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Basavaraju2004.PDF
Source: PIPING HANDBOOK
CHAPTER B4
STRESS ANALYSIS OF
PIPING SYSTEMS
C. Basavaraju, P.E.
Senior Engineering Specialist
Bechtel Power Corporation
Frederick, Maryland
William Saifung Sun, P.E.
Senior Engineering Specialist
Bechtel Power Corporation
Frederick, Maryland
Piping stress analysis is a discipline which is highly interrelated with piping layout
(Chap. B3) and support design (Chap. B5). The layout of the piping system should be
performed with the requirements of piping stress and pipe supports in mind (i.e.,
sufficient flexibility for thermal expansion; proper pipe routing so that simple and
economical pipe supports can be constructed; and piping materials and section
properties commensurate with the intended service, temperatures, pressures, and
anticipated loadings). If necessary, layout solutions should be iterated until a
satisfactory balance between stresses and layout efficiency is achieved. Once the piping
layout is finalized, the piping support system must be determined. Possible support
locations and types must be iterated until all stress requirements are satisfied and
other piping allowables (e.g., nozzle loads, valve accelerations, and piping movements)
are met. The piping supports are then designed (Chap. B5) based on the selected
locations and types and the applied loads.
This chapter discusses several aspects of piping stress analysis. The discussion is
heavily weighted to the stress analysis of piping systems in nuclear power plants,
since this type of piping has the most stringent requirements. However, the discussion
is also applicable to the piping systems in ships, aircraft, commercial buildings,
equipment packages, refrigeration systems, fire protection piping, petroleum
B.107
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STRESS ANALYSIS OF PIPING SYSTEMS
B.108
GENERIC DESIGN CONSIDERATIONS
refineries, and so on. Each of these types of piping must meet the requirements of its
applicable code.
FAILURE THEORIES, STRESS CATEGORIES,
STRESS LIMITS, AND FATIGUE
Failure Theories
The failure theories most commonly used in describing the strength of piping systems
are the maximum principal stress theory and the maximum shear stress theory (also
known as the Tresca criterion).
The maximum principal stress theory forms the basis for piping systems governed
by ASME B31 and Subsections NC and ND (Classes 2 and 3) of Section III of the
ASME Boiler and Pressure Vessel Codes. This theory states that yielding in a piping
component occurs when the magnitude of any of the three mutually perpendicular
principal stresses exceeds the yield strength of the material.
The maximum shear stress theory is more accurate than the maximum principal
stress theory for predicting both yielding and fatigue failure in ductile metals. This
maximum shear stress theory forms the basis for piping of Subsection NB (Class 1) of
ASME Section III.1
The maximum shear stress at a point τmax is defined as one-half of the algebraic
difference between the largest and the smallest of the three principal stresses σ1, σ2,
and σ3. If σ1 > σ2 > σ3 (algebraically), then τmax = (σ1 - σ3)/2. The maximum shear stress
theory states that failure of a piping component occurs when the maximum shear
stress exceeds the shear stress at the yield point in a tensile test. In the tensile test, at
yield, σ1 = Sy (yield stress), σ2 = σ3 = 0. So yielding in the component occurs when
(B4.1)
Equation (B4.1) has an unnecessary operation of dividing both sides by 2 before
comparing them. For the sake of simplicity, a stress defined as 2τmax and equal to σmax
-σmin of the three principal stresses has been used for Class 1 piping. This stress is
called the equivalent intensity of combined stresses, or stress intensity. Thus the stress
intensity S is directly comparable to the tabulated yield stress values Sy from tensile
tests with some factor of safety.
Stress Categories
There are various failure modes which could affect a piping system. The piping engineer
can provide protection against some of these failure modes by performing stress analysis
according to the piping codes. Protection against other failure modes is provided by
methods other than stress analysis. For example, protection against brittle fracture is
provided by material selection. The piping codes address the following failure modes:
excessive plastic deformation, plastic instability or incremental collapse, and highstrain–low-cycle fatigue. Each of these modes of failure is caused by a different kind
of stress and loading. It is necessary to place these stresses into different categories
and set limits to them.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.109
The major stress categories are primary, secondary, and peak. The limits of these
stresses are related to the various failure modes as follows:
1. The primary stress limits are intended to prevent plastic deformation and bursting.
2. The primary plus secondary stress limits are intended to prevent excessive plastic
deformation leading to incremental collapse.
3. The peak stress limit is intended to prevent fatigue failure resulting from cyclic
loadings.
Primary stresses which are developed by the imposed loading are necessary to
satisfy the equilibrium between external and internal forces and moments of the piping
system. Primary stresses are not self-limiting. Therefore, if a primary stress exceeds
the yield strength of the material through the entire cross section of the piping, then
failure can be prevented only by strain hardening in the material. Thermal stresses
are never classified as primary stresses. They are placed in both the secondary and
peak stress categories.
Secondary stresses are developed by the constraint of displacements of a structure.
These displacements can be caused either by thermal expansion or by outwardly
imposed restraint and anchor point movements. Under this loading condition, the
piping system must satisfy an imposed strain pattern rather than be in equilibrium
with imposed forces. Local yielding and minor distortions of the piping system tend
to relieve these stresses. Therefore, secondary stresses are self-limiting. Unlike the
loading condition of secondary stresses which cause distortion, peak stresses cause
no significant distortion. Peak stresses are the highest stresses in the region under
consideration and are responsible for causing fatigue failure. Common types of peak
stresses are stress concentrations at a discontinuity and thermal gradients through a
pipe wall.
Primary stresses may be further divided into general primary membrane stress,
local primary membrane stress, and primary bending stress. The reason for this division
is that, as will be discussed in the following paragraph, the limit of a primary bending
stress can be higher than the limit of a primary membrane stress.
Basic Stress Intensity Limits
The basic stress intensity limits for the stress categories just described are determined
by the application of limit design theory together with suitable safety factors.
The piping is assumed to be elastic and perfectly plastic with no strain hardening.
When this pipe is in tension, an applied load producing a general primary membrane
stress equal to the yield stress of the material Sy results in piping failure. Failure of
piping under bending requires that the entire cross section be at this yield stress. This
will not occur until the load is increased above the yield moment of the pipe multiplied
by a factor known as the shape factor of the cross section. The shape factor for a
simple rectangular section in bending is 1.5.
When a pipe is under a combination of bending and axial tension, the limit load
depends on the ratio between bending and tension. In Fig. B4.1, the limit stress at the
outer fiber of a rectangular bar under combined bending and tension is plotted against
the average tensile stress across the section. When the average tensile stress Pm is zero,
the failure bending stress is 1.5 Sy. When Pm alone is applied (no bending stress Pb),
failure stress is yield stress Sy.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.110
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.1 Limit stress for combined tension and
bending (rectangular section). (ASME,“Criteria.” 1
Courtesy of ASME.)
It also can be seen in Fig. B4.1 that a design limit of 2/3 Sy for general primary
membrane stress Pm and a design limit of Sy for primary membrane-plus-bending
stress Pm + Pb provide adequate safety to prevent yielding failure.
For secondary stresses, the allowable stresses are given in terms of a calculated
elastic stress range. This stress range can be as high as twice the yield stress. The
reason for this high allowable stress is that a repetitively applied load which initially
stresses the pipe into plastic yielding will, after a few cycles, “shake it down” to
elastic action.
This statement can be understood by considering a pipe which is strained in tension
to a point e1 somewhat beyond its yield strain, as shown in Fig. B4.2. The calculated
elastic stress at this point would be equal to the product of the modulus of elasticity E
and the strain ε1, or S1 = Eε1. The path OABC is considered as cycling the strain from
0 to ε1 (loading) and back to 0 (unloading). When the pipe is returned to its original
position O, it will retain a residual compressive stress of magnitude S1 - Sy. On each
subsequent loading cycle, this residual compression must be overcome before the pipe
can go into tension; thus the elastic range has been extended by the value S1 - Sy.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.111
FIGURE B4.2 Strain history beyond yield. (ASME, “Criteria.”1
Courtesy of ASME.)
Therefore, the allowable secondary stress range can be as high as 2Sy when S1 =
2Sy. When S1 > 2Sy, the pipe yields in compression and all subsequent cycles generate
plastic strain EF. For this reason 2Sy is the limiting secondary stress which will shake
down to purely elastic action.
Fatigue
As mentioned previously, peak stresses are the highest stresses in a local region and
are the source of fatigue failure. The fatigue process may be divided into three stages:
crack initiation resulting from the continued cycling of high stress concentrations,
crack propagation to critical size, and unstable rupture of the remaining section.
Fatigue has long been a major consideration in the design of rotating machinery,
where the number of loading cycles is in the millions and can be considered infinite for
all practical purposes. This type of fatigue is called high-cycle fatigue. High-cycle fatigue
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STRESS ANALYSIS OF PIPING SYSTEMS
B.112
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.3 Typical relationship among stress, strain, and cycles to failure.
(ASME, “Criteria.”1 Courtesy of ASME.)
involves little or no plastic action. Therefore, it is stress-governed. For every material, a
fatigue curve, also called the S–N curve, can be generated by experimental test2 which
correlates applied stress with the number of cycles to failure, as shown in Fig. B4.3. For
high-cycle fatigue, the analysis is to determine the endurance limit, which is the stress
level that can be applied an infinite number of times without failure.
In piping design, the loading cycles applied seldom exceed 105 and are frequently
only a few thousand. This type of fatigue is called low-cycle fatigue. For low-cycle
fatigue, data resulting from experimental tests with stress as the controlled variable
are considerably scattered. These undesirable test results are attributable to the fact
that in the low-cycle region the applied stress exceeds the yield strength of the material,
thereby causing plastic instability in the test specimen.
However, when strain is used as the controlled variable, the test results in this lowcycle region are consistently reliable and reproducible.
As a matter of convenience, in preparing fatigue curves, the strains in the tests are
multiplied by one-half the elastic modulus to give a pseudostress amplitude. This
pseudostress is directly comparable to stresses calculated on the assumption of elastic
behavior of piping. In piping stress analysis, a stress called the alternating stress Salt is
defined as one-half of the calculated peak stress. By ensuring that the number of load
cycles N associated with a specific alternating stress is less than the number allowed
in the S–N curve, fatigue failure can be prevented. However, practical service conditions
often subject a piping system to alternating stresses of different magnitudes. These
changes in magnitude make the direct use of the fatigue curves inapplicable since the
curves are based on constant-stress amplitude. Therefore, to make fatigue curves
applicable for piping, it is necessary to take some other approach.
One method of appraising the fatigue failure in piping is to assume that the cumulative
damage from fatigue will occur when the cumulative usage factor U equals unity, i.e.,
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.113
(B4.2)
where Ui = usage factor at stress level i
ni = number of cycles operating at stress level i
Ni = number of cycles to failure at stress level i per fatigue curve
CLASSIFICATION OF LOADS, SERVICE LIMITS,
AND CODE REQUIREMENTS
Classification of Loads
Primary loads can be divided into two categories based on the duration of loading.
The first category is sustained loads. These loads are expected to be present throughout
normal plant operation. Typical sustained loads are pressure and weight loads during
normal operating conditions. The second category is occasional loads. These loads
are present at infrequent intervals during plant operation. Examples of occasional
loads are earthquake, wind, and fluid transients such as water hammer and relief
valve discharge.
In addition to primary loads, there are expansion loads. Expansion loads are those
loads due to displacements of piping. Examples are thermal expansion, seismic anchor
movements, thermal anchor movements, and building settlement.
Service Limits
Service levels and their limits are defined for nuclear power plant safety-related piping
by the ASME Boiler and Pressure Vessel Code, Section III.3 They are described in the
following list:
1. Level A service limits. The piping components or supports must satisfy these sets
of limits in the performance of their specified service function. Examples of level
A loadings are operating pressure and weight loadings.
2. Level B service limits. The piping component or support must withstand these
loadings without damage requiring repair. Examples of level B loadings are fluid
transients such as water hammer and relief valve discharge, and operating-basis
earthquake (OBE), defined as the maximum likely earthquake postulated to occur
during plant design life or one-half of the safe shutdown earthquake (see definition
below), whichever is higher.
3. Level C service limits. The occurrence of stress up to these limits may necessitate
the removal of the piping component from service for inspection or repair of
damage. An example of level C loading is the combination of fluid transient
loads occurring simultaneously with the operating-basis earthquake.
4. Level D service limits. These sets of limits permit gross general deformations with
some consequent loss of dimensional stability and damage requiring repair, which
may require removal of the piping component from service. An example of level
D loading is the loading associated with a loss-of-coolant accident or a
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STRESS ANALYSIS OF PIPING SYSTEMS
B.114
GENERIC DESIGN CONSIDERATIONS
safe-shutdown earthquake (SSE), which is defined as the maximum possible
earthquake postulated to occur at the site of the plant at any time.
Code Requirements
There are various ASME and ANSI codes which govern the stress analysis of diferent
kinds of pressure piping. These codes contain basic reference data, formulas, and
equations necessary for piping design and stress analysis.
Each power plant is committed to a particular edition of a code for different types
of piping. For example, the nuclear Class 1, 2, and 3 piping of a power plant may be
committed to comply with the ASME Boiler and Pressure Vessel Code, Section III,
1974 edition, while the nonnuclear piping may be committed to ANSI B31.1 Power
Piping Code, 1973 edition.
The following sections provide summaries of the ASME and ANSI codes.
ASME Boiler and Pressure Vessel Code, Section III, Subsection NB.3 This subsection
provides the code requirements of nuclear piping designated as Class 1. The loadings
required to be considered for this subsection are the effects of pressure, weight (live and
dead loads), thermal expansion and contraction, impact, earthquake, and vibrations.
The stress limits which must be met are as follows:
1. Primary stress intensity. The primary stress intensity must meet the following
requirement:
(B4.3)
where B1, B2 = primary stress indices for specific piping components under
investigation
P = design pressure, psi
Do = outside diameter of pipe, in
t = nominal wall thickness, in
Mi = resultant moment due to combination of design mechanical loads,
in · lb
I = moment of inertia, in4
kSm = 1.5Sm for service level A; 1.8Sm for service level B but not greater
than 1.5Sy; 2.25Sm for service level C but not greater than 1.8Sy;
and 3.0Sm for service level D but not greater than 2.0Sy
Sm = allowable design stress intensity, psi
Sy = yield strength value taken at average fluid temperature under
consideration, psi
2. Primary plus secondary stress intensity range. The following equations are used to
evaluate a stress range as the piping system goes from one service load set (pressure,
temperature, and moment) to any other service load set which follows in time. For
each specified pair of load sets, the stress range Sn is calculated:
(B4.4)
where C1, C2, C3 = secondary stress indices for specific component under consideration
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.115
P0 = range of service pressure, psi
Mi = resultant range of moment, in · lb
Eab = average modulus of elasticity of two sides of a gross structural
discontinuity or material discontinuity at room temperature , psi
αa, αb = coefficient of thermal expansion on side a or b of gross structural
discontinuity or material discontinuity at room temperature , in/
(in · °F)
Ta, Tb = range of average temperature on side a or b of gross structural
discontinuity or material discontinuity, °F
And Sn has the following limit: Sn ≤ 3Sm.
If this requirement is not met for all pairs of load sets, then the piping
component may still be qualified by using the simplified elastic-plastic
discontinuity analysis described below; otherwise, the stress analyst may proceed
to the fatigue analysis.
3. Simplified elastic-plastic discontinuity analysis. If Sn > 3Sm for some pairs of load
sets, a simplified elastic-plastic analysis may be performed if the thermal stress
ratchet is not present. This analysis is required only for the specific load sets
which exceeded 3Sm. The following two equations must be satisfied:
(B4.5)
where Se = nominal value of expansion stress, psi
= resultant range of moments due to thermal expansion and thermal
anchor movements, in · lb
Mi = resultant range of moment excluding moments due to thermal expansion
and thermal anchor movements, in · lb
C3 = stress index for specific component under consideration
For later editions of the code, if Sn > 3Sm, the thermal stress ratchet must be
evaluated and demonstrated to be satisfactory before a simplified elastic-plastic
discontinuity analysis can be done. This ratchet is a function of the |∆T1| (see
definition below) range only. The following requirement must be met:
(B4.6)
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STRESS ANALYSIS OF PIPING SYSTEMS
B.116
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.4 Decomposition of temperature distribution range. (Figure NB-3653.2(b)-1, Section
III, Division 1, ASME B & PV Code, 1989. Courtesy of ASME.)
