results for

external pressure.

Keywords: Hoop stresses, pressure vessels, design formulae

_______________*To receive correspondence. Telephone: (225)767-5786

Home e-mail: [email protected]


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1. Introduction

Distinguished by the subscript c, the classical formulae for the elastic hoop stress, ,

produced by an internal gauge pressurepacting within thin-walled pressure vessels have

= , = ,2

c c

pr pr

t t (1)

for cylindrical and spherical vessels, respectively. In eqns (1), rand t are corresponding inner

radii and thicknesses.

Possibly the earliest development of an expression like the first of eqns (1) is that

obtained experimentally by Mariotte circa 1670, [1]. Mariotte tested closed thin-walledcylindrical tanks by connecting them to standpipes placed on top of them. He found that the

height of the water in the standpipes when cylinders burst was proportional to the cylinders wall

thickness and inversely proportional to its radius. That is, in effect, p t/r at rupture. Thus

Mariotte experimentally confirmed the essential elements of the first of eqns (1).

It is not clear to us when the explicit expressions for the hoop stresses in eqns (1) were

developed. An alternative form for the first of eqns (1) was provided by Barlow circa 1830, [2].

Identified with the subscriptB, this has

,BpR

t= (2)

for cylinders, where R = r + t is the external radius. Otherwise we are unaware of just who

developed eqns (1) and when. It would seem likely, however, that this occurred before Lams

derivation of the more complex formulae for hoop stresses in thick-walled cylindrical and

spherical vessels. This derivation was reported in [3] in 1852.

The classical formula for the hoop stress in cylinders, the first of eqns (1), has been given

in strength of materials texts, and occasionally in statics texts, for over a century. Examples of

the former are [4,5]. An example of the latter is [6]. For more than 50 years this formula has

been given in introductory elasticity books [7,8], texts on shell theory [9], design books [10], andengineering handbooks [11].

This formula continues to be cited in modern mechanics of materials texts right up to the

present time. In chronological order, examples are [12-19]. It is also given in current

introductory solid mechanics books [20,21] and machine design books [22-25], as well as

current/recent handbooks [26-28]. At this time it would appear to be fair to say that the classical

formula for the hoop stress in cylinders under internal pressure has gained long-standing and

widespread acceptance.

Similarly the classical formula for the hoop stress in spheres, the second of eqns (1), has

been given in texts of yore [4], [6,7], [9], [29]. It too continues to be given in current/recent texts


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including all of the modern mechanics of materials texts previously cited, as well as an

introductory solid mechanics text and some handbooks, [20], [26, 27]. It is not usually given in

machine design books. Nonetheless it would seem reasonable to view the classical formula forthe hoop stress in spheres under internal pressure as having also gained long-standing and

widespread acceptance.

The formulae of eqns (1) only apply away from any discontinuities that cause stress

concentrations, and provided pressure vessel walls are sufficiently thin. The most commonly

accepted range of sufficiently small wall thicknesses is

110.t r (3)

That is, that the thickness be at least one order of magnitude smaller than the inner radius:

requirements of this nature are common in engineering when defining relatively small

dimensions (see, e.g., [12]). On occasion, further justification for the range in inequality (3) is

offered by noting that it leads to no more than about a 5% deviation from the maximum hoop

stresses, [13],[18]. Though less frequent than inequality (3), there are some other upper limits

for applying eqns (1) given in the literature. Examples are t/r1/20 in [24], t/r1/5 in [22],

[27]. When upper limits are exceeded, presumably formulae for thick-walled vessels are to be

used.

TheseLam formulaefor the maximum hoop stress, max , produced by internal pressure

within thick-walled pressure vessels have

( )( )

( )( )

2 2 3 3

max max 2 2

+ + 2= , = ,

+ 2 + +

p R r p R r

t R r t R Rr r (4)

for cylindrical and spherical vessels, respectively. Derivations of the formulae in eqns (4) may

be found in [30]: therein it is also explicitly pointed out that in the limit as t 0 the stresses in

eqns (4) recover their counterparts in eqns (1). Clearly, therefore, there has to be some

sufficiently small range of thicknesses such that eqns (1) suffice. Within this range, then, the

relative simplicity of eqns (1) facilitates design by enabling the ready determination of design t

given allowable and specifiedp and r. This simplicity is a key reason for the continued useof formulae like eqns (1) today. A detailed and precise explanation of the role of these formulae

in the design of pressure vessels can be found in the ASME code [31], or the exposition of this

code in Megyesy [32].

