- Email: [email protected]
Uncertanties in expansion stress evaluation criteria in piping codes
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Uncertanties in expansion stress evaluation criteria in piping codes
T
Nikola Jaćimović Danieli & C. Officine Meccaniche S.p.A, Via Nazionale 41, 33042, Buttrio, UD, Italy
ABSTRACT
ASME B31 and EN 13480-3 piping design codes are considered as the state-of-the-art codes for many piping engineers. However, as shown in this paper, there seem to be inconsistencies between ASME B31 code and more elaborate calculation codes, such as ASME Section III. In fact, it is shown in the paper that the ASME B31 approach can produce potentially non-conservative results. This seemingly important flaw in the B31 (and other major) piping codes is based on the allowable stress calculation based on equation (1b) of both ASME B31.1 and B31.3 and has been shown in detail in the paper. Finally, the stated flaw is verified by means of a finite element analysis which corroborates the stated concerns.
1. Introduction
in which:
ASME B31 and EN 13480-3 piping design codes are considered as the state-of-the-art codes for many piping engineers. Although the general idea behind the modern B31 codes (which is copied to other Codes, such as EN 13480) is more than six decades old, throughout the years these Codes have been evolving and incorporating new breakthroughs in the field piping calculation and analysis, thus keeping them in line with the requirements of the modern age. However, this author believes that there might be an issue with the code philosophy concerning thermal expansion stress range. As it is shown in further text, it may happen that by following the code provisions even an overstressed piping element might be deemed acceptable, which is not true for the other, more elaborate piping codes (e.g. ASME Section III, which is based on polished bar fatigue curves). This issue is verified using advanced numerical simulations (by means of finite element analysis) which seem to confirm the stated doubts.
Although the following discussion is based on ASME B31 Codes, it is also applicable to all other major piping codes which have adopted the same design philosophy. First of all, it is beneficial to provide an overview of the limits set in ASME B31 codes for thermal expansion stress range. ASME B31 equations for allowable stress range (i.e. equations (1a) and (1b) in both references [1] and [2]) state that (1)
or if preceding equation is not fulfilled
SA = f [1.25 (SC + SH )
SL]
A
(2)
6 N 0.2
m
m
C
H
L
2
Conceptually, it can be considered that SC = SH = 3 SY (basic concept in definition of the allowable stress), and so equation (1) yields (3)
SA = f SY in which SY , MPa , denotes material yield stress. On the same basis, equation (2) yields
SA = f
2. ASME B31 stress range limits
SA = f (1.25 SC + 0.25 SH )
• S , MPa , allowable stress range; f , stress range factor; • f= f , maximum value of stress range factor (typically 1.2 for ferrous • materials); • N , number of cycles; • S , MPa , basic allowable stress at minimum metal temperature; • S , MPa , basic allowable stress at maximum metal temperature; • S , MPa , longitudinal stress due to sustained loads.
5 SY 3
SL
(4)
2 3
If SL = SH = SY , equation (4) becomes equivalent to equation (3), However, since the basic idea behind equation (2) is to increase the allowable stress range, in further discussion it will be considered that SL < SH , which yields (5)
SA > f SY which, for N
SA > SY
7000 (f = 1) yields (6)
If more elaborate stress terminology is used (i.e. as in Ref. [3] or [4]), this limit may be acceptable for peak stresses, while it falls short of the requirements for primary plus secondary stresses. This fact will
E-mail address: [email protected]. https://doi.org/10.1016/j.ijpvp.2019.01.003 Received 31 October 2018; Received in revised form 17 December 2018; Accepted 5 January 2019 Available online 10 January 2019 0308-0161/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
become important, as discussed in further text.
then the nominal stress caused by this moment equals
3. ASME B31 expansion stress range
SNOM =
C N 0.2
(8)
in which:
Calculation of expansion stress range according to ASME B31 philosophy is based on stress intensification factors which are correlating the real stress in a piping component with a nominal stress in the girth butt weld of a straight pipe. Stress intensification factors (i ) are first introduced in ASME Codes in 1955. (ASA B31.1), and are a direct result of tests and publications of A.R.C. Markl [5–8]. They represent the first take on the fatigue design of piping components. New standards (such as [9]) and research in the field material fatigue ([10–12]) and stress intensification factors ([13,14]) show that important improvements can be made to the calculation method of the stress intensification factors. Although the use of correct stress intensification factors is of utmost importance for safe operation and economic sizing of piping ([15,16]), it is not the topic of this paper. For further analysis it will be considered that the stress intensification factors published in Refs. [1,2] are correct. Benchmark for all stress intensification factors in Markl's tests is girth butt weld of a straight pipe. Seen as girth butt weld in a pipe in itself represents a stress concentration zone (stress concentration factor according to Ref. [4] ranges from 1 to 4, but is typically considered 2), the obtained stress intensification factors and consequent calculated stresses are approximately one half of the real stress values, as it is shown in further text. This is a well-known and accepted fact within the pipe stress community. S-N curve which was used for derivation of stress intensification factors is originally written in the following form
iS=
M Z
• S , MPa , nominal stress caused by application of bending moment M; • Z,mm , pipe section modulus. NOM
3
By its definition, the stress intensification factor represents the ratio of the actual stress (SACT , MPa ) caused by the applied bending moment which would cause failure in a girth butt weld after the measured number of cycles (NACT ) to the nominal stress induced in a straight pipe by the same bending moment.
i=
SACT CZ = 0.2 SNOM N ACT M
(9)
Since the above derivation is made for girth butt weld in a pipe, it must be remembered that if the failure occurs away from some stress concentration (i.e. stress concentration factor is 1), the real value of stress intensification factor would be approximately two times higher than the one obtained by the preceding equation, or
iREAL =
SACT 2CZ = 0.2 =2i SNOM N ACT M
(10)
in which:
•i •S
REAL ,
real value of stress intensification factor; MPa , stress caused by the applied bending moment which would cause failure in a straight pipe (not girth butt weld) after NACT cycles. ACT ,
(7)
where constant C = 245000psi (=1689.22MPa) and stress intensification factor of i = 1 correspond to a girth butt weld in a straight pipe. It must be noted that the N in equation (7) denotes the “number of cycles of complete reversal”, which means that this equation is valid for expansion stress range. By comparison, the same equation can be used for not clamped pipe (i.e. pipe with no stress concentration factor) using the stress intensification factor of i = 0.64 as reported in Ref. [7]. Equation (7) for both girth butt weld and straight pipe is shown in Fig. 1. Although the stress intensification factor of i = 0.64 is reported in Ref. [7] as governing for straight pipe, the following discussion will be based on the conservative and yet simplifying assumption that the stress concentration factor of straight pipe is 0.5. If a piping component is loaded with a bending moment M, N mm ,
A rough connection between S-N curves of straight pipe and of girth butt weld is represented in the following form
SACT = 2 SACT = 2
C 0.2 N ACT
(11)
Consequently, it may occur that the stresses calculated by applying code stress intensification factor represent only one half of the real stress in the component. This is a well-known fact, and B31 allowable stresses are seemingly adjusted to take this fact into consideration. However, as it will be shown, using equation (1b) from ASME B31 codes (either [1] or [2]) may yield non-conservative results for components in which the typical failure occurs away from stress
Fig. 1. S-N curve of girth butt weld. 231
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 2. Allowable stress comparison.
concentration areas (such as welding tees or elbows).
Equations (16) and (17) are similar to equations (1a) and (1b) of ASME B31 codes [1] or [2]. However, the fact that using the stress intensification factors proposed by Markl [8] misses the actual stress by a factor of ∼2 seems to be disregarded in equation (17), which becomes important if equation (16) is disregarded and only equation (17) used. It is interesting to note that even the author himself has stated the following doubt in relation to equation (17): “selection of a proper value for the factor on the right hand side of the equation is open for discussion and review”. Considering that by definition SA = SE , it can be shown that there are two extremes of equation (17):
3.1. ASME B31 approach basis This following text provides a brief overview of the proposed expansion stress range limits set in Ref. [8], which establish the basis for the modern ASME B31 codes. It serves only to show that the doubt outlined in the preceding section draws its roots from the original paper (i.e. reference [8]). Given the number of full reversals N , the S-N curve in a girth butt weld of a straight pipe according to Ref. [5] can be accurately represented by
S=
1689.22 N 0.2
5 SY 8
SE
SE , REAL
(13)
(19)
[1.875, 3.125] SY
(20)
in which SE , REAL, MPa , denotes the real stress range (i.e. without stress concentration). As it will be shown later, this criterion would be considered as inacceptable by the more elaborate piping and pressure vessel codes (references [3,4]). 3.2. Comparison of allowable stresses
(14)
Fig. 2 shows the allowable stress given by equation (2) considering the following assumptions:
(15)
- material is A106 grade B pipe (typical carbon steel); - there are no longitudinal stresses (i.e. SL = 0 MPa ); - allowable stresses (SC and SH ) are as defined in Refs. [1,2].
