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.

**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:

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.

In general, expansion joints are used for the following applications:

Where thermal movements would induce excessive stress in normal piping arrangements

Where space is inadequate

Where reactions transmitted by pipe supports or anchors create large loads on supporting structures

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.

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.

**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.

**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 hydro-static test. When a steam or gas piping system is to be hydro-tested, the effects of the weight of the water on the system and its supports must be considered. A hydro-weight 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 hydro-test loads. If permanent supports cannot withstand these hydro-test loads, temporary supports may be added. Spring supports are available with hydro-static 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 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 decay in the piping. One simple approach to this temperature attenuation problem is as follows:

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.

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.

**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 deg 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 deg C) in the winter, negative coefﬁcients of expansion should be considered in evaluating the stress range.

**Occasional Loads: Seismic**

The code of Federal Regulation 10 CFR 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. 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 earth-quake (OBE).

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:

The integrity of the reactor coolant pressure boundary

The capability to shut down the reactor and maintain it in a safe shutdown condition

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.

The equation of motion for a piping system subjected to an externally applied loading (seismic excitation) may be expressed as

This equation could be solved by time-history analysis.

**Time-History Analysis.**

Time-history analysis is based on hypothetical earth-quake data in the form of ground displacement, velocity, or acceleration versus time. The piping system is represented by lumped masses connected by mass-less elastic members. The analysis is performed on this mathematical model by the direct numerical integration method. 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 earth-quake 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 mass-less 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:

The modal spectral acceleration taken from the appropriate response spectrum is then used to ﬁnd the maximum response of each 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:

These modal components are then combined by the appropriate method (see later subsection ‘‘Methods for Combining System Responses’’) to obtain the total dis-placements, 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.

**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 A58.1 is also one form of static seismic analysis. The code recommends that a lateral seismic force will be assumed to act non-concurrently in the direction of each of the main axes of the structure in accordance with the formula:

**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, 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, 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 N-411-1 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.

Mass Point Spacing. In a seismic analysis, the piping is represented by lumped masses connected by mass-less 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 ‘‘Cookbook-Type 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 cantilevered vents and drains. At least two mass points should be placed between supports in the same direction.

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) (seeFig. 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, could also occur in the following cases:

On pipe runs with axial restraint (support, anchor, or nozzle) where the longitudinal frequency could be higher than the cutoff frequency

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 missing-mass effect.

**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 ofthe 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.