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A method of computing stress range and fatigue damage in a nuclear piping system
NUCLEAR ENGINEERING AND DESIGN 22 I1972) 318-325. NORTtt-ttOLLANI) PUBLISHING COMI'ANY
A METHOD OF COMPUTING STRESS RANGE AND F A T I G U E D A M A G E IN A N U C L E A R P I P I N G S Y S T E M Wallace B. WRIGHT Westinghouse Electric Corporation. Nu clear Energy Systems, PWR Systems Division. Pittsburgh, Pennsylvania 15230, USA and Everett C. RODABAUGH Battelle Memorial Institute, Applied Mechanics Section, Columbus, Ohio 43201, USA Received 10 January 1972
The ASME Boiler and Pressure Vessel Code, Section 111 (1971) employs stress range and fatigue evaluation as part of the total stress evaluation of piping systems. One method of computing this stress range and the resultant fatigue damage using the basic equations of the Code is presented here. This method is then illustrated by examples and discussed in detail.
1. Introduction The ASME Boiler and Pressure Vessel Code, Section III (1971) [1 ], is written as a guide to cover the following subjects: design, analysis, fabrication, erection, arid inspection of nuclear components. The Code treats each o f these subjects in separate sections. The piping analysis section of the Code is written to be used for the evaluation o f primary, secondary, and peak stresses at selected points in the piping system. The secondary and peak (fatigue) stresses are based on a stress range concept and a comprehensive discussion o f this subject may be found in the foreword of ref. [2]. The purpose of this paper is twofold; to present a method for computing stress range and, in turn, using the stress range to compute fatigue damage. To accomplish the stated purpose, this paper is divided into several sections. The first section discusses power plant operating transients and their effect on stresses in a plant's piping system. The cyclic stresses in the piping system are then combined to give a range of stress as illustrated by a simple example of
two stress transients. The technique of combining stress ranges and computing fatigue damage in a manner compatible with Article NB3600 of the Code is then illustrated by formula and a detailed computation illustrating this formula.
2. Relation of plant operation to transient stress 2.1. Plant operations relating to piping loading Fig. 1 is an illustration of the operating cycles experienced by a plant during its operational life. The numbers one through six on this figure represent operating cycles that are hypothesized to be experienced by the plant in a random manner during its operational life. Number 1 in fig. 1 represents a hydrotest operational cycle on the plant piping that will occur 100 times during the total life o f plant operation. Number 2 represents a hypothetical earthquake that will occur five times with each occurrence having ten significant vibrational cycles which will occur during plant shutdown. The sloping line that rises to number 3 repre-
W.B. Wright and E.C. R odabaugh, Stress range and fatigue damage in nuclear pipeworks
First, the values of temperature and pressure resulting from the various plant operating cycles are presented in the Design Specification (which is the basis for the analysis) as a series of time history plots. An example of one of these plots is illustrated by fig. 2 which represents operational cycle number 4 of fig. 1. Additionally, the seismic disturbance that may occur at the plant site is defined in the Design Specification as a set of response spectra to be used in computing dynamic loads.
@
0.0
I,
I,
POSTULATEDCONDITIONS Fig. 1. Plant hypothetical operating history.
sents plant startup and the leveling off at number 3 indicates steady state operation. There are 60 of these startups which are one-half of the total operational start up-shutdown cycle. The surge up and down and back to normal operation at number 4 represents a turbine trip that occurs 38 times. The sharp slope to number 5 and return to normal operation represents an overpressure of the system from which the plant returns to normal operation and there are 13 such operational cycles in the plant's history. The slope from number 6 to plant shutdown occurs sixty times and is the other half of the start up-shut down cycle. These operational cycles are assumed to represent the total plant hypothetical operating history. Associated with this operating history will be a stress history of the plant. For illustrative purposes we will examine the stress history at one point in one of the plant's main piping systems.
2.2. Stress transient history Prior to the evaluation of metal fatigue from stress caused by the plant operational cycles it is instructive to examine three possible types of plant stress history, each resulting from a separate type of operational cycle. First, examining fig. 2 note that it illustrates an appropriate sine wave response of one cycle duration. Superimposed on the plant transient is the stress time history at a point in the piping system that is shown in fig, 3. Alternately, looking at an operational plant transient cycle (that includes both temperature and pressure) with the superimposed stress transient as shown in fig. 4, note that the plant operational cycle has the characteristic sine waveresponse and consequently, there is a stress response that takes the approximate form of the transient as before but now the stress transient contains several stress cycles. It is impractical to compute the total stress time history of each instant in time, therefore, a judgment is made as to maximum and minimum loads within the
300
100
I 100
319
LNORMAL OPERATION
t.~
300
TiME Fig. 2. Typical temperature-pressure response to an operational cycle.
320
W.B. Wright and b~:C. R odabaugh, Stress range and fatigue damage in n.clear pipeworks
/f"••//--T
RANSIENT STRESS /
ATUR[
-PRESSOR
S~RESS [RANSIFN[
/
~ PLANTTRANSIENJ
TIME I.
