Beam displacement
Beam displacement
(OP)
Hello
Is there moderately easy way for hand-calcuations of the beam dispacement after plasticity occurs? My model is beam clamped on left and simply supported on right side. Force P in the middle of beam and plastic moment is Mpl (plastic resistance). I use elastic perfectly plastic material model. I used unit-load method to determi displacement in the middle of beam but there is a big diffrence with FEM results.
My calculations and model are in attachment or under link:
https://www.dropbox.com/s/5vzo994gjtrzwli/Scan%20j...
I will appreaciate any help
Is there moderately easy way for hand-calcuations of the beam dispacement after plasticity occurs? My model is beam clamped on left and simply supported on right side. Force P in the middle of beam and plastic moment is Mpl (plastic resistance). I use elastic perfectly plastic material model. I used unit-load method to determi displacement in the middle of beam but there is a big diffrence with FEM results.
My calculations and model are in attachment or under link:
https://www.dropbox.com/s/5vzo994gjtrzwli/Scan%20j...
I will appreaciate any help






RE: Beam displacement
1) The concentrated load.
2) Mpl as an end moment.
Use your favourite method for each case and add the results.
I think that should do it.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
a propped cantilever with a plastic hinge ... sounds like a standard plastic hinge problem ?
i think your unit-load solution is good up to the onset of plasticity. after the plastic hinge develops i think you have to approach the problem differently.
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
but the LH portion is cantilevered ? i don't think you can superimpose the fixed end moment ... what moment to use ??
maybe you have to use displacement compatibility between the LH cantilever and the RH SS span to determine how much of the load is reacted at the RH (and LH) ends ?
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
No plastic hinge at mid span. After hinge formation, I see it as an elastic, simple span beam subjected to a point load and an end moment equal to the plastic hinge yield moment.
No cantilever. And not the fixed end moment. The moment that you'd superimpose would be the plastic hinge moment (Zx x Fy).
I'm only about 75% confident in my answer here. It's been a while since mechanics of material and I don't see problems like this in practice.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
so then it's a simple SS beam, no?
but back to your 1st post, no superposition, no? ...
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
i see the propped cantilever working as load is applied, developing it's proper FE moment and it's proper reactions (as calc'd by unit load). as the fixed end starts to yield the moment reacted will fall as the beam approaches a SS span.
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
so it'd start to yield at the wall, but later a second hinge would develop at the mid-span ?
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
The end moment doesn't fall. Rather, it remains constant at the plastic value (Zx x Fy).
The end moment will be the maximum moment in the beam right up until the midspan plastic hinge forms. Then those two moments will be equal and the beam would collapse.
Yes, that's how I see it. Of course, once the second hinge forms, there would be a collapse mechanism and there would be no need for deflection calcs.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
but have we helped the OP ? now, i can see your superposition ... the FE moment is the maximum elastic moment (rather than the moment due to the reduncancy, so unit load doesn't work). so it should be a simple matter of calc'ing displacements due to SS beam with a point load and a SS beam with a moent load at on end.
another day in paradise, or is paradise one day closer ?
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
...and I trust the OP will get an 'A' on this homework...
RE: Beam displacement
Even before MA reaches full plasticity, MB will increase beyond its elastic value 5PL/32 and the mid section of beam will not be fully elastic either.
To calculate deflections of a beam which has reached full plasticity at Point A, one must account for the plastic deformation at the end and the middle. Offhand, I don't know how to do that but the method of superposition described above will not yield correct results.
I doubt that this is a homework problem, but I could be wrong.
BA
RE: Beam displacement
My proposal wasn't an "elastic solution" BA. Rather, it pays explicit homage to plasticity in two ways:
1) It caps the end moment rather than letting it increase without bound.
2) It applies elastic treatment to a final stage model with boundary conditions different from the original beam (simply supported versus fixed one end).
Both of these modifications to a classic elastic solution are necessary to adapt it to this plastic problem. Plasticity has not been overlooked.
The OP specified the use of an elastic / perfectly plastic model at the top. The state of interest is full plasticity at point A with full elasticity elsewhere. This is part of what suggests this may be a homework assignment or an FEM software verification exercise.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
Finding deflection of a beam after the onset of plastification is not something we normally do in an engineering practice. I am wondering why it is required.
BA
RE: Beam displacement
You certainly can if that condition is part of the problem statement BA. Binary, on/off plasticity is precisely what this means:
Theoretical exercises aside, the elastic / perfectly plastic simplification has historically been the "go to" model for design problems involving plasticity. While we all acknowledge the phenomenon of distributed plasticity, explicit consideration of it has mostly remained the domain of academia. The one modern exception is some of the fancy seismic work that is being done.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
1) In the solution that I have proposed, I chose to consider binary "point" section plasticity only and ignore the effects of distributed plasticity. Within the bounds of the assumptions that I've made, I believe my proposed solution to be correct.
2) If the OP's FEM model includes distributed plasticity effects, then my proposed solution will serve as merely a lower bound estimate on expected deflection output.
3) Within the realm of what I would consider to be "practical hand calculations", I contend that my solution is both reasonable and valuable as a benchmark.
4) My proposed solution ought to yield the same answer as the unit load method given that both are correctly applied. To my knowledge, the unit load method would also not account for distributed plasticity. My method might serve as a check that the OP's unit load calculation was done properly.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
If a plastic hinge has formed at Point A, the sections immediately adjacent to point A will be nearly plastic with only a small portion of the web behaving elastically; thus the effective elastic section is severely reduced resulting in increased curvature. Sections further away will have a reduction in effective section until the point is reached where the entire section is elastic.
