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Elastic buckling of cone-cylinder intersection under localized circumferential compression J. G. Teng Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (Received July 1994; revised version accepted February 1995)

Conical shells are often joined to cylindrical shells asend closures, reducers or roofs. Under a variety of loading conditions, the intersection between the large end of a cone and a cylinder is subject to a large circumferential compressive force which can lead to its failure by buckling. The problem may be idealized as a cone-cylinder intersection under a radial inward ring load. This paper first investigates the elastic buckling strength of thin cone-cylinder intersections under a radial inward ring load and develops simple and accurate equations for the prediction of buckling mode and strength. The ability of ring-loaded intersections to conservatively represent intersections under a variety of other loading conditions for their buckling behaviour is then explored. The ring load idealization is shown to be generally conservative, but may become rather conservative for some loading conditions such as uniform internal pressure. The strength of cone-cylinder intersections under uniform internal pressure is examined in detail in the final part of the paper and approximate strength equations are also developed, as this loading condition is important for pressure vessel and piping applications.

Keywords:cone-cylinder intersection, conical head, buckling, bifurcation, elastic, pressure vessel, cone, shell

I.

Introduction

Conical shells are commonly used in many engineering structures, for example, as reducers in piping, end closures for pressure vessels and liquid storage tanks, discharge hoppers for storage silos, and roofs for tanks and silos. The conical shell is often joined directly to a cylindrical shell (Figure 1), leading to a slope discontinuity in the shell meridian. This slope discontinuity can lead to local high bending and circumferential membrane stresses. Under many loading conditions, a vertical force is imposed on the cone, pulling the cone away from the cylinder (Figure 1). Examples include uniform internal pressure as in pressure vessels, linearly varying internal pressure as in liquid storage tanks, complex nonuniform internal pressure and meridional traction as in silos, and wind uplift pressure on roofs of silos and tanks. This vertical force is balanced by the meridional tension in the cone which often reaches its maximum at the point of intersection. At the point of intersection, the vertical component of the meridi-

onal tension is transferred to a vertical support directly or through the cylinder, whilst the radial component needs to be resisted by the intersection itself. This radial inward force leads to circumferential compressive stresses at the cone-cylinder intersection (Figure 1), which can cause the intersection to fail by buckling. Closely similar circumferential compression at the intersection can be set up by applying a radial inward ring load at the cone-cylinder intersection (Figure 2). Such intersections are referred to as 'ring-loaded intersections' for brevity. Because of the local weakening effect of the intersection, reinforcement of the intersection by local thickening of the shell wall or by a ring is generally required. When local thickening of the shell wall is used, both the cylinder and the cone generally have the same increased thickness near the point of intersection, though away from this area their thicknesses may differ. The size of the stiffened zone needs to be larger than that involved in intersection failure, guidance on which may be found in pressure vessel codesL For the purpose of this study, it is assumed that buckling failure

Elastic buckling of cone-cylinder intersection: J. G. Teng

42 Vertical Force W

The first part of this paper is an extension of Greiner and Ofner's 7 work to intersections with a cone apex half angle greater than 50 ° under a radial ring load. The numerical results are used to develop simple and accurate equations for the prediction of buckling mode and strength. The ability of ring-loaded intersections to conservatively represent intersections under a variety of other loading conditions for their buckling behaviour is then explored. The strength of cone-cylinder intersections under uniform internal pressure is examined in detail in the final part of the paper and approximate strength equations are also developed, as this loading condition is important for pressure vessel and piping applications.

