John G Pulos. # Structural analysis and design considerations for cylindrical pressure hulls online

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D^-^-H -P5_iJ5L+ _Eh_ w . -p(i-i//2) ^^.

dx^ 2 dx2 r2 ^'^

where the necessary nomenclature and definitions used here and in the

equations to follow are indicated in the Notation. A derivation of Equation

(1) may be found in either References 8 or 9.

The termi pR d_w ^hi^h renders the solution of Equation (1) to be a

2 dx2

nonlinear function of the pressure is the "beam-column effect" which was

not considered in the analyses of References 6 and 7. The importance of

this term is further emphasized by the fact that it is the necessary in-

gredient for extracting a criterion for axisymmetric elastic buckling of a

19

cylindrical shell supported by ring frames of finite stiffness; this is dis-

cussed more fully in Reference 8.

It can be shown that the general solution of Equation (1), which is a

linear, ordinary differential equation with constant coefficients, can be

written in the form

w(x) = A sinhX,x + Bcosh X, x+C sinhX-X + Fcosh X,x-^(1- v/Z) (2)

1 1 3 J -tyn

The characteristic roots \-^ and X^ of the fourth degree auxiliary equation

which results when w(x)'Â«'e'^''' is substituted into Equation (1) are given by

^-^-^^-(7)^(^)

-1

1/2

1 1/2

(3)

w^here

and

e = \| 3(1-1/2)

\lRh

* _ 2E(h/R)'

^3(1-1/2)

is the critical load for the axisymmetric elastic buckling of an unstiffened

cylindrical shell under the action of uniform axial pressure (see Refer-

ence 9).

An examination of the roots Xj and X3 reveals that four possibilities

for the ratio p/p exist and that they all influence the nature of Equation

(2) . These are

(4)

(5)

20

^=1-0:^=0 (6)

The case of greatest importance and the only one which will be considered

here is that defined by p/p <1.0. A complete discussion of all possible

solutions defined by Equation (6) is given in Reference 8.

The integration constants A, B, C, and F appearing in Equation (2) are

evaluated from the following boundary conditions:

(a) Evenness of the function w(x) dictates A = C =

(b) Zero slope at the ring frames: -, â€” = at x = -zr-

(c) Compatability of the shell and frame radial deflections:

-Â±- (Aeff + bh)w = 2D -pRâ€”â€” -pb(l-i//2) at x = â€”

R^ dx

The boundary condition (c) results from the fact that the total load sup-

ported by a ring frame per unit circumferential length is

|Q*1= 2Qo + pb(l-://2) (7)

and the transverse shear force Qq is that transmitted by the shell to the

frame at their juncture and is given by

Qo =

3

d w pR dw

dx3 2 dx

^ (8)

2

21

Also, the effective frame cross-sectional area A^ff. appearing in boundary-

condition (c) is defined by the following for internal and external framing,

respectively:

^eff

^eff

(9)

where R is the radius to the median surface of the shell and R^g is the

radius to the centroid of the frame cross section.

The total longitudinal stress dy in the shell is given as the sum of the

longitudinal bending comiponent o^b pl^s the longitudinal membrane com-

ponent "^xM* i'^M

. =Â±â€” ^^_P^ (10)

X 2(1-1/'^) dx'^ 2h

so that enforcing boundary condition (a) in Equation (2) results, in

Eh

2 2

BXi coshXiX+FX3COshX.3X

pR

fc- (11)

Xi 2(1-1/'^)

The total circumferential stress 0^ in the shell is given as the sum

of the circumferential bending component i/d plus the circumferential

membrane component c ,i.e.,

2(1-1/2) dx

I/Eh d w E pR nz^

22

so that

<I> h

E â– ^^"^1

B cosh X X +

zd-i' )_

FcoshXx (13)

where the subscripts o and i in Equations (11) and (13) refer to the outer

and inner fibers of the shell plating in conjunction with the plus and minus

signs, respectively.

The total radial ring -load Q* is found from Equations (7), (8) and (2)

to be

Eh3

Q^

6(1-1/2)

3 , L .3 . ii

BXiSinhXi ~2 ^ F>Â«^sinhA.o Â£

-pb(l-i//2) (14)

When boundary conditions (b) and (c) are invoked, the integration con-

stants B and F are determined and the following expressions for the more

important shell stresses are derived in terms of the convenient notation

of Reference 10: at midbay between adjacent ring frames (i.e., at x = o)

"^"^^ 1 + ^xbm

(15)

$Â°m

_J_= 1

.(.W)

F, Â±

z-i'-s-

xbm

(16)

23

at a ring frame (i.e., at x = )

^XÂ°f

2 \ " ci^ >

0.91

l-V

(17)

'^y -1-/1 - %MfV,Y, . ^4>MÂ£\

2 3

Ml-l/

(18)

and where

â€¢5^ = - -2 â€” is the circumferential stress in an unstiffened cylinder

h

of infinite length,

"^xbm ^^ ^^^ longitudinal bending stress in the shell at midbay

between adjacent ring frames, and

^ (h TsAi ^^ *^^ circumferential membrane stress in the shell at a

ring frame.

These latter two stresses are given by

xbm

0.91

cJu \ll-i/2

(l-l//2)c(

_o(+p + (l-p)Fi _

(19)

/ ^ _ ^4>Mf \ _ (l-l//2)c(

\ ' c^u / " c<+p+(l-p)Fi

(20)

and where:

"â€¢eff

Lfh

Jb_

"^Lf

24

In Reference 11, Lunchick and Short modify the theory of Reference 8

to include the effect of initial cixisymmetric deviations from straightness

of the cylinder generators. If it is assumed that this deviation possesses

constant curvature between adjacent ring frames so that it can be express-

ed analytically by a second-degree parabola and assumes a maximum

amplitude A ( + inward), then the stress expressions, i.e., Equations (15)

through (18), respectively, become in the present notation

X m

1

2 \li_i,

0.91

4RA /o(+l\

2(1-^/2) V Â°^ /.

