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BUCKLING OF MULTIPLE-BAY RING-REINFORCED

CYLINDRICAL SHELLS SUBJECT TO

HYDROSTATIC PRESSURE

by

W.A. Nash

Reprint fromJournal of Applied Mechanics

Vol. 20, No. 4, pp. 469-474 (1953)

Report 785

IYIIIIIIY 1 II IIIIIIII --1 - -

April 1954

, III III illilill lullimil, I'll IIII ,

iii

FOREWORD

In the strength analysis of submarine structures consideration has been

given to the possible elastic buckling of the shell plating between-ring stif-

feners. Re-evaluation of the existing theoretical and experimental results has

indicated that the desired accuracy of calculation is not often obtained. As a

consequence further research was initiated at the Taylor Model Basin under a

project designated "Buckle."

The experimental phase of this research program has been discussed in

TMB Reports C-439 and 822. Theoretical phases have been under study based

on preliminary analyses developed by Salerno and Levine, working at the Poly-

technic Institute of Brooklyn.

In this report a mathematical treatment is presented for the shell insta-

bility of ring-reinforced cylindrical shells which is believed more general and

more in accord with recent experimental observations of buckling configuration

than the previously available theory of von Mises. At the same time, the follow-

ing restrictions inherent in recent analyses of Salerno and Levine have been re-

moved:

(a) The expressions for the work done by the external forces acting upon

the shell during the buckling process appear to be incorrect. Although the im-

portance of these errors upon the numerical results obtained for the particular

buckling configurations considered in their analyses appears to be small, it is

of course desirable to correct the analysis, particularly if other buckling con-

figurations are to be considered.

(b) The reinforcing ring was considered to be a line element. This leads

to some ambiguities regarding the unsupported length of the cylindrical shell.

Further, the energy contained in that portion of the shell directly under the

frame is not considered.

(c) The treatment applies only to rings consisting of thin-walled "open"

cross sections. It is not strictly applicable to rings of rectangular cross

sections.

(d) The assumed displacement configurations did not permit any radial

displacement of the rings during buckling.

The analysis presented herein is considered significant primarily in

reporting a rigorous treatment of the shell with fixed ends as contrasted with

the condition of hinged ends assumed by von Mises. Results computed by

_

iv

applying this theory to several special geometries are found in even poorer

agreement with experimental observations than were the less valid von Mises

analysis. There are new and definite indications, however, that discrepancies

result from initial imperfections in the test specimens. This analysis should

thus be considered as applicable only to a perfect structure and thus in need

of further extension to accommodate initial out-of-roundness and residual

welding stresses.

_ ~- II. II ~L --- I --

Paper No.53-APM-29

Buckling of Multiple-Bay Ring-ReinforcedCylindrical Shells Subject to

Hydrostatic PressureBY W. A. NASH,1 WASHINGTON, D. C.

An analytical solution is presented for the problemof the elastic instability of a multiple-bay ring-reinforcedcylindrical shell subject to hydrostatic pressure appliedin both the radial and axial directions. The method usedis that of minimization of the total potential. Expres-sions for the elastic strain energy in the shell and also inthe rings are written in terms of displacement com-ponents of a point in the middle surface of the shell.Expressions for the work done by the external forces actingon the cylinder likewise are written in terms of thesedisplacement components. A displacement configurationfor the buckled shell is introduced which is in agreementwith experimental evidence, in contrast to the arbitrarypatterns assumed by previous investigators. The totalpotential is expressed in terms of these displacement com-ponents and is then minimized. As a result of thisminimization a set of linear homogeneous equations isobtained. In order that a nontrivial solution to thissystem of equations exists, it is necessary that the deter-minant of the coefficients vanish. This condition deter-mines the critical pressure at which elastic buckling of thecylindrical shell will occur.

INTRODUCTIONT HE first analytical treatment of the problem of the buck-

ling of a cylindrical shell of infinite length subject toexternal hydrostatic pressure was carried out by Bresse

(1)2 in 1859. In 1888 Bryan (2) published his classical paperderiving the same expression as did Bresse. Bryan's work wasbased upon the energy criterion for stability. In 1913 South-well (3) published the first of a series of three papers treatingthe elastic instability of a geometrically perfect cylindrical shell.In the first of these papers he rederived Bryan's expression forthe tube of infinite length in a manner different from that of the

original author. In this same paper Southwell discussed theeffect of circumferential reinforcing rings upon the hydrostaticbuckling pressure of a shell of finite length and derived anexpression for the minimum length of tube for which the effectof the rings can be neglected. He used an energy method toobtain the buckling pressure for a shell of finite length, neglecting

1 Structural Research Engineer, David Taylor Model Basin, NavyDepartment. Mem. ASME.

2 Numbers in parentheses refer to the Bibliography at the end ofthe paper.

For presentation at the National Conference of the AppliedMechanics Division, Minneapolis, Minn., June 18-20, 1953, of THEAMERICAN SOCIETY OF MECHANICAL ENGINEERS.