Fig. 3. Stress transient superimposed on plant operational cycle.
operational cycle. In the first instance shown by fig. 3 there is one maximum and one minimum value. In the second instanc e shown by fig. 4 there are several significant stress cycles, two maximum and two minimum. A third possibility is that only one maximum and one minimum value may exist, but they are embedded in the transient and are not readily discernable. This third type of transient will be included and illustrated by the detailed example on fatigue evaluation.
3. Example of transient stresses converted to stress range 3.1. Combining cyclic" stress To illustrate the basic concept of combining stresses to establish the range of stresses and determining the number of stress cycles for fatigue evaluation a first example involving two operational cycles will be used. Assume for the discussion in this section that the maximum and minimum stress associated with the two operational cycles is known. The stress values and number of cycles are presented by table 1 with the maximum and minimum stress values assigned a number for bookkeeping purposes. To establish the range of stress between the several values given in table I a computation of stress differences is made by subtracting combinations of stress as shown by table 2. This table introduces the notation i and ] in the column "Number combination". The individual i and] values permit a record to be kept during computation as the values of stress are formed in all possible combinations. Additionally, the i and j
Fig. 4. Stress transient superimposed on plant's multiple cycle.operating transient.
terms will be used shortly in table 4 to identify stress values and the number of occurrences of various stress cycles as range-transient combinations are formed. To best operate with these stress ranges they are reordered in descending value and are presented by table 3. A new table, table 4, is now constructed that associates the sequence number with its related number of cycles by use of the letters i and j. The number of occurrences of each stress range is computed by determining the maximum number of times each set i, j can occur in combination. From each of these maximum numbers of combinations, progressing from top to bottom, one number (i orj) is eliminated and the other is adjusted downward to represent the number of times it can now form in new combination with other i,] values. There is an orderly progression of reducing the number of cycles until they have each formed in combination with others until all significant
Table 1 Maximum and minimum cyclic stress values. Operational cycle
Number
Stress (psi)
1
1 2 3 4
15 60 35 10
2
000 000 000 000
Number of cycles of occurrence 1000 10 000
W.B. Wrightand E.C. R odabaugh, Stress range and fatigue damage in nuclearpipeworks
321
Table 3 Stress ranges in descending magnitude.
Table 2 Stress ranges. Number combination
Stress range (Absolute values)
Number combination
i/
Stress range (Absolute values)
ij
1-2 1-3 1-4 2-3 2-4 3-4
45 000 20 000 5 000 25 000 50 000 25 000
2-4 1-2 2-3 3-4 1-3 1-4
50 000 45 000 25 000 25 000 20 000 5 000
PoD0
combinations are eliminated. The combination o f maximum possible stress ranges will form a conservative approximation o f the usage factor because this method gives " w o r s t " combinations o f stress cycles.
S n = C 1- - ~
4. Detailed computation of loads to give stress range
Sp = K1C 1~
DO
1
+ C2 - ~ ' M i +
Ec~IAT1 [ (10)
+ C3Eab [¢~ara - % r b l and peak stresses are computed by: PoD0
and fatigue evaluations
DO
1
+ K2C2~ 7 M i + 2(----i-~-~)K3EaIATI [
1
+~_uEalAT2l+K3C3EablaaTa--abTbl,
(11)
4.1. Computation of stress range using equations of where:
ASME Section 111(1971) Section NB 3650 o f Section III (1971) presents the equations for computing secondary and peak stresses. These equations are repeated here for completeness. Secondary stresses are computed by:
C 1, C 2, C 3 = secondary stress indices for the specific component under investigation, K1, K2,K 3 = peak stress indices for the specific component under investigation,
Table 4 Sequence and number of occurrences of stress ranges. Number combination
Stress
range
ij 2-4
50 000
1-2 2-3 3-4
45 000 25 000 25 000
1-3
20 000
Number of transients
i 1 000
Number of occurrences of stress range j 10 000
1000
1 000
1 000
0
1 000 10 000
10 000 9 000
1 000
1 000
Number of occurrences of transient left i 2-none
J 4-9000
0 9000
3-1000
4-none
1000
1-none
3-none
Comments
2 has formed in maximum number and is eliminated
4 has formed in maximum number and is eliminated 1 and 3 have formed in maximum number and are eliminated
W.B. Wrightand E. C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
322
P0 = range of operating pressure [psi], D O = outside diameter of pipe [in] t = nominal wall thickness of component [in], I = moment of inertia [in 4 ], M i = range of moment loading [in--lb], v = Poisson's ratio = 0.3, Ec~ = modulus of elasticity (E) times the mean coefficient of thermal expansion (c¢) [psi/°Fl, AT 1 = range of absolute value (without regard to sign) of the temperature difference assuming moment-generating equivalent linear temperature distribution [°F], AT 2 = range of absolute value (without regard to sign) for that portion of the nonlinear thermal gradient through the wall thickness [°F], Eab = the average modulus of elasticity of the two parts of the gross discontinuity [psi], c~a = mean coefficient of expansion of side a of a gross discontinuity [in/in ° F], Ta = range of average temperature on side a of a gross discontinuity [°F], % = mean coefficient of expansion on side b of a gross discontinuity [in/in °F], S m = allowable stress [psi], S n = primary plus secondary stress range [psi], Sp = peak stress range [psi]. Examination of these two equations, (10) and (11), show the variables PO, Mi, ATI, Ta and Tb are common to both equations. Their stress values vary only by a constant (K) value. Additionally, eq. (11 ) has a AT 2 term which is similar to AT 1 for determining stress ranges. Therefore, we may discuss stress range using eq. (I 1 ) and this discussion will be applicable to eq. (10). For the stress range computation eq. (11) is modified to read:
Sp
K1CIDo [p, - PI] + K2C2Do 2t [(gx* - mxl)2 + (M.~ - Myi)2 + (m~ - mz])2 ] 1/2.
