Barber J.R. Intermediate Mechanics of Materials

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9.6 Shell transition regions 435

cylinder

;

cone

2t cos

(9.31)

and axial equilibrium is satisﬁed.

To restore radial equilibrium, we would need to

add a force F

per unit circumference as in Figure 9.11 (c), where

cone

t sin

pa tan

. (9.32)

In the actual shell, this force is not present and the transition zone therefore tends

to pull radially inwards under the inﬂuence of the membrane stresses until addi-

tional circumferential stresses are developed sufﬁcient to oppose the unbalanced ra-

dial force.

The bending stresses near the discontinuity can be estimated by ﬁrst assuming

that the force F

is provided. In this case, there will be some bending associated with

the discontinuity in u

between the cylinder and the cone, but it will be small. We

then remove the force F

, which is equivalent to applying an inward concentrated

radial load of the same magnitude. We have already solved this ‘corrective’ problem

in §9.5 and the maximum bending moment produced is

(0) =

pa tan

corresponding to a maximum bending stress of

max

3pa tan

Unless

is very small, this will be substantially larger than the membrane stresses,

which are of order pa/t.

One interpretation of the function of the toroidal transition is that is serves to dis-

tribute the force F

of equation (9.32) over the toroidal region, thereby reducing the

maximum bending moment it produces. Thus, we might regard the toroidal segment

as a sequence of inﬁnitesimal changes of

at each of which an inﬁnitesimal radial

force is required. As long as

is not too large, the change in

per unit axial length

is d

/dz= 1/R and the required inward radial force corresponds to pressure

p(z) = −

, (9.33)

where the negative sign reﬂects the fact that the required distributed force is equiv-

alent to an external pressure. The bending moment due to this load can be written

down as

(z) = −

(

|z −z

′

|)dz

′

, (9.34)

using equation (9.30). This method can also be used for transitions that are not

toroidal (i.e. that have variable radius R) always provided the maximum value of

As indeed it must be, since this condition was used to determine equations (9.31).

436 9 Axisymmetric Bending of Cylindrical Shells

is not too large. From a design perspective, a good rule of thumb is that a tran-

sition will have a signiﬁcant effect in moderating bending stresses if its length (e.g.

the arc length R

for the toroidal transition of Figure 9.10) is about twice the decay

length l

9.6.2 Reinforcing r ings

If a discontinuous change in slope

is unavoidable in a shell design, the stress

concentration at the transition can be reduced by using a reinforcing ring, as shown

in Figure 9.12 (a). The ring resists the tendency for the shell to pull in locally at the

discontinuity under the inﬂuence of the axial loading of the shell and thereby reduces

both the circumferential membrane stress

and the local bending stresses.

reinforcing

ring

u , F

θ, M

(a) (b)

Figure 9.12: (a) Cone/cylinder transition reinforced by a ring; (b) deformation of the

ring due to a radial force and a moment

A rigorous analysis of the geometry of Figure 9.12 (a) requires that the ring be

treated as an elastic member whose rotation

and radial expansion u

are functions

of applied moment and radial load. For example, if the ring has a rectangular cross

section b×c as shown in Figure 9.12 (b), elementary calculations

show that

Ebc

;

12M

, (9.35)

where F

are respectively the radial force per unit circumference and the moment

per unit circumference.

However, if the ring is fairly substantial compared with the shell (b,c ≫t), it can

reasonably be approximated as rigid, in which case both shell segments have built in

end conditions [u

(0)=

(0)= 0].

Reinforcing rings can also be used to reduce the circumferential membrane stress

in a cylindrical shell by restraining radial expansion. Figure 9.13 shows the deformed

shape of a shell stiffened in this way.

see Problems 9.12, 9.13 below.

9.7 Thermal stresses 437

internal pressure p

Figure 9.13: Thin cylindrical shell stiffened by rings

To be effective, the rings must be sufﬁciently closely spaced to ensure that the

maximum radial displacement between rings is signiﬁcantly lower than the unre-

stricted membrane displacement u

. It can be shown

that this condition is satisﬁed

s< 1 and hence the spacing s< 0.78

√

at.

9.7 Thermal stresses

Thin-walled vessels are often used for chemical processes and as components of

thermal machines (e.g. boilers and heat exchangers). In these applications, additional

loading will generally result from temperature gradients in the shell. Two distinct

forms of thermal loading can be identiﬁed in an axisymmetric shell:-

(i) If there is heat ﬂow through the wall thickness (e.g. as in a heat exchanger tube)

there will be a radial temperature gradient which will induce additional shell

bending stresses.

(ii) Even if the temperature is uniform through the thickness, axial temperature gra-

dients will affect the membrane radial displacement u

in equation (9.24) and

hence change the bending stress distribution.

These effects can be characterized by a temperature function of the form

∆

T (

,z) = T

(z) +

(z) . (9.36)

With this notation, the outward heat ﬂux per unit area through the shell wall is

q(z) = −K

∂

∂η

= −KT

(z) , (9.37)

where K is the thermal conductivity of the shell material.

