I read in "Piping Flexibility Analysis Course Notes" by COADE, SECTION 1, Code Stress Equation Derivation, the next paragraph:
" SE = i/Z[Mx**2+My**2+Mz**2]**1/2
....
The above equation for maximum shear stress in a piping element is easy to apply. Adding hoop and radial stresses to the plane stress equation detailed above makes the application much more difficult."
To compute the expansion stress, only those varying component of the stress should be included in the calculations. The question is: hoop and radial stresses are varying? pressure causes only sustained stress?

The piping codes address longitudinal flexibility stresses. Hoop and radial stresses are not longitudinal, and are therefore not components of your typical (B31.1 / B31.3) code stress equations.

In CAESAR II, the Expansion load case (for alternating loads) is addressed by taking the difference between two previously solved load cases. Hence the nomenclature "DS5 = DS3 - DS4", which means "the displacement vector for load case 5 is equal to the displacement vector for load case 3 minus the displacement vector for load case 4". Elemental forces and moments are computed based on DS5, yielding a "force difference" or a "force range". The applicable code equation is applied to this force vector, yielding a "stress difference" or "stress range".

In the above "DS5 = DS3 - DS4" equation, neither DS3 or DS4 will include hoop or radial components (for B31.1 / B31.3 codes). The code equations typically include bending and longitudinal pressure, and depending on the code and load case, perhaps axial and torsional effects.

Hoop stress is very important, but should be addressed before any flexibility analysis is attempted. This occurs in sizing the wall thickness of the pipe.

Radial and shear stresses are local effects and should be addressed on a case by case basis as necessary.


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Regards,
Richard Ay (COADE, Inc.)

Hi Edgar,

If I might add a "few" words to what Rich has already written.

It is important to recognize that the B31 Pressure Piping Codes have always tried to provide the most simplified rules for design consistent with safety. At the same time, the Code rules provide protection against the failure mechanisms most commonly affecting piping systems. With the exception of wall thickness calculations ("pressure design"), the B31 Codes employ classic beam theory for bending (Sb = Mb / Z) and torsion (St = Mt / 2 x Z) for the structural design of piping. While it is incumbent upon the designer to address all phenomena which might potentially be deleterious to the system under design (challenge its structural integrity), the Code does not necessarily address every specific design issue. The Codes do not provide "instructions" on how to design piping systems. Reference is made to the Introduction to B31.3. Caesar II (CAESAR II) endeavors to follow and satisfy the specific requirements of the Codes.

There is a little variation in what the B31 Codes require you to calculate, however the requirements are basically the same. As Rich points out earlier in this thread, B31 does not ask you to calculate circumferential stress directly. Rather, B31 requires you to calculate the required pipe wall thickness, with consideration of the temperature/pressure conditions and the method of pipe manufacture (B31.1 - para. 104.1.2, B31.3 - para.304.1.2). It is obvious that the classic circumferential stress calculation (Scirc = P x D / 2t) has been rearranged to calculate the required wall thickness (t = P x D / 2[Sh x E = (P x Y)]. It should be noted that B31.3 employs a different wall thickness calculation (employing Von Mises failure theory) for high-pressure design (B31.3 para. K304.1.2) and this equation adequately addresses radial stresses. It is important to recognize that circumferential stress is a sustained primary stress and that it is adequately addressed in the cited B31 required wall thickness paragraphs. Further, the B31 Codes require you to calculate other primary and secondary stresses and to compare them to Code prescribed maximum allowable stresses. You are required to calculate the primary (bending and torsional) longitudinal stresses (due to sustained loadings of weight and longitudinal pressure - B31.1 para. 104.8.1., - B31.3 para. 302.3.5(c)). Also, you are required to calculate the secondary (bending and torsional) longitudinal stresses "Expansion" (B31.1) or "Displacement" (B31.3) Stress Range (B31.1 para. 104.8.3, - B31.3 para. 319.4.4). The various maximum allowable stresses (stress ranges) are given by the Codes (B31.1 - para's. 102.3.2, 102.3.3, 1.4.1.2, and for Equations 11,12, and 13 in para 104.8; B31.3 - para's. 302.3.5, 302.3.6, and 304.1.1)

The Code treats primary and secondary stresses (stress ranges) differently and prescribes specific allowable stresses for each of them. Primary stresses are caused by loadings that are "non-self-limiting", in that the stresses remain as long as the load is sustained. Primary stresses are not diminished by the deformation of the pipe due to the sustained loading. Primary stresses have the potential to cause the collapse of a piping system with a single application of load. Secondary stresses are caused by loadings that are "self-limiting" or limited by boundary restraints. Secondary stresses are diminished as (assuming) the piping system deflects under thermal loading (e.g., as the system deforms under thermal expansion or contraction). The stress magnitudes only increase if the piping system is restrained from expanding or contracting ("boundary restraints"). A secondary (or thermal expansion or displacement) stress will not typically fail the pipe on a single application of loading - rather the pipe may eventually fail due to accumulated fatigue caused by cyclic load applications. If you can obtain a copy of the old B31.7 Code (now out of print), the "forward" of this document will provide a detailed and well-presented explanation of these concepts. Also, if you can find a copy of "Design Methods for Power Plant Structures" by David Burgreen (Arcturus Publishers) there is a lucid and detailed treatment of the above topic there.

