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27 FLEXIBILITY, STRESS ANALYSIS AND SUPPORT DESIGN

# 27 FLEXIBILITY, STRESS ANALYSIS AND SUPPORT DESIGN

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111

3.27.2

AS 4041—1998

Flexibility

3.27.2.1 General The preferred method of absorbing displacements is by designing a
piping layout which has inherent flexibility to deflect in bending and torsion.
An alternative method, which may provide more economical plant layouts makes use of
expansion fittings. (See Clause 3.18).
3.27.2.2 Need for flexibility analysis A formal flexibility analysis (to satisfy stress
range requirements) is generally not required in piping which —
(a)

is a duplicate of piping with a satisfactory service record;

(b)

can readily be validated by comparison with previously analysed piping; or

(c)

is of uniform size, fixed at not more than two points, has no intermediate restraints,
and is non-critical piping within the limitations of the Equation 3.27.2.2.
Dy
< 208
(L − U )2

. . . 3.27.2.2

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where
D

=

outside diameter of pipe in millimetres

L

=

developed length of the pipe route, in metres

U

=

the length of the straight line joining the anchor points, in metres

y

=

resultant thermal expansion and terminal point movement to be
absorbed by the piping system, in millimetres

NOTES:
1

This equation is empirical, and cannot be relied on to give consistently conservative
results. It is not applicable to piping used under severe cyclic conditions (e.g. where
number of significant stress cycles exceeds 7 000). It is used with caution where L/U
> 2.5, where stress intensification factors exceed 5, or where displacements which are
not in the direction of the line joining the anchor points are a significant part of the
total displacement.

2

There is no assurance that
Equation 3.27.2.2 is satisfied.

3

The equation assumes that a pipe is free to expand without constraints due to weight
and a supporting system. A flexibility analysis may be required to provide data
(movements and loads) for the design of a support system which satisfies sustained
weight stress requirements.

4

This equation does not apply if flexible joints or expansion fittings are used in the
system.

terminal

reactions

will

be

acceptably

low

if

3.27.2.3 Self and cold spring When of sufficient magnitude, stresses caused by thermal
expansion, relax in the hot condition as a result of yielding or creep and reappear as
stresses of the opposite sign in the cold condition. This is known as self-springing.
The amount of self-springing which takes place can be reduced by the application of cold
spring during the erection of the piping. This is discussed in Appendix Q.
Where cold spring is applied, see Clause 3.27.6 for end reactions.
3.27.2.4 Balanced design All commonly used methods of piping flexibility analysis
assume elastic behaviour of the entire piping system. This assumption is sufficiently
accurate where plastic straining occurs at many points over relatively wide regions, but
fails to reflect the actual strain distribution in unbalanced systems where only a small
portion of a system undergoes plastic strain, or where, in piping operating in the creep
range, the strain distributions are uneven. In these cases, the weaker or higher stressed

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AS 4041—1998

112

portions will be subjected to strain concentrations due to elastic follow-up of the stiffer or
lower stressed portions.
Unbalance can be produced —
(a)

by the use of pipes with significantly different stiffness in series; or

(b)

in a system of uniform pipe size, by the use of a line configuration for which the
neutral axis or thrust line is situated close to the major portion of the line, with only
a very small offset portion of the line absorbing most of the expansion.

Conditions of this type should be avoided, particularly where materials of relatively low
ductility are used; if unavoidable, the effects may be mitigated by the judicious
application of cold spring and limit travel stops.
3.27.3 Stress analysis The flexibility of a piping system can be influenced
significantly by the geometry of fittings (e.g. bends, reducers) which change
cross-sectional shape under the action of bending moments and thus provide greater
flexibility than the same length of straight pipe. This action also increases the stress levels
in the fittings. In a flexibility analysis, these phenomena are covered by the use
‘flexibility’ and ‘stress intensification’ factors.

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Methods of flexibility analysis progress in complexity and accuracy through the
following:
(a)

Simple ‘structural’ methods which ignore the effects of pipe fittings.

(b)

More comprehensive methods which use similarity methods to evaluate terminal
reactions and stress levels from flexibility charts.

(c)

Rigorous methods based upon strain energy theory and the use of both flexibility
and stress intensification factors to give an accurate assessment of stresses in a
piping system.

