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Elastic curve
3

Elastic curve

Elastic curve

(OP)
What will be the elastic curve of this beam

RE: Elastic curve

If I read the symbol at "B" correctly, the joint can develop moment and horizontal force, but no vertical.  Correct?  Sorry, but I have not seen this notation before.

Mike McCann
MMC Engineering

RE: Elastic curve

You could treat it as a plain catilever with A fully fixed, a downward point load W at B and an upward point load W at mid-span.   

RE: Elastic curve

Pardon my ignorance but wouldn't this be a mechanism? at least for small deflections?

RE: Elastic curve

(OP)
msquared48
Mike, I also haven't seen this notation before. I just realized doing work from softwares only made me loose o basics of statics. So I  started to study own my own to take the rust off. I am trying to solve this with moment area method. Can you draw out elastic curve for this or try to solve it  ! !
KootenayKid
How can you change this to fixed end etc? I do not buy that

RE: Elastic curve

dhkahn i have great respect for you.  Too many engineers forget the basis of their computer calculations and end up losing the grasp on the real core aspects on which everything else is built.

As with the beam restraint conditions i think they are both s/s roller support allowing lateral movement (no moment at this point due to it being simply supported.

Try drawing out the deflected shapes and approximates of the bending moments to see whats really happening.
lt.

this may shed some light on the problem:

http://stommel.tamu.edu/~esandt/Teach/Summer00/CVEN305/Examples/Set9/example_set9.html


As with the beam restraint conditions i think the one on the left is a s/s roller support allowing lateral movement (no moment at this point due to it being simply supported), and the other end is  fixed end roller support.

Try drawing out the delfected shapes and approximates of the bending moments to see whats really happening.

Please keep us posted on your progress i shall also have a go at finding a solution

RE: Elastic curve

for some reason my old post posted as part of my last post please ignore the text after the web link

PST

RE: Elastic curve

(OP)
Thanks prex.
The spread sheet gave me the shape of elastic curve and after that it was easy. Got all answers matched.

RE: Elastic curve

dgkhan,

You don't buy my method????

Your problem is statically determinate.  You can work out the moment diagram in about 30 seconds flat.  If you work out the moment diagrams for your example, and for the simplification that I suggested, you will find that they are identical.  And, if the moment diagrams are identical, then so are the deflected shapes.

You could have solved this by the superposition of a couple of formulas straight out of the steel manual.



 

RE: Elastic curve

(OP)
Please show me.
End A is pinned so theta A has value and matches the answer.
If I make it fixed than as per your method theta A = 0
How do you justify that?

RE: Elastic curve

dgkhan,

See the sketch attached.  At least I think that I've attached a sketch (I've never done it before).

I reversed the node lettering when I described the method in my intial post.  You are correct of course, point A will have a rotation.

The trick to the simplification is to realise that it is only the relative displacement between ends that matters.  Whether it's the pinned or guided end that does the actual travelling, is irrelevant.

RE: Elastic curve

(OP)
KootenayKid,
Thx.
BMD is same for all these.
But elastic curve for your cantilever will not match the elsatic curve of my problem. In my problem it becomes horizontal ar support B. U can x-check with prex excel sheet above. But in typ cantilever the elsatic slope is sloping away ! !  

RE: Elastic curve

The structure as illustrated by the OP is a mechanism and has no solution.

BA

RE: Elastic curve

(OP)
BAretired

I was able to solve it with moment area method. I was also able to verify my BMD with STAAD once I placed the right D.O.F's

RE: Elastic curve

dgkahn,

In a flexural bending problem, if the moment diagrams are the same, the elastic curve is the same.  There's just no getting around that.

Just imagine: you could also solve the problem by double integration.  And, when you performed that double integration, what you'd be integrating is the moment diagram.  Same moment diagram --> same deflected shape

My method results in zero slope at node B as well.  How could it not, I've got that node completeley fixed??



 

RE: Elastic curve

dgkhan,

I misinterpreted the support at point B.  Using Mike's interpretation, the problem is easy.

BA

RE: Elastic curve

dgkahn,

One more thing: in a typical cantilever, the slope is most definately not sloping away from the support.  It is, by definition, zero slope at the support.

RE: Elastic curve

(OP)
Moment at point B is same (WL/2).
In your beam point B is fixed so not going anywhere.
In my problem this point rolls down and gives deflection.
See ! same BMD but different elastic curve.
  

RE: Elastic curve

dgkhan,

That's my whole point.  It doesn't matter whether point B rolls down or point A tranlates up.  It's only the relative deflection between the beam ends that matters for defining the elastic curve.

If you solve the problem by our respective methods, you will find that my cantilever tip deflection at point A exacly matches your guided end vertical translation point B.  And, more importantly, the slopes and relative deflections will be identical at all points along the beam.
 

RE: Elastic curve

(OP)
KootenayKid

OK I will do it when I have time. Got to do some office work also. To me it looks more of a coincidence like 2x2 = 2+2 but not in general form. The defelction of cantilever tip will not match with guided rotation.

RE: Elastic curve

It is not a coincidence.  The elastic curves are identical except for a rigid body translation.  

 

BA

RE: Elastic curve

Nicely stated BA.  I should have said it that way.  Either way dgkahn, it sounds as though you've got your problems sovled.

RE: Elastic curve

(OP)
KootenayKid,

I checked and answers of your beam are correct.
How did you figure out this beam ... conjugate beam concept. was never a big fan of it. Any way tip of cantilever still point down contrary to guided roller but perhaps that is ignorable difference here.

RE: Elastic curve

dgkahn,

Thanks for going to the trouble of checking.

The solution doesn't use the conjugate beam concept, at least not as I understand it.  The method simply uses superposition of two basic load cases combined with the recognition that my simplified model and your original one are statically equivalent.

If you load the cantilever with an upward force at point A and a downward force at midspan (to match your model exactly), you will find that the cantilevered tip points upwards, just as you would expect it to.  There should be no difference, negligable or otherwise.

That's basically what I showed in the middle diagram of the sketch that I attached to my previous post.  I only flipped the loads around in my third sketch to make it look more familiar.  Perhaps that muddled my explanation rather than clarifying it.

 

RE: Elastic curve

I think I did the same mistake as BAretired.  End B can slide but still support moment, interesting, have never come across one of these supports before at school or out of school.

RE: Elastic curve

I was not familiar with the OP's representation of a "FREE BUT GUIDED" support.  I have seen it represented as a moment with no vertical support.

A simple beam of length 2L with load W placed L/2 from each end is symmetrical and each half has precisely the same elastic curve as the beam in this thread.

BA

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