**Allowable Stress For Thermal Expansion**

The allowable stress for thermal expansion and other deformation-induced stresses is substantially higher than for **sustained loads**. This is due to the difference between load-controlled conditions, such as weight and pressure, and deformation-controlled conditions, such as thermal expansion or end displacements (e.g., due to thermal expansion of attached equipment).

When a load-controlled stress is calculated, it is an actual stress value. It is governed by equilibrium. For example, the stress in a bar when a tensile force is applied to it is the force divided by the area of the bar. This is not the case for thermal stresses. In the case of thermal stresses, it is the value of strain that is known. The elastically calculated stress is simply the strain value times the elastic modulus. This makes essentially no difference until the stress exceeds the yield strength of the material. In that case, the location on the stress-strain curve for the material is determined based on the calculated stress for load-controlled, or sustained, loads. The location on the stress-strain curve for the material is determined based on the calculated strain (or elastically calculated stress divided by elastic modulus) for deformation-controlled (e.g., thermal expansion) loads. This is illustrated in Fig. 7.1. Because the stress analyses are based on the assumption of elastic behavior, it is necessary to discriminate between deformation-controlled and load-controlled conditions in order to properly understand the post-yield behavior.

It is considered desirable for the piping system to behave in a substantially elastic manner so that the elastic stress analysis is valid. Furthermore, having plastic deformation every cycle carries with it uncertainties with respect to strain concentration and can be potentially far more damaging than calculated to be in the elastic analysis. One way to accomplish this would be to limit the total stress range to yield stress. However, this would be overly conservative and result in unnecessary expansion loops and joints. Instead, the concept of shakedown to elastic behavior is used in the Code. The basis for the Code equations is described by Markl (1960d). Rossheim and MarkI (1960) also provide an interesting discussion on some of the thinking behind the rules.

The allowable thermal expansion stress in the Code is designed to result in shakedown to elastic behavior after a few operating cycles. The equation provided in the Code is (ASME B31.3, Eq. (1a)).

where

SA = allowable displacement stress range Sc = basic allowable stress at the minimum metal temperature expected during the displacement cycle under analysis Sh = basic allowable stress at the maximum metal temperature expected during the displacement cycle under analysis f = stress range reduction factor

This equation assumes that the sustained stress consumes the entire allowable sustained stress, and it is simplified, in that it is not necessary to know the sustained stress in order to determine the allowable thermal expansion stress.

Note that the values of Sc and Sh, do not include weld joint quality factors or strength reduction factors. However, casting quality factors, Ec, must be included.

Equation (lb) of the Code is a more detailed equation, and considers the magnitude of the sustained longitudinal stress.

where

SL = longitudinal stress due to sustained loadings The allowable thermal expansion stress range can exceed the yield strength for the material, because both Sc and Sh, may be as high as two-thirds of the yield strength. However, it is anticipated that the piping system will shake down to elastic behavior if the stress range is within this limit.

This behavior is illustrated in Fig. 7.2, which is based on the assumption of elastic, perfectly plastic material behavior. Consider, for example, a case where the elastically calculated thermal expansion stress range is two times the yield strength of the material. Because it is a deformation-controlled condition, one must actually move along the strain axis to a value of stress divided by elastic modulus. In the material, assuming elastic, perfectly plastic behavior, the initial start-up cycle goes from point A to B (yield) to C (strain value of twice yield). When the system returns to ambient temperature, the system returns to zero strain and the piping system will unload elastically until it reaches yield stress in the reverse direction. If the stress range is less than twice yield, there is no yielding on the return to ambient temperature. On returning to the operating condition, the system returns from point D to point C elastically. Thus, the cycling will be between points D and C, which is elastic. The system has essentially self-sprung and is under stress due to displacement conditions in both the ambient and the operating conditions.