On the other hand, as t in eqns (4), max , 2, p p respectively, in marked

contrast to eqns (1) as t which has 0 for both cylinders and spheres. Hence there has

to be some upper limit or limits on t/rfor eqns (1) to provide reasonable respective estimates of

eqns (4). One objective of the present review is to check inequality (3) in this role and, if

needed, to augment it so that upper limits are set in a clear and consistent way.


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While eqns (1) contain by far the most common simple expressions for hoop stresses in

the literature, there are also some variations in these expressions themselves. One such is the

already noted expression of Barlow for cylinders, eqn (2). Another for cylinders under internalpressure has

= ,Spr

t (5)

where r = r + t/2 is the mean radius. As far as we can discern, this formula was first given in

Shigleys Mechanical Engineering Design circa 1972, thus the subscript S. It continues to be

given in the current version of Shigley [23]. It is also provided in [24], [32-34]. A further

alternative for spheres under internal pressure has

= ,2

R

pr

t (6)

the analogue of eqn (5) in effect. As far as we can discern, this formula is only given in Roarks

Formulas for Stress and Strain [28], thus the subscriptR.

Two other simple formulae are furnished in the ASME code [31]. Distinguished by the

subscript dfor design code, these have

= + 0.6 = + 0.1 , d c d c

p, p (7)

for cylinders and spheres under internal pressure, respectively. Precise ranges of application for

these formulae are set out in [31] (t/r0.500, 0.356, respectively).

While it seems certain that the various alternatives of eqn (5), eqn (6) and eqns (7) would

not have been put forward unless they offered some advantages over their counterparts in eqns

(1), it is not obvious from any of the references listed here just exactly what these advantages

are. Accordingly, as this review continues we seek to try and clarify the relative merits of eqns

(5) (7) versus eqns (1).

All of the foregoing applies to internal pressure. With external pressure, buckling is

possible. Consequently most discussions of classical formulae in the literature preclude their usewith external pressure because of a sense that, in the limited thickness ranges in which formulae

like eqns (1) could be physically appropriate, buckling occurs. There are instances in the

literature, however, which, recognizing that buckling may not always so dominate, suggest that

then eqns (1) apply with a sign change. That is, the hoop stress with external pressure, ,e is

simply given by

= - ,e (8)


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where is as in eqns (1) or eqn (5) for cylinders, eqns (1) or eqn (6) for spheres. This view of

the hoop stress produced by external pressure on cylinders is stated in [15], [28], [35,36] for the

first of eqns (1) and [32] for eqn (5) . For spheres, it is stated in [35, 36] for the second of eqns(1), and [28] for eqn (6). Here, for instances in which buckling does not preclude their use, we

also seek to review the accuracy of eqns (1), eqn (5) and eqn (6) in conjunction with eqn (8)

when pressure is applied externally.

In the remainder of this review, we begin (Sect. 2) with a recap of the traditional statics

derivations underlying eqns (1), followed by systematic improvements afforded by synthetic

division. Shigleys formula eqn (5) is a natural outcome. The resulting formulae for hoop

stresses are lower bounds. Thus next (Sect. 3) we develop upper bounds. Barlows formula (2)

is one outcome of this exercise. Thereafter (Sect. 4) we combine lower and upper bounds. This

leads to design formulae, including those of ASME in eqns (7). Then (Sect. 5) we examine what

formulae apply with external pressure. In light of this review, we conclude (Sect. 6) bysummarizing what simple formulae can be applied when.

2. Lower bounds

A straightforward way to obtain lower bounds is to determine average hoop stress values

using equilibrium: being averages, these hoop stresses have to be lower bounds for hoop stress

maxima.

To this end, first we view the free-body diagram in Fig. 1 as being for half of a cross

section of a cylindrical pressure vessel under internal pressure. Then balancing forces per unit

out-of-plane length gives

= =c

pr,

t (9)

where is the average cylindrical hoop stress and c is as in the first of eqns (1). Accordingly,

as is well recognized, thiscis an average value and so a lower bound.

Second we view Fig. 1 as being for half of a cross section of a spherical pressure vessel.

Then, as shown in [12], balancing forces on an inner circle with those on an outer annulus gives

2

= ,2

pr

rt (10)

where is now the average spherical hoop stress.

As noted in [30], the lower bound in eqn (9) coincides with the dominant term in the first

of eqns (4) as t 0. This lower bound can therefore be improved by including the next tterms

from eqns (4) as t0.


