Pipe stress analysis is the study of stresses in a piping system. It is a term applied to calculations, which addresses the static and dynamic loads such as deadweight (self-weight of the pipe including fluid, fittings and its associated components), internal and external pressure, thermal loads (due to change in temperature), seismic loads, wind loads, vibration, water hammer, steam hammer, slug force, PSV reaction force, etc. Not all the type of loads are applicable in every system e.g. if the system temperature is expected to stay at ambient conditions, thermal loads will not apply. The applicable loads derive from the design basis.

We will cover separately an article on the â€œDesign Basisâ€ (both for nuclear and non-nuclear applications). The objective of the stress analysis is to calculate these loads and strategically design the layout which allows required flexibility (e.g. expansion loops or change in the direction of pipe) and provide the constraints (directional and rotational) using piping supports, anchors, and hangers. The type of supports can be rigid, spring, snubbers, etc based on the stresses and forces in the piping system.

We have practically seen many failures where the stress analysis is not performed properly and/or the required supporting system is not installed in the field as per the design. Some of our personal experiences are:

1. Steam Pipe header: Complete block of the foundation shifted. You can imagine the amount of forces generated in the pipe header due to thermal expansion which shifted the complete concrete block. In this case, the importance of stress analysis was completely ignored.

2. All loads from the design basis not considered during the analysis.

3. Earthquake anchor movement for Seismic loads response spectrum not properly analyzed.

4. Cold pull suggested by design based on the analysis not properly implemented. (Note: sometimes the cold pull of pipe is needed due to higher expansion expected because of high temperature and a long run of pipe. In those situations a cold pull is recommended to accommodate the full expansion when the pipe is heated and expands).

5. The travel stop pin from the spring hanger is not taken out.

6. Missing nuts and bolts in the hangers. A very common problem in a running station and ignored.

7. Allowable forces at the equipment nozzle exceed the allowable provided by the manufacturer of equipment such as Pumps, compressors, heat exchangers, vessels. This is a very common situation for a stress engineer when it is not practically possible to limit the forces at the equipment nozzle provided by the equipment manufacturer. A solution to this is to work together with the equipment manufacturer engineer as they generally have more margins than defined in their standard specifications. I have faced and resolved this type of issue on many occasions.

The main purpose and objective of piping stress analysis is to ensure structural integrity and to maintain system operability. It can be elaborated as below:

• To ensure adequate flexibility in the piping system for absorbing the thermal expansion of the pipe.

• To ensure that the stresses in the piping components are within the allowable limits with respect to applicable codes and standards.

• To keep the piping connected equipment nozzle loads within specified allowable limits with respect to relevant standards (e.g. NEMA SM 23, API 610, API 661, API 667, ASME sec III, etc.) and Vendor allowable.

• To select the correct type of pipe support based on piping stress calculation.

• To check and ensure pipe displacements at maximum and minimum temperatures for clash-free routing.

• To calculate and ensure loads and moments on the supporting structures are in the acceptable range with respect to civil requirements.

• To solve dynamic problems in piping, such as those due to seismic loads, mechanical vibration, acoustic vibration, fluid hammer, pulsation, transient flow, relief valve discharge.

• To ensure leakage proof flange joints.

• To find out unintentional disengagement of pipe from support.

One of the prerequisites for satisfactorily performing piping stress analysis is to become familiar with the principles of strength of materials. Below is a brief understanding of pipe material behavior under stresses in a piping system.

Typical stress-strain behavior of a ductile material under tensile loading is shown in the famous stress-strain curve below, taking A 53 Gr B â€“ a commonly used CS material.

The various terms associated with the stress-strain curve of material is briefly discussed below and same is shown at different stages of the curve:

**• Stress:** - The resistance developed in the material
per unit area against the applied force is the stress in the material.
It can be simply specified as force per unit area of the material.

Stress (s) = Force / Cross-sectional area

**• Strain:** - A component subjected to load undergoes
deformation. The deformation is quantified by strain defined as the
change in length per unit length of the material.

Strain (e) = dl / L (Lateral strain)

**• Modulus of elasticity (E):** - Up to a certain limit
of loading known as the proportional limit, the strain developed in the
material is in direct proportion to the stress. This law is called
Hookeâ€™s law and the constant of proportionality, E, is the modulus of
elasticity or Youngâ€™s modulus, which is a definite property of the
material.

Mathematically: E = s / e>

**• Yield strength:** The point at which the specimen or
material under tension or compression generates a large deformation
without the addition of any load is called the yield point. The
corresponding stress is called the yield stress, or yield strength, Sy.
The yield point is easy to recognize for materials with a stress-strain
curve similar to that shown in the above figure.