Computer programs using algorithms based upon Item (c) are available for the analysis of
multi-anchor systems and the evaluation of specified stress limitations in accordance with
the methods of ANSI/ASME B31.3 and BS 806 which are covered by this Standard.
It is the designer’s responsibility to ensure that the method of analysis used for specific
piping system ensures that all categories of stress limitations are satisfied.
A piping system is analysed between points of constraint which control thermal expansion
in a predictable manner. In general these include one or more anchor points (stops
movement and rotation of pipe ends, e.g. pumps, vessels and heat exchangers) plus partial
restraints (stops less than six degrees of freedom of a pipe).
The system between points of constraint shall be treated as a whole and the effects of any
movement at anchor points and of any partial restraints shall be included.
Computer programs which carry out flexibility analysis to ANSI/ASME B31.3, including
flexibility and stress intensification factors but with design strengths as per Clause 3.11
also satisfy this Standard.
3.27.4
(a)

(b)

Data for stress analysis

Material properties
for—

The following data are given for stress analysis:

The more commonly used mechanical properties of materials

(i)

values of thermal expansion — refer Appendix E ;

(ii)

values of Young modulus — refer Appendix F; and

(iii)

Poisson ratio — the value may be taken as 0.3 for all metals at all
temperatures although a more accurate value may be used.

Dimensions

Nominal dimensions of piping and fittings shall be used.

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113

(c)

calculated in accordance with AS 1170.2 using permissible stress methods and static
analysis rules. To determine the drag factor, piping may be regarded as ‘a smooth
cylindrical shape’.

(d)

divided by 1.4). Unless otherwise agreed, piping shall be regarded as mechanical
components and treated as follows:
(i)

Class 1 and 2A piping shall be categorized with ‘Boilers, furnaces
incinerators, water heaters, and other equipment using combustible energy
sources or high-temperature energy sources, chimneys, flues, smokestacks,
vents and pressure vessels’, i.e. with the Cc2 factor equal to 2 for all piping
DN32 and greater. The exemptions listed in Clause 5.3.2 of
AS 1170.4 — 1993 are not applicable

(ii)

Class 2P and Class 3 piping shall be categorized as ‘ducts and piping
distribution systems’, i.e. with the Cc2 factor equal to 1.

3.27.5

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AS 4041—1998

Stress limitations

3.27.5.1 General The methods of evaluation of specified categories of stress in this
Section are based upon the methods of ANSI/ASME B31.3. They do not give any credit
for cold spring.
Alternative methods based upon the methods of BS 806 are given in Appendix R and
these make concessions on allowable hot stress for cold spring.
3.27.5.2 Displacement stress range The displacement stress range, being the stress
caused by the thermal expansion of the piping plus terminal movement shall satisfy
Equation 3.27.5.2.
fe < fa M

. . . 3.27.5.2

where
fe is the displacement stress range, in megapascals
fa is defined in 3.11.7
3.27.5.3

Sustained longitudinal stress

The sustained longitudinal stress (defined in Clause 3.11.5) shall satisfy Equation 3.27.5.3
fL =

P Do
+ fs < f M
4 tn

. . . 3.27.5.3

where
Do = nominal outside diameter of the pipe, in millimetres
fL = the sustained longitudinal stress, in megapascals
P = internal pressure, in megapascals
tn = the nominal thickness of the pipe, in millimetres
f

= the design strength of the material at the temperature under consideration (see
Appendix D), in megapascals

fs = the longitudinal stress cause by dead weight and other sustained loads, in
megapascals
M = piping class design factor (see Table 3.12.3)

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AS 4041—1998

114

3.27.5.4 Stress due to sustained occasional loads
satisfy Equation 3.27.5.4
fes = P

Do
4 tn

The sustained occasional stress shall

+ fs + fo < 1.33 f M

. . . 3.27.5.4

where
P, Do, tn, fs and f are defined in Clause 3.27.5.3
fo is the longitudinal flexural stress caused by occasional loads such as safety valve
thrust, wind and earthquake loads. Wind and earthquake loads need not be
considered concurrently, in megapascals.
Refer to Clause 3.11.6 for recommended limitations at flanged joints.

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3.27.5.5 Flexibility and stress intensification factors The factors known to apply to
components other than straight pipe shall be included in the analysis. The flexibility
factors (k) and stress intensification factors (i) shown in Table 3.27.5 and
Figure 3.27.5(A) Charts A and B may be used in the absence of more reliable data. The
values of the latter factor for tees are based on tests on equal outlet intersections and may
be used for unequal outlet intersections until more appropriate ones are developed, but it
is recommended that moments at these intersections be minimized.
3.27.5.6 Calculation of flexural stresses The magnitude of the flexural stresses fe, fs
and fo shall be calculated from Equation 3.27.5.6(1).
fx = ((fb)2 + 4 (ft)2)½