•

Notation Figure I

Schematic of a cone-cylinder intersection

E l~c,lec,,

is confined to the thickened zone adjacent to the point of intersection and the shell thickness is constant in this zone. Therefore, only uniform thickness cone-cylinder intersections are considered. Most previous studies on the strength of cone-cylinder intersections were concerned with uniform internal pressure, intended for applications in the design of pressure vessels and piping. Axisymmetric collapse has been studied by several researchers =-5. Teng 5 presented a thorough investigation of the axisymmetric failure strength and developed simple design equations. Teng has also presented the first study of the plastic buckling behaviour of cone-cylinder intersections under uniform internal pressure in which simple design equations were also developed 6. This study showed that elastic buckling can be critical for thin intersections with a large apex angle. Greiner and Ofner 7 examined in some detail the elastic and plastic buckling strength of cone-cylinder intersections under a radial ring load. The radial ring load was used by them as a simple and conservative representation of the real loading condition in silos. Their numerical results are restricted to cone-cylinder intersections with an apex half angle c~ -< 50 °. The studies of the buckling strength of ringbeams (ring stiffeners) at cone-cylinder-skirt intersections in steel silos carried out by Rotter and his associates s-j° are also relevant. They also employed the equivalent radial ring load concept and characterized the buckling strength of ringbeams using the maximum compressive membrane stress in the ringbeam.

I//i

//N\

\\

I ,I

Figure 2 Cone-cylinder intersection under a radial ring load

X~ ~cr

F

F~ P q R t

W OL

Young's modulus effective lengths of cylinder and cone, respectively, in resisting circumferential compression meridional membrane force in cone at point of intersection (force per unit circumference) number of circumferential buckling waves radial ring load at point of intersection (force per unit circumference) ring load at intersection in direction of cone meridian (force per unit circumference) internal pressure meridional traction (force per unit area) radius of cylinder thickness of shell total vertical force on cone cone apex half angle

Subscripts and superscripts

nl eq

at buckling linear nonlinear equivalent

2.

Finite element analysis

cr

l

The finite element results presented in this paper were obtained using program NEPAS for the nonsymmetric bifurcation buckling analysis of axisymmetric shells. The formulation implemented in this program has been described in detail elsewhere ~. The program has been shown to give accurate predictions by extensive comparison with existing theoretical and experimental results and has been applied widely to study collapse and buckling problems in shells of revolution. Both linear elastic buckling analysis which is based on a small deflection prebuckling analysis, and nonlinear elastic buckling analysis which is based on a large deflection prebuckling analysis, were carried out for each intersection. The material of the cone-cylinder intersection was assumed to have properties typical of steel: a Young's modulus of 2 × l0 s MPa and a Poisson's ratio of 0.3. However, the results can also be applied to other materials with a Poisson's ratio close to 0.3 (i.e. aluminium shells), as the elastic buckling strength is proportional to the Young's modulus E. All buckling loads presented here are normalized by the Young's modulus E. The end boundary conditions of the cylindrical shell were chosen to represent continuity in the shell (free to displace radially and restrained against mer-

Elastic buckling of cone-cylinder intersection: J. G. Teng

either linear or nonlinear elastic buckling analysis, only the nonlinear buckling mode for a = 30 ° and R/t = 250 has a symmetric meridional shape (i.e. symmetric mode). All the other buckling modes are antisymmetric. It should be noted that none of the buckling modes is either symmetric or antisymmetric in the strict sense, so these terms should not be taken to imply their exact shape. Intersections with deep cones and relatively small R/t ratios are likely to buckle with a symmetrical mode, and their buckling behaviour is different from those with an antisymmetric buckling mode 7. A comparison between the linear and the nonlinear buckling mode indicates that the large deflection effect leads to a slightly more localized meridional buckling mode and a greater number of circumferential buckling waves. As the cone becomes shallower, the buckling deformations reduce in the cylinder but increase in the cone. The meridional shape of the buckling mode does not change much with R/t ratio except when a change occurs in the symmetry nature of the meridional shape.

idional rotation and circumferential displacement). A vertical support was applied either at the end of the cylinder or at the point of intersection as specified later. The cylindrical shell modelled in the analysis had a height equal to its radius, ensuring that the cylinder end boundary conditions had little influence on the buckling deformations at the intersection.

3.

