One of the main duties of piping stress engineers is how to calculate stresses resulting from various loadings and conditions in which combined stresses lead to the worst case. Based on stress types, each stress combinations, presented by ASME BPVC section VIII div 2, is limited to an allowable value which is known as an allowable stress depending on various pipe failure modes. No method is presented in B31.3 for calculating and combining local stresses. Also there are not any suggested limitations and criteria for that. However, in ASME BPVC Section VIII Div2 a method is presented for combining stress categories and their related allowable ones. The three requirements which are mentioned in the code are as follows:

Primary stress requirements

Secondary stress requirements

Peak stress requirements
Each of above requirements implies one failure mode of pipe. So, each requirement needs to be met for preventing collapse. In order to put a stress in a right category, a stress engineer should pay attention to stress categories definitions, their imposing hazard to a system and consequently their related limitations. The characteristic of stresses depends on the load types. For better understanding of the code requirements, the bellow path is going to be followed:
At first, stress categories will be introduced. Then failure modes will be discussed and finally the code requirements for stress checking will be presented with a sample.
Stress Combinations Based on ASME BPVC SEC VIII Div 22010
1Introduction
2Stress categories
21Primary stresses
Primary stresses are generated by unrelenting loads which remain constant after the stresses in a pipe cross section go beyond the yielding point. Sustained stresses caused by internal pressure and weight of system are in this category.
22Secondary stress
Secondary stresses are generated by relenting loads which get relaxed beyond yield stress. So they are considered as selflimiting loads and stresses. For instance, an axial thermal load which is developed in a two fixed pipe increases by any increasing in temperature difference. If the load (and its related resulting stress) increases more than yielding point, the pipe material goes to a plastic region and so it starts relaxation and consequently the load disappears. Therefore it seems that relenting loads are selflimited.
23General Stress
This type of stresses is resulted from loads which are applied more or less to all pipe cross sections. Circumferential hoop stress and longitudinal stresses due to internal pressure and weight of system are considered in this category. Generally they are more dangerous and impose a higher risk than local ones. Because local stresses affect a small area of a pipe they are redistributed to the adjacent places where the local stresses generated.
24Local Stress
Local stresses are generated in a small area of a pipe. They are named as discontinuity stresses which are presented in “Discontinuity Stresses” section of the technical papers. These types of stress which exist in a discontinuity area such as welded points of a trunnion or lug can be primary or secondary stresses depending on the related loads. Moreover, local stresses can be categorized as membrane or bending stresses.
Regardless of loading types which produce local primary bending stresses, they are categorized as a secondary stress due to similarity of failure mechanism between it and secondary loading. After the maximum stress which occurs in outer edge of a cross section reaches to its yielding point, the stress are redistributed to the adjacent points. So it could be considered selflimited which is an apparent characteristic of a secondary stress.
25Membrane stress
Membrane stresses are uniformly applied on cross section of a pipe. For instance, primary longitudinal stresses resulting from weight of system is a membrane type stress. (Figure 1)
Figure 1: Membrane stress due to an internal bending moment
26Bending stress
As shown in figure 2, the distribution of local longitudinal stresses exerted on pipe wall is not uniform. Actually the maximum stress takes place on outer edge of the pipe cross section. Therefore it the load increases the outer edge is the first place which senses yield stress. As mentioned above the stresses are redistributed to the points placing bellow the outer edge. Because of the increased load carrying capacity a higher allowable stress has been assigned by the code.
Figure 2: Bending stress distribution due to local longitudinal stress in chocking model
3Failure Modes
The code considers three mentioned requirements for stress checking. These requirements have been designed based on failure modes of a pressure vessel and they also can be used for a pipe. Obtaining the knowledge about the source and basis of the code stress requirements enables an engineer to make a better decision when confronting a controversial issue such as defining of a proper category of a settlement stresses: primary or secondary stresses.
Because each stress categories has different failure behavior which implies various safety factors, stress category selection affects on safety design of a system. The failure modes are as follows:
31Collapse due to sustained loads (internal or external pressure, weight,…)
The main factor of this type of collapse is primary membrane stresses which may be resulted from an internal/external pressure or sustained loads such as weight of a system. As mentioned above membrane stresses are uniformly applied on a pipe cross section in way that all points on the area feel the same stress value. As a result, all points of cross section reach to yield stress. This characteristic of a membrane stress makes the section more vulnerable to failure. Besides, a primary load does not limit itself (it is not selflimiting) as mentioned before and so it can lead to a dire consequence in comparison with the same value of a secondary load. This is the reason why a low allowable stress value of 2/3 Sy is considered for this category.
32Elastic shakedown in a cyclic loading
For a ductile piping system which is under a cyclic thermal load, it can be proved that if the alternating component of stress is less than 2Sy, the system will go back to a stress range between –Sy and Sy although, at the beginning cycles, the stress goes beyond yield point [1]. This behavior which is known as shakedown guarantees the system integrity providing that the whole system endures a stress value of 2Sy at the beginning cycles. In certain circumstances, some parts of a system may collapse due to an excessive plastic strain. This phenomena is known as followup[3]. If a system which comprises parts with different stiffness goes under a secondary load which causes a stress more than Sy in the stiffer part, the part with a lower stiffness may go to plastic area whereas the stiffer part is still in an elastic zone. So the load won’t be relaxed effectively and consequently any increase in the load leads to an excessive plastic strain in the lower stiffness part. If the stiffness of different parts of a system is close enough, the possibility of this kind of failure decreases. There is a factor in FEPipe or Nozzle Pro results, named as strain concentration factor, which is defined as plastic strain to elastic strain ratio, may give a degree of followup possibility to an engineer.
33Ratcheting
Ratcheting occurs in a system which is under a notcyclic load such as axial load and a cyclic load such as a bending moment. In this case, the outer edge of a pipe cross section experiences a plastic deformation which accumulates in each cycle. Bree diagram, figure 3, presents stress limitations for ratcheting phenomena: a limitation of 2Sy and 2/3 Sy for alternating stress component and for primary stress respectively.
Figure 3: Bree diagram
The stress limitations for the two previous failure modes: primary membrane stress and shakendown stress, satisfies the ratcheting criteria.
34Fatigue
Repeated loads cause crack propagation in a piping system. Especially in an imperfect material and places such as welds and notches. Peak stress is defined as summation of primary and secondary stresses as well as stress intensification factor (SIF) at those places. The allowable stresses which are used in fatigue are based on fatigue curves known as SN diagrams presented in SAME BPVC Sec VIII Div 2 Annex 3F.
4Combined stress categories
Stress categories and their allowable stresses are mentioned in ASME BPVC Sec VIII div 2, 2010. The combinations of stress categories as well as their stress limitation may be summarized as follows: Please be aware that general primary bending stress has not been considered in the following formula. It is rare in piping but engineers should pay attention to it by referring to the relevant codes.
41General primary membrane stress requirement
(1)
As mentioned above, this kind of stress is categorized as a membrane primary stress and imposes the most dangerous situation to a pipe because of its membrane and primary characteristic. So the code considers a minimum value for its limitation.
42Local primary membrane stress requirement
(2)
As mentioned above, local stresses occur in a local area known as discontinuity area. Redistribution of a local stress to its adjacent places enables engineers to use a higher stress limitation.
43Primary and secondary stress
(3)
(4)
Where:
is allowable limit on the primary plus secondary stress range (see paragraph 5.5.6 of ASME BPVC Sec VIII Div 2, 2010).
may be assumed as larger quantities of 2Sy and.
This criterion satisfies ratcheting requirements as mentioned in 33.
44Peak stress
(6)
(5)
As mentioned in section 34, this criterion satisfies fatigue requirements.
The following steps illustrate how to combine the stress categories for an element shown in the following figures.