Discussion of this paper should be addressed to the Secretary,ASME, 29 West 39th Street, New York, N. Y., and will be accepteduntil one month after final publication of the paper itself in theJOURNAL OF APPLIED MECHANICS.

NOTE: Statements and opinions advanced in papers are to beunderstood as individual expressions of their authors and not thoseof the Society. Manuscript received by ASME Applied MechanicsDivision, July 2, 1952. Paper No. 53-APM-29.

the effect of the reinforcing rings. This solution consideredonly radial pressure.

This paper also presented for the first time an analyticaltreatment of the problem of the number of lobes that would formon a shell subject to external hydrostatic pressure. In 1914von Mises (4) published his classical analysis of the bucklingof a thin elastic shell of finite length subject to uniform radialpressure. However, this study failed to take into account anyaxial component of pressure. In 1929 von Mises (5) extendedhis original analysis so as to cover the case of hydrostatic pres-sure applied to the ends as well as to the curved wall of the shell.In neither of his analyses was there considered the effect of theelastic restraint of the ends (or any reinforcing rings in the caseof a ring-reinforced shell) upon the structure.

In 1934 Windenburg (6) presented a simplification of vonMises equation for the buckling pressure of a cylinder subjectto hydrostatic pressure. This result is independent of thenumber of lobes formed upon buckling and differs on the averagefrom the von Mises value by about 1 per cent. Batdorf (7)min 1947 presented a new method of determining the bucklingstresses of cylindrical shells under various loading conditions.By this method he obtained a solution to the problem of thecollapse of a cylinder of finite length loaded by hydrostaticpressure on all surfaces. His solution is almost identical withthat presented by von Mises.

In a recent series of four papers Salerno and Levine (8, 9,10, 11) treated the elastic instability of a circular cylindricalshell reinforced by evenly spaced circumferential rings havingan I-type cross section. The scope of their work is restrictedby the assumption that the rings must be thin-walled opensections.

All the foregoing investigations, as well as that presentedhere, are predicated upon the classical small deformation theory ofelastic thin shells as presented by Love (12). This theoryassumes:

1 The shell is composed of a material which is elasticallyhomogeneous and isotropic.

2 The material follows Hooke's law.3 The thickness of the shell at any point is small compared

to either of the principal radii of curvature at that point.4 The normals to the middle surface of the shell before de-

formation also are normal to the middle surface after deforma-tion.

In addition to these theoretical analyses of the buckling of acylindrical shell subject to hydrostatic pressure numerousexperimental investigations have been conducted within thepast hundred years. The principal investigators were Fair-bairn (13), Carman (14), Steward (15), Carman and Carr (16),Southwell (3), Windenburg (17), Sturm (18), and Kirkby (19).

ANALYSIS

Fundamental Equations. The problem of the buckling of amultiple-bay ring-reinforced cylindrical shell subject to hydro-

JOURNAL OF APPLIED MECHANICS

FIG. 1

static pressure may be attacked by the method of minimumpotential described by Timoshenko (20). Let us consider a

cylinder of mean radius R and thickness h with stiffening ringsspaced a distance L apart. Such a shell is shown in Fig. 1.

The rings under consideration here are assumed to be spaced

equally along the axis of the cylinder and of uniform cross

section around the circumference of any ring. Further, it is

assumed that all rings are identical. In the first part of the

analysis rings of infinite rigidity are discussed and they, ofcourse, may be of arbitrary cross section. In the second part

of the analysis, rings of finite rigidity are discussed and theircross sections must be rectangular.

