John G Pulos.

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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

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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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-

Online LibraryJohn G PulosStructural analysis and design considerations for cylindrical pressure hulls → online text (page 2 of 8)