(16)
In addition, equation (12) and yield stress are plotted in the same figure for comparison purposes. As it is well known and will be shown later in text, the minimum limit for the actual secondary stress range (i.e. expansion stress range) should be twice the material yield strength. Considering that the actual stresses calculated by using stress intensification factor as defined in Refs. [1,2] is approximately only one half of the real stress value, it follows that in this case the limit of expansion stress range should be set to material yield strength. As it can be seen, if equation (1b) from either ASME B31.1 or B31.3 is adopted in calculation, both references [1,2]
or
SA + SPW = 1.25 (SC + SH )
0.9375 SY
(18)
Considering that as calculated by the Code 2 SE = SE , REAL , and considering equation (14), it turns out that
The final limit set by the author in the proposed rules for piping fatigue analysis in Ref. [8] is
SA = f (1.25 SC + 0.5 SH )
= SH equation (17) becomes
1.5625 SY PW
it follows that
Sav = 2 SY
• if S
SE
(12)
in which Sav, MPa , denotes available stress range. Considering that at the time of publication of this proposal the allowable stresses were conceptually defined as
SC = SH =
= 0 equation (17) becomes
PW
It should be noted that this equation is taken as benchmark value for derivation of the original stress intensification factors, some of which are still used in their original form today. Conservative estimate of allowable stress presented in Ref. [8], which ensures that there is no plastic flow either at maximum or minimum temperature, states that
Sav = 1.6 (SC + SH )
• if S
(17)
in which SPW , MPa , represents longitudinal pressure and weight stresses (i.e. roughly corresponding to SL, MPa as defined in equation (2)). It must be noted that the author in Ref. [8] never did indicate expressly that the two equations (16) and (17) are interchangeable. In fact, in the originally proposed rules the equation (16) is indicated as mandatory, while equation (17) is proposed by M.W. Kellogg Company, rather than the Task Force on Flexibility. 232
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
can deem overstressed components as acceptable. Similarly, it can be shown that even if the less conservative straight pipe stress intensification factor is used (i.e. 0.64 instead of 0.5) the rules set in ASME B31.3 may still result in deeming the overstressed components acceptable. Similar figure to Fig. 2 is give in Ref. [17]. However, the major difference is that in Ref. [17] the author had used an allowable stress for SA-106 Grade B of 15 ksi, which is far less than the value of 17.1÷20ksi given in Refs. [1,2].
stress (as in equations (23) and (24)) the system will undergo elastic shakedown, and upon every consecutive load application will remain within the elastic region (i.e. in the region [ SY , SY ]). If the stress surpasses the aforementioned limit, the system will undergo small additional plastic deformation upon every load application and will consequently fail. As discussed beforehand, if expansion stress range as defined in Refs. [1,2] is considered as secondary stress (i.e. excluding stress concentrations) in which case the Code equations calculate roughly one half of the real stress value, it can be concluded that
4. Stress categories and their limits
iREAL MC 2 i MC = Z Z
Below is presented a quick overview of the various stress categories and failure mechanisms in connection with thermal (secondary) stresses. It is only supposed to give an insight into the code philosophy, and to present a comparison between ASME B31 stresses with the more realistic stresses calculating using the more elaborate codes. Main focus should be put on the discussion part presented in the subsequent chapters.
SL +
SE =
2 SY
(29)
Sa
• in one counted cycle there are two stress amplitudes (as defined in Refs. [1,2]), and • based on the facts that equation (12) is derived for the stress range
(22)
(i.e. twice the stress amplitude) and that the stress concentration factor of welds is approximately 2, Sa can be approximated with C equation (12) as Sa N 0.2 (which can be verified using values from Appendix I, Figure I-9.1 of [3]) equation (29) becomes
and
PL + PB + PE + Q
(28)
It must be noted that the Sa in the preceding equation denotes the allowable stress amplitude obtained for a polished bar and is not directly comparable to equation (12). Knowing that equation (12) is derived for full range of stress, its right-hand side can be correlated to the Sa , as shown below. Considering that:
(23)
2 SY
SY
(PL + PB + PE + Q + F ) AMPLITUDE
in which Sm, MPa , denotes mean allowable stress considering both cold and hot design temperatures. This limit is helping to prevent low-cycle fatigue or ratcheting failure. Considering that conceptually the mean allowable stress corresponds to two thirds of yield stress, the preceding equations become
PE
(27)
As defined in Refs. [3,4], peak stresses represent the incremental stress in a component added to primary or secondary stress which is caused by a stress concentration (i.e. a notch). Failure mechanism connected to peak stresses is high-cycle fatigue, which causes crack initiation and growth until the through-wall crack is generated and catastrophic failure occurs. Typically, the limit of peak stresses is derived from S-N curve, or
(21)
3 Sm
SY
4.2. Peak stresses
and
PL + PB + PE + Q
(26)
As it can be seen, the criterion represented by equation (28) is more conservative than the one represented by equation (27). However, it is important to note that in Ref. [3] the criteria represented by equations (27) and (28) are not mutually exclusive, but must be applied together.
In further analysis it will be considered that the longitudinal stresses defined in both [1] and [2] are comparable to the primary stresses defined in Ref. [3] (i.e. SL = PL + PB ). This fact is elaborated in Appendix 1. Considering that local membrane stress always includes general membrane stress contribution, the Code limits for this stress category are set as
3 Sm
i MC Z
SL S i MC + SE = L + 2 2 Z
m
mechanical loads (excluding discontinuities, and local stress concentrations); PL, MPa , local membrane stress produced by pressure and other mechanical loads (excluding discontinuities, and local stress concentrations); PB, MPa , bending stress produced by pressure and other mechanical loads (excluding discontinuities, and local stress concentrations); PE , MPa , expansion stress produced by constraint of free end displacement (including discontinuities, but not local stress concentrations); Q, MPa , secondary membrane and bending stress necessary to satisfy continuity of the structure (including discontinuities, but not local stress concentrations).
PE
3 Sm
It should be noted that this criterion is conceptually identical to equation (1a) of B31 Codes. Based on the same reasoning, equation (26) becomes
• P , MPa , general membrane stress produced by pressure and other
•
iREAL MC 2 i MC = SL + Z Z
Considering that, conceptually, the mean allowable stress corresponds to two thirds of yield stress, equation (25) becomes
According to Ref. [3], this stress category includes the following stresses:
• •
(25)
and
4.1. Primary plus secondary stresses
•
3 Sm
(PL + PB + PE + Q + F ) AMPLITUDE
(24)
1 i MC SL + 2 2 Z
=
SL i MC + 2 Z
C N 0.2
(30)
which is a well-known limit against progressive plastic deformation (PPD) and low-cycle fatigue failure. This failure mechanism and logic behind the aforementioned limit are shown in Fig. 3. As it is shown in Fig. 3, in case that the limit is set to twice the yield
For comparison purposes only, equation (30) can be conservatively written in the form of equation (28). As shown in Fig. 2, for any number C cycles under 10000 for carbon steels it can be considered that N 0.2 > SY , 233
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 3. Low-cycle fatigue (a) no PPD (b) PPD exists.
• S and S – elastic shakedown zones; • P – plastic cycling zone; • E − elastic zone.
which in term means that the following estimate resulting from equation (30) is conservative
SL i MC + 2 Z
SY
1
(31)
Obviously, the acceptable zones are E, S1 and S2, while all the others will lead to system failure. As it can be seen, both equations (1) and (2) may yield unacceptable results. Equation (1) may lead to ratcheting if the primary stress is sufficiently high and the limit of twice the yield strength is reached by secondary stress range. Equation (2) is even worse since it ranges from plastic cycling zone through both ratcheting zones R1 and R2.
or
SL + 2 SE = SL + 2
i MC Z
2 SY
(32)
Comparing equations (28) and (30) it is for the low number of cycles equation (28) is predominant in setting the load limit while equation (30) tends to become governing for larger number of cycles.
5. Discussion
4.3. Bree diagram
Although most of the time the stresses and derivation performed by Markl are considered as peak stresses (i.e. including local discontinuities and stress concentrations), which is not untrue, a question is raised what are the stress types when failure did not occur at the weld location but at some other location (e.g. typical failures in bends loaded with in-plane moment). In these cases, it is author's belief that the correct interpretation is that primary plus secondary stress category has been measured rather than peak stresses. In fact, even in the original paper [5] the author has acknowledged that welding elbows loaded with in-plane bending moment the crack had invariably started at the center of the side wall and has progressed towards both ends of specimens. Considering that there are no notches or other concentrations at this location, this kind of failure can only be classified as occurring due to secondary stress. In essence, two different failure mechanisms are encompassed by the calculation of thermal expansion stress range given by Refs. [1,2], and consequently there are two different verification paths:
Bree diagram [18] depicts the regions of ratcheting, shakedown, plastic cycling and elastic cycling. It is based on a plot of primary stress versus secondary stress range. Although it is derived only for a very simple example problem, the Bree diagram can be considered quintessential. Consequently, it can be used to further fortify the preceding points. Bree diagram is shown in Fig. 4, alongside acceptance criteria given in form of equations (1) and (2). Fig. 4 is derived based on the following two cases:
• real thermal stress range is two times higher than the one calculated •
by using the ASME B31 equations ([1,2]), corresponding to i = 0.5 in equation (7) – Fig. 4a; real thermal stress range is 1.56 times higher than the one calculated by using the ASME B31 equations ([1,2]), corresponding to i = 0.64 in equation (7) – Fig. 4b. In Fig. 4 the following characteristic zones are shown:
•R
1
2
1. Failure occurs due to primary and secondary stress; 2. Failure occurs due to peak stress (at the stress concentration area).
and R2 – ratcheting zones; 234
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Derivation of relationships between various stresses is given in Appendix 1.
6. Conclusion ASME B31 codes have long been one (and still are) of the most wide spread and useful piping codes. Their practical approach has made them indispensable in everyday engineering. However, it must be acknowledged that they are based on reasoning that was adopted over six decades ago. Following more recent findings and definition of stress categories and their respective allowable values, the author has tried to show that one of the Code requirements may lead to non-conservative results. This non-conservatism has been a verified in an example shown at the end of the paper using finite element analysis which seems to have confirmed the stated doubts. In order to keep consistency between various piping code philosophies, it could be suggested to update the acceptance criteria of references [1,2]. Furthermore, it would seem reasonable to perform a careful study to evaluate the influence on secondary and peak stresses, as shown in equations (34)-(37), for various piping materials. The discussion presented herewith does not necessarily mean that every piping component calculated using the disputed approach is overstressed (since there are various safety factors inherent to the Codes), but only to point out what seems to be an inaccuracy in the Code philosophy which may or may not lead to nonconservative results. Considering the constant breakthroughs and accumulation of knowledge and computational power, it is inevitable that the piping codes will evolve, making calculations more accurate and eliminating excessive safety factors. However, relatively simple calculation procedures, such as the ones adopted in B31 codes, will always have their immense practical use. One certainty is that B31 codes and their philosophy are here to stay and will for sure be used for years to come. This paper aims to simply provide an insight into possible uncertainties in Code philosophy which may lead to non-conservative results.
