A mechanism approach for the prediction of the collapse strength of ring-stiffened cylinders under axial compression and external pressure

Thin-Walled Structures 2 (1984) 325-353

A Mechanism Approach for the Prediction of the Collapse Strength of Ring-Stiffened Cylinders Under Axial Compression and External Pressure S. K. Tsang and J. E. H a r d i n g Department of CivilEngineering,ImperialCollegeof Scienceand Technology, London SW7 2BU, UK

ABSTRACT A mechanism approach is presented for determining the collapse behaviour of ring-stiffened cylindrical shells subjected to axial compression and external pressure. It extends previous work of Alexander, 5 and Andronicou and Walker, 6 incorporating the effect of pressure and calculates failure loads as an upper bound of elastic and mechanism responses. Comparisons with recent experiments and a lower bound analysis are given. The paper also discusses the relevance of the results to offshore design.

NOTATION D E f h H K L M N

Diameter of cylinder. Young's modulus. Von Mises yield function. Length of small element along leg of mechanism. Length of leg of buckling mechanism. Loading ratio (=N/p). Ring spacing. Bending moment per unit length. Axial loading per unit length (=o-at).

No

N/No. orot.

p

Pressure loading.

325 Thin-Walled Structures 0263-8231/84/$03.00 O ElsevierApplied SciencePublishersLtd, England, 1984. Printed in Great Britain

326

S. K. Tsang, J. E. Harding

po

trot/R. P/Po. R Radius of cylinder. t Thickness of cylinder. u Stress distribution factor (o-e = upR/t). w Out-of-plane displacement. w0 Initial out-of-plane imperfection. A E n d shortening. Strain. 0 Angle of mechanism. 00 Initial angle. v Poisson's ratio. O-o Yield stress. Subscripts

a e f 0

Axial direction. Elastic. Failure. H o o p direction.

Sign convention

Compression: + ve Tension: - ve Inward radial deflection: O u t w a r d radial deflection:

+ ve - ve

Terms not shown here are defined in the text and figures.

1 INTRODUCTION The behaviour of ring-stiffened shells, such as those used in offshore structures, is extremely complex. Data on the behaviour of these elements under interactive loading are negligible and as a result design rules are generally ill-founded and potentially conservative. The recent availability of elasto-plastic nonlinear computer packages has resulted in some data becoming available in this area but such packages are generally very expensive to run and require a background knowledge of shell

Collapse strength of ring-stiffened cylinders

327

response to allow their use with realistic choices of imperfection modes and other parameters. They are therefore not generally considered appropriate as design tools. In the treatment of ring-stiffened cylinders, offshore design codes assume the rings to be stiff and initiation of instability is intended to occur in the shell panel between the stiffeners. Many elastic solutions for the instability of cylindrical shell panels are available and research related to the aerospace industry has also provided much experimental data to validate these solutions. However, the geometries of the shells used in the construction of offshore structures are quite stocky and, as a result, failure of these shells generally occurs in the plastic range. In the past 30 years or so, plastic methods of analysis have become used in estimating the ultimate load carrying capacity of various structural forms. The plastic mechanism approach, for example, has been widely researched for the analysis of steel-plated structures. In recent times notable contributions on this approach have been made by Walker and Murray, ~ Murray, 2 Sherbourne and Korol, 3 Dean 4 and Alexander. 5 Recently, a plastic mechanism approach based on Ref. 5 was successfully modified by Andronicou and Walker 6 to analyse the post-failure behaviour of unstiffened cylinders under uniaxial end compression. This work has been extended in this paper to incorporate the analysis of the inter-ring panels of ring-stiffened cylinders under any arbitrary combination of axial and external pressure loadings. The failure loads obtained from this analysis have been correlated with results from a series of experiments conducted recently at Imperial College and Surrey University. 7 In addition, the analytical results have been compared with lower bound predictions of Croll 8 and calculations based on the DnV design code. 9 These comparisons have led to indications of areas in the interactive behaviour where code predictions are overconservative or even possibly nonconservative, and where more investigation and correlation are needed.