. . . 3.27.5.6(1)

where
fx = the stress being evaluated (fe, fs or fo as the case may be), in megapascals
fb = the resultant bending stress, in megapascals
ft = torsional stress, in megapascals
Mt × 10 3
=
2Z
Mt = torsional moment, in newton metres
Z = section modulus of pipe, in millimetres cubed.
For the calculation of —
fb and ft are evaluated using moments derived from the load cases which are
fs
the result of sustained load only using values of Young modulus for the
operating temperature.
fo
fb and ft are evaluated using moments derived from the load cases which are
the result of occasional loads only using values of Young modulus for the
operating temperature.
fe
fb and ft are evaluated using moments derived from the load cases involving
thermal expansion and displacement only, and values of Young modulus for
the as-installed temperature and coefficient of expansion, which is the
algebraic difference between the coefficients derived from Appendix E for the
design maximum and minimum temperatures.
The resultant bending stress (fb) for elbows and mitre bends shall be calculated from
Equation 3.27.5.6(2) with the moments as shown in Figure 3.27.5(B).
fb =
where
fb

1
[(iiMi)2 + (ioMo)2]½ × 10 3
Z

. . . 3.27.5.6(2)

= resultant bending stress, in megapascals

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115

AS 4041—1998

ii = in-plane stress intensification factor
Mi = in-plane bending moment, in newton metres
io = out-of-plane intensification factor
Mo = out-of-plane bending moment, in newton metres
Z = section modulus of pipe, in millimetres cubed.
The resultant bending stress (fb) for branch connections shall be calculated using
Equations 3.27.5.6(2) and 3.27.5.6(3) with moments as shown in Figure 3.27.5(C).
For header (legs 1 and 2) Equation 3.27.5.6(2) applies.
For branch (leg 3):
fb =

1
[(iiMi)2 + (ioMo)2]½ × 10 3
Zc

. . . 3.27.5.6(3)

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where
Ze =
=
r2 =
ts =
tnh =

effective section modulus of branch, in millimetres cubed
π(r2)2ts
mean branch cross sectional radius, in millimetres
effective branch wall thickness (lesser of tnh and tnb), in millimetres
thickness of pipe matching run of tee or header exclusive of reinforcing
elements, in millimetres
tnb = thickness of pipe matching branch, in millimetres
fb, ii, Mi, io, Mo have the meanings in Equation 3.27.5.6(2).

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AS 4041—1998

116

TABLE 3.27.5
FLEXIBILITY FACTOR AND CHARACTERISTIC, AND STRESS
INTENSIFICATION FACTORS FOR FITTINGS AND JOINTS

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Description

Flexibility
Flexibility Stress intensification factor
characteristic
factor (k)
ii*
io†
(h)

Welding elbow or pipe
bend (Notes 1, 2, 3, 6, 7)
re ≥ 114 Rb, r ≥ 1.5 t n

1.65
h

0.9
h

0.75
h

t nR

Closely spaced mitre
bend (Notes 1, 2, 3)
s < r (1 + tan θ)

1.52
h 0.83

0.9
h

0.90
h

Cot θt nS

Widely spaced mitre
bend (Notes 1, 2, 4)
s ≥ r (1 + tan θ)

1.52
h 0.83

0.90
h

0.9
h

1 + cot θ t n
2
r

0.75i o + 0.25

0.9
h

Welding tee to ANSI
B16.9 (Notes 1, 2)

Reinforced fabricated tee
(Notes 1, 2, 5)

Unreinforced fabricated
tee (Notes 1, 2)

Extruded welding tee
(Notes 1, 2)
re ≥ 0.25rb
tr < 1.5 tn

Welded-in contour insert
(Notes 1, 2)
rc > 0.25 r b
tr > 1.5 tn

1

1

0.75i0 + 0.25

0.9
h

Sketch

r2

2r 2

4.4

tn
r

t n + 0.5t p 5/2
tn r
3/2

1

1

1

0.75i o + 0.25

0.9
h

tn

0.75i o + 0.25

0.9
h

rc  tn

1 +
r  r2

0.75i o + 0.25

0.9
h

4.4 t n

r

r

(continued)

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117

Description

Flexibility
Flexibility Stress intensification factor
characteristic
factor (k)
ii*
io†
(h)

Branch welded-on fitting
(integrally reinforced)
(Notes 1, 2)

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AS 4041—1998

Sketch

1

0.9
h

0.9
h

3.3 t n

Butt-welded joint,
reducer, or welding neck
flange

1

1.0

1.0

Double-welded slip-on
flange

1

1.2

1.2

Fillet-weld joint (singlewelded), or singlewelded slip-on flange

1

1.3

1.3

Lapped flange (with
ANSI/ASME B16.9 lapjoint stub)