4. Peak stress intensity range and fatigue analysis. For each specified loading
condition, peak stress is calculated as follows:
(B4.7)
where K1, K2, K3 = local stress indices for specific component under consideration
v = Poisson’s ratio of material
= absolute value of range of temperature difference between
temperature of outside surface and inside surface of pipe wall,
assuming moment generating equivalent linear temperature
distribution, °F (see Fig. B4.4)
= absolute value of range of that portion of nonlinear thermal
gradient through wall thickness not included in ∆T1, °F (see
Fig. B4.4)
For each Sp, an alternating stress intensity Salt is determined by
(B4.8)
m, n = material parameters given in Table B4.1
The alternating stress intensities are used to evaluate the cumulative effect of
the stress cycles on the piping system. This evaluation is performed as follows:
a. The number of times each stress cycle of type 1, 2, 3, etc., is repeated during
the life of the system shall be called n1, n2, n3, and so on. Cycles shall be
superimposed such that the maximum possible peak stress ranges are
developed.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.117
TABLE B4.1 Values of m, n, and Tmax for Various
Classes of Permitted Materials
Source: Table NB-3228.5(b)-1, Section III, Division 1, ASME B
& PV Code, 1998. (Courtesy of ASME.)
Note: (°F - 32)/1.8 = °C).
b. For each type of stress cycle, determine the alternating stress intensity Salt.
c. For each value of Salt, use the applicable design fatigue curve from the code to
determine the maximum number of cycles permitted if this were the only cycle
occurring. These numbers shall be designated N1, N2, N3, and so on.
d. For each type of stress cycle, calculate the usage factor:
e. The cumulative usage factor U is the sum of the individual usage factors:
ASME Boiler and Pressure Vessel Code, Section III, Subsections NC and ND.3 These
two subsections give the code requirements of nuclear piping designated as Class 2
and Class 3, respectively. The loadings required to be considered for Subsections NC
and ND are the effects of pressure, weight, other sustained loads, thermal expansion
and contraction, and occasional loads. The stress limits to be met are as follows:
1. Stresses due to sustained loads. The calculated stresses due to pressure, weight,
and other sustained mechanical loads must meet the allowable 1.5Sh, that is,
(B4.9)
where P
Do
t
Z
MA
=
=
=
=
=
internal design pressure, psi
outside diameter of pipe, in
nominal wall thickness, in
section modulus of pipe, in3
resultant moment loading on cross section due to weight and other
sustained loads, in · lb
Sh = basic material allowable stress at design temperature, psi
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.118
2. Stresses due to occasional loads. The calculated stress due to pressure, weight, other
sustained loads, and occasional loads must meet the allowables as follows:
(B4.10)
where MB = resultant moment loading on cross section due to occasional loads,
such as thrusts from relief and safety valves, loads from pressure and
flow transients, and earthquake, if required. For earthquake, use only
one-half the range. Effects of anchor displacement due to earthquake
may be excluded if they are included under thermal expansion
Pmax = peak pressure, psi
kSh = 1.8Sh for service level B (upset condition) but not greater than 1.5Sy;
2.25Sh for service level C (emergency condition) but not greater than
1.8Sy; and 3.0Sh for service level D (faulted condition) but not greater
than 2.0Sy
Sh = material allowable stress at temperature consistent with loading under
consideration, psi
Sy = material yield strength at temperature consistent with loading under
consideration, psi
3. Stresses due to thermal expansion
a. Thermal expansion stress range must meet the allowable SA, that is,
(B4.11)
where SA =
=
f =
Mc =
allowable stress range for expansion stresses
f(1.25Sc + 0.25Sh), psi
stress range reduction factor, as in Table B4.2
range of resultant moment due to thermal expansion, in · lb; also
include moment effects of anchor displacements due to earthquake
if anchor displacement effects were omitted from occasional
loadings
Sc = basic material allowable stress at minimum (cold) temperature, psi
TABLE B4.2 Stress-Range Reduction
Factors
Source: Table NC-3611.2(e)-1, Section III,
Division 1, ASME B & PV Code, 1998. (Courtesy of
ASME.)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.119
Sh = basic material allowable stress at maximum (hot) temperature, psi
i = stress intensification factor
b. If Eq. (B4.11) is not met, the piping may be qualified by meeting the following
equation:
(B4.12)
where 0.75i shall not be less than 1.0.
4. Stresses due to nonrepeated anchor movement. The effect of any single nonrepeated
anchor movement (such as building settlement) must meet 3.0Sc,
(B4.13)
where MD = resultant moment due to any single nonrepeated anchor movement
(e.g., predicted building settlement), in · lb.
5. The stress-intensification factor (SIF) is defined as the ratio of the maximum
stress intensity to the nominal stress, calculated by the ordinary formulas of
mechanics. It is used as a safety factor to account for the effect of localized stresses
on piping under a repetitive loading. In piping design, this factor is applied to
welds, fittings, branch connections, and other piping components where stress
concentrations and possible fatigue failure might occur. Usually, experimental
methods are used to determine these factors.
It is recognized that some of the SIFs for the same components are different
for different codes. In some cases, different editions of the same code provide
different SIFs for a given component. The way that the SIFs are applied to moment
loadings is also different for different codes. The B31.1 and ASME Section III
codes require that the same SIF be applied to all the three-directional moments
while the B31.3, B31.4, B31.5, and B31.8 codes require that different SIFs be
applied to the in-plane and out-of-plane moments, with no SIF required for torsion
(see Fig. B4.5a and figure note 10).
Therefore, the stress analyst has to ensure that the appropriate SIFs from the
applicable code (i.e., committed code) are used. The formulas for SIFs in the
ASME Section III code (1989 edition) are given in Fig. B4.5c for reference.
Recommended SIFs for some piping components which are not addressed in
the code are listed below:
a. Weldolets or sockolets4
(1) If r/R > 0.5,
(B4.14)
where r = mean radius of branch pipe, in
t = wall thickness of run pipe, in
R = mean radius of run pipe, in
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STRESS ANALYSIS OF PIPING SYSTEMS
B.120
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.5a Flexibility factor n and stress intensification factors ii and io per ASME B31.3,
B31.4, B31.5, and B31.8 codes.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.121
FIGURE B4.5a (Continued) Flexibility factor n and stress intensification factors ii and io per
ASME B31.3, B31.4, B31.5, and B31.8 codes.
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.122
FIGURE B4.5b Flexibility and stress intensification
factors for those cases where result is a power of the
characteristic h, defined in Fig. B4.5a. Key: a = no
flanges; b = one flange; c = two flanges; i = in plane; o =
out of plane.
(2) If r/R ≤ 0.5,
(B4.15)
or
(B4.16)
whichever is less and
where R =
T=
r =
t =
rp =
run pipe mean radius, in
run pipe wall thickness, in
branch pipe mean radius, in
branch pipe wall thickness, in
outer radius of weldolet, in
SIF values for typical weldolet branch connections with r/R ≤ 0.5 are
tabulated in Tables B4.3a to B4.3l.
b. Half-Couplings (Welding Boss). For half-couplings with r/R ≤ 0.5, use the above
branch connection Eq. (B4.15) or the unreinforced fabricated tee equation,
whichever is less. For half-coupling with r/R > 0.5, use the unreinforced fabricated
tee formula. Tables B4.4a to B4.4f give SIFs for commonly used half-coupling
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.123
FIGURE B4.5c Flexibility and stress intensification factors (Do/tn ≤ 100). (Figure NC-3673.2(b)1, Section III, Division 1, ASME B & PV Code, 1998. Courtesy of ASME.)
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STRESS ANALYSIS OF PIPING SYSTEMS
B.124
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.5c (Continued) Flexibility and stress intensification factors (Do/tn ≤ 100). (Figure
NC-3673.2(b)-1, Section III, Division 1, ASME B & PV Code, 1998. Courtesy of ASME.)
configurations. If the half-coupling rating is not known, assume a Class 3000
half-coupling, since this will give the more conservative value.
c. Sweepolets. For branch:
(B4.17)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.125
FIGURE B4.5c (Continued) Flexibility and stress intensification factors (Do/tn ≤ 100). (Figure
NC-3673.2(b)-1, Section III, Division 1, ASME B & PV Code, 1998. Courtesy of ASME.)
For run:
(B4.18)
(B4.19)
where F1 =
=
F2 =
Fs =
R=
F2 = 1.0 for flush or dressed insert welds
1.6 for as-welded insert welds
(0.5 + r/R), but not less than 1.0 for as-welded insert welds
1 + 0.05(r - 3), but not less than 1.0
mean radius of run pipe, in
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.126
TABLE B4.3a SIFs for Typical Weldolet Branch Connections (Run Pipe
Size: NPS 1½ or DN 40)
1 in = 25.4 mm
r = mean radius of branch pipe, in
T = nominal wall thickness of run pipe, in
t = nominal wall thickness of branch pipe, in
If a more detailed analysis is desirable, see Ref. 5 for the equations to be
used for moment separation.
d. Lateral branch connections (45°).6,7 For rb/r > 0.5,
(B4.20)
TABLE B4.3b SIFs for Typical Weldolet Branch Connections (Run Pipe
Size: NPS 2 or DN 50)
1 in = 25.4 mm
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1 in = 25.4 mm
TABLE B4.3c SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 2½ (DN 65) and NPS 3 (DN 80)
STRESS ANALYSIS OF PIPING SYSTEMS
B.127
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1 in = 25.4 mm
TABLE B4.3d SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 4 (DN 100) and 6 or (DN 150)
STRESS ANALYSIS OF PIPING SYSTEMS
B.128
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.129
TABLE B4.3e SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 8 or
DN 200)
1 in = 25.4 mm
(B4.21)
where t = run pipe wall thickness, in
r = run pipe mean radius, in
rb = branch pipe mean radius, in
These equations are for integrally reinforced branch connections such as latrolets. By
the analogy used in Fig. NC-3673.2(b)-1 in Section III of the ASME Code, the SIF for
unreinforced 45° branch connections (stub-ins) can be obtained by multiplying the
factors obtained above by (4.4)2/3 = 2.685.
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.130
TABLE B4.3f SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 10
(DN 250))
1 in = 25.4 mm
e. Pipet
(B4.22)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.131
TABLE B4.3g SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 12 or
DN 300)
1 in = 25.4 mm
f. Branchlet
(B4.23)
g. Reducing elbow
(B4.24)
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.132
TABLE B4.3h SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 14 or
DN 350)
1 in=25.4 mm
where T = wall thickness of large end, in
B = actual bend radius, in
R = mean radius of large end, in
ASME B31.1 Power Piping Code.8 This code concerns nonnuclear piping such as
that found in the turbine building of a nuclear plant or in a fossil-fuel power plant.
Piping services include steam, water, oil, gas, and air. Design requirements of this
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STRESS ANALYSIS OF PIPING SYSTEMS
B.133
TABLE B4.3i SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 16 or
DN 400)
1 in = 25.4 mm
code cover those for pipe, flanges, bolting, gaskets, valves, relief devices, fittings, and
the pressure-containing portions of other piping components. It also includes hangers
and supports and other equipment items necessary to prevent overstressing the
pressure-containing components.
The loadings required to be considered are pressure; weight (live, dead, and
under test loads); impact (e.g., water hammer); wind; earthquake (where
applicable); vibration; and those loadings resulting from thermal expansion and
contraction.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.134
GENERIC DESIGN CONSIDERATIONS
TABLE B4.3j SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 18 or
DN 450)
1 in = 25.4 mm
The design equations and stress limits are as follows (terms are the same as those
for Class 2 and 3 piping except for those defined below):
1. Stress due to sustained loads. The effects of pressure, weight, and other sustained
mechanical loads must meet the requirements of Eq. (B4.25):
(B4.25)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.135
TABLE B4.3k SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 20 or
DN 500)
1 in = 25.4 mm
where S L = sum of longitudinal stresses due to pressure, weight, and other
sustained loads, psi.
2. Stress due to occasional loads. The effects of pressure, weight, other sustained
loads, and occasional loads including earthquake must meet the requirements of
Eq. (B4.26):
(B4.26)
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STRESS ANALYSIS OF PIPING SYSTEMS
B.136
GENERIC DESIGN CONSIDERATIONS
TABLE B4.3l SIFs for Typical Weldolet Branch Connections (Run Pipe Size: NPS 24 or
DN 600)
1 in = 25.4 mm
where k = 1.15 for occasional loads acting less than 10 percent of operating period;
1.2 for occasional loads acting less than 1 percent of operating period
3. Thermal expansion stress range. The effects of thermal expansion must meet the
requirements of Eq. (B4.27):
(B4.27)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.137
TABLE B4.4a SIFs for Class 3000 Half-Couplings (Branch Pipe Schedule 40) (Run Pipe Size:
NPS 1½ to 8 (DN 40 to 200))
1 in = 25.4 mm
4. The requirement for the effects of any single nonrepeated anchor movement is
not specified.
ASME B31.3 Process Piping Code.9 This code governs all piping within the property
limits of facilities engaged in the processing or handling of chemical, petroleum, or
related products. Examples are a chemical plant, petroleum refinery, loading terminal,
natural gas processing plant, bulk plant, compounding plant, and tank farm. Excluded
from the B31.3 code are piping carrying nonhazardous fluid with an internal gauge
pressure less than 15 psi (103.5 kPa) and a temperature below 366°F (186°C);
plumbing; sewers; fire protection systems; boiler external piping per B31.1 as well as
pipelines per B31.4 or B31.8.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.138
GENERIC DESIGN CONSIDERATIONS
TABLE B4.4b SIFs for Class 3000 Half-Couplings (Branch Pipe Schedule 40) (Run Pipe Size:
NPS 10 to 24 (DN 250 to 600))
1 in = 25.4 mm
The loadings required to be considered are pressure, weight (live and dead loads),
impact, wind, earthquake-induced horizontal forces, vibration, discharge reactions,
thermal expansion and contraction, temperature gradients, and anchor movements.
The governing equations are as follows:
1. Stresses due to sustained loads. The sum of the longitudinal stresses SL due to
pressure, weight, and other sustained loads must not exceed Sh (basic allowable
stress at maximum metal temperature). The thickness of pipe used in calculating
SL shall be the nominal thickness minus mechanical, corrosion, and crosion
allowances.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.139
TABLE B4.4c SIFs for Class 3000 Half-Couplings (Branch Pipe Schedule 80) (Run Pipe Size:
NPS 1½ to 8 (DN 40 to 200))
1 in = 25.4 mm
2. Stresses due to occasional loads. The sum of the longitudinal stresses due to
pressure, weight, and other sustained loads and of the stresses produced by
occasional loads such as earthquake or wind shall not exceed 1.33Sh. Earthquake
and wind loads need not be considered as acting simultaneously.
3. Stress range due to expansion loads. The displacement stress range SE shall not
exceed SA:
(B4.28)
where SE =
Sb = resultant bending stress, psi
= [(iiMi )2 +(ioMo)2]1/2/Z
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.140
TABLE B4.4d SIFs for Class 3000 Half-Couplings (Branch Pipe Schedule 80) (Run Pipe Size:
NPS 10 to 24 (DN 250 to 600))
1 in = 25.4 mm
Mi = in-plane bending moment, in · lb
Mo = out-of-plane bending moment, in · lb
ii = in-plane stress intensification factor obtained from Fig. B4.5a (see also
figure note 10)
io = out-of-plane stress intensification factor obtained from Fig. B4.5a (see
also figure note 10)
St = torsional stress, psi
= Mt/(2Z)
Mt = torsional moment, in · lb
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.141
TABLE B4.4e SIFs for Class 6000 Half-Couplings (Branch Pipe Schedule 160) (Run Pipe Size:
1½ to 8 in (DN 40 to 200))
1 in = 25.4 mm
SA =
=
=
Sc =
f=
allowable displacement stress range
f(1.25Sc + 0.25Sh)
f[1.25(Sc + Sh) - SL] when Sh > SL
basic allowable stress at minimum metal temperature, psi
stress range reduction factor per Table B4.2
ASME B31.4 Liquid Transportation Systems for Hydrocarbons, Liquid Petroleum
Gas, Anhydrous Ammonia, and Alcohols Piping Code.10 The scope of ASME B31.4,
Liquid Transportation Systems for Hydrocarbons, Liquid Petroleum Gas, Anhydrous
Ammonia, and Alcohols, governs piping transporting liquids such as crude oil,
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STRESS ANALYSIS OF PIPING SYSTEMS
B.142
GENERIC DESIGN CONSIDERATIONS
TABLE B4.4f SIFs for Class 6000 Half-Couplings (Branch Pipe Schedule 160) (Run Pipe Size:
10 to 24 in (DN 250 to 600))
1 in = 25.4 mm
condensate, natural gasoline, natural gas liquids, liquefied petroleum gas, liquid
alcohol, liquid anhydrous ammonia, and liquid petroleum products between producers’
lease facilities, tank farms, natural gas processing plants, refineries, stations, ammonia
plants, terminals, and delivery and receiving points. Excluded from B31.4 are auxiliary
piping such as water, air, steam, lubricating oil, gas, and fuel; piping with an internal
gauge pressure at or below 15 psi (103.5 kPa) regardless of temperature; piping with
an internal gauge pressure above 15 psi (103.5 kPa) and a temperature below -20°F
(-29°C) or above 250°F (121°C); and piping for petroleum refinery, gas transmission
and distribution, ammonia refrigeration, and so on, that is covered by other ASME
B31 sections.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.143
The limits of calculated stresses are as follows:
1. Stresses due to sustained loads. The sum of the longitudinal stresses due to pressure,
weight, and other sustained external loads shall not exceed 0.75SA, where SA =
0.72Sy (specified minimum yield strength).