5.1. Failure due to primary and secondary stress In case that the failure occurs away from stress concentration area (i.e. notch), then the expansion stress range calculated by using ASME B31 approach equals to secondary expansion stress range as defined in Ref. [3]. In this case, there are two criteria which are to be applied. First one is in the form of equation (27) and equals to
SE
PE 2
SY
(33)
which is conceptually equal to
SE
0.75 (SC + SH )
(34)
and is very similar to equation (1). Second criterion which should be considered is given by equation (28) and states that
SL + SE 2
0.75 (SC + SH )
(35)
Seen as the latter equation is more conservative, it seems logical for it to be set as a limit. However, seen as both references [3,4] provide special rules for exceeding the set limit, it may be sufficient to use equation (34) as the Code limit as equation (35) would probably result in overly conservative design. Equation (35) is applicable to the components for which the expected failure is away from the weld area (i.e. K2 = 1, as defined in Ref. [3]). These are typically elbows/bends and welding tees. For other components, such as fabricated tees, the stress concentration factor defined in ASME B31 codes already includes effects of stress concentration due to weld. In these cases a following less conservative criterion can be used
0.8 SL + SE
1.15 (SC + SH )
Example Below example will show influence of the preceding discussion on a typical pipe elbow stresses. For this example, a pipe elbow with the following characteristics has been selected:
(36)
• ¯D = 273mm; • t = 9.27mm; • R = 381mm (ASME B16.9 long radius elbow); • Material A234 WPB (S = 241.3MPa ); • Number of cycles: 7000; • Hot/cold allowable considering maximum operating temperature
Equation (36) is somewhat similar to equation (2), but is not always applicable to all piping components.
out
5.2. Failure due to peak stress
Y
If failure occurs at the stress concentration area (i.e. notch), then the expansion stress range calculated by using ASME B31 approach equals to primary plus secondary plus peak stress as defined in Ref. [3]. In this case, the criterion which should be employed is
SL + SE 2
C N 0.2
under 140 °C equals SH = SC = 137.9MPa according to Refs. [2,3] (even lower value than the conceptual limit of 2/3 SY used in the main discussion.
Calculation according to ASME B31.3
(37)
Assuming that SL = 2 SH and 7000 cycle limit, according to equation (2), the limit for expansion range would be SA = 2 SH = 275.8 MPa . Flexibility characteristic 1
in which C = 245000psi (=1690MPa) , as discussed beforehand. Since the effect of stress concentration is already included in the ASME B31 stress intensification factors, there is no need to include it explicitly in equation (37). It should be noted that equation (37) represents calculation and comparison of stress amplitudes, and not stress ranges. Also, the obvious shortcoming of this equation is its independence on the material. It is derived only for carbon steels and a detailed study should be performed to correlate the proposed equation to the other materials (for example using polished bar curves reported in Ref. [3]) and expected failure locations (e.g. considering that typical failure location on elbows occurs away from stress concentration areas, this criterion is obsolete). Finally, seen as employing the calculation procedure of references [1] or [2] it remains unclear whether secondary or peak stresses are calculated, it seems reasonable to use the worst one of the two.
h=
4 t¯ R = 0.203 (Dout t¯)2 Flexibility factor
k=
1.65 = 8.123 h In-plane stress intensification factor
ii =
0.9 = 2.605 h2/3 Out-plane stress intensification factor
235
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
io =
Calculation according to ASME section III NB3600
0.75 = 2.171 h2/3
Above results will be compared against the more elaborate ASME Section III Code. For the verification, it has been assumed that the component is Class 1 (i.e. Section NB applicable) under level A (normal) service limits. This selection has been made since both sections NC and ND will yield approximately the same results as shown above for ASME B31.3. In any case, the Code states that also components classified under classes 2 and 3 can be calculated using Section NB. Stress indices in curved pipe or elbows at points remote from girth or longitudinal welds or other local discontinuities are given as
Assuming that the bending stress is predominant (i.e. axial and torsion stresses can be neglected), based on equations from Ref. [2] and aforementioned assumptions the following limit can be set
(ii Mi)2 + (io Mo)2 Z
SA = 2 SH
which yields maximum in-plane moment between operating and ambient conditions
Mi max =
K2 = 1 (Table NB
2 SH Z = 51800N m ii
C2 =
and maximum out-plane moment
Mo max =
2 SH Z = 62200N m io
3681(a)
1.95 = 5.643 (Para. NB h2/3
1)
3683.7)
Primary plus secondary stress intensity range, as defined in Para. NB-3653.1 is calculated as
Even higher allowable moment is obtained if it is assumed that there is no longitudinal stress (purely theoretical possibility since the sustained stress will always exist). In this case the allowable stress would be SA = 2.5 SH = 344.75 MPa and the maximum in-plane and out plane moments of 64800 N m and 77800 N m, respectively. In fact, this case can be easily verified using the commercial beam analysis software. The image below shows the verification of the bend using the Caesar II software (in accordance with ASME B31.3 Code and by using the so-called “liberal allowable stress” – i.e. equation (2)). As it can be seen, applying the above in-plane moment yields a stress which is very close, but still within the Code limits (i.e. no overstress).
Sn = C1
Po Dout D + C2 out Mi + C3 Eab 2t 2I
a
Ta
b
Tb
3 Sm
If through-wall temperature gradient does not exist, preceding equation becomes
Sn = C1
Po Dout D + C2 out Mi 2t 2I
3 Sm
Furthermore, if pressure stress is considered negligible (for discussion purposes only; it will exist in the reality thus lowering the effective maximum allowable moment)
Image 1 236
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Figure 4. Bree diagram alongside equations (1) and (2).
Sn = C2
Dout Mi 2I
K2 C2
3 Sm
Dout Mi 2I
2 Sa
and finally, maximum bending moment equals 6 Sm I 3 Sm Z Mi max 1 = = = 35900N m Dout C2 C2
and finally, maximum bending moment
If simplified elastic-plastic analysis allowed by the code under Section NB-3653.6, it would result in the identical result due to the assumptions listed above. Peak stress intensity range, as defined in Para. NB-3653.2 is calculated as
Maximum allowable resultant range of moment which occurs when the system goes from one service load set to another equals to
Sp = K1 C1 a
Po Dout 2 t
Ta
b
+ K2 C2
Dout 2 I
Mi +
1 2 (1
)
K3 E
Mi max 2 =
Mi max = min(Mi max 1, Mi max 2) = 35900N m
Calculation according to the proposed criteria
T1 + K3 C3 Eab
Tb + +
1 1
E
According to equation (35) the maximum allowed in plane moment would be limited by
T2
If through-wall temperature gradient does not exist, preceding equation becomes
Sp = K1 C1
SL + SE = 2
Po Dout D + K2 C2 out Mi 2t 2I
(ii Mi)2 + (io Mo) 2 Z
0.75 (SC + SH )
which would result in the maximum allowable in-plane and out-plane moments due to thermal expansion of 38900 N m and 54000 N m, respectively. On the other hand, based on equation (37), the maximum moments would be limited to
Furthermore, if pressure stress is considered negligible (as already stated above, it will exist in the reality)
Sp = K2 C2
4 Sa I 2 Sa Z = = 47700N m Dout K2 C2 K2 C2
Dout Mi 2I
SL + SE = 2
where Mi, N m , denotes resultant range of moment which occurs when the system goes from one service load set to another. Considering that the previous equation denotes peak stress range, and not peak alternating stress, its limit shall be taken as twice the appropriate amplitude from S-N curve. Considering that at 7000 cycles the allowable alternating stress (Section III Appendix I, Fig. I-9.1) is Sa = 275MPa ( 43 ksi) , it follows that
(ii Mi)2 + (io Mo) 2 Z
C N 0.2
which would result in the maximum allowable in-plane and out-plane moments due to thermal expansion of 46700 N m and 64900 N m, respectively. These values are obviously much closer to the values calculated according to the rules from Ref. [3]. 237
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 5. Maximum primary plus secondary stress range.
Comparison of the results
in accordance with ASME B31 approach would be M max=38900 ÷ 46700N m which much closer and comparable to the values obtained by using ASME Section III NB3600 approach. If the stress intensification factors and stress indices are compared, it follows that according to ASME B31.3 the in-plane (greater) stress intensification factor (ii) is 2.605 while secondary stress index (C2) according to ASME Section III NB3600 is 5.643. Ratio if these two values is 2.17, which is in line with the previous statements that ASME B31 approach underestimates the stresses by approximately 2. On the other hand, the allowable stress according to ASME B31.3 ranges from 207 MPa (equation (1) in this paper) to 345 MPa (equation (2) with no longitudinal stress), while the allowable stress according to ASME Section III NB3600 is 414 MPa. Ratio of these two values ranges from 0.5 to 0.83. It is evident that in the latter case a much higher moment would be required to lead to overstress (approximately 75% higher value would be deemed admissible in case of ASME B31.3 calculation).
According to ASME B31 approach (under the above assumption that the longitudinal stress equals half of its respective allowable), the maximum allowable bending moment between two stress states would be M max=51800 ÷ 62200N m , while the comparable value obtained using ASME Section III NB3600 approach would be Mi max = 35900N m . This in term means that a bending moment which is producing 44÷73% overstress (according to Ref. [3]) will still be considered acceptable using B31 approach. As shown above, in case extreme that SL = 0 then the limit for expansion range would be SA = 2.5 SH , which is even less conservative. In this case, the allowable bending moment in accordance with ASME B31 approach would be M max=64800 ÷ 77800N m (more than 80% higher than the values obtained by using reference [3]). Other extreme case is that SL = SH then the limit for expansion range would be SA = 1.5 SH . In this case, the allowable bending moment
Fig. 6. Maximum primary plus secondary plus peak stress value. 238
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Finite element analysis
applied (calculated in accordance with [3]), the bend is no longer overstressed.