1 +

L L'-(l-i//2)

aF4

i_ = 1 - aFo.. 4RA

1 __iÂ±tiHi_n.

c^ +p+(l-p)FiJ

-V.

0.91

l-V

1 +

4RA

L'^(l-l//2)

(^)1

(15')

aF4 (16')

^X9f

1Â±

0.91

1-1/2

4RA /Â°(+l'

l2(1-j//2)V^

aF:

(17')

09f r

L_ = 1 -a 1

4RA (1-P)(

L'^(l-l//2)

limy

079i |7+ 4RA / Â°<+l\

aF3 (18')

25

in which the short-hand notation

(l-l//2)c(

^~c<+p+(l-p)Fi

has been introduced. For the case of no eccentricity, i.e., A= 0,

Equations (15') through (18') reduced to the corresponding stress ex-

pressions given previously. Also, in the same convenient notation, the

following expression for the total radial load acting on a ring frame per

unit circumferential length is obtained:

Q* = -pb(l-i//2)^l + ^â€” 1 k (21)

(+P+(l-p)Fi J

which corresponds to the case of zero initial axisymmetric eccentricity.

In the case of a ring- stiffened cylinder under some loading, such as

hydrostatic pressure which is of interest in the case here, a portion of the

deformed shell between stiffeners will act effectively with each ring frame

to resist direct stress and bending moment caused by the interaction

between the shell and the frames. A knowledge of this "effective width"

is of particular interest in a study of the buckling strength of the ring

itself and in the elastic and inelastic general-instability analyses of the

entire stiffened cylinder (this problem is considered in a later section).

It is also important in calculating the stresses in the frame flanges of

innperfectly circular cylindrical shell structures.

In Reference 8, Pulos and Salerno discuss the many "effective width"

formulas developed by earlier investigators, and they present a formal

derivation of a new forinula to include the "beam-column" effect. Details

26

of this may be found in Reference 8; however it is of interest here to give

the end result which can be expressed by the following convenient formula;

Lg = LFi+b (22)

In Equations (15) through (22) the following F functions (graphical

solutions for which were first developed by Krenzke and Short in Reference

10) have been introduced for ease of calculation:

â– ^1^(0)1^Â°^^ T]^9 - cos Ti^O

/2>

(23)

^2 =

cosh T] sin T) sinh tj cos tj

â– â€” â€” +

/2>

:24j

"3%

0.91

cosh T] sinh r\ cos t] sin t]

/2>

(25)

NO. 91

cosh:]! 6sinq2Â® sinhtij^Scosij^Q

^2

T]l

/57

(26)

where:

D

coshT]j6sinhT)-^e cosT]2esinq2^

L ^1 ^2 -

-^ = i^f^^'(-r)

(27)

e = ^3(l.v2)^

27

Curves of the functions Fj, F2, F3, and F4 may be found in either of

References 8 or 10.

The elastic analysis developed in Reference 8 is intended for the

determination of the deformations and stresses in a typical bay of a pres-

surized ring- stiffened cylinder composed of many identical bays as shown

in Figure 11. This longitudinal identity and symmetry between adjacent

bays is disturbed by the introduction of rigid bulkheads, intermediate deep

frames, cone and sphere-cylinder junctures, and other contiguous structure

which goes to make up the pressure hull of a submersible. In these more

complicated configurations, a more general analysis of the axisymmetric

behavior is needed.

Short and Bart have given a general analysis for determining the

12

stresses in stiffened cylindrical shells near structural discontinuities.

The formulation includes the possibility that the shell thickness may differ

in adjacent bays, the stiffness properties of adjacent ring-frames may be

different, and the spacing between ring frames may vary along the length

of the cylindrical compartment. The general equations developed by these

investigators are given in the form of frame and shell matrices to better

identify the stiffness and response of each element and to facilitate

numerical calculations. This form of the solution lends itself very con-

veniently to high-speed digital computers and also permits immediate

identification of those geometric and material properties which can be

varied to produce desirable changes in the static response. All the

28

i

i

,Undeformed

Shell

DÂ« formed

Shell

-Frame

-P

UHIU

yy^^^/ '

r

eg

^i

W7^?^V77?i

YZA

IT

e.g.

Figure 11 - Symmetrically Loaded Cylindrical Shell with Equally Spaced

Reinforcing Ring Frames

29

necessary equations and details of the formulation may be found in Refer-

ence 12; they are not given here because of their length and rather formi-

dable nature. However, it is worth mentioning here that extensive exper-

imental evaluation has been obtained of the Short-Bart analysis. For

1 3

example, Keefe and Overby present the results of structural model

tests undertaken to check the "end-bay" theory Reference 12. Also,

Keefe and Short present a method for eliminating the effect of end con-

ditions on the static collapse strength of stiffened cylindrical pressure

hulls and give experimental verification of the suggested procedure.

Another special problem of interest to pressure hull designers, and

one worth mentioning here, is that concerned with the discontinuity

stresses which arise at the juncture of axisymmetric shells possessing

1 5

dissimilar meridional shape. Raetz and Pulos present an analysis for

determining the elastic deformations occurring at either cone-cylinder or

cone-cone junctures and discuss several other analyses developed by

earlier investigators, notably Wenk and Taylor.

Conical transition sections are used rather extensively to join

cylindrical hull components of different diameter, and not only is the

problem of the edge effects on static collapse strength of the "weakened"

bays of concern, but also, the occurrence of high, localized longitudinal

stresses in these juncture regions is of great concern from the point of

view of low-cycle fatigue in way of welded joints. Raetz discusses this

problem and suggests the use of flexible, tapered ring-segments at these

30

junctures to reduce the high longitudinal stresses; he also presents an

analysis for determining the elastic behavior of these structural elements

and gives results to indicate the degree of reduction which can be realized

in these high stresses.