Radial ring load

Only intersections with a cone apex half angle c~ ~ 30 ° and an RIt ratio -----250 are considered here as the strength of intersections with a thicker wall or a deeper cone is unlikely to be controlled by elastic buckling 6. The intersection was vertically supported at the end of the cylinder for all cases discussed in this section. Variations of the buckling strength and buckling mode with both R/t ratio and cone apex half angle a are examined. A number of the meridional buckling modes are shown in Figure 3. Among all the buckling modes obtained from

n

lcr ~-23

nLcr =

ntcr n{

=

n=,=

--

nLcr = 40

n =r = 45

e=30 °

46

(b) R/f=600,

n

c~=30 °

nlcr

nL n

o~=30 °

Undeformed Shape,

¢~ =

o~=45 °

"....... Linear Buckling Mode,

Figure 3 Buckling modes of intersections under a radial ring load

- -

(f)

e=50

°

---- 1 6

nnler =

5g

(e) R / f = I O 0 0 ,

n4 cr ~" 5 5

(c) R / t = 7 5 0 ,

nl©r ---- 1-3

63

(d) R/f=IO00,

~3

ni

nn{cr=36

(e) R/t=250,

43

25

R/t=IO00,

Nonlinear Buckling

Mode

¢<=85 °

Elastic buckling of cone-cylinder intersection: J. G. Teng

44

2.5

,q.

%. ,r.

v

= 30 k z~

c~ = 30"

.,~2.0 d=

c~ = 45*

t-

c~ = 30" "I:3 ¢3 O

Cl

°

"~'""~

C~ ¢°--

I

°

o

r~

= 45*

3=

~x = 45*

u

•-= 1.0

o o o o o Linear Buckling A,,,,z~,, Nonlinear Buckllng

--Equation

O

O)

&

2k

,3,

D

1 or 2

a = 85*

(lB

~o.5

'~'

= 85*

@ m

w

t-

0

Radial Ring Load

Jo

.9 z

Radial Ring Load

@

E

200

i5

400 600 800 Radius-to-Thickness Ratio R/I

1000

200

o o o o o Linear Buckling z~zxt,z,A Nonlinear Buckling

Equation 3 or 4

400 600 800 Radius-to-Thickness Ratio R/I

1000

Figure 6 Variation of buckling mode with cylinder radius: radial ring load

Figure4 Variation of buckling strength with cylinder radius: radial ring load

Similarly, the following equation predicts the nonlinear buckling strength closely B a s e d on numerical results from nonlinear buckling analysis of c o n e - c y l i n d e r intersections under a radial ring load, Greiner and Ofner 7 observed that for intersections with an antisymmetric buckling mode, the buckling load F J E t is proportional to (RIt) ]~, and the number of circumferential buckling waves depends on (R/t) °-~. These observations are confirmed by the present results from both linear and nonlinear elastic buckling analyses (Figures 4-7). Figure 4 shows that the finite element dimensionless buckling strength [F,,fl(Et)][R/t] L~ does not vary with R/t except for an intersection with c~ = 30 ° and R/t = 250 which has a symmetric buckling mode. The variation of the dimensionless buckling strength [FJ(Et)][R/t] ]~ with cosc~ is plotted in Figure 5. The effect of prebuckling deflections is seen to raise the buckling strength significantly (Figure 5). Figures 4 and 5 indicate that simple approximate buckling strength equations may be found. The linear elastic buckling strength was found to be well approximated by

--

Et

= 3.4 cos° 56c~

( 1)

--

= 4.3 cos°52a

Et

(2)

The variations of the number of circumferential buckling waves with radius-to-thickness ratio R/t and cone apex half angle a are shown in Figures 6 and 7, respectively. Except for the nonlinear buckling result for a = 30 ° and R/t -- 250, the number of buckling waves can be closely predicted using n~., = 1.6 cos°a%t

(3)

for linear elastic buckling results and using

~cr

(4)

~

for nonlinear elastic buckling results. A simpler version of equations ( 1 ) - ( 4 ) which still provides a good approximation to the finite element results is given in equations ( 5 ) - ( 8 ) together with their error ranges 2.5

rv =n

Radial Ring Load

~2.o

u_~ x~

g

)

2;

~1.5

._1

o e-

=

2

.E 1.0 .2£ 0

C13

g

,500°

~o.5 __ i5

0.0

0.2

__

Equation 1 or 2

0.4

0.6

0.8

1.0

Cos=

Z

°'°o.o . . . . . . . . . 0.2" "

