What in the World is Sustained Stress Anyway?

By Thomas J. Van Laan, PE President/CEO of COADE, Inc.

Houston, TX

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 (which may open during the operating case but, when closed, may have a significant effect on the distribution of the weight loads) or other non-linear 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 should, as one code (the French petrochemical code CODETI) explicitly states, be treated as secondary, or 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 W+T1+P1+D1 (OPE) L2 T1+D1 (EXP) L3 W+P1 (SUS) L4 L1-L3 (EXP) L5 L1-L2 (SUS)

In the above set of load cases, L3 represents the “Cold Sustained”, L4 represents the Expansion case (the stress range between the two extremes of operating and installed), and L5 represents the “Hot Sustained”. (L2 represents a potentially fictitious load case where the piping system grows thermally prior to weight being applied.) 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 challenge anybody to 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, then certainly they envelope the actual distribution. The L3 (SUS) and L4 (EXP) results sum to the L1 (OPE) results – these results represent a 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 medical technicians are carrying a 250-pound patient in a stretcher, holding the ends of the stretcher roughly three feet above the ground – 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 ways:

1) Cold Sustained – the stretcher was lying on the ground when the patient got onto it. The displaced shape at “installation” (continuously supported, so there are no real weight stresses) is calculated:

"Cold Sustained"

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Then imposed displacements are applied (lifting the ends of the stretcher 3 feet), taking us to the operating state.

"Operating"

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The Expansion Case is the difference between these two shapes (and thus will look virtually the same as the Operating shape) -- so the Sustained Case and the Expansion Case will always add up to the Operating Case.

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.

Lifting the stretcher…

"Pre-Weight Thermal"

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…and then the patient gets on:

"Operating"

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In this case the Sustained response is the difference from the original displaced position and the fina l operating (sagged from the original) position. So the “Hot Sustained” in this case would look like:

"Hot Sustained"

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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 medical technicians 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 35 inches under the weight of the patient, shoves a stool underneath that reduces the sag from 35 inches to only 22 inches.

"Operating with Stool"

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The stool would serve as an active support. Does that mean that all of the weight stress goes away simply because we no longer have a lift-off problem? COADE’s concept of Hot Sustained says that the weight stresses can be calculated by subtracting “Operating with Stool” shape from the “Pre-weight Thermal” case. The traditional method of handling non-linear supports in for the weight case (including active supports and removing inactive supports but ignoring displaced condition in the operating case) would show virtually no weight stresses since there would be anchors at the ends and a support in the middle. The Traditional displaced shape would be:

"Traditional Sustained with Stool"

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It is clear that in this case, the Weight and Thermal cases do NOT sum to the Operating Case!