Let x, s, and z be orthogonal co-ordinates in the axial, tangen-

tial, and radial directions, respectively. The positive directionsof these co-ordinates are shown in Fig. 1. The shell is subject

to a hydrostatic pressure p acting both radially and axially.Let uo, vo, and wo be the components of displacement (in the

x, s, and z-directions) of any point in the middle surface of the

shell at an instant before buckling and further let u, v, and w

denote the additional displacements during the buckling process.For the purpose of this analysis it is sufficient to isolate and

study one bay of the shell, i.e., the portion of the structurebetween the centers of two adjacent rings. In the minimumpotential method it is first necessary to obtain values of theadditional extensional, bending, and shearing elastic strainenergies stored in one bay of the shell during the buckling process.If the axial and tangential strains existing in the shell immedi-ately before buckling are denoted by Exo and E,0, then, for thecase of uniform radial pressure p and axial pressure P, thesequantities are given by

e.o E -(P . .pR Il[

h ) . . . . . . .

eao = ( - - + VPP .............. [2]E h

where E denotes the modulus of elasticity of the material andv represents Poisson's ratio. In Equations [1] and [2] the

effect of the rings on the strains near the ends of the bay areneglected. The axial and tangential strains existing immedi-ately after buckling are given by

[(1 + Eo ± u.)(dx) + w)2 (dx)] '/ - dx

1= EXo + u + - w.'

. . . . . . . . [3]2

and

es =

= +, v.- + -+ +w ........ [4]1R 2 R

The increase in the extensional elastic energy during the bucklingprocess is given by

AU, Eh 2) J J [(E2 + 2VEE, + E.)

2(l - v1) o f(Eo

2 + 2 veze ,o + e, 02 )]dx ds.... [5]

With the values of these strains given by Expressions [1] through[4 ], the increase in extensional energy is found to be

Eh f2,r L + W

AU, - 2 0 [u.2 + V.2 - + 2vuxv.

2(1 - p2) R2

2puw 2vw JL[ +w L W2,uw 2 dxds + pR fR R o R

V +2] P 2R fL

- j+ w dx ds J ws2dx ds2 R 2 0 O

- 2rR

[uL - uo]ds........ [6]

The bending energy immediately before buckling is given byLove (12) and Salerno and Levine (8) to be

Ehs 2rR fLF

Ub 24R2(1 - v

2) O 0

1+ - (R 2

wo., + wo)2 + 2pwozz(2 wo,, W)R 2 xRW' O

S2(1 - v) (Rwo. + 2 2 ) dx ds........[7]

The bending energy immediately after buckling is obtained byreplacing uo by (uo + u), v0 by (v0 + v), and w0 by (w0 + w).

Consequently the increase in strain energy of bending during

the buckling process is

Eh3 IAUb = 2r IzzX82 + W8. 2 + 2PwXX 8w 8

24(1 - Vp2) Jo J0 L.

w2 2 1+ 2(1 - p)(w,,)

2 + 2 + 2 w,,w dx ds ........ [8]

This expression for the increase in bending energy was obtained62 62

by neglecting certain terms differing by h2/R 2 , h2 bx2, or h2 -

from corresponding terms appearing in A U,.Prior to buckling the cylindrical shell is in a state of uniform

compression and the membrane shear stresses are zero. After

buckling, the shear stress is given by Love (12) to be

U, - (,) yx)dA ........ [9]Us=2(1 - p2) fl(2

8where -

R

[(1 + Eo + v, - w/R)2(ds)2 + (v/R + w,)2(ds)2]'/ ' - ds

- -- -- ' lllilil

_ ------ 'I-- 1~-I II I I

.. 1 dlillibmil1i

NASH-BUCKLING OF RING-REINFORCED CYLINDRICAL SHELLS

But 74, = .................. [10]

Eh f2wRfLHence U, = E - (u, + v.)'dx ds . . . . [11]

4(1 + P) Joc

Knowing the elastic strain energy stored in the shell, it isnext necessary to calculate the work done by the external forcesactingon the shell. The total work done by the external forces act-

ing upon the cylinder may be calculated as the hydrostaticpressure p multiplied by the decrease of volume of the cylinder.This decrease of volume may be considered to consist of a

shoftening of the radius in any fixed radial plane togetherwith a sliding of the ends of the cylinder along a plate. Anyradius is shortened by an amount.

v2w - UWz - VWs - 2R

(in a fixed radial plane) and hence the decrease in volume be-tween the initial end planes is

