Stiff plate on elastic supports

[be2]

I´m trying to develop some simple equations for the resulting axial bolt loads for a STIFF plate on elastic supports(=bolts). Question is: Are there any ready made such equations? I looked up my books and the internet but found only really advanced analytical treatments for elastic plates on soft supports such as concrete slabs on soil. I know there are limitations for regarding a plate as being stiff.

I developed a formula for the axial loads on a symmetrical or semi-symmetrical bolt pattern when the plate is loaded by moments and forces, but it doesn´t work for assymetrical cases. I use the same method as for the in plane shear loads on thin plate fittings as described in Bruhn and others,(Bolt number n load is equal to the moment times the bolts radial distance to the bolt pattern cg, divided by the sum of the squares of all bolts radial distances). The derivation of that formula is rather simple and applying it to the axial loads for a semi-symmetrical bolt pattern is not difficult.You just look at the(very thick) plate from the side and use the same equations, only that the "X-distance" comes to play.

The difficulty is when it comes to assymetry since I found I got wrong answers and an unbalanced moment around the symmetry axis. I do bolt calculations at work on rather "stiff" cast iron housings for brakes bolted to trains. I checked the "Bruhn" results with my FEM calcs and found surprisingly good agreement for the in plane (shear) loads in the elastic range. Also the axial loads were close enough for design but only in the symmetrical or semi-symetrical cases.(By semi symmetrical I mean symmetry only on one coordinate axis.

The reason I want to develop some simple equations (easily adaptable to Excel) is that I´m tired of FEM calcs for such matters that really shouldn´t need FEM. Only in cases where I recognize that elasticity plays a major role, or where greatest precision is needed I intend to use FEM. Analytical solutions if they´re not too involved also mean a good check of results.

I live in Sweden, its evening here and when I wake up next morning I expect someone in the US to have worked through this all day to save me a lot of work (just joking).

 

[be2]

Hi again.. I am checking my formulas now. I found that the force on one bolt is a constant times its x-distance to the cg plus another constant times the y-distance (sign conventions important). To find the constants you have to solve a simple two-row equation system involving the sum of the squared x-distances, the sum of the y-distances and the sum of x times y.

If anybody has another method or point of view, or is maybe interested in my results please tell me. You can also tell me if I´m all wrong, but give some reasons why that is so..

I found that it is very important to define the directions and signs of the rotations and forces (I mixed this up a couple of times during the process). I´m surprised that there is no such derivation in the text books. It is simple and can be used in the special case of a stiff plate with relatively "springy" bolts. In the absence of FEM it can be handy to get some idea of the forces on an assymetrical sturdy fitting. Of course, the elasticity of the supporting ground has influence, but we´re talking about reasonable assumptions all the time, whether "exact" FEM calcs or hand calcs.

 

[TERIO]

be2,

I think you have gotten it figured out now that you also have the sum of x times y terms. For the symmetric case the sum of x times y terms would come out to be zero. Is there also a third constant in you equation for the bolt force due to a force normal to the plate?

This problem seems similar to unsymmetrical bending of beams (sum x^2 -> integral of x^2dA, sum y^2 -> integral of y^2dA, sum xy -> integral of xydA, and the plate not bending is like plane sections remain plane). I bet if you looked at a derivation for that you might find it looks pretty similar to what you have.

 

[be2]

Hi Terio, thanks a lot for your comments, very interesting that there is a similarity with assymetrical beam bending, never thought about that.
Yes,if it is an axial force that created some of the moments the direct contribution from it will be divided equally on all bolts, just as in shear load cases with a horizontal and a vertical load in addition to the moment at the cg of the bolt group.
I derived the case for reaction of moments only but thanks for pointing out the axial force contribution, it´s very easy to forget some terms.
This problem destroyed my weekend completely since I found out late Friday afternoon that an excel sheet I proudly had introduced at work (and used in a loads report) didn´t give correct answers. Fortunately the loads report covered only a mildly assymetrical case, so I won´t mention it because nobody really understands my calcs anyway.