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?

 

[IDS]

I assumed the theoretical question was related to a real design situation.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[prex]

The answer is here Beam under simple bending: left end fixed: right end fixed: temperature gradient: across depth: linearly varying for ends axially unrestrained, and here Beam under bending: with held ends: left end fixed: right end fixed: temperature gradient: across depth: linearly varying with full restraint. In both cases, if the average temperature doesn't change from initial state, there is no deflection: the two cases are the same as there is no net elongation of the beam when the average temperature stays the same.

prex

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[ToadJones]

Well something has me wondering....

Derivative of Deflection = slope
Derivative of Slope = moment
derivative of moment = shear
derivative of shear = load

how can you have a resulting deflection without the requisite slope, moment, shear, load and stress?

 

[BAretired]

prex,

You stated:

In both cases, if the average temperature doesn't change from initial state, there is no deflection: the two cases are the same as there is no net elongation of the beam when the average temperature stays the same.

I agree but the statement that there is no deflection and no change in length is true whether or not the average temperature changes.

BA

 

[prex]

The first relationship is a matter of geometry and is always true.
The third one and the fourth come from the law of equilibrium and are also always true (with due consideration of some limit cases).
The second one, instead, is a relationship between elastic strains and the loading that originated them. Thermal strain is not included in this derivation, as it is not an elastic strain (as correctly noted by someone above, we can have thermal strain and no stress at all).
That second relationship is still valid in presence of thermal strain, but only considering the pure elastic strain component, obtained by withdrawing the thermal strain from the total one.

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes and launchers for fun rides

http://www.levitans.com

: Air bearing pads

 

[BAretired]

prex,

That's as clear as mud.

BA

 

[prex]

Sorry BAretired, my answer was directed to ToadJones.
And of course I agree with your statement above (not the one about mud), as a uniform temperature increase, added to whatever preceding state of stress, will give no stress with a free end and an axial stress with held ends (as Monsieur de la Palice said).

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes and launchers for fun rides

http://www.levitans.com

: Air bearing pads

 

[BAretired]

Ci gît Monsieur de La Palice: Si il' n'était pas mort, il ferait encore envie.
("Here lies Sir de la Palice: If he wasn't dead, he would still be envied.")

BA

 

[ToadJones]

Or as Bruce Dickinson said:
"at the gates and the walls of Montsegur, blood on the stones of the Citadel"

 

[IDS]

Hello BAretired. The other day you pointed me in the direction of conjugate beams, and now thanks to you and Prex I have learned of the life and tautological death of Monsieur de la Palice.

The things you learn on an engineering forum


(For the benefit of others who didn't know of Ms de la Palice, the last word of the epitaph can also be interpretted as en vie, translating as "if he wasn't dead, he would still be alive")

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[TERIO]

If you are bored go to the link below and read through Section D.

NASA Astronautic Structures Manual:

http://trs.nis.nasa.gov/archive/00000177/

Specifically, in Section D.3.2.3 (

http://trs.nis.nasa.gov/archive/00000177/01/asm-D323.pdf)

starting at the bottom of page 74 there is an example calculation for fixed-fixed beam with linearly varying temperature distribution.

 

[BAretired]

Doug,

The things you learn on an engineering forum

Stick around...it gets better when the snow starts to fly.

BA

 

[rapt]

Prex & ToadJones,

Just making sure I understand that you are both saying that there will be no deflection of a member with a linear temperature differential over its depth. Is this what you are saying?

If so, then I am sorry but you are incorrect.

If we have a member of zero weight and no restraint and apply a temperature differential over its depth, say with the top hotter than the bottom. Then the top extends compared to the bottom (simple physics!). As long as there is no restraint etc, the beam shape will hog upwards at the centre. There is no stress induced (as long as the differential is linear) but there is strain as the top surface is longer than the bottom surface. So the member is deflected upwards. It is no longer straight.

To simulate this member being cast into fixed ends, the ends of the member must remain horizontal. To achieve this, we need to apply Moments at the ends to rotate them back to horizontal. This gives the bending moment full length that the calculations in prex's calc sheets showed.

 

[BAretired]

rapt,

I can't speak for prex or ToadJones but for any consistent variation in temperature, linear or non-linear, over the depth of beam, the deflection is zero provided that buckling is not induced.

BA

 

[Ron]

BA is correct...again.

 

[IDS]

I think everyone agrees that there is no deflection if the ends are fully fixed against rotation.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[prex]

Of course BAretired is correct, as stated by others above.
It is however worth repeating something that has also been already said: zero deflection is true, but only with a uniform temperature gradient over the beam length. If the gradient varies, or is present in a portion only of the beam, then there is a deflection.

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes and launchers for fun rides

http://www.levitans.com

: Air bearing pads

 

[paddingtongreen]

This is a fairly simple concept, perhaps we are making it too complicated.
If the ends are unrestrained and there is a linear temperature gradient, the beam will take up a circular curve, with no bending moment induced. The ends will rotate through equal and opposite angles.
To fix the ends, they must both be rotated back to their original position, taking equal and opposite moments, yielding circular bending in the opposite direction. There being no change of moment along the beam, there is no change of curvature, it must be a straight line.

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

 

[BAretired]

I hate to beat a dead horse, but the fact is that the temperature gradient need not be uniform (prex) or linear (paddington) for zero deflection. It can be non uniform, non linear, zigzag or a step function but it must be consistent throughout the length of the beam.

BA

 

[rapt]

Gentlemen,

You have not read my response completely.

ToadJones said that there is no deflection is there is no moment. The first part of my response pointed out that, for a weightless beam with no restraints, there is no moment but there is strain and deflection, and it is upwards if the top is hotter than the bottom.

I then went on to discuss the situation with end restraints where I did not discuss deflection, only pointed out that in this case there is moment. And, as pointed out by others, if the ends are full restrained against rotation, there is no deflection as the deflection caused by the rotational restraint exactly counters the free deflection from the temperature differential.

So,
- in the unrestrained case we have deflection and strain without moment and load
- in the second case with full restraint, we have moment and stress without deflection or applied load.
- in any other case, with partial restraint, which would be normal in a building structure, there will be differing amounts of moment, stress, strain and deflection somewhere between the 2 preceeding extremes.