Buckling of a simple Plate HELP! 4

[Preach]

Hi,

I am trying to work out the deflection of a plate when at a certain loading force. There are a number of end conditions so a general formual would be much appreciated.

The plate is 3mm x 410mm in cross section and then 1640mm long. The load is appled to the top the far end is definately fixed. Other possible conditions for the top end are guided or pinned - depending on final design.

I just need a simple or complex way of calculating the max deflection (assuming I know material characteristics) at a load less than the critical buckling load.

Thanks

 

[JStephen]

Are the long sides supported? If not, it's a column.

 

[profengmen]

It's always scary when I see these posts. I mean no real disrespect; but, if you don't have a book with these formulas, and you don't know how to apply the correct formula, then why are you doing this?

 

[IRstuff]

Sounds like you should get a copy of Roark and Young

TTFN

 

[Tobalcane]

May be you should start with Shigley's Mechanical Engineering Design first…

Go Mechanical Engineering
Tobalcane

 

[pso311]

I recommend Applied Strength of Materials by Mott. JStephen is correct that this would be a column problem, and you must use the appropriate steps (end fixity, radius of gyration,..) to find whether this is a "long" (Euler equation) or "short" (Johnson equation)column.

 

[JStephen]

It's a column problem only if the long sides are unsupported.

If the long sides are supported, you'll probably find a buckling load in Roark & Young, but won't find anything giving the deflection of a semi-buckled shape.

 

[Preach]

Thanks for your help, the design is a column - unsupported sides. Both ends are clamped (fixed) and the force is concentric (800N). I can work out the critical buckling force but have found little guidance as to work out the max deflection at a force lower than critical. Roark and Young and a number of other mech eng books are fine for calcualting loads but offer little insight into the deflection.

I have only recently joined a design company - and hence want to get the calcs right. Apologies for any offense caused to those seeing it as a trivial question.

 

[zekeman]

For the concentric load and built-in end condition you describe, there is no deflection for loads less than critical. This can be seen from the solution to the fundamental differential equation, namely
Y=d(1-cospx)
where d is the offset at the point of application and
p=sqrt(P/E)
Y= deflection
P=load
E= modulus of elasticity
I= moment of inertia of crossection
Since Y=d at Y=L i.e. Y(L)=d, cospL must be 0 which leads to
pL=(2n+1)*pi/2
the smallest P would coincide with n=0 and is the critical value.
If P is any other value less than critical, cospL would be different from zero and therefore the only remaining possibility is that d=0 i.e. no deflection.

 

[bellman]

Zekeman is correct. Once the column starts to deflect, you have reached critical loading, and your load is no longer concentric. In technical terms, fall down, go boom may soon follow.

 

[JStephen]

Once the plate starts deflecting laterally, it should still be possible to calculate deflection as a function of force.

You can envision the problem as stringing a bow, where the bow is initially straight. In the tank world, come-alongs are used to bend steel plate in a similar manner.

The problem is that the moment cannot be easily integrated along the length of the plate because it is a function of the deflection.

A similar problem is that of beams with combined axial and transverse loads. This is discussed in Roark and Young, but the information given there assumes that the transverse load is always non-zero, so there's no way to get a solution for the column effect only.

And FYI, note that when a plate bends like a beam, the stiffness varies slightly from that of a beam due to restraint of transverse strain.

 

[prex]

See thread727-102600 for the discussion of a similar problem.
Unfortunately I have no ready to work solutions for the clamped ends condition, only hinged ends.
And JStephen the deflection would not be calculated as a function of force, but only as a function of end axial displacement: as what we are talking about is a post buckling condition, the load cannot increase beyond the buckling value (or failure would occur).

prex

http://www.xcalcs.com

Online tools for structural design

 

[Preach]

Thanks for all the help.

preach

 

[JStephen]

prex, I assume the post-buckled load is LESS than the buckling load, but still varies with displacement. The more you bow the plate, the more force it requires to do it.