If twice the yield is exceeded, shakedown to elastic cycling does not occur. An example is if the elastically calculated stress range is three times the yield strength of the material. In this case, again referring to Fig. 7.2, the startup goes from point A to point B (yield) to point E. Shutdown results in yielding in the reverse direction, going from point E to F to D. Returning to the operating condition again results in yielding, from point D to C to E. Thus, each operating cycle results in plastic deformation and the system has not shaken down to elastic behavior.

This twice-yield condition was the original consideration. Since the yield strengths in the operating and the ambient conditions are different, the criterion becomes that the stress range must be less than the hot yield strength plus the cold yield strength, which, due to the allowable stress criterion, must be less than 1.5 times the sum of Sc and Si, (Note that the original ASME B31 criterion limited the allowable stress to 62.5% of yield, so the original factor that was considered was 1.6.) This 1.5 (1.6 originally) factor was conservatively reduced to 1.25. This total permissible stress range is then reduced by the magnitude of sustained longitudinal stress in order to calculate the permissible thermal expansion stress range. The resulting equation is Eq. (lb) of the Code [Eq. (7.2) here]. Equation (la) of the Code [Eq. (7.1) here] simply assumes that SL = Sh the maximum permitted value, and assigns the remainder of the allowable stress range to thermal expansion.

The same equation (7.2) also works in the creep regime. Deformation controlled stresses relax to a stress value sufficiently low that no further creep occurs. This stress value is the hot relaxation strength, Sh- Stress-strain behavior under the condition of creep, is illustrated in Figure 7.3. The initial start-up cycle, which can include some yielding, goes from point A to point B. During operation, the stresses relax to the hot relaxation strength, Sh, at which point no further relaxation occurs, point C. When the system returns to ambient temperature, the system returns to zero strain and the piping system will unload elastically until it reaches yield stress in the reverse direction. If the stress range is less than Sh plus to cold yield strength, there is no yielding on the return to ambient temperature. This is illustrated by going from point C to point D. On returning to the operating condition, the system returns from point D to point C elastically. Thus, if the stress range is less than the cold yield strength plus the hot relaxation strength, shakedown to elastic behavior also occurs at elevated temperature. If Sh is considered to be 1.25 Sh, then elevated temperature shakedown also is achieved with the Code allowable for displacement stresses, SA. The anticipated behavior over time, with multiple shut downs, and a gradual relaxation process, is illustrated in Figure 7.4.

Figure 7.3 also shows the behavior when the allowable stresses are exceeded at elevated temperatures. In this case, the startup goes from A to E. Stresses relax to point F. When the system returns to ambient temperature, yielding in the reverse direction occurs, going from point F to G to D. Returning to operating condition again results in yielding, from point D to H to E. Since high stresses are re-established, another relaxation cycle then must occur. The behavior of this system over time is illustrated in Figure 7.5. Even though the stress range is limited so as to result in shakedown to elastic behavior, there remains the potential for fatigue failure if there is a sufficient number of cycles. Therefore, the f factor is used to reduce the allowable stress range when the number of cycles exceeds 7000. This is about once per day for 20 years.

Figure 7.6 provides the basic fatigue curve for butt-welded pipe, developed by Markl (1960a) for carbon steel pipe. A safety factor of two on stress was applied to this curve, giving a design fatigue curve. It can be observed that the allowable thermal expansion stress range, prior to application of an f factor, intercepts the design fatigue curve at about 7000 cycles. For higher numbers of cycles, the allowable stress is reduced by the f factor to follow the fatigue curve, per the equation

where

N = equivalent number of full displacement cycles during the expected service life of the piping system (see the next section for how to combine different cycles into an equivalent number of cycles)

fm = maximum value of stress range factor: 1.2 for ferrous materials with a specified minimum tensile strength <517 MPa (75 ksi), and at metal temperatures <371°C (700°F); otherwise fm = 1-0.