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To obtain such an improved estimate, we use synthetic division rather than binomial

expansions because then there is no ambiguity as to the sign of the entire remaining terms. Thus

for cylinders from the first of eqns (4) and eqn (9),

( )

2

max = 1+ + .2 2 2 +

t t

r r r t

(11)

Hence the first-order correction results in

= = ,Spr

t (12)

wherein is the consequent improved lower boundfor cylindrical hoop stresses, r continues as

the average radius, andS is as in eqn (5). That is still a lower bound is confirmed by the

positive sign of the last term in the brackets in eqn (11). This improved hoop stress estimate is

explicitly observed to be so in Shigley, [23].

To gain an appreciation of the improvement afforded by , we compare errors with eqn

(12) with those of the first of eqns (1). If we adopt the conventional range of application of

inequality (3), eqn (1) underestimates cylindrical hoop stress maxima on averaging by

integrating over this range by 2.5%, and at most by 5.0%. The latter is a confirmation for

cylinders of the observations to this effect in [13], [18]. In contrast, for the same range, eqn (12)underestimates hoop stress maxima on average by 0.08% and at most by 0.23%: clearly

markedly more accurate. Moreover, suppose instead we choose to be somewhat more cautious

because errors with lower bounds are not conservative and limit underestimates to 1%. Then the

first of eqns (1) is only applicable for 0 < t/r1/50: eqn (12), on the other hand, is applicable

for 0 < t/r 2/9, or eleven times the range for eqns (1). Tradition and familiarity aside,

therefore, there would seem to be little if any reason to use the first of eqns (1) for cylindrical

hoop stresses instead of eqn (12).

Proceeding similarly for spheres, the second of eqns (4) and eqn (10) have

( )

( )

22

max 2 2 2

2 += 1+ + .

2 2 3 + 3 +

t r tt

r r r rt t

(13)

Now the first-order correction results in

= = ,2

c

pr

t (14)


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wherein is now the improved lower bound for spherical hoop stresses, and c is as in the

second of eqns (1). That is still a lower bound is confirmed by the remainder in eqn (13).

The fact that thisc realizes an improved estimate of max over that of the average value is

generally recognized in the literature (a good specific example of this being the case is given in

[12]). In fact, of eqn (14) is about an order of magnitude more accurate than of eqn (10)

in the conventional range of inequality (3).

If, as previously, we adopt the range of inequality (3), thec

of eqn (14) underestimates

spherical hoop stress maxima on average by 0.21% and at most by 0.63%. These are

approximately an order of magnitude lower than corresponding percentages forc for

cylindrical hoop stresses. Accordingly it is not consistent to give a single range of applicable

thicknesses for both classical formulae. If instead we adopt a limit on nonconservative errors of1%, then

cof eqn (14) is applicable for 0 < 1 8.t r This range is quite distinct from that for

eitherc

or for cylinders.

3. Upper bounds

There are quite a number of pressure vessels that have thicknesses outside of the

applicable ranges given for the improved lower bounds in the preceding section (see, e.g., the

cylinder in [27] that has t/r= 0.9). With a view to avoiding the nonconservative errors that limit

the applicability of lower bound estimates of hoop stresses, here we obtain upper bounds instead.

For cylindrical pressure vessels, recognizing that (R2

+ r2

)/ (R+ r) Rin the first of eqns(4), we have max . pR t Hence for an upper bound for cylindrical hoop stresses

+ = = ,BpR

t (15)

whereinB

continues to be from Barlows formula (2). Now the correct limit results for t ,

namely p. In terms of the errors produced by estimating corresponding hoop stress maxima

using eqn (15), these are now conservative for all thickness. However, these errors can have

significant magnitudes (up to 21%).For spherical pressure vessels, recognizing that (R

3+ 2r

3)/(R

2+ Rr + r

2) R in the

second of eqns (4), we havemax

2 . pR t Hence for an upper bound for spherical hoop

stresses

+ = .2

pR

t (16)


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Again the correct limit results for t , namely p/2, and errors produced by estimating

corresponding maxima with eqn (16) are conservative for all thickness. Again, too, these errors

can have significant magnitudes (up to 40%).The upper bounds in eqn (15) and eqn (16) both admit to being expressed as

+= + 2, p where is as in eqn (12) for cylinders, eqn (14) for spheres. This relationship

between upper and lower bounds raises the possibility that by adjusting the coefficient ofpin the

foregoing expression we may be able to obtain hoop stress estimates that are superior to

estimates from either upper bounds or lower bounds by themselves. We explore this possibility

next.

