What in the World is your Sustained Stress Anyway? Stating the Problem: Since the beginning of time, up until relatively recently, all pipe stress calculations have been done assuming a linear world. The implication of this is, if no boundary conditions change during the operation of the pipe, one assumed that all applied loads act against the same model, so therefore they can be analyzed against the same model, in isolation of each other, and either evaluated separately or superimposed upon each other, as necessary (this is the same as the “Cold Sustained” analysis). Therefore there were no problems, nor arguments among pipe stress analysts regarding how sustained stress calculations need be done (although this is not really the case). However, a few years back, somebody noticed that the real world isn’t linear (or what really happened is that pipe stress programs began to give non-linear capabilities, so somebody started to notice that piping models weren’t always linear). For somewhat obvious reasons, this caused the greatest concern when supports that were previously considered to be weight-bearing lifted off in the operating case, but in reality, this is an equal concern for all non-linear restraints (gaps, friction, one-way restraints, bi-linears, etc.) which change state during operation. The question is – how does one calculate sustained stress during the various operating states of the piping system, since the weight loads may act against a different set of boundary conditions during different operating conditions? Traditional Solution vs. COADE’s Solution: One traditional solution has been to remove from the model any restraints designated as “weight supports” that lift off during any operating case and then reanalyze the Cold Sustained case. It is easy to misapply this method, as typically only +Y supports are removed, but almost never horizontal gapped restraints (that may open during the operating case but, when closed, may have a significant effect on the distribution of the weight loads) or other nonlinear restraints. COADE does not believe that this is the correct way to analyze the weight loads. Since 1984, we have provided the Cold Sustained case (W+P1) as the basics for the sustained calculation. Our arguments were (1) as seen in the first paragraph, this is how the calculation has traditionally been done, and (2) this is “sustained” stress, isn’t it – there should only be one – any redistribution of sustained stresses due to operating displacements are, as the CODETI code agrees, expansion effects. As boundary condition changes during operation became more of a concern, we decided to look into this issue more carefully. About 7-8 years ago, we came upon a solution that we believe is correct. And what’s more, it’s defensible – it doesn’t suffer from a lot of the problems that the “Traditional Solution” (removing restraints and rerunning the Cold Sustained) will be shown to suffer from. This solution was described as building a Hot Sustained load case: L1 L2 L3 L4 L5
W+T1+P1+D1 (OPE) T1 (EXP) W+P1 (SUS) L1-L3 (EXP) L1-L2 (SUS)
In the above set of load cases, L3 represents the “Cold Sustained”, the greater of L2 or L4 represents the Expansion case (the stress range between the two extremes of operating and installed, and L5 represents the “Hot Sustained”. We expect that load cases L3 and L5 most likely envelope any of the sustained stress distributions that may occur during the boundary condition changes due to changes in the operating state. Theory: Our Solution is based upon two statements that we doubt anybody can dispute: 1) The distribution of forces, moments, and stresses in a system is a direct reflection of the displaced shape of the system under that load. 2) The sum of the Sustained response and the Expansion response (in terms of forces and moments and most importantly displaced shape) at any given time must be exactly equal to the Operating response. So once we know the displaced shape of the system in the Operating state, all we have to do is subtract the displaced shape of the system due to the Expansion loads from that Operating shape, and voila we will have the Sustained displaced shape, from which we