We recall from equations (1.14–1.16) that the effect of a local increase of temper-

ature

∆

T is to produce normal strains in all three directions of magnitude

α∆

T . We

can therefore generalize the preceding derivations by adding in these thermal strains

at each point where Hooke’s law was invoked — notably in equations (9.7, 9.10).

The new version of (9.7) is

The problem is analogous to Problem 7.23.

438 9 Axisymmetric Bending of Cylindrical Shells

= e

θθ

−

νσ

α∆

−

νσ



12(M

−

)



The condition that u

be independent of

then yields

−

(9.38)



−

νσ



a . (9.39)

Equation (9.10) is generalized to

−

νσ

θθ

(

−

νσ

)



12(M

−

)



. (9.40)

Equating coefﬁcients of

and using (9.38), we then have

(

−

νσ

)

(9.41)

= −D



(1 +



. (9.42)

A similar derivation to that in §9.4 then leads to the results

νσ

−E

(9.43)

V = −D



(1 +

)



(9.44)

+ 4

= 4

−

(1 +

)

, (9.45)

where the membrane displacement is

−

νσ

a . (9.46)

We see from these results that the term T

(z) in equation (9.36) affects the bend-

ing solution only through the membrane displacement u

. Once this has been cal-

culated, the solution procedure is identical to that described in §9.4. In particular,

bending stresses will be induced only at ends or discontinuities if T

is a constant or

a linear function of z.

By contrast, the through-thickness temperature gradient T

(z) inﬂuences both

the governing equation (9.45) and the moment-curvature relation (9.42) and it will

always give rise to thermally induced bending stresses.

9.8 The ASME pressure vessel code 439

Example 9. 3

A long cylindrical heat exchanger tube of radius a and thickness t transmits a con-

stant heat ﬂux q

per unit area throughout its length. The temperature is independent

of the axial coordinate z. Find the maximum bending stress in the tube if the material

has Young’s modulus E, Poisson’s ratio

, thermal conductivity K, and coefﬁcient of

thermal expansion

. There is no other loading of the shell.

In this case, we have

(z) = −

from equation (9.37) and there is no axial temperature variation (T

=0), so the mem-

brane displacement u

=0, from (9.46).

Substituting for T

into (9.45), we ﬁnd that the right hand side of this equation is

zero. It follows that u

= 0 is a particular integral of this equation and it is also the

complete solution, since the tube is ‘long’, i.e the ends are far enough away not to

inﬂuence the solution. We then have

(1 +

from (9.42) and

(1 +

)

12K

from (9.38).

Substituting for D and simplifying, we have

= M

12K(1 −

)

and

max

2K(1 −

)

9.8 The ASME pressure vessel code

The consequences of pressure vessel failure can be extremely serious, particularly

where toxic and/or high temperature ﬂuids are involved. Designs are therefore usu-

ally based on standardized procedures described in the American Society of Me-

chanical Engineers’ pressure vessel code.

Conformity to the code is also usually a

requirement for obtaining liability insurance for the design. The recommendations

of the code have evolved over many years, but in the area of stress analysis they are

largely based on shell bending calculations such as those described in the present

chapter.

ASME Boiler and Pressure Vessel Code — An American National Standard, American

Society of Mechanical Engineers, New York.

440 9 Axisymmetric Bending of Cylindrical Shells

All common shell conﬁgurations are described in the code, but it is dangerous to

use it as a substitute for thinking through the calculation process for a design. This

is particularly true if the design is in any respect unconventional. It is a sobering

thought that many code revisions have their origins in serious accidents involving

vessels that were considered by their designers and insurers to be within the code at

that time.

9.9 Summary

In this chapter, we have developed methods for determining the bending stresses

in cylindrical shells subjected to axisymmetric radial, axial or thermal loading. The

resulting governing equation has the same form as that for the bending of a beam on

an elastic foundation and, as in that case, bending stresses are generally associated

with discontinuities of geometry or loading.

Shell bending effects are important at the junction between two shells or at a point

of localized loading, such as a support. Bending stresses are typically comparable

with (and additive to) the membrane stresses and hence give stress concentration

factors of the order of 2.

The effects are extremely localized, with an axial decay length signiﬁcantly

smaller than the shell radius. In design, the ﬁrst step is to calculate the character-

istic decay length given by equation (9.22). This enables us to determine whether a

particular feature can be analyzed in isolation (as is usually the case).

Bending stresses at discontinuities can be reduced by using appropriate geomet-

ric transitions. The ASME Pressure Vessel Code provides detailed guidance for these

cases, but a good rule of thumb is to use a transition whose axial length is twice the

characteristic decay length.

Further reading

A.P. Boresi, R.J. Schmidt, and O.M. Sidebottom (1993), Advanced Mechanics of

Materials, John Wiley, New York, 5th edn., §10.7.