2. Stresses due to occasional loads. The sum of the longitudinal stresses produced
by pressure, live and dead loads, and those produced by occasional loads, such as
wind or earthquake, shall not exceed 0.8Sy.
3. Stresses due to expansion loads
a. Restrained lines. The net longitudinal compressive stress due to the combined
effects of temperature rise and fluid pressure shall be computed from the equation.
(B4.29)
where SL
SH
T1
T2
E
α
v
=
=
=
=
=
=
=
longitudinal compressive stress, psi
hoop stress due to fluid pressure, psi
temperature at time of installation, °F
maximum or minimum operating temperature, °F
modulus of elasticity, psi
linear coefficient of thermal expansion, in/(in · °F)
Poisson’s ratio = 0.30 for steel
Then the equivalent tensile stress is calculated as
(B4.30)
where Seqiv = the equivalent tensile stress, psi. Beam bending stresses shall be
included in the longitudinal stress for those portions of the restrained line
which are supported aboveground.
b. Unrestrained lines. Stresses due to expansion for those portions of the piping
without substantial axial restraint shall be combined in accordance with the
following equation:
(B4.31)
where SE = stress due to expansion, psi
Sb = [(iiMi)2 +(ioMo2) ]1/2/Z
= equivalent bending stress, psi
St = Mt/(2Z) = torsional stress, psi
Mi = in-plane bending moment, in · lb
Mo = out-of-plane bending moment, in · lb
Mt = torsional moment, in · lb
ii = in-plane stress intensification factor obtained from Fig. B4.5a
io = out-of-plane stress intensification factor obtained from Fig. B4.5a
Z = section modulus of pipe, in3
ASME B31.5 Refrigeration Piping Code.11 The scope of this code covers refrigerant
and secondary coolant piping for temperatures as low as -320°F (196°C). Excluded
from this code are piping designed for external or internal gauge pressure not exceeding
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STRESS ANALYSIS OF PIPING SYSTEMS
B.144
GENERIC DESIGN CONSIDERATIONS
15 psi (103.5 kPa) regardless of size; water piping; and any self-contained or unit
systems subject to the requirements of Underwriters’ Laboratories or other nationally
recognized testing laboratory.
The limits of calculated stresses are as follows:
1. Stresses due to expansion loads. The expansion stress range SE shall not exceed
the allowable stress range SA:
(B4.32)
where SE = expansion stress range =
psi
Sb = resultant bending stress
= [(iiMi)2 + (iσ Mo)2]1/2/Z, psi
St = torsional stress = Mt/(2Z), psi
Mi = in-plane bending moment, in · lb
Mo = out-of-plane bending moment, in · lb
Mt = torsional moment, in · lb
ii = in-plane stress intensification factor obtained from Fig. B4.5a
io = out-of-plane stress intensification factor obtained from Fig. B4.5a
Z = section modulus of pipe, in3
Sc = basic material allowable stress at minimum (cold) normal
temperature , psi
Sh = basic material allowable stress at maximum (hot) normal
temperature , psi
f = stress range reduction factor obtained from Fig. B4.5d
FIGURE B4.5d Stress range reduction factors. (Extracted
from Refrigeration Piping Code, ASME B 31.5 1992.
Courtesy of ASME.)
2. Stresses due to sustained loads. The sum of the longitudinal stresses due to pressure,
weight, and other sustained external loading SL shall not exceed Sh. Where SL >
Sh, the difference Sh - SL may be added to the term in parentheses in Eq. (B4.32).
3. Stresses due to occasional loads. The sum of the longitudinal stresses produced
by pressure, live and dead loads, and occasional loads, such as wind or earthquake,
may not exceed 1.33Sh. It is not necessary to consider wind and earthquake as
occurring concurrently.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.145
TABLE B4.5 Temperature Derating Factor T for Steel Pipe
Note: For intermediate temperatures, interpolate for derating
factor.
Source: ASME B31.8, 1995. Gas Transmission and Distribution
Piping Systems. (Courtesy of ASME.)
ASME B31.8, Gas Transmission and Distribution Piping Code.12 This code governs
most of the pipelines in gas transmission and distribution systems up to the outlet of
the customer’s meter set assembly. Excluded from this code are piping with metal
temperatures above 450°F (232.2°C) or below -20°F (-29°C); piping beyond the
outlet of the customer’s meter set assembly; piping in oil refineries or natural gas
extraction plants, gas treating plants, and so on, which is covered by other ASME
B31 codes; waste gas vent pipe operating at atmospheric pressures; and liquid
petroleum transportation piping. The governing equations are as follows:
1. Stresses due to pressure and external loads. The sum of the longitudinal pressure
stress and the longitudinal bending stress due to external loads such as weight,
wind, and so on, SL, shall not exceed 0.75Sy FT:
(B4.33)
where Sy = specified minimum yield strength, psi
T = temperature derating factor obtained from Table B4.5
F = construction-type design factor obtained from Table B4.6. The
construction types are associated with the population density of the
surrounding area as follows:
Type A: Sparsely populated areas such as deserts, mountains, and
farmland
Type B: Fringe areas around cities or towns
TABLE B4.6 Values of Design Factor F
Source: ASME B31.8, 1995. (Courtesy of
ASME.)
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.146
Type C: Cities or towns with no buildings over three stories tall
Type D: Areas with taller buildings
2. Stress range due to expansion loads. The maximum combined expansion stress
range SE shall not exceed 0.72Sy:
(B4.34)
where Sb
St
Mb
Mt
i
=
=
=
=
=
resultant bending stress = iMb/Z, psi
torsional stress = Mt/(2Z), psi
resultant bending moment, in · lb
torsional moment, in · lb
stress intensification factor obtained from Fig. B4.5a (see figure
note 10)
Z = section modulus of pipe, in3
3. Stresses due to pressure, external loads, and expansion loads. The sum of the
longitudinal pressure stress, the longitudinal bending stress due to external loads,
and the combined stress due to expansion shall not exceed Sy.
STRESS INTENSIFICATION FACTORS (SIF) FOR
NONSTANDARD FITTINGS
Stress intensification factors for fittings such as Wye (Y) connections and latrolets are
not available in the current editions of codes. The following subsections provide data
for obtaining appropriate SIFs for various nonstandard fittings.
SIF for 90-Degree Wye Connection
The following SIF (i) which can be used is based on a comparative finite element
analysis study13 of a Wye connection and a Tee connection. See Fig. B4.5i.
i for forged Wye connection = 1.0 (i of WTEE)
for all moments except for torsion and out-of-plane bending moment components
from branch side
iob for forged Wye connection = 3.3 (i of WTEE)
for out-of-plane bending moment components from branch side
itb for forged Wye connection = 1.1 (i of WTEE)
for torsional moment components from branch side
Use UTEE (unreinforced tee) instead of WTEE (welding tee) in the above expressions
if the Wye connection is a fabricated Wye.13
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.147
SIF for Weldolets, Sockolets, and Half-Couplings
For branch-side stresses, branch-side section modulus should be used.
For insert or contour weldolets, the coefficients in the above equations 0.8, 1.5,
and 0.9 are replaced by 0.4, 0.75, and 0.45, respectively.
R, T = mean radius and wall thickness of run pipe
r, t = mean radius and wall thickness of branch pipe
rp = outside radius of the reinforcement on nozzle or branch
The above equations are based on Refs. 14 and 15.
SIF for 45-Degree Latrolets
Forged latrolet: i = 0.5727 (R/T)2/3 based on h = 1.97 (T/R)
Fabricated lateral: i = 1.5378(R/T)2/3 obtained by multiplying the SIF of forged
latrolet by (4.4)2/3
See Ref. 16.
SIF for Reducers
Even though the reducer is a standard fitting and the formula for SIF is available in
the codes,17 the cone angle is not readily available. The following SIF formula from
the codes can be used for large-bore concentric and eccentric reducers. Using data
from Ref. 18, the following expressions can be used for the cone angle, α:
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STRESS ANALYSIS OF PIPING SYSTEMS
B.148
GENERIC DESIGN CONSIDERATIONS
where α = cone angle in degrees
D, d = mean diameter of large and small ends of reducer, respectively
d2, t2 = outside diameter and thickness of small end of reducer, respectively
See Refs. 17 and 18.
LOCAL STRESSES
In addition to the general pipe stresses (the pressure stress and the moment stress) as
described in the previous sections, there are certain local pipe wall stresses produced
by (1) restraint of the pipe radial thermal and internal pressure expansion of pipethrough-structural-steel type of anchors, (2) the transfer of load from the supporting
surface to the pipe surface over a contact length along the axis of the pipe, or (3)
attachments welded to pipe (e.g., lugs and trunnions).
Local Stresses and Code Requirements
The local stresses SL,
,
, and
can be expressed as follows:
(B4.35)
where SL = local stress due to deadweight, psi
= local stress due to deadweight, seismic inertia, and other dynamic
loads, psi
= local stress due to thermal expansion and seismic anchor movement, psi
= local stress due to concurrently acting loads, psi
σ1 = longitudinal membrane stress, psi
σ2 = circumferential membrane stress, psi
= circumferential bending stress, psi
= longitudinal bending stress, psi
Strictly speaking, the present piping codes give no specific limits for local stresses. As
an industry practice, the calculated local stress is added to the general pipe stress and
then compared with the pipe stress allowables specified by the applicable code. As an
example, the total (general plus local) pipe stresses for ASME Class 2 and 3 piping
shall satisfy the following equations [see Eqs. (B4.9), (B4.10), (B4.11), (B4.12) for
definitions of symbols]:
1. Design loading
(B4.36)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.149
2. Service loadings
(B4.37)
3. Sustained and thermal expansion loading: Either of the following equations shall
be satisfied:
(B4.38)
or
(B4.39)
4. Local stress limit loading
(B4.40)
Local Stress Due to Restraint of Pipe Radial Expansion
The membrane and flexural stresses can be calculated as follows19:
(B4.41)
(B4.42)
(B4.43)
(B4.44)
(B4.45)
where P =
R=
E=
t=
α=
∆T =
v=
internal pressure of pipe, psi
pipe outside radius, in
modulus of elasticity of pipe, psi
pipe wall thickness, in
coefficient of thermal expansion of pipe, in/(in · °F)
range of thermal expansion temperatures, °F
Poisson’s ratio
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.150
In the case of the local stress produced by restraint to the pipe radial expansion,
For fillet weld, i = 2.1 should be used. In addition, the stress check on
limit loading is required. Here,
.
Local Stress Due to Contact
The membrane and flexural stresses can be calculated as follows19:
1. For line contact
(B4.46)
(Note:
is considered to be included in this equation.)
(B4.47)
(B4.48)
where F
Rm
L
t
=
=
=
=
support load, lb
mean radius of pipe, in
contact length, in
pipe wall thickness, in
2. For point contact. σ1 and
are negligible compared to
:
(B4.49)
(B4.50)
3. In the case of contact stress, the minimum nominal general pipe stress (i.e., the
unintensified general pipe stress) may be used in Eqs. (B4.36) to (B4.39) to calculate
the total pipe stress. In addition, the stress check on limit loading is not required.
Other Types of Local Stresses
The two types of local stresses previously described are commonly encountered by
stress analysts. Detailed descriptions and analysis methods for other types of local
stresses such as the local stresses at integral welded attachments to pipe (e.g., lugs and
trunnions) can be found in technical publications, Welding Research Council Bulletins
107 and 198, and ASME Code Cases.21-23
ANALYSIS OF INTEGRAL WELDED
ATTACHMENTS (IWA)
Integral Welded Attachments are often used to support piping systems. The local
stresses in the piping at IWA locations are commonly evaluated using the Welding
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.151
Research Council (WRC) Bulletin #107 approach,21 which is based on Bijlaard’s20
work. Generally, the various methods for local stress evaluations can be categorized
in accordance with the following list. (Friction-induced loads due to weight and thermal
expansion, if applicable, should be included.)
1. Stress intensification factor (SIF) approach for certain configurations
2. WRC Bulletin #107 approach with limitation on ß (attachment parameter) and γ
(shell parameter) parameters
3. ASME Code cases approach
4. Approach based on utilization of any available finite element analysis (FEA) results
or published data
5. Rigorous FEA approach
SIF Approach
In this method, the local stresses are not evaluated directly but are indirectly accounted
for by applying a SIF to the general piping stresses.
The SIF approach has the following limitations:
•
Applicable to some specific IWA configurations only
•
Not applicable to lugs, irregular shapes, and so forth
The SIF approach is applicable in the following situations:
•
360° (full) wrapper plates. This configuration is no longer a local stress problem.
A SIF (i) of 2.1 or 1.3 can be applied, depending on the applicable code.
•
180° wrapper plates. The following SIFs are recommended by Rodabaugh (see
Fig. B4.5e):
i = 4.2 for the run pipe torsional moment (Mtr) component
i = 2.1 for the run pipe out-of-plane bending moment (Mobr) component
i = 1.3 for the run pipe in-plane bending moment (Mibr) component
•
Circular trunnion/stanchions on straight pipe. Consider the configuration as a
reinforced tee (RTEE) and intensify the general piping stresses using a SIF (i) of
FIGURE B4.5e 180° wrapper plate (IWA).
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STRESS ANALYSIS OF PIPING SYSTEMS
B.152
GENERIC DESIGN CONSIDERATIONS
RTEE per the applicable code requirements. Since there is no hole made in the
pipe’s pressure boundary, the run pipe thickness can be considered as a
reinforcement. If there is a pad, the pad thickness can be considered as an additional
reinforcement. Of course, the codes limit the effective thickness of reinforcement
[(te)max = 1.5 × t].
•
Attachments on fittings. Cross multiplication of SIFs (for example, a round
attachment on elbow or on a tee) can be used (elbow SIF × RTEE SIF or TEE SIF
× RTEE SIF).
WRC Bulletin #107 Approach
This is based on the analytical work performed by Professor P.P. Bijlaard of Cornell
University and subsequently generated experimental data. The results are presented
in WRC Bulletin #107 (WRC-107) as nondimensional curves based on ß and y
parameters for three different types of loading. The local stresses can be evaluated by
hand by filling out the computation sheets for local stresses as given in the WRC-107
bulletin or utilization of commercially available computer program software which
has stored the nondimensional curves as digitized data. The WRC-107 method
generally yields conservative results. The following additional information relates to
this approach:
•
Basis. Bijlaard’s approach is based on shell theory and some simplifications for
radial load (P), longitudinal moment (ML), and circumferential moment (MC).
•
Strength-of-materials formulas. Simple strength-of-materials formulas are used
to compute shear stresses due to longitudinal shear load (VL), circumferential
shear load (VC), and torsional moment (MT) loadings.
•
Shell and attachment parameters and loadings. The shell, attachment parameters,
and loadings are as follows (see Figs. B4.5f and B4.5g):
shell parameter γ = Dm/(2T) where Dm = Do - T
FIGURE B4.5f IWA notations.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.153
FIGURE B4.5g IWA notations (continued).
attachment parameter (ß):
For circular attachment ß = 0.875 (do/Dm)
For square or rectangular attachment
•
Caution: WRC-107 uses C 1 and C2 to represent one-half of the attachment
dimensions in circumferential and longitudinal directions, while here C 1 and
C2 are used for the full attachment dimensions.