4. Design formulae

Motivated by the preceding observation, we consider a class of design estimates of hoop

stress maxima given by

= +d p, (17)

wherein continues to be as in eqn (12) and eqn (14) for cylinders and spheres, respectively.

In eqn (17), is a nondimensional parameter with a range of 0 . When = 0, just the

improved lower-bound estimates of hoop stress maxima are recovered: when = , the upper-

bound estimates.

To consider other values, we introduce an error measure. In line with the commonpractice of designing against yield, we choose our error measure, e, to be the relative error

between the maximum shear stress calculated with , ,d d and that calculated with max max, .

This choice is readily implemented. However, it is not critical. If instead the von Mises stress is

used, maximum errors for cylindrical vessels are reduced by less than 4% of their values, while

errors for spherical vessels are unchanged. Furthermore, if as earlier the relative error in hoop

stress maxima is chosen, while error measure magnitudes are altered, the ordering of the

performance of the various hoop stress estimates entertained remains the same.

With this choice, the percentage error is given by

( ) ( )max max= - x 100 % , de (18)

where ( ) ( )max max= + + . d d p , = p These maximum shear stresses occur on the inner

walls of pressure vessels whereonpacts.

For cylinders, from the first of eqns (4), eqn (12) and eqn (17), eqn (18) has

( )0 for 4 1- 2 .e t r That is, errors are conservative in this range of thicknesses. Within

this conservative thickness range, the maximum error occurs when de/dt= 0. Thus


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( ) ( )

2

max = x 100 % ,4 +

te

r r t (19)

when t/r= 2 .

For spheres, from the second of eqns (4), eqn (14) and eqn (17), eqn (18) has

( ) ( )20 for 1-3 + 3 -1 1- 2 .e t r Within this conservative thickness range, themaximum error is

( )

( )( )

2 4

max 22

6 + += x 100 % ,

9 +

rt r t t e

r r t (20)

when t/r = 6 + 4 - 2. Turning to specific choices for , we first note the choice effectively made by the ASME

Boiler and Pressure Vessel Code, [31]. Now and henceforth reserving the subscript d to

distinguish these design code estimates of hoop stress maxima, from [31]

= + 0.1 ,d p (21)

wherein continues to be as in eqn (12) for cylinders and eqn (14) for spheres. The d of eqn

(21) are consistent with the earlier reporting of these estimates usingc in eqn (7). This

common choice of = 0.1 for both cylinders and spheres realizes low conservative errors in

hoop stress maxima over quite wide thickness ranges. These conservative error ranges extend up

to t/r= 0.500 for cylinders and t/r = 0.356 for spheres: these thickness limits are precisely the

ones prescribed in [31], so that there the code restricts the use of eqn (21) to thicknesses that only

result in conservative errors. Within these thickness ranges, the average value of the error

measure eis 0.5% and the maximum value is 0.8% at t/r= 0.20 for cylinders, while the average

value is 0.8% and the maximum value is 1.2% at t/r= 0.14 for spheres.

When designing pressure vessels in accordance with code, clearly one has to comply

completely with the applicable code (in the U.S., [31]). Nonetheless, during preliminary design,

wider use of simple formulae can facilitate exploring a range of configurations before settling onone to subject to the full requirements of the governing code. With this in mind, we seek to

develop formulae for preliminary design that extend over yet larger thickness ranges by

entertaining larger, but still conservative, percentage errors of about five percent. We note that

because such formulae are more conservative than their code counterparts, any design

thicknesses determined with them has to be in compliance with the thickness ranges permitted by

code formulae.


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To this end we merely select = , its mid-range value. Then from eqn (12) and eqn

(17), we have, for a preliminary design estimate of cylindrical hoop stress maxima distinguished

from its code counterpart by the subscriptD,

( )+= ,

2D

p R r

t (22)

for 0 t/r2.0. The average error with eqn (22) is 2.5% while the maximum error is 4.2% at t/r

= 0.50. Likewise from eqn (14) and eqn (17), we have, for a preliminary design estimate of

spherical hoop stress maxima,

= ,2D R

pr

=t

(23)

for 0 t/r1.3. The average error with eqn (23) is 3.6% while the maximum error is 6.0% at t/r

= 0.35. As indicated in eqn (23), this design estimate is the same as the formula for spherical

vessels given in eqn (6) and furnished in Roark [28].