K3E~ +2(-71-u) IT~I - Till
+ E~ ( 1 - v ) lT~ - T2il
+K3C3Eablaa(T*-Taj )
. . . . .
%(T~
Tbj)[.
(11")
The * denotes the pivotal value and tile subscript 1 denotes the values that are formed in combination with the pivotal value. For the fatigue evaluation that follows, the load values given in table 5 will be used in combination with constants that represent the product terms in eq. (11 *). The value of these product constants is taken to be:
K 1C1Do/(2t) = 20.773, K2C2Do/(21 ) = 3.071, K 3 k a / 2 ( 1 - u ) = 520.714, Ea/(1 v) = 347.143. 4.2. Fatigue damage One method of evaluating fatigue damage in a structure is by use of the theory of Linear Cumulative Damage (LCD), the method adopted by Section III. This theory was first proposed by Palmgren [3] and I.anger [4] and subsequently adapted to structures by Miner [5]. In other words, this theory states that a random series of stress cycles of variable amplitude contributes to the fatigue damage of a structure in a linear manner. Therefore, if a stress range, S 1 , occurs n 1 times but could occur N 1 times, then the contribution of S! to the fatigue damage is computed by taking the ratio n l / N 1 . The value o f N 1 comes from the Code's SIN cyclic stress fatigue curve. Similar stress-fatigue-evaluations may be made for each stress range (S) from 1 to k and the sum of the ratios of ni/N i produces the total LCD of the piping system at a point. This total summation is to be equal to or less than 1.0. Symbolically this may be written k
LCD = ~
ni/N i < 1.0,
i=1 where the summation is over the repeated index i. The value of the upper limit of the summation, k, represents the number of combinations of stress resulting from plant operating transients. 4.3. Structural load history to stress evaluation We are now in a position to begin the evaluation of
323
W.B. Wright and E.C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
Table 5 Components of loading at a selected point in a piping system resulting from the operation cycles. Sequence number 1
2 3 4 5 6 7 8 9 10 11 12
Operational cycle
Number of cycles
(1)
100
(2)
50
(3)
60
(4)
38
(5)
13
Mx
M
Mz
P
AT 1
AT 2
E a ( T a - T b)
[in-lb]
[inY-lb]
[in- lb]
[psi]
[°F]
[°F]
[psi]
0
0
0
0
0
0
0
500 1 000 -1 000 0 8 000 8 500 8 900 7 500 500 10 000 8 500
300 500 -500 0 6000 6300 6400 6000 6300 8000 6300
-400 750 -750 0 -1700 -1800 -1890 -1000 -1800 -2000 -1800
2000 0 0 0 1000 1900 2000 1700 1900 2300 1900
0 0 0 0 32 60 63 -60 25 90 -90
0 0 0 0 18 38 41 -38 14 120 -120
0 0 0 0 0 0 0 0 0 0 0
secondary and peak loads, the resulting stress ranges and the fatigue damage in a manner compatible with Section III (1971) definitions. The series o f loads resuiting from the various operational cycles are computed and put in tabular form (see table 5) for a selected point in the piping system. The selected point for this set o f computed loads is not at a gross discontinuity, therefore, the value in the right-hand column of table 5 is zero. The first column in table 5 is numbered from 1 to 12 for convenience in identifying the loads during selected times in the various operational cycles. The operational cycles (1), (2), (3), and (5) in table 5 each have a maximum and minimum value listed that represents the extremum loadings o f each operational cycle. The operational cycle (4) is a notable exception to the above. This operational cycle is complex enough in loading that it is not possible to pick a clear cut set of extreme loading values, hence this complex operational cycle is judged to require four separate loads in time to be representative o f the operating cycle. These will be formed in combinations with other loads described by the sequence number. Their sequential combination is arbitrary, but they are limited in number to 38 cycles for any sequence number (7) through (10) and a total of 76 cycles of stress for any combination. The loads associated with the sequence numbers are formed in combination 1 - 2 , I - 3 , 1 - 4 . . . . . 1 - 1 2 using eq. (11 *) to give stress values for each combination. In turn the sequence numbers 2 - 3 , 2 - 4 , 2 - 5 ,
. . . . 2 - 1 2 are formed in combination, then 3 - 4 , . . . , 3 - 1 2 , etc. until the total of 1 1+10+9+...+1 =66 stress combinations has been formed. Due to the large amount of computation required, the calculations of stress combinations were made by computer with the results noted in table 6. The full set of information is presented in table 6 to establish a maximum stress range for a transient and each range is the maximum value denoted in a particular computer pass. In this example, 11 and 12 happen to be the maximum and minimum of the stress range of all combinations and they are both generated in the same transient. The next maxima and minima occur in combination of the second and the fourth transient, sequence numbers 4 and 8. This operation then continues through a total of eight computer passes, as noted in table 6. An interesting consequence of making the judgment that more than two (actually an infinite number of loading occurs) significant points of loading occur during a plant operational cycle is that intermediate load points may contribute to cyclic fatigue. For example, we have four load sets that have been calculated for transient number 4. However, this transient will occur 38 times. Therefore, in this transient we can have 38 minimum and 38 maximum stress values. Which 76 of the 152 (4 × 38) cycles that we have listed in this transient do we use? The answer to this question must come from the detailed analysis by
324
W.B. Wright and E.C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks Table 6 Results of computer search for maximum stress range combinations.
