Strength of Materials Basics 53
2.6.5
Effective Stress (von Mises Stress)
The maximum distortion energy failure theory agrees with the nature of ductile materials the most.
This theory is very popular in the European piping community. Based on distortion energy theory, the
condition for yielding is [7]
(S1 - S2)2 + (S2 - S3)2 + (S1 - S3 )2 = 2Sy2p
where Syp is the yield point tensile stress in the tension test. To make a stress directly comparable
to the yield point stress and other data obtained in the tension test, the effective stress is coined and
defined as
Se =
1
[(S1 - S2) 2 + (S2 - S3 ) 2 + (S1 - S3 ) 2]
2
(2.20)
For a two-dimensional stress field, as is commonly the case in practical piping system, S3 can be set
to zero, and we have
Se =
S12 - S1 S2 + S22
The principal stresses S1 and S2 can be found using Eqs. (2.17a) and (2.17b), respectively, for twodimensional stress systems. Substituting Eqs. (2.17a) and (2.17b) to the above, the effective stress for
two-dimensional stress field becomes
Se =
2
(Sx - Sy )2 + Sx Sy + 3txy
(2.21)
By comparing Eq. (2.21) with Eq. (2.19), it is difficult to assess whether the effective stress is larger
or smaller than the stress intensity. In other words, it is not possible to say which theory is more
conservative. However, one thing is clear: when either one of the normal stresses is zero, the stress
intensity is somewhat larger than the effective stress. The degree of difference depends on the ratio of
the normal stress and the shear stress. At the extreme condition when both normal stresses are zero,
¾¾
the stress intensity is bigger than the effective stress by a factor of Ö 4/3 = 1.155. However, in most
practical piping applications, both stress intensity and effective stress can be considered equivalent to
each other.
2.7
BASIC BEAM FORMULAS
A piping system is essentially a group of beams connected together to form the shape required for
transporting fluids from one point to another. Therefore, the behavior of the beam is the basic component of the pipe stress analysis. Table 2.1 shows some basic beam formulas that stress engineers are all
familiar with. The following are a few important notes derived from these basic formulas:
(1) With a given loading, the bending moment is proportional to the length of the beam, but the
displacement is proportional to the cube of the length. A slight increase in length creates a large
increase in displacement, which translates to a large increase in flexibility.
(2) For a given configuration, the displacement is inversely proportional to EI. This EI is generally
referred to as the stiffness coefficient of the cross-section. If a simulation is required for a nonstandard cross-section such as refractory lined or concrete lined pipe, the EI of the simulating
pipe has to match the combined EI of the pipes being simulated.
54
Chapter 2
TABLE 2.1
BASIC BEAM FORMULAS
(3) The displacement formulas include only the term EI. The terms involving shear modulus G
and cross-section area A, associated with shear deformation, are not included. The formulas
are good for practical lengths of beams. However, they are not accurate for short beams whose
length is shorter than ten times the cross-sectional dimension of the beam. The formulas used
by most computer programs include the shear deformation, thus making a short beam more
flexible than that calculated by the formulas in Table 2.1. This is one of the potential discrepancies between a computer result and a hand calculation result. In such cases, the computer result
is more accurate.
(4) Items (a) and (c) can be used to determine the stress due to weight under normal supporting
spans. Because the actual piping in the plant is supported somewhere between simple support
and fixed support [8], the following average formulas are generally used, instead, for evaluating
weight supports.
S =
M
wL2
=
,
Z
10Z
D =
3wL4
384EI
Strength of Materials Basics 55
FIG. 2.14
DIFFERENT TYPES OF CANTILEVER BEAMS
2.7.1
Guided Cantilever
In piping flexibility analysis, the guided cantilever method is one of the often-mentioned approximate approaches [8]. Figure 2.14 shows two different types of cantilever beams that may be used
in piping stress analysis. The regular cantilever beam shown in Figure 2.14(a) is actually half of the
simple beam subject to a concentrated load as given in Table 2.1(b). The fixed end is equivalent to
the mid-point of the simple beam. The guided cantilever beam shown in Figure 2.14(b) is actually
half of the fixed beam subject to a concentrated load as given in Table 2.1(d). These beam models are
often used to calculate the forces and moments, approximately, in a given length of pipe subject to a
displacement resulting from thermal expansion.
The regular cantilever model requires an accompanying rotation, θ, while absorbing the displacement, D. This rotational relief reduces the forces and moments generated by a given displacement.
The problem is that an actual piping system does not freely offer this rotational relief. It does offer
some relief, but much less than the amount shown in Fig. 2.14(a). A more realistic and conservative
approach is to assume that no rotation is taking place, as shown in Fig. 2.14(b). This is the so-called
guided cantilever approach. For a given expansion or displacement, the force created by the guided
cantilever is four times as great as that by a regular cantilever, and the moment, thus the stress is twice
as large as that by a regular cantilever.