I L 2wR L

J Rd - JR-w +uw v , + d ds

o 2RJO o 2R

f2,rRfL [ V2 W2_U. Vj j2 L 2- - - w - VW, dx ds. .. . [12]

where terms of the third and higher orders have been neglected.The correction to the decrease of volume in the vicinity of the

ends of the cylinder is of the form

S2rR ds

2 (R - w)2 Rf 2

7R R

Jo 2- w) ds

Accordingly, the correction for both ends of the cylinder is

R f2wR

2 0

(UL -U) d - 2R

(UL - Uo) ds - f [(UW)L - (uw)o] ds

Eh f2,RfL [ 2UT = 2 2 Lu .2 v+±7 + 2vuv,

2(1 - 0') o.2 + -. + 2

2vu~w 2v,wl Eh3 (2TR (L

,- Rvw dx ds + w W.2 + W,,'R R 24(1 -V 2)J Jo0

W2 + 2 1+ 2Pww., + 2(1 - v)(w,,) + - - w,,w dx ds

R4 R2 j

Eh 2wR L

4(1 +V) J O (u, + v)' dx ds

2rRf L 2 +- pR 2 2- 4 dx ds .... [14]o JO 2 2R2 R 4

Displacement Pattern. For the case of infinitely rigid rein-forcing rings let us take the additional displacements duringbuckling to be

Insu = A cos - sin 28x

R

v = B sin - (1 -cos 2x)

w = C cos (1 - cos 26x)

......... [15]

where A, B, and C are arbitrary constants, m is the number ofwaves (lobes) in the circumferential direction, L/n representsthe period of the function defining the deformed generator inone bay of the shell, and 5 = nr/L. The elastic strain energiesand the potential due to the external loads may now be ex-pressed in terms of the constants A, B, and C by substitutingthe displacements, Equations [15], in the energy expressionsand integrating. For brevity, let us introduce the followingnotation

The work done by the hydrostatic pressure p is thus given by

f 2,R fL (W V2 W2V) N dP 2#o JO 2R - -w - w, dx deo 2R 2R

p R 2 l R f2 TR

2J (urL - uo)ds + p [(uw)L - (uw)o] d8

........... [13]1

For the cylindrical shell, then, the total potential (which isdefined to be the algebraic sum of the strain energies and thenegative of the work done by the external loads) may be foundby use of Equations [6], [8], [11], and [13]. Equation [6]may be simplified by use of the statics relation

pRP-2h

Also, Equation [13] may be simplified by using the followingresult obtained by integration by parts

0L uwdx = (uw)L - (uw)o - fo uw dx

Let us first consider the case of a cylindrical shell reinforcedby infinitely rigid reinforcing rings. In this case the ringscontain no elastic strain energy and the total potential (i.e.,the sum of the strain energies and the potential of the externalloads) is

Eh2(1 - v')

k2 = 2627'RL

3m'irLk3 = 2R

3trL

k4 = 3r2R

kh = 2vbmrL

ko = 2vbrL

3mrLRk 7 = R?

Eh= 24(1 - p2)

kg = 8547rRL

3m 4 rLko - 2R'

4p52m'rLS

4(1 - v)65m'rLR

Ehk13 = 4(1 + P)m~rL

k 14 mrL2R

ki5 = 2mwrL

kle = 262iRL

3pm'rL4

k1 3prL4

k = pRSrL

p5'rR'LknR - 2

37rLk21 - 2R

2R1

3m rLk22

3R3

.. [16]

IIYIIYIIIIIYIYIYIIII ~_.~ -


Minimization of Potential. The variation of the total ptial with respect to each of the constants A, B, and Cvanish for equilibrium. This means that

DUT _ Ur UT- 0; - 0; - 06A 6B aC

When the differentiations indicated in Equations [171carried out they lead to the three linear homogeneous equa

k23A + k 2 4B + k 2 5C = 0 ..............

k24A + k26B + k27C = 0.............

k25A + k27B + k2sC = 0 ..............

where

k2 3 = 2kik2 + 2k1 3k14

k24 = -kik 5 - k 13k1 5

k25 = k1k 6 + k19

k26 = 2kk3 + 2k13sk1 6 ..

k2 7 = -klk7

k28 = 2kak 4 + 2ksk9 + 2k8ko + 2k8 k21- 2kak22 + 2k 8ku + 2ksk 12 - 2k17 + 2k1 s - 2k 20