Finite element analysis of evaluated bend for in-plane moment Analysis is performed using FE Bend (PRG 2018) under the rules of ASME Section VIII Division 2 (design by analysis). Mesh multiplier was set at 2, thus ensuring a good numerical model. An arbitrary section of straight pipe is included both upstream and downstream of elbow in order to minimize the effect of boundary conditions (fix point on one end and loading applied at the other). In this way more flexibility is given to the model and the resulting stresses will even be underestimated with respect to the DBF approach used above. First, an in-plane bending moment of 64800 N·m has been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 32%. Peak stress value (PL + PB + Q + F ) shows overstress of 82%. These are shown in Figs. 5 and 6. As it can be evidenced from the results the maximum stress is generated at locations away from the welds. It is interesting to note that the high-stress regions indicated in these pictures coincide with actual failure locations shown in Ref. [7]. Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
639 528
483.0 (= 2∙Sy) 289.5
132 182
Finite element analysis of evaluated bend for combined moment Same analysis as above has been performed by applying both inplane and out-of-plane moment acting concurrently. Both moments have been assumed to be equal so that their resultant moment equals the lower of the two limits for in-plane and out-plane moment (i.e. their resultant is 64800 N m). Both moments have been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 17% while peak stress value (PL + PB + Q + F ) shows overstress of 32%. Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
567 382
483.0 (= 2∙Sy) 289.5
117 132
Finite element analysis of evaluated bend for out-plane moment Finally, only out-plane moment of 77800 N m has been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 9%. Peak stress value (PL + PB + Q + F ) shows overstress of 7%.
Obviously, if the actual allowable stress defined in Ref. [3] for PL + PB + Q is used (3 Sm = 413.7 MPa ), the overstress shown above (and in subsequent analyses) would be even greater. By means of trial and error it has been established that an in-plane moment of approximately 7500 N m would result in a stress of approximately 70 MPa (50% of the allowable stress for the material as defined in Ref. [2]). Concurrent action of this moment with the allowable in-plane moment calculated above for this load case (51800 N m), results in overstress of 21% primary and secondary stress range and overstress of < 1% for peak stress. Sure enough, when maximum moment of Mi max = 35900N m is
Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
526 311
483.0 (= 2∙Sy) 289.5
109 107
The fact that the stresses decrease as the moment changes from inplane to out-plane are explained by the fact that elbows are typically more flexible in the out-plane bending, resulting in lower stress intensification factor. This fact is not taken into consideration in either references [1] or [3], but is given credit for in Ref. [2].
APPENDIX 1. Comparison of B31 stresses with different stress categories Primary stresses vs. sustained stresses Definition of primary stresses in accordance with reference [3] is as follows
PL + PB = B1
p Dout D + B2 out Mi, mech 2t 2I
in which:
• B and B , primary stress indices; • p, MPa , internal design pressure; mm , resultant moment due to a combination of resultant mechanical loads; • MI, mm ,,Npipe moment of inertia. • 1
2
i, mech 4
On the other hand, the definition of sustained stresses as per reference [2] is
(|SAX | + SB )2 + 4 ST2 =
SL =
p Dout F + AX + 4t ACS
(Ii Mi )2 + (Io Mo)2 Z
2
+
in which:
• S , MPa , axial stress caused by sustained loads; • S , MPa , bending stress caused by sustained loads; • S , MPa , torsional stress caused by sustained loads; • F , N, axial force due to sustained loads (other than pressure); AX
B
T
AX
239
IT MT Z
2
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
• A , mm , pipe cross-sectional area; • I , sustained in-plane moment index; • M , N mm, in-plane bending moment due to sustained loads; • I , sustained out-plane moment index; • M , N mm, out-plane bending moment due to sustained loads; • I , sustained torsional moment index; • M , N mm, torsional moment due to sustained loads. 2
CS
i
i
o
o
T
T
If axial force and torsional moment due to sustained loads other than pressure are neglected (i.e. FAX = 0N and MT = 0N mm ) and sustained indices are considered equal (i.e. Ii = Io = IT ), the preceding equation becomes
p Dout I M + i R 4t Z
SL =
in which MR = Mi2 + Mo2 denotes resultant moment due to sustained loads. Considering that, as per reference [3], the maximum value of stress index B1 is typically 0.5 and of stress index B2 is typically 1, and that MR Mi, mech , it follows that
SL
PL + PB
In the preceding equation it is assumed that the sustained stress index in ASME B31 equals Ii = B2 = 1 (in other words it is assumed that the values shown in Ref. [3] are correct). Considering the uncertainty in the relationship between the sustained stress indices and stress intensification factors, the stated assumption is justified. For discussion purposes the influence of the primary or sustained stress indices can be disregarded. Secondary stress range vs. expansion stress range Definition of primary plus secondary stress range as per reference [3] is the following
PL + PB + PE + Q = C1
p Dout D + C2 out Mi + C3 Eab 2t 2I
a
Ta
b
Tb
or if through-wall temperature gradient is neglected
PL + PB + PE + Q = C1
p Dout D + C2 out Mi 2t 2I
in which:
• C and C , primary plus secondary stress indices; • M , N mm, resultant range of moment which occurs when the system goes from one service load set to another. 2
1
i
In the preceding equation the only primary load is pressure, and if secondary stresses due to primary loads are neglected (i.e. primary loads are assumed to generate only primary stresses), it follows that
PE = C2
Dout Mi 2I
On the other hand, definition of expansion stress range as per reference [1] is
SE =
i MC Z
Considering that Mi = MC , as well as the relationship from paragraph NC-3673.2 of [3] which states that
i=
K2 C2 2
the final relationship between expansion stresses calculated in accordance with [1] or [2] and secondary stresses becomes
2 SE = PE K2 It is important to define the peak stress index, or more accurately stress concentration factor K2 in the above equation. Away from welds the value of stress index is K2 = 1, which is in line with the expected failures indicated in Ref. [7] for fabricated tees and elbows/bends. For unreinforced and reinforced fabricated tees and other components for which the failure is expected to occur at the location of the weld the value of stress index will range K2 = 1.1 ÷ 1.8. Logical value to be adopted in these cases is K2 = 1.6 1/0.64 which can be derived from Markl's original experiments (reported stress intensification factor for straight pipe of 0.64). It should be noted that according to Figure NB-3222-1 of reference [3], the limit for the abovementioned stress is
PE =
2 SE K2
3 Sm = 1.5 (SC + SH )
and
PL + PB + PE + Q = SL +
2 SE K2
3 Sm = 1.5 (SC + SH )
240
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Peak stress vs. expansion stress range Definition of peak stress intensity range as per reference [3] is the following
Sp = PL + PB + PE + Q + F = K1 C1 + K3 C3 Eab
a
Po Dout 2 t
Ta
b
+ K2 C2 Tb +
1 1
Dout 2 I
Mi +
E
1 2 (1
)
K3 E
T1 +
T2
or if through-wall temperature gradient is neglected
Sp = K1 C1
D Po Dout + K2 C2 out Mi 2t 2I
in which K1 and K2 denote peak stress indices. In the preceding equation the only primary load is pressure, and if secondary stresses due to primary loads are neglected (i.e. primary loads are assumed to generate only primary stresses), it follows that
PE + F = K2 C2
Dout Mi 2I
On the other hand, definition of expansion stress range as per reference [1] is
SE =
i MC Z
Considering that Mi = MC , it can be concluded that
PE + F =
K2 C2 SE i
Finally, knowing that in paragraph NC-3673.2 of [3] it is stated that
i=
K2 C2 2
the relationship between expansion stress range and peak stress amplitude becomes
2 SE = PE + F
References [11]
[1] ASME B31.1, Power Piping, American Society of Mechanical Engineers, New York, USA. [2] ASME B31.3, Process Piping, American Society of Mechanical Engineers, New York, USA. [3] ASME III, Rules for Construction of Nuclear Facility Components, American Society of Mechanical Engineers, New York, USA. [4] ASME VIII Division 2 Alternative Rules, Rules for Construction of Pressure Vessels American Society of Mechanical Engineers, New York, USA. [5] Markl, A.R.C., Fatigue tests of welding elbows and comparable double-mitre bends, Transactions of ASME, Vol. 69, No. 8. [6] A.R.C. Markl, H.H. George, Fatigue tests on flanged assemblies, Transactions of ASME 72 (1950). [7] A.R.C. Markl, Fatigue tests of piping components, Transactions of ASME 74 (3) (1952). [8] A.R.C. Markl, Piping-flexibility analysis, Transactions of ASME 77 (1955). [9] ASME B31J, Stress Intensification Factors (I-Factors), Flexibility Factors (KFactors), and Their Determination for Metallic Piping Components, American Society of Mechanical Engineers, New York, USA. [10] P. Dong, J.K. Hong, A.P. De Jesus, Analysis of recent fatigue data using the
[12] [13] [14] [15] [16] [17] [18]
241
structural stress procedure in ASME div 2 rewrite, ASME Journal of Pressure Vessel and Technology 129 (3) (2006). P. Dong, X. Pei, S. Xing, H.M. Kim, A structural strain method for low-cycle fatigue evaluation of welded components, Int. J. Pres. Ves. Pip. 119 (2014). P. Dong, et al., The master SN curve method: an implementation for fatigue evaluation of welded components in the ASME B&PV code, section VIII, division 2 amd API 579-1/ASME FFS-1, Weld. Res. Counc. Bull. 523 (2010). P.Q. Vu, Revision of B31 code equations for stress intensification factors and flexibility factors for intersections, 2011 ECTC Proceedings, ASME Early Career Technical Conference, 2011. N. Jacimovic, Analysis of piping stress intensification factors based on numerical models, Int. J. Pres. Ves. Pip. 163 (2018). S. Genic, B. Jacimovic, V. Genic, Economic optimization of pipe diameter for complete turbulence, Energy Build. 45 (2012). S. Genic, B. Jacimovic, A shortcut to determine optimal steam pipe diameter, Chem. Eng. Prog. 114 (8) (2018). E.C. Rodabaugh, NUREG/CR-3243 Comparisons of ASME Code Fatigue Evaluation Methods for Nuclear Class 1 Piping with Class 2 or Class 3 Piping, USA, (1983). J. Bree, Elastic-plastic behaviour of tubes with application to nuclear reactor fuel elements, J. Strain Anal. 2 (3) (1967).