FAILURE CRITERIA FOR AXISYMAIETRIC

COLLAPSE PRECIPITATED BY YIELDING

We will now consider the question of how the biaxial stresses (defined

by Equations (15) through (20)) in a pressurized ring- stiffened cylinder can

combine to produce axisymmetric collapse precipitated by yielding of the

shell plating. Although these stresses are based on equilibrium con-

siderations only and do not reflect any buckling state, they can and do

predict good results when used in conjunction with appropriate theories

of failure even though, strictly speaking, axisymmetric collapse is

associated with an instability phenomenon. Formulas for predicting

axisymmetric collapse precipitated by yielding based on various theories

of failure are summarized in Reference 8.

The simplest formula devised for the design of pressure vessels is

the so-called "boiler formula". This formula may have some merit in

predicting the bursting strength of internally pressurized unstiffened

cylindrical tubes, but it is unsatisfactory (for other than comparative pur-

poses) in the design of pressure hulls in which instability and the influence

of reinforcing ring frames play a dominant role.

Equation (28) gives the pressure at which the circumferential

membrane stress in an unstiffened cylinder of mean radius R and

31

thickness h just reached the yield stress o of the material.

Pbz'^yh/R (28)

Equation (28) does not reflect in any way the strengthening effect of the

transverse ring frames on the average circumferential stress. However,

an estimate of this effect can be found by assuming that the cross- section-

al area of the frames is spread out and its orthotropic stiffness effect is

"felt" in the form of a thicker unstiffened cylindrical shell. This requires

that the actual thickness h in Equation (28) be replaced by

SO that we now get the following modified boiler formula:

Pel r <^yh(l+Aeff/Lfh)/R (29)

From the theory of Salerno and Pulos outlined earlier, the maLximum

stresses occur in the circumferential direction on the outside surface of

the shell plating midway between adjacent ring frames, and in the longitu-

dinal direction on the inside surface of the shell plating at a frame; these

stresses can be determined from Equations (16) and (17), respectively.

Which of the two stresses is the larger depends upon the geometry of the

cylindrical shell and the reinforcing ring frames, but in most cases of

interest, it turns out that ^X-f-^ "^ffjOm- However, extensive Model Basin

tests have shown that the stress '^^Oj^ is determinative in precipitating

axisymmetric collapse. Application of the maximum principal stress

17

theory of Rankine to this stress, i.e.,

32

leads to the following expression for the pressure at which yielding begins

on the outside fiber of the shell plating midway between adjacent ring

frames:

P,3= p.-^^ ,===-=r (31)

-^& - ^JS]

_ (l-l//2)Â°(

where a = â€¢

o(+p+{l-p)Fi

If the uniaxial criterion of Rankine is applied to the circumferential

membrane (midfiber) stress "^(bi^. i.e., Equation (16) with ^xbm ^^^ equal

to zero, the following expression for the pressure at which yielding has

penetrated through the plating thickness is obtained:

oâ€žh/R

In deriving the expressions for Pj^j, Pc3> ^.nd Pc5j it has tacitly been

assumed that a uniaixial state of stress exists in the shell plating whereas,

in reality, a biaxial state exists. More realistic criteria for axisymmetric

17

collapse can be derived from the energy-of-distortion theory which grew

out of the analytical work of Huber, Von Mises, and Hencky. Since the

octahedral shear- stress theory gives the same results as the energy-of-

distortion theory and permits the use of a more familiar quantity, such as

stress, the former theory will be used in what follows. For a biaxial state

of stress at midbay, i.e., midway between adjacent ring frames, defined

by the principal stresses "^Xm a-^d '^$rn. the octahedral shear-stress is

33

given by

'^G = Tn'^Xm- Om) + Xm+ cj-mj (33)

However, since according to this theory inelastic action at any point in a

body under any combination of stresses begins only when the octahedral

shear-stress T becomes equal to (v[2"/3)<3 then Equation (33) leads to the

following:

2 2

a + c - a ^ 6

Xm Om Xm Oin

- % (34)

Essentially two distinct criteria can be derived from Equation (34)

depending upon whether the outer-fiber stresses or mid-fiber (membrane)

stresses at midbay are used. For yielding on the outer surface of the

shell plating, when the appropriate stresses xÂ°m^^^'^<5Â°m from

Equations (15) and (16), respectively, are substituted into Equation (34),

the following criterion is obtained:

Pc6

%h/R (35)

If it is assumed that axisymmetric collapse is precipitated by the

yield zone having penetrated through the shell thickness, when the approp-

riate stresses ^Xm ^^^ % m. f^^orn Equations (15) and (16) with '^xbm ^^*-

equal to zero therein, respectively, are substituted into Equation (34), the

following criterion is obtained:

'cl

^h/R (36)

+a2rZ-3,P^]l/Z

1 8

Lunchick working at the Model Basin derived another criterion of

failure for predicting axisymmetric collapse which is based on the plastic-

hinge concept. He made use of the Hencky-Huber-Von Mises theory of

yielding, i.e., Equation (34), and allowed for the plastic reserve strength

after yielding begins in the shell plating at midbay. Lunchick developed a

formula for the pressure at which a complete plastic hinge has formed at

midbay. Since the combined stress gradients at the frame locations are

steeper than those at midbay so that complete plastic hinges form much

earlier at the frames, this TMB plastic-hinge theory, in reality, gives the

pressure at which the shell fails as a three-hinge mechanism. Although

this mechanism is not physically possible in the case of cylindrical shells

as it is for beams, it does lead to predictions of a collapse pressure (p^-g)

which agree well with experiment in certain ranges of geometry. A

complete discussion of this theory together with some comparisons to

experimental data may be found in Reference 18. For our purposes here,

it suffices to give the salient results which can be used for computation.

