Fixed end beam flexural stress under differential temperature
Fixed end beam flexural stress under differential temperature
(OP)
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?






RE: Fixed end beam flexural stress under differential temperature
RE: Fixed end beam flexural stress under differential temperature
By the way, what temperature differential are you dealing with?
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
If:
tT1 > t0 then top is in compression
tB1 > t0 then bottom is in compression
tB1 = t0 then bottom is unstressed
tB1 < 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
RE: Fixed end beam flexural stress under differential temperature
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/
RE: Fixed end beam flexural stress under differential temperature
BA
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
BA
RE: Fixed end beam flexural stress under differential temperature
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/
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
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/
RE: Fixed end beam flexural stress under differential temperature
BA
RE: Fixed end beam flexural stress under differential temperature
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
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?
RE: Fixed end beam flexural stress under differential temperature
You stated:
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
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
That's as clear as mud.
BA
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
("Here lies Sir de la Palice: If he wasn't dead, he would still be envied.")
BA
RE: Fixed end beam flexural stress under differential temperature
"at the gates and the walls of Montsegur, blood on the stones of the Citadel"
RE: Fixed end beam flexural stress under differential temperature
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/
RE: Fixed end beam flexural stress under differential temperature
NASA Astronautic Structures Manual: http://trs.nis.nasa.gov/archive/00000177/
Specifically, in Section D.3.2.3 (http:/
RE: Fixed end beam flexural stress under differential temperature
Stick around...it gets better when the snow starts to fly.
BA
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
RE: Fixed end beam flexural stress under differential temperature
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end beam flexural stress under differential temperature
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
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
BA
RE: Fixed end beam flexural stress under differential temperature
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.
RE: Fixed end beam flexural stress under differential temperature
As I see it at the moment, if it is non linear, there will be shear plane(s) for the length of the beam. That shear will cause complimentary vertical shear, causing bending moment. I have to convince myself that that moment causes circular bending.
I am trying to think my way through a case where all of the upper half of a beam is heated to an even temperature, above that of the uniformly cool lower half.
Michael.
Timing has a lot to do with the outcome of a rain dance.
RE: Fixed end beam flexural stress under differential temperature
Say that the upper half is heated to 100o above starting temperature and the lower half is unchanged. Now, the upper half would like to expand, but it can't because there is a fixed support at each end. So it is stressed to 100*C*E where C is the coefficient of expansion and E is the modulus of elasticity.
The temperature in the lower half has not changed, so the stress in the lower half is zero.
The stress over the beam depth is a step function, having a large value in the top half and zero in the bottom half. But there is no bending because none of the fibers can change in length. There are no horizontal shear stresses developing between the top and bottom half because there is zero strain in both halves.
BA
RE: Fixed end beam flexural stress under differential temperature
Separating and then adding loading effects has helped me considerably over the years, but in this case it was a complication.
Thanks
Michael.
Timing has a lot to do with the outcome of a rain dance.
RE: Fixed end beam flexural stress under differential temperature
BA
RE: Fixed end beam flexural stress under differential temperature
I think the procedure of splitting the beam into statically determinate sections is the right way to approach this problem, but it needs to be done in the right order:
1) Split the member into statically determinate sections.
2) Apply non-load related strains (i.e. temperature and shrinkage)
3) Apply any real external restraints
4) If necessary, apply imaginary forces to each section to restore strain compatibility.
5) Apply equal and opposite forces to the composite section to restore equilibrium
For the example in the original post, consider three different end conditions:
1) Fully fixed both ends
2) Fully fixed left, restrained against rotation only right
3) Simply supported both ends, laterally restrained at bottom left
and two different strain conditions
A) Top half of beam at uniform +10 degrees relative to the bottom half.
B) Left hand end of top half at +10 degrees relative to the bottom half and right hand end
1) Split the member into statically determinate sections.
A) Split the beam along the Neutral Axis (NA) and remove all end restraints
B) Split the top section into two halves
2) Apply non-load related strains (i.e. temperature and shrinkage)
A) Top half expands uniformly
B) Top Left hand section expands uniformly
3) Apply any real external restraints
1A) Apply restraint force to end of top half to return to original position
1B) Slide the top left section so that its left hand end is in the original position. No force required since it is unrestrained at the right
2) and 3) No external restraint to horizontal movement
4) If necessary, apply imaginary forces to each section to restore strain compatibility.
1A) External restraints have already restored strain compatibility, so nothing to do
1B, 2B) Apply force to right hand end of top left section
2A) Apply force to right hand end of top half
3A) Apply forces to both ends of top half
3B) Apply forces to both ends of top left section
5) Apply equal and opposite forces to the composite section to restore equilibrium
1A) Nothing to do
1B, 2B, 3B) Apply restoring force to right hand end of top left section at mid height (i.e. top quarter point at mid span of the composite beam)
3A) Apply restoring force to both ends of top half at mid height (i.e. top quarter point at both ends of the composite beam)
End result:
1A) No rotation or deflection, top half of beam in compression, outward force applied to external restraints at the level of the top quarter point.
1B) No rotation at the ends, but a counter-clockwise moment is applied at the mid point, with resulting reverse curvature, deflections and stresses.
2A) As 1A (no rotation or deflection) except bottom half of beam in tension, and compression in top half reduced. Moment applied to external restraints.
2B) As 1B except stresses, rotations and deflections modified by lack of translational restraint at ends.
3A, 3B) Beam hogs with no nett moment or axial force on any section.
So the main point of all that is: apply the real restraints before messing around with imaginary restraint forces and equal and opposite eqilibrium forces.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end beam flexural stress under differential temperature
Which is of course clockwise in the Northern Hemisphere :)
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end beam flexural stress under differential temperature