Th t i l l ti A B

oten-must

[17]

aretions

11 Iv lVa sou on = = U 1o no consequence.

For a nontrivial solution to exist, the determinant of the coef-ficients of the unknowns must vanish. When this is the case,there exist deformed equilibrium configurations in addition tothe original one. Such a situation corresponds to neutralequilibrium and the loads under which it exists are the bucklingloads. Here, the stability determinant is

k23 k24 k25k2 4 k2o k27 = 0................. [22]k25 k27 k2s

The expansion of this determinant leads to a quadratic equationin the unknown buckling pressure p. The buckling pressure ppredicted by this theory is consequently the minimum positiveroot of this quadratic equation. It is of interest to note that forthe case of an irifinitely long cylindrical shell, the bucklingpressure obtained by this analysis is

Eh 3

P = 4(1 - V2 )R 3 . . . . .. . . . . . .. [23]

which agrees with the well-known Bryan-Bresse result (12).Other Displacement Patterns. It might be thought that the

somewhat more general displacement pattern

u = (A sin - + B cos sin 26x +R R) L

msv = Dsin - (1 - cos 26x) . [24]

w=Fi m s Rs

W = (F sin - + G cos (1 - cos 23x)R R

would lead to a lower buckling pressure since it contains sixconstants instead of three as used in Equations [15]. How-ever, by the minimum potential method it may be shown thatthe same stability determinant, Equations [22], is obtained asin the case of the configuration given by Equations [15].

It is of interest to investigate a slight variation in the con-figuration given by Equations [15]. Let us assume that theadditional displacements during buckling are given by

msu = A cos - sin 46x

R

msv = B sin - (1 - cos 26x)

w = Ccos- (1 - cos 26x)

..... [25]

Using the minimum potential method, three equations analo-[18] gous to Equations [18], [19], and [201 may be written and a

[19 stability determinant formed. However, this configurationwill usually indicate a slightly higher buckling pressure than that

[20] found by use of the piattern, Equations [15]. This is illustrated inthe numerical example given later for one particular geometry.For an infinitely long cylindrical shell the buckling pressurefound by use of the Configuration [25] again agrees with theBryan-Bresse expression.

The displacement configurations discussed thus far are for[21] the case of infinitely rigid reinforcing rings. In all previous

analyses investigations have assumed displacement componentscorresponding to arbitrarily selected boundary conditions with-out any reference to experimental evidence. As the configura-tion occurring during buckling of a cylindrical shell reinforcedby rings having finite rigidity let us take

ms msu = A sin - sin 26x + B sin - cos Sx

s ms Fx+ C cos msin 26 x + D cos - cos 6x + -

R R LMS

v = sin - (G + H sin 23x ± J cos 23x)R

w=(K sin ms + s (W K sin - + M cos s (1 - cos 26x) + N

. [26]

This configuration permits the ring to (a) bend out of its plane,(b) undergo a uniform radial compression, (c) undergo tangentialdisplacements, and (d) translate along the axis of the cylinder.Here, the form of the w-component of displacement was ob-tained from radial-displacement measurements taken shortlybefore and immediately after the formation of lobes in two ring-stiffened cylinders each subject to hydrostatic pressure. (Thereinforcing rings were of rectangular cress section.) Thesemeasurements indicated that the ring did not undergo any bend-ing in its plane and further that the profile of the shell along agenerator was given very closely by the expression (1 - cos 26x)with n = 1. This experimental evidence has been incorporatedinto the expressions given in Equations [26] in such a way thatall other types of deformation of the ring, namely, (a) through(d), are possible.

In calculating the total potential of a cylindrical shell rein-forced by rings of finite rigidity it is necessary to compute thepotential of the rings. This may be done in a manner analogousto that used for the derivation of the potential of the shell andthe result, for a ring of rectangular cross section is

UR = bhoE 2 'rR 2 W2 2vw d

2 o R R,2 R z=o

+ pbR1 f 2rR[2] Eo J2rR

SpbR L2R ds + 2 o [u,,2

] =O ds

........ [27]

where b denotes the width of the ring as shown in Fig. 1, h,denotes the depth of the composite ring-shell section (i.e.,

I- II _ -, I I _ _ I I - -II r~

NASH-BUCKLING OF RING-REINFORCED CYLINDRICAL SHELLS

the ring and the portion of the shell immediately contiguous to

it is regarded as monolithic), R1 represents the radius to the

centroid of the composite ring-shell section, and 1I0 denotes the

moment of inertia of the ring-shell cross section with respect to

an axis through the centroid of the cross section and coinciding

with a diameter of the shell.