It can be shown that the ratio of circumferential bending stress to

circumferential membrane stress and the ratio of longitudinal membrane

stress to circumferential membrane stress (all stresses considered at

midbay) can be expressed in the convenient- notation of Reference 8 and

35

that adopted here as follows:

0.91

ct)bm/ OM - ^

1-1/2 (37)

l-aF2

l-aF7

In Figure 12, the pressure ratio Pc8^Pc6 ^^^ been plotted as a function of

the stress ratios, Equations (37) and (38). Thus, Equations (37) and (38)

can be used in conjunction with these curves and Equation (35) for p^^ to

determine values of the plastic-hinge pressure p^g for different geome-

tries.

The formulas for predicting axisymmetric collapse precipitated by-

yielding, and given in this section, represent explicit expressions for

collapse pressure only for the special case of zero "beam column" effect,

i.e.,Y= 0, since in this case only are the F functions given by Equations

(23) through (26) independent of pressure. For the general case in which

Y i 0, the stresses become nonlinear functions of the pressure, and Equa-

tions (31), (32), (35), and (36) are transcendental in the pressure. How-

ever, a numerical iteration procedure can be used in which the collapse

pressures p^2>> Pc5' ^^^- ^.re first calculated for Y = 0, and these values

are then used as the first approximation in the last of Equations (27) to

determine a value of Y- Then, with this value of Y in each corresponding

case, new values of the pressures Pc3Â» Pc5> â€¢ â€¢ â€¢ ^'c. can be found. This

36

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SS381S 3NV^gtN3y\l "IVIlN3tJ3jy\in3dl3

ss3ais 3NvaaiM3w nvNianiioNOi

^ o

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2d

Oif> O *"

h- <D U) ID

odd d

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in

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^

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\\^

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s\

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^^

^x

V

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\

^^

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^"^

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^

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V

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CM jO-

â€” O

CO

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_1

<

(-

u>

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37

process can be repeated again, but probably one or two, or at most, three

iterations are required to obtain satisfactory convergence.

One important point to be made is that the collapse criteria reflected

by the expressions for P^3, Pc5Â» Pc6Â» Pc7Â» ^^^ Pc8 ^^^ inherently based on

a material which exhibits elastic-perfectly plastic characteristics, so that

any effects of strain-hardening are neglected. The assumption is made

that once the yield strength is reached, at the critically stressed mid-

bay locations according to each of the respective criteria, collapse then

occurs.

Another solution for axisymmetric yield collapse of a ring- stiffened

pressurized cylinder in the presence of the "beam-column" effect was

19

developed by Kempner and Salerno. Their elastic analysis was pre-

dicated on the Hencky-Huber-Von Mises failure criterion as applied to the

critically stressed fiber at the midbay location. For the usual range of

geometries the yield strength of the material is first reached at the ring

frames due to the high longitudinal stress on the inner fiber. The authors

then considered that for loading beyond this initial yielding pressure, a full

plastic-hinge exists at the stiffening rings. Thus, the additional pressure

which the shell can support is calculated on the assumption that the shell

plating is "hinged" to the ring frames. The maximum additional pressure

which the shell can support is then based on applying the aforementioned

failure criterion to the total stress at the midbay location.

38

AXISYMIVIETRIC ELASTIC BUCKLING BETWEEN RING FRAMES

In reference 8, Salerno and Pulos derive the following criterion

which represents an exact solution to the axisymmetric elastic buckling

problem of a thin cylindrical shell reinforced by ring frames of finite

rigidity and loaded by hydrostatic pressure:

where

Â°(+ p + (1-P)F5 =

cos^Ti^e - cos^T] e

COST] 'esinri' e cost] esinT]26

^1

^2

(39)

(40)

and

(41)

â– ^~ 2E \

3(1-1/^)

(r)'

Equation (39) is a transcendental equation to be solved for the critical

pressure p = p^.^. for a given shell and ring-frame geometry defined by the

nondimensional parameters o(, p, and 6. A graphical representation of

Equation (40) is given in Reference 8 to facilitate the iteration calculations

required to find the value of p which satisfies the buckling criterion,

Equation (39).

In small-displacement theory, the condition for instability of a

39

structure may be obtained by requiring that the displacements become in-

finitely large at buckling. It is on this basis that Salerno and Pulos arriv-

ed at the criterion of Equation (39). Since Von Sanden and Gunther, and

later Viterbo, omitted the beam-column term in the basic differential

equation, Equation (1), then this would correspond to y = in the more

complete formulation of Reference 8. In such a case, Equation (39) would

no longer have any meaning because the pressure p would not appear.

A more complete analytical development and discussion of the

axisymmetric elastic buckling problem is given by Short and Pulos in

Reference 20.

AXISYMMETRIC INELASTIC BUCKLING BETWEEN RING FRAMES

The collapse criteria which were developed earlier and were identi-

fied by the pressures p^^, p^3, P(,5, p^^, p^y, and p^Q are, strictly speak-

ing, valid only for an elastic-perfectly plastic material. As it has already

been stated, the axisymmetric collapse mode is in reality associated with

the phenomenon of buckling at a reduced modulus, so that the strain-hard-

ening characteristics as reflected by the secant (Eg) and tangent (E^)

modulii must be considered.

2 1

On the basis of the deformation theory of plasticity, Gerard de-

veloped a general set of differential equations of equilibrium for cylin-

drical shells in which the coefficients reflect the plasticity or state-of-

D^-^-H -P5_iJ5L+ _Eh_ w . -p(i-i//2) ^^.

dx^ 2 dx2 r2 ^'^

where the necessary nomenclature and definitions used here and in the

equations to follow are indicated in the Notation. A derivation of Equation

(1) may be found in either References 8 or 9.