2 D E R I V A T I O N OF PLASTIC M E C H A N I S M E Q U A T I O N S W h e n a ring-stiffened cylindrical shell is subjected to the simultaneous action of axial compression and external pressure, the shell panel between the rings may buckle either inward or outward. Which mode occurs is determined to a large extent by the ratio of the two types of

328

S. K. Tsang, J. E. Harding

loading (K), but the shape of the initial imperfections may also exert a significant influence. In the following analysis an axisymmetric mechanism has been assumed and the governing equations for both outward and inward directions have been developed to ensure that the correct failure mechanism is obtained. The assumption of an axisymmetric failure limits the applicability of the m e t h o d to cylinders which have ring spacings of the order of the elastic critical buckling wavelength and this should be borne in mind in its use. First, the analysis is developed by assuming clamped boundary conditions for the shell at the ring sections. Let M be the m o m e n t of resistance per unit length at each plastic hinge. In Fig. 1 the work done dWn in bending at the three plastic hinges during a virtual change of deformation is dWll

=

2(TrDMdO) + 2rcM(D - 2HsinO)dO

= 47rM(D - HsinO) dO

(l)

For a wide strip of material, M is given by M - X/~ 4

~

(2)

The factor of 2/X/3 has been introduced because the material has been assumed to behave under conditions of plane strain. The full derivation can be found in textbooks such as Ref. 10. Combining eqns (1) and (2), d W . becomes 2 2 dWi1 = ~--~o-0t [1 - ]V2][D - HsinO]dO

(3)

where N = N / N o . In any virtual change of geometry, circumferential and longitudinal deformations must be compatible. Consider an annulus of material of length d h , as shown in Fig. 2. Initial length (before virtual deformation) = 27r(R - h sin0) Final length (after virtual deformation) = 2rr[R - h sin(0 + dO)] Hence, shortening in length = 27rh[sin(0 + dO) - sin0] = 2~-hcos0 dO (since sin(O+ dO) - sin0 -~ cos0d0)

(4)

Collapse strength of ring-stiffened cylinders

329

N

P

P

+~

f N [3

lrg. 1. Kinematics of mechanism.

Taking tre as the average value for the panel, i.e. effectively constant, the work done dW~2during this shortening in length for the whole mechanism is

dW~2= 2zrtTetcosOdO( 2 ~ ~hdh ) = 2zrtretHEcosOdO

(5)

330

S. K. Tsang, J. E. Harding

2

,_]_

Fig. 2. Material stretching component of deformation. Von Mises yield function f m a y be expressed as f ~ O"2--0-a0-O-kO " 2 - 0 - o2 = 0

(6)

Solving for tro gives O~0 = ½[~a -+- Xv/(4 -- 3 ~ ) ]

(7)

where if0 = o'0/cr0 and ffa = O'a/tr0. Compatibility of strains implies that deformation must occur in the longitudinal direction. Since such deformations take place under yield conditions, a flow rule must be adopted to determine the change in the plastic strain increment

of

dea = h - d0-a = X(2Ora--0-0)

(8)

and d~o = h d f ° =

h(2o-0 - O'a)

(9)

Collapse strength of ring-stiffened cylinders

331

w h e r e h is a constant of proportionality. Combining eqns (8) and (9) gives dE~ - 2 ~ a - ¢?Od~° 2 ~ 0 - ffa

(10)

The strain in the circumferential direction during a virtual change is d~0 -

h cos0 dO R - h sin(9

This may be approximated to d~0 ~

hcos0 dO R

(11)

T h e change in length of both segments of the mechanism All is

2ft.-fro fon h CosO dOdH 2ff o - f f ~2 R

An-

2ff ~- ~o H z - - cos0 d0 2oVo- ff~ R

(12)

Assuming the axial stress o'a to be uniform (=N/t) along the length of the mechanism, the corresponding virtual work done dWl3 is dWz3 = 7rDN (2~Ya- ~0)H 2cos0d0 (2~e - ~a) R

(13)

The total external work done comprises two parts: the work done by the axial loading, dWE~ and the work done by the external pressure loading dWE2. During a virtual change in angle, neglecting any strain deformation, the axial displacement AE~ and the radial displacement AE2 are given, respectively, by (see Fig. 3) AE1 = HCOS(O+ dO) - Hcos0

=HsinOdO

(14)

and AE2 =

Hsin(0 + dO) - Hsin0

= HcosOdO

(15)

S. K. Tsang, J. E. Harding

332

H

\

f Fig. 3. Rotational component of deformation.