It may be of interest to consider the various stresses that can occur in pipe and to discuss their causes. Of course stresses are caused by "loadings". It may be useful to think of a stress as a material’s internal resistance to deforming in response to applied loadings (for cases where conditions of stable deformation prevail). Loadings such as internal pressure can cause a pipe to become larger in diameter and longer in the axial direction. The material's resistance to the pipe becoming larger in diameter due to internal pressure can be expressed in terms of the resulting circumferential stress (or as we popularly call it, "hoop" stress). The material's resistance to the pipe becoming longer due to internal pressure can be expressed in terms of the resulting longitudinal pressure stress. Notice that both of these are tensile stresses. Also, both of these are primary stresses. Internal pressure will also cause a compressive stress (another primary stress) on the inside surface (wall) of the pipe in a direction radial from the pipe centerline and this is called (who would have guessed), radial stress. This is the materials's resistance to the pipe wall being compressed (it is really only significant in thick wall pipe (t/d ratio greater or equal to about 2, opinions vary) with high internal pressure). Obviously, other loadings (e.g., bending due to weight and bending due to thermal expansion/contraction) can also cause various stresses.

What are some of the typical stresses that occur in piping systems? And equally important, how do they combine to structurally challenge the system. Well, from undergraduate school we all know about principal stresses (normal and shear), unit cubes and Mohr's circle of stress - so we will skip that part (Glynn Woods covers this well and concisely in his (and Roy Baguley's) book "A Practical Guide to B31.3 Process Piping Systems", CASTI Publishing). In the B31 Codes the individual stresses are combined by employing the Tresca theory of failure. So lets look at the salient stresses we will have to address.

Longitudinal (bending and torsional) stresses (due to sustained loadings of weight and longitudinal pressure). These primary stresses are sometimes referred to as "additive stresses" because the tensile stresses due to longitudinal pressure are superposed upon the bending tensile stresses which occur on the bottom extreme fiber (of course, the top extreme fiber is under compression - but we are looking for the maximum) of our "beam" (pipe) under weight loadings. That last sentence may set some record for length. B31.3 tells us we have to calculate these "additive" stresses and compare them to the allowable stress at temperature, Sh. But they do not give us an explicit equation to use in this calculation. No problem though, B31.1 does give us an equation (para. 104.8.1 - equation 11), and most of us use it for both B31.1 and B31.3 applications. There are other equations provided by other sources; some of them also include consideration of purely axial loadings (i.e., P / A). The B31.1 equation for calculating "additive" stresses has 2 terms, a pressure term (P x D / 4 x t) and a beam bending term (0.75 i MA / Z). It is important to recognize that the "bending term" moment, MA, is a SRSS of the 3 moments acting about the 3 principal (mutually perpendicular) axes. B31.1 applies an SIF to the resultant moment, thereby also intensifying the torsional moment(!). The pressure term of the B31.1 equation is the classic equation for longitudinal pressure stress. So, we do have guidance within the B31 Code structure on the evaluation of this stress. The cited equations employ the Tresca (maximum shear) failure theory (see Burgreen).

Longitudinal (bending and torsional) stress range (due to thermal expansion/contraction and other cyclic displacements). It is important to recognize that now we are addressing a range of secondary stress (see paragraph 3, above). As an example, just looking at stresses in the pipe due to temperature excursions, we want to consider the total range of longitudinal stress which is the sum of the stress caused by the pipe temperature increasing (expansion) from the installed temperature (theoretically zero thermal expansion/contraction stress) to the design maximum temperature, added to the stress caused by the pipe temperature decreasing (contraction) from the installed temperature to the design minimum temperature. That sentence may break the previous record. The thermal excursions as stated would be one complete thermal cycle - we are considering delta-T (hot) added to delta-T (cold) to comprise the total delta-T. Other cyclic loadings are possible (probable) and these are described in the B31 Codes. So, if that is how we develop the required loading for this case, how do the B31 Codes calculate the stress range? Remember again that we are addressing combined stresses, so bring up a mental picture of unit cubes and Mohr's circles. In B31.1 (para. 104.8.3) we find an equation for thermal expansion (but really total delta-T) stress range (SE) given as SE = i MC / Z. And again, the resultant bending moment, MC, is a SRSS of the 3 moments acting about the 3 principal (mutually perpendicular) axes. Once again B31.1 applies an SIF to the resultant moment, thereby also intensifying the torsional moment. The B31.3 Code provides an equation (Eq. 17) for this "Flexibility Stress" (SE) in B31.3 - para. 319.4.4 as SE = (Sb**2 + St**2)**1/2. Further B31.3 defines the bending stress (Sb) as Sb = [(ii x Mi)**2 / Z + (io x Mo)**2 / Z]**1/2 . There are important differences here; B31.3 uses in-plane SIF's and out-of-plane SIF's and does not intensify the torsional stress (St). Partial temperature excursions and other partial cycling loadings are provided for within the B31 Codes (by including a power function for calculating their effect). The allowable stress range for this case is interesting and it was discussed previously on the COADE CAESAR II board (a direct link to that thread is ( http://www.coade.com/ubb/Forum1/HTML/000057.html ).