Limitations. The WRC Bulletin #107 approach has the following limitations:
•
Shear stresses. The formulas for shear stress calculations are as follows:
•
Attachments on elbows. To evaluate local stresses in elbows with attachments,
the following approach can be utilized. Attachment loads
can be resolved
at the elbow/attachment interface to components P, VL, Mc, MT, ML. An equivalent
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STRESS ANALYSIS OF PIPING SYSTEMS
B.154
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.5h IWA on elbow.
straight pipe with the attachment and resolved loads can be considered in local
stress evaluation using WRC-107 (see Fig. B4.5h).
Approach Based on ASME Code Cases
The Welding Research Council (WRC) Bulletin #19822 fitted equations with some
inherent conservatism to curves of WRC Bulletin #107. The results are published as
ASME Code cases. Interested readers can refer to the ASME Code cases listed in the
following table23 for details and limitations of their applicability. Generally speaking,
the local stress results from Code cases are more conservative than WRC Bulletin
#107 results.
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STRESS ANALYSIS OF PIPING SYSTEMS
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B.155
Approach Based on Available FEA Results
For some commonly utilized configurations and sizes, FEA was performed and the
results were compared with WRC-107 results. Reduction factors are supplied relative
to WRC-107 results. See Ref. 24.
Based on extensive finite element analysis studies of certain sizes, shapes, and
configurations of attachments, the factors in the following tables are generated. The
P, ML, Mc loads can be reduced by dividing them with the applicable factors, and then
the WRC-107 approach can be used. The γ value should be limited to the following
range: 3.5 ≤ γ ≤ 31.5.
Factors for Attachments on Straight Pipe
Notes: 1. Factors for rectangular attachments when 3.5 ≤ γ < 5 and ß ≤ 0.5.
2. Apply these factors to the loads when actual ß > 0.5 and evaluate the local stress
based on WRC-107 utilizing artificially reduced attachment size to correspond to
ß = 0.5.
Notation: S: square attachments
R: rectangular attachments
C: circular attachments
σm: membrane stress
σb: bending stress
P: radial load
ML: longitudinal moment
MC: circumferential moment
See ASME paper listed in Ref. 24.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.156
GENERIC DESIGN CONSIDERATIONS
Factors for Circular Attachments on Long-Radius Elbows
Notes: Apply these factors to the loads (divide the loads by these factors) and perform
WRC-107 evaluation using attachment size reduced to correspond to ß = 0.5.
Notation: do: outside diameter of circular attachment
Do: outside diameter of pipe
See ASME paper listed in Ref. 24.
Rigorous FEA Approach
For this approach, a 3-D finite-element model of the shell and attachment has to be
built. A portion of the pipe, attachment, and pipe/attachment interface can be modeled
and loadings applied. FEA analysis can be performed and the local stresses can be
evaluated.
Although cumbersome and time-consuming, FEA is becoming a viable option due
to the availability of commercial FEA software which can run on personal computer
platforms. Personal computers have become very powerful tools.
FIGURE B4.5i Wye connection.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.157
TYPES OF PIPE LOADING CONDITIONS
1. Types of loads. As previously mentioned in the subsection “Classification of
Loads,” piping loads are classified into three types: sustained loads, occasional
loads, and expansion loads. These three types of loads and the corresponding
analysis will be discussed in this section in detail.
2. Method of analysis. The piping stress analysis to be performed could be a simplified
analysis or a computerized analysis. The choice of the proper analysis depends on
the pipe size and the piping code. For small (nominal diameter 2 in and under)
pipe except nuclear Class 1 pipe, a cookbook-type, simplified analysis could be
performed. For nuclear Class 1 piping, since the requirements are more stringent,
a computerized analysis is required. A detailed description of a cookbook-type,
simplified analysis and a brief description of a computerized analysis are given in
the section that follows, “Methods of Analysis.” Generally, before computerized
analysis is performed, pipe supports may be located using the cookbook method.
Sustained Load: Pressure
Internal pressure in piping usually induces stresses in the pipe wall rather than loads
on the pipe supports. This is because pressure forces are balanced by tension in the
pipe wall, resulting in zero pipe support loadings. A discussion of unbalanced forces
in the pipe created by pressure waves during fluid transients is given in the subsection
“Dynamic Loads.”
Pressure Stress. The longitudinal stress developed in the pipe due to internal pressure
can be calculated as follows:
or
(B4.51)
where SLP =
P =
D =
d =
Af =
Am =
t =
longitudinal stress, psi
internal design pressure, psig
outside diameter, in
inside diameter, in
flow area, in2
metal area, in2
pipe wall thickness, in
The second equation gives pressure stress in terms of the ratio of pipe flow area to
metal area. It also provides a more accurate result. Both equations are acceptable to
the code.
Expansion Joint. In piping design, elbows, bends, and pipe expansion loops normally
provide adequate flexibility for piping thermal expansion and contraction. However,
in some cases this flexibility may not be adequate. As a solution, expansion joints
may be used to absorb the expansion and contraction of pipe.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.158
GENERIC DESIGN CONSIDERATIONS
In general, expansion joints are used for the following applications:
1. Where thermal movements would induce excessive stress in normal piping
arrangements
2. Where space is inadequate
3. Where reactions transmitted by pipe supports or anchors create large loads on
supporting structures
4. Where reactions to equipment terminals are in excess of allowables
When expansion joints are used in piping, the pressure forces can no longer be balanced
by tension in the pipe wall, and the pressure forces will be resisted by pipe supports
and anchors.
There are many types of expansion joints available, ranging from a piece of rubber
hose to metal bellows. The metal bellows expansion joint is most commonly used for
power or process piping. Figure B4.6 shows the various components of a bellows
expansion joint.
FIGURE B4.6 Bellows expansion joints.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.159
Expansion joints do not have the capability to transmit large pressure forces.
Restraints are usually installed on both sides of the expansion joint to prevent the
pressure force from pulling the joint apart. The pressure force developed in the expansion
joint is equal to the internal pressure times the maximum cross-sectional area over
which it is applied. Since an expansion joint increases the flexibility of a piping system,
the flexibility (spring rate) of the expansion joint should be incorporated in the piping
stress analysis. Typical axial spring rates of bellows can be found in Ref. 25.
Sustained Load: Weight
The total design weight load of pipe supports includes the weight of the pipe, fittings,
insulation, fluid in pipe, piping components such as valves, valve operators, flanges,
and so on, and the supports themselves. Supports should be located as specified in
Chap. B5.
Hydrotest and Other Occasional Loadings. To assure the integrity and leak tightness
of a piping system designed to Section III of ASME Boiler and Pressure Vessel Code
or ASME B31.1, the codes require that a pressure test be performed prior to placing
the system in service. The most commonly used test is the hydrostatic test. When a
steam or gas piping system is to be hydrotested, the effects of the weight of the water
on the system and its supports must be considered. A hydroweight stress analysis
should be performed to assure that the pipe supports, which have been designed for
the normal operating condition, are able to withstand the hydrotest loads. If permanent
supports cannot withstand these hydrotest loads, temporary supports may be added.
Spring supports are available with hydrostatic test stops, which, in effect, transform
the units into rigid supports.
Whether or not required by code, other conditions, such as the added weight of a
cleaning medium of density greater than that of the process fluid, must be considered
in a manner similar to that discussed above. Both dynamic and static loading analyses
may be impacted by flushing and blowing-out activities during construction or after
major rework.
Thermal Expansion Loads
For weight analysis, the more pipe supports installed, the lower the stress developed
in the pipe. However, the opposite is true for the case of piping thermal expansion.
When thermal expansion of the piping due to fluid or environmental temperature is
restrained at supports, anchors, equipment nozzles, and penetrations, large thermal
stresses and loads are caused.
Thermal Modes. Piping systems are generally analyzed for one thermal condition or
mode, that is, the maximum operating temperature. However, piping systems that
have more than one operating mode with different operating temperatures concurrently
in different parts of the piping system should be analyzed for these operating thermal
modes.
With the aid of system flow diagrams or piping and instrumentation drawings
(P&ID), the stress analyst can determine the thermal modes required for a particular
piping system. For B31.1 piping and ASME Class 2 and 3 piping, the required
thermal modes can be determined by using good engineering judgment in selecting
the most severe thermal conditions. For ASME Class 1 piping, the required thermal
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STRESS ANALYSIS OF PIPING SYSTEMS
B.160
GENERIC DESIGN CONSIDERATIONS
modes can be determined by examining the load histograms specified in the design
specification.
Free Thermal Analysis. During the initial stage of piping analysis, an unrestrained
(i.e., no intermediate pipe supports) or free thermal analysis may be performed. This’
analysis is performed for the worst thermal mode and includes only terminal points
such as penetrations, anchors, and equipment nozzles. The result of this free thermal
analysis usually gives useful information, which can be utilized by the stress analyst
in the later stages of the piping analysis. Generally, a resulting thermal expansion
stress < 10 ksi (68,948 kPa) means adequate flexibility exists in the piping system.
The piping locations with low resulting thermal displacements would be good locations
where rigid supports may be installed without adversely affecting the flexibility of the
piping system. The resulting equipment nozzle loads could be used to evaluate the
capabilities of the equipment for meeting the equipment manufacturer’s nozzle
allowables.
Imposed Thermal Movements. Thermal expansion of equipment causes displacements
in the attached piping. Thermal stresses may also be caused due to thermal anchor
movements at terminal ends and intermediate restraints. Therefore, appropriate
thermal analysis for thermal anchor movements relating to the respective thermal
modes should also be performed. Sometimes, it is possible for thermal anchor
movements to exist when the piping is cold. In such cases, analysis in the cold condition,
with only the thermal anchor movements as input, may be required.
LOCA Thermal Analysis. In nuclear power plants, following a loss-of-coolant accident
(LOCA), the containment (the building structure designed to contain fission products)
expands due to the rise in temperature and pressure inside the containment. This
containment thermal growth results in large containment penetration anchor
movements which affect the connected piping. It is not required to qualify the piping
for this faulted condition. Thermal analysis for these LOCA anchor movements is
used only for the evaluation of flanges, equipment nozzle loads, and pipe support
loads.
Temperature Decay. For piping systems having a portion of the system with stagnant
branch lines (dead legs) as shown in Fig. B4.7, it is necessary to consider the temperature
FIGURE B4.7 Temperature decay at dead leg.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.161
decay in the piping. One simple approach to this temperature attenuation problem is
as follows:
1. For a piping system with water, the temperature of the branch pipe is assumed to
be the same as the run pipe up to a length equivalent to 10 times the inside pipe
diameter. The remaining portion of the branch pipe may be considered at ambient
temperature.
2. For a piping system with steam or gas, the temperature of the branch pipe is
considered the same as the run pipe up to the closed valve.
For cases such as thermal transient analysis of ASME Class 1 piping, where a more
accurate temperature profile along the branch pipe may be required, the approach
described in Ref. 26 should be used.
Stress Ranges. The thermal stresses developed in the pipe are in fact “stress ranges,”
that is, the difference between the unit thermal expansion for the highest operating
temperature and for the lowest operating temperature.
For piping systems that do not experience temperatures below ambient temperature,
the stress range is the difference between the unit expansion for the maximum thermal
mode and that for 70°F (21°C). (See later subsection “Seismic Anchor Movement
and Building Settlement Analysis.”)
For systems with supply from a pool or river which might go below 70°F (21°C)
in the winter, negative coefficients of expansion should be considered in evaluating
the stress range.
Occasional Loads: Seismic
The code of Federal Regulation 10CFR Part 50 requires that safety-related piping in
nuclear power plants be designed to withstand seismic loadings without loss of
capability to perform their function.27 For nonnuclear piping in regions of high seismic
activity, this design requirement should also be considered.
OBE and SSE. Nuclear piping systems and components classified as Seismic Category
I are designed to withstand two levels of site-dependent hypothetical earthquakes:
the safe-shutdown earthquake (SSE) and the operational-basis earthquake (OBE).28
For conservatism, the OBE must usually be equal to at least one-half of the SSE.
Their magnitudes are expressed in terms of the gravitational acceleration g. Their
motions are assumed to occur in three orthogonal directions: one vertical and two
horizontal.
Seismic Category I systems are defined as those necessary to assure:
1. The integrity of the reactor coolant pressure boundary
2. The capability to shut down the reactor and maintain it in a safe shutdown
condition
3. The capability to prevent or mitigate potential off-site radiation exposure
Types of Seismic Analysis. Generally, piping seismic analysis is performed through
one of three methods: time-history analysis, modal response spectrum analysis, or
static analysis.
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.162
The equation of motion for a piping system subjected to an externally applied
loading (seismic excitation) may be expressed as
(B4.52)
:
where M
C
K
x
.
x
x
f
=
=
=
=
=
=
=
mass matrix of system
damping matrix
stiffness matrix
acceleration vector
velocity vector
displacement vector
external loading vector, function of time
This equation could be solved by time-history analysis.
Time-History Analysis. Time-history analysis is based on hypothetical earthquake
data in the form of ground displacement, velocity, or acceleration versus time. The
piping system is represented by lumped masses connected by massless elastic members.
The analysis is performed on this mathematical model by the direct numerical
integration method.29,30 At each time step, the piping stresses, displacements, and
restraint loads are calculated. Time history simulates the behavior of the piping system
during the seismic excitation. The main advantage of time-history analysis is that
analytically it is more accurate and less conservative compared to other approaches.
The main disadvantages of time-history analysis are the excessive computational time
required and the difficulty of obtaining a realistic earthquake input time function.
Modal Response Spectrum Analysis. The seismic response spectrum is a plot of
the maximum acceleration response of a number of idealized single-degree-of-freedom
oscillators attached to the floor (structure) with certain damping.
These response spectra are based on design response spectra and specified maximum
ground accelerations of the plant site. Usually, a series of curves with different damping
values for operating and design basis earthquakes for each orthogonal direction are
generated, as shown in Fig. B4.8.
In the modal response spectrum analysis, the piping system is idealized as lumped
masses connected by massless elastic members. The lumped masses are carefully located
to adequately represent the dynamic properties of the piping system.
After the stiffness and mass matrix of the mathematical model are calculated, the
natural frequencies of the piping system and corresponding mode shapes for all
significant modes of vibration are also determined using the following equation:
(B4.53)
where K = stiffness matrix
Wn = natural circular frequency for the nth mode
M = mass matrix
= mode shape matrix for the nth mode
The modal spectral acceleration taken from the appropriate response spectrum is
then used to find the maximum response of each mode:
(B4.54)
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.163
FIGURE B4.8 Response spectrum curves.
where San = spectral acceleration value for the nth mode
D = earthquake direction coefficient
= transpose of the nth mode shape
Mn = generalized mass of the nth mode
Yn = generalized coordinate for the nth mode
Using the maximum generalized coordinate for each mode, the maximum
displacements, the effective inertia forces, the effective acceleration, and the internal
forces and moments associated with each mode are calculated as follows:
(B4.55)
where Xn = displacement matrix due to nth mode
Fn = effective inertia force matrix due to nth mode
an = effective acceleration matrix due to nth mode
M-1 = the inverse of mass matrix
Ln = internal force and moment matrix due to nth mode
b = force transformation matrix
These modal components are then combined by the appropriate method (see later
subsection “Methods for Combining System Responses”) to obtain the total
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STRESS ANALYSIS OF PIPING SYSTEMS
B.164
GENERIC DESIGN CONSIDERATIONS
displacements, accelerations, forces, and moments for each point in the piping
system.
Two types of response spectrum analyses can be performed depending on the pipe
routing and attachments to buildings and structures.
Single-Response Spectrum Analysis. This type of analysis is performed using an
enveloped response spectrum curve that covers all buildings and elevations to which
the piping system is attached.
Multiple-Response Spectrum Analysis. This type of analysis is used where the
piping is attached to various buildings or structures that have a wide variation in the
amplitude or frequency of accelerations. In such cases, various response spectra curves
may be applied at corresponding support and anchor points in the piping system.31,32
Static Analysis. Static analysis may be used to evaluate power piping or some piping
systems in nuclear power plants. It is performed by analyzing a piping system for the
statically applied uniform load equivalent to the site-dependent earthquake accelerations
in each of the three orthogonal directions. All rigid restraints and snubbers supporting
the pipe in the direction of the earthquake acceleration are included in the analysis. The
total seismic effect is obtained by combining the results of the three directions.