In Fig. 2, a graphical comparison of the errors in the maximum shear stress, eof eqn (18),

is shown ford

of eqn (21) andD

of eqn (22) and eqn (23). We reiterate that [31] prohibits the

use ofd

outside the ranges given previously: above the upper limits of these ranges, this code

uses Lam formulae and so has no error. What Fig. 2 shows, however, is that, if as earlier we

admitted a 1% nonconservative error in preliminary design, increases in the applicable ranges of

d are modest (where d intersect the horizontal dashed lines).

Shown for comparison in Fig. 2 are of eqn (12) and eqn (14) ( = ,S c for

cylinders, spheres respectively), as well as+ of eqn (15) and eqn (16) ( )+ = for cylinders .B

For these curves, to be consistent we continue to use eof eqn (18) but now withd replaced by

the relevant hoop stress estimate. Consequently the magnitudes of maximum errors for+ are

different from those stated in Sect. 3 because of the use of the relative error in the maximum

shear stress instead of that in the maximum hoop stress. What Fig. 2 clearly illustrates is that, as

expected, either of the design estimates perform better than or + for both cylinders and

spheres.

Also for comparison, e of eqn (18) forc

of eqns (1) is included in Fig. 2(a) for

cylinders, as a broken line. This comparison shows the markedly higher errors that occur with

the traditional classical formula for cylinders.

5. External pressure

With external pressure instead of internal, buckling is the key phenomenon to be guarded

against in pressure vessels. Marks [27] furnishes a simplified approach to meet this objective,


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while the ASME code [31] provides complete procedures for designing against buckling.

Nonetheless, when vessels become thick enough, stresses also need to be considered. Marks

notes that there exist instances of yielding before buckling, and ASME code recognizes thepossibility that allowable stresses instead of buckling criteria may set design limits when

1/4.t r > As thickness ranges for conservative errors for the simple formulae in eqns (15),

(16), (21-23) all extend past t/r = 1/4, here we consider corresponding formulae for external

pressure.

The development of such formulae follows immediately from superposition. For

example, for a cross section of a cylinder (Fig. 3), the combined application of a uniform

pressure applied externally and internally leads to an all-round uniform applied pressure that

induces a hydrostatic state of uniform, normal, compressive stresses of magnitude equal to the

applied pressure. This hydrostatic state includes the hoop stress. Hence for hoop stressesunder

external pressure, ,e we simply have

( )= - + ,e ep (24)

whereinpeis the applied external pressure and is the hoop stress under an internal pressure of

magnitude pe. That is, all we have to do to obtain hoop stress formulae for external pressure is

take the corresponding result for internal pressure, exchangep for pe, add pe, then change the

sign. Moreover, this result holds not only for cylinders but also for spheres, and for that matter,

for any pressure vessel. This is because, when any pressure vessel is subjected to an all-round

uniform applied pressure, the same hydrostatic stress state is induced.1

Accordingly, the corresponding upper bound formulae for external pressure are

( )+ += - + ,e ep where + is as in eqn (15) for cylinders and eqn (16) for spheres. Here what is

being bounded is the magnitude of the maximum compressive hoop stress, and these stresses

continue to be bounded for all thicknesses. However, these formulae can be expected to be

precluded from applying for smaller thicknesses because buckling dominates. Corresponding

maximum errors for eof eqn (18) continue to be as for eqn (15) and eqn (16). This is because

the maximum shear stresses are the same for external pressure as they are for internal pressure.

Further, design code estimates of hoop stress maxima with external pressure are, from

eqns (21), (24),

( )= - +1.1 ,ed p (25)

where is as in eqn (12) for cylinders and eqn (14) for spheres, with respective thickness

ranges of up to t/r= 0.500 and t/r= 0.356 so as to maintain conservative errors. Lower ends of

1Equation (24) also holds for hoop stress maxima in thick-walled vessels.


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these ranges are set by when buckling dominates response (t/r1/4). The formulae from eqn

(25) are not currently included in the ASME code [31].

Likewise preliminary design estimates of hoop stress maxima with external pressureare,from eqns (22-24),

7 5= - + , = - + ,

4 2 4eD e eD e

r rp p

t t

(26)

for cylinders and spheres, respectively, with thickness ranges up to t/r= 2.0 and t/r= 1.3 so as to

maintain conservative errors. Lower ends of these ranges are set by when buckling dominates.