Equation 3 or 4

0.8 ..... 6.'~ ....... 6.'6 .................. Cosc~

1.0

Figure 5 Variation of buckling strength with cone apex angle:

Figure 7 Variation of buckling mode with cone apex angle:

radial ring load

radial ring load

Elastic buckling of cone-cylinder intersection: J. G. Teng 0.983 < FIe" < 1.127

Et

Fapp

--

= 4.2

Et

;

(5)

45

tO as the equivalent radial ring load here, and may be accurately estimated by 12

Feq = Ntsina - 0.5(pl + p2)lec

1.006 < Fy~ < 1.054 Fapp

(6)

-- 0.5(p 3 + p4)lecoCOSOt + 0.5(q3 + q4)lecosina

ncr-- 1.6 - -

;

0.990 < nfea < 1.073

n,:.-'r ---

;

0.902 < nfea < 1.136

2.4 ( R - c- t s a ) " 2

napp

napp

(7)

(8)

where Ere a and nle,, are the finite element predictions, and F,pp and napp the simplified approximations. The way the apex half angle a appears in the above equations can be explained by noting that, in the vicinity of the point of intersection, the 'equivalent' radius of curvature of the conical shell is equal to its second principal radius of curvature here.

4.

Other loading conditions

4.1. Equivalent radial ring load Figure 8 shows a cone-cylinder intersection subject to linearly varying internal pressure and meridional traction. The meridional tension in the cone at the point of intersection can be found using the membrane theory of shells. If the total vertical force on the cone W is known, global equilibrium consideration of the cone leads to W

N~, - 27rRcosa

(9)

in which R is the radius of the cylinder. At the point of intersection, the vertical component of the cone meridional tension is balanced by the meridional tension in the cylinder, whilst the radial inward component loads the intersection in circumferential compression. The effect of this radial component, together with the influence of the normal pressure and meridional traction on the effective circumferential compression region (Figure 8), can be simulated by applying a radial inward ring load of appropriate magnitude at the intersection. This ring load is referred

"1,

(lO)

,--qt Pl

'

Figure 8 E q u i l i b r i u m a t intersection including local p r e s s u r e s

where le,. and le, v a r e the effective lengths of the cylinder and the cone, respectively, (Figure 8) in resisting the circumferential compression and are given by /e,. = 0.778~/-~

(11)

I.... =0.778( Rt ]1/2 \cosa/

(12)

A linear variation of pressure has been assumed in deriving equation (10), but a more complex variation can be readily incorporated either accurately or by a linear approximation for this local region. For uniformly distributed internal pressure and meridional traction, the radial force F divided by shell wall thickness t is given by R Feq - 0.5(p tana + q) t t -0.778p

+ 0.778q

(1 + ,~,co~)

{R~

'/2 "

s,n.

(13)

4.2. Buckling strength The buckling strength of an intersection with R/t = 500 and a = 60 ° under a radial ring load and four other loading conditions was studied. The five loading conditions are numbered 1 to 5 and shown i n Figure 9. For each loading condition, two support conditions were considered: vertical support at the end of the cylinder (case A), and vertical support at the point of intersection instead of the end of the cylinder (case B). The five loading conditions combined with case A support are referred to as cases A1-A5, and those with case B support cases B I - B 5 (Table 1). Finite element buckling analyses were combined with equation (10) to find the equivalent radial ring loads at buckling. The equivalent ring loads at buckling are listed in Table 1 together with the numbers of circumferential buckling waves. For a ring-loaded intersection (case A1), the only membrane stresses are the circumferential compressive stresses at the intersection (the meridional membrane stresses in the cone are negligible). For case A2, meridional tension exists in the cylinder, but this seems to change the buckling strength only slightly. For cases A 3 A5, apart from the local circumferential compression at the intersection, the meridional membrane force in the circumferential compression region and the membrane forces elsewhere in the shell are tensile or zero. These tensile membrane forces are stabilizing, so both linear and nonlinear buckling loads of each of these three cases are significantly higher than those of the ring-loaded intersections. Clearly, the radial load idealization is conservative for these cases. It represents a close conservative idealization for cases A2

46

Elastic buckling o f cone-cylinder intersection: J. G. Teng ¢

..l-0

E E >,,.