The minimum earthquake force for structures described in ANSI A58.133 is also
one form of static seismic analysis. The code recommends that a lateral seismic force
will be assumed to act nonconcurrently in the direction of each of the main axes of
the structure in accordance with the formula:
(B4.56)
where V = lateral seismic force, lb
Z = numerical coefficient, dependent upon the earthquake zone (see Fig.
B4.9), 0.1 for Zone 0, 0.25 for Zone 1, 0.50 for Zone 2, and 1.00 for
Zone 3
FIGURE B4.9 Map for seismic zones, contiguous 48 states. (ANSI A58.1, 1982. Courtesy of
ANSI.)
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STRESS ANALYSIS OF PIPING SYSTEMS
B.165
TABLE B4.7 Dampling Values (Percent of Critical Damping)
Source: U.S. Nuclear Regulatory Commission. Regulatory Guide 1.61.
I = occupancy importance factor, usually between 1.0 and 1.5
K = horizontal force factor, dependent upon the arrangement of lateral
force-resisting elements, usually between 0.67 and 2.50
C = 1/(15T1/2) but not to exceed 0.12
T = fundamental period of structure, s
S = soil factor, dependent upon the soil profile type, usually between 1.0
and 1.5
W = total dead weight of structure, lb
Damping. Damping is the phenomenon of dissipation of energy in a vibrating system.
Each damping value expressed as a percentage of the critical damping is represented
in the seismic response spectrum by a separate curve. The higher the damping value,
the lower would be the effects of the seismic excitation. The damping values to be
used for different levels of the earthquake are given by the NRC (U.S. Nuclear
Regulatory Commission) Regulatory Guide 1.61,34 as shown in Table B4.7.
When a system has both categories of pipe sizes mentioned in the table, dual
damping values should be considered in the analysis.
Alternative damping values for response spectrum analysis of ASME Classes 1, 2,
and 3 piping are given in ASME Code Case N-411-1,35,36 as shown in Fig. B4.10.
These damping values are applicable to both OBE and SSE. They are also independent
of pipe size. As can be seen from Fig. B4.10, the damping values of Code Case N411-1 are generally higher than the damping values given in Regulatory Guide 1.61.
The industry has been applying these higher damping values to existing piping systems
to reduce the number of snubbers installed in the plants in order to save snubber
maintenance cost. The use of Code Case N-411-1 is acceptable to the NRC subject to
the conditions described in the NRC Regulatory Guide 1.84.37
Mass Point Spacing. In a seismic analysis, the piping is represented by lumped masses
connected by massless elastic members. The locations of these lumped masses are
referred to as the mass points. In order to accurately represent the piping, the mass
points on straight runs of pipe should be no farther apart than a length of pipe which
would have a fundamental frequency of 33 Hz (see the later subsection “CookbookType Analysis”). Mass points should also be located at all supports, concentrated
weights such as valves, valve operators, flanges, and strainers, and at the end of
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STRESS ANALYSIS OF PIPING SYSTEMS
B.166
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.10 Code Case N-411 damping values. (ASME
B & PV Code, Case N-411-1, February 1989. Courtesy
of ASME.)
cantilevered vents and drains. At least two mass points should be placed between
supports in the same direction.38
Cutoff Frequency, Rigid Range, Zero Period Acceleration, and Missing-Mass Effect.
Generally, the piping response spectrum analysis is terminated at a frequency called
the cutoff frequency. The cutoff frequency is usually specified as the frequency beyond
which the spectral acceleration remains constant, and this constant spectral acceleration
is known as the zero period acceleration (ZPA) (see Fig. B4.8).
Supposing a piping system is so designed and supported that the first mode is
higher than the cutoff frequency; then as far as the computer program is concerned,
this piping system does not receive seismic excitation at all. Consequently, the result
of this seismic analysis is invalid because of the artificial constraint specified by the
stress analyst.
This phenomenon, known as the missing-mass effect,39,40 could also occur in the
following cases:
1. On pipe runs with axial restraint (support, anchor, or nozzle) where the
longitudinal frequency could be higher than the cutoff frequency
2. Concentrated masses in a piping system supported in such a manner that the
frequency of that portion of piping is high
Most of the computer programs normally used for piping stress analysis have the
capability to evaluate the missing-mass effect. These programs usually utilize the
acceleration from the spectrum at the cutoff frequency (ZPA) to calculate the missingmass effect.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.167
Methods for Combining System Responses. In general, there are two approaches for
the combination of system responses. One approach, the absolute sum method (ABS),
adds the peak system responses. The second approach, square root-sum-of-squares
method (SRSS), gives a combined response equal to the square root of the sum of the
squares of the peak responses. The SRSS method is preferred over the ABS method
because not all the peak responses occur simultaneously.
In a response spectrum modal analysis, if the modes are not closely spaced (two
consecutive modes are defined as closely spaced if their frequencies differ from each
other by less than 10 percent of the lower frequency), responses could be combined
by taking the SRSS method. For closely spaced modes, the NRC suggests that the
method of combining the responses by the SRSS method may not be conservative. An
acceptable method of grouping the closely spaced modes of vibration and combining
the responses is described in the NRC Regulatory Guide 1.92.41,42
Seismic Anchor Movement and Building Settlement Analysis
A piping system, supported from two seismically independent structures that move
out of phase during a seismic event, will experience stresses due to the differential
displacement of the supports.
Buried pipe could be considered as supported by the soil. A differential movement
during a seismic event between the soil and the building to which the pipe is routed
could also cause stresses in the pipe.
Similarly, the differential settlements between two structures or between a building
and the adjacent soil will induce stresses in piping which is routed between them.
Seismic Anchor Movement (SAM) Analysis. A seismic anchor movement analysis is
required on a piping system where:
1. The piping is supported from two seismically independent structures, or
2. The piping is attached to large equipment having its own modes of vibration
(e.g., steam generator, pressurizer, reactor vessel, or reactor coolant pump).
SAM analysis is performed by applying the corresponding seismic displacements
of the building and structures at the pipe support and anchor locations. It is usually
analyzed by a static method. However, dynamic supports such as snubbers and
rigids (including anchors and nozzles) will be active while spring supports remain
passive.
SAM displacements from the same building or structure are generally in phase,
while those from different buildings or structures are considered out of phase.
When a terminal end of a piping system being analyzed is at a large pipe, the
seismic movements from the large pipe analysis should be applied as a SAM
displacement in the analysis.
The code allows the consideration of the stress due to SAM as either primary
stress [see Eq. (B4.10)] or secondary stress [see Eq. (B4.11)]. However, it will usually
be evaluated as secondary stress. Since the stress due to SAM is a cyclic type of stress,
it should be combined with other cyclic-type secondary stresses such as thermal
expansion stresses.
The total secondary stress range should include the thermal and SAM stress range.
If the SAM stress is less than the thermal stress range, the effective secondary stress
range is the sum of the SAM stress and the thermal stress range, as shown in Fig.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.168
GENERIC DESIGN CONSIDERATIONS
(b)
FIGURE B4.11 Effective secondary stress range.
B4.11a. If the SAM stress is higher than the thermal stress range, the effective secondary
stress range then equals twice the SAM stress, as shown in Fig. B4.11b.
Building Settlement Analysis. ASME Boiler and Pressure Vessel Code Section III
requires that the stresses due to building settlement be evaluated and be considered as
secondary stresses. However, the stress due to building settlement is a one-time (single
nonrepeated) anchor movement. Therefore, it is not required to combine it with other
stresses. From Subsection NC-3653.2(b) of the code, the effects of any single
nonrepeated anchor movement shall meet Eq. (B4.13).
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.169
Dynamic Loads
The dynamic loads discussed herein are limited to occasional loads (other than seismic
loads) frequently encountered in piping stress analysis.
Safety Relief Valve Discharge Analysis. Safety-relief valves are installed for the purpose
of protecting the fluid system from accidental overpressure, or venting the fluid
generated in excess of requirement.
The general requirements pertaining to the design of the pressure relief discharge
piping are provided in Appendix II of ASME B31.1, Subsections NC-3677 and NB3677 of the ASME Code for different pipe classes.
There are two types of pressure relief valve discharge, namely, open discharge and
closed discharge, as shown in the figures of Chap. B3.
Open Discharge. A typical open discharge is the transient due to discharging of
steam from a steam line to the atmosphere through relief valves or safety valves.
When the steam line pressure reaches the valve set point, the valve opens and
decompression waves will travel both upstream and downstream of the valve. This
flow transient sets up pressure imbalances along each pipe segment (a straight run of
pipe bounded by elbows). The transient forces can be calculated by a computerized
method as described in the later subsection “Steam Hammer-Water Hammer Analysis,”
while the reaction force at the valve exit due to steady-state flow is determined relatively
easily.
Closed Discharge. In a closed-discharge system, the fluid is transmitted to its
terminal receiver through continuous discharge piping. A typical closed discharge is
the transient induced by a sudden opening of the relief and safety valves located on
top of the pressurizer in a power plant. A water seal, which is maintained upstream
of each valve to minimize leakage, driven by this high discharge pressure, generates a
transient thrust force at each pipe segment. The water seal is discharged ahead of the
steam as the valve disk lifts. For discharge piping with a water seal, only the first cycle
of each event has a transient force based on water in the seal. The remaining cycles
would be based on steam occupying the seal piping, and the transient forces would
be reduced in magnitude.
Static Analysis. The static method of open discharge described in Appendix II of
ASME B31.1 can be summarized as follows:
1. The reaction force F due to steady-state flow following the opening of the valve
may be computed by
(B4.57)
where F
W
V
g
=
=
=
=
=
P=
Pa =
A=
reaction force at exit, lbf
mass flow rate, lbm/s
exit velocity, ft/s
gravitational constant
32.2 lbm · ft/lbf · s2
static pressure at exit, psia
atmospheric pressure, psia
exit area, in2
2. The dynamic load factor (DLF) is used to account for the increased load caused
by the sudden application of the discharge load. The DLF value will range between
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STRESS ANALYSIS OF PIPING SYSTEMS
B.170
GENERIC DESIGN CONSIDERATIONS
1.1 and 2.0, depending on the time history of the applied load and the natural
frequency of the piping.43 If the run pipe is rigidly supported and the applied load
could be assumed to be a single ramp function, the DLF may be determined in the
following manner:
a. Calculate the safety valve installation period T:
(B4.58)
where T = safety valve installation period, s
W = weight of safety valve, installation piping, flanges, attachments,
etc., lb
h = distance from run pipe to centerline of outlet piping, in
E = Young’s modulus of inlet pipe, psi, at design temperature
I = moment of inertia of inlet pipe, in4
b. Calculate the ratio to/T where to is the time the safety valve takes to go from
fully closed to fully open (seconds).
c. For the ratio to/T, determine the DLF from data given in Appendix II of ASME
B31.1, as shown in Fig. B4.12.
3. The moment due to valve reaction force is calculated by simply multiplying the
force times the distance from the point in the piping system being analyzed, times
a suitable DLF. The stress is then calculated accordingly.
FIGURE B4.12 Hypothetical dynamic load factor (DLF).
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STRESS ANALYSIS OF PIPING SYSTEMS
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B.171
Dynamic Analysis. The reaction force effects are dynamic in nature. A time-history
dynamic analysis of the discharge piping is considered to be more accurate.
Furthermore, closed-discharge systems do not easily lend themselves to simplified
analysis techniques. A time-history analysis (such as the one described in the following
subsection) is required to achieve realistic results.
Steam Hammer-Water Hammer Analysis. The steam hammer-water hammer event is
often initiated by intentional actuation of certain flow control devices (main steam
stop valve closure, feedwater pump trip, etc.), but in other cases a transient event
could be introduced unintentionally as the result of some unforeseen operating
condition, component malfunction, or accident (e.g., feedwater line check valve slam
following a pipe break upstream of the check valve).44,45 While these events may
produce very complex transient fluid flow, the pipe stress analyst is interested in just
the unbalanced force along the pipe segment tending to induce piping vibration.
Calculation of Unbalanced Forces. These time-history unbalanced forces are usually
generated through a two-step computerized calculation. The fluid system is modeled
as an assemblage of control volumes (e.g., piping volumes or steam generator)
interconnected by junctions (e.g., valves, pump, or break). Piping fluid flow data,
such as flow area, friction losses, valve closing-opening time, feed pump characteristics,
or break characteristics, together with fluid initial conditions (flow rate, pressure,
temperature, and mixture quality) are supplied as input to a thermal hydraulic finite
difference computer program.46
Using this input information and a built-in steam table (fluid thermodynamic state),
the first step solves the three equations of conservation (mass, momentum, and energy)
at each time step for fluid properties such as pressure, velocity, internal energy, and
mixture quality. A typical stop valve closure time history and its associated dynamic
pressure time history are shown in Fig. B4.13. The second step utilizes a postprocessor.
This postprocessor then accepts the output information from the first step and
computes the unbalanced forces in piping segments by applying the momentum
theorem.
Static Analysis. Static analysis is simple and saves computer time. It is used when
the unbalanced forces are small and the total transient time is long. In the analysis,
the peak values of the time-history fluid forcing functions at pipe segments are applied
statically to the piping. The piping stress, deflections, and support-nozzle loads are
then calculated by the computer program.
To obtain a conservative result for the static analysis, care must be taken in applying
a proper dynamic load factor to the unbalanced forces.
Dynamic Analysis. The dynamic analysis generally utilizes either the direct stepby-step integration method (as described in the subsection “Pipe Break Analysis”) or
the modal-superposition method. In the dynamic analysis, the piping system is idealized
as a mathematical model consisting of lumped masses connected by weightless elastic
members. These lumped masses are carefully located to adequately represent the
dynamic characteristics of the piping system. For computer programs utilizing the
modal-superposition method, enough modes (or appropriate cutoff frequency) should
be specified in the computer input such that the dynamic response of the piping
system can be truly represented. There are no specific guidelines to damping values
used in piping fluid transient dynamic analysis in the ASME Code or NRC published
material. Therefore, it is recommended to use the OBE damping values prescribed in
the NRC Regulatory Guide 1.61. Alternative damping values of Code Case N-411-1
are not applicable to the dynamic analysis.
The time-history unbalanced forces are applied to all pipe segments. Snubbers and
rigid supports are effective restraints for transient forces. However, axial supports
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STRESS ANALYSIS OF PIPING SYSTEMS
B.172
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.13 Steam hammer flow transients.
should be avoided in general. An axial support not only requires welded attachments
on the pipe but also a pair of supports, which doubles the cost. To support the pipe
axially, lateral supports can be used around the elbows. In addition, snubbers should
not be located in the immediate vicinity of equipment nozzles. Snubbers located in
such areas may not be activated during a fluid transient because of the dead band
(built-in manufacturing tolerance) of the snubber hardware.
Stress Allowables. For the steam hammer–water hammer (e.g., feed pump trip)
event, the pipe stress from the analysis is combined with stresses due to pressure,
deadweight, and OBE in meeting the upset stress allowable. For piping in the turbine
building, OBE stress is not included in the stress combination. For some water hammer
(e.g., check valve slam) events, the stress from the analysis is combined with stresses
due to pressure and deadweight in meeting the faulted stress allowable.
LOCA Analysis. LOCA (loss-of-coolant accident) is a postulated accident that results
from the loss of reactor coolant, at a rate in excess of the capability of the reactor
coolant makeup system, from breaks in the reactor coolant pressure boundary. Analyses
should be performed by the nuclear steam supply system (NSSS) vendor to confirm
the structural design adequacy of the reactor internals and reactor coolant piping
(unbroken loop) to withstand the loadings of the most severe LOCA in combination
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STRESS ANALYSIS OF PIPING SYSTEMS
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B.173
with SSE per the requirements of 10CFR Part 50, Appendix A,20 and the NRC Standard
Review Plan 3.9.2.47
The integrity of the secondary system piping (main steam, feedwater, blowdown
lines) off the steam generators also has to be assured by the architect-engineer (AE).
Additional analyses to demonstrate the structural adequacy of some of the branch
piping attached to the broken loop may be required by the NSSS vendor.48 The
information provided herein is limited to the secondary system piping off the steam
generators.
Static Analysis. If a substantial separation between the forcing frequencies of the
LOCA loading and the natural frequencies of the piping system can be demonstrated,
a static analysis may be performed. In the static analysis, the maxima of each of the
LOCA displacement components (three deflections and three rotations) are separately
applied to the junctions of the reactor coolant loop (RCL) and the secondary system
piping. The results should be combined absolutely and multiplied by an appropriate
dynamic load factor.