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 demonstrate 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 Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L31 -0.2436 -0.2758 -0.2954 -0.3118 -0.3188 -0.3148 -0.3036 -0.2954 COADE Exp L13 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 Traditional Sus L10 -0.0367 -0.0167 0 0.0002 -0.0045 -0.0065 -0.0037 0 Exp #1 L13 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 0.2954 Exp #2 L22 0.0885 0.0363 0 -0.0166 -0.0189 -0.0128 -0.0045 0 COADE Sus + Exp 0.0518 0.0196 0 -0.0164 -0.0234 -0.0194 -0.0082 0 MatchTrad Sus + Exp #1 0.2587 0.2787 0.2954 0.2956 0.2909 0.2889 0.2917 0.2954 ?? Trad Sus + Exp #2 0.0518 0.0196 0 -0.0164 -0.0234 -0.0193 -0.0082 0 MatchOpe L1 0.0518 0.0196 0 -0.0164 -0.0234 -0.0193 -0.0082 0 150 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L32 -0.4573 -0.5433 -0.6005 -0.634 -0.6434 -0.633 -0.6134 -0.6005 COADE Exp L14 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 Traditional Sus L10 -0.0367 -0.0167 0 0.0002 -0.0045 -0.0065 -0.0037 0 Exp #1 L14 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 0.6005 Exp #2 L23 0.1798 0.0738 0 -0.0337 -0.0383 -0.0261 -0.0092 0 COADE Sus + Exp 0.1432 0.0572 0 -0.0335 -0.0429 -0.0325 -0.0129 0 MatchTrad Sus + Exp #1 0.5638 0.5838 0.6005 0.6007 0.596 0.594 0.5968 0.6005 ?? Trad Sus + Exp #2 0.1431 0.0571 0 -0.0335 -0.0428 -0.0326 -0.0129 0 MatchOpe L2 0.1431 0.0571 0 -0.0335 -0.0429 -0.0326 -0.0129 0 196.68 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L33 -0.7152 -0.8662 -0.9686 -1.0228 -1.035 -1.0172 -0.9871 -0.9686 COADE Exp L15 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 Traditional Sus L10 -0.0367 -0.0167 0 0.0002 -0.0045 -0.0065 -0.0037 0 Exp #1 L15 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 0.9686 Exp #2 L24 0.29 0.1191 0 -0.0544 -0.0618 -0.0421 -0.0149 0 COADE Sus + Exp 0.2534 0.1024 0 -0.0542 -0.0664 -0.0486 -0.0185 0 MatchTrad Sus + Exp #1 0.9319 0.9519 0.9686 0.9688 0.9641 0.9621 0.9649 0.9686 ?? Trad Sus + Exp #2 0.2533 0.1024 0 -0.0542 -0.0663 -0.0486 -0.0186 0 MatchOpe L3 0.2534 0.1024 0 -0.0542 -0.0664 -0.0486 -0.0186 0 196.7 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L34 -0.7153 -0.8663 -0.9687 -1.0229 -1.0351 -1.0173 -0.9873 -0.9688 COADE Exp L16 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 Traditional Sus L11 -0.3105 -0.3332 -0.3174 -0.2669 -0.1916 -0.1068 -0.0339 0 Exp #1 L16 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 0.9688 Exp #2 L25 0.2901 0.1191 0 -0.0544 -0.0618 -0.0421 -0.0149 0 COADE Sus + Exp 0.2535 0.1025 1E-04 -0.0541 -0.0663 -0.0485 -0.0185 0 MatchTrad Sus + Exp #1 0.6583 0.6356 0.6514 0.7019 0.7772 0.862 0.9349 0.9688 ?? Trad Sus + Exp #2 -0.0204 -0.2141 -0.3174 -0.3213 -0.2534 -0.1489 -0.0488 0 ??