Radial Ring Load F"

Uniform Meridional Traction q

Inclined

Ring Load F~,

O3

.m X <[

(a)

Loading

Condition

1

(b)

Loading

Condition

2

(c)

Loading

Condition

3

¢

Uniform Internal Pressure

Linearly Varying Internal

ts77pS

J (d)

Figure 9 Table 1

Loading

Condition

H= R

Pressure

p

4

(e)

Loading

Condition 5

Various loading conditions Equivalent radial ring load at buckling for various loading conditions

(Figure 9)

Linear buckling Equivalent radial load

Nonlinear buckling

Support

Loading

F~q/(Et)

Number of buckling waves ncr

A: Vertical support at end of cylinder

1 2 3 4 5

211 213 263 394 361

× × × × ×

10.6 10.6 10.6 10-6 10-6

26 26 29 37 36

280 290 324 546 489

× 10 6 × 10 6 × 10 6

× 10 6 × 10 s

37 37 37 44 43

B: Vertical support at point of intersection

1 2 3 4 5

211 211 244 386 354

x x × × x

10-6 10-6 10 6 10-6 10-6

26 26 27 37 35

280 280 279 504 457

× × x × ×

37 37 35 43 42

and A3, but becomes quite conservative for cases A4 and A5 where tensile circumferential membrane stresses are large. The radial ring load approximation is more conservative for case 4 than for case 5 because, at the same equivalent ring load, the internal pressure present in the circumferential compression region is smaller for case 5. There is no difference between case At and case B1 because no vertical force is present here. Case B2 is basically the same as case BI (or case A1 ) because the vertical component of the applied force is transferred to the vertical support directly, so their buckling loads are little different. For case B3, whilst the meridional force in the cone is tensile, that in the cylinder is compressive, the predicted buck-

Equivalent radial load

FeJ(Et)

Number of buckling waves ncr

10 10 10 10 10

6 6 6 6 6

ling load is close to the radial ring load idealization because the stabilizing effect of the meridional tension in the cone is cancelled by the destabilizing effect of the meridional compression in the cylinder. The linear buckling load for case B3 is significantly higher than that predicted from the equivalent load approach, but the nonlinear buckling load is very slightly lower than that from the equivalent load approach. It may be envisaged that the degree of conservativeness of the equivalent radial load approach for case B3 depends on how close the cylinder is to the buckling condition under axial compression. For cases B4 and B5, the equivalent radial load approach is conservative because no compressive membrane forces are present in the shell apart

Elastic buck/ing of cone-cy/inder intersection: J. G. Tang from the local circumferential compression at the intersection. Moving the vertical support from the end of the cylinder to the point of intersection leads to reduction in the buckling strength as the tensile meridional membrane force in the cylinder is reduced to zero or changed to being compressive. The ability of the radial ring load idealization to represent other loading conditions is further explored by studying the buckling strength of cone-cylinder intersections under combined internal pressure p and meridional traction q. A practical structure in which the shell is subject to combined internal pressure and meridional traction is a steel silo for the storage of bulk solids. There, the ratio between the meridional traction and the internal pressure is around 0.5. Both linear and nonlinear buckling loads are presented in Figure 10 as equivalent radial ring loads, together with the linear and nonlinear radial ring loads at buckling of ring-loaded intersections. Clearly, the radial ring load idealization provides a close conservative approximation when the qlp ratio is large, but becomes progressively conservative as the ratio drops. As is expected, the approach is less conservative for intersections vertically supported at the point of intersection. The difference between linear and nonlinear buckling results becomes larger as qlp becomes smaller. The general conclusion from the discussion presented here is that the equivalent radial ring load approach always provides a conservative, sometimes, very conservative approximation to the buckling strength of cone-cylinder intersections with the vertical support in the cylinder. If the vertical support is provided at the point of intersection, whilst the equivalent radial ring load approach is generally conservative, this cannot be guaranteed if the axial compression in the cylinder becomes high.

