# Calculating Bending Force on a Fixed Metal Plate

At work we've had a steel pipe support break on us, and I can't for the life of me figure out what happened. Here's some background info:

A long (125m) steel heating pipe (ø51mm) is suspended, resting on 27 identical supports over its total length. The very last support fixes the pipe in place there (to prevent it from expanding through a wall), but it is free to move in all the other ones.

This 27th support is the one that broke, and my best guess is that it might have had something to do with the sum of the frictional forces of the 26 other supports being greater than the force required to bend/break the 27th support.

The pipe is heated (from 20 to 90 °C) and expands as a result. A quick google yielded me the following equation for thermal expansion of a steel pipe:

F = E * α * dt * A = 200 GPa * 0,012 * 70 * 7.1e-4 m2 ~ 117 MN


Assuming the pipe is fixed on one end, the force of thermal expansion would be 117 Mega Newtons, which to me pretty much just means there's absolutely no stopping the expansion.

Now comes the part I'm confused about. The pipe is supported by 26 equally spaced supports, through which it is 'free' to move. I say 'free', because I assume it needs to beat static friction first.

The formula for static friction I found was as follows:

f_s = μ * m * a


I found a value for the static friction coefficient μ being 0.75 for steel-on-steel, but this seemed to vary a lot depending on where I looked. The mass of the pipe is approximately 4.57kg/m, so about 570kg in total. The acceleration in this case is equal to g, so 9.81 m/s2. Interestingly, friction is contact area independent.

Now, filling in the equation, I am left with 0.75x570x9.81 = 4194 kg.m.s^-2 so about 4200 N.

Am I correct in assuming this means the total friction force experienced by the pipe when heating up is 4200N? And that the fixed end of the pipe would have to be strong enough to resist that force in order for it to expand only in the other direction?

The next part of my question is, how can I calculate what the maximum amount of force on that 27th steel plate can be before it bends/breaks? I can calculate that the torque on the steel plate at the pipe's location is 4200N * 50mm = 210 Nm, but that doesn't really mean anything to me, nor do I know if that's even remotely close to the truth. All steel involved is s235 structural steel.

Either way, I don't think it broke because of the friction force, but rather because it was mistakenly also fixed somewhere else. The thermal expansion force is more than strong enough to break steel if it has to, so that seems more likely to me.

If anyone has any idea how to tackle this problem, help would be greatly appreciated!

Edit: As a requested clarification, here's a simple isometric sketch of the situation.

• Make sure the 26 supports have sliding supports. Sep 10 '20 at 8:00
• This is very interesting, I was not aware of these things. Definitely worth considering for future applications where expansion is expected. Thank you! Sep 10 '20 at 9:27
• can you draw an isometric sketch (with distances) or something so we can figure out the layout? I assume the pipe is not simply a straight run. For the slding supports: It can be as simple as a band of teflon in the arch the pipe is lying in.
– mart
Sep 10 '20 at 12:28
• Are you saying some young engineer built this with no expansion loops ? Sep 10 '20 at 16:17
• You can't stop thermal expansion, you accommodate it. Sep 10 '20 at 16:19

In my view, you are attacking this from the wrong angle. The pipe will expand by about 150 mm. You need to give your pipe run the required space to do so.

Look if you can use existing bends (this is what we need an isometry of the pipe run for) to account for the expansion. There's a wall, there has to be a bend. If not possible, see if you can incorporate an expansion loop:

(For a thorough explanation, check out the image source) This allows the pipe to bend somewhat. Peng and Peng (Pipe stress engineering) give the minimum length of the elbow as $$2w=66\sqrt{D\Delta}$$ (for steel pipes) with pipe diameter $$D$$ and change of length per elbow $$\Delta$$. Just use consistent units.

Lastly, you can use expansion joints:

Note that you get about 50 mm of expansion per joint at most so you'd need three. Probably 40 m pipe runs with expansion joints in between and fixed supports in the middle, all else sliding supports - tbh I've only ever designed pipe runs with elbows anyway so I never really needed to plan something like you appear to have.

Of course you could have a combination (expansion joint at one end, elbow at the other).

• Thank you for your reply! The pipe does have expansion room, but not in the middle of it. We've left free space at one end (the one where it ISN'T fixed) for it to expand. Now though, the fixed end broke. My question was, could the friction on the supports have caused the fixed support to break, or did something else go wrong? Sep 11 '20 at 6:05
• Added one. There's a total of 26 supports that it simply 'rests' on between the free space to expand into and the fixed support. Each support is a simple steel plate with holes to hold the pipe. Each support has an area of about 20mm to support the pipe. Sep 11 '20 at 6:19
• the sketch is good & IMO the design of the system seems ok. Now we need to know if the design was followed: Is the pipe run indeed straight? Is the pipe movement jammed in any way? are there notches or so? Does the pipe hang? 5 m from support to support is rather long for DN50.
– mart
Sep 11 '20 at 6:39