can calculate the Sustained forces, moments, and stresses. The only problem with this is that there are potentially an infinite number of combinations of Expansion and Sustained displaced shapes that might make up this Operating displaced shape. The question then, is which one is correct? It is our belief that the load cases shown above most likely match the Expansion/Sustained distribution exactly (first in the installed case and then in the Operating case) – if not, they certainly they envelope the actual distribution. The L3 (SUS) and L4 (EXP) results sum to the L1 (OPE) results – these results represent the scenario where the pipe weight is applied first, and then the pipe expands to the operating position from that condition. The L5 (SUS) and L2 (EXP) results also sum to the L1 (OPE) results – but these results represent the possible scenario that the pipe expands thermally from its neutral position first, and then the weight is applied, causing it to sag back to its operating position. Now most likely, scenario 1 above (weight applied first, then thermal expansion) is the correct order of loading, but that isn’t what is really important. What we are trying to decide is what is the implication of the thermal expansion growing from a fully weight loaded system, and then again, what is the impact of the system sagging under weight from its fully expanded position. Consider an example: Two EMTs are carrying a 250-pound patient in a stretcher, holding the ends of the stretcher roughly three feet above the ground – and it is very likely that the stretcher is sagging in the middle. What is the displaced shape of the stretcher due to weight vs. displacement (lifting it up). We can model this in these two manners: 1) Cold Sustained – the stretcher was lying on the ground when the patient got onto it. The displaced shape at “installation” (continuously supported, so no real weight stresses) is calculated. Then imposed displacements are applied (lifting the ends of the stretcher 3 feet) are applied, taking us to the operating state. 2) Hot Sustained – the stretcher had imposed displacements by lifting its ends 3 feet high (resulting in a nearly horizontal shape) and then the patient gets on it, causing it to sag from
that original displaced position. In this case the Sustained response is the difference from the original displaced position and the final operating (sagged from the original) position. Now, in order to estimate the distribution of the Expansion vs. Sustained stresses, do I have to know exactly when the patient got on the stretcher? No, because the exact timing is irrelevant – what is important is the effect. In other words, I would be willing to make two bets here: (1) the patient got on the stretcher BEFORE the EMTs lifted it (i.e., case #1) and (2) the actual distribution of the stresses is best represented by case #2! That is the same principle behind COADE’s Hot Sustained solution. (Oh, and now, let’s complicate things a little further – let’s say that somebody, after seeing that the stretcher is sagging 6 inches under the weight of the patient, shoves a stool underneath that reduces the sag to only 4 inches. Does that make all of the weight stress go away simply because we no longer have a lift-off problem? No, because there still is some sag under weight from the displaced position.) Example: In the provided example, I have modeled a system with 3 +Y supports (at nodes 45, 70, and 95) and 10 different operating temperatures (ranging from 70 to 1100 degrees F). I have also provided displacements at the +Y restraints which permit me to remove them from the Sustained load case to simulate the “Traditional Method” described above (Displacement Vector D1 removes the supports at 45 and 95, Displacement Vector D2 removes the support at node 70). I have run a number of load cases here, to represent the Sustained response as the system temperature increases from 70 degrees through 1100 degrees, over the course of which first two and then the third support lifts off. The most interesting temperatures are around 196.7 degrees and around 1091.3 degrees, where supports 45-95 and support 70 lift off, respectively (T3 and T4 are a hair