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Uncertanties in expansion stress evaluation criteria in piping codes
T
Nikola Jaćimović Danieli & C. Officine Meccaniche S.p.A, Via Nazionale 41, 33042, Buttrio, UD, Italy
ABSTRACT
ASME B31 and EN 13480-3 piping design codes are considered as the state-of-the-art codes for many piping engineers. However, as shown in this paper, there seem to be inconsistencies between ASME B31 code and more elaborate calculation codes, such as ASME Section III. In fact, it is shown in the paper that the ASME B31 approach can produce potentially non-conservative results. This seemingly important flaw in the B31 (and other major) piping codes is based on the allowable stress calculation based on equation (1b) of both ASME B31.1 and B31.3 and has been shown in detail in the paper. Finally, the stated flaw is verified by means of a finite element analysis which corroborates the stated concerns.
1. Introduction
in which:
ASME B31 and EN 13480-3 piping design codes are considered as the state-of-the-art codes for many piping engineers. Although the general idea behind the modern B31 codes (which is copied to other Codes, such as EN 13480) is more than six decades old, throughout the years these Codes have been evolving and incorporating new breakthroughs in the field piping calculation and analysis, thus keeping them in line with the requirements of the modern age. However, this author believes that there might be an issue with the code philosophy concerning thermal expansion stress range. As it is shown in further text, it may happen that by following the code provisions even an overstressed piping element might be deemed acceptable, which is not true for the other, more elaborate piping codes (e.g. ASME Section III, which is based on polished bar fatigue curves). This issue is verified using advanced numerical simulations (by means of finite element analysis) which seem to confirm the stated doubts.
Although the following discussion is based on ASME B31 Codes, it is also applicable to all other major piping codes which have adopted the same design philosophy. First of all, it is beneficial to provide an overview of the limits set in ASME B31 codes for thermal expansion stress range. ASME B31 equations for allowable stress range (i.e. equations (1a) and (1b) in both references [1] and [2]) state that (1)
or if preceding equation is not fulfilled
SA = f [1.25 (SC + SH )
SL]
A
(2)
6 N 0.2
m
m
C
H
L
2
Conceptually, it can be considered that SC = SH = 3 SY (basic concept in definition of the allowable stress), and so equation (1) yields (3)
SA = f SY in which SY , MPa , denotes material yield stress. On the same basis, equation (2) yields
SA = f
2. ASME B31 stress range limits
SA = f (1.25 SC + 0.25 SH )
• S , MPa , allowable stress range; f , stress range factor; • f= f , maximum value of stress range factor (typically 1.2 for ferrous • materials); • N , number of cycles; • S , MPa , basic allowable stress at minimum metal temperature; • S , MPa , basic allowable stress at maximum metal temperature; • S , MPa , longitudinal stress due to sustained loads.
5 SY 3
SL
(4)
2 3
If SL = SH = SY , equation (4) becomes equivalent to equation (3), However, since the basic idea behind equation (2) is to increase the allowable stress range, in further discussion it will be considered that SL < SH , which yields (5)
SA > f SY which, for N
SA > SY
7000 (f = 1) yields (6)
If more elaborate stress terminology is used (i.e. as in Ref. [3] or [4]), this limit may be acceptable for peak stresses, while it falls short of the requirements for primary plus secondary stresses. This fact will
E-mail address: [email protected]. https://doi.org/10.1016/j.ijpvp.2019.01.003 Received 31 October 2018; Received in revised form 17 December 2018; Accepted 5 January 2019 Available online 10 January 2019 0308-0161/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
become important, as discussed in further text.
then the nominal stress caused by this moment equals
3. ASME B31 expansion stress range
SNOM =
C N 0.2
(8)
in which:
Calculation of expansion stress range according to ASME B31 philosophy is based on stress intensification factors which are correlating the real stress in a piping component with a nominal stress in the girth butt weld of a straight pipe. Stress intensification factors (i ) are first introduced in ASME Codes in 1955. (ASA B31.1), and are a direct result of tests and publications of A.R.C. Markl [5–8]. They represent the first take on the fatigue design of piping components. New standards (such as [9]) and research in the field material fatigue ([10–12]) and stress intensification factors ([13,14]) show that important improvements can be made to the calculation method of the stress intensification factors. Although the use of correct stress intensification factors is of utmost importance for safe operation and economic sizing of piping ([15,16]), it is not the topic of this paper. For further analysis it will be considered that the stress intensification factors published in Refs. [1,2] are correct. Benchmark for all stress intensification factors in Markl's tests is girth butt weld of a straight pipe. Seen as girth butt weld in a pipe in itself represents a stress concentration zone (stress concentration factor according to Ref. [4] ranges from 1 to 4, but is typically considered 2), the obtained stress intensification factors and consequent calculated stresses are approximately one half of the real stress values, as it is shown in further text. This is a well-known and accepted fact within the pipe stress community. S-N curve which was used for derivation of stress intensification factors is originally written in the following form
iS=
M Z
• S , MPa , nominal stress caused by application of bending moment M; • Z,mm , pipe section modulus. NOM
3
By its definition, the stress intensification factor represents the ratio of the actual stress (SACT , MPa ) caused by the applied bending moment which would cause failure in a girth butt weld after the measured number of cycles (NACT ) to the nominal stress induced in a straight pipe by the same bending moment.
i=
SACT CZ = 0.2 SNOM N ACT M
(9)
Since the above derivation is made for girth butt weld in a pipe, it must be remembered that if the failure occurs away from some stress concentration (i.e. stress concentration factor is 1), the real value of stress intensification factor would be approximately two times higher than the one obtained by the preceding equation, or
iREAL =
SACT 2CZ = 0.2 =2i SNOM N ACT M
(10)
in which:
•i •S
REAL ,
real value of stress intensification factor; MPa , stress caused by the applied bending moment which would cause failure in a straight pipe (not girth butt weld) after NACT cycles. ACT ,
(7)
where constant C = 245000psi (=1689.22MPa) and stress intensification factor of i = 1 correspond to a girth butt weld in a straight pipe. It must be noted that the N in equation (7) denotes the “number of cycles of complete reversal”, which means that this equation is valid for expansion stress range. By comparison, the same equation can be used for not clamped pipe (i.e. pipe with no stress concentration factor) using the stress intensification factor of i = 0.64 as reported in Ref. [7]. Equation (7) for both girth butt weld and straight pipe is shown in Fig. 1. Although the stress intensification factor of i = 0.64 is reported in Ref. [7] as governing for straight pipe, the following discussion will be based on the conservative and yet simplifying assumption that the stress concentration factor of straight pipe is 0.5. If a piping component is loaded with a bending moment M, N mm ,
A rough connection between S-N curves of straight pipe and of girth butt weld is represented in the following form
SACT = 2 SACT = 2
C 0.2 N ACT
(11)
Consequently, it may occur that the stresses calculated by applying code stress intensification factor represent only one half of the real stress in the component. This is a well-known fact, and B31 allowable stresses are seemingly adjusted to take this fact into consideration. However, as it will be shown, using equation (1b) from ASME B31 codes (either [1] or [2]) may yield non-conservative results for components in which the typical failure occurs away from stress
Fig. 1. S-N curve of girth butt weld. 231
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 2. Allowable stress comparison.
concentration areas (such as welding tees or elbows).
Equations (16) and (17) are similar to equations (1a) and (1b) of ASME B31 codes [1] or [2]. However, the fact that using the stress intensification factors proposed by Markl [8] misses the actual stress by a factor of ∼2 seems to be disregarded in equation (17), which becomes important if equation (16) is disregarded and only equation (17) used. It is interesting to note that even the author himself has stated the following doubt in relation to equation (17): “selection of a proper value for the factor on the right hand side of the equation is open for discussion and review”. Considering that by definition SA = SE , it can be shown that there are two extremes of equation (17):
3.1. ASME B31 approach basis This following text provides a brief overview of the proposed expansion stress range limits set in Ref. [8], which establish the basis for the modern ASME B31 codes. It serves only to show that the doubt outlined in the preceding section draws its roots from the original paper (i.e. reference [8]). Given the number of full reversals N , the S-N curve in a girth butt weld of a straight pipe according to Ref. [5] can be accurately represented by
S=
1689.22 N 0.2
5 SY 8
SE
SE , REAL
(13)
(19)
[1.875, 3.125] SY
(20)
in which SE , REAL, MPa , denotes the real stress range (i.e. without stress concentration). As it will be shown later, this criterion would be considered as inacceptable by the more elaborate piping and pressure vessel codes (references [3,4]). 3.2. Comparison of allowable stresses
(14)
Fig. 2 shows the allowable stress given by equation (2) considering the following assumptions:
(15)
- material is A106 grade B pipe (typical carbon steel); - there are no longitudinal stresses (i.e. SL = 0 MPa ); - allowable stresses (SC and SH ) are as defined in Refs. [1,2].