**• Ultimate Tensile Strength:** The maximum stress in
the stress-strain curve of the material is the Ultimate Tensile Strength
of the material. This is the point beyond which the material becomes
unstable under load and breaks after uncontrolled yielding. This point
signifies the beginning of the reduction in the cross-section area
(Necking). This explains why the curve shows a drop in stress near the
breakpoint toward the end of the curve.

**• Allowable stress:** The yield strength or Ultimate
tensile strength of a material, as obtained from standard property
charts, is divided by a factor of safety to reach the allowable stress
of the material.

Mathematically: Allowable stress, s = Yield Strength (or UTS) / Factor of safety

Factor of safety or safety factor is a factor widely followed by the method of design accounts for the uncertainties in the loading and material behavior.

The stress-strain curve presents us with the different characteristics of a ductile material, that piping or material engineer needs to know. Once after getting know-how about the major terms in the stress-strain curve, now letâ€™s get an idea about the main stresses which are developed in a piping system during the various stages of its life cycle. All these types are briefed below.

**• Axial stress (S _{axial}):** Under external loads in the axial direction, axial stresses are developed in the pipe.

**S _{axial} = F / A**

**• Bending Stress (S _{b}):** Bending stresses
are developed in a pipe under loads acting in a plane normal to the axis
of the pipe. These may be caused due to temperature, weight of the
pipe, weight of contents, snow and ice, wind, or earthquake.

**S _{b} = M / Z**

Where,

M = Bending moment

Z = Section modulus of pipe

**• Circumferential stress \ Hoop Stress:** Under
internal pressure loading, this stress is developed tangential to the
cross-section (This stress acts in a direction parallel to the pipe
circumference).

**S _{h} = PD / 2t**

**• Longitudinal Stress (S _{L}):** This stress is developed normally to the cross-section of the pipe.

Longitudinal stress because of pressure

**= PD / 4t**

**• Radial Stress:** Stresses in the radial direction across the wall thickness of the pipe are called radial stresses.

**• Torsional Stress (S _{t}):** This is the
resultant stress caused by the rotational moment or torsional moment
around the pipe axis and is caused by body forces.

The basic allowable stresses are also called code stresses as they are tabulated in the codes such as ASME 31.1, 31.3, sec III. These are basic allowable stresses as the design stresses for the basic sustained loads. For other loadings, the design stresses are modified from these basic allowable stresses by applying factors and/or combinations.

The basic allowable stress values at a given temperature for materials other than bolting materials, cast iron, and malleable iron are taken as the lowest of the following:

• The lower of one-third of ultimate strength at room temperature and one-third of ultimate strength at temperature: B31.1 and Class 2 nuclear piping uses a 1/3.5 factor instead of 1/3.

• The lower of 2/3 of yield strength at room temperature and 2/3 of yield strength at temperature: Except for austenitic stainless steel and nickel alloys having similar stress-strain behavior.

• 100% of the average stress for a creep rate of 0.01% per 1000 hours.

• 67% (2/3) of the average stress for rupture at the end of 100,000 hours.

• 80% of the minimum stress for rupture at the end of 100,000 hours.

**Commonly used Codes for Piping Design and Stress analysis**

There is a complete series of ASME B31 codes for pressure piping. The most common codes used are ASME B31.1 and B31.3. For nuclear applications, ASMEâ€™s Boiler and Pressure Vessel Code (BPVC) sec III Div 1 is followed.

Subsection NB for class 1 piping and components.

Subsection NC for class 2 piping and components.

Subsection ND for class 3 piping and components.

Subsection NF for supports.

**ASME B31.1:**

ASME B31.1 prescribes minimum requirements for the design, materials, fabrication, erection, test, inspection, operation, and maintenance of piping systems typically found in electric power generating stations, industrial and institutional plants, geothermal heating systems, and central and district heating and cooling systems. It also covers boiler-external piping for power boilers and high-temperature, high-pressure water boilers in which steam or vapor is generated at a pressure of more than 15 psig; and high temperature water is generated at pressures exceeding 160 psig and/or temperatures exceeding 250 degrees F.

**ASME B31.3:**

ASME B31.3 contains requirements for piping typically found in petroleum refineries; chemical, pharmaceutical, textile, paper, semiconductor, and cryogenic plants; and related processing plants and terminals. It covers materials and components, design, fabrication, assembly, erection, examination, inspection, and testing of piping.

** **

**ASME B31.1:**

This Subsection contains requirements for the material, design, fabrication, and examination of supports that are intended to conform to the requirements for Classes 1, 2, 3, and MC construction. Nuclear power plant supports for which rules are specified in this Subsection are those metal supports which are designed to transmit loads from the pressure retaining barrier of the component or piping to the load-carrying building structure.