H e n c e dWE~ m a y b e written as dWEI = 7rDN(2AEI + All)

2 f r o - ff a ] R

j

dO

(16)

and dWE2 may be written as dWE2 = prcD2H½AE2 = pTrDH2cosOdO

(17)

To maintain equilibrium, the total internal work must be equal to the total external work. Hence dWII + dWl2 + dWl3 = dWEl + dWE2 •

2zr

X/---~ Orot2(1 --/~ 2) (D = 2rrDNH

- H sin0) dO + 2 r c t r o t H =cos0 dO

sin0 dO + p T r D H 2 cos0 dO

(18)

Collapse strength of ring-stiffened cylinders

333

by writing K = /V/p, and o-0 = (poD)/(2t), eqn (18) may be written as 2

t[1 -(Kfi) 2] (D - Hsin0)dO + 2~sH2cosOdO = 2DKfiHsinOdO + 2fiH2cosOdO

(19)

Noting that ~0 = ~[ffa+ ~/(4--3~2)] for the inward mechanism and ~ e = ~[ffa- ~ / ( 4 - 3~2a)] for the outward mechanism, the equation of equilibrium for each mechanism may be obtained. It should be remembered that for the case of an outward mechanism, the d W . term must be p r e c e d e d by a - v e sign to take account of the fact that the direction of dO is opposite to the direction of + M . Hence for an inward mechanism 2t ~/3(1 - (Kff) 2)(D - Hsin0) + HZcos0 [Kfi + x/(4 - 3K2fi2)]

(2(I)

= 2DKfiHsinO + 2fiH2cosO and for an outward mechanism 2t

~3(1 -

(Kfi) 2)(D - Hsin0)+ H2cosO[Kfi - ~/(4 - 3KZfiz)]

= 2DK~HsinO + 2/~H2cos0

(21)

T h e length of the buckling mechanism H may be obtained by minimisation of eqns (20) and (21) with respect to H. Hence for an inward mechanism

H =

~ x/a 4/~ - 2[K~ + X/(4 -

(22)

3K2/~2)]

and for an outward mechanism tan0 {~_~[K2fiz_2t 11+ 2DKff } H=

)

2[Kfi - x/(4 - 3K2fi2)] - 4fi

(23)

Substitution of H from eqn (22) into eqn (20) and from eqn (23) into eqn (21) will give the required relationship between p and 0. If 2 H is

S. K. Tsang, J. E. Harding

334

greater than the length of the cylinder (or the ring-stiffener spacing) L, then the value of L/2 should be adopted instead of H. It is interesting to note that although H is d e p e n d e n t on 0 (see eqns (22) and (23)), its value remains fairly constant except in the very early stages of deformation. This zone of behaviour is removed by the presence of a significant initial imperfection. In the pressure-dominated region of the interaction, H generally takes the fixed value of the ring spacing. If required, the out-of-plane displacement w at the middle hinge may be obtained from w = Hsin0

(24)

and the total end shortening A is given by A = 2H( 1 - cos0) -~

2 ~ a -- ~ 0 H e

- - sin0 2~Yo- Na R

(25)

For the case of simply supported boundary conditions, the formulation is identical to the clamped boundary conditions except that in this case only one or two plastic hinges are required. If H < L/2 two hinges are n e e d e d and d W . should be modified to 4MTr(ad9 - H sin 0) dO. If H is equal to L/2, only one plastic hinge is required and the magnitude ofdWn should be further modified to 4Mrr(kD - H sin 0) dO. The effects of initial geometric imperfections may be incorporated into the analysis by introducing an initial angle 00. 0 should then be measured from the initially distorted configuration. The analysis is modified by replacing 0 by (0 + 00). The expresSions for w and A now become (26)

w = H(sin0- sin0o) n 2

A = 2H[cos0,,- cos(0 + 0o)] -~ 2 4 . - 4o R- [sin(0 + 0o) - sin0o] 2 ~ o - (Ta

(27)

3 PREDICTION OF FAILURE LOADS A n upper b o u n d estimate of the failure load may be obtained by determining the point of intersection of the plastic unloading curve for a particular value of K and the elastic loading curve for the same value of K.