R.D. Cook and W.C. Young (1985), Advanced Mechanics of Materials, Macmillan,

New York, §§6.1–6.4.

S.P. Timoshenko and S. Woinowsky-Krieger (1959), Theory of Plates and Shells,

McGraw-Hill, New York, 2nd edn., §§114–119.

Problems 441

Problems

Sections 9.1–9.4

9.1. Develop a more rigorous version of the analysis of equations (9.7–9.8) noting

that

(i) The circumferential strain

θθ

a +

and this expression should be used in place of u

/a in the left-hand side of equa-

tion (9.7).

(ii) The membrane stresses

will generate Poisson’s ratio strains

∂

∂η

= −

(

)

Show that this leads to the modiﬁed equation

−

) = −

(1 +

)

12a

in place of (9.8), and explain why this correction is negligible in view of the assump-

tion t ≪ a.

9.2. Figure P9.2 shows the design for the attachment of an 8 in diameter pipe to

a cylindrical tank. The end of the pipe and the aperture in the tank are stiffened

sufﬁciently for this region to afford a rigid (built-in) support to the end of the pipe.

Find the maximum tensile stress in the pipe when the internal pressure is 300 psi

(E = 30×10

psi,

=0.3).

diameter

thick

Figure P9.2

442 9 Axisymmetric Bending of Cylindrical Shells

9.3. The vertical cylindrical water tank of Figure P9.3 is made from galvanized steel

(E = 30×10

psi,

= 0.3) of thickness 1/16 inch. The bottom of the tank is a ﬂat

steel plate which is sufﬁciently rigid in extension to prevent radial displacement at

the bottom, but which affords no restraint against rotation,

Find the location and magnitude of the maximum radial displacement of the tank

when it is just full of water of density 62.5 lbs per ft

. Find also the maximum tensile

stress in the tank.

Figure P9.3

9.4. A cylindrical pressure vessel of radius 700 mm and wall thickness 8 mm has

rigid end plates that can be considered to give built-in end conditions to the shell.

Find the maximum tensile stress, if the internal pressure is 1.2 MPa and the material

is steel (E = 210 GPa,

=0.3).

9.5*. The analysis of the bending of circular plates shows that a moment M

per unit

circumference applied to the edge of a plate of radius a and thickness t, as shown in

Figure P9.5, produces a local rotation

D(1 +

)

where the stiffness of the plate D is deﬁned by equation (9.14). A similar plate,

simply supported around the edge and loaded by a uniform pressure p

experiences

an edge rotation

8D(1 +

)

See S.P. Timoshenko and S. Woinowsky-Krieger (1959) Theory of Plates and Shells,

McGraw-Hill, New York, 2nd edn., §15, §16.

62.5 lb/ft

4 ft

3 ft

Problems 443

, θ

Figure P9.5

Two such plates are used to close the ends of a cylindrical shell of the same

radius, thickness and material properties. Find an expression for the bending moment

at the shell/plate junction when the vessel is subjected to an internal pressure p

Assume that the end plates completely resist radial expansion at the junction.

9.6. A rigid conical wedge is driven into a cylindrical tube by an axial force F

, as

shown in Figure P9.6. The wedge angle

≪1 and the coefﬁcient of friction at the

contact point is f .

Figure P9.6

Assuming that contact occurs only at the end of the tube as shown, ﬁnd an expres-

sion for the radial displacement of the tube u

as a function of z and F

. Hence show

that there is a minimum angle

for which contact is restricted to the end and ﬁnd

it’s value. The cylinder has a radius a and thickness t and the material has Young’s

modulus E and Poisson’s ratio

Sections 9.5, 9.6

9.7. A compressed air storage tank comprises a cylindrical segment and two hemi-

spherical ends, each of radius 300 mm and wall thickness 4 mm. Find the maximum

tensile stress in the tank when the internal pressure is 25 bar (1 bar = 100 kPa, E =210

GPa,

=0.3).

444 9 Axisymmetric Bending of Cylindrical Shells

9.8. A toroidal shell of radius R is joined to a cylindrical shell of radius a, as shown

in Figure P9.8. Both segments have thickness t and the shell is loaded by an internal

pressure p

. Find the required value of R as a function of a,t, p

, if the maximum

axial stress

max

is to be just equal to the membrane stress

in the cylindrical

segment, distant from the transition.

internal pressure p

Figure P9.8

9.9. Find the maximum tensile stress for the cylindrical/toroidal transition of Figure

P9.9 if the vessel is loaded by an internal pressure of 800 kPa. The shell thickness is

7 mm and the material is steel (E =210 GPa,

=0.3).

7 mm

250 mm

800 mm

Figure P9.9

9.10. To reduce the bending stresses at a transition between a cylindrical shell of

radius a and a hemispherical shell, a transition region is proposed in which the radius

as shown in Figure P9.10.