(16)
In addition, equation (12) and yield stress are plotted in the same figure for comparison purposes. As it is well known and will be shown later in text, the minimum limit for the actual secondary stress range (i.e. expansion stress range) should be twice the material yield strength. Considering that the actual stresses calculated by using stress intensification factor as defined in Refs. [1,2] is approximately only one half of the real stress value, it follows that in this case the limit of expansion stress range should be set to material yield strength. As it can be seen, if equation (1b) from either ASME B31.1 or B31.3 is adopted in calculation, both references [1,2]
or
SA + SPW = 1.25 (SC + SH )
0.9375 SY
(18)
Considering that as calculated by the Code 2 SE = SE , REAL , and considering equation (14), it turns out that
The final limit set by the author in the proposed rules for piping fatigue analysis in Ref. [8] is
SA = f (1.25 SC + 0.5 SH )
= SH equation (17) becomes
1.5625 SY PW
it follows that
Sav = 2 SY
• if S
SE
(12)
in which Sav, MPa , denotes available stress range. Considering that at the time of publication of this proposal the allowable stresses were conceptually defined as
SC = SH =
= 0 equation (17) becomes
PW
It should be noted that this equation is taken as benchmark value for derivation of the original stress intensification factors, some of which are still used in their original form today. Conservative estimate of allowable stress presented in Ref. [8], which ensures that there is no plastic flow either at maximum or minimum temperature, states that
Sav = 1.6 (SC + SH )
• if S
(17)
in which SPW , MPa , represents longitudinal pressure and weight stresses (i.e. roughly corresponding to SL, MPa as defined in equation (2)). It must be noted that the author in Ref. [8] never did indicate expressly that the two equations (16) and (17) are interchangeable. In fact, in the originally proposed rules the equation (16) is indicated as mandatory, while equation (17) is proposed by M.W. Kellogg Company, rather than the Task Force on Flexibility. 232
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
can deem overstressed components as acceptable. Similarly, it can be shown that even if the less conservative straight pipe stress intensification factor is used (i.e. 0.64 instead of 0.5) the rules set in ASME B31.3 may still result in deeming the overstressed components acceptable. Similar figure to Fig. 2 is give in Ref. [17]. However, the major difference is that in Ref. [17] the author had used an allowable stress for SA-106 Grade B of 15 ksi, which is far less than the value of 17.1÷20ksi given in Refs. [1,2].
stress (as in equations (23) and (24)) the system will undergo elastic shakedown, and upon every consecutive load application will remain within the elastic region (i.e. in the region [ SY , SY ]). If the stress surpasses the aforementioned limit, the system will undergo small additional plastic deformation upon every load application and will consequently fail. As discussed beforehand, if expansion stress range as defined in Refs. [1,2] is considered as secondary stress (i.e. excluding stress concentrations) in which case the Code equations calculate roughly one half of the real stress value, it can be concluded that
4. Stress categories and their limits
iREAL MC 2 i MC = Z Z
Below is presented a quick overview of the various stress categories and failure mechanisms in connection with thermal (secondary) stresses. It is only supposed to give an insight into the code philosophy, and to present a comparison between ASME B31 stresses with the more realistic stresses calculating using the more elaborate codes. Main focus should be put on the discussion part presented in the subsequent chapters.
SL +
SE =
2 SY
(29)
Sa
• in one counted cycle there are two stress amplitudes (as defined in Refs. [1,2]), and • based on the facts that equation (12) is derived for the stress range
(22)
(i.e. twice the stress amplitude) and that the stress concentration factor of welds is approximately 2, Sa can be approximated with C equation (12) as Sa N 0.2 (which can be verified using values from Appendix I, Figure I-9.1 of [3]) equation (29) becomes
and
PL + PB + PE + Q
(28)
It must be noted that the Sa in the preceding equation denotes the allowable stress amplitude obtained for a polished bar and is not directly comparable to equation (12). Knowing that equation (12) is derived for full range of stress, its right-hand side can be correlated to the Sa , as shown below. Considering that:
(23)
2 SY
SY
(PL + PB + PE + Q + F ) AMPLITUDE
in which Sm, MPa , denotes mean allowable stress considering both cold and hot design temperatures. This limit is helping to prevent low-cycle fatigue or ratcheting failure. Considering that conceptually the mean allowable stress corresponds to two thirds of yield stress, the preceding equations become
PE
(27)
As defined in Refs. [3,4], peak stresses represent the incremental stress in a component added to primary or secondary stress which is caused by a stress concentration (i.e. a notch). Failure mechanism connected to peak stresses is high-cycle fatigue, which causes crack initiation and growth until the through-wall crack is generated and catastrophic failure occurs. Typically, the limit of peak stresses is derived from S-N curve, or
(21)
3 Sm
SY
4.2. Peak stresses
and
PL + PB + PE + Q
(26)
As it can be seen, the criterion represented by equation (28) is more conservative than the one represented by equation (27). However, it is important to note that in Ref. [3] the criteria represented by equations (27) and (28) are not mutually exclusive, but must be applied together.
In further analysis it will be considered that the longitudinal stresses defined in both [1] and [2] are comparable to the primary stresses defined in Ref. [3] (i.e. SL = PL + PB ). This fact is elaborated in Appendix 1. Considering that local membrane stress always includes general membrane stress contribution, the Code limits for this stress category are set as
3 Sm
i MC Z
SL S i MC + SE = L + 2 2 Z
m
mechanical loads (excluding discontinuities, and local stress concentrations); PL, MPa , local membrane stress produced by pressure and other mechanical loads (excluding discontinuities, and local stress concentrations); PB, MPa , bending stress produced by pressure and other mechanical loads (excluding discontinuities, and local stress concentrations); PE , MPa , expansion stress produced by constraint of free end displacement (including discontinuities, but not local stress concentrations); Q, MPa , secondary membrane and bending stress necessary to satisfy continuity of the structure (including discontinuities, but not local stress concentrations).
PE
3 Sm
It should be noted that this criterion is conceptually identical to equation (1a) of B31 Codes. Based on the same reasoning, equation (26) becomes
• P , MPa , general membrane stress produced by pressure and other
•
iREAL MC 2 i MC = SL + Z Z
Considering that, conceptually, the mean allowable stress corresponds to two thirds of yield stress, equation (25) becomes
According to Ref. [3], this stress category includes the following stresses:
• •
(25)
and
4.1. Primary plus secondary stresses
•
3 Sm
(PL + PB + PE + Q + F ) AMPLITUDE
(24)
1 i MC SL + 2 2 Z
=
SL i MC + 2 Z
C N 0.2
(30)
which is a well-known limit against progressive plastic deformation (PPD) and low-cycle fatigue failure. This failure mechanism and logic behind the aforementioned limit are shown in Fig. 3. As it is shown in Fig. 3, in case that the limit is set to twice the yield
For comparison purposes only, equation (30) can be conservatively written in the form of equation (28). As shown in Fig. 2, for any number C cycles under 10000 for carbon steels it can be considered that N 0.2 > SY , 233
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 3. Low-cycle fatigue (a) no PPD (b) PPD exists.
• S and S – elastic shakedown zones; • P – plastic cycling zone; • E − elastic zone.
which in term means that the following estimate resulting from equation (30) is conservative
SL i MC + 2 Z
SY
1
(31)
Obviously, the acceptable zones are E, S1 and S2, while all the others will lead to system failure. As it can be seen, both equations (1) and (2) may yield unacceptable results. Equation (1) may lead to ratcheting if the primary stress is sufficiently high and the limit of twice the yield strength is reached by secondary stress range. Equation (2) is even worse since it ranges from plastic cycling zone through both ratcheting zones R1 and R2.
or
SL + 2 SE = SL + 2
i MC Z
2 SY
(32)
Comparing equations (28) and (30) it is for the low number of cycles equation (28) is predominant in setting the load limit while equation (30) tends to become governing for larger number of cycles.
5. Discussion
4.3. Bree diagram
Although most of the time the stresses and derivation performed by Markl are considered as peak stresses (i.e. including local discontinuities and stress concentrations), which is not untrue, a question is raised what are the stress types when failure did not occur at the weld location but at some other location (e.g. typical failures in bends loaded with in-plane moment). In these cases, it is author's belief that the correct interpretation is that primary plus secondary stress category has been measured rather than peak stresses. In fact, even in the original paper [5] the author has acknowledged that welding elbows loaded with in-plane bending moment the crack had invariably started at the center of the side wall and has progressed towards both ends of specimens. Considering that there are no notches or other concentrations at this location, this kind of failure can only be classified as occurring due to secondary stress. In essence, two different failure mechanisms are encompassed by the calculation of thermal expansion stress range given by Refs. [1,2], and consequently there are two different verification paths:
Bree diagram [18] depicts the regions of ratcheting, shakedown, plastic cycling and elastic cycling. It is based on a plot of primary stress versus secondary stress range. Although it is derived only for a very simple example problem, the Bree diagram can be considered quintessential. Consequently, it can be used to further fortify the preceding points. Bree diagram is shown in Fig. 4, alongside acceptance criteria given in form of equations (1) and (2). Fig. 4 is derived based on the following two cases:
• real thermal stress range is two times higher than the one calculated •
by using the ASME B31 equations ([1,2]), corresponding to i = 0.5 in equation (7) – Fig. 4a; real thermal stress range is 1.56 times higher than the one calculated by using the ASME B31 equations ([1,2]), corresponding to i = 0.64 in equation (7) – Fig. 4b. In Fig. 4 the following characteristic zones are shown:
•R
1
2
1. Failure occurs due to primary and secondary stress; 2. Failure occurs due to peak stress (at the stress concentration area).
and R2 – ratcheting zones; 234
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Derivation of relationships between various stresses is given in Appendix 1.
6. Conclusion ASME B31 codes have long been one (and still are) of the most wide spread and useful piping codes. Their practical approach has made them indispensable in everyday engineering. However, it must be acknowledged that they are based on reasoning that was adopted over six decades ago. Following more recent findings and definition of stress categories and their respective allowable values, the author has tried to show that one of the Code requirements may lead to non-conservative results. This non-conservatism has been a verified in an example shown at the end of the paper using finite element analysis which seems to have confirmed the stated doubts. In order to keep consistency between various piping code philosophies, it could be suggested to update the acceptance criteria of references [1,2]. Furthermore, it would seem reasonable to perform a careful study to evaluate the influence on secondary and peak stresses, as shown in equations (34)-(37), for various piping materials. The discussion presented herewith does not necessarily mean that every piping component calculated using the disputed approach is overstressed (since there are various safety factors inherent to the Codes), but only to point out what seems to be an inaccuracy in the Code philosophy which may or may not lead to nonconservative results. Considering the constant breakthroughs and accumulation of knowledge and computational power, it is inevitable that the piping codes will evolve, making calculations more accurate and eliminating excessive safety factors. However, relatively simple calculation procedures, such as the ones adopted in B31 codes, will always have their immense practical use. One certainty is that B31 codes and their philosophy are here to stay and will for sure be used for years to come. This paper aims to simply provide an insight into possible uncertainties in Code philosophy which may lead to non-conservative results.