Dynamic Analysis. The dynamic analysis can be performed in one of the following
two ways:
Time-History Analysis. The LOCA displacement time history is applied dynamically
to the junctions of the RCL and the secondary system piping. The damping value
prescribed in the NRC Regulatory Guide 1.61 for SSE is suggested for this dynamic
LOCA analysis. The detailed analysis method is similar to that described in the
following subsection “Pipe Break Analysis.”49
Response Spectrum Analysis. Compared to the time-history analysis, the response
spectrum method is favorable for its low computer cost. However, this method may
be unnecessarily conservative since the same loading has to be applied to the entire
piping system. Because of the nature of the LOCA break and the impacting of the
gapped RCL supports, the LOCA motion has much higher frequency content than
the seismic excitation. The ZPA of a typical LOCA motion spectrum for a RCL
junction is usually higher than that of a typical SSE response spectrum. Therefore, a
higher cutoff frequency should be used in the analysis.50
Stress Allowables. The resulting stress from the LOCA analysis for the secondary
piping system is combined with the stresses due to pressure, dead weight, and SSE in
meeting the faulted stress allowables.
Pipe Break Analysis. Although it is extremely improbable that a pipe break will
occur as postulated, public safety and the NRC licensing requirements make it
necessary that such events must be considered in the design of high-energy piping
systems.
A high-energy piping system is a piping system that, during normal plant conditions,
is maintained at a temperature > 200°F (93.3°C), or a pressure > 275 psig (1896.1
kPa).
Pipe Break Locations. Pipe breaks are postulated in high-energy piping based on
the primary plus secondary stresses and the cumulative usage factor.
1. ASME Section III, Class 1 Pipe: Pipe breaks are postulated to occur at terminal
ends (the extremities of piping connected to structures, components, or anchors)
and at all intermediate locations where:
a. The primary plus secondary stress intensity range, as calculated by Equation
(10) of Subsection NB-3653 [i.e., Eq. (B4.4) of this chapter], exceeds 2.4Sm
and either Equation (12) or (13) [i.e., Eq. (B4.5) of this chapter] exceeds
2.4Sm.
b. Cumulative usage factor exceeds 0.1.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.174
GENERIC DESIGN CONSIDERATIONS
2. ASME Section III, Class 2 and 3 Pipe: Pipe breaks are postulated to occur at
terminal ends and at all intermediate locations where the primary plus secondary
stresses, as calculated by the sum of Equations (9) and (10) of Subsection NC3653 [i.e., Eqs. (B4.10) and (B4.11) of this chapter], exceed 0.8(1.2Sh + SA).
3. Nonnuclear piping: If a rigorous analysis, including seismic loading condition, is
done on a high-energy ASME B31.1 piping, the requirements of the Class 2 and
3 piping mentioned above will apply. If no analysis is performed, breaks are
postulated at the following locations:
a. Terminal ends
b. At all fittings, welded attachments, and valves
The detailed pipe break design criteria and guidelines are given in the NRC Standard
Review Plan No. 3.6.1 and 3.6.2.51,52
No-Break Zone. In the design of nuclear power plants, the region of piping in the
containment building penetration areas between the isolation valves requires extra
protection so that neither the leak-tight integrity of the containment nor the operability
of the containment isolation valves is jeopardized. The extra protection consists of
the following:
1. Installing special whip restraints, called isolation restraints, to mitigate the effects
of the postulated pipe breaks located beyond this region
2. Keeping the primary plus secondary stresses and the cumulative usage factor below
certain conservative values
3. Holding the piping stress, the isolation valve acceleration, and the stress at the
valve-pipe weld below specified limits during a postulated pipe break outside this
region
4. Special construction (welding) requirements and in-service inspection procedures
Because of the stringent design requirements, no pipe breaks are assumed to occur in
this region. This area of piping is often referred to as the no-break zone, the break
exclusion region, or the superpipe area.
No-Break Zone Piping Analysis. An analysis is required to determine the stresses in
no-break zone piping and the accelerations of isolation valves due to a postulated break
located beyond this region. During the pipe break event, a portion of the piping and the
isolation restraints may enter the inelastic region because of the large pipe break loads
imposed on the piping system. A static method, or the energy balance method, is
acceptable but usually not used because necessary information on the no-break zone
such as isolation valve acceleration is impossible to determine. Therefore, a nonlinear
dynamic analysis utilizing the direct step-by-step integration method is necessary for
the no-break zone analysis.53 Computer programs based upon the direct integration
method with linear elastic and nonlinear inelastic capabilities are often used for this
type of analysis.54,55
In the analysis, the piping structural model is similar to that described in the
subsection “Steam Hammer–Water Hammer Analysis.” The nonlinear effects are
accounted for by updating the system stiffness matrix at the end of each time step.
The integration time step must be short enough to permit a reliable and stable solution.
In addition, suitable system damping values should be used to obtain numerical
stability. The time-history pipe break forcing function can be calculated by thermalhydraulic computer programs as described in the subsection “Steam Hammer–Water
Hammer Analysis,” or obtained from the acceptable simplified method specified in
Appendix B of ANSI/ANS 58.2.56
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B.175
Wind Loads. The wind possesses kinetic energy by virtue of the velocity and mass of
the moving air. If an obstacle is placed in the path of the wind so that the moving air
is stopped or is deflected, then all or part of the kinetic energy of the wind is transformed
into the potential energy of pressure.
A piping system which is located outdoors is usually designed to withstand the
maximum wind velocity expected during the system operating life.
Dynamic Pressure. The intensity of wind pressure depends on the shape of the
obstacle, the angle of incidence of the wind, and the velocity and density of the air.
For standard air (density of the air = 0.07651 lb/ft3, temperature = 59°F), the
expression for the wind dynamic pressure could be adapted from Bernoulli’s equation
for fluid flow as follows33,57:
(B4.59)
where p = dynamic pressure, lb/ft2
V = basic wind speed, mi/h
CD = drag coefficient, dimensionless
For the case of piping under wind loading, Eq. (B4.59) can be rewritten as
(B4.60)
where F = linear dynamic pressure loading on projected pipe length, lb/ft
D = pipe diameter, including insulation, in
FIGURE B4.14 Basic wind speed (miles per hour). (ANSI A58.1, 1982. Courtesy of ANSI.)
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B.176
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.15 Drag coefficients for spheres and long cylinders. (Task Committee on Wind
Forces, “Wind Forces.”30 Courtesy of ASCE.)
Basic Wind Speed. The basic wind speed V is the fastest wind speed at 33 ft above
the ground in open terrain with scattered obstructions having heights less than 30 ft,
as given in Fig. B4.14 for the United States.33 The basic wind speed used for design
shall be at least 70 mi/h.
Drag Coefficient. The drag coefficient CD is a function of the shape of the structure
and a fluid flow factor called the Reynolds number. The Reynolds number R is the
ratio of the inertial force to the viscous force which a fluid stream exerts on an object.
For standard air, the Reynolds number R could be expressed as
(B4.61)
The drag coefficient CD for a cylinder (i.e., a pipe) is given versus the Reynolds
number in Fig. B4.15.
Wind Loading Analysis. The piping wind loading analysis is usually performed
by a static method. In the analysis, the wind loading F is modeled as a uniform load
acting over the projected length of the pipe, parallel to the direction of the wind. Two
horizontal directions of wind loads (north-south and east-west) are included in the
analysis. The design loads are based on the worst case of the two directions. Similar
to the case of earthquake, the wind loading is considered reversing. For load
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B.177
TABLE B4.8 Gust Response Factor G
Source: ANSI A58.1, 1982. (Courtesy of ANSI.) (1 ft = 0.3048 m)
combination, the wind and the earthquake are assumed to not happen at the same
time. A safety factor, the gust response factor G, should also be considered in the
analysis. This factor is used to account for the fluctuating nature of wind and its
interaction with structures. Its value depends on the exposure categories as shown in
Table B4.8, where:
1. Exposure A: Large city centers with at least 50 percent of the buildings having a
height in excess of 70 ft
2. Exposure B: Urban and suburban areas, wooded areas, or other terrain with
numerous closely spaced obstructions having the size of single-family dwellings
or larger
3. Exposure C: Open terrain with scattered obstructions having heights generally
less than 30 ft
4. Exposure D: Flat, unobstructed coastal areas directly exposed to wind flowing
over large bodies of water
METHODS OF ANALYSIS
Cookbook-Type Analysis
The following cookbook-type method is mainly for supporting 2-in and smaller
Nuclear Class 2, 3, and B31 piping under gravity, thermal expansion, and seismic
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GENERIC DESIGN CONSIDERATIONS
B.178
loadings. This method is based on standard support span tables. It covers a simplified
weight analysis, a simplified thermal analysis, as well as a simplified seismic analysis.
A simplified seismic analysis often requires many pipe supports that are designed
to large loads. The cost saving in engineering is offset by increased fabrication and
installation cost. The current approach is to analyze the nonseismic piping by simplified
methods and all seismic piping by computerized analysis. This greatly reduces the
number of required seismic supports and gives an overall cost saving.
Simplified Weight Analysis. A simplified weight analysis is performed by locating
the gravity supports based on gravity pipe spans. The maximum gravity pipe spans
can be calculated from the following formula:
(B4.62)
where L
Z
W
S
=
=
=
=
maximum gravity pipe span, ft
section modulus of pipe, in3
distributed weight, lb/ft
pipe stress due to gravity, psi
and the corresponding stress S is
(B4.63)
where M = bending moment, in · lb.
Alternatively, the bending stress in empty pipe may be read from Fig. B4.16, and
the bending stress in water-filled pipe from Fig. B4.17. The deflection of empty pipe
can be read from Fig. B4.18.
The distributed weight of pipe includes the weight of metal, the weight of pipe
contents, and the weight of insulation. Pipe material weights are subject to tolerance
of applicable manufacturing specifications.
Weights of insulation depend on the composition of insulation material and should
be obtained from the insulation manufacturer. Weights of weatherproof protection,
if specified, must be added. Insulation thicknesses recommended by insulation
manufacturers do not necessarily agree with insulation specifications for a particular
job. Insulation specifications should be reviewed prior to development of final weights
of piping.
Weights of insulation should be added to weights of flanges, valves, and fittings.
Flange, flanged valve, and flanged fitting weights should include weights of bolts and
nuts.
Valve weights vary among particular manufacturers’ designs and should include
weights of electric-motor operators (if any) or other devices which may be specified
for particular valves. It is suggested that, wherever possible, valve weights should be
obtained from the manufacturer of the particular valves which are to be installed in
the piping.
Equation (B4.62) is based on the combination of a simply supported beam model
and a fixed-end beam model because the behavior of pipe lies somewhere between
these two models.
A table of suggested maximum spans between supports of pipe based on a formula
similar to Eq. (B4.62) is given in ASME Codes,58,59 as shown in Fig. B5.1 of Chap. B5.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.179
FIGURE B4.16 Bending stress in empty pipe.
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B.180
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.17 Bending stress in water-filled pipe.
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B.181
FIGURE B4.18 Deflection of empty pipe.
These spans have been calculated by considering insulated, standard wall thickness
and heavier pipe, limited to a maximum stress of 1500 psi (10,350 kPa) and maximum
pipe sag of 0.1 in (0.25 mm). For small pipe where socket welds are used, Eq. (B4.63)
can be rewritten as
(B4.64)
and
(B4.65)
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STRESS ANALYSIS OF PIPING SYSTEMS
B.182
GENERIC DESIGN CONSIDERATIONS
where i = the stress intensification factor (SIF), 2.1 for socket welds per ASME
B31.1 and ASME Section III piping codes.
Figures B5.2 and B5.3 of Chap. B5 give the maximum spans for water and steam,
air, or gas filled steel pipe, respectively. These tables are based on a pipe stress S of
2000 psi (13.790 MPa) and a socket weld SIF of 2.1. When these suggested weight
spans are adhered to, the stress in the piping system due only to gravity load usually
need not be explicitly calculated.
Load Calculation by Weight Balance. The following example is used to illustrate a
method by which hanger loadings may be determined. The method consists of locating
the center of gravity of the specific piping configuration and then, by equating moments,
to determine the resultant loads at particular hangers.
A single-plane bend is shown in Fig. B4.19. Hangers are indicated as H-1, H-2, H3, and H-4. The effects of uniform and concentrated loads are indicated at the points
at which these loads act; it is noted that the weight of the 90° bend acts at the centroid
of a quarter circle which, in this example, is located 1.8 ft distant from the centerline
of the pipe run. The straight pipe length between hangers H-3 and H-4 is not included
in this calculation because it can be analyzed by simple straight-beam theory.
For the piping section which lies between equipment flange F and hanger H-3,
moments are taken about the Y-Y and Z-Z axes. As an example, let the center of
gravity of this configuration be located Y ft from the Y-Y axis. Then, from equilibrium
considerations, the following equation may be written:
A solution to this equation results in Y = 7.75 ft.
Similarly, the distance from the Z-Z axis to the center of gravity is found to be
6.43 ft.
For convenience, the calculations are made frequently in a tabular fashion as shown
in Fig. B4.19.
FIGURE B4.19 One-line piping diagram for illustration of load calculation by weight balance.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.183
FIGURE B4.20 Hanger load calculations for system of Fig. B4.19.
Three hangers with zero reaction at flange F.
Let it be now required to determine hanger loadings for the piping configuration
of Fig. B4.19 with the stipulation that no load due to weight be imposed on the
equipment flange F. This is accomplished easily by use of simple geometrical
relationships, and the solution is as indicated in Fig. B4.20.
If it were desired to support the piping with two, rather than three, hangers, it
would be convenient to eliminate H-1 and to relocate H-2 to a position at which it
would be colinear with the center of gravity and hanger H-3. The construction for
this arrangement and the associated hanger-load calculations are shown in Fig. B4.21.
In each of the two above cases, one-half of the 2320-lb load between H-3 and H4 has been included in the calculations for hanger loading on H-3. Thus H-4 would
be required to support 1160 lb plus, of course, any additional piping load to the right
of H-4 in Fig. B4.19.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.184
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.21 Hanger load calculations for system of Fig.
B4.19 except that one hanger has been eliminated.
Simplified Thermal Expansion Analysis. This simplified analysis is based on the
guided cantilever method. The guided cantilever is a cantilever beam restrained in
such a way that its free end will not rotate when it is deflected in a direction
perpendicular to the longitudinal axis of the beam, as shown in Fig. B4.22.
For piping systems under thermal expansion loads, the behavior of the piping
approximates that of a guided cantilever. The thermal growth forces the pipe leg to
FIGURE B4.22 Guided cantilever.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.185
translate while pipe rotations are restricted by piping continuity. Therefore, this method
can be used to check the flexibility of a piping system.
For a guided cantilever, the moment induced by an imposed deflection is
(B4.66)
where M
E
I
∆
L
=
=
=
=
=
induced moment, in · lb
modulus of elasticity, psi
moment of inertia, in4
deflection, in
length of pipe leg perpendicular to deflection, in
The corresponding stress is then
(B4.67)
where S
D
Z
i
=
=
=
=
induced stress, psi
outside diameter of pipe, in
section modulus of pipe, in3
stress intensification factor
Solving the equation for the beam length L gives
(B4.68)
By determining the proper allowable stress and taking into account the appropriate
stress intensification factor, Eq. (B4.68) gives an estimate of the minimum allowable
offset pipe span L required to sustain a piping thermal movement ∆ normal to the
piping.
Tables B4.9 and B4.10 give the minimum allowable offset span for steel piping (E
= 27.9 × 106 psi) with socket welds (i = 2.1) and without socket welds, respectively.
These tables are based on allowable stresses S of 22,500 psi.
Thermal Movement Calculations. The simplified method shown below is one which
gives satisfactory approximations of the piping movements. Whenever differences
occur between the approximations and actual movements, the approximation of the
movement will always be the greater amount.
Step 1. The piping system of Fig. B4.23 is drawn, and on it are shown all known
vertical movements of the piping from its cold to hot, or operating, position. These
movements will include those supplied by the equipment manufacturers for the terminal
point connections. For the illustrated problem, the following vertical movements are
known:
Point A—2 in up, cold to hot
Point B—1 1/6 in up, cold to hot
Point C—1/8 in down, cold to hot
H-4—0 in cold to hot
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STRESS ANALYSIS OF PIPING SYSTEMS
GENERIC DESIGN CONSIDERATIONS
B.186
TABLE B4.9 Thermal Expansion Minimum Allowable Offset Span (Feet-Inches),
Straight Steel Pipe with Socket Welds
1 in = 25.4 mm
1 ft = 0.3048 m
The operating temperature of the system is given as 1050°F (566°C), and the
coefficient of expansion for low-chrome steel at 1050°F (566°C) is 0.0946 in/ft.