Collapse strength of ring-stiffenedcylinders

335

H o w e v e r , in order to simplify the construction of the elastic loading curve, all geometrical nonlinear effects have been neglected. Let p~ and N, be, respectively, the pressure loading and axial loading in the elastic range. (re and o-~may then be written as o'0

upeR

-

(28)

t

o'a -

Ne

(29)

t

w h e r e u is a factor introduced to account for the uneven distribution of h o o p stress in the panel. By defining 0' = 00+ 0 and w' = w0+ w, the hoop strain E. may be written as w H E8 - R - R (sin0' - sin00)

(30)

But since ~0 may also be expressed as 1

(31)

Expressions for ~o and/~/e may be obtained by combining eqns (28), (29), (30) and (31) /~ =

H(sin0' - sin00) Et

R (upoR - uKNo)

(32)

and

Ale

-

-~ol 1. [ up-epoR - --REtH(sin0' - sin00) ]

(33)

w h e r e po = pJP0, and/V~ = NJNo. For an inward mechanism, combining eqns (20) and (32) and assuming tan 0' = sin 0' gives

Et (1~_..~ [ 2 t K2_ 2p - 1 ]

-2DKfflsin20'-EtsinOo( -~-g[KF-1]-2DK~2t-22

- fi~R(upoR - vKNo) {4p - 2 [K/5 + X/(4 - 3KZ,ff:)]} = 0

}sin0' (34)

S. K. Tsang,J. E. Harding

336

This is a quadratic equation in sin 0'. The positive root of this equation gives the angle Of at which p, = ~ = pf where pf denotes the failure pressure pf divided by p0. By substituting the above value of Of into eqn (20), the following implicit expression for pf may be obtained 2t ~3[1 - (Kil0 z][O - H sin0f] + + x / ( 4 - 3K2fi2)] =

HZcosOf[Kfif

2OKfifHsinOf

+ 2fifHZcos0f

(35)

Solution for pf may be obtained by an iterative procedure. In a similar manner, an equation for the outward mechanism may be obtained by replacing 0 and ~ in eqn (21) by Of and pf. Of in this case is the negative root of the following quadratic in sin O'

Et{~[Kfi2-1]+2DKfi}sin20'-EtsinOo l-~[K2t-2"fi'-l]+2DKfi} -fieR(upoR - vKNo) {2[Kfi -

x/(4 - 3KZfi2)]- 4fi} = 0

sin0' (36)

In the derivation of the failure load the coincidence of the elastic and mechanism out-of-plane peak deflections has been considered in order to define the failure loads. It is an approximation inherent in the m e t h o d that unless the postulated elastic shape is similar to that of the mechanism the axial shortening of the cylinder corresponding to elastic and mechanism solutions will be slightly inconsistent at failure. By referring the failure only to the maximum lateral deflection no definition of elastic deflection mode is required.

4 EXPERIMENTAL DATA Experimental data on interactive behaviour have recently been obtained at Imperial College and Surrey University. 7These data have been used to provide some correlation of the results obtained from the plastic mechanism analysis. The test series involved testing five ring-stiffened models to failure under various loading combinations to enable an interaction curve

Collapse strength of ring-stiffened cylinders

337

d e s c r i b i n g the u l t i m a t e s t r e n g t h o f the m o d e l s u n d e r axial c o m p r e s s i o n a n d e x t e r n a l p r e s s u r e to b e o b t a i n e d . T h e m o d e l s c o n s i s t e d o f five b a y s a n d w e r e d e s i g n e d to fail b y p a n e l b u c k l i n g o f the c e n t r e b a y . D e t a i l s o f all m o d e l g e o m e t r y a n d loading c o n d i t i o n s are g i v e n in T a b l e 1. P r i o r to t e s t i n g , all t h e m o d e l s w e r e s c a n n e d in c o n s i d e r a b l e detail to e n a b l e the s h a p e a n d m a g n i t u d e o f the initial i m p e r f e c t i o n s to b e o b t a i n e d . T h e m a x i m u m v a l u e s o f t h e s e i m p e r f e c t i o n s are given in T a b l e 2 t o g e t h e r w i t h TABLE 1

Details of Cylinder Geometry and Loadings a

Cylinder number

Radius, R Thickness of central panel, t Depth of stiffener, d~ Thickness of stiffener, t~ Length of central panel, L