This potential must be added to the potential of the shell

given in Equation [14]. The elastic strain energies and the

potential of the external loads may now be expressed in terms

of the constants A, B, C,ments [26 ] in the energyintroduce the notation

Ehhr -= 2(1 - 0)

h2 = 28 7rRL

h3 = - 22

h = 27rRL

8brR3

m7rLh6 = R

R

mh7 2 rLh 2R

3irL2R

27rLh9= R

ho = 4v rm

hn = 26mrLv

4vrmhl,) -

3

h3 = 2&irvL

ha =h 14 --

3

2wrLh35 -

R3

2m7rLhie -Rh1 = R

mrLh17 = RR

Eh= 24(1 - V2)

h = 854rRL

3m rL

2R34Pt2m rLL

h~l - --2M7r

R

3rLh22 - 2R 3

. . . N by substituting the Displace-expressions and integrating. Let us

3m'rLh23 -

R3

4(1 - v)522'Lh24

=

R

Eh= 4(1 + v)

8m2Lh26= 3R3R

h27 = 262rRL

h2 = 26mrL

16m7rS3

h129 = 3

3m2 Lph30

-

4

3rLph31 -

4

h32 = 2rLpha33 = rLpR

85pRLh34

-----

3

h 3 5 = 27'Rp

6 rR2L6ph3 6 = 2

2

bhoEha7 -2

h rM 2R,3

R4

2m 21rR1

h39 - R 4

h40 = 27rRi

h4 = brp

Eloh42 = Eo

2R4

h43 = mirRi

8

9 h 46 h48

10

11

where

Again, the variation of the total potential with respect to each ofthe constants A, B, . . . N must vanish for equilibrium. Thiscondition leads to the system of equations

A B C D F G H J K

1 h44 h45 h4

2 h45 h47 h4s

h44 h 4 5 h49

h45 h47 h51 hso

M

h 46

h 48

N

=0

=0

=0

=0

h35 = 0

h5 h53 h54 h55 = 0h6o = 0

h49 hso h54 h56 h57 = 0

h58 = 0

h 4 6 h48 h55 h57 hss = 0

h35 h59 = 0

h44 = 2hzh2 + 2h25 7

h45 = hlhb + ho5h 26

h46 = hih 3 +- h3 3

h47 = 2hih3 + 2h26h7 + 2h42h43

h4s = hh14 -+- h34

h49 = hhi + h25/28

ho = hlhl2 + h25h29

h5i = - hihio

hb2 = 2hh4h53 = 2hih + 2h37h38s

h54 = ha37ha39

h.b = -hh

ha56 = 2hh7 + 2h25h27 + 2hah3s

hA57 = hih17

h=s = 2hzh + 2hshi- + 2hsh2o + 2his8h2l + 2hish22- 2hsh23 + 2hsh24 - 2h3o + 2ha31- 2h36

h59 = 2hh9 + 2h,5hs + 2h32 + 2h37h4o + 2h4l

h6o = 2hh + 2h25h27

.. [291

.. [301

These equations show that H = 0 for all geometries of ring-reinforced cylindrical shell. Also, the constants A, B, and Kand no others each appear in only three equations and thestability determinant of these three may be solved to obtaina value of p. Also, the constants C, D, G, J, and M and noothers each appear in five equations and the stability determinantof these five may be solved to determine another value of p.Lastly, a stability determinant may be formed of the twoequations containing F and N and a third value of p determined.(This last corresponds to the case of axial symmetric buckling.)The critical buckling pressure is, of course, the minimum of thesethree values of p. For the particular geometries investigated inthis paper the minimum value of p is given by the second of thesedeterminants.

NUMERICAL EXAMPLE

Let us consider a multiple-bay ring-reinforced cylinder havingthe following dimensions:

2R = 26.753 in.

h = 0.065 in.

L= 5.08 in.

b = 0.1875 in.

ho= 0.71875 in.

AN


If the rings are first assumed to be infinitely rigid and the dis-placement pattern, Equations [15],, adopted, the stabilitydeterminant, Equations [22], becomes

3.497362 X 108 -1.374697 X 107m 6.344761 x 106+4.476262 X 10'ml +1.320211 X 10

2p

-1.374697 X 107m

6.344761 X 108

+1.320211 X 102p

3.836798 X 106m2

+1.224077 X 10'

-3.836799 X 106m

-3.836799 X 106m

4.024995 X 106+1 361312 X 10am'

+7.519691m'4

-2.395072 X 10m2p

-1.067641 X 10p

In this form n has been taken equal to unity, as found by the testsof the two ring-reinforced cylinders, but it is necessary to mini-mize the value of p with respect to the variable m. It is foundthat the minimum value of p occurs with m = 17 and is 189psi. It is of interest to note that if the p 2-term of the quadraticequation is neglected (with m = 17) the same pressure is ob-tained.