The termi pR d_w ^hi^h renders the solution of Equation (1) to be a

2 dx2

nonlinear function of the pressure is the "beam-column effect" which was

not considered in the analyses of References 6 and 7. The importance of

this term is further emphasized by the fact that it is the necessary in-

gredient for extracting a criterion for axisymmetric elastic buckling of a

19

cylindrical shell supported by ring frames of finite stiffness; this is dis-

cussed more fully in Reference 8.

It can be shown that the general solution of Equation (1), which is a

linear, ordinary differential equation with constant coefficients, can be

written in the form

w(x) = A sinhX,x + Bcosh X, x+C sinhX-X + Fcosh X,x-^(1- v/Z) (2)

1 1 3 J -tyn

The characteristic roots \-^ and X^ of the fourth degree auxiliary equation

which results when w(x)'Â«'e'^''' is substituted into Equation (1) are given by

^-^-^^-(7)^(^)

-1

1/2

1 1/2

(3)

w^here

and

e = \| 3(1-1/2)

\lRh

* _ 2E(h/R)'

^3(1-1/2)

is the critical load for the axisymmetric elastic buckling of an unstiffened

cylindrical shell under the action of uniform axial pressure (see Refer-

ence 9).

An examination of the roots Xj and X3 reveals that four possibilities

for the ratio p/p exist and that they all influence the nature of Equation

(2) . These are

(4)

(5)

20

^=1-0:^=0 (6)

The case of greatest importance and the only one which will be considered

here is that defined by p/p <1.0. A complete discussion of all possible

solutions defined by Equation (6) is given in Reference 8.

The integration constants A, B, C, and F appearing in Equation (2) are

evaluated from the following boundary conditions:

(a) Evenness of the function w(x) dictates A = C =

(b) Zero slope at the ring frames: -, â€” = at x = -zr-

(c) Compatability of the shell and frame radial deflections:

-Â±- (Aeff + bh)w = 2D -pRâ€”â€” -pb(l-i//2) at x = â€”

R^ dx

The boundary condition (c) results from the fact that the total load sup-

ported by a ring frame per unit circumferential length is

|Q*1= 2Qo + pb(l-://2) (7)

and the transverse shear force Qq is that transmitted by the shell to the

frame at their juncture and is given by

Qo =

3

d w pR dw

dx3 2 dx

^ (8)

2

21

Also, the effective frame cross-sectional area A^ff. appearing in boundary-

condition (c) is defined by the following for internal and external framing,

respectively:

^eff

^eff

(9)

where R is the radius to the median surface of the shell and R^g is the

radius to the centroid of the frame cross section.

The total longitudinal stress dy in the shell is given as the sum of the

longitudinal bending comiponent o^b pl^s the longitudinal membrane com-

ponent "^xM* i'^M

. =Â±â€” ^^_P^ (10)

X 2(1-1/'^) dx'^ 2h

so that enforcing boundary condition (a) in Equation (2) results, in

Eh

2 2

BXi coshXiX+FX3COshX.3X

pR

fc- (11)

Xi 2(1-1/'^)

The total circumferential stress 0^ in the shell is given as the sum

of the circumferential bending component i/d plus the circumferential

membrane component c ,i.e.,

2(1-1/2) dx

I/Eh d w E pR nz^

22

so that

<I> h

E â– ^^"^1

B cosh X X +

zd-i' )_

FcoshXx (13)

where the subscripts o and i in Equations (11) and (13) refer to the outer

and inner fibers of the shell plating in conjunction with the plus and minus

signs, respectively.

The total radial ring -load Q* is found from Equations (7), (8) and (2)

to be

Eh3

Q^

6(1-1/2)

3 , L .3 . ii

BXiSinhXi ~2 ^ F>Â«^sinhA.o Â£

-pb(l-i//2) (14)

When boundary conditions (b) and (c) are invoked, the integration con-

stants B and F are determined and the following expressions for the more

important shell stresses are derived in terms of the convenient notation

of Reference 10: at midbay between adjacent ring frames (i.e., at x = o)

"^"^^ 1 + ^xbm

(15)

$Â°m

_J_= 1

.(.W)

F, Â±

z-i'-s-

xbm

(16)

23

at a ring frame (i.e., at x = )

^XÂ°f

2 \ " ci^ >

0.91

l-V

(17)

'^y -1-/1 - %MfV,Y, . ^4>MÂ£\

2 3

Ml-l/

(18)

and where

â€¢5^ = - -2 â€” is the circumferential stress in an unstiffened cylinder

h

of infinite length,

"^xbm ^^ ^^^ longitudinal bending stress in the shell at midbay

between adjacent ring frames, and

^ (h TsAi ^^ *^^ circumferential membrane stress in the shell at a

ring frame.

These latter two stresses are given by

xbm

0.91

cJu \ll-i/2

(l-l//2)c(

_o(+p + (l-p)Fi _

(19)

/ ^ _ ^4>Mf \ _ (l-l//2)c(

\ ' c^u / " c<+p+(l-p)Fi

(20)

and where:

"â€¢eff

Lfh

Jb_

"^Lf

24

In Reference 11, Lunchick and Short modify the theory of Reference 8

to include the effect of initial cixisymmetric deviations from straightness

of the cylinder generators. If it is assumed that this deviation possesses

constant curvature between adjacent ring frames so that it can be express-

ed analytically by a second-degree parabola and assumes a maximum

amplitude A ( + inward), then the stress expressions, i.e., Equations (15)

through (18), respectively, become in the present notation

X m

1

2 \li_i,

0.91

4RA /o(+l\

2(1-^/2) V Â°^ /.

1 +

L L'-(l-i//2)

aF4

i_ = 1 - aFo.. 4RA

1 __iÂ±tiHi_n.

c^ +p+(l-p)FiJ

-V.

