**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 (Class1) of **ASME Section III**.

**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 high-strain–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.

The major stress categories are primary, secondary, and peak. The limits of these stresses are related to the various failure modes as follows:

The primary stress limits are intended to prevent plastic deformation and bursting.

The primary plus secondary stress limits are intended to prevent excessive plastic deformation leading to incremental collapse.

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.

It also can be seen in Fig. B4.1 that a design limit of ²⁄₃Sy 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.

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 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 test 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 10^5 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 low-cycle 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.,