5.1. Failure due to primary and secondary stress In case that the failure occurs away from stress concentration area (i.e. notch), then the expansion stress range calculated by using ASME B31 approach equals to secondary expansion stress range as defined in Ref. [3]. In this case, there are two criteria which are to be applied. First one is in the form of equation (27) and equals to
SE
PE 2
SY
(33)
which is conceptually equal to
SE
0.75 (SC + SH )
(34)
and is very similar to equation (1). Second criterion which should be considered is given by equation (28) and states that
SL + SE 2
0.75 (SC + SH )
(35)
Seen as the latter equation is more conservative, it seems logical for it to be set as a limit. However, seen as both references [3,4] provide special rules for exceeding the set limit, it may be sufficient to use equation (34) as the Code limit as equation (35) would probably result in overly conservative design. Equation (35) is applicable to the components for which the expected failure is away from the weld area (i.e. K2 = 1, as defined in Ref. [3]). These are typically elbows/bends and welding tees. For other components, such as fabricated tees, the stress concentration factor defined in ASME B31 codes already includes effects of stress concentration due to weld. In these cases a following less conservative criterion can be used
0.8 SL + SE
1.15 (SC + SH )
Example Below example will show influence of the preceding discussion on a typical pipe elbow stresses. For this example, a pipe elbow with the following characteristics has been selected:
(36)
• ¯D = 273mm; • t = 9.27mm; • R = 381mm (ASME B16.9 long radius elbow); • Material A234 WPB (S = 241.3MPa ); • Number of cycles: 7000; • Hot/cold allowable considering maximum operating temperature
Equation (36) is somewhat similar to equation (2), but is not always applicable to all piping components.
out
5.2. Failure due to peak stress
Y
If failure occurs at the stress concentration area (i.e. notch), then the expansion stress range calculated by using ASME B31 approach equals to primary plus secondary plus peak stress as defined in Ref. [3]. In this case, the criterion which should be employed is
SL + SE 2
C N 0.2
under 140 °C equals SH = SC = 137.9MPa according to Refs. [2,3] (even lower value than the conceptual limit of 2/3 SY used in the main discussion.
Calculation according to ASME B31.3
(37)
Assuming that SL = 2 SH and 7000 cycle limit, according to equation (2), the limit for expansion range would be SA = 2 SH = 275.8 MPa . Flexibility characteristic 1
in which C = 245000psi (=1690MPa) , as discussed beforehand. Since the effect of stress concentration is already included in the ASME B31 stress intensification factors, there is no need to include it explicitly in equation (37). It should be noted that equation (37) represents calculation and comparison of stress amplitudes, and not stress ranges. Also, the obvious shortcoming of this equation is its independence on the material. It is derived only for carbon steels and a detailed study should be performed to correlate the proposed equation to the other materials (for example using polished bar curves reported in Ref. [3]) and expected failure locations (e.g. considering that typical failure location on elbows occurs away from stress concentration areas, this criterion is obsolete). Finally, seen as employing the calculation procedure of references [1] or [2] it remains unclear whether secondary or peak stresses are calculated, it seems reasonable to use the worst one of the two.
h=
4 t¯ R = 0.203 (Dout t¯)2 Flexibility factor
k=
1.65 = 8.123 h In-plane stress intensification factor
ii =
0.9 = 2.605 h2/3 Out-plane stress intensification factor
235
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
io =
Calculation according to ASME section III NB3600
0.75 = 2.171 h2/3
Above results will be compared against the more elaborate ASME Section III Code. For the verification, it has been assumed that the component is Class 1 (i.e. Section NB applicable) under level A (normal) service limits. This selection has been made since both sections NC and ND will yield approximately the same results as shown above for ASME B31.3. In any case, the Code states that also components classified under classes 2 and 3 can be calculated using Section NB. Stress indices in curved pipe or elbows at points remote from girth or longitudinal welds or other local discontinuities are given as
Assuming that the bending stress is predominant (i.e. axial and torsion stresses can be neglected), based on equations from Ref. [2] and aforementioned assumptions the following limit can be set
(ii Mi)2 + (io Mo)2 Z
SA = 2 SH
which yields maximum in-plane moment between operating and ambient conditions
Mi max =
K2 = 1 (Table NB
2 SH Z = 51800N m ii
C2 =
and maximum out-plane moment
Mo max =
2 SH Z = 62200N m io
3681(a)
1.95 = 5.643 (Para. NB h2/3
1)
3683.7)
Primary plus secondary stress intensity range, as defined in Para. NB-3653.1 is calculated as
Even higher allowable moment is obtained if it is assumed that there is no longitudinal stress (purely theoretical possibility since the sustained stress will always exist). In this case the allowable stress would be SA = 2.5 SH = 344.75 MPa and the maximum in-plane and out plane moments of 64800 N m and 77800 N m, respectively. In fact, this case can be easily verified using the commercial beam analysis software. The image below shows the verification of the bend using the Caesar II software (in accordance with ASME B31.3 Code and by using the so-called “liberal allowable stress” – i.e. equation (2)). As it can be seen, applying the above in-plane moment yields a stress which is very close, but still within the Code limits (i.e. no overstress).
Sn = C1
Po Dout D + C2 out Mi + C3 Eab 2t 2I
a
Ta
b
Tb
3 Sm
If through-wall temperature gradient does not exist, preceding equation becomes
Sn = C1
Po Dout D + C2 out Mi 2t 2I
3 Sm
Furthermore, if pressure stress is considered negligible (for discussion purposes only; it will exist in the reality thus lowering the effective maximum allowable moment)
Image 1 236
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Figure 4. Bree diagram alongside equations (1) and (2).
Sn = C2
Dout Mi 2I
K2 C2
3 Sm
Dout Mi 2I
2 Sa
and finally, maximum bending moment equals 6 Sm I 3 Sm Z Mi max 1 = = = 35900N m Dout C2 C2
and finally, maximum bending moment
If simplified elastic-plastic analysis allowed by the code under Section NB-3653.6, it would result in the identical result due to the assumptions listed above. Peak stress intensity range, as defined in Para. NB-3653.2 is calculated as
Maximum allowable resultant range of moment which occurs when the system goes from one service load set to another equals to
Sp = K1 C1 a
Po Dout 2 t
Ta
b
+ K2 C2
Dout 2 I
Mi +
1 2 (1
)
K3 E
Mi max 2 =
Mi max = min(Mi max 1, Mi max 2) = 35900N m
Calculation according to the proposed criteria
T1 + K3 C3 Eab
Tb + +
1 1
E
According to equation (35) the maximum allowed in plane moment would be limited by
T2
If through-wall temperature gradient does not exist, preceding equation becomes
Sp = K1 C1
SL + SE = 2
Po Dout D + K2 C2 out Mi 2t 2I
(ii Mi)2 + (io Mo) 2 Z
0.75 (SC + SH )
which would result in the maximum allowable in-plane and out-plane moments due to thermal expansion of 38900 N m and 54000 N m, respectively. On the other hand, based on equation (37), the maximum moments would be limited to
Furthermore, if pressure stress is considered negligible (as already stated above, it will exist in the reality)
Sp = K2 C2
4 Sa I 2 Sa Z = = 47700N m Dout K2 C2 K2 C2
Dout Mi 2I
SL + SE = 2
where Mi, N m , denotes resultant range of moment which occurs when the system goes from one service load set to another. Considering that the previous equation denotes peak stress range, and not peak alternating stress, its limit shall be taken as twice the appropriate amplitude from S-N curve. Considering that at 7000 cycles the allowable alternating stress (Section III Appendix I, Fig. I-9.1) is Sa = 275MPa ( 43 ksi) , it follows that
(ii Mi)2 + (io Mo) 2 Z
C N 0.2
which would result in the maximum allowable in-plane and out-plane moments due to thermal expansion of 46700 N m and 64900 N m, respectively. These values are obviously much closer to the values calculated according to the rules from Ref. [3]. 237
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Fig. 5. Maximum primary plus secondary stress range.
Comparison of the results
in accordance with ASME B31 approach would be M max=38900 ÷ 46700N m which much closer and comparable to the values obtained by using ASME Section III NB3600 approach. If the stress intensification factors and stress indices are compared, it follows that according to ASME B31.3 the in-plane (greater) stress intensification factor (ii) is 2.605 while secondary stress index (C2) according to ASME Section III NB3600 is 5.643. Ratio if these two values is 2.17, which is in line with the previous statements that ASME B31 approach underestimates the stresses by approximately 2. On the other hand, the allowable stress according to ASME B31.3 ranges from 207 MPa (equation (1) in this paper) to 345 MPa (equation (2) with no longitudinal stress), while the allowable stress according to ASME Section III NB3600 is 414 MPa. Ratio of these two values ranges from 0.5 to 0.83. It is evident that in the latter case a much higher moment would be required to lead to overstress (approximately 75% higher value would be deemed admissible in case of ASME B31.3 calculation).
According to ASME B31 approach (under the above assumption that the longitudinal stress equals half of its respective allowable), the maximum allowable bending moment between two stress states would be M max=51800 ÷ 62200N m , while the comparable value obtained using ASME Section III NB3600 approach would be Mi max = 35900N m . This in term means that a bending moment which is producing 44÷73% overstress (according to Ref. [3]) will still be considered acceptable using B31 approach. As shown above, in case extreme that SL = 0 then the limit for expansion range would be SA = 2.5 SH , which is even less conservative. In this case, the allowable bending moment in accordance with ASME B31 approach would be M max=64800 ÷ 77800N m (more than 80% higher than the values obtained by using reference [3]). Other extreme case is that SL = SH then the limit for expansion range would be SA = 1.5 SH . In this case, the allowable bending moment
Fig. 6. Maximum primary plus secondary plus peak stress value. 238
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Finite element analysis
applied (calculated in accordance with [3]), the bend is no longer overstressed.