Because of the increased curvature near A, the moment at B will tend to increase beyond the theoretical value of 5PL/32 (83% Mp) which was determined assuming elastic conditions throughout. But even if MB remains at 83% Mp, any section in the immediate vicinity of B will be only partially within the elastic range.
Deflections can be calculated for this ideal material, but it will not be by superposition as described above. Offhand, I would not like to suggest how it could be done.
BA
RE: Beam displacement
@OP: if this is an FEM verification study -- as I suspect -- rerun your FEM with a beam that is long and slender. This will result in more of the beam remaining truly elastic and better correlation to hand calculations that do not account for distributed plasticity.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
"Everything should be simplified as far as possible, but no further".
An approach that should give a reasonable estimate of FEA output deflections:
- Find a bi-linear-plastic moment-curvature diagram for the beam; i.e. linear1 up to first yield, linear2 up to full plasticity, then fully plastic.
- Find the moment distribution along the beam from a linear combination of the simply supported moments and the plastic moment applied at the fixed end.
- Divide the beam into short sections and find the effective EI at the mid-point of each section.
- Find the deflection as suggested by KookT from combination of simply supported deflections, and deflections due to plastic moment at the fixed end.
I think I might give that a go and see how close I get.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
I was using beam elements in the FEA, so it could be that it was using a simplified moment-curvature response as well, but I don't have time for anything more elaborate right now.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Beam displacement
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
I should correct an erroneous statement I made earlier. The formation of a plastic hinge at A will not affect the moment at B unless the load is increased beyond P (the load required to produce Mp at A).
Doug's method of dividing the beam into short sections is good for a computer solution; the OP asked for a moderately easy hand calculation; I would just say that Doug's method could be used in a hand calculation (similar to the Newmark Method of calculating deflections in an elastic beam). It would be prudent to take smaller sections in the immediate vicinity of points A and B.
BA
RE: Beam displacement
Oh no, I'm going to party like it's 1999. Any minute now someone will surface with a contradictory FEM result and I'll be back to eating humble pie.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
» In the former case KootK's simplifications are fully valid.
» In the latter, the complications raised by BA are highly influential.
» IDS proposes a compromise approach, but in a really extreme case (a "+" cross-section with a fat "-" and a thin "|"?) I suspect his approach might fail his own Einstein test.
RE: Beam displacement
RE: Beam displacement
In keeping with the goal of keeping things simple, I think that it would be expeditious to have an upper bound estimate of the deflection as well. If the two solutions suitable bookend the FEM result, then all is well.
I propose that that one could make a simple estimate of deflection assuming that the plastic hinge has a finite length of twice the member depth. This should be manageable as a hand calculation and suffice as an upper bound deflection estimate.
The greatest trick that bond stress ever pulled was convincing the world it didn't exist.
RE: Beam displacement
I looked at a 10 m long beam, fixed at the left hand end and simply supported at the right, with a point load at mid-span, and plotted deflections under the load from just before first yield of the top and bottom surfaces at the fixed support up to near full plastic moment at both the support and mid-span.
The top graph is for an I-beam approx. 300x150 mm with 10 mm flange and 5.5 mm web. The green and blue lines are from a frame analysis with either a linear-perfectly plastic stress-strain curve for the steel (green line), or a moment-curvature curve calculated for the same steel properties (blue line). These two analyses gave near-identical results. The purple line is from a spreadsheet based linear-elastic analysis up to the support moment reaching the full plastic moment, then a simply supported analysis with the plastic reaction moment applied at the left support. This has given pretty good results, up until the load at which yielding at mid span becomes significant.
The bottom graph is for a rectangular beam with approximately the same moment at first yield as the I beam (approx. 300 deep x 25 mm). In this case the red line is from a frame analysis program with calculated moment-curvature diagram. The green line is from a spreadsheet analysis with 10 segments, each with a stiffness based on the same moment-curvature diagram, and the moment at the start of the segment. This has given reasonably good agreement up until the mid-span moment is close to the full plastic moment. The two blue lines are using the applied end moment technique, using the full plastic moment for the light blue line, and the first yield moment for the light blue line. The former gives a reasonably good estimate of the deflection up to about 85% of the maximum load, and the latter gives a conservative upper bound estimate up to about 95% of the maximum load, which is really as good as can be expected for a simple analysis.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Beam displacement
Consider a RECTANGULAR cross-section, width b and depth d. The moment-curvature relationship is linear until the curvature reaches
Cy = 2σy/(dE)
at which point the bending moment is
My = (bd2/6) * σy
At larger curvatures, it can be shown that the moment M when the curvature is C is given by
M = (bd2/4)*σy - b/(3E2C2)*σy3
This can be made simpler and more meaningful by using Cy and My to achieve non-dimensional measures for the curvature and the moment. Define
Cn = C/Cy as the non-dimensional curvature
and
Mn = M/My as the non-dimensional bending moment.
The above moment-curvature relationship then becomes
Mn = 3/2 - 1/(2Cn2)
Now consider a DIAMOND-SHAPED cross-section, width b and depth d. With this shape the width of the cross-section decreases linearly with distance from the neutral axis. The algebra is slightly more voluminous than for the rectangular case, but leads to the following results:
Cy = 2σy/(dE) (same as before)
My = (bd2/24) * σy
Mn = 2 - (2Cn-1)/Cn3
Now consider an HOURGLASS-SHAPED cross-section, width b and depth d. (This shape comprises a downwards-pointing isosceles triangle sitting atop an upwards pointing, congruent, isosceles triangle.) With this shape the width of the cross-section increases linearly with distance from the neutral axis. The algebra now leads to the following results:
Cy = 2σy/(dE) (same as before)
My = (3bd2/24) * σy
Mn = 4/3 - 1/(3Cn3)
The shapes of the three resulting moment-curvature graphs are shown in the attachment.