47

internal pressure, so more accurate strength equations are developed here. Cone-cylinder intersections with 250 - R/t -< 1000 and 30 ° <- a <- 85 ° were examined for their linear and nonlinear elastic buckling strengths. The linear and nonlinear buckling loads are presented in Figures 11 and 12, respectively, where a filled circle is shown if the meridional buckling mode is symmetric. Most of the meridional buckling modes are antisymmetric, but in a few cases where R/t and/or a is small, the symmetric mode is critical. For three intersections, the antisymmetric mode was found to be critical by a linear buckling analysis, but the symmetric mode was predicted to be critical by a nonlinear buckling analysis (Figures 11 and 12). The characteristics of the buckling modes are similar to those observed in ring-loaded conecylinder intersections (Figure 3). Linear elastic buckling loads have been found to be reasonably well approximated by

E t - tan~Sa

(14)

and nonlinear elastic buckling loads by

E t

~I.=JI O

tanZSc~ -2_ ""'"

•

.~... ~

%

1 .... = - 3 0 ~

~

(15) " Linear Buckhng

o

Loads

-1

~1 0 - ~ a.

.E ~ 10 -~

5.

Uniform internal pressure

Uniform internal pressure is a well-defined loading condition important to pressure vessel and piping applications. A recent study 6 has shown that elastic buckling may be critical for thin intersections with a large cone apex angle. The previous section shows that the radial ring load idealization is very conservative for intersections under uniform

- "~ 10 -s. .... ~

I

Equation 14 x Equa~on.l. 4

@

.,

E

.--

~

~..

~

Cylinder Radlus-to-Thickness

I

~.,

~ I 1000 Ratia R/t

(a) -2

6E-4-

g

,•

RIt :500

"~ 5E-4 ~

I i

-~1o ~v

F

~ . .

cz=40 o

Linear Bu.ckling

1.6 -5

~o 4E-4i

""'-.

._= Q.

-..~......M.-.2---~2:22-.~.22~_~2222 ~

3E-4

Radial Ring Load

a

~ 2E-4 E _o 1 E - 4 1

.T:

..............................................................................

........ Linear Buckling Analysis Nonlinear Buckllng Analysis o o o o o Vertical Support at Cylinder Top Edge r, oanr, Vertical Support at Intersection

w OE+O 0.0

1.0 0.5 0.0 q / p --I ~ p/q I Ratio between Infernal Pressure and Meridional Traction

Figure 10

0.5

Buckling strength of intersections under c o m b i n e d internal pressure and meridional traction

i10 nn

-E OOOO0 FE R e s u l t / ~ o "~10 -=" - Equation 14 co ........ 1.2 x Equation 14 ,~

Cylinder Radius-to-Thickness

1000 Ratio R/t

(b)

Figure 11 Linear buckling strength of intersections under uniform internal pressure

48

Elastic buckling of cone-cylinder intersection: J G Teng ~ ' "[. . . .

10

¢<=30" ~ . " .

.........

60*

¢-

..... --..

-....

~ 10 _,.

6.

Nonlinear Buckling Loads

...

ou

-~-

.

:~oooo FE BucklinQ Load Equation 1~ •~ - . . E¢10 E

- ....... 1.2 x Equation

•

15

Symmetric" Buckling Mode i ~ ~ : ~ :

1000 Cylinder Radius-to-Thickness Ratio R/t

(a)

Conclusions

This paper has examined in detail the elastic buckling strength of cone-cylinder intersections under a radial ring load and a uniform internal pressure. Simple accurate formulae have been developed to approximate the finite element buckling loads for both cases. The equivalent ring load approach has also been shown to be a reasonable, and generally conservative, first approximation to the buckling load for intersections under other more complex loading conditions. Both linear and nonlinear buckling results have been discussed to illustrate the effect of prebuckling large deflections. A linear buckling analysis leads to significant underestimation of the real buckling strength, so a nonlinear buckling analysis is required for an accurate stability assessment. For the same reason, the approximate equations based on nonlinear buckling results should be used in design.

Acknowledgment ~

......