To close this consideration of external pressure, we review the performance of eqn (8)

instead of eqn (24). When, as suggested in [15], [28], [35] and [36], eqn (8) is applied to c of

eqns (1) for cylinders in the usual range of inequality (3), the maximum value of eof eqn (18) is

13%. Even if external pressures were light enough so as not to induce buckling, it is difficult to

see such a large nonconservative error being acceptable. If instead we were to limit such

nonconservative errors to 1%, then the range of applicability is 1 150,t r a range of

thicknesses within which buckling would almost certainly preclude application ofc

of eqns (1)

with eqn (8) for anything but the lightest of external pressures.

For spheres, performance is not significantly different. When, as suggested in [35, 36],

eqn (8) is applied toc

of eqns (1) for spheres in the range of inequality (3), then the

corresponding maximum value of e from eqn (18) is 17%. If such nonconservative errors wereto be limited to 1%, the range of applicability is 1 200.t r

While the situation is improved somewhat by using eqn (8) withS of eqn (5), [32], for

cylinders and with R of (6), [28], for spheres, it is still not good. Corresponding maximum

errors within inequality (3) are both 9%, while corresponding ranges with limited 1% errors are

both t/r1/100. All told, the performance of eqn (8) for external pressure in conjunction with

any of the suggested simple formulae is less than satisfactory.

6. Concluding remarks

Throughout the summary of formulae for hoop stresses that follows, estimates are firstgiven for cylindrical pressure vessels then for spherical pressure vessels. In these formulae for

, pcontinues as the internal gauge pressure, tas a vessels wall thickness, and r, r andRas its

inner, mean and outer radius. The errors reported are relative percentage errors in maximum

shear stresses.

Consistent classical formulae for hoop stresses are

= , = .2

S c

pr pr

t t (27)


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The first of these is apparently due to Shigley circa 1972 and should be used instead of the

traditional expression,pr/t, because it is far more accurate. The second has been in existence formore than 120 years [4], and hence has the subscript c for truly classical. These formulae are

lower bounds on corresponding hoop stress maxima. Hence errors are nonconservative.

Because they are so, here we limit their magnitude to 1%. Then applicable respective ranges are

0 < 0.25t r and 0 < 0.14.t r2While these precise ranges are a consequence of the limiting

error chosen as well as the error measure, with any consistent basis applicable ranges for

cylinders are uniformly markedly more extensive than those for spheres.

Upper bounds for hoop stresses are

+= , = .2BpR pR

t t (28)

The first of these is given in Barlow [2]. Errors are conservative for all thickness but can be

quite large: up to 12.5% and 17.6%, respectively.

Hoop stress estimates permitted by the ASME code [31] are

= + 0.1 , = + 0.1 .2

d d

pr prp p

t t (29)

In effect, these are the consistent classical formulae with a constant correction term added. Bothformulae are conservative provided 0 < t/r 0.500 and 0 < t/r 0.356, respectively:

corresponding maximum errors are 0.8% and 1.2%.

Possible hoop stress estimates for preliminary design are

( )+= , = .

2 2D R

p R r pr

t t (30)

These are simply the average of respective lower and upper bounds. The second formula for

spheres is given in Roark [28]. Both formulae are conservative for broader ranges than those for

eqn (9), namely 0 < t/r2.0 and 0 < t/r1.3, respectively: corresponding maximum errors are

4.2% and 6.0%. Because these hoop stress estimates are more conservative than hoop stresses in

the ASME code, any preliminary design of thicknesses using them within the given ranges is

guaranteed to be in compliance with [31].

For external pressurepe, replacepwithpe in the chosen formula for internal pressure, add

pe, and change the sign. That is,

2By way of comparison, the applicable range with this error limit for the traditional expression for cylinders is but

0 < 0.02.t r


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( )= - +

=

e e

e

p ,

p p

(31)

wheree

is the hoop stress with external instead of internal pressure. Then upper limits on

applicable thickness ranges remain the same, but lower limits are replaced by those set by

buckling. Thus classical formulae can be expected not to apply, but other formulae can. When

applicable, errors are the same as for internal pressure.

Acknowledgment

We are grateful for comments from Dr. S.F. Hoysan of Stress Engineering Services Inc.,

Houston.

References

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Figure captions

Fig. 1. Cross section of half a pressure vessel including pressure exerting fluid.

Fig. 2 Comparison of percentage errors for hoop stress formulae: (a) cylindrical pressure

vessels, (b) spherical pressure vessels.

Fig. 3. Superposition of cylindrical pressure vessel configurations.


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A Review of Simple Formulae for Elastic Hoop Stress

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