1

1.6

1.6

1

2.3

2.3

r

* In-plane, see Figure 3.27.5(B) and (C).
† Out-of-plane, see Figure 3.27.5(B) and (C).
NOTES TO TABLE 3.27.5:
1 For fittings and mitre bends, the flexibility factors (k) and stress-intensification factors (i) in the table apply to bending
in any plane and shall be not less than unity. Factors for torsion shall be unity. Both factors apply over the effective arc
length (shown by heavy centre lines in the sketches) for curved and mitre elbows, and to the intersection point for tees.
2 The values of k and i can be read directly from Figure 3.27.5(A) (Chart B) by entering the flexibility characteristic h
calculated from the equations given in this Table,
where
R = bend radius of welding elbow or pipe bend, in millimetres
r = mean radius of matching pipe bend, in millimetres
rb = mean radius of branch, in millimetres
re = radius of external contour of tee, in millimetres
s = mitre spacing at centre-line, in millimetres
tn = nominal wall thickness, in millimetres.
NOTE: Where the nominal wall thickness of a fabricated tee is greater than that of the adjoining pipes, this
thickness is to be maintained on each side of the branch, for a length not less than the pipe diameter.
tr = thickness of crotch of a tee, in millimetres
θ = one-half angle between adjacent mitre axes, in degrees.
3 Where flanges are attached to one or both ends, the values of h and i in the table are to be corrected by the factor C
given below, which can be read directly from Figure 3.27.5(A) (Chart B), for the calculated h:
One end flanged C = h1/6 ≥ 1
Both ends flanged C = h1/3 ≥ 1
4 Also includes single mitre joint.
5 When tp ≥ 1.5th, use h = 4tn/r.
6 Cast butt-welding elbows may have considerably heavier walls than those of the pipe with which they are used. Large
errors may be introduced unless the effect of these greater thicknesses is considered.

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AS 4041 — 1998

118

7 In large diameter, thin-wall elbows and bends, pressure can significantly affect the magnitude of the flexibility and
stress intensification factors. The values obtained from the table for the pressure effect, are to be corrected as follows —
(a) flexibility factor (k) to be divided by a number equal to 1 + [6(p/Ea)(r/tn)7/3(R/r)1/3]
(b) stress intensification factor (i) to be divided by a number equal to 1 + [3.25(p/Ea)(r/tn)5/2(R/r)2/3]
where
Ea = modulus of elasticity, in megapascals
p

= pressure, in megapascals.

3.27.6

Reactions

3.27.6.1 General Reaction forces and moments to be used in the evaluation of the
effects of piping displacements on connected plant and in the design of restraints shall be
obtained by modifying the reaction range (R), taken from the stress range computation, to
make allowance for cold spring.

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3.27.6.2 Maximum reactions for a simple system
may be derived as follows:

The reactions for a simple system

(a)

For a two anchor system without intermediate restraints The maximum
instantaneous values of reaction forces and moments may be estimated from
Equations 3.27.6(1) and 3.27.6(2).

(b)

For the design operating conditions

2  E
Rm = R 1 − C m
3  Ea

. . . 3.27.6(1)

where

(c)

Rm

= estimated instantaneous maximum reaction force or moment at design
operating temperature, in newton metres

R

= the range of reaction forces and moments used to determine the stress
range in Equation 3.27.5.6(1), in newton metres

C

= cold spring factor varying between zero for no cold spring and 1.0 for
100% cold spring. (The 2/3 factor is based upon experience which
show that specified cold spring cannot be assured even with elaborate
precaution.)

Em

= Young modulus at design operating temperature, in megapascals

Ea

= Young modulus at installation temperature, in megapascals

For the as-installed condition
Ra

= CR or C1.R whichever is greater

Ra

= estimated instantaneous reaction forces
installation temperature, in newton metres

. . . 3.27.6(2)

where

C1 = 1 −

and

moments

at

the

f Ea
f eE m

= estimated self spring or relaxation factor
= 0 when C1 is negative

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119

AS 4041 — 1998

where
f

= design strength at maximum metal temperature during the cycle, in
megapascals

fe = computed displacement
megapascals

stress

range

Equation 3.27.5.6(1),

in

3.27.6.3 Maximum reaction for complex systems For multi-anchor systems with
intermediate restraints, Equations 3.27.6(1) and 3.27.6(2) do not apply. Each case shall be
studied to estimate location, nature and extent of local overstrain, and its effect upon
stress distribution and reactions.
3.27.6.4 Reaction limits The calculated reactions shall not exceed the limits which the
connected equipment can safely sustain. Special consideration should be given to rotating
machinery.
3.27.6.5 Calculation of pipe movements Calculations of displacements and rotations at
specific locations may be required for the following purposes:
To check clearances from adjacent plant.

(b)

To design piping supports.

(c)

For the analysis of branch lines which are being designed separately.

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(a)

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AS 4041 — 1998

FIGURE 3.27.5(A)

120

FLEXIBILITY AND STRESS-INTENSIFICATION FACTORS

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121

FIGURE 3.27.5(B)

AS 4041 — 1998

MOMENTS IN BENDS — NOTATION AND SIGN CONVENTION

FIGURE 3.27.5(C) MOMENTS IN BENDS AND BRANCHES —
NOTATION AND SIGN CONVENTION

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