to either side of 196.7 degrees, while T7 and T8 are a hair to either side of 1091.3 degrees). Using COADE’s Hot Sustained concept, I can show that the Sustained plus Expansion results add up to the Operating results in all cases (as we all agree they should, right?). However, if we use the Traditional Method described above, we would assume that the Sustained response is the same as the Cold Sustained (L10) up to 196.7 degrees (L1 through L3), is the Cold Sustained with supports 45-95 removed (L11) between 196.7 and 1091.3 degrees (L4-L7), and the Cold Sustained with supports 45-95 and 70 removed (L12) above 1091.3 degrees (L8 and L9). I can show that these results do not add up, regardless of which Expansion case one combines these results with: 110 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
150 F
L31 L13 L10 L13 L22
L1
Node 35 -0.2436 0.2954 -0.0367 0.2954 0.0885 0.0518 0.2587 0.0518 0.0518
Node 40 -0.2758 0.2954 -0.0167 0.2954 0.0363 0.0196 0.2787 0.0196 0.0196
Node 45 -0.2954 0.2954 0 0.2954 0 0 0.2954 0 0
Node 50 -0.3118 0.2954 0.0002 0.2954 -0.0166 -0.0164 0.2956 -0.0164 -0.0164
Node 55 -0.3188 0.2954 -0.0045 0.2954 -0.0189 -0.0234 0.2909 -0.0234 -0.0234
Node 60 -0.3148 0.2954 -0.0065 0.2954 -0.0128 -0.0194 0.2889 -0.0193 -0.0193
Node 65 -0.3036 0.2954 -0.0037 0.2954 -0.0045 -0.0082 0.2917 -0.0082 -0.0082
Node 70 -0.2954 0.2954 0 0.2954 0 0 0.2954 0 0
Node 35
Node 40
Node 45
Node 50
Node 55
Node 60
Node 65
Node 70
Match ?? Match
COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
196.68 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
196.7 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
500 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
800 F COADE Sus COADE Exp
L32 L14 L10 L14 L23
-0.4573 0.6005 -0.0367 0.6005 0.1798 0.1432 0.5638 0.1431 0.1431
-0.5433 0.6005 -0.0167 0.6005 0.0738 0.0572 0.5838 0.0571 0.0571
-0.6005 0.6005 0 0.6005 0 0 0.6005 0 0
-0.634 0.6005 0.0002 0.6005 -0.0337 -0.0335 0.6007 -0.0335 -0.0335
-0.6434 0.6005 -0.0045 0.6005 -0.0383 -0.0429 0.596 -0.0428 -0.0429
-0.633 0.6005 -0.0065 0.6005 -0.0261 -0.0325 0.594 -0.0326 -0.0326
-0.6134 0.6005 -0.0037 0.6005 -0.0092 -0.0129 0.5968 -0.0129 -0.0129
-0.6005 0.6005 0 0.6005 0 0 0.6005 0 0
Node 35 -0.7152 0.9686 -0.0367 0.9686 0.29 0.2534 0.9319 0.2533 0.2534
Node 40 -0.8662 0.9686 -0.0167 0.9686 0.1191 0.1024 0.9519 0.1024 0.1024
Node 45 -0.9686 0.9686 0 0.9686 0 0 0.9686 0 0
Node 50 -1.0228 0.9686 0.0002 0.9686 -0.0544 -0.0542 0.9688 -0.0542 -0.0542
Node 55 -1.035 0.9686 -0.0045 0.9686 -0.0618 -0.0664 0.9641 -0.0663 -0.0664
Node 60 -1.0172 0.9686 -0.0065 0.9686 -0.0421 -0.0486 0.9621 -0.0486 -0.0486
Node 65 -0.9871 0.9686 -0.0037 0.9686 -0.0149 -0.0185 0.9649 -0.0186 -0.0186
Node 70 -0.9686 0.9686 0 0.9686 0 0 0.9686 0 0
Node 35 -0.7153 0.9688 -0.3105 0.9688 0.2901 0.2535 0.6583 -0.0204 0.2534
Node 40 -0.8663 0.9688 -0.3332 0.9688 0.1191 0.1025 0.6356 -0.2141 0.1024
Node 45 -0.9687 0.9688 -0.3174 0.9688 0 1E-04 0.6514 -0.3174 0
Node 50 -1.0229 0.9688 -0.2669 0.9688 -0.0544 -0.0541 0.7019 -0.3213 -0.0542
Node 55 -1.0351 0.9688 -0.1916 0.9688 -0.0618 -0.0663 0.7772 -0.2534 -0.0664
Node 60 -1.0173 0.9688 -0.1068 0.9688 -0.0421 -0.0485 0.862 -0.1489 -0.0486
Node 65 -0.9873 0.9688 -0.0339 0.9688 -0.0149 -0.0185 0.9349 -0.0488 -0.0186
Node 70 -0.9688 0.9688 0 0.9688 0 0 0.9688 0 0
L5
Node 35 -1.8243 3.6223 -0.3105 3.6223 1.8347 1.798 3.3118 1.5242 1.7981
Node 40 -2.3267 3.6223 -0.3332 3.6223 1.3123 1.2956 3.2891 0.9791 1.2956
Node 45 -2.753 3.6223 -0.3174 3.6223 0.8693 0.8693 3.3049 0.5519 0.8693
Node 50 -3.0938 3.6223 -0.2669 3.6223 0.5283 0.5285 3.3554 0.2614 0.5285
Node 55 -3.3458 3.6223 -0.1916 3.6223 0.2811 0.2765 3.4307 0.0895 0.2765
Node 60 -3.5114 3.6223 -0.1068 3.6223 0.1174 0.1109 3.5155 0.0106 0.1109
Node 65 -3.5989 3.6223 -0.0339 3.6223 0.0271 0.0234 3.5884 -0.0068 0.0234
Node 70 -3.6223 3.6223 0 3.6223 0 0 3.6223 0 0
L36 L18
Node 35 -3.111 6.7014
Node 40 -4.0213 6.7014
Node 45 -4.8233 6.7014
Node 50 -5.4967 6.7014
Node 55 -6.0269 6.7014
Node 60 -6.4054 6.7014
Node 65 -6.6292 6.7014
Node 70 -6.7014 6.7014
L2
L33 L15 L10 L15 L24
L3
L34 L16 L11 L16 L25
L4
L35 L17 L11 L17 L26
Match ?? Match
Match ?? Match
Match ?? ??
Match ?? ??
Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
1091.28 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
1091.31 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
1100 F COADE Sus COADE Exp Traditional Sus Exp #1 Exp #2 COADE Sus + Exp Trad Sus + Exp #1 Trad Sus + Exp #2 Ope
L11 L18 L27
L6
L37 L19 L11 L19 L28
L7
L38 L20 L12 L20 L29
L8
L39 L21 L12 L21 L30
L9
-0.3105 6.7014 3.627 3.5904 6.3909 3.3165 3.5904
-0.3332 6.7014 2.6967 2.6801 6.3682 2.3635 2.6801
-0.3174 6.7014 1.8781 1.8781 6.384 1.5607 1.8781
-0.2669 6.7014 1.2045 1.2047 6.4345 0.9376 1.2047
-0.1916 6.7014 0.679 0.6745 6.5098 0.4874 0.6745
-0.1068 6.7014 0.3025 0.296 6.5946 0.1957 0.296
-0.0339 6.7014 0.0758 0.0722 6.6675 0.0419 0.0721
0 6.7014 0 0 6.7014 0 0
Node 35 -4.4629 9.9363 -0.3105 9.9363 5.5101 5.4734 9.6258 5.1996 5.4734
Node 40 -5.8017 9.9363 -0.3332 9.9363 4.1513 4.1346 9.6031 3.8181 4.1346
Node 45 -6.9985 9.9363 -0.3174 9.9363 2.9379 2.9378 9.6189 2.6205 2.9379
Node 50 -8.0212 9.9363 -0.2669 9.9363 1.9149 1.9151 9.6694 1.648 1.9151
Node 55 -8.8439 9.9363 -0.1916 9.9363 1.0971 1.0924 9.7447 0.9055 1.0925
Node 60 -9.446 9.9363 -0.1068 9.9363 0.4969 0.4903 9.8295 0.3901 0.4904
Node 65 -9.813 9.9363 -0.0339 9.9363 0.127 0.1233 9.9024 0.0931 0.1233
Node 70 -9.9363 9.9363 0 9.9363 0 0 9.9363 0 0
Node 35 -4.463 9.9367 -4.463 9.9367 5.5103 5.4737 5.4737 1.0473 5.4737
Node 40 -5.8018 9.9367 -5.8018 9.9367 4.1516 4.1349 4.1349 -1.6502 4.1349
Node 45 -6.9987 9.9367 -6.9987 9.9367 2.9381 2.938 2.938 -4.0606 2.9381
Node 50 -8.0214 9.9367 -8.0214 9.9367 1.9151 1.9153 1.9153 -6.1063 1.91153
Node 55 -8.8441 9.9367 -8.8441 9.9367 1.0972 1.0926 1.0926 -7.7469 1.0926
Node 60 -9.4462 9.9367 -9.4462 9.9367 0.497 0.4905 0.4905 -8.9492 0.4905
Node 65 -9.8133 9.9367 -9.8133 9.9367 0.1271 0.1234 0.1234 -9.6862 0.1234
Node 70 -9.9366 9.9367 -9.9366 9.9367 0.0001 1E-04 1E-04 -9.9365 0.0001
Node 35 -4.463 10.0363 -4.463 10.0363 5.6099 5.5733 5.5733 1.1469 5.5733
Node 40 -5.8018 10.0363 -5.8018 10.0363 4.2511 4.2345 4.2345 -1.5507 4.2344
Node 45 -6.9987 10.0363 -6.9987 10.0363 3.0377 3.0376 3.0376 -3.961 3.0377
Node 50 -8.0214 10.0363 -8.0214 10.0363 2.0147 2.0149 2.0149 -6.0067 2.0149
Node 55 -8.8441 10.0363 -8.8441 10.0363 1.1968 1.1922 1.1922 -7.6473 1.1922
Node 60 -9.4462 10.0363 -9.4462 10.0363 0.5966 0.5901 0.5901 -8.8496 0.5901
Node 65 -9.8133 10.0363 -9.8133 10.0363 0.2267 0.223 0.223 -9.5866 0.223
Node 70 -9.9366 10.0363 -9.9366 10.0363 0.0997 0.0997 0.0997 -9.8369 0.0997
One more point works in favor of the COADE method here – there should be continuity in the Sustained stress as the temperature creeps up by each tenth of a degree. Below is a plot of the calculated sustained stress using COADE’s Hot Sustained vs. the Traditional Method. COADE’s method shows a continuous curve as one would expect as the temperature moves up an
Match ?? ??
Match ?? ??
Match Match ??
Match Match ??
imperceptible amount, whereas the Traditional Method shows a step function with massive jumps at temperature points where the pipe goes from “not able to slide a sheet of paper under it” to “just able to slide a sheet of paper under it”. Which response seems more realistic? Maximum Sustained Stress vs. Temperature 35000 30000
Stress
25000 20000
COADE Method Traditional Method
15000 10000 5000 0 1
76
151 226 301 376 451 526 601 676 751 826 901 976 Temp
Does it make any sense to notify the user about the lift-off at support 70 in Load Case L38 (SUS), where the maximum stress is 31564 psi, but not about the basically equivalent stress condition (but without lift off) in Load Case L37 (SUS), where the maximum stress “only” 31563 psi?