Number* combination
Stress range
Computer pass No. 6
Cycles
1-2 2-3 3-4
176 132 173 179 176 115 133 138 155 131 192
100 6OO 300 200 100 300 000 900 700 000 300 < < <
1-2 2-3
1-
8
122 79 119 125 122 72 97 102 88
700 250 900 8O0 < < < 7OO 060 060 900 120
1-2
116 78 114 119 116 72 97 88
900 450 100 900 < < < 900 060 O6O 120
116 78 114 116 72 97 88
900 < < < 450 100 900 060 060 120
91 56 88 91
450 < < < 270 660 450
36"~
forming all possible c o m b i n a t i o n s o f loads. At this point it suffices to say that, o f the four conditions, we must exercise care n o t to use any one value o f stress over 38 times and no c o m b i n a t i o n o f stresses f r o m the total loads o f this transient over 76 times. 38"~ 4.4. Fatigue evaluation
12",,
Table 7 presents the final results o f m a x i m u m stress ranges shown in table 6 plus the individual usage factors. These usage factors were c o m p u t e d assuming stainless steel material and the fatigue curve for this metal f r o m ASME Section III ( 1 9 7 1 ) w a s applied. As an interesting sidelight n o t e that a s u m m a t i o n of the n u m b e r o f cycles in table 5 is 261. When ranges are f o r m e d in table 7 and the n u m b e r o f cycles summed the value is 249. This clearly indicates the variance that may exist b e t w e e n operational cycles and stress cycles. Table 7 LCD computation from stress ranges.
26'~
Computer pass No. 5 1-5 2-5 3-5 4-5
43 720 < < <
13,
Computer pass No. 4 1-6 2-7 3-6 4-6 5-7 6-7 7-8
14~
*Reordering the number combinations from one computer pass to the next to reflect updating means no correspondence exists in the numbering between computer passes.
Computer pass No. 3 1-7 2-8 3-7 4-7 5-7 6-8 7-8 8-9
43 720 < < < 43 720 Computer pass No. 8
Computer pass No. 2 2-,8 3-8 4-8 5-8 6-9 7-9 8-9 9-10
50~
Computer pass No. 7
Computer pass No. 1 1-11 2-11 3-11 4-11 5-11 6-12 7-12 8-12 9-11 10-12 11-12
43 720 45 450 < < < 4 130
60~
Number combination
Stress range
Design cycles Ni
Operating cycles ni
LCD value
11-12 4- 8 4- 7 1- 6 1- 5 2- 3 1- 2 1- 2
192 125 119 116 91 45 43 43
188 648 752 815 1 816 29 250 35 280 35 280
13 38 12 26 60 50 14 36
0.0691 0.0586 0.0160 0.0139 0.0330 0.0017 0.0004 0.0010
300 800 900 900 450 450 720 720
I¢.B. Wright and E.C Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
5. Summary Two examples of transient loading at a point in a piping system have been presented. The first was a simple example to illustrate the concept of transients forming in combination to produce stress ranges.The second example dealt with more complex transients and included the use of eq. (11 *). This second example then used the results of eq. (11") to compute the linear cumulative damage (LCD). The method of computing LCD uses the hypothesis of linear damage causing fatigue failure for a random series o f stress cycles.
Acknowledgement The authors thank Mr. Tom Williams o f Westing-
325
house Nuclear Energy Systems for generating the computer code necessary to make the detailed logical computation presented here.
References [1] Nuclear Power Plant Components, ASME Boiler and Pressure Vessel Code, Section IlI, 1971, published by the American Society of Meohanical Engineers, N.Y., N.Y. [2] Nuclear Power Piping, ANSI B31.7, 1969, published by the American Society of Mechanical Engineers, N.Y., N.Y. [3] A. Palmgren, "Die Lebensdauer yon Kugellagern, Vet. deut. Ingenieure 14 (1924) 339. [4] B.F. Langer, Fatigue Failure from Stress Cycles of Varying Amplitude, Trans. ASME 59 (1937) A150. [5] M.A. Miner, Cumulative Damage in Fatigue, J.A.M. 12 (September, 1945) A159.
