Equation Needed 1

[Violator]

I am trying to find an equation that will give me the deflection of a plate that is held at its Center of Gravity (CG). The plate 1"x8'x12' is hung from a cable and what I am trying to figure out is the deflection. Given that the ends are free and the only weight is its own what equation can I use or manipulate others to get the oveall deflection. Needs some help. Violator

 

[Curmudgeon]

Sounds like you should be using cantilever beam equation. Fairly simple stuff. Unless you've got other things going on.

Curmudgeon

 

[Qshake]

Visit the age old text Roarks Formulas for stress and strain. Check out the section on plates and try to fit a case to yours. You may have to be inventive, but there are many, many, cases available

 

[prex]

Unfortunately you'll find no help in the Roark: only distributed loads and supported sides.
The closest results to your problem I have found are for a square plate, so I hope this will be sufficient for you to guess what you want.
Now for a square plate of sides a with 4 supported corners and a distributed load P=qa² the center deflection and the deflection at mid sides are respectively:
f_c=0.025Pa²/D
f_s=0.017Pa²/D
where as usual D=Et³/12(1-ni²)
If the load is concentrated at the center the above deflections become:
f_c=0.039Pa²/D
f_s=0.022Pa²/D

The original reference for the above results is a german book published by Springer in 1924. The author is H. Marcus.

Now, by superposition, you'll find that the deflection at the corners and at mid sides (both with respect to center) are (of course the corners are now unsupported, as the same total load is used in the superposition):
f_c=0.014Pa²/D
f_s=0.009Pa²/D

I guess that to find the full solution to your problem with Fourier series shouldn't be too difficult, but a non negligible effort would be required.
prex
motori@xcalcsREMOVE.com

http://www.xcalcs.com

Online tools for structural design

 

[LPPE]

I agree with Curmudgeon. Sounds like a cantilever deflection equation to me with uniformly distributed load (self weight).

Deflection = wL^4 / 8EI

which would occur at the free end.

 

[Conex]

I agree with Curmudgeon and pylko, simple cantilever deflection theory. Consider half the plate as a cantilever with the central support as the fixed end. Beam length L is then half the plate length (or distance from support to end if not equal distance each side of support).

 

[Qshake]

I understand the tendency to reduce the evaluation but considering the middle part of the plate as fixed seems unrealistic. Depending on the support conditions (hung from a cable) the rotations at the support locations will serve to increase the end deflections.

I suggest that you look into what PREX has noted or work out a solution from plate theory.

 

[prex]

Just to be a bit academic, if you work out the proposed cantilever scheme, using a square plate of sides a as in my previous post, you'll find that the end deflection may be written as
f=Pa²/128D(1-D²)=0.0086Pa²/D.
So Curmudgeon, pylko and Conex can see that the beam theory underestimates deflections quite a bit (as Qshake intended to say, of course).
As always, it all depends on what Violator is looking for: a rule of thumb? Beam theory is OK. A check for measured values (different at corners and at mid sides)? Then plate theory is a must.
prex
motori@xcalcsREMOVE.com

http://www.xcalcs.com

Online tools for structural design

 

[Violator]

What I need is the deflection from the center to the ends. The reason is that I need to see how much bowing is ocurring during the lifting and how much compression and tension is being created during this action. I ultimately need to find the impact force generated during the lifting. I really want to thank all of you because y'all have been a great help, just curious where are all of you from STATE, CITY. I am from Downey, CA. Once again thanks.

 

[Imagineer]

Why would there be an impact force? Do you plan on dropping this?
Ok, now I'm really curious. What are you planning on doing? Could you describe your project?
Imagineer

 

[Violator]

What I intend on doing is calculating how much force is generated on threads, that are located at the CG of a plate 12'x 8' x 1". This force is to be generated by the plates own weight, the plate is on a 1' cable (the cable length will play no major factor since the downward force is at a point) which is to be hanging from a crane. But I also needed to know how much deflection is actually seen on the plate before the jerking is to performed. Ultimately I need to know the force generated, so if anyone knows how to find this we can start there. Tha Violator

 

[RiBeneke]

It seems you have bi-axial bending, with a single support point. I do not know of simple equations to solve this.

It should be quite easy to get it modelled by someone with finite element analysis software. The model is simple to create, and the modelled support node must be located just above the plate for stability.