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I'm misunderstanding something about how geometry affects yield stress. I think this example best illustrates my confusion:

Say you have two pipes with the same internal diameter, same material grade, and same internal pressure (The internal pressure is right at the yield stress). The only difference is pipe A is thicker than pipe B.

In my mind, since both pipes are experiencing the yield stress on their internal diameters, then both should deform plastically. How I also know that the thicker a pipe is the more stress it can withstand. This seems like a contradiction to me.

Yield stress is supposed to be independent of geometry, however geometry can clearly increase or decrease a parts ability to withstand stress and deformation.

BoddTaxter
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  • So you are suggesting that a 3mm sheet of aluminium can be folded over a cooking dish by hand just like the aluminium cooking foil? – Solar Mike Dec 30 '22 at 18:31
  • "*Yield stress is supposed to be independent of geometry*" Says who? If someone actually did say that you're probably missing some context. Because if define stress as force per area, it can't be independent of geometry. – DKNguyen Dec 30 '22 at 19:28
  • For thick-walled pipes, stress is not uniform across the wall, rather it is linearly varying. When the entire wall yields, F = yield stress * t, it is evident that F is larger for the pipe with a thicker wall. – r13 Dec 30 '22 at 19:29
  • @DKNguyen I am referencing ASME B31.3 page 186 where is gives yield stresses for different material grades, independent of geometry. Yield stress is often given as a material property. – BoddTaxter Dec 30 '22 at 20:00
  • @BoddTaxter You, or perhaps other people, are misusing the same term "yield stress" to represent two different things: the yield stress of a material, and the yield *force* of a part. Either that or people are using the term and expecting the listener to understand the context between the material property and the property versus the part properties. – DKNguyen Dec 30 '22 at 20:05
  • @r13 If you have the yield stress right at the internal diameter, will you have deformation there regardless of overall pipe thickness? – BoddTaxter Dec 30 '22 at 20:06
  • @DKNguyen Thank you. Can you explain more where I am misusing the terms, I don't see it. – BoddTaxter Dec 30 '22 at 20:09
  • 'The internal pressure is right at the yield stress' is not a realistic condition - the pipe will have been torn apart by tensile hoop stress long before the internal pressure reaches that level. – Daniel Hatton Dec 30 '22 at 22:44
  • @BoddTaxter At yield, the internal material tends to deform but the outer layer won't allow it. So there is no deformation until the final stage of the stress redistribution that causes the entire cross-section to yield. – r13 Dec 31 '22 at 00:27

4 Answers4

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You have two things floating around:

  • (a) the yield stress as a material property listed as force per
    unit area
  • (b) the yield strength at which the part itself yields listed as
    force

However, you have four different names that can be assigned to these two things:

  1. yield stress of a material
  2. yield strength of a material
  3. yield strength of a part
  4. yield stress of a part

So when you think about a material or part withstanding "stress" are you really thinking about stress? Or are you actually thinking about strength?

In normal language, when you "stress" a part, are you stressing the material? Or are you stressing the part? Or are you "forcing" the material? Or are you "forcing" the part? Or are you doing all of them so it doesn't really matter what word you use? Of course, that doesn't mean stress and strength are the same thing so don't fall into the trap of thinking that they are or taking the terms at face value when you hear other people use them without accounting for context.

Technically, (1) and (2) are both equivalent to (a) because nothing else makes sense. Materials don't inherently have geometry so you cannot use units of force alone to specify the yield strength of a material. You must refer to yield stress.

However, parts do have inherent geometry and in this case part strength and part stress mean different things. (3) means (b) since strength is units of force, whereas (4) means (a) since stress is force per unit area distributed in the part which is the same as force per unit area distributed in the material that makes up the part.

In any case regardless of the name of things, force/area is geometry independent but when translated to force it must be geometry dependent.

DKNguyen
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Pipe does not withstand just internal pressure, but pressure difference between internal and external. With this in mind, you can think of the thicker pipe like it is the thinner pipe with extra pressure from the outside. So effectively, the thin pipe and "thin" part of the thick pipe experience different pressure differences when both pipes are subject to the same internal pressure.

By the way, elastic limit pressure of a pipe is at half of the material yield strength, so if you consider situation, where pressure is at yield strength of the material, at least partial plasticity will be involved. For more information on pipe pressure limits, you can check out my answer to a related question.

Tomáš Létal
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  • Thank you, this is starting to make sense to me. If you look at a cube stress element of each pipe, both get the same pressure from the inside of the pipe. But on the thicker pipe the extra thickness is "pushing back" more than the ambient air is for the thin pipe. Is that right and is there a resource I can learn more about this? – BoddTaxter Dec 31 '22 at 00:14
  • Classic way of seeing this effect would be using "shrink-fit" of two pipes using Lamé equations as shown in r13's answer (recently I have also used it [here](https://engineering.stackexchange.com/a/53593/21214)). However, easier way is to imagine, that you can separate the pipe thickness into layers and control the pressures between them. You can use almost any thickness design or maximum pressure formulae for individual layers. Using more and more layers with each bearing maximum pressure difference will approach maximum pressure the pipe can take in full plasticity. – Tomáš Létal Dec 31 '22 at 22:42
  • A good book for this is [Pressure Vessel Engineering Technology](https://www.worldcat.org/cs/title/pressure-vessel-engineering-technology/oclc/139223). – Tomáš Létal Dec 31 '22 at 22:45
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Maybe this thread can help you to get started. When calculating the deformed radius of a pressurized thick-walled cylinder, why is the hoop strain used rather than the radial strain?

enter image description here

Another practical reading material - https://www.mydatabook.org/solid-mechanics/stress-for-thick-walled-cylinders-and-spheres-using-lames-equations/

r13
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So one interesting example is the pre-stressing of a gun barrel so it can withstand higher pressures.

A good example is when large gun barrels have the rifled inner surrounded by an outer. The outer is heated to expand then shrink over the inner.

See enter link description here

Solar Mike
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