0.91

l-V

1 +

4RA

L'^(l-l//2)

(^)1

(15')

aF4 (16')

^X9f

1Â±

0.91

1-1/2

4RA /Â°(+l'

l2(1-j//2)V^

aF:

(17')

09f r

L_ = 1 -a 1

4RA (1-P)(

L'^(l-l//2)

limy

079i |7+ 4RA / Â°<+l\

aF3 (18')

25

in which the short-hand notation

(l-l//2)c(

^~c<+p+(l-p)Fi

has been introduced. For the case of no eccentricity, i.e., A= 0,

Equations (15') through (18') reduced to the corresponding stress ex-

pressions given previously. Also, in the same convenient notation, the

following expression for the total radial load acting on a ring frame per

unit circumferential length is obtained:

Q* = -pb(l-i//2)^l + ^â€” 1 k (21)

(+P+(l-p)Fi J

which corresponds to the case of zero initial axisymmetric eccentricity.

In the case of a ring- stiffened cylinder under some loading, such as

hydrostatic pressure which is of interest in the case here, a portion of the

deformed shell between stiffeners will act effectively with each ring frame

to resist direct stress and bending moment caused by the interaction

between the shell and the frames. A knowledge of this "effective width"

is of particular interest in a study of the buckling strength of the ring

itself and in the elastic and inelastic general-instability analyses of the

entire stiffened cylinder (this problem is considered in a later section).

It is also important in calculating the stresses in the frame flanges of

innperfectly circular cylindrical shell structures.

In Reference 8, Pulos and Salerno discuss the many "effective width"

formulas developed by earlier investigators, and they present a formal

derivation of a new forinula to include the "beam-column" effect. Details

26

of this may be found in Reference 8; however it is of interest here to give

the end result which can be expressed by the following convenient formula;

Lg = LFi+b (22)

In Equations (15) through (22) the following F functions (graphical

solutions for which were first developed by Krenzke and Short in Reference

10) have been introduced for ease of calculation:

â– ^1^(0)1^Â°^^ T]^9 - cos Ti^O

/2>

(23)

^2 =

cosh T] sin T) sinh tj cos tj

â– â€” â€” +

/2>

:24j

"3%

0.91

cosh T] sinh r\ cos t] sin t]

/2>

(25)

NO. 91

cosh:]! 6sinq2Â® sinhtij^Scosij^Q

^2

T]l

/57

(26)

where:

D

coshT]j6sinhT)-^e cosT]2esinq2^

L ^1 ^2 -

-^ = i^f^^'(-r)

(27)

e = ^3(l.v2)^

27

Curves of the functions Fj, F2, F3, and F4 may be found in either of

References 8 or 10.

The elastic analysis developed in Reference 8 is intended for the

determination of the deformations and stresses in a typical bay of a pres-

surized ring- stiffened cylinder composed of many identical bays as shown

in Figure 11. This longitudinal identity and symmetry between adjacent

bays is disturbed by the introduction of rigid bulkheads, intermediate deep

frames, cone and sphere-cylinder junctures, and other contiguous structure

which goes to make up the pressure hull of a submersible. In these more

complicated configurations, a more general analysis of the axisymmetric

behavior is needed.

Short and Bart have given a general analysis for determining the

12

stresses in stiffened cylindrical shells near structural discontinuities.

The formulation includes the possibility that the shell thickness may differ

in adjacent bays, the stiffness properties of adjacent ring-frames may be

different, and the spacing between ring frames may vary along the length

of the cylindrical compartment. The general equations developed by these

investigators are given in the form of frame and shell matrices to better

identify the stiffness and response of each element and to facilitate

numerical calculations. This form of the solution lends itself very con-

veniently to high-speed digital computers and also permits immediate

identification of those geometric and material properties which can be

varied to produce desirable changes in the static response. All the

28

i

i

,Undeformed

Shell

DÂ« formed

Shell

-Frame

-P

UHIU

yy^^^/ '

r

eg

^i

W7^?^V77?i

YZA

IT

e.g.

Figure 11 - Symmetrically Loaded Cylindrical Shell with Equally Spaced

Reinforcing Ring Frames

29

necessary equations and details of the formulation may be found in Refer-

ence 12; they are not given here because of their length and rather formi-

dable nature. However, it is worth mentioning here that extensive exper-

imental evaluation has been obtained of the Short-Bart analysis. For

1 3

example, Keefe and Overby present the results of structural model

tests undertaken to check the "end-bay" theory Reference 12. Also,

Keefe and Short present a method for eliminating the effect of end con-

ditions on the static collapse strength of stiffened cylindrical pressure

hulls and give experimental verification of the suggested procedure.

Another special problem of interest to pressure hull designers, and

one worth mentioning here, is that concerned with the discontinuity

stresses which arise at the juncture of axisymmetric shells possessing

1 5

dissimilar meridional shape. Raetz and Pulos present an analysis for

determining the elastic deformations occurring at either cone-cylinder or

cone-cone junctures and discuss several other analyses developed by

earlier investigators, notably Wenk and Taylor.

Conical transition sections are used rather extensively to join

cylindrical hull components of different diameter, and not only is the

problem of the edge effects on static collapse strength of the "weakened"

bays of concern, but also, the occurrence of high, localized longitudinal

stresses in these juncture regions is of great concern from the point of

view of low-cycle fatigue in way of welded joints. Raetz discusses this

problem and suggests the use of flexible, tapered ring-segments at these

30

junctures to reduce the high longitudinal stresses; he also presents an

analysis for determining the elastic behavior of these structural elements

and gives results to indicate the degree of reduction which can be realized

in these high stresses.

