Plane strain bending
Plane strain bending
(OP)
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?





RE: Plane strain bending
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.
RE: Plane strain bending
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?
RE: Plane strain bending
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.
RE: Plane strain bending
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
RE: Plane strain bending
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....
RE: Plane strain bending
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.
RE: Plane strain bending
corus
http://www.corusresearch.com
RE: Plane strain bending
(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.)
RE: Plane strain bending
your integral is incorrect, it should be σzxdA or σzydA (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
RE: Plane strain bending
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
RE: Plane strain bending
My proposal was to adopt T2-T1=1.5(∫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
RE: Plane strain bending
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
RE: Plane strain bending
corus