16426P108-116

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16587 Pressurised Systems

108

9.49.4 SHELLS UNDER EXTERNAL PRESSURESHELLS UNDER EXTERNAL PRESSURE

This topic involves quite different problems from those arising in the design of internally

pressurised vessels. It is important to give attention to :-

(a) Elastic and Plastic Buckling

(b) Shape imperfections

(c) Residual Stresses due to the manufacturing processes.

The Design Approach in PD 5500

The aim of the method is to :

predict the pressure at which buckling will occur in each part.

If this pressure is less than the required working pressure then:-

(i) the shell may be thickened;

(ii) stiffening rings can be added;

(iii) if rings are already present they can be placed more

frequently or increased in size;

(iv) for dished ends, the thickness could be increased, or the

geometry changed.

To illustrate the approach and give some background, to at least part of the method, a series

of figures are presented;

Vessel with various types of stif feners located in the cylindrical & conical parts

In PD 5500, there is a margin of at least 50%between the design pressure specified and the

pressure at which signs of buckling might first appear - providing the vessel is circular to

within 0.5% of the radius.

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109

In Cylinders, there is a multitude of elastic buckling modes which can occur and describe,

analytically, the behaviour of the vessel. The technique in BS 5500 is to design the cylinders

by considering three simple cases only.

(a) interstiffener buckling(b) overall buckling

(c) stiffener tripping

The photograph shows interstiffener buckling as

exhibited in a test model. Note the wrinkles occur

between those parts which remain circular. This can

be clearly seen on the above diagram.

In this course, only interstiffener

buckling will be dealt with in detail.

However, be aware of the existence of

the other two and be capable of

describing them in some detail.

The following photograph and

diagram show overall buckli ngwherethe whole cylinder has collapsed. The

light stiffeners have given way and

only those major portions of shell with

substantial stiffening remain circular.

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110

Elastic Analysis of Equally Spaced Stiffeners in a Cylindrical VesselElastic Analysis of Equally Spaced Stiffeners in a Cylindrical Vessel

As a first step to understanding the distribution of the stresses in such a vessel an elastic

analysis can be carried out.

The cylinder equation:-

d

dx

d

dx D z

4

4

2 22

2

44 4 1

+ + =

can be used. This can be solved for the cylinder shown.

There are 4 stresses of importance

(1) the circumferential stress on the outer flange of the stiffener s

(2) the axial stress on the vessel 7(3) the circumferential stress on the outside surface between the stiffeners s

(4) The circumferential stress at the centre of the wall, between the stiffeners 5

It is therefore necessary to find that the value of the external pressurewhich will cause each

of these four stresses to each reach the material yield point, . In general, the lowest stress

is 5 . It is a compressive membrane stressand thus seems a good candidate for being used

as the stress to cause buckling. This value is called the Yield Pressure,pyand is given by the following expression,

( )p

sfe

R Gy = 1

The value sfis a modified design stress, and can be thought of as a reduced yield value.

It relatesfto an effective yield point: s= 1.4 for carbon steels

s= 1.1 for stainless steels

s

5

7

s

R

External pressure

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One could argue fors= 1.5, so thatsf=y. A lowersvalue introduces a factor of safety.Thein the above equation is a rather complicated value, which is a function of the stiffeners. Itis generally taken as 0 as a first approximation. Such an assumption produces an

underestimation for the allowable pressure, and therefore is safe to do so. Therefore,

p sfeR

= (Note: = =f s pRe)

Recall, this is the membrane equation again.

Elastic Buckling

This graph is a pressure-deflection response for cylinders. The line OA shows a uniform

response, single value of deflection and only slight non-linearity. Displacement is

axisymmetric.At A, the response becomes unstable - bifurcation point. From A to B - In

theory the shell displacement would continue along AB and buckle axi-symmetrically at high

pressure. However, the slightest disturbance at A would cause a dynamic motion. A fly

breathing at the other end of the Universe ! Note: the pressure at A - Elastic Buckling

Pressurepm

Shape of vessel in cir cumferential direction

From point A, the cylindrical shell develops into a series of sine waves in the circumferentialdirection, n= 2, 3, 4 etc. There will be a similar behaviour in the axial direction. There are

C

CB

A

2

1

imperfect

cylinders

perfect cylinders

axisymmetric

behaviour

bifurcation

occurs

Deflection

Pressure

Pm

non-axisymmetric

behaviour

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112

thus innumerable elastic buckling modes. For each n(circumferential), there will be an

infinite number of longitudinal mode shapes. In practice number is finite, because the wave

lengths must be several times the wall thickness. It has been found that one can design

cylinders by considering the Three Simple Cases- as noted above.

The possibility is passible because, for a given vessel of radiusRand wall thickness e,

(1) Interstiffener buckling is governed by stiffener spacing,Ls.

(2) Overall buckling is governed in the main by the size of the stiffeners, and

(3) Stiffener tripping by the proportions of the stiffener.

Interstiffener BucklingInterstiffener Buckling

In this course, only the analysis of interstiffener buckling is dealt with. The other forms of

buckling are presented in the standard and in detail in BS Document PD 6550 Part:2 1989.

