How to Calculate Thin Plate Deflection for a Point Load at the Center?

[damo03]

Hi. I have been doing some FEA modelling with solid works and am trying to calculate my deflection for a point load at the centre of a clamped periphery (not simply supported) circular thin plate. I need to calculate the theoretical values to make sure that my FEA is correct

The formula I have found is this :

w=(-W/16pieD)*(a^2-r^2*(1+2*ln(a/r)))
for r not = to 0

w= deflection
W= load in N
a= fixed max radius (m)
r = variable radius (m)

D=flexual rigidity = Eh^3/(1-v^2)
E=Young's modulus (Pa)
h=plate thickness (m)
v=poissons ratio

When I plug my relevant data into the formula I get stuck because i am using a point load at the centre...therefore my r=0...I cannot find the formula for when the load is at the centre. Can anyone please help?

I have been having no trouble doing this with a distruited load (pressure) but it's the point load that I have been having trouble with.

Thanks in advance.

 

[minger]

You will get a problem using concentrated analytic loadings with erroneously high answers. Roark says in Chapter 11: Flat Plates, Section 1: Common Case:

Concentrated Loading
It will be noted that all formulas for maximum stress due to a load applied over a small area give very high values when the radius of the loaded area approaches zero. Analysis by a more precise method (Ref 12) shows that the actual maximum stress produced by a load concentrated on a very small area of radius $$r_0$$ can be found by replacing the actual $$r_0$$ by a so-called equivalent radius $$r'_0$$, which depends largely upon the thickness of the plate $$t$$ and to a lesser degree on its least transverse dimension. Holl (Ref. 13) shows how $$r'_0$$ varies with the width of a flat plate. Westergaard (Ref. 14) gives an approximate expression for this equivalent radius:

$$r'_0 = \sqrt{ 1.6 r^2_0 + t^2} - 0.675t$$

This formula which applies to a plate of any form, may be used for all values of $$r_0$$ less than $$0.5t$$; for larger values the actual $$r_0$$ may be used.

Use of the equivalent radius makes possible the calculation of the finite maximum stresses produced by a (nominal) point loading whereas the ordinary forumula would indicate that these stresses were infinite.

So, the application of a concentrated loading physically is erroneous. You can try to apply the loading in your FEA as a concentrated surface loading over a finite area. Then use the formula given to get an equivalent radius, thereby which you can get the stresses and deflections.

For uniform loading over a very small central circular area of radius r0, those are Roark cases 16 and 17 depending on the boundary conditions. I can supply those if you would like. I think you have case 17 though (edges fixed rather than simply supported). In that case, the maximum deflection at r=0 is:

$$y_{max} = \frac{ -W a^2}{16 \pi D}$$

Where:

$$W = q \pi r^2_0$$

q being the "pressure", and a being the radius of the flat plate.

 

[damo03]

thanks for that minger. That solves my problem. :)

 

[minger]

Glad I could help.

 

[smentos]

Damo03,

Where did you find that formula you first were trying? Could you give me a reference, as I am doing analysis on a microplate and am only finding things for calculating the nodal frequencies.

That could really be useful, although seems to only give the deflection for one state.