FAILURE CRITERIA FOR AXISYMAIETRIC

COLLAPSE PRECIPITATED BY YIELDING

We will now consider the question of how the biaxial stresses (defined

by Equations (15) through (20)) in a pressurized ring- stiffened cylinder can

combine to produce axisymmetric collapse precipitated by yielding of the

shell plating. Although these stresses are based on equilibrium con-

siderations only and do not reflect any buckling state, they can and do

predict good results when used in conjunction with appropriate theories

of failure even though, strictly speaking, axisymmetric collapse is

associated with an instability phenomenon. Formulas for predicting

axisymmetric collapse precipitated by yielding based on various theories

of failure are summarized in Reference 8.

The simplest formula devised for the design of pressure vessels is

the so-called "boiler formula". This formula may have some merit in

predicting the bursting strength of internally pressurized unstiffened

cylindrical tubes, but it is unsatisfactory (for other than comparative pur-

poses) in the design of pressure hulls in which instability and the influence

of reinforcing ring frames play a dominant role.

Equation (28) gives the pressure at which the circumferential

membrane stress in an unstiffened cylinder of mean radius R and

31

thickness h just reached the yield stress o of the material.

Pbz'^yh/R (28)

Equation (28) does not reflect in any way the strengthening effect of the

transverse ring frames on the average circumferential stress. However,

an estimate of this effect can be found by assuming that the cross- section-

al area of the frames is spread out and its orthotropic stiffness effect is

"felt" in the form of a thicker unstiffened cylindrical shell. This requires

that the actual thickness h in Equation (28) be replaced by

SO that we now get the following modified boiler formula:

Pel r <^yh(l+Aeff/Lfh)/R (29)

From the theory of Salerno and Pulos outlined earlier, the maLximum

stresses occur in the circumferential direction on the outside surface of

the shell plating midway between adjacent ring frames, and in the longitu-

dinal direction on the inside surface of the shell plating at a frame; these

stresses can be determined from Equations (16) and (17), respectively.

Which of the two stresses is the larger depends upon the geometry of the

cylindrical shell and the reinforcing ring frames, but in most cases of

interest, it turns out that ^X-f-^ "^ffjOm- However, extensive Model Basin

tests have shown that the stress '^^Oj^ is determinative in precipitating

axisymmetric collapse. Application of the maximum principal stress

17

theory of Rankine to this stress, i.e.,

32

leads to the following expression for the pressure at which yielding begins

on the outside fiber of the shell plating midway between adjacent ring

frames:

P,3= p.-^^ ,===-=r (31)

-^& - ^JS]

_ (l-l//2)Â°(

where a = â€¢

o(+p+{l-p)Fi

If the uniaxial criterion of Rankine is applied to the circumferential

membrane (midfiber) stress "^(bi^. i.e., Equation (16) with ^xbm ^^^ equal

to zero, the following expression for the pressure at which yielding has

penetrated through the plating thickness is obtained:

oâ€žh/R

In deriving the expressions for Pj^j, Pc3> ^.nd Pc5j it has tacitly been

assumed that a uniaixial state of stress exists in the shell plating whereas,

in reality, a biaxial state exists. More realistic criteria for axisymmetric

17

collapse can be derived from the energy-of-distortion theory which grew

out of the analytical work of Huber, Von Mises, and Hencky. Since the

octahedral shear- stress theory gives the same results as the energy-of-

distortion theory and permits the use of a more familiar quantity, such as

stress, the former theory will be used in what follows. For a biaxial state

of stress at midbay, i.e., midway between adjacent ring frames, defined

by the principal stresses "^Xm a-^d '^$rn. the octahedral shear-stress is

33

given by

'^G = Tn'^Xm- Om) + Xm+ cj-mj (33)

However, since according to this theory inelastic action at any point in a

body under any combination of stresses begins only when the octahedral

shear-stress T becomes equal to (v[2"/3)<3 then Equation (33) leads to the

following:

2 2

a + c - a ^ 6

Xm Om Xm Oin

- % (34)

Essentially two distinct criteria can be derived from Equation (34)

depending upon whether the outer-fiber stresses or mid-fiber (membrane)

stresses at midbay are used. For yielding on the outer surface of the

shell plating, when the appropriate stresses xÂ°m^^^'^<5Â°m from

Equations (15) and (16), respectively, are substituted into Equation (34),

the following criterion is obtained:

Pc6

%h/R (35)

If it is assumed that axisymmetric collapse is precipitated by the

yield zone having penetrated through the shell thickness, when the approp-

riate stresses ^Xm ^^^ % m. f^^orn Equations (15) and (16) with '^xbm ^^*-

equal to zero therein, respectively, are substituted into Equation (34), the

following criterion is obtained:

'cl

^h/R (36)

+a2rZ-3,P^]l/Z

1 8

Lunchick working at the Model Basin derived another criterion of

failure for predicting axisymmetric collapse which is based on the plastic-

hinge concept. He made use of the Hencky-Huber-Von Mises theory of

yielding, i.e., Equation (34), and allowed for the plastic reserve strength

after yielding begins in the shell plating at midbay. Lunchick developed a

formula for the pressure at which a complete plastic hinge has formed at

midbay. Since the combined stress gradients at the frame locations are

steeper than those at midbay so that complete plastic hinges form much

earlier at the frames, this TMB plastic-hinge theory, in reality, gives the

pressure at which the shell fails as a three-hinge mechanism. Although

this mechanism is not physically possible in the case of cylindrical shells

as it is for beams, it does lead to predictions of a collapse pressure (p^-g)

which agree well with experiment in certain ranges of geometry. A

complete discussion of this theory together with some comparisons to

experimental data may be found in Reference 18. For our purposes here,

it suffices to give the salient results which can be used for computation.

