The routing of piping is heavily influenced by the stresses generated due to different loadings such as temperature differences, operating pressure, weight of components, dynamic loadings and others. Long piping operating under high temperature gradients tends to move significantly due to thermal movements, and this can potentially generate considerable stresses and even failures. Another key concern is the allowable nozzle loads of machineries and equipment.

It is important to guarantee safety against piping or connected equipment failure, as well as to protect anchor, support structures, and other parts from overstress. On the other hand, optimization is necessary and excessive factors or massively oversized systems or components have been discouraged. Therefore, accurate piping stress analysis is required for proper optimization.

This article offers practical notes and useful guidelines of piping stress analysis in optimization, revamp, renovation, expansion, and debottlenecking of plants. Some piping systems have been assessed as simple and just a simple check and review is needed for them. However, many piping systems need proper stress analysis.

Stress analysis should be undertaken for the following piping:

- Piping operating at low or high temperatures or extreme conditions. This includes piping that is subjected to extreme conditions only during special or alternative cases (such as de-pressurization or re-pressurization). This group also includes piping subjected to pressure relief or safety valve (PRV/PSV) loads, relatively large displacements, and those that are susceptible to flow-induced vibration and other extreme situations.
- Relatively medium and large sized piping. These pipes need stress analysis because the piping has inherently low flexibility, and even small deformation (such as a small thermal movement) is associated with high stresses. As a very rough indication, 4-inch (DN100) pipe could be the limit. Many 3-inch (DN80) pipes might need stress analysis due to their configuration or operating conditions.
- Piping connected to equipment and machineries. These pipes nearly always need stress analysis as nozzle loads should be compared with allowable values.
- Piping subject to transition. For example, those at transition from above-ground piping to underground piping.

### 1. Thermal stresses and extreme conditions

Thermal stresses are due to thermal movements of piping and induced stresses by supports and by the surrounding facilities. Piping expands or contracts due to the extreme temperature of the fluid being transported, and due to the temperature difference imposed. This thermal movement creates high loads and moments on points with limited displacement, such as nozzles of the equipment, supports, or anchors, and this results in high stresses.

Ordinary codes or specifications cover thermal ranges from the installation (or ambient) temperature to the hottest and coldest temperatures generated in the system. Another approach is to consider the full thermal stress range of the piping, from minimum temperature to maximum temperature. In this approach, it is assumed that there might be a credible case, whereas piping operating at one extreme operating case is suddenly switched to another extreme operating case, and this subjects the piping to the full thermal stress range.

### 2. Wall thickness of piping

An increase in piping wall thickness reduces the stresses due to pressure and weight loadings; however, the effect of increased thickness on thermal movement cases is an important part of the discussion. An increase in wall thickness increases the modulus of the piping section, but also proportionally increases the moment of inertia, and the section modulus is directly proportional to the moment of inertia. Therefore, the first consequence of increasing the wall thickness is an increase in bending moment under a given thermal movement. This increased moment divided by the proportionally increased section modulus ends up with more or less similar stresses as were present prior to the increase of the thickness. The thicker pipe wall does not usually reduce the thermal movement stresses; instead, it usually unfavorably increases the forces and moments in the piping, supports, and at the connecting equipment.

The basic allowable stresses are those directly used in the calculation of the pipe wall thickness based on rated pressure. These are called basic allowable stresses because they are the allowable stresses for the basic sustained loads. For other loadings, these allowable stresses are modified by applying factors or combinations. As a very rough indication, basic allowable stress might be 33% (1/3) of the ultimate strength at indicated temperature or 67% (2/3) of yield strength at the intended temperature. In general, allowable stresses might be determined based on creep, rupture, or fatigue stresses, if this is the lowest allowable stress at the operating condition or service.

### 3. Dynamic loads on connected piping

A piping system can respond far differently to a dynamic load than it would to a static load of the same magnitude. Static loads are those which are applied slowly enough that the piping system has time to react and internally distribute the loads, thus remaining in equilibrium.

A dynamic load changes quickly with time, and the piping system does not have time to internally distribute the loads. Forces and moments are not usually resolved, resulting in unbalanced loads and dynamic movements of the piping. Because the sum of forces and moments are not in equilibrium, the internally-induced loads can be different (either higher or lower) than the applied loads.