Nonlinear

I

,,o

References

o~

:<~z--~

70* - " : : ~ . . .

ol

: o o o o IrE Bucklinq

- -

=,~-5 ¢c

The author is grateful to the Australian Research Council (ARC) for its financial support.

:

"""-.. L 0 0 d ~

Equation 15 - ~ ........ 1 2 x Eauafion 15 • Symmetric Buckling Mode 1000

Cylinder Rodius-fo-Thlckness Ratio R/f

(b) Figure 12 N o n l i n e a r b u c k l i n g s t r e n g t h o f i n t e r s e c t i o n s u n d e r u n i f o r m internal p r e s s u r e

as s h o w n in F i g u r e s 11 and 12, respectively. Equations (14) and (15) w e r e d e v e l o p e d to give close or c o n s e r v a t i v e approximations in most cases ( F i g u r e s 11 and 12). The values predicted by the a b o v e two equations increased by 20% are s h o w n in F i g u r e s 11 and 12 as dashed lines to indicate the accuracy o f the two equations. For the four cases associated with a s y m m e t r i c b u c k l i n g mode, the finite e l e m e n t results are all significantly underestimated by the a p p r o x i m a t e equations. There are also several other cases associated with the antisymmetric m o d e that are underestim a t e d by m o r e than 20%. Nevertheless, the error is generally smaller than 20% and m u c h smaller than 20% in m a n y cases especially for very thin intersections for which elastic b u c k l i n g is mostly likely to be the controlling failure m o d e 6.

1 BS 5500: Specification of unfired fusion welded pressure vessels, British Standards Institution, London, UK 1988 2 Davie, J., Elsharkawi, K. and Taylor, T. E. 'Plastic collapse pressures for conical heads of cylindrical pressure vessels and their relation to design rules in two British Standard specifications'. Int. J. Pres. Ves. Piping 1978, 6, 131-145 3 Taylor, T. E. and Polychroni, G. Y. 'Optimum reinforcement of uniform thickness cone-cylinder intersections in vessels subject to internal pressure', Int. J. Pres. Ves. Piping 1983, 11, 33-46 4 Myler, P. and Robinson, M. 'Limit analysis of intersecting conical vessels', Int. J. Pres. Ves. Piping 1985, 18, 209-240 5 Teng, J. G. 'Cone-cylinder intersection under internal pressure: axisymmetric failure', J. Engng Mech., ASCE 1994, 120 (9), 1896-1912 6 Teng, J. G. 'Cone-cylinder intersection under internal pressure: nonsymmetric buckling', J. Engng Mech. Div., ASCE 1995, 121 (12) 7 Greiner, R. and Ofner, R. 'Elastic plastic buckling at cone-cylinder junctions of silos', in Buckling of shell structures on land, in the sea and in the air (J. F. Jullien Ed.), Elsevier Applied Science, London, UK, 1991, pp. 304-312 8 Rotter, J. M. 'The buckling and plastic collapse of ring stiffeners at cone/cylinder junctions', Proc. of Int. Colloquium on Stability of Plate and Shell Structures, Gent, Belgium, 1987, pp. 449-456 9 Jumikis, P. T. and Rotter, J. M. 'Buckling of simple ringbeams for bins and tanks', Proc. of lnt. Conf. on Bulk Materials Storage, Handling and Transportation, IEAust, Newcastle, Australia, 1983, pp. 323-328 10 Sharma, U. C., Rotter, J. M. and Jumikis, P. T. 'Shell restraint to ringbeam buckling in elevated steel silos', Proc. of 1st National Structural Engineering Conference, IEAust, Melbourne, Australia, 1987, pp. 604-609 11 Teng, J. G. and Rotter, J. M. 'Non-symmetric bifurcation of geometrically non-linear elastic-plastic axisymmetric shells subject to combined loads including torsion', Comput. Struct. 1989, 32, 453-477 12 Rotter, J. M. 'Analysis and design of ringbeams', in Design of steel bins for the storage of bulk solids, School of Civil and Mining Engineering, University of Sydney, Australia, 1985, pp. 164-183

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