Piping stresses and equations based on ASME B31:

The types of stresses considered by ASME B31 codes are basically categorized as:

**• Sustained stresses or Longitudinal stresses (SL):** Sustained stresses are the sum of the longitudinal stresses in any
component due to pressure, weight, and other sustained loadings.

**• Displacement Stress or Expansion Stress Range (SE):** Expansion stresses are the stresses developed by the secondary loads like thermal expansion or displacement of the pipe.

**• Occasional Stress:** The stresses which develop in
the pipe due to seismic and wind effects, PSV pop up, steam hammer,
etc., and are occasional in nature are known as occasional stresses.

**Equations for stress allowable as per ASME B31.1 (Clause 104.8.1):**

**ASME B31.1** is considered the mother of all piping codes. Many
of the other piping codes follow B31.1 stress principles. So, below are
the equations that are used in the code for stress evaluation in a pipe.

Expansion Stress Allowable equation is,

Expansion Stress Allowable equation is,

Expansion Stress Allowable equation is,

Here, SE = (Sb2 + 4St2) Â½ Sb = resultant bending stress,psi = [(Ii Mi)2 + (Io Mo)2] / Z

Mi = in-plane bending moment, in.lb

Mo = out-plane bending moment, in.lb

Ii = in-plane stress intensification factor

Io = out-plane stress intensification factor

St = Torsional stress, psi

= Mt / (2Z)

Mt = Torsional moment, in.lb

Z = Section modulus of pipe

Mb = Resultant moment due to occasional loads

k = An allowable factor. Depends on the duration of occasional loads.

i = Stress intensification factor. The product 0.75i shall never be taken as less than 1.0.

MA = Resultant moment loading on cross-section due to weight and other sustained loads, in.-lb (mm-N)

Z = Section modulus, in.3 (mm3)

PD/4t is the stress due to internal pressure of the pipe.

Allowable expansion stress range

SA = f(1.25Sc + 0.25Sh) [ASME B31.3 Clause 302.3.5 or ASME B31.1 Clause 102.3.2]

Here,

Sc is the basic allowable stress at the minimum metal temperature known as cold allowable stress.

Sh is the basic allowable stress at the maximum metal temperature known as hot allowable stress.

f is the stress range factor and derived by the equation,

f = N, an equivalent number of full displacement cycles during the expected service life of the piping system.

ASME B31.3 specifies only the allowable for the longitudinal stress, without giving any specific formula for the calculation. The code stipulates that the sum of longitudinal stresses, SL, in any component in a piping system, due to pressure, weight, and other sustained loadings shall not exceed the basic hot allowable stress, Sh.

The above sustained, expansion, and occasional stress equations are the allowable conditions for a pipe material as per ASME B31 codes. The code equations are the minimum requirement for qualifying a piping system in question.

Pressure design basis:

The pressure resisting capacity of a pipe is characterized by its thickness.

The equation for the minimum required thickness of a straight pipe to resist the internal pressure in question is calculated as per ASME B31.1 (Clause 104.1.1) or ASME B31.3 (Clause 304.1.2), which is:

Where,

t = minimum required net thickness

P = design pressure

D = outside diameter of the pipe

d = inside diameter

S = allowable stress of the pipe material at design temperature

E = longitudinal joint efficiency or quality factor

W = weld strength reduction factor

c = allowance for corrosion, erosion, and others

y = coefficient value as given respective tables of ASME B31 codes

**Nuclear Piping stress equations based on ASME BPVC Sec III, Subsection NB:**

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.

**• Primary stress intensity:** The primary stress intensity must meet the following requirement:

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, in^{4}

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 an average fluid temperature under consideration.

**• 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.

For each pair of load, the stress range Sn is calculated as,

Where:

C1, C2, C3 = secondary stress indices for specific component under consideration

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

ð°, ð›ƒ = 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. Otherwise, the stress analyst
may proceed to the fatigue analysis.

**• 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 that exceed 3Sm. The following two equations must be satisfied:

Where,

Se = nominal value of expansion stress, psi

Mi* = 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

NC-3600 and ND-3600 nuclear piping codes are used for Class 2 and Class 3 piping, respectively, are similar to ASME B31.1 power piping. However, since the 1983 edition of the codes, evaluations of sustained stress and occasional stress have diverged from that of B31.1. Currently, these stresses are evaluated in accordance with the procedure similar to the one used in Class1 nuclear piping.

This article covers only the basics of stress analysis and gives the readers a flavor of it. Piping stress analysis is much more than a simple piping flexibility check. A good basic design and layout is the foundation for an optimized piping system.

For more details please contact us.

MECS Engineering specializes in Piping Design and Stress Analysis.