R/t L/R Compressive yield strength, or0(N/mm2) K

1

2

3

4

5

160 0.6 12.8 1.6 24 267 0.15

160 0.6 12-8 1.6 24 267 0-15 375 0.5

160 0.6 12.8 1.6 24 267 0.15 375 0

160 0.6 12.8 1.6 24 267 0.15 375 1.5

160 0.6 9-6 0-6 24 267 0.15 375 0-5

375 oo

aAll dimensions in millimetres. TABLE 2

Comparison of Initial Imperfections with DnV Tolerances a

Perfect radius Cylinder (ram) 1 2 3 4 5

159-55 159-69 159.68 159-61 159"63

Maximum outward reading,w6

Maximum inward reading,w~

Out-ofstraightnessb

Deviation from circle

-0-23 -0-26 -0-47 -0-26 -0-33

0.18 0"25 0.42 0.34 0"35

0"15 (0"33) 0.21 (0"33) 0.44 (0"33) 0.40 (0"33) 0.22 (0"33)

0-07 0-12 0-14 0-15 0-14

(0.48) (0-48) (0.48) (0-48) (0-48)

aValues in millimetres; DnV tolerances in parentheses. bModel out-of-straightness values have been measured relative to ring-stiffener positions. Ring spacing is less than the DnV gauge length.

338

S. K. Tsang, J. E. Harding

DnV tolerance values. To keep the effects of residual stresses to a minimum, all the models were stress-relieved. Further details about the experimental procedures are given in Ref. 7. From the experiments it was discovered that the failure modes ranged from a single half-wave outward bulge for the case of pure axial compression (K -- oo) to a single half-wave inward bulge for the case of pure lateral pressure (K -- 0). Models 2 and 5 were both subjected to pure hydrostatic pressure (K = 0.5) and failed at practically identical load levels. The failure modes were also single half-wave inward bulges. Model 4 (K = 1-5) generated the failure mode of the greatest interest with both inward and outward deflections occurring simultaneously, but its axial buckling mode was still essentially a single half-wave. With the exception of this model, all the failure modes may be regarded as essentially axisymmetric.

5 EXPERIMENTAL CORRELATION Before any comparisons with the experimental results can be made, it is necessary to relate the level of hoop stress present in the models to the pressure applied. A stress distribution factor (u) has been introduced in the analysis so that the effect of the rings in reducing the hoop stress level in their vicinity can be allowed for. u[(pR)/t] represents the average hoop stress level in the panel. The influence of u is illustrated in Fig. 4 where interaction curves have been plotted for various values of u for both simply supported and clamped boundary conditions. The results of this figure assume zero imperfections. It may be seen that the strengths obtained from the analytical model generally increase as the value of u is decreased. This is consistent with the physical behaviour since a lower value of u implies that a higher proportion of the applied pressure is being resisted by forces associated with the ring response. As expected, for the case of pure axial compression, the variation of u does not have any influence on the analytical results. However, for the case of pure lateral pressure, the load carrying capacity of the panel also does not vary significantly with u. The reason for this is not immediately apparent. The mechanism length is effectively independent of u for this case and the only dependence of peak strength on u results from the effect of u on the slope of the elastic loading curve. In this instance the unloading mechanism line is very flat and hence

Collapse strength of ring-stiffened cylinders

339 0

E

IE

O_

Z

~._>,,

6 Ii

II

E

ii

~-

~o~ fy,

--I

I

I 0

•

I

.=

,~%.- -%-.;'> f

/\.

l/

°-1

B"

c~'/,,/ u = 0.?.~

"-'~ 0

I/ ! h

?,//

o

!O

kO

cb

0

6

340

S. K. Tsang, J. E. Harding ~5 {K : o o )

~o

l..~o (K..
o

o

o

B1"8

o

o

o

I. 02

I 04

I 06

o.-- K = O

-ri ~1"6

+

i

0.2

outward ~1 -08

reward I -0.6

_014

I -0 2

i

w°/t

0'8

o clamped A simply supported

Fig. 5. Variation of load carrying capacity with initial imperfection level (u = 1-0).

approximately the same value of collapse pressure for any value of u is produced. The flatness of the unloading curve for the case of lateral pressure only (K = 0) also makes this load case insensitive to variation of initial imperfections, as may be seen from Fig. 5 in which the predicted failure loads for each of the experimental cylinders have been plotted against imperfection level for u = 1.0. The mode of the imperfection incorporated into the analysis is a single longitudinal half-wave, either outward or inward. For the load cases K = 0-5 and K = 1.5, initial inward bulges have the more adverse effect since the failure modes for these load cases are also