A METHOD OF COMPUTING STRESS RANGE AND F A T I G U E D A M A G E IN A N U C L E A R P I P I N G S Y S T E M Wallace B. WRIGHT Westinghouse Electric Corporation. Nu clear Energy Systems, PWR Systems Division. Pittsburgh, Pennsylvania 15230, USA and Everett C. RODABAUGH Battelle Memorial Institute, Applied Mechanics Section, Columbus, Ohio 43201, USA Received 10 January 1972
The ASME Boiler and Pressure Vessel Code, Section 111 (1971) employs stress range and fatigue evaluation as part of the total stress evaluation of piping systems. One method of computing this stress range and the resultant fatigue damage using the basic equations of the Code is presented here. This method is then illustrated by examples and discussed in detail.
1. Introduction The ASME Boiler and Pressure Vessel Code, Section III (1971) [1 ], is written as a guide to cover the following subjects: design, analysis, fabrication, erection, arid inspection of nuclear components. The Code treats each o f these subjects in separate sections. The piping analysis section of the Code is written to be used for the evaluation o f primary, secondary, and peak stresses at selected points in the piping system. The secondary and peak (fatigue) stresses are based on a stress range concept and a comprehensive discussion o f this subject may be found in the foreword of ref. [2]. The purpose of this paper is twofold; to present a method for computing stress range and, in turn, using the stress range to compute fatigue damage. To accomplish the stated purpose, this paper is divided into several sections. The first section discusses power plant operating transients and their effect on stresses in a plant's piping system. The cyclic stresses in the piping system are then combined to give a range of stress as illustrated by a simple example of
two stress transients. The technique of combining stress ranges and computing fatigue damage in a manner compatible with Article NB3600 of the Code is then illustrated by formula and a detailed computation illustrating this formula.
2. Relation of plant operation to transient stress 2.1. Plant operations relating to piping loading Fig. 1 is an illustration of the operating cycles experienced by a plant during its operational life. The numbers one through six on this figure represent operating cycles that are hypothesized to be experienced by the plant in a random manner during its operational life. Number 1 in fig. 1 represents a hydrotest operational cycle on the plant piping that will occur 100 times during the total life o f plant operation. Number 2 represents a hypothetical earthquake that will occur five times with each occurrence having ten significant vibrational cycles which will occur during plant shutdown. The sloping line that rises to number 3 repre-
W.B. Wright and E.C. R odabaugh, Stress range and fatigue damage in nuclear pipeworks
First, the values of temperature and pressure resulting from the various plant operating cycles are presented in the Design Specification (which is the basis for the analysis) as a series of time history plots. An example of one of these plots is illustrated by fig. 2 which represents operational cycle number 4 of fig. 1. Additionally, the seismic disturbance that may occur at the plant site is defined in the Design Specification as a set of response spectra to be used in computing dynamic loads.
@
0.0
I,
I,
POSTULATEDCONDITIONS Fig. 1. Plant hypothetical operating history.
sents plant startup and the leveling off at number 3 indicates steady state operation. There are 60 of these startups which are one-half of the total operational start up-shutdown cycle. The surge up and down and back to normal operation at number 4 represents a turbine trip that occurs 38 times. The sharp slope to number 5 and return to normal operation represents an overpressure of the system from which the plant returns to normal operation and there are 13 such operational cycles in the plant's history. The slope from number 6 to plant shutdown occurs sixty times and is the other half of the start up-shut down cycle. These operational cycles are assumed to represent the total plant hypothetical operating history. Associated with this operating history will be a stress history of the plant. For illustrative purposes we will examine the stress history at one point in one of the plant's main piping systems.
2.2. Stress transient history Prior to the evaluation of metal fatigue from stress caused by the plant operational cycles it is instructive to examine three possible types of plant stress history, each resulting from a separate type of operational cycle. First, examining fig. 2 note that it illustrates an appropriate sine wave response of one cycle duration. Superimposed on the plant transient is the stress time history at a point in the piping system that is shown in fig, 3. Alternately, looking at an operational plant transient cycle (that includes both temperature and pressure) with the superimposed stress transient as shown in fig. 4, note that the plant operational cycle has the characteristic sine waveresponse and consequently, there is a stress response that takes the approximate form of the transient as before but now the stress transient contains several stress cycles. It is impractical to compute the total stress time history of each instant in time, therefore, a judgment is made as to maximum and minimum loads within the
300
100
I 100
319
LNORMAL OPERATION
t.~
300
TiME Fig. 2. Typical temperature-pressure response to an operational cycle.
320
W.B. Wright and b~:C. R odabaugh, Stress range and fatigue damage in n.clear pipeworks
/f"••//--T
RANSIENT STRESS /
ATUR[
-PRESSOR
S~RESS [RANSIFN[
/
~ PLANTTRANSIENJ
TIME I.
Fig. 3. Stress transient superimposed on plant operational cycle.
operational cycle. In the first instance shown by fig. 3 there is one maximum and one minimum value. In the second instanc e shown by fig. 4 there are several significant stress cycles, two maximum and two minimum. A third possibility is that only one maximum and one minimum value may exist, but they are embedded in the transient and are not readily discernable. This third type of transient will be included and illustrated by the detailed example on fatigue evaluation.