Longitudinal (bending and torsional) stresses (due to weight, longitudinal pressure, and "occasional" loads). This combination of primary stresses superposes another "bending term" upon the load combination considered in "additive stresses" above. What constitutes these "occasional" loadings? Such things as seismic events, relief valve "blow-down", extreme (but creditable) weather conditions and other infrequent design loadings. Although B31.3 tells us to consider these loadings and supplies us with an allowable stress to compare (B31.3 - para 302.3.6(a)), they do not provide a specific equation. The B31.1 Code also requires that these loadings be considered (and they give us an allowable stress) in B31.1 - para 104.8.2. The equation includes the longitudinal pressure stress term (P x D / 4 x t) and the same "bending" term (0.75 i MA / Z) as the "sustained longitudinal stress" equation but it add another "bending" term which calculates the stress due to the "occasional" load (0.75 i MB / Z). Again the moment, MB in this case, is a SRSS with the additional complication that multiple vector load applications must be computed (earthquakes shake from all directions) so you might do an "SRSS of the SRSS's". The sum of these 3 terms must be less than the allowable stress. The allowable stress (with frequency of application considered) is given in B31.1 - para. 104.8.2 and B31.3 - para. 302.3.6(a) and they are not the same, reflecting the 2 committee’s difference in philosophy.

The B31 Codes address pure tension, pure compression, shear and bearing loading (B31.1 - para. 102.3.1, B31.3 - para. 302.3.2) and prescribe allowable stresses for each. There will not be many times when the piping designer will have to consider these loadings individually. Shear caused by a heavy valve close to a rigid support comes to mind.

The B31 Codes do not yet address the combination of pressure, weight and temperature loadings taken together (operating case) and an appropriate allowable stress. However, this subject is currently a work in progress at the B31.3 Section Committee. Radial stresses (primary compressive) are only addressed in B31.3 high-pressure design (required wall thickness), and again, stress calculations are not mandated. Local geometric, metallurgical, and other discontinuity stresses are not specifically addressed. Through wall transient thermal stresses (thick wall phenomenon) are not specifically addressed. Supporting large diameter, relatively thin wall piping on saddles and similar supports (e.g., pentstocks) may result in large local membrane stresses and this issue is also not specifically addressed by the B31 Codes. You can only do so much with beam theory. However the B31 Codes never limit the degree of conservatism that can be applied. Also, the B31 Codes allow more rigorous analyses methods (e.g., FEA) to be applied to the design. It should be noted however, that the analyst must look beyond B31 Code methodologies when employing FEA. It would be more appropriate to look to the ASME B&PV Code, Section VIII, Division 2 for guidance in finite element analyses (never try to mix the Codes). Again, it is incumbent upon the designer to address all phenomena which might potentially be deleterious to the system under design (challenge its structural integrity), but the Code does not necessarily address every specific design issue. If rules addressing these issues are included in the B31 Codes in the future, I feel confident that COADE will follow the Codes and address these rules in Caesar II.

Of course all the above is only my opinion and it is not necessarily the opinion of ASME International or any ASME B31 Code Committee.

Thanks for posting an interesting question.
Best regards, John

This NOTE is added on 8-16-2006:
Because this old thread has been resurrected please note the dates on the postings. These discussions predate changes that were made to B31.3-2004 when Appendix "P" was added.


[Modified referenced thread's URL.]

[This message has been edited by rich_ay (edited November 11, 2000).]

[This message has been edited by John Breen (edited December 18, 2000).]

Dear Firend,

i could not undertsand one thing... Why in caesar II report the code stress and allowable stress at start of flange or at start of valve and end of a flange or end of a vlave is "0".

Is it becasue the element is "rigid". If it is so; but still there must be some stresses being exerted on a flange or valve....Why not caesar II is reporing such stresses.

Thanks

How would you suggest that the stress be computed, since we don't know the cross section, and therefore we don't know "Z"?