If the displacement pattern, Equations [25], be adopted it isfound that the minimum value of p occurs when m = 17 and is199 psi. If the displacement pattern, Equations [26], is adoptedit is found that the fifth-order determinant with m = 16 yieldsa critical buckling pressure of 179 psi, which is a lower valuethan may be obtained by use of either the second or the third-order determinants that also arose in conjunction with configura-tion, Equations [26]. Consequently, for a cylinder having thegeometry stated it is apparent that the deformations of the ringpermitted by the buckling configuration, Equations [26], reducethe buckling pressure 5.3 per cent from that given for a cylin-drical shell reinforced by infinitely rigid rings. (It is of interestto note that the theory of Reference (10) predicts a buckling pres-sure of 175 psi for a model having this geometry and reinforcedby infinitely rigid rings.)

CONCLUSIONS

1 A theoretical analysis is presented for the problem ofdetermining the pressure necessary to cause an elastic instability

mode of failure of a geometrically perfect thin cylindrical shellreinforced by equally spaced rings and subject to hydrostaticpressure.

2 The buckling pressures given by this, as well as by allother existing theories, are higher than those found during test.Possible causes of this are initial out-of-roundness or residualwelding stresses.

ACKNOWLEDGMENT

The author expresses his appreciation to Dr. E. H. Kennardof the David Taylor Model Basin for suggesting Equations[6] and [13]. Further, the author would like to express histhanks to Mrs. L. V. Peugh, Mr. R. C. Slankard, and Mr. M.Borg of the David Taylor Model Basin staff, for checking the

mathematical analysis and for carrying out the calculationsoccurring in this paper.

BIBLIOGRAPHY

1 "Cours de M6chanique Applique," by M. Bresse, Paris,France, 1859, pp. 323-338.

2 "Application of the Energy Test to the Collapse of a LongThin Pipe Under External Pressure," by G. H. Bryan, Proceedingsof the Cambridge Philosophical Society, Cambridge, England, vol. 6,1888, pp. 287-292.

3 "On the Collapse of Tubes by External Pressure," by R. V.Southwell, Philosophical Magazine, vol. 25, part 1, May, 1913, pp.687-698; vol. 26, part 2, September, 1913, pp. 502-511; vol. 29,part 3, January, 1915, pp. 67-77.

4 "Der kritische Aussendruck Zylindrischer Rohre," by R. vonMises, Zeitschrift des Vereines deutscher Ingenieure, vol. 58, 1914,pp. 750-755.

5 "Der kritische Aussendruck fOr allseits belastete zylindrischeRohre," by R. von Mises, Fest. zum 70 Geburstag von Prof. Dr. A.Stodola, Zurich, Switzerland, 1929, pp. 418-430.

6 "Collapse by Instability of Thin Cylindrical Shells UnderExternal Pressure," by D. F. Windenburg and C. Trilling, Trans.ASME, vol. 56, 1934, pp. 819-825.

7 "A Simplified Method of Elastic-Stability Analysis for ThinCylindrical Shells," by S. B. Batdorf, NACA Report 874, 1947, 25pages.

8 "Buckling of Circular Cylindrical Shells With Even Spaced,Equal Strength Circular Ring Frames, Part I," by V. L. Salerno andB. Levine, PIBAL Report No. 167, 1950, 30 pp.

9 "Buckling of Circular Cylindrical Shells With Evenly Spaced,Equal Strength Circular Ring Frames, Part II," by V. L. Salernoand B. Levine, PIBAL Report No. 169, 1950, 21 pp.

10 "Charts for the Determination of the Upper and LowerLimit of Hydrostatic Buckling Pressures for Reinforced CircularCylindrical Shells," by V. L. Salerno, B. Levine, and J. G. Pulos,PIBAL Report 177, 1950, 25 pp.

11 "The Determination of the Hydrostatic Buckling Pressuresfor Circular Cylindrical Shells Reinforced With Rings," by V. L.Salerno and B. Levine, PIBAL Report No. 182, 1951, 25 pp.

12 "A Treatise on the Mathematical Theory of Elasticity," byA. E. H. Love, Dover Publications, New York, N. Y., 1944.