3. Example of transient stresses converted to stress range 3.1. Combining cyclic" stress To illustrate the basic concept of combining stresses to establish the range of stresses and determining the number of stress cycles for fatigue evaluation a first example involving two operational cycles will be used. Assume for the discussion in this section that the maximum and minimum stress associated with the two operational cycles is known. The stress values and number of cycles are presented by table 1 with the maximum and minimum stress values assigned a number for bookkeeping purposes. To establish the range of stress between the several values given in table I a computation of stress differences is made by subtracting combinations of stress as shown by table 2. This table introduces the notation i and ] in the column "Number combination". The individual i and] values permit a record to be kept during computation as the values of stress are formed in all possible combinations. Additionally, the i and j
Fig. 4. Stress transient superimposed on plant's multiple cycle.operating transient.
terms will be used shortly in table 4 to identify stress values and the number of occurrences of various stress cycles as range-transient combinations are formed. To best operate with these stress ranges they are reordered in descending value and are presented by table 3. A new table, table 4, is now constructed that associates the sequence number with its related number of cycles by use of the letters i and j. The number of occurrences of each stress range is computed by determining the maximum number of times each set i, j can occur in combination. From each of these maximum numbers of combinations, progressing from top to bottom, one number (i orj) is eliminated and the other is adjusted downward to represent the number of times it can now form in new combination with other i,] values. There is an orderly progression of reducing the number of cycles until they have each formed in combination with others until all significant
Table 1 Maximum and minimum cyclic stress values. Operational cycle
Number
Stress (psi)
1
1 2 3 4
15 60 35 10
2
000 000 000 000
Number of cycles of occurrence 1000 10 000
W.B. Wrightand E.C. R odabaugh, Stress range and fatigue damage in nuclearpipeworks
321
Table 3 Stress ranges in descending magnitude.
Table 2 Stress ranges. Number combination
Stress range (Absolute values)
Number combination
i/
Stress range (Absolute values)
ij
1-2 1-3 1-4 2-3 2-4 3-4
45 000 20 000 5 000 25 000 50 000 25 000
2-4 1-2 2-3 3-4 1-3 1-4
50 000 45 000 25 000 25 000 20 000 5 000
PoD0
combinations are eliminated. The combination o f maximum possible stress ranges will form a conservative approximation o f the usage factor because this method gives " w o r s t " combinations o f stress cycles.
S n = C 1- - ~
4. Detailed computation of loads to give stress range
Sp = K1C 1~
DO
1
+ C2 - ~ ' M i +
Ec~IAT1 [ (10)
+ C3Eab [¢~ara - % r b l and peak stresses are computed by: PoD0
and fatigue evaluations
DO
1
+ K2C2~ 7 M i + 2(----i-~-~)K3EaIATI [
1
+~_uEalAT2l+K3C3EablaaTa--abTbl,
(11)
4.1. Computation of stress range using equations of where:
ASME Section 111(1971) Section NB 3650 o f Section III (1971) presents the equations for computing secondary and peak stresses. These equations are repeated here for completeness. Secondary stresses are computed by:
C 1, C 2, C 3 = secondary stress indices for the specific component under investigation, K1, K2,K 3 = peak stress indices for the specific component under investigation,
Table 4 Sequence and number of occurrences of stress ranges. Number combination
Stress
range
ij 2-4
50 000
1-2 2-3 3-4
45 000 25 000 25 000
1-3
20 000
Number of transients
i 1 000
Number of occurrences of stress range j 10 000
1000
1 000
1 000
0
1 000 10 000
10 000 9 000
1 000
1 000
Number of occurrences of transient left i 2-none
J 4-9000
0 9000
3-1000
4-none
1000
1-none
3-none
Comments
2 has formed in maximum number and is eliminated
4 has formed in maximum number and is eliminated 1 and 3 have formed in maximum number and are eliminated
W.B. Wrightand E. C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
322
P0 = range of operating pressure [psi], D O = outside diameter of pipe [in] t = nominal wall thickness of component [in], I = moment of inertia [in 4 ], M i = range of moment loading [in--lb], v = Poisson's ratio = 0.3, Ec~ = modulus of elasticity (E) times the mean coefficient of thermal expansion (c¢) [psi/°Fl, AT 1 = range of absolute value (without regard to sign) of the temperature difference assuming moment-generating equivalent linear temperature distribution [°F], AT 2 = range of absolute value (without regard to sign) for that portion of the nonlinear thermal gradient through the wall thickness [°F], Eab = the average modulus of elasticity of the two parts of the gross discontinuity [psi], c~a = mean coefficient of expansion of side a of a gross discontinuity [in/in ° F], Ta = range of average temperature on side a of a gross discontinuity [°F], % = mean coefficient of expansion on side b of a gross discontinuity [in/in °F], S m = allowable stress [psi], S n = primary plus secondary stress range [psi], Sp = peak stress range [psi]. Examination of these two equations, (10) and (11), show the variables PO, Mi, ATI, Ta and Tb are common to both equations. Their stress values vary only by a constant (K) value. Additionally, eq. (11 ) has a AT 2 term which is similar to AT 1 for determining stress ranges. Therefore, we may discuss stress range using eq. (I 1 ) and this discussion will be applicable to eq. (10). For the stress range computation eq. (11) is modified to read:
Sp
K1CIDo [p, - PI] + K2C2Do 2t [(gx* - mxl)2 + (M.~ - Myi)2 + (m~ - mz])2 ] 1/2.