Ope L4 0.2534 0.1024 0 -0.0542 -0.0664 -0.0486 -0.0186 0 500 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L35 -1.8243 -2.3267 -2.753 -3.0938 -3.3458 -3.5114 -3.5989 -3.6223 COADE Exp L17 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 Traditional Sus L11 -0.3105 -0.3332 -0.3174 -0.2669 -0.1916 -0.1068 -0.0339 0 Exp #1 L17 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 3.6223 Exp #2 L26 1.8347 1.3123 0.8693 0.5283 0.2811 0.1174 0.0271 0 COADE Sus + Exp 1.798 1.2956 0.8693 0.5285 0.2765 0.1109 0.0234 0 MatchTrad Sus + Exp #1 3.3118 3.2891 3.3049 3.3554 3.4307 3.5155 3.5884 3.6223 ?? Trad Sus + Exp #2 1.5242 0.9791 0.5519 0.2614 0.0895 0.0106 -0.0068 0 ?? Ope L5 1.7981 1.2956 0.8693 0.5285 0.2765 0.1109 0.0234 0 800 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L36 -3.111 -4.0213 -4.8233 -5.4967 -6.0269 -6.4054 -6.6292 -6.7014 COADE Exp L18 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 Traditional Sus L11 -0.3105 -0.3332 -0.3174 -0.2669 -0.1916 -0.1068 -0.0339 0 Exp #1 L18 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 6.7014 Exp #2 L27 3.627 2.6967 1.8781 1.2045 0.679 0.3025 0.0758 0 COADE Sus + Exp 3.5904 2.6801 1.8781 1.2047 0.6745 0.296 0.0722 0 MatchTrad Sus + Exp #1 6.3909 6.3682 6.384 6.4345 6.5098 6.5946 6.6675 6.7014 ?? Trad Sus + Exp #2 3.3165 2.3635 1.5607 0.9376 0.4874 0.1957 0.0419 0 ?? Ope L6 3.5904 2.6801 1.8781 1.2047 0.6745 0.296 0.0721 0 1091.28 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L37 -4.4629 -5.8017 -6.9985 -8.0212 -8.8439 -9.446 -9.813 -9.9363 COADE Exp L19 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 Traditional Sus L11 -0.3105 -0.3332 -0.3174 -0.2669 -0.1916 -0.1068 -0.0339 0 Exp #1 L19 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 9.9363 Exp #2 L28 5.5101 4.1513 2.9379 1.9149 1.0971 0.4969 0.127 0 COADE Sus + Exp 5.4734 4.1346 2.9378 1.9151 1.0924 0.4903 0.1233 0 MatchTrad Sus + Exp #1 9.6258 9.6031 9.6189 9.6694 9.7447 9.8295 9.9024 9.9363 ?? Trad Sus + Exp #2 5.1996 3.8181 2.6205 1.648 0.9055 0.3901 0.0931 0 ?? Ope L7 5.4734 4.1346 2.9379 1.9151 1.0925 0.4904 0.1233 0 1091.31 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L38 -4.463 -5.8018 -6.9987 -8.0214 -8.8441 -9.4462 -9.8133 -9.9366 COADE Exp L20 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 Traditional Sus L12 -4.463 -5.8018 -6.9987 -8.0214 -8.8441 -9.4462 -9.8133 -9.9366 Exp #1 L20 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 9.9367 Exp #2 L29 5.5103 4.1516 2.9381 1.9151 1.0972 0.497 0.1271 0.0001 COADE Sus + Exp 5.4737 4.1349 2.938 1.9153 1.0926 0.4905 0.1234 1E-04 MatchTrad Sus + Exp #1 5.4737 4.1349 2.938 1.9153 1.0926 0.4905 0.1234 1E-04 MatchTrad Sus + Exp #2 1.0473 -1.6502 -4.0606 -6.1063 -7.7469 -8.9492 -9.6862 -9.9365 ?? Ope L8 5.4737 4.1349 2.9381 1.91153 1.0926 0.4905 0.1234 0.0001

1100 F Node 35 Node 40 Node 45 Node 50 Node 55 Node 60 Node 65 Node 70 COADE Sus L39 -4.463 -5.8018 -6.9987 -8.0214 -8.8441 -9.4462 -9.8133 -9.9366 COADE Exp L21 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 Traditional Sus L12 -4.463 -5.8018 -6.9987 -8.0214 -8.8441 -9.4462 -9.8133 -9.9366 Exp #1 L21 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 10.0363 Exp #2 L30 5.6099 4.2511 3.0377 2.0147 1.1968 0.5966 0.2267 0.0997 COADE Sus + Exp 5.5733 4.2345 3.0376 2.0149 1.1922 0.5901 0.223 0.0997 MatchTrad Sus + Exp #1 5.5733 4.2345 3.0376 2.0149 1.1922 0.5901 0.223 0.0997 MatchTrad Sus + Exp #2 1.1469 -1.5507 -3.961 -6.0067 -7.6473 -8.8496 -9.5866 -9.8369 ?? Ope L9 5.5733 4.2344 3.0377 2.0149 1.1922 0.5901 0.223 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 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

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Does it make any sense that we would consider it to be a problem when there is lift-off of support 70 in Load Case L38 (SUS), where the maximum stress is 31564 psi, but not with the basically equivalent stress condition (but without lift off) in Load Case L37 (SUS), where the maximum stress “only” 31563 psi?