Collapse strength of ring-stiffened cylinders

341

inward, being dominated by the external pressure loading. For the case of pure axial compression (K = ~) failure occurs by outward bulging and, not surprisingly, the outward imperfection mode is the more critical. In general, analyses for simply supported boundaries lie below those of clamped boundaries. Figures 6 and 7 illustrate the influence of initial imperfection on the predicted ultimate strengths for the experimental parameters for various u values. Two sets of imperfection levels have been chosen. In Fig. 6 DnV tolerance levels have been factored down to take account of the fact that the ring spacings or the lengths of the mechanism may be less than the gauge length specified in DnV. The experimental ring spacings were only 24.0 m m with the corresponding gauge length specified in DnV being 39.2 mm. The lengths of the buckling mechanism, in some cases, were less than half the ring spacing. Using a linearly reduced value of DnV tolerance is is found that the shape of the interaction curve is similar to that of Fig. 4. However, when the actual (unfactored) DnV tolerance level was incorporated into the analysis in the critical mechanism mode, the load carrying capacity for the case of K = 1.5 showed a significant drop for all u values as illustrated in Fig. 7. This discovery is in agreement with the experimental observation. The reason for such a significant drop is that the lengths of the buckling mechanisms were considerably smaller than those for the other K values and hence larger initial angles 00 were included in the analysis to produce the equivalent imperfection. This seems to imply that significant critical mode imperfections were also present in the experimental model. The effect of a change in u is to change the failure pressure pf because of the variation in the slope of the elastic curve. As the length H of the mechanism is dependent on pf, this in turn causes a variation in 00 for a given w0. In other words, a different part of the unloading mechanism curve is being considered since 00 is varied. Thus, fixing the w0 value irrespective of H results in a larger drop in collapse load for certain ranges of the parameters. The choice of the value of u to use for the experimental correlation can be derived approximately from the DnV specification which suggests values of 0.3 lpR/t for the hoop stresses at the ring stiffener and 1.0pR/t at the panel centre-line for the case of pure lateral pressure. A value of 0-75 for u is therefore a reasonable estimate for the stress distribution factor for pressure, particularly bearing in mind that the hoop stress is more uniform across the central area of the panel. It also tends to form lower

342

S. K. Tsang, J. E. Harding

tS 0 r~ r~ u3

t.. 0

§~_

t'-

E~

C,4

E E

I k_ 0

Z

i}

ii

Ii

0

I I I

o

>

ii

~D

0

..;; ms/,," / / I"

~

o

/ / ~ @

I

c~

i I i I

/ I

'X !

0 4-

°~

/

,'/,',//

d~

0

*d d~

N

l l'l

0

t

0

ete

Collapse strength of ring-stiffened cylinders

343

,r-

2

,""

C',.I

E E

-.-.

tm

m

Z L~

~

M

~

II

II

L~

LO

0

0

II

'1

IS

0

I I I

>

./ /////7/ _ u:> //// /

/

-,:_ o~

/1"/

I

t_.(/I

"•"l•

: Itl~~

°

Ii

0

8

.m

I 04

I

I

0

I

I

CO

I

tJD

~4"

Cxl

0

344

S. K. Tsang, J. E. Harding ~D t~

~D

O £5_

7ca_

E E

c~.

E

c~

O._

E

ca_

E

E

X

(-,4

6q

Z

I I I I I I

x IL

o

II

II

II

0

I I

I I

I

I

I

I

<~

u c-

(z)

t.9 C

e.,

E

'r"

e., ~D

o

G ,,>

/

\ \

I

I

~

L (2)

"4

,,:-

',:--

O

O

O

O

Collapse strength of ring-stiffenedcylinders

345

b o u n d analytical strengths for other loading ratios corresponding to other u values. Such a value of u has therefore been adopted for correlation with the experimental results. This is shown in Fig. 8 where the analytical results for the two different approaches to the use of the D n V imperfections have been included. It is seen that imperfection values based on actual D n V tolerance limits in the critical mode give remarkable agreem e n t with the experimental results and certainly much better than is obtained by using factored D n V values. Use of D n V imperfections in general seems justified as measured peak values are comparable to the tolerance level. It is interesting to note that for the range of K of 0.5-0, the plastic mechanism method is slightly conservative even with clamped boundaries. This could have been caused by strain hardening which is not accounted for in the present analysis or possibly some model interaction between panel and ring. For K < 0.5, therefore, the analysis for clamped boundaries gives a good conservative estimate of the failure load. For K > 0-5, the analyses with simply supported boundaries result in moderately conservative predictions of the failure loads. Curves are presented later in the paper combining the two above limits for comparison with design formulations.

