Plane strain bending

[corus]

In a 2D cross-section (defined in the x-y plane) stresses have been calculated from a calculated temperature distribution based upon generalised plane strain across the section. From the general stress distribution how do you calculate the bending deformation of the section as if it had been modelled in 3D? Does this involve calculating the equivalent linear bending moment across the x-x axis?

 

[EdR]

corus:

If I understand the question properly (and I'm not sure I do) the answer is yes....You would have to compute the bending moment from the stress distribution and then use EI y'' = M to integrate for the deflections....This means you would need M (from the stresses) at several points along the length...You also have to think about the value to use for "I" since plane strain implies infinite z dimension...

Hope that helps

Ed.R.

 

[corus]

Ed,
Thanks but I don't think I've explained it properly.
Suppose you have an I beam and there is a general temperature distribution across the cross-section which applies along the full length of the beam so that for any section of the beam the same temperature distribution is seen. The stresses in the beam arise from the temperature differences and as the beam is infinite, plain strain applies. Because of the temperature distribution there is some temperature gradient across the depth of the beam cross-section which will cause bending. For a rectangular cross-section with a simple delta T there is a formula in Roark. For a general temperature distribution and a general shaped section, how do you calculate this bending and hence the arc curvature?
For axi-symmetric shells I've seen methods by which the equivalent linear temperature gradient across the wall thickness is calculated from which the primary bending stress is calculated and separated from the peak stress. Is a similar method applied?

 

[Denial]

It's not all that hard to solve this problem by hand.

Initially assume that the beam is locked totally rigid (restrained against both bending and axial expansion). For this condition the stress at any point in the cross-section is proportional to the temperature at that point. Calculate these stresses. Then numerically integrate them to determine the net axial force in the beam (P), and numerically evaluate their first moment about the section's centroid to determine the net bending moment on the cross-section (M).

However, the beam is not fully restrained but is completely free to deform as it wishes. The force P will relieve itself through axial expansion, and the moment M will relieve itself through the beam taking up a curvature. Simple beam theory (now ignoring everything thermal) relates these action to the resulting deformations.

HTH.

 

[prex]

corus,

Roark's formula is not limited to rectangular sections, but is only applicable for a linear temperature distribution over beam depth.
In my opinion the correct procedure is to first extract from the actual temperature distribution the equivalent linear distribution (possibly you could do this in two orthogonal directions, or choose a transverse direction with respect to beam main directions where the temperature change is max) and then use this equivalent linear gradient to enter Roark's formulae. Of course for an isostatic beam there will be no stresses at all.
Using the stresses calculated with a plain strain model and linearizing them to extract an equivalent bending moment seems to me a less correct procedure, but possibly will give similar results (apart from the influence of Poisson, that is normally not very important): could you adopt both methods and report the results?

prex

http://www.xcalcs.com

Online tools for structural design

 

[EdR]

corus:

Now that I understand better I would add the following comments to what has been said.....

1. Essentially this is a beam subjected to axial load plus constant moment...
2. The stresses we are talking about are the confining stresses (sig-z) which act along the length of the beam...
3. Plot the sig-z stresses over the face and reognise that the stress at any point must be P/A +/- MY/I to conform to conventional beam theory....
4. Use this to establish the bending component of stress and thus the moment as well as the neutral axis (remember the N.A. can move when subjected to axial load..only the bending component is symmetric about the centroid...)
5. Having established the moment use the actual beam end conditions and EI y''=M to find the deflection. (I'd ignore axial load effects at least to start)

The next question is what to do if the stresses don't behave in a linear manner over the x-sect....I'd make the best linear assumption based on the plotted values and write the remaining non-linear behaviour off to the fact that plain strain conditions don't know anything about beam behaviour....If you want better than this will give you you will have to go to a full 3-D analysis.....

Hope this gives you some ideas

Ed.R.

P.S. I did look at Roark and I think his formula's are developed from the same basic ideas outlined above....

 

[corus]

Thanks all,
Fortunately I can afford the luxury of doing the analysis with a 3D model as well as comparing the result from a hand calculation of results from a 2D model. I had thought that Denial's solution was correct but have trouble believing the answer for, say, a hot spot on the edge of the section which is why I thought an equivalent linear temperature gradient/bending moment from the 2D results would be correct. The only way to satisfy my mind is to test it and, to satisfy any one else facing this problem, let you know the outcome.

 

[corus]

Using Denial's method appears to give the wrong answer when looking at results from a 3D model and the resulting curvature. The results from the 3D model give an axial stress distribution that you'd expect from plane strain. If the beam had been assumed to be locked, however, and the stress, Sz=EaT (presumably), integrated across the section as Sz.z.dA doesn't appear to give the moment to give the correct curvature (using an approximate hand calculation). Ed's method appears to be more correct but how do you integrate the axial stresses resulting from plane strain in a 2D model without looking at each separate element, it's area, and it's distance from the neutral axis, without doing it by hand?

corus

http://www.corusresearch.com

 

[Denial]

The approaches taken by EdR and by me should be the same. If they differ, then I expect it is a result of differing assumptions about what is happening to the beam in the direction perpendicular to the plane in which the bending is occurring.

(Similar to the difference between
E*t^3/12
and
E*t^3/[12*(1-n^2)]
in unidirectional slab bending, where n is Poisson's ratio.)

 

[prex]

corus,
your integral is incorrect, it should be &[ignore]sigma[/ignore];_zxdA or &[ignore]sigma[/ignore];_zydA (the two integrals being the same only for a round section or for a square section if the temperature distribution has sufficient symmetry): I propose that you better explain your assumptions for us to evaluate.
The FE code should have commands to make table operations on calculated results: in Ansys this is ETAB if I remember correctly.

prex

http://www.xcalcs.com

Online tools for structural design

 

[corus]

Thanks prex, I meant Sz.y.dA, and not Sz.z.dA, where z is into the paper and the section is defined in the xy plane.
The temperature distribution and shape of the section is only symmetric about the y axis, ie. the bending of the section is about the x-x axis due to asymmetric temperatures about the x axis.

corus

http://www.corusresearch.com

 

[prex]

That a first order approximation of the curvature in your 3D model is obtained by taking M=&[ignore]int[/ignore];&[ignore]sigma[/ignore];_zydA for an unsupported beam is obvious: the point is how much this will be close to the 3D result and whether there are better (first order) approximations.
My proposal was to adopt T₂-T₁=1.5(&[ignore]int[/ignore];T(x,y)ydA)/h where T₂-T₁ is as used in Roark's formulae and h is the section depth along y.
I'm not excluding that this will give a result very close to the method based on M, though a difference should exist due to Poisson effect.

prex

http://www.xcalcs.com

Online tools for structural design

 

[corus]

Apologies to Denial. I checked the integration over a rectangular beam assuming that the stress was 0.5Ea(T1-T2) at the extreme point as given by Roark, and also assuming that the stress was EaT1 at the extremity, both varying linearly over the section. The integral and hence the moment and deflection is the same for both cases.
For a general case where the temperature isn't linear though the method doesn't appear to work and using the equivalent linear temperature, given by prex, would give the reason for the difference, if there was an easy way of calculating it. Unfortunately Abaqus doesn't appear to do that integration. The method would save having to do a full 3D analysis where a 2D plane strain model would do.

corus

http://www.corusresearch.com

 

[corus]

As a follow up question: If a circular section has a general temperature distribution across it, which is not symmetric about a vertical or horizontal plane, how can you find the plane in which bending will occur (due to the equivalent temperature gradient), or alternatively the direction in which the centroid will deflect, assuming the section represents an infinitely long beam?

corus