The movements at points D and E are calculated by multiplying the coefficient of
expansion by the vertical distance of each point from the position of zero movement
on the riser DE:
Step 2. A simple drawing is made of the piping between two adjacent points of
known movement, extending the piping into a single plane as shown for the portion
of the system between A and D.
The vertical movement at any hanger location will be proportional to its distance
from the endpoints:
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STRESS ANALYSIS OF PIPING SYSTEMS
B.187
TABLE B4.10 Thermal Expansion Minimum Allowable Offset Span (Feet-Inches),
Straight Steel Pipe, No Socket Weld
1 in = 25.4 mm
1 ft = 0.3048 m
The vertical movement at H-1 = 0.41 in + 2 in:
The vertical movement at H-2 = 2.27 in + 2 in:
∆H-2 = 4.27 in up
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GENERIC DESIGN CONSIDERATIONS
B.188
Step 3. To calculate the vertical movement at H-3, multiply
its distance from H-4 by the coefficient of expansion:
Step 4. The next section of pipe on which there are two
points of known movement is the length E-J. The movement
at E was calculated as 1.89 in down:
The movement at J is equal to the movement at the terminal
point C (1/8 in down) plus the amount of expansion of the
leg C-J:
∆J = 0.125 in + 3.5 ft × 0.0946 in/ft
∆J = 0.46 in down
∆7 =
3.5
/42 × 1.43 = 0.12 in
∆H-7 = 0.12 in + 0.46 in
∆H-7 = 0.58 in down
∆6 =
17
/42 × 1.43 = 0.58 in
∆H-6 = 0.58 + 0.46 in
∆H-6 = 1.04 in down
∆f =
30
/42 × 1.43 = 1.02 in
∆F = 1.02 + 0.46
∆F = 1.48 in down
∆5 =
32
/42 × 1.43 = 1.09 in
∆H-5 = 1.09 + 0.46
∆H-5 = 1.55 in down
Step 5. In the section G-H, the movement at G is
equal to the movement at F minus the expansion
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STRESS ANALYSIS OF PIPING SYSTEMS
B.189
FIGURE B4.23 One-line piping diagram for calculation of hanger movements. Points A,
B, and C are equipment connections. H-1, H-2, and so on, represent hanger locations.
of the leg GF:
∆G = 1.48 in down – 4 ft × 0.0946 in/ft
∆G = 1.10 in down
The movement at H is equal to the movement of the terminal point B (1/16 in up)
plus the expansion of the leg B-H:
∆H = 0.0625 in up + 9 ft × 0.0946 in/ft
∆H = 0.91 in up
Since H-9 is located at point H,
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STRESS ANALYSIS OF PIPING SYSTEMS
B.190
GENERIC DESIGN CONSIDERATIONS
After calculating the movement at each hanger location it is often helpful, for easy
reference when selecting the appropriate type hanger, to make a simple table of hanger
movements.
1 in = 25.4 mm
Calculation of Hanger Loads. A 6-in medium-temperature steam piping system is
shown in Fig. B4.24. Terminal movements at equipment flanges A and B are indicated;
dimensions of system components and physical data are also given. It is required to
determine hanger loadings and also to determine movements at each of the hangers
H-1 through H-7.
It is noted that hanger H-3 on the vertical leg has been located 20 ft (6.0 m) above
the lower horizontal pipe run. Calculations would indicate that the center of gravity
of the vertical leg is 16.16 ft (5.0 m) above the lower horizontal run. It would not be
desirable to place the hanger at the center of gravity because the hanger would then
act as a pivot point and would not resist sway. If the hanger H-3 were placed below
the center of gravity, an unstable turnover condition would result. The most desirable
location is above the center of gravity; hanger H-3 has thus been placed arbitrarily a
distance of 20 ft (6.0 m) above the lower horizontal piping run.
Starting with equipment flange A, the system is broken up into component parts
between hangers and hanger reactions are calculated. The procedure is indicated in
Figs. B4.25a to B4.25g, and the results are listed in Table B4.11. Hanger deflections,
or movements, are determined as shown in Figs. B4.26a and B4.26b.
Simplified Seismic Analysis. A simplified seismic analysis utilizing simple beam formulas
and response spectrum curves is given here. The maximum support spacings are selected
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.191
FIGURE B4.24 One-line piping diagram for calculation of hanger loadings and deflections.
from Tables B4.12 to B4.14 so that the fundamental frequency of the span is in the
rigid range of the response spectrum.
These seismic spans are based on the following formula:
(B4.69)
where L
f
g
E
I
W
=
=
=
=
=
=
maximum seismic spacing, ft
desired frequency, cycles/s
386 in/s2
modulus of elasticity, psi
moment of inertia, in4
distributed weight, lb/ft
For a system with seismic supports designed in the rigid range, the seismic acceleration
of the system is low and consequently the design loads for the system decrease.
The corresponding seismic stress is then
(B4.70)
where Z = section modulus of pipe, in3
G = seismic acceleration (OBE or SSE) in gs
i = stress intensification factor
The number 1.5 in Eq. (B4.70) is a factor to account for the contribution from the
higher modes.47,60
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STRESS ANALYSIS OF PIPING SYSTEMS
B.192
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.25 (a) Distribution of weight between
equipment flange A and H-1. (b) Distribution of weight
between H-1 and H-2.
Computerized Method
Types of Computer Programs. The microcomputer has become the daily tool and
workstation of the piping stress analyst. Files which contain data for piping stress
analysis are created, edited, and saved at this workstation. These files are later
transferred to the mini- or mainframe computer for the calculation of piping stresses
and support loads.
Most of the computer programs for piping stress analysis such as ADLPIPE,
NUPIPE, and SUPERPIPE were developed for use on mainframe computers. With
the introduction of many powerful microcomputers in the mid-1980s, microcomputerbased programs for piping stress analysis were also developed such as AU-TOPIPE
and CAESAR II. Some of these new programs are menu driven and user friendly.
Refer to Appendix E9. They help save engineering time and cost. In general, these
computer programs may be divided into four classes:
1. Programs that can perform pressure, thermal expansion, deadweight, and external
forces (e.g., wind) analyses for ASME Section III, Class 2, 3, ASME B31.1, B31.3,
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.193
FIGURE B4.25 (c) Distribution of weight between H-2 and
H-3. (d) Distribution of weight between H-3 and H-4.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.194
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.25 (e) Distribution of weight between H-4 and
H-5. (f) Distribution of weight between H-5 and H-6.
B31.4, B31.5, B31.8, NEMA, API-610, and API-617 piping. Programs such as
TRIFLEX, AUTOPIPE, and CAESAR II are in this class. (AUTOPIPE and CAESAR
II have response spectrum and SAM analysis capability. However, there is a limit
on the number of analyses which can be performed in the same computer run
because of the memory capability of microcomputers.)
2. Programs that can perform seismic, independent support motion, thermal transient,
and time-history analyses in addition to those mentioned in item 1 for ASME Section
III, Class 1, 2, 3, ASME B31.1, and B31.3 piping. Programs such as ADLPIPE,
ME101, NUPIPE, PIPESD, and SUPERPIPE are in this class.
3. General-purpose programs, such as ANSYS. ANSYS is a general-purpose finite
element analysis program which can perform static and dynamic analysis; elastic
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.195
FIGURE B4.25 (g) Distribution of weight between H-6 and H-7 to maintain zero
reaction on flange B.
TABLE B4.11 Summary of Hanger Loadings
1 lb = 4.448 N
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STRESS ANALYSIS OF PIPING SYSTEMS
B.196
GENERIC DESIGN CONSIDERATIONS
FIGURE B4.26 (a) Deflections of vertical leg of Fig. B4.24. (b)
Determination of deflections at H-1 and H-2 of Fig. B4.24.
and plastic analysis; steady-state and transient heat transfer; steady-state
fluid flow analyses; and nonlinear time-history analyses. There are 40 different
finite elements available for static and dynamic analysis. Dynamic analyses
can be performed either by modal superposition or direct integration.54
4. Specialized programs such as PIPERUP. PIPERUP performs nonlinear
elasticplastic analyses of piping systems subjected to concentrated static or
dynamic time-history forcing functions. These forces result from fluid jet
thrusts at the location of a postulated break in high-energy piping. PIPERUP
is an adaptation of the finite element method to the specific requirements of
pipe rupture analysis. 55
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* Insulation type. 1 in = 25.4 mm, 1 ft = 0.3048 m
TABLE B4.12 Maximum Support Spacing for Seismic Stress for a Frequency of 20 Hz (Feet -Inches), Steel Pipe
STRESS ANALYSIS OF PIPING SYSTEMS
B.197
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* Insulation type. 1 in = 25.4 mm, 1 ft = 0.3048 m
TABLE B4.13 Maximum Support Spacing for Seismic Stress for a Frequency of 20 Hz (Feet -Inches), Steel Pipe
STRESS ANALYSIS OF PIPING SYSTEMS
B.198
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* Insulation type. 1 inch = 25.4 mm, 1 ft = 0.0348 m
TABLE B4.14 Maximum Support Spacing for Seismic Stress for a Frequency of 20 Hz (Feet -Inches), Steel Pipe
STRESS ANALYSIS OF PIPING SYSTEMS
B.199
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STRESS ANALYSIS OF PIPING SYSTEMS
B.200
GENERIC DESIGN CONSIDERATIONS
Method of Analysis. The piping system is modeled as a series of masses connected by
massless springs having the properties of the piping. The mathematical model should
include the effects of piping geometry changes, elbow flexibilities, concentrated weights,
changes in piping cross sections, and any other parameters affecting the stiffness matrix
of the model. Mass point spacing should follow the guidelines specified above. Valves
should be modeled as lumped masses at valve body and operator, with appropriate
section properties for valve body and valve topworks. Rigid supports, snubbers, springs,
and equipment nozzles should be modeled with appropriate spring rates in particular
degree of freedom. Stress intensification factors should be input at the appropriate
locations (elbows, tees, branch connections, welds, etc.). Piping distributed weight should
include pipe weight, insulation weight, and entrained fluid weight.
Once an accurate model is developed, the loading conditions are applied
mathematically:
1. Statically applied loads (deadweight, wind loads, pressure thrust, etc.)
2. Thermal expansion
3. Statically applied boundary condition displacements (seismic anchor
movement, LOCA containment displacement, etc.)
4. Response spectrum analysis (seismic, etc.)
5. Dynamically applied boundary condition displacements (LOCA motion, etc.)
6. Dynamically applied forcing functions (steam hammer, etc.)
The results of the analyses should be examined in order to determine if all allowables
are met (i.e., piping stress, valve acceleration, nozzle loads, etc.). The loads must be
combined using the appropriate load combinations and submitted to structural
designers for their analysis.
PROCEDURES FOR THE DESIGN OF RESTRAINED
UNDERGROUND PIPING
This section is reproduced by the courtesy of ASME B31.1.
Foreword
The B31.1 Code contains rules governing the design, fabrication, materials, erection,
and examination of power piping systems. Experience over the years has demonstrated
that these rules may be conservatively applied to the design and analysis of buried
piping systems. However, the ASME B31.1 rules were written for piping suspended
in open space, with the supports located at local points on the pipe. Buried piping, on
the other hand, is supported, confined, and restrained continuously by the passive
effects of the backfill and the trench bedding. The effects of continuous restraint
cannot be easily evaluated by the usual methods applied to exposed piping, since
these methods cannot easily accommodate the effects of bearing and friction at the
pipe/soil interface. Accordingly, this section has been prepared to illustrate and clarify
the application of B31.1 Code rules to restrained buried piping.
All components in the buried piping system must be given consideration, including
the building penetrations, branches, bends, elbows, flanges, valves, grade penetrations,
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.201
and tank attachments. It is assumed that welds are made in accordance with the
B31.1. Code and that appropriate corrosion protection procedures are followed for
buried piping.
This section provides analytic and nomenclature definition figures to assist the
designer, and is not intended to provide actual design layout.
Scope
The scope of this section is confined to the design of buried piping as defined in Pa+
expansion in buried piping affects the forces, the resulting bending moments and
stresses throughout the buried portions of the system, particularly at the anchors,
building penetrations, buried elbows and bends, and branch connections, and it is
the designer’s responsibility to consider these forces. This section, however, deals
only with the buried portions of the system, and not the complete system.
The design and analysis of buried piping requires that careful attention be paid to:
1. All loads acting on the system
2. The forces and the bending moments in the piping and piping components
resulting from the loads
3. The loading and stress criteria
4. General design practices
Definitions
Confining Pressure the pressure imposed by the compacted backfill and overburden
on a buried pipe. Confining pressure is assumed to act normal to the pipe circumference.
Flexible Coupling a piping component that permits a small amount of axial or angular
movement while maintaining the pressure boundary
Friction the passive resistance of soil to axial movement. Friction at the pipe/soil
interface is a function of confining pressure and the coefficient of friction between
the pipe and the backfill material. Friction forces exist only where there is actual or
impending slippage between the pipe and soil.
Influence Length that portion of a transverse pipe run which is deflected or
“influenced” by pipe thermal expansion along the axis of the longitudinal run
Modulus of Subgrade Reaction the rate of change of soil bearing stress with respect
to compressive deformation of the soil. It is used to calculate the passive spring rate
of the soil.
Penetration the point at which a buried pipe enters the soil either at grade or from a
wall or discharge structure
Settlement the changes in volume of soil under constant load which result in the
downward movement, over a period of time, of a structure or vessel resting on
the soil
Virtual Anchor a point or region along the axis of a buried pipe where there is no
relative motion at the pipe/soil interface
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STRESS ANALYSIS OF PIPING SYSTEMS
B.202
GENERIC DESIGN CONSIDERATIONS
Nomenclature
a, b, c = quadratic equation functions
A = cross-sectional metal area of pipe, in2
Ac = surface area of a 1-in long pipe segment, in2
Bd = trench width at grade, in
CD = soil bearing parameter from Table B4.15, dimensionless
Ck = horizontal stiffness factor for backfill [61],2 dimensionless
D = pipe outside diameter, in
dL = length of pipe element, in
E = Young’s modulus for pipe, psi
f = unit friction force along pipe, lb/in
fmin, fmax = minimum, maximum unit friction force on pipe, lb/in
Ff = total friction force along effective length, lb
Fmax = maximum axial force in pipe, lb
H = pipe depth below grade, in
I = pipe section moment of inertia, in4
k = soil modulus of subgrade reaction, psi
kh = soil horizontal modulus of subgrade reaction, psi
ki,j = orthogonal soil springs on pipe, lb/in
kv = soil vertical modulus of subgrade reaction, psi
L1 = length of transverse pipe run, in
L2 = length of longitudinal pipe run, in
Lm = minimum slippage length of pipe, in
L⬘ = effective slippage length for short pipes, in
L⬙ = effective slippage length for long pipes, in
n = number of modeling elements for pipe springs, dimensionless
Nh = horizontal force factor,61 dimensionless
P = maximum operating pressure in pipe, psi
Pc = confining pressure of backfill on pipe, psi
SA = allowable expansion stress range, psi
SE = expansion stress, psi
Sh = basic material allowable stress at T degrees fahrenheit, psi
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STRESS ANALYSIS OF PIPING SYSTEMS
B.203
t = pipe wall thickness, in
T = maximum operating temperature, °F
To = ambient temperature of pipe, °F
w = soil density, pcf, pci
Wp = unit weight of pipe and contents, lb/in
α = coefficient of thermal expansion of pipe, in/in/°F
ß = pipe/soil system characteristic,62 in-1
ε = pipe unit thermal expansion, in/in
µ = coefficient of friction, dimensionless
Ω = effective length parameter, in
1 inch = 25.4 mm
1 lb = 4.448 N
1 psi = 6894.8 Pa
deg F = 1.8 deg C + 32
Loads
Thermal Expansion. Thermal displacements at the elbows, branch connections, and
flanges in a buried piping system and the forces and moments resulting from the
displacements may be determined by analyzing each buried run of pipe by the method
described in this section.
Installations with Continuous Runs. For buried piping installations that
contain continuous runs without flexible couplings, the passive restraining effects
of soil bearing on the transverse legs at the ends of long runs subject to thermal
expansion may be significant and result in high axial forces and elbow or branch
connection bending moments.