6 EFFECTS OF STEEL G R A D E In Fig. 9 interaction curves for two different grades of steel (o'0 = 245 and 350 N / m m 2) and for various R/t ratios have been plotted to assess the influence of yield stress on the ultimate load carrying capacity. It can be observed that the curves for each R/t ratio do not vary significantly and hence it may be assumed that the ratio of failure pressure to yield pressure does not depend on or0. The curves for o-0 = 245 N/mm 2 provide a slight lower bound for most of the interaction and for this reason, a value of 245 N / m m 2 for o'0 has been adopted in the construction of the subsequent curves given in this paper. It should be noted that the results presented in Fig. 9 and those of subsequent figures were produced using a u value of 0-75. u is dependent on both ring-stiffener size and ring spacing. The results presented are therefore indicative of behaviour but should be recalculated for a particular geometry. The DnV code gives guidance on the variation of hoop stress with geometry and this could be used to give a value of u forming the basis of a particular calculation.

346

S. K. Tsang, J. E. Harding E E Z

iS

E E Z

('o o4 II

II

bobo u C

0

"U

O~ il

I'

I I I

0

d~

o-- d II

--J

0

~

O0

,,x/

II

~

h5

r" 0 o

=

~J

0

C

¢

0

o

0

c5

Collapse strength of ring-stiffenedcylinders

347

In general, u becomes larger (but never exceeds unity) as the ring spacing increases but reduces with increasing ring stiffness. The trends shown in the various figures, however, are not significantly affected by variations in u for stiffeners satisfying the DnV requirements and for geometries within the range considered.

7 V A R I A T I O N OF S T R E N G T H WITH C Y L I N D E R G E O M E T R Y Strength curves produced using the mixed boundary condition approach are presented in Figs 10 to 13. The geometrical parameters given cover the range of parameters found in the main legs of typical floating offshore structures. D n V design strength predictions have been included for comparison. For the pressure-dominated area of the interaction, the present plastic mechanism approach indicates that the DnV code is extremely conservative. For the lowest L/R ratio (= 0.1), curves for R/t = 50 and 100 give surprisingly high failure pressures. Similar results are obtained from Croll's 8 lower bound analysis. The high results are due to the restraint provided by the ring stiffeners. This is only of major significance in the case of stocky shells with low ring spacings although these extreme geometries are of little practical relevance. In the more imperfection-sensitive area of the interaction, particularly for K = ~ and K = 1.5, there are considerable differences between the analytical curves and the DnV predictions. Results from Croll's lower b o u n d analysis corresponding to the fully plastic state have been included in Figs 10 and 13, and provide reasonable agreement with the present mechanism approach for the K = ~ case. Croll's analysis does not produce the inflection in the interaction curve for K = 1-5 observed in the experimental curve. However, it should be pointed out that Croll's results are for a yield stress of 315 N/mm 2 and assume a u value of 1-0, and are therefore not strictly comparable. It has to be admitted that the present analysis takes a possibly overconservative attitude to the correlation of DnV tolerance levels and short wavelength mechanisms in the K = 1.5 area but it is not immediately obvious why this trend is not followed by Croll's work which also considers critical imperfection modes. The point of concern is that this approach does give excellent experimental correlations, albeit for limited experimental results, and if the present analytical predictions are likely to be encountered in practice, the conclusion must be drawn that the design

348

S. K. Tsang, J. E. Harding

~o

o~ u C

E

~8

~

Z .,,.1. r" C,4 C2~ li

~o

I

c5

ii

ii

0 CD Lf)

\,

il

6 II

rY

/ .i /

.-.--...:~ "lZI

cl

" ~

"?z.~

--..-/" ~ '/


~-

~-

0

,

i

I~1 'it i iX Ilk

<,o

i'->. I.

0

<~

0

0

,#.