K3E~ +2(-71-u) IT~I - Till
+ E~ ( 1 - v ) lT~ - T2il
+K3C3Eablaa(T*-Taj )
. . . . .
%(T~
Tbj)[.
(11")
The * denotes the pivotal value and tile subscript 1 denotes the values that are formed in combination with the pivotal value. For the fatigue evaluation that follows, the load values given in table 5 will be used in combination with constants that represent the product terms in eq. (11 *). The value of these product constants is taken to be:
K 1C1Do/(2t) = 20.773, K2C2Do/(21 ) = 3.071, K 3 k a / 2 ( 1 - u ) = 520.714, Ea/(1 v) = 347.143. 4.2. Fatigue damage One method of evaluating fatigue damage in a structure is by use of the theory of Linear Cumulative Damage (LCD), the method adopted by Section III. This theory was first proposed by Palmgren [3] and I.anger [4] and subsequently adapted to structures by Miner [5]. In other words, this theory states that a random series of stress cycles of variable amplitude contributes to the fatigue damage of a structure in a linear manner. Therefore, if a stress range, S 1 , occurs n 1 times but could occur N 1 times, then the contribution of S! to the fatigue damage is computed by taking the ratio n l / N 1 . The value o f N 1 comes from the Code's SIN cyclic stress fatigue curve. Similar stress-fatigue-evaluations may be made for each stress range (S) from 1 to k and the sum of the ratios of ni/N i produces the total LCD of the piping system at a point. This total summation is to be equal to or less than 1.0. Symbolically this may be written k
LCD = ~
ni/N i < 1.0,
i=1 where the summation is over the repeated index i. The value of the upper limit of the summation, k, represents the number of combinations of stress resulting from plant operating transients. 4.3. Structural load history to stress evaluation We are now in a position to begin the evaluation of
323
W.B. Wright and E.C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
Table 5 Components of loading at a selected point in a piping system resulting from the operation cycles. Sequence number 1
2 3 4 5 6 7 8 9 10 11 12
Operational cycle
Number of cycles
(1)
100
(2)
50
(3)
60
(4)
38
(5)
13
Mx
M
Mz
P
AT 1
AT 2
E a ( T a - T b)
[in-lb]
[inY-lb]
[in- lb]
[psi]
[°F]
[°F]
[psi]
0
0
0
0
0
0
0
500 1 000 -1 000 0 8 000 8 500 8 900 7 500 500 10 000 8 500
300 500 -500 0 6000 6300 6400 6000 6300 8000 6300
-400 750 -750 0 -1700 -1800 -1890 -1000 -1800 -2000 -1800
2000 0 0 0 1000 1900 2000 1700 1900 2300 1900
0 0 0 0 32 60 63 -60 25 90 -90
0 0 0 0 18 38 41 -38 14 120 -120
0 0 0 0 0 0 0 0 0 0 0
secondary and peak loads, the resulting stress ranges and the fatigue damage in a manner compatible with Section III (1971) definitions. The series o f loads resuiting from the various operational cycles are computed and put in tabular form (see table 5) for a selected point in the piping system. The selected point for this set o f computed loads is not at a gross discontinuity, therefore, the value in the right-hand column of table 5 is zero. The first column in table 5 is numbered from 1 to 12 for convenience in identifying the loads during selected times in the various operational cycles. The operational cycles (1), (2), (3), and (5) in table 5 each have a maximum and minimum value listed that represents the extremum loadings o f each operational cycle. The operational cycle (4) is a notable exception to the above. This operational cycle is complex enough in loading that it is not possible to pick a clear cut set of extreme loading values, hence this complex operational cycle is judged to require four separate loads in time to be representative o f the operating cycle. These will be formed in combinations with other loads described by the sequence number. Their sequential combination is arbitrary, but they are limited in number to 38 cycles for any sequence number (7) through (10) and a total of 76 cycles of stress for any combination. The loads associated with the sequence numbers are formed in combination 1 - 2 , I - 3 , 1 - 4 . . . . . 1 - 1 2 using eq. (11 *) to give stress values for each combination. In turn the sequence numbers 2 - 3 , 2 - 4 , 2 - 5 ,
. . . . 2 - 1 2 are formed in combination, then 3 - 4 , . . . , 3 - 1 2 , etc. until the total of 1 1+10+9+...+1 =66 stress combinations has been formed. Due to the large amount of computation required, the calculations of stress combinations were made by computer with the results noted in table 6. The full set of information is presented in table 6 to establish a maximum stress range for a transient and each range is the maximum value denoted in a particular computer pass. In this example, 11 and 12 happen to be the maximum and minimum of the stress range of all combinations and they are both generated in the same transient. The next maxima and minima occur in combination of the second and the fourth transient, sequence numbers 4 and 8. This operation then continues through a total of eight computer passes, as noted in table 6. An interesting consequence of making the judgment that more than two (actually an infinite number of loading occurs) significant points of loading occur during a plant operational cycle is that intermediate load points may contribute to cyclic fatigue. For example, we have four load sets that have been calculated for transient number 4. However, this transient will occur 38 times. Therefore, in this transient we can have 38 minimum and 38 maximum stress values. Which 76 of the 152 (4 × 38) cycles that we have listed in this transient do we use? The answer to this question must come from the detailed analysis by
324
W.B. Wright and E.C. Rodabaugh, Stress range and fatigue damage in nuclear pipeworks Table 6 Results of computer search for maximum stress range combinations.