Installations with Flexible Couplings. For buried piping installations that
incorporate flexible couplings into the pipe runs subject to thermal expansion,
the bending moments and stresses may be substantially reduced. However, the
flexible couplings must be chosen carefully to accommodate the thermal
expansion in the pipe, and the friction forces or stiffness in the coupling must be
considered.
Installations with Penetration Anchors. For buried piping systems in which
the building penetration provides complete restraint to the pipe, it is necessary
to calculate the penetration reactions to thermal expansion in the initial buried
run. If this run incorporates flexible couplings, piping reactions at the
penetration resulting from unbalanced forces due to internal pressure must be
considered.
Installations with Flexible Penetrations. For buried piping systems in which
the building penetrations permit some axial or angular movements, the interaction
between the buried run outside the penetration and the point-supported portion
of the system inside the building must be considered.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.204
GENERIC DESIGN CONSIDERATIONS
Pressure. Pressure loads in buried piping are important for two primary reasons:
1. In pipe runs which incorporate flexible couplings, there is no structural tie
between the coupled ends, with the result that internal pressure loads must be
reacted externally. External restraint may be provided by thrust blocks, external
anchors, soil resistance to elbows or fittings at each end of the pipe run, or by
control rods across the coupling. Where one or both of the ends terminate at a
penetration or an anchor, or at connected equipment such as a pump or vessel,
the pressure forces can be quite high and must be considered in the anchor or
equipment design.
2. For discharge structures, the reaction forces due to upstream pressure and mass
flow momentum in the discharge leg may be high and must be considered in the
design of the last elbow or bend before the discharge.
Earthquake. An earthquake subjects buried piping to axial loads and bending moments
from soil strain due to seismic waves, or from ground faulting across the axis of the
pipe. The seismic soil strain can be estimated for a design earthquake in a specific
geographical region, from which design values for forces and moments in buried
piping can be calculated. However, consideration of the magnitude and effects of
seismic ground faulting on buried piping is beyond the scope of this section.
Calculations
The calculations for stresses in restrained underground piping are carried out in four
steps, as follows.
Assembling the Data. The pipe material and dimensions, soil characteristics, and
operating conditions must be established:
Pipe Data
1.
2.
3.
4.
5.
Pipe outside diameter D, in
Wall thickness t, in
Length of pipe runs L1 (transverse) and L2 (longitudinal), in
Young’s modulus E, psi
Pipe depth below grade H, in
Soil Characteristics
1.
2.
3.
4.
Soil density w, pcf (from site tests)
Type of backfill
Pipe trench width at grade Bd, in
Range of coefficient of friction µ between pipe and backfill
Operating Conditions
1.
2.
3.
4.
Maximum operating pressure P, psi
Maximum pipe temperature T, °F
Ambient pipe temperature T o, °F
Pipe coefficient of thermal expansion α, in/in/°F
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.205
Calculations of Intermediate Parameters. The following parameters must be
calculated:
Maximum Relative Strain ε at the Pipe/Soil Interface, in/in. For thermal expansion,
this is the unit thermal elongation of the unrestrained pipe,
(B4.71)
where α = coefficient of thermal expansion
T - To = difference between operating and installation temperatures
Modulus of Subgrade Reaction k, psi. This is a factor which defines the resistance
of the soil or backfill to pipe movement due to the bearing pressure at the pipe/soil
interface. Several methods for calculating k have been developed in recent years by
Audibert and Nyman, Trautmann and O’Rourke, and others.63–67 For example,61 for
pipe movement horizontally, the modulus of subgrade kh may be found by
(B4.72)
where Ck = a dimensionless factor for estimating horizontal stiffness of compacted
backfill. Ck may be estimated at 20 for loose soil, 30 for medium soil,
and 80 for dense or compacted soil.
w = soil density, lb/in3
D = pipe outside diameter, in
Nh = a dimensionless horizontal force factor from Fig. 8 of Ref. 61. For a
typical value where the soil internal friction angle is 30°, the curve from
Ref. 61 may be approximated by a straight line defined by
where H = the depth of pipe below grade at the pipe centerline, in
For pipe movement upward or downward, the procedures recommended in Ref.
63 may be applied. Conservatively, the resistance to upward movement may be
considered the same as for horizontal movement with additional consideration for
the weight of the soil. Resistance to downward movement may conservatively be
considered as rigid for most expansion stress analysis.
Unit Friction Force at the Pipe/Soil Interface f.
(B4.73)
where µ
Pc
Ac
Wp
=
=
=
=
coefficient of friction between pipe and soil
confining pressure of soil on pipe, psi
surface area of a pipe segment, in2
unit weight of pipe and contents, lb/in
For piping which is buried within 3 pipe diameters of the surface, confining pressure
Pc may be estimated by
where w = the soil density, lb/in
H = the depth below grade, in
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STRESS ANALYSIS OF PIPING SYSTEMS
B.206
GENERIC DESIGN CONSIDERATIONS
For piping which is buried more than 3 pipe diameters below grade, confining
pressure Pc is found by using the modified Marston equation67:
where CD = a dimensionless parameter obtained from Table B4.15
BD = the trench width, with a maximum value of 24 in plus the pipe diameter
TABLE B4.15 Approximate Safe Working Values of CD for Use in
Modified Marston Formula
Pipe/Soil System Characteristic62
(B4.74)
where k = soil modulus of subgrade reaction kh or kv, psi
E = Young’s modulus for pipe, psi
I = area moment of inertia for pipe, in4
Minimum Slippage Length Lm68
(B4.75)
where A = pipe cross-section area
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.207
Maximum Axial Force Fmax in the Longitudinal Pipe Run. The maximum axial
force in a pipe long enough for friction force to develop to the point where a region
of the pipe is totally restrained longitudinally by the soil is found by
(B4.76)
Classification of the Pipe Runs
Purpose. The classification and subclassification of the buried pipe elements is
used in choosing the proper equation for effective slippage length L⬘ or L⬙ which is
then used in calculating piping forces and stresses. The pipe segment identified by the
dimension L⬘ or L⬙ always begins at either an elbow, bend, tee, or branch connection
and terminates at the point (described below as the virtual anchor) at which there is
no slippage or relative movement at the pipe/soil interface.
Classification of the Pipe Elements. It is in the bends, elbows, and branch
connections that the highest stresses are found in buried piping subject to thermal
expansion of the pipe. These stresses are due to the soil forces that bear against the
transverse run (the run running perpendicular or at some angle to the direction of the
pipe expansion). The stresses are proportional to the amount of soil deformation at
the elbow or branch connection.
Piping elements are divided into three major categories depending upon what type
of transverse element is being analyzed, as follows:
Category A. Elbow or bend (see Fig. B4.27)
FIGURE B4.27 Element category A, elbow or bend.
Category B. Branch pipe joining the longitudinal run (see Fig. B4.28)
FIGURE B4.28 Element category B, branch pipe joining
the P leg.
Category C. Longitudinal run ending in a tee (see Fig. B4.29)
FIGURE B4.29 Element category C, tee on end of P leg.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.208
GENERIC DESIGN CONSIDERATIONS
Category D. Straight pipe, no branch or transverse run (see Fig. B4.30)
FIGURE B4.30 Element category D, straight pipe.
Categories A, B, and C are further divided into three subcategories depending on
the configuration of the pipe run at the end opposite that being analyzed. The piping
elements are classified as follows:
A1, B1, C1. Other end free or terminating in a flexible coupling or joint
A2, B2, C2. Other end contains an elbow or tee
A3, B3, C3. Other end is anchored
Category D elements include straight runs between an anchor (either actual or
virtual) and a free end or a pipe section that is connected to an expansion joint.
The elements are further broken down into subtypes depending upon whether the
longitudinal run (the pipe or P leg) and the transverse run (called the T leg) are long
or short with respect to certain criteria. The transverse or T leg is the run against
which the soil bears, producing an in-plane bending moment in elbow, branch, or tee.
(Category D elements have no transverse leg.)
The strict criterion for a long or short transverse leg is whether the length of the
transverse run L1 is longer or shorter than 3π/4ß, the length at which the hyperbolic
functions in Hetenyi’s equations,62 approach unity. The critical value for L1 is often
called the influence length, or that portion of transverse or T run which is deflected
or “influenced” by seismic soil strain or pipe thermal expansion along the axis of the
longitudinal or P run. In practice, a critical influence length L1 of 25,983 to 1.2/ß
may often be used, since there is very little deformation or load in that portion of the
transverse run which exceeds this length. This implies that the vast majority of the
bearing load on the transverse or T leg occurs in the first several feet of the pipe at the
bend or branch. In summary, a transverse pipe is “long” if
or
The criterion for a short or long P leg is whether its length L2 is sufficiently long to
experience the maximum force that can develop at the friction interface. For full
maximum friction force (Fmax = εAE) to occur in a straight pipe axially free at each
end, its length L2 would have to equal or exceed 2Lm with Lm calculated by Eq.
(B4.75). If one end terminates in an elbow or a tee, with the other end remaining
axially unrestrained, the total length L2 necessary for full friction to develop is L⬙ +
Lm; the friction force over Lm is equal to the soil bearing force S plus the friction force
acting on the length L⬘ or L⬙, which is called the effective slippage length. The effective
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.209
slippage length is the maximum length along which slippage occurs at the pipe/soil
interface of a pipe with a transverse leg or branch. The effective slippage length L⬙ for
long pipes with long transverse legs is calculated by
(B4.77)
where Ω = AEß/k and Fmax is calculated by Eq. (B4.76).
Equation (B4.77) applies to bends, tees, and branches. Although Eq. (B4.77) was
developed for the case where L2 = L⬙ + Lm, it applies also for any case where L2 > L⬙ +
Lm, since the length of the region where there is zero slippage at the friction interface
is immaterial.68 Using L⬙ as calculated by Eq. (B4.77), it can now be established that
a P leg is classified long if it meets these criteria:
1. For Types A1, B1, C1, L2 ≥ L m + L⬙;
2. For Types A2, B2, C2, L2 ≥ 2 L⬙;
3. For Types A3, B3, C3, D, L2 ≥ L⬙.
That point which is located a distance L⬘ or L⬙ from the bend, branch, or tee, is
called the virtual anchor, since it acts as if it were a three-axis restraint on the pipe.
Locating the Virtual Anchor. Calculation of the forces and moments in buried
piping at the changes in direction requires that the location of the virtual anchor (the
effective slippage length L⬘ away from the bend or branch element) in the P run and
the deformation δ of the soil at the buried element be established. For elements of all
types with long P legs, L⬙ may be calculated by Eq. (B4.77).
For Types A1, B1, and C1 elements (with one end of the P leg free or unrestrained
axially) with “short” P legs, L⬘ must be found by a less direct method as follows68:
(B4.78)
where a = 3f/(2AE)
b = ε - fL2/(AE) + 2fß/k
c = -fßL2/k
However, the most highly stressed runs in a buried piping system typically are
restrained at both ends, either by a combination of transverse runs or a transverse
run and an anchor (either real or virtual).
For Types A2, B2, and C2 elements with short P legs, L⬘ is expressed by
(B4.79)
For Types A3, B3, C3, and D elements with short P legs, L⬘ is expressed by
(B4.80)
Computer Modeling of Buried Piping
Determination of Stresses. With f, k, and L⬘ or L⬙ established, the stresses in a buried
pipe due to thermal expansion can be determined with a general purpose pipe stress
computer program. A buried piping system can be modeled with a typical mainframe
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STRESS ANALYSIS OF PIPING SYSTEMS
B.210
GENERIC DESIGN CONSIDERATIONS
or microcomputer pipe stress program by breaking the buried portions into elements
of convenient length and then imposing a transverse spring at the center of each
element to simulate the passive resistance of the soil. The entire pipe can be divided
into spring-restrained elements in this manner; however, the only regions of the pipe
that really need to be modeled in this manner are the lengths entering and leaving
elbows or tees. The analyst should refer to the program users’ manual for guidance in
modeling soil springs.
All pipe stress computer programs with buried piping analysis options require
that the following factors be calculated or estimated:
1. Location of the virtual anchor (dimension L⬘ or L ⬙)
2. Soil spring rate ki, j, which is a function of the modulus of subgrade reaction k.
3. Influence length, also a function of k.
Some programs ignore the friction at the pipe/soil interface; this is conservative
for calculating bending stresses on the buried elbows and branch connections, but
may be unconservative for calculating anchor reactions.
Determination of Element Lengths. The element lengths and transverse soil spring
rates for each element are calculated by the following procedure:
1. Establish the element length dL and the number n of elements, as follows:
(A) Set the element length to be equal to between 2 and 3 pipe diameters.
For example, dL for a NPS 6 may be set at either 1 ft or 2 ft, whichever is more
convenient for the analyst.
(B) Establish the number n of elements by:
(B4.81)
This gives the number of elements, each being dL inches in length, to which springs
are to be applied in the computer model. The number n of elements is always rounded
up to an integer.
2. Calculate the lateral spring rate ki, j to be applied at the center of each element.
(B4.82)
where k = the modulus of subgrade reaction calculated from Eq. (B4.72).
3. Calculate the equivalent axial load necessary to simulate friction resistance to
expansion. The friction resistance at the pipe/soil interface can be simulated in the
computer model by imposing a single force Ff in a direction opposite that of the
thermal growth.
(B4.83)
4. Incorporate the springs and the friction force in the model. The mutually
orthogonal springs ki, j are applied to the center of each element, perpendicular to the
pipe axis. Shorter elements, with proportionally smaller values for the springs on
these elements, may be necessary in order to model the soil restraint at elbows and
bends. The friction force Ff for each expanding leg is imposed at or near the elbow
tangent node, opposite to the direction of expansion.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.211
Determination of Soil Parameters. Soil parameters are difficult to establish accurately
due to variations in backfill materials and degree of compaction. Consequently, values
for elemental spring constants on buried pipe runs can only be considered as rational
approximations. Stiffer springs can result in higher elbow stresses and lower bending
stresses at nearby anchors, while softer springs can have the opposite effects. Backfill
is not elastic; testing has shown that soil is stiffest for very small pipe movements, but
becomes less stiff as the pipe movements increase. References 61, 63, and 66 discuss
soil stiffness and recommend procedures for estimating values for k which are
consistent with the type of soil and the amount of pipe movement expected. The
analyst should consult the project geotechnical engineer for assistance in resolving
any uncertainties in establishing soil parameters, such as the modulus of subgrade
reaction k, confining pressure pc, and coefficient of friction µ.
Pipe with Expansion Joints. An expansion joint must be considered as a relatively
free end in calculating stresses on buried elbows and loads on anchors. Since
incorporation of expansion joints or flexible couplings introduces a structural
discontinuity in the pipe, the effects of the unbalanced pressure load and the axial
joint friction or stiffness must be superimposed on the thermal expansion effects in
order to determine the maximum pipe stresses and anchor loads.
Pipe Stresses at Building Penetrations. Stresses at building penetrations can be
calculated easily after the reactions due to thermal expansion in the buried piping
have been determined. If the penetration is an anchor, then the stress due to the axial
force Fmax and the lateral bending moment M can be found by
(B4.84)
If the penetration is not an anchor, but is instead a simple support with a flexible
water seal, it is necessary to determine the stiffness affects of the water seal material in
order to calculate the stress in the pipe at the penetration. Differential movement due to
building or trench settlement can generate high theoretical stresses at piping penetrations
to buildings. Calculation of such stresses is beyond the scope of this section.
Allowable Stress in Buried Pipe
Buried piping under axial stress can theoretically fail in one of two ways: either by
column buckling (pipe pops out of the ground at midspan) or local failure by crippling
or tensile failure (much more serious than column buckling). Since buried piping
stresses are secondary in nature, and since the piping is continuously supported and
restrained (see Fig. B4.31), higher total stresses may be permitted
FIGURE B4.31 Plan of example buried pipe.
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STRESS ANALYSIS OF PIPING SYSTEMS
B.212
GENERIC DESIGN CONSIDERATIONS
as follows:
(B4.85)
where SA and Sh are as defined in Para. 102.3.2 of B31.1 Code.
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STRESS ANALYSIS OF PIPING SYSTEMS
STRESS ANALYSIS OF PIPING SYSTEMS
B.213
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STRESS ANALYSIS OF PIPING SYSTEMS
B.214
GENERIC DESIGN CONSIDERATIONS
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exceeding 2300 psi.
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