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# Shear internal forces in Kirchoff element plate

## Shear internal forces in Kirchoff element plate

(OP)
Hi,
I am trying to calculate shear internal forcesin the plate (Kirchoff triangle elements with three nodes), but I do not know how to get them.
I meshed simple rectangular plate with triangles,
I solved global equation system, so I have displacements in each node (vertical movement and two rotation angles).
Then I got bending moments (m=-D*B*q). (displacements and moments are correct - I compared with commercial FEM software)
But how to get shear forces in nodes?
Anyone could send a clue?

### RE: Shear internal forces in Kirchoff element plate

Well first off, your software isn't giving you any output (i.e. shear stresses, strains, reactions)? Secondly, what are "Kirchoff triangle elements with three nodes"? If you meshed a Kirchhoff [quad] plate to get CST elements.....you went from bad to worse: a pure Kirchhoff plate is built around the assumption of no transverse shear. And a pure CST element is terrible for out of plane forces. (Can't be used.)

All this is assuming your software is not doing something it isn't showing you. For example, with STAAD, it has 3 noded triangle elements.....but you can tell there are some shape functions being used that are not consistent with a CST.

I'd go back to the single quad and mesh it (into smaller quads). The shear at supports should be easy enough to estimate.....if the software is indeed using a "pure" Kirchhoff plate (which I would think is unlikely at this point), you can figure the shears at intermediate locations with the equations of equilibrium. (Assuming you have displacements, moments, etc from the software.)

### RE: Shear internal forces in Kirchoff element plate

(OP)
Hi WARose.
There is no software. I am trying to write my own software - I wrote triangular mesher on the region, and that is why I am trying to use triangle element based on Kirchoff's theory.
Quad meshing is too tricky for me.
I am using shape functions from CKZ triangle. In FEM (Kