Fixed end beam flexural stress under differential temperature 3

[Yolac]

Hi guys, for a both end fixed beam subject to differential temperature, say top part hotter than the bottom part, what sort of bending moment diagram will we be getting? I think it would be constant tension at top (rectangular moment diagram) while my colleague says its constant tension at bottom of beam. Who is correct?

 

[Ron]

Hotter at the top, with both ends fixed, puts the top in compression from material expansion. The bottom would be either neutral, compressed, or potentially in tension, depending on the self-weight of the beam.

 

[CTW]

How is the top hotter than the bottom? Without a thermal break, wouldn't the heat transfer to the bottom as well?

By the way, what temperature differential are you dealing with?

 

[BAretired]

If the sun shines on the top of a beam, it will be hotter than the shaded bottom. There will presumably be a temperature gradient from top to bottom.

If the ends are unrestrained, i.e. the beam is a free body, it will act as a thermocouple and will tend to arch upward from end to end.

End restraint will tend to straighten the beam out by exerting a constant positive moment throughout the beam. Neglecting self weight, the top fibers will be in compression. If the bottom fibers remain at the same temperature throughout, they will be unstressed.

BA

 

[hokie66]

If the ends are truly fixed, the beam will be in double curvature, so there has to be change in the stress state on each face.

 

[BAretired]

hokie,

It will be in double curvature if the weight of beam is taken into account, but if only temperature effects are considered, it will not.

BA

 

[paddingtongreen]

I'm not sure I agree.
With free ends, one end a slider, the beam will take up circular curvature, up in the middle (hogging) with zero bending moment, provided that the temperature gradient is linear. Going to pinned/pinned will introduce an axial force that will cause hogging moment, greatest at the center because deflection is greatest there, zero at the ends. Introducing fixity at the supports causes compression on top and tension at the bottom, I think this brings circular bending in the opposite direction, removing the original deflection.

Have at it.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[rapt]

If the beam is unrestrained there will be no stress induced by a linear temperature differential.

The only stress from temperature differential is induced by the restraint to rotation by the fixed ends.

For top hotter than bottom and no restraint, the top expands compared to the bottom so it hogs upwards and no stress is induced.

Fixing the ends will induce rotational moment restraints at each end causing a positive moment in the beam constant full length. So bottom is in tension and top in compression due to the top hotter than the bottom. Then the vertical loading effects need to be added to this.

 

[BAretired]

Say the initial temperature is t₀ throughout the entire beam and say the beam is weightless. After heating the top, say the temperature varies uniformly from t_T1 at the top to t_B1 at the bottom.

If:

t_T1 > t₀ then top is in compression

t_B1 > t₀ then bottom is in compression

t_B1 = t₀ then bottom is unstressed

t_B1 < t₀ then bottom is in tension

Of course, if the beam is not weightless, the moments at all sections must be adjusted to account for the load.

BA

 

[IDS]

This is covered in bridge design codes. In practice the temperature gradient is not uniform. Typical design curves are hot on top (about +20 C)reducing to zero over about 300 mm, and much smaller increment on the bottom face (about +5 C) reducing to zero over about 200 mm. Reverse differential temperatures (with the top and bottom face cooler than the interior) also need to be considered.

The result is that there is an induced bending moment even if the ends of the beam are not restrained laterally.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

If both ends of the beam are fixed, there is zero deflection throughout the beam as a result of temperature change irrespective of the temperature gradient. Fibers at any depth will be in tension if the temperature falls and compression if it rises.

BA

 

[STH003]

BA is right. both ends fixed gives no deflection, but gives a constant moment throughout the beam due to equal fixed end actions: apha*deltaT*E*I.

No matter how hard you try you want to say that the beam bows up because it has stresses in it. But in our case, we have stresses with no strains.

You can go through the direct stiffness method for a single frame element with a midpoint to see that there are no deflections....

or try analysis software

 

[Ron]

STH003...well, not exactly. You have both stresses and strains (can't have one without the other in an elastic material). In this case you have compressive stress and compressive strain in the top flange. It would only bow up if the top flange to bottom flange temperature gradient were sufficient to induce buckling stress beyond Pcr.

 

[STH003]

this is not true...

an element that has no supports subjected to a uniform temperature gradient undergoes strain with no internal stress.... when there is no restraint to elongation then no stresses are picked up in the member.

the reverse is true for the fixed-fixed case with a uniform thermal loading. the only reason the member has compressive stress in it is because it is restrained from deforming, or straining.

thermal effects are the only case where stress and strain don't go hand in hand. i agree that the thing would move due only to buckling or ltb

 

[BAretired]

Buckling is an important consideration which I should have addressed earlier. A concrete member should be permitted to grow without restraint in order to avoid buckling.

BA

 

[IDS]

Looking at the Australian Standard Loading Code, it certainly isn't as clear as the Bridge Code, but is does require consideration of a non-linear temperature gradient in members subject to different conditions on opposite faces.

The procedure for analysis of a section subject to these thermal strains without longitudinal restraint is to apply stresses to the individual layers to return the strain across the section to zero, then apply an equal and opposite moment and axial load to the whole section to maintain equilibrium.

With end restraint it is more complicated, but you need to include whatever is providing the restraint in the analysis.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rapt]

BAretired/STH003

What you suggest is correct for the following case
- temperature change over the whole depth
- axial restraint to shortening

The subject under discussion is temperature differential with the top hotter than the bottom and a linear temperature differential (I know bridge designers make different assumptions about variation of temperature with depth but most building slabs are less than 300 thick and then the linear assumption is made).
Also in buildings, the restraint is normally rotational and not purely axial.

In this case, as I explained in an earlier post,

"If the beam is unrestrained there will be no stress induced by a linear temperature differential. Only strain will be induced.

The only stress from temperature differential is induced by the restraint to rotation by the fixed ends.

For top hotter than bottom and no restraint, the top expands compared to the bottom so it hogs upwards and no stress is induced (as long as the temperature gradient is linear).

Fixing the ends will induce rotational moment restraints at each end causing a positive moment in the beam constant full length. So bottom is in tension and top in compression due to the top hotter than the bottom. Then the vertical loading effects need to be added to this. "

If there is overall expansion or shortening of the member due to the temperature changes and there is axial restraint also, extra stresses will be induced.

 

[BAretired]

I'm not sure why we are debating the issue. It seems pretty clear to me. If the beam is fully fixed at each end, there can be no change in length of any fiber.

If there is no change in length, the stress in every fiber is strictly a function of its change in temperature. It does not matter what the temperature gradient is throughout the section provided it is the same at every section.

If the heat is sufficient to buckle the beam, that would be a different story.

BA

 

[IDS]

If the beam is fully fixed at each end, there can be no change in length of any fiber.

Because the beam will not be fully fixed at both ends, and even if the rotation at the ends was negligible you would still need to check the effect of the temperature stresses on the beam and whatever is restraining it.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

The OP posed a theoretical question. A fixed end is fully fixed by definition. The rotation at each end is not negligible, it is zero. The change in length at every fiber is zero. Where is the problem?

BA