2.8
ANALYSIS OF PIPING ASSEMBLY
The analysis of a piping assembly is very complicated and is accomplished mostly by using computer programs. In general, an engineer is not required to have knowledge of the computing methods
implemented in the computer programs in order to use the program. However, some common sense
regarding general analytical approaches can help analysts to better understand the procedure and better interpret the results.
A piping system consists of many components laid out in all directions. Analysis of the piping
system is normally idealized into a combination of straight pipe segments and pipe bend elements.
In other words, the analysis is performed using a finite element method consisting of two types of elements: the straight pipe and the pipe bend.
158
Chapter 6
excessive displacements will be discovered during stress calculations, it is better to limit them when
the piping is being routed.
3. Avoid Support Rotations
Excessive pipe rotations twist support shoes making the supports ineffective. In general, limiting the
sagging displacements can prevent excessive rotations.
4. Provide Rigidity
A certain amount of piping system rigidity is needed to reduce undesirable vibration. This, in part,
is related to limiting the displacements.
The allowable support spans are determined by the stress and displacement criteria. These can be
evaluated by the semi-fixed beam approach as discussed in Section 2.7. Based on the allowable bending stress criterion, the spacing is limited to
L1 =
10ZS
w
(6.1)
where S is the design weight bending stress {B31.1 [2] uses S = 2300 psi (15.86 MPa), whereas CEN [3]
uses S = 10 MPa (1450 psi), and Kellogg [1] uses S = 2750 psi (18.96 MPa)}; Z is the section modulus
of the pipe cross-section; and w is the weight force per unit length of pipe.
Based on the allowable sagging criterion, the spacing is limited to
L2 =
4
128EID
w
(6.2)
D is the design sagging displacement. B31.1 uses D = 0.1 in. (2.5 mm) for power plants, whereas Kellogg suggests a D of 0.5 in. to 1.0 in. (12.5 mm to 25 mm) for process plants.
The allowable span, LS, is therefore taken as the smaller of L1 and L2. Table 6.1 shows the suggested
pipe support spacing by B31.1 for power plants. A similar table can be constructed accordingly for
each type of service with different allowable stress and displacement criteria.
The suggested pipe support spacing is applicable only for uniform pipe without any attached concentrated weight, such as a valve or flange. It is not applicable for the overhanging span either. For
overhanging spans, the quarter circle approach as shown in Fig. 6.7 can be used. The curve shown is
a quadrant of a circle having a radius equal to the allowable spacing. The two coordinates of the point
TABLE 6.1
SUGGESTED PIPE SUPPORT SPACING
FOR POWER PIPING [2]
Suggested maximum
span, LS
Nominal
pipe sizes
Liquid
service
Gas service
in.
mm
ft
m
ft
m
1
2
3
4
6
8
12
16
20
24
25
50
80
100
150
200
300
400
500
600
7
10
12
14
17
19
23
27
30
32
2.1
3.0
3.7
4.3
5.2
5.8
7.0
8.2
9.1
9.8
9
13
15
17
21
24
30
35
39
42
2.7
4.0
4.6
5.2
6.4
7.3
9.1
10.7
11.9
12.8
Pipe Supports and Restraints 159
FIG. 6.7
OVERHANGING PIPING SPANS
on the curve determine the two allowable overhanging lengths. For instance, if LB = 0.45LS, then the
allowable LA is 0.16LS.
6.3
ANALYSIS OF PIPING SYSTEMS RESTING ON SUPPORTS
The most common and economical approach in dealing with the countless piping in a process
plant or power plant is to rest the piping on pipe racks or other support structures [4]. The piping is
supported either directly on the pipe wall or through pipe shoes. These types of supports are generally called resting supports. These supports are single-acting, because they only stop the pipe from
moving downward but allow the pipe to move up freely. Due to the nature of this non-linearity, exact
solutions are not expected for piping that goes through various temperature cycles. Therefore, three
major schools of thought have developed in the pipe stress software community with regard to resting
supports and temperature cycles. The analysis results and the qualities of the system designed differ
considerably among the methods used.
A simple example shall be used to demonstrate the merits and pitfalls of some analysis approaches
designed to satisfy the code requirements and philosophy. Figure 6.8 shows a typical piping system
resting on the support structure. The piping has one end connected to a process tower and the other
end connected to another process equipment. The system is supported at three locations. As the temperature of the process fluid rises, the tower grows upward and the pipe expands. With the tower connection moving up gradually, the piping system also goes through the following sequence of changes:
• With a small tower movement, the piping is held down on all supports by the weight of the
piping including fluid and attachments. Some thermal expansion (displacement) stress is generated, but the weight stress remains the same as in the cold condition.
• As the temperature rises gradually and the tower movement increases somewhat, the piping
will lift from the first support (support 20). A further increase of the tower movement will lift
the pipe off support 30, thus making a large portion of the piping unsupported. This substantially increases the sustained weight stress.
• As the system reaches the maximum operating temperature, the tower connection moves up
some more, but the pipe is still being held down at support 40. The expansion stress increases