Finite element analysis of evaluated bend for in-plane moment Analysis is performed using FE Bend (PRG 2018) under the rules of ASME Section VIII Division 2 (design by analysis). Mesh multiplier was set at 2, thus ensuring a good numerical model. An arbitrary section of straight pipe is included both upstream and downstream of elbow in order to minimize the effect of boundary conditions (fix point on one end and loading applied at the other). In this way more flexibility is given to the model and the resulting stresses will even be underestimated with respect to the DBF approach used above. First, an in-plane bending moment of 64800 N·m has been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 32%. Peak stress value (PL + PB + Q + F ) shows overstress of 82%. These are shown in Figs. 5 and 6. As it can be evidenced from the results the maximum stress is generated at locations away from the welds. It is interesting to note that the high-stress regions indicated in these pictures coincide with actual failure locations shown in Ref. [7]. Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
639 528
483.0 (= 2∙Sy) 289.5
132 182
Finite element analysis of evaluated bend for combined moment Same analysis as above has been performed by applying both inplane and out-of-plane moment acting concurrently. Both moments have been assumed to be equal so that their resultant moment equals the lower of the two limits for in-plane and out-plane moment (i.e. their resultant is 64800 N m). Both moments have been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 17% while peak stress value (PL + PB + Q + F ) shows overstress of 32%. Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
567 382
483.0 (= 2∙Sy) 289.5
117 132
Finite element analysis of evaluated bend for out-plane moment Finally, only out-plane moment of 77800 N m has been applied as operating load (all other loads are neglected). Result is overstress in primary and secondary stress range (PL + PB + Q ) of 9%. Peak stress value (PL + PB + Q + F ) shows overstress of 7%.
Obviously, if the actual allowable stress defined in Ref. [3] for PL + PB + Q is used (3 Sm = 413.7 MPa ), the overstress shown above (and in subsequent analyses) would be even greater. By means of trial and error it has been established that an in-plane moment of approximately 7500 N m would result in a stress of approximately 70 MPa (50% of the allowable stress for the material as defined in Ref. [2]). Concurrent action of this moment with the allowable in-plane moment calculated above for this load case (51800 N m), results in overstress of 21% primary and secondary stress range and overstress of < 1% for peak stress. Sure enough, when maximum moment of Mi max = 35900N m is
Stress
Stress, MPa
Theoretical allowable stress, MPa
Utilization ratio, %
PL + PB + Q PL + PB + Q + F
526 311
483.0 (= 2∙Sy) 289.5
109 107
The fact that the stresses decrease as the moment changes from inplane to out-plane are explained by the fact that elbows are typically more flexible in the out-plane bending, resulting in lower stress intensification factor. This fact is not taken into consideration in either references [1] or [3], but is given credit for in Ref. [2].
APPENDIX 1. Comparison of B31 stresses with different stress categories Primary stresses vs. sustained stresses Definition of primary stresses in accordance with reference [3] is as follows
PL + PB = B1
p Dout D + B2 out Mi, mech 2t 2I
in which:
• B and B , primary stress indices; • p, MPa , internal design pressure; mm , resultant moment due to a combination of resultant mechanical loads; • MI, mm ,,Npipe moment of inertia. • 1
2
i, mech 4
On the other hand, the definition of sustained stresses as per reference [2] is
(|SAX | + SB )2 + 4 ST2 =
SL =
p Dout F + AX + 4t ACS
(Ii Mi )2 + (Io Mo)2 Z
2
+
in which:
• S , MPa , axial stress caused by sustained loads; • S , MPa , bending stress caused by sustained loads; • S , MPa , torsional stress caused by sustained loads; • F , N, axial force due to sustained loads (other than pressure); AX
B
T
AX
239
IT MT Z
2
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
• A , mm , pipe cross-sectional area; • I , sustained in-plane moment index; • M , N mm, in-plane bending moment due to sustained loads; • I , sustained out-plane moment index; • M , N mm, out-plane bending moment due to sustained loads; • I , sustained torsional moment index; • M , N mm, torsional moment due to sustained loads. 2
CS
i
i
o
o
T
T
If axial force and torsional moment due to sustained loads other than pressure are neglected (i.e. FAX = 0N and MT = 0N mm ) and sustained indices are considered equal (i.e. Ii = Io = IT ), the preceding equation becomes
p Dout I M + i R 4t Z
SL =
in which MR = Mi2 + Mo2 denotes resultant moment due to sustained loads. Considering that, as per reference [3], the maximum value of stress index B1 is typically 0.5 and of stress index B2 is typically 1, and that MR Mi, mech , it follows that
SL
PL + PB
In the preceding equation it is assumed that the sustained stress index in ASME B31 equals Ii = B2 = 1 (in other words it is assumed that the values shown in Ref. [3] are correct). Considering the uncertainty in the relationship between the sustained stress indices and stress intensification factors, the stated assumption is justified. For discussion purposes the influence of the primary or sustained stress indices can be disregarded. Secondary stress range vs. expansion stress range Definition of primary plus secondary stress range as per reference [3] is the following
PL + PB + PE + Q = C1
p Dout D + C2 out Mi + C3 Eab 2t 2I
a
Ta
b
Tb
or if through-wall temperature gradient is neglected
PL + PB + PE + Q = C1
p Dout D + C2 out Mi 2t 2I
in which:
• C and C , primary plus secondary stress indices; • M , N mm, resultant range of moment which occurs when the system goes from one service load set to another. 2
1
i
In the preceding equation the only primary load is pressure, and if secondary stresses due to primary loads are neglected (i.e. primary loads are assumed to generate only primary stresses), it follows that
PE = C2
Dout Mi 2I
On the other hand, definition of expansion stress range as per reference [1] is
SE =
i MC Z
Considering that Mi = MC , as well as the relationship from paragraph NC-3673.2 of [3] which states that
i=
K2 C2 2
the final relationship between expansion stresses calculated in accordance with [1] or [2] and secondary stresses becomes
2 SE = PE K2 It is important to define the peak stress index, or more accurately stress concentration factor K2 in the above equation. Away from welds the value of stress index is K2 = 1, which is in line with the expected failures indicated in Ref. [7] for fabricated tees and elbows/bends. For unreinforced and reinforced fabricated tees and other components for which the failure is expected to occur at the location of the weld the value of stress index will range K2 = 1.1 ÷ 1.8. Logical value to be adopted in these cases is K2 = 1.6 1/0.64 which can be derived from Markl's original experiments (reported stress intensification factor for straight pipe of 0.64). It should be noted that according to Figure NB-3222-1 of reference [3], the limit for the abovementioned stress is
PE =
2 SE K2
3 Sm = 1.5 (SC + SH )
and
PL + PB + PE + Q = SL +
2 SE K2
3 Sm = 1.5 (SC + SH )
240
International Journal of Pressure Vessels and Piping 169 (2019) 230–241
N. Jaćimović
Peak stress vs. expansion stress range Definition of peak stress intensity range as per reference [3] is the following
Sp = PL + PB + PE + Q + F = K1 C1 + K3 C3 Eab
a
Po Dout 2 t
Ta
b
+ K2 C2 Tb +
1 1
Dout 2 I
Mi +
E
1 2 (1
)
K3 E
T1 +
T2
or if through-wall temperature gradient is neglected
Sp = K1 C1
D Po Dout + K2 C2 out Mi 2t 2I
in which K1 and K2 denote peak stress indices. In the preceding equation the only primary load is pressure, and if secondary stresses due to primary loads are neglected (i.e. primary loads are assumed to generate only primary stresses), it follows that
PE + F = K2 C2
Dout Mi 2I
On the other hand, definition of expansion stress range as per reference [1] is
SE =
i MC Z
Considering that Mi = MC , it can be concluded that
PE + F =
K2 C2 SE i
Finally, knowing that in paragraph NC-3673.2 of [3] it is stated that
i=
K2 C2 2
the relationship between expansion stress range and peak stress amplitude becomes
2 SE = PE + F
References [11]
[1] ASME B31.1, Power Piping, American Society of Mechanical Engineers, New York, USA. [2] ASME B31.3, Process Piping, American Society of Mechanical Engineers, New York, USA. [3] ASME III, Rules for Construction of Nuclear Facility Components, American Society of Mechanical Engineers, New York, USA. [4] ASME VIII Division 2 Alternative Rules, Rules for Construction of Pressure Vessels American Society of Mechanical Engineers, New York, USA. [5] Markl, A.R.C., Fatigue tests of welding elbows and comparable double-mitre bends, Transactions of ASME, Vol. 69, No. 8. [6] A.R.C. Markl, H.H. George, Fatigue tests on flanged assemblies, Transactions of ASME 72 (1950). [7] A.R.C. Markl, Fatigue tests of piping components, Transactions of ASME 74 (3) (1952). [8] A.R.C. Markl, Piping-flexibility analysis, Transactions of ASME 77 (1955). [9] ASME B31J, Stress Intensification Factors (I-Factors), Flexibility Factors (KFactors), and Their Determination for Metallic Piping Components, American Society of Mechanical Engineers, New York, USA. [10] P. Dong, J.K. Hong, A.P. De Jesus, Analysis of recent fatigue data using the
[12] [13] [14] [15] [16] [17] [18]
241
structural stress procedure in ASME div 2 rewrite, ASME Journal of Pressure Vessel and Technology 129 (3) (2006). P. Dong, X. Pei, S. Xing, H.M. Kim, A structural strain method for low-cycle fatigue evaluation of welded components, Int. J. Pres. Ves. Pip. 119 (2014). P. Dong, et al., The master SN curve method: an implementation for fatigue evaluation of welded components in the ASME B&PV code, section VIII, division 2 amd API 579-1/ASME FFS-1, Weld. Res. Counc. Bull. 523 (2010). P.Q. Vu, Revision of B31 code equations for stress intensification factors and flexibility factors for intersections, 2011 ECTC Proceedings, ASME Early Career Technical Conference, 2011. N. Jacimovic, Analysis of piping stress intensification factors based on numerical models, Int. J. Pres. Ves. Pip. 163 (2018). S. Genic, B. Jacimovic, V. Genic, Economic optimization of pipe diameter for complete turbulence, Energy Build. 45 (2012). S. Genic, B. Jacimovic, A shortcut to determine optimal steam pipe diameter, Chem. Eng. Prog. 114 (8) (2018). E.C. Rodabaugh, NUREG/CR-3243 Comparisons of ASME Code Fatigue Evaluation Methods for Nuclear Class 1 Piping with Class 2 or Class 3 Piping, USA, (1983). J. Bree, Elastic-plastic behaviour of tubes with application to nuclear reactor fuel elements, J. Strain Anal. 2 (3) (1967).