It can be shown that the ratio of circumferential bending stress to

circumferential membrane stress and the ratio of longitudinal membrane

stress to circumferential membrane stress (all stresses considered at

midbay) can be expressed in the convenient- notation of Reference 8 and

35

that adopted here as follows:

0.91

ct)bm/ OM - ^

1-1/2 (37)

l-aF2

l-aF7

In Figure 12, the pressure ratio Pc8^Pc6 ^^^ been plotted as a function of

the stress ratios, Equations (37) and (38). Thus, Equations (37) and (38)

can be used in conjunction with these curves and Equation (35) for p^^ to

determine values of the plastic-hinge pressure p^g for different geome-

tries.

The formulas for predicting axisymmetric collapse precipitated by-

yielding, and given in this section, represent explicit expressions for

collapse pressure only for the special case of zero "beam column" effect,

i.e.,Y= 0, since in this case only are the F functions given by Equations

(23) through (26) independent of pressure. For the general case in which

Y i 0, the stresses become nonlinear functions of the pressure, and Equa-

tions (31), (32), (35), and (36) are transcendental in the pressure. How-

ever, a numerical iteration procedure can be used in which the collapse

pressures p^2>> Pc5' ^^^- ^.re first calculated for Y = 0, and these values

are then used as the first approximation in the last of Equations (27) to

determine a value of Y- Then, with this value of Y in each corresponding

case, new values of the pressures Pc3Â» Pc5> â€¢ â€¢ â€¢ ^'c. can be found. This

36

1^0

\NX

SS381S 3NV^gtN3y\l "IVIlN3tJ3jy\in3dl3

ss3ais 3NvaaiM3w nvNianiioNOi

^ o

O CO

2d

Oif> O *"

h- <D U) ID

odd d

o

in

d

\

^^

^>

\\

\

^

^>

\

^

\\^

^

\

^^

k

s\

\

^^

^x

V

^

^

\\

\

^^

^^

\

\

^

^\

\

^"^

\

\

^

\\\,

\

^

1\

^

^

A

i

V

^

1^

A

CM jO-

â€” O

CO

C/5

LJ

o

o

C/)

O

2

Q

CO

2

o

LJ

fT)

o

_1

<

(-

u>

2

o

UJ

(r

u

bJ

u.

2

3

-vl-

U

o

oc

o

o

a,

-)

()

tn

oc

>

(.)

u

d

U

CVJ

93

LA

83

3dnSS3dd Qn3IA

3ynSS3ad 3SdV1100

37

process can be repeated again, but probably one or two, or at most, three

iterations are required to obtain satisfactory convergence.

One important point to be made is that the collapse criteria reflected

by the expressions for P^3, Pc5Â» Pc6Â» Pc7Â» ^^^ Pc8 ^^^ inherently based on

a material which exhibits elastic-perfectly plastic characteristics, so that

any effects of strain-hardening are neglected. The assumption is made

that once the yield strength is reached, at the critically stressed mid-

bay locations according to each of the respective criteria, collapse then

occurs.

Another solution for axisymmetric yield collapse of a ring- stiffened

pressurized cylinder in the presence of the "beam-column" effect was

19

developed by Kempner and Salerno. Their elastic analysis was pre-

dicated on the Hencky-Huber-Von Mises failure criterion as applied to the

critically stressed fiber at the midbay location. For the usual range of

geometries the yield strength of the material is first reached at the ring

frames due to the high longitudinal stress on the inner fiber. The authors

then considered that for loading beyond this initial yielding pressure, a full

plastic-hinge exists at the stiffening rings. Thus, the additional pressure

which the shell can support is calculated on the assumption that the shell

plating is "hinged" to the ring frames. The maximum additional pressure

which the shell can support is then based on applying the aforementioned

failure criterion to the total stress at the midbay location.

38

AXISYMIVIETRIC ELASTIC BUCKLING BETWEEN RING FRAMES

In reference 8, Salerno and Pulos derive the following criterion

which represents an exact solution to the axisymmetric elastic buckling

problem of a thin cylindrical shell reinforced by ring frames of finite

rigidity and loaded by hydrostatic pressure:

where

Â°(+ p + (1-P)F5 =

cos^Ti^e - cos^T] e

COST] 'esinri' e cost] esinT]26

^1

^2

(39)

(40)

and

(41)

â– ^~ 2E \

3(1-1/^)

(r)'

Equation (39) is a transcendental equation to be solved for the critical

pressure p = p^.^. for a given shell and ring-frame geometry defined by the

nondimensional parameters o(, p, and 6. A graphical representation of

Equation (40) is given in Reference 8 to facilitate the iteration calculations

required to find the value of p which satisfies the buckling criterion,

Equation (39).

In small-displacement theory, the condition for instability of a

39

structure may be obtained by requiring that the displacements become in-

finitely large at buckling. It is on this basis that Salerno and Pulos arriv-

ed at the criterion of Equation (39). Since Von Sanden and Gunther, and

later Viterbo, omitted the beam-column term in the basic differential

equation, Equation (1), then this would correspond to y = in the more

complete formulation of Reference 8. In such a case, Equation (39) would

no longer have any meaning because the pressure p would not appear.

A more complete analytical development and discussion of the

axisymmetric elastic buckling problem is given by Short and Pulos in

Reference 20.

AXISYMMETRIC INELASTIC BUCKLING BETWEEN RING FRAMES

The collapse criteria which were developed earlier and were identi-

fied by the pressures p^^, p^3, P(,5, p^^, p^y, and p^Q are, strictly speak-

ing, valid only for an elastic-perfectly plastic material. As it has already

been stated, the axisymmetric collapse mode is in reality associated with

the phenomenon of buckling at a reduced modulus, so that the strain-hard-

ening characteristics as reflected by the secant (Eg) and tangent (E^)

modulii must be considered.

2 1

On the basis of the deformation theory of plasticity, Gerard de-

veloped a general set of differential equations of equilibrium for cylin-

drical shells in which the coefficients reflect the plasticity or state-of-

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