There have been different methods for analyzing piping system response under dynamic loads. Some common methods are modal calculation, harmonic analysis, response spectrum analysis, and time-history analysis. Modal analysis is an extremely important first step. This determines modal natural frequencies and associated mode shapes, which enable the study of the piping system under different excitation regimes. In other words, this method measures the tendency of a piping system to respond to dynamic loads.

The modal natural frequencies of a system typically should not be too close to excitation frequencies. As a general rule, higher natural frequencies usually cause less trouble than low natural frequencies. In addition, it is possible to check excitation forces and moments can excite a specific modal shape or not.

**Common forms of dynamic excitation.** Equipment vibration is known as a common excitation source for connected piping systems. Rotating equipment and machineries attached to piping can impose a cyclic displacement onto the piping at the points of attachment (mainly nozzles). The displacement at the pipe connection can be imperceptibly small, but it could cause significant dynamic loading problems.

During the operation of a reciprocating machine (i.e., a reciprocating pump or compressor), the fluid is compressed by pistons driven by a rotating shaft. This causes a cyclic change over time in the fluid pressure at any specified location in the piping system. Pulsation of piping is a major issue in piping connected to reciprocating machines, particularly reciprocating compressors. Unequal fluid pressures at opposing elbow pairs or closures create an unbalanced pressure load in the piping system. Because the pressure balance changes with the cycle of the reciprocating machine, the unbalanced force also changes. The frequency of the force cycle is likely to be some multiple of the operating cycle of the machine, because multiple pistons cause a corresponding number of force variations during each shaft rotation.

The pressure variations continue to move along through the fluid. In a steady state flow condition, unbalanced forces may be present simultaneously at any number of elbow pairs in the piping system. Load magnitudes can vary, and load cycles may or may not be in phase with each other, depending upon the pulse velocity, the distance of each elbow pair from the reciprocating machine, and the length of the piping legs between the elbow pairs.

Another major source of dynamic excitation is acoustic vibration. If fluid flow characteristics are changed within a piping, slight lateral vibrations may be established within the piping. For example, this could happen when flow conditions change from laminar to turbulent as the fluid passes through an orifice. These vibrations often fit harmonic patterns, with predominant frequencies somewhat predictable based upon the flow conditions.

There have been other types and forms of dynamic problems, such as those caused by wind passing over the piping and internal pressure transients. To reduce the risk of detrimental vibrations caused by internal flow, pressure transients, and vortex shedding oscillations from wind passing over the piping, a modal analysis should be conducted for many susceptible piping systems. With a good pipe support arrangement in place, the resulting lowest natural frequency should be higher than a specified limit.

Care should be taken to achieve a high value of the lowest natural frequency. Usually a stiff support scheme with many supports are needed, and this contradicts the requirements of flexible piping for low thermal stresses. As a very rough indication, such a frequency limit could be somewhere between 5 Hz and 14 Hz.

**Friction and gap.** Many piping supports are rests or guides, and friction plays a major role on piping system stress, especially friction at sliding supports. Therefore, the impact of friction should be modelled accurately in the piping stress analysis. For steel on steel, the friction factor is usually considered as 0.3 or 0.35. In reality, the steel-steel friction factor, under special circumstances could be 0.4 or even higher. In some special supports, low friction pad supports (such as those with PTFE pads) have been used, and the friction factor of such supports could be as low as 0.1. (The above-mentioned friction factors are just rough estimates, and friction factors have been known to be variable and uncertain.)

Many piping systems also have guide supports with gaps. Some other types of restraint support can be designed and implemented with gaps. However, such a gap combined with friction introduce a great deal of nonlinearities and difficulties to the piping stress analysis.

### 4. Piping transitions

Above-ground – under-ground (A/G–U/G) transitions have been known as a classic special case of piping stress analysis. Some piping systems go below-ground at a point, and their stress analysis usually includes some portion of under-ground piping, in order to establish virtual anchor length and to provide a proper boundary condition. The objective is to determine the range of axial movement of the piping in the partially restraint section under the combined loading due to thermal effects, internal pressure and others. As an indication, a transient angle of 20° is applied to reduce the lift off and bending moment at the A/G – U/G transition location.

In some piping systems, as a result of thermal movements or bending moments, a flange can leak. Flange leakage calculations should be carried out on flanged joints susceptible to high bending moments using proper methods. For example, ASME NC3658.3 and ASME Section VIII Div.1 Mandatory Appendix 2 can be used. Such a calculation is part of piping stress analysis and can show if there is a risk of flange leakage.

**About the Author: Amin Almasi**