Beam displacement 1

[marmilew]

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 job (3).pdf?dl=0

I will appreaciate any help

 

[KootK]

Superimpose two load cases on a simply supported beam model:

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.

 

[rb1957]

i don't think you can use superposition to solve a redundant problem.

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 ?

 

[KootK]

When all is said and done, you're left with an elastic, simple span beam subjected to a point load and an end moment. Nothing fancier. Load history and plasticity no longer enter into the equation. Of course, OP must be sure that the load history is such that one, and only one, plastic hinge has formed in the beam. I took that to be a given condition of the problem.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[KootK]

After the hinge has formed, I believe it to be a statically determinate problem.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[rb1957]

true enough, after the plastic hinge forms i think it is determinate.

another day in paradise, or is paradise one day closer ?

 

[rb1957]

so you're saying it'd be a SS beam with a load and a plastic hinge at the mid-span ?

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 ?

 

[KootK]

so you're saying it'd be a SS beam with a load and a plastic hinge at the mid-span ?

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.

but the LH portion is cantilevered ? i don't think you can superimpose the fixed end moment ... what moment to use ??

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.

 

[rb1957]

oh, sorry i assumed the hinge would be at the load, but thinking about it for a second, the max moment is at the support.

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 ?

 

[KootK]

No worries. Yes, a simple span beam. And I still contend that superposition applies to the two loads present on that simple span beam. Superposition is valid so long as you're linear elastic. Determinacy is irrelevant. Post hinge, I believe that simple span beam to indeed be linear elastic.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[rb1957]

i'm not seeing the superposition ...

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 ?

 

[rb1957]

that's an interesting thought ... the max moment for a SS span is higher than the FEM of the proped cantilever.

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 ?

 

[KootK]

as the fixed end starts to yield the moment reacted will fall as the beam approaches a SS span.

The end moment doesn't fall. Rather, it remains constant at the plastic value (Zx x Fy).

the max moment for a SS span is higher than the FEM of the proped cantilever.

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.

so it'd start to yield at the wall, but later a second hinge would develop at the mid-span ?

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.

 

[rb1957]

i can buy that.

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 ?

 

[KootK]

I think that we're in agreement now rb1957. And I do think that we've helped OP. We've reduced his problem to something very simple and manageable. Short of spoon feeding him the particular equations for the two cases to be superimposed, I think that our work here is done.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[Ingenuity]

Short of spoon feeding him the particular equations for the two cases to be superimposed, I think that our work here is done.

...and I trust the OP will get an 'A' on this homework...

 

[BAretired]

When the load P is gradually increased until a plastic hinge forms at Point A, the beam is no longer in the elastic range. The outer fibers of sections adjacent to Point A are stressed beyond the yield point, so an elastic solution is not accurate.

Even before M[sub]A[/sub] reaches full plasticity, M[sub]B[/sub] 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

 

[KootK]

e load P is gradually increased until a plastic hinge forms at Point A, the beam is no longer in the elastic range. The outer fibers of sections adjacent to Point A are stressed beyond the yield point, so an elastic solution is not accurate.

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.

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.

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.

 

[BAretired]

KootK, you cannot have full plasticity at Point A and full elasticity elsewhere. The calculation of deflection after the onset of plasticity must take into account plastic deformation in portions of the beam which are only partially elastic. I suspect the solution is not simple but could likely be found with the proper theory.

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

 

[KootK]

KootK, you cannot have full plasticity at Point A and full elasticity elsewhere.

You certainly can if that condition is part of the problem statement BA. Binary, on/off plasticity is precisely what this means:

I use elastic perfectly plastic material model.

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.