13 "On the Resistance of Tubes to Collapse," by W. Fairbairn,Philosophical Transactions, vol. 148, 1858, pp. 389-413.

14 "Resistance of Tubes to Collapse," by A. P. Carman, PhysicalReview, vol. 21, 1905, pp. 381-387.

15 "Collapsing Pressures of Bessemer Steel Lap-Welded Tubes,Three to Ten Inches in Diameter," by R. T. Stewart, Trans.ASME, vol. 27, 1906, pp. 730-822.

16 "Resistance of Tubes to Collapse," by A. P. Carman andM. L. Carr, University of Illinois Engineering Experiment StationBulletin No. 5, 1906, 23 pp.

17 "Strength of Thin Cylindrical Shells Under External Pres-sure," by D. F. Windenburg and H. E. Saunders, JOURNAL OFAPPLIED MECHANICS, Trans. ASME, vol. 53, 1931, pp. A-207-218.

18 "A Study of the Collapsing Pressure of Thin-Walled Cylin-ders," by R. G. Sturm, University of Illinois Engineering Experi-ment Station Bulletin No. 329, 1941, 80 pp.

19 "The Collapse of Cylinders Subject to External Pressure,"by J. M. Kirkby, Admiralty Mining Establishment Report No.1032/47, 1947.

20 "Sur la Stabilit6 des Systemes lastiques," by S. Timoshenko,Annales des Ponts et Chaussees, vol. 13, 1913, pp. 496-566; 73-132;372-412.

~_~ ~__~_ -- -- .~~xlY rii i ii i

II I - -- a_, -----------------~

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3 . CJS

NAVY-DPPO PRNC WASH 0 C

IN, i'', , , i d11111

David W. Taylor Model Basin. Rept. 785.BUCKLING OF MULTIPLE-BAY RING-REINFORCED CYLIN-

DRICAL SHELLS SUBJECT TO HYDROSTATIC PRESSURE,by W.A. Nash. April 1954. iv, 7 p. incl. fig., bibl. (Reprintfrom Journal of Applied Mechanics, Vol. 20, No. 4, pp. 469-474(1953) UNCLASSIFIED

An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure applied in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy intheshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Ex-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is



An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure appnlieca in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy intheshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Fx-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is

1. Cylindrical shells -Stresses

2. Submarines - Pressurehulls

3. RingsI. Nash, William A.






An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure applieo in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy in theshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Fx-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is



An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure applied in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy in theshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Ex-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is







- I --------

I

introduced which is in agreement with experimental evidence, in contrast to the arbitrary pat-terns assumed by previous investigators. The total potential is expressed in terms of thesedisplacement components and is then minimized. As a result of this minimization a set oflinear homogeneous equations is obtained. In order that a nontrivial solution to this systemof equations exists, it is necessary that the determinant of the coefficients vanish. Thiscondition determines the critical pressure at which elastic buckling of the cylindrical shellwill occur.


introduced which is in agreement with experimental evidence, in contrast to the arbitrary pat-terns assumed by previous investigators. The total potential is expressed in terms of thesedisplacement components and is then minimized. As a result of this minimization , set oflinear homogeneous equations is obtained. In order that a nontrivial solution to this systemof equations exists, it is necessary that the determinant of the coefficients vanish. Thicondition determines the critical pressure at which elastic buckling of the cylindrical shellwill occur.







An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure nanplieca in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy in theshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Fx-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is









An analytical solution is presented for the problem of the elas-tic instability of a multiple-bay ring-reinforced cylindrical shellsubject to hydrostatic pressure appliea in both the radial andaxial directions. The method used is that of minimization of thetotal potential. Expressions for the elastic strain energy intheshell and also in the rings are written in terms of displacementcomponents of a point in the middle surface of the shell. Fx-pressions for the work done by the external forces acting on thecylinder likewise are written in terms of these displacement com-ponents. A displacement configuration for the buckled shell is










_ _



introduced which is in agreement with experimental evidence, in contrast to the arbitrary pat-terns assumed by previous investigators. The total potential is expressed in terms of thesedisplacement components and is then minimized. As a result of this minimization a set oflinear homogeneous equations is obtained. In order that a nontrivial solution to this systemof equations exists, it is necessary that the determinant of the coefficients vanish. Thiscondition determines the critical pressure at which elastic ouckling of the cylindrical shellwill occur.


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