Collapse strength of ring-stiffened cylinders

349 0

Cxl

E

6~

t.) t'-

>~

'--

0 Z

tO>= ',4 t'~ 0'4 II

II

tO

I

0

i i

II

0

Ii

0

gO

kO

f

..,/

f

0

,J

/./

/

/ .,9,Z~... ",7>.. "/ ~

, ,~ x I i~ ~" , , ~

/ " 'cb

oO

0

/.z~ ./,-? 0

e

tl

6

i

i

ii

i

t

Ii i

O,4

0

J

i ,

i

,~0

kO

~

0'4

0

cb II

S. K. Tsang, J. E. Harding

350

o

>,

u

r-

"0 6) L.) r-

('~ E

E

E~

r-

J!

oLE~

z

2

L~

~

C',J

r-~

It

H

~_

I I

LC~ 0

i

I1

o

r,i il

i-

e-,

z.

// ./ I. ~./

./

/

i

i1 j ~ f

i.~'1

,11

I

;i

~ "\ \ \ \

c.4 I

"1

,~

Collapse strength of ring-stiffenedcylinders

351

T

t'~

O

t-(.3 c"

C~l

+=2 E

(',4

6~

Z

~>~ il

O~

ii

I I

i'

11

C)

I

~3 CO

~5 lid

II

r~

~4 ¢)

q~

/, ,b -'/

o

/,, e ] +*

f

O

s

\ I,\I

q+D O

/

I \'~

#

\\\

i I \\\\ ii, • \,', \\/ O

+-

aO

o

6

~O

c5

.,,.,1.

c5

O4

<:5

352

s. K. Tsang, J. E. Harding

code predictions could be significantly nonconservative. Insufficient data are available at present to confirm or discount this postulate.

8 CONCLUSIONS By extending the work of Ref. 6, a concise approach to the prediction of the ultimate load for axisymmetric panel failure of ring-stiffened cylinders under any combination of axial compression and external pressure has been presented. The results obtained are found to provide excellent agreement with the limited experimental data available. Analysis predictions indicate that the DnV design code is very conservative for cylinders loaded by pure lateral pressure but there are also indications that the DnV code may be nonconservative in the axially dominated area of the interaction, especially in one particularly imperfection-sensitive zone where axial compression is combined with a small level of pressure. Insufficient evidence is available to prove or disprove the latter possibility and the approach to imperfections used in the analysis may be conservative. Some concern must, however, be expressed at the present time.

ACKNOWLEDGEMENT The sponsorship provided for S. K. Tsang by the SERC is gratefully acknowledged.

REFERENCES 1. Walker, A. C. and Murray, N. W., A plastic collapse mechanism for compressed plates, International Association for Bridge and Structural Engineers, 35 (1975) 127-34. 2. Murray, N. W., Buckling of stiffened panels loaded axially and in bending, The Structural Engineer, 51 (1973) 285-301. 3. Sherbourne, A. N. and Korol, R. M., Ultimate strength of plates in uniaxial compression, ASCE Nat. Strut. Eng. Meeting, 1971. 4. Dean, J. A., The collapse behaviour of steel plating subject to complex loading, PhD Thesis, University of London, 1975. 5. Alexander, J. M., An approximate analysis of the collapse of thin cylindrical shells under axial loading, Quart. J. Mech. Appl. Math., 13 (1960) 10-15.

Collapse strength of ring-stiffened cylinders

353

6. Andronicou, A. and Walker, A. C., A plastic collapse mechanism for cylinders under uniaxial end compression, J. Constructional Steel Research, 1 (1981) 23-34. 7. Tsang, S. K., Harding, J. E., Walker, A. C. and Andronicou, A., Buckling of ring stiffened cylinders subjected to combined pressure and axial compressive loading, A S M E Conference on Pressure Vessels, Portland, Oregon, June, 1983. 8. Croll, J. G. A., Axisymmetric collapse of cylinders including deformations of ring stiffening, In: Behaviour of thin-walled structures, Ed. by J. Rhodes and J. Spence, Elsevier Applied Science Publishers Ltd, London, 1984. 9. Det Norske Veritas (DnV). Rules for the design, construction and inspection of offshore structures--Appendix C: steel structures, Hovik, Det norske Veritas, 1979. 10. Save, M. A. and Massonnet, C. E., Plastic analysis and design of plates, shells and disks, Amsterdam, North Holland Publishing Co., 1972, pp. 434--69.