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Consider a bar which is fixed at one of the ends. I increase the temperature of the bar by $\Delta T$. The sources I'm referring to say that stress developed in the bar will be zero because the bar is free to expand.

However the bar is free to expand only in the longitudinal direction not in the lateral ones. So how the stress can be zero.

If the bar would've rested on a frictionless surface then the stress would've been zero. But how in this case?enter image description here

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The meral bar is not likely to be "fixed" on a base that has less thermal expansion coefficient than it, when the heat reaches the support, it expands too.

The source was mainly focused on the longitudinal displacement (1D), but you have a valid point for 3D volume expansion.

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  • $\begingroup$ So you suggest a larger thermal expansion rate for the bracket - is there a danger it could fall out or worse droop? $\endgroup$
    – Solar Mike
    Oct 24, 2021 at 14:24
  • $\begingroup$ I don't suggest, I explain. $\endgroup$
    – r13
    Oct 24, 2021 at 14:28
  • $\begingroup$ @r13 Interesting, but won't the bar fall off on to the ground if its not fixed. For a case where we have a bar between two supports, supports expanding slower than the bar will prevent it from falling, but how for the case of one end "fixed" to the support? $\endgroup$
    – Harshit Rajput
    Oct 24, 2021 at 17:29
  • $\begingroup$ Think of usually how a fixed supported for metal is constructed - it has to be welded or connected by a metal rod to a metal base plate, or arguably to the concrete, note that all metals in the process are usually the same type or with similar properties for compatibility, also the concern of chemical reactions due to dissimilarity. If a cantilever bar (as shown) is heated to the extent that the metal softens, then it will develop internal stress due to deformation (deflection), however, the bar will deform before the support starts to yield because it is less stiff than the support materials. $\endgroup$
    – r13
    Oct 24, 2021 at 17:49
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    $\begingroup$ Note, all I wanted to point out is the "no stress" statement is a generalization for ease of understanding the phenomenon and capturing the concept. It is not necessarily the "exact depiction of what was happening", which will be the topic of higher studies. Let's make it simple by getting rid of the support, and letting the bar lay on the frictionless table with one end against a wall, and unrestricted on the other end, where is the elongation will be; is there any developed internal stress after it is fully elongated? $\endgroup$
    – r13
    Oct 24, 2021 at 18:09
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Perhaps the support bracket accepts radial expansion.

Or you could consider that the stress is only being evaluated over the free section and measurement starts a value of x distance from the support.

At least that was how we set up our examination of the change in length of a copper bar. I think we had point zero about 2cm from the support and not only was change in length measured but the change in temperature along the bar as the heated point was the other end. Again we started measuring about 2cm for the point heat was applied.

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  • $\begingroup$ Can you share some images where the support accepts radial expansion? I'm not able to visualize such a support. Actually, I came across another case where the bar was restricted from both it's ends and still there was lateral expansion. I had the same doubt here also, that how the bar is able to laterally expand. Probably a support which accepts radial expansion will solve both my doubts - of one end fixed and both end fixed $\endgroup$
    – Harshit Rajput
    Oct 24, 2021 at 14:16
  • $\begingroup$ @HarshitRajput, a lag bolt from the wall into the center of the beam $\endgroup$
    – Tiger Guy
    Oct 24, 2021 at 15:30
  • $\begingroup$ @TigerGuy Don't have any knowledge about lag bolts but is this what you're somewhat referring to? - drive.google.com/file/d/1G6w6HgrxL5ZwQmS5GkxqvWwmph3tMiVC/… $\endgroup$
    – Harshit Rajput
    Oct 24, 2021 at 17:18
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I guess nobody knows the analytical answer but I hazard to say the stresses if the support is really kept cool or has a largely smaller thermal index, will be very significant.

A large bar with a sudden increase in temperture can even crack at the support or damage the support

Roark's formulas for stress and strain had some empirical formulas. I am flying cross country. when I get back home I check it.

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In practical engineering, a "bar" is something which only transmits axial and (maybe) torsional loads.

You are right that there are some local stresses where the bar is fixed to the plate. That is true however the bar was fixed (e.g. welded, bolted, etc). The local stresses are self-equilibrating.

If you care about the local stresses, you don't model the object as a bar. You model it in full 3-dimensional detail - probably including the nonlinear effects of geometric tolerances in the connection, the pre-stress caused by torque and local plastic deformation in bolted joints, etc.

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  • $\begingroup$ Say I dont want to model in 3D and am treating it as a bar, can I assume the support to be like this - drive.google.com/file/d/1G6w6HgrxL5ZwQmS5GkxqvWwmph3tMiVC/view with the help of a bolt I have fixed the bar. The material of the bolt expands faster than the material of bar, thus not losing a contact with it. This eliminates the problem of constrained lateral expansion, bar will be free to expand laterally. This was also highlighted in one of the comments of one of the answers. $\endgroup$
    – Harshit Rajput
    Oct 25, 2021 at 14:23

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