In this, the treatment assumes the following:-

(a) effect of stiffeners on pre-buckling stress in shell is neglected

(b) rotational and axial restraint to shell buckling due to stiffeners is neglected

(c) stiffeners remain circular during buckling.

Governing equation is given as:

( ){ } ( )[ ] ( )[ ]p

Ee

R n n

e

Rnm

RL

LR

RL=

+ ++

+

1

1

1

112 1

12 1

2

22 2

2

2

2 2

2 22

( )

This is an expression for pm- the Elastic Buckling Pressure)

pe

Rm=

(notice this is another membrane type equation, since is the circumferential buckling strain

and Eis a stress)

If this function is graphed, for a specific cylinder with the following dimensions;

R= 2700mm,Ls= 750mm,e= 25mm,E= 207,000N/mm2, = 0.3.

The equation above can be used with a range of n values to findpm. There is a minimum

value at pm= 7.31N/mm2and n = 14.

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The fuller version of this graph appears as Figure 3.6(3) of the standard. Values of n at

which the minimum occurs are given on this figure from the BS Code. The plot enables -the Circumferential Buckling Strain to be found and from this, pm.

The cusps on Fig 3.6(3) represent the points at which the mode associated with the minimum

pressure occurs. This figure can be been marked up to show the values of n corresponding

to the minimum nodal buckling pressures.

This value of pm is a theoretical value for infinitely long ring stiffened cylinders which are

perfectly circular.

In real shells, however, the Shape Imperfectionslower the collapse pressure. Also, as theshell thickness increases, or small diameter vessels are employed, plastic buckling occurs

and the collapse pressure is close to py, which is much less than pm.

Because of these effects, an empirical method is employed which uses pyand pmbut also a

large number of results from well documented tests.This graph has been prepared from data

from 700 well documented tests which have taken place over the last 100 years.

Actual Measured Collapse Pressurep

p

p

p

c m

~

These parameters minimised the scatter.

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The values were bounded, then divided by 1.5 to give the design curve in PD 5500. Its Fig

3.6(3) in PD 5500.

Curve (a) for cylinders and cones and (b) for spheres or cylinders to subject to axial stress.

The reason curve (b) is lower is because spheres and cylinders subject to axial stress are very

sensitive to imperfections.

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Example of Designing for External Pressure -Example of Designing for External Pressure - Interstiffener Buckling OnlyInterstiffener Buckling Only

Design data:

Design External PressureMean radius of shell

Stiffener spacing

Material for shell and stiffeners

Design stress

Effective yield,

Modulus of elasticity

pR

Lss

f

sf

E

= 6.9 bar= 2500 mm

= 1000 mm

= 1.4

= 165 N/mm2

= 231 N/mm2

= 2.07105N/mm2

This is trial and error procedure, so a range of values must be taken. Assume a thickness of

12.5mm as an initial value. This value would be at least the minimum required internal

pressure thickness for an equivalent internal pressure loading.

2R/e 400 300 250 200

e 12.5 16.7 20 25

py=sfe/R 1.155 1.540 1.848 2.310

forL/2R = 0.2 from Fig 3.6(2) 0.00088 0.0013 0.0018 0.0026

pm =Ee/R 0.911 1.798 2.981 5.382

K = pm/py 0.789 1.167 1.613 2.330

= p/py from Fig 3.6(3) 0.263 0.380 0.470 0.525

Allowable pressure,p (N/mm2) 0.306 0.585 0.869 1.213

From this table, 12.5mm and 16.7mm provide insufficient allowable pressures. Thicknesses

of 20mm and 25mm yield adequate results. Linearly interpolating in the table for an

allowable pressure of 0.69N/mm2gives a required thickness of 17.9mm.

Spheres under External PressureSpheres under External Pressure

Spheres are designed to prevent either yielding or buckling from occurring due to the applied

external pressure loading. As with the design of cylindrical shells, the aim is to ensure the

geometry can carrying sufficient pressure loading to prevent either of the two failure

mechanisms arising. Typical failures can be seen in the photographs shown overleaf.

The pressure to cause yield of a sphere is found from the membrane expression

psfe

Ryss=

2

(subscript yssdenotes yield of a spherical shell.)

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The theoretical pressure to cause elastic buckling is given by the following expression;

pEe

Re=

1 21 2

2

.

However, since this value is only the theoretical value, and shape imperfections must be

considered, then it is possible to useFigure 3.6(3)to evaluate the actual allowable pressure.

This is a non-dimensional curve which relates the theoretical elastic buckling pressure to the

actual buckling pressure with a safety factor of at least 50%. The curve axes values are non-

dimensionalised by dividing through by the yield pressure value.

Photographs of collapsed dished ends.

Hemispherical Ends under External PressureHemispherical Ends under External Pressure

As far as PD 5500 is concerned, hemi-spherical ends are designed as spherical shells.

Torispherical Ends under External PressureTorispherical Ends under External Pressure

Tori-spherical ends are designed as spherical shells of a mean radiusRequal to the external

crown radius.

Ellipsoidal Ends under External PressureEllipsoidal Ends under External Pressure

Ellipsoidal ends are designed as spherical shells with a mean radiusRequal to the maximum

crown radius i.e.D2/4h.

16426P108-116

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