Number* combination
Stress range
Computer pass No. 6
Cycles
1-2 2-3 3-4
176 132 173 179 176 115 133 138 155 131 192
100 6OO 300 200 100 300 000 900 700 000 300 < < <
1-2 2-3
1-
8
122 79 119 125 122 72 97 102 88
700 250 900 8O0 < < < 7OO 060 060 900 120
1-2
116 78 114 119 116 72 97 88
900 450 100 900 < < < 900 060 O6O 120
116 78 114 116 72 97 88
900 < < < 450 100 900 060 060 120
91 56 88 91
450 < < < 270 660 450
36"~
forming all possible c o m b i n a t i o n s o f loads. At this point it suffices to say that, o f the four conditions, we must exercise care n o t to use any one value o f stress over 38 times and no c o m b i n a t i o n o f stresses f r o m the total loads o f this transient over 76 times. 38"~ 4.4. Fatigue evaluation
12",,
Table 7 presents the final results o f m a x i m u m stress ranges shown in table 6 plus the individual usage factors. These usage factors were c o m p u t e d assuming stainless steel material and the fatigue curve for this metal f r o m ASME Section III ( 1 9 7 1 ) w a s applied. As an interesting sidelight n o t e that a s u m m a t i o n of the n u m b e r o f cycles in table 5 is 261. When ranges are f o r m e d in table 7 and the n u m b e r o f cycles summed the value is 249. This clearly indicates the variance that may exist b e t w e e n operational cycles and stress cycles. Table 7 LCD computation from stress ranges.
26'~
Computer pass No. 5 1-5 2-5 3-5 4-5
43 720 < < <
13,
Computer pass No. 4 1-6 2-7 3-6 4-6 5-7 6-7 7-8
14~
*Reordering the number combinations from one computer pass to the next to reflect updating means no correspondence exists in the numbering between computer passes.
Computer pass No. 3 1-7 2-8 3-7 4-7 5-7 6-8 7-8 8-9
43 720 < < < 43 720 Computer pass No. 8
Computer pass No. 2 2-,8 3-8 4-8 5-8 6-9 7-9 8-9 9-10
50~
Computer pass No. 7
Computer pass No. 1 1-11 2-11 3-11 4-11 5-11 6-12 7-12 8-12 9-11 10-12 11-12
43 720 45 450 < < < 4 130
60~
Number combination
Stress range
Design cycles Ni
Operating cycles ni
LCD value
11-12 4- 8 4- 7 1- 6 1- 5 2- 3 1- 2 1- 2
192 125 119 116 91 45 43 43
188 648 752 815 1 816 29 250 35 280 35 280
13 38 12 26 60 50 14 36
0.0691 0.0586 0.0160 0.0139 0.0330 0.0017 0.0004 0.0010
300 800 900 900 450 450 720 720
I¢.B. Wright and E.C Rodabaugh, Stress range and fatigue damage in nuclear pipeworks
5. Summary Two examples of transient loading at a point in a piping system have been presented. The first was a simple example to illustrate the concept of transients forming in combination to produce stress ranges.The second example dealt with more complex transients and included the use of eq. (11 *). This second example then used the results of eq. (11") to compute the linear cumulative damage (LCD). The method of computing LCD uses the hypothesis of linear damage causing fatigue failure for a random series o f stress cycles.
Acknowledgement The authors thank Mr. Tom Williams o f Westing-
325
house Nuclear Energy Systems for generating the computer code necessary to make the detailed logical computation presented here.
References [1] Nuclear Power Plant Components, ASME Boiler and Pressure Vessel Code, Section IlI, 1971, published by the American Society of Meohanical Engineers, N.Y., N.Y. [2] Nuclear Power Piping, ANSI B31.7, 1969, published by the American Society of Mechanical Engineers, N.Y., N.Y. [3] A. Palmgren, "Die Lebensdauer yon Kugellagern, Vet. deut. Ingenieure 14 (1924) 339. [4] B.F. Langer, Fatigue Failure from Stress Cycles of Varying Amplitude, Trans. ASME 59 (1937) A150. [5] M.A. Miner, Cumulative Damage in Fatigue, J.A.M. 12 (September, 1945) A159.