In the 2004 edition, the maximum permissible value of/was increased from 1.0 to 1.2, with certain limitations. A value of 1.2 corresponds to 3125 cycles. The rationale for allowing a factor as high as 1.2 is that stresses are permitted to be as high as two times yield when f = 1.2. Thus, the desired shakedown behavior is maintained. This change reduces the conservatism introduced when the original criteria were developed. The limitations are that

the specified minimum tensile strength of the material must be less than 517 MPa (75 ksi);

the maximum value of Sc and Si, are limited to 138 MPa (20 ksi) when using an f factor greater than 1.0;

the material must be ferrous, and

the metal temperature must be less than or equal to 371°C (700°F).

The first and second limitations address a concern regarding the conservatism of the present f factors for high strength steel. There is a concern that the present rules overestimate the fatigue life for high strength steel components, so the limitations avoid further reducing the conservatism. The third limitation addresses a similar concern, but for non-ferrous alloys. The/factors were originally developed based on fatigue testing of carbon steel and austenitic stainless steel piping components; their application to other alloys such as aluminum and copper are not necessarily conservative. The fourth limitation is included because the rationale for increasing / to 1.2 does not apply to components operating in the creep regime.

In the 2004 edition, the equation for / was also extended from a maximum of 2,000,000 cycles to an unlimited number of cycles. The minimum value is/= 0.15, which results in an allowable displacement stress range for an indefinitely large number of cycles. The term endurance limit was not used, as it is associated with stress amplitude, rather than stress range. The background on the derivation of the value of 0.15 is provided in **Insert 7.1.**

The development of this methodology is described by Markl (1960a-d). It has been shown (Hinnant and Paulin, 2008) that the slope of the fatigue curve developed by Markl, from which the/factors were developed, is significantly flatter than has been observed in more recent fatigue tests of welded structures. The newer data includes much more data in the higher cycle regime, and the difference is apparent. A new fatigue curve has been developed in the cited reference, and the two fatigue curves cross at about 850,000 cycles. At higher cycles, fatigue design using the Markl equation becomes less conservative, and potentially unconservative. This data is being considered by the committee. Further, a more comprehensive treatment of high cycle fatigue is being considered.

**Insert 7.1 What About Vibration?**

ASME B31.3 is a new design code, and typically, piping systems are not analyzed for vibrating conditions during design. While the Code does require that piping be designed to eliminate excessive and harmful effects of vibration (para. 301.5.4), this is typically done by attempting to design systems to not vibrate, rather than by performing detailed vibration analysis. However, there are cases, such as certain reciprocating compressor piping systems, for which detailed vibration assessments are performed, but these are exceptions to the general rule. As such, evaluation of vibration tends to be a post-construction exercise.

Excellent guidance for the evaluation of vibrating piping systems can be found in Part 3 of ASME Standard OM-S/G, ASME OM-3, Requirements for Preoperational and Initial Start-Up Vibration Testing of Nuclear Power Plant Piping Systems. The procedures contained therein were developed based on experience with reciprocating compressor systems for gas pipelines. **Vibration** can be evaluated via the procedures contained therein based on peak measured velocity of the vibrating pipe or calculated stress. A screening velocity criterion that is generally very conservative, 0.5 in./sec peak velocity, is provided. Endurance limit stress ranges are also provided.

Based on the endurance limit stress ranges, an f factor for an unlimited number of cycles may be derived. The "endurance limit" stress ranges are provided for carbon steel and stainless steel. Including all the factors provided in the document gives the "endurance limit" stress range of 106 MPa (15.4 ksi) for carbon steel. Assuming a typical SA of 345 MPa (50 ksi) (1.25 Sc + 1.255h), and considering that the Code flexibility analysis equations calculate about one-half of the actual peak stress, we find an "endurance limit" f factor of 0.15 (at about 10 cycles). The "endurance limit" f factor for stainless steel would be higher, if calculated by the same procedure. However, the same "endurance limit" f factor is applied to all **materials**, consistent with existing rules.