Determination of stress categories including general and local primary stresses, general and local secondary stresses. For fatigue requirements a proper SIF may be considered.

Calculation of equivalent VonMises stress by using VonMises failure theory.

Comparison of the equivalent stresses to the corresponding allowable ones.
5A sample of combined stresses
Note: In this example fatigue stress has not been considered.
51Stress categories
511General Primary membrane stresses
The stresses in this category have been shown in longitudinal and circumferential directions (figure 4).
Figure 4: General longitudinal and circumferential primary membrane stresses
: Total general longitudinal primary membrane stress caused by internal pressure and sustained loads.
: General circumferential primary membrane stress caused by internal pressure (Hoop stress)
512Local primary Stresses
5121Local primary bending stresses
Figure 5: Local longitudinal and circumferential primary bending stresses
: Local longitudinal primary bending stress such as a stress due to a sustained load, calculated by equation (28) mentioned in choking model section.
: Local circumferential primary bending stress such as a stress due to a sustained load calculated by equation (7) mentioned in zick’s model
: Local primary nonuniform shear stress such as stresses due to a torsional moment
5122Local primary membrane stresses
Figure 6: Local longitudinal and circumferential primary membrane stresses
: Local longitudinal primary membrane stress
: Local circumferential primary bending stress such as a stress due to sustained load, calculated by equation (34) mentioned in choking model section.
: Local primary uniform shear stress such as stresses due to a shear force
513 General secondary stresses
Figure 7: General longitudinal and circumferential secondary membrane and bending stresses
: General longitudinal secondary membrane stress such as a stress due to thermal load which may be calculated by a FE software (CAESAR II)
: General longitudinal secondary bending stress; in most cases is not available
: General circumferential secondary membrane stress; in most cases is not available
: General circumferential secondary bending stress; in most cases is not available
: General secondary nonuniform shear stress such as stresses due to a torsional moment
: General secondary uniform shear stress such as stresses due to a shear force
514 Local secondary stresses
Figure 8 : Local longitudinal and circumferential secondary membrane and bending stresses
: Local longitudinal secondary membrane stress
: Local longitudinal secondary bending stress such as a stress due to a thermal load, calculated by equation (28) mentioned in choking model section.
: Local circumferential secondary membrane stress such as a stress due to a thermal load, calculated by equation (34) mentioned in choking model section.
: Local circumferential secondary bending stress such as a stress due to a thermal load calculated by equation (7) mentioned in zick’s model
: Local secondary nonuniform shear stress such as stresses due to a torsional moment
: Local secondary uniform shear stress such as stresses due to a shear force
53 Combined stress calculation
The stresses which are mentioned in sections 51 should be combined based on the code stress requirements explained in sections 41 to 44. The total longitudinal and circumferential combined stresses are calculated as follows:
Figure 9 : Combined stresses
531 General primary membrane stress requirement
(7)
(8)
(9)
532 Local primary membrane stress requirement
(12)
(11)
(10)
533 General Primary membrane and secondary stress requirement
(12)
(13)
(14)
534 local Primary and secondary stress requirement
(17)
(16)
(15)
54 Equivalent VonMises stress
By using VonMises failure theory, the equivalent VonMises stress is calculated as follows:
(18)
Where are principal stresses. For pipes usually ratio is big enough to ignore radial stress, so Referring to Mohr’s circle can be calculated for the stresses shown in figure 9 as following:
(19)
(20)
55 Comparison the equivalent stress to its allowable stress
Referring to the code stress requirements which are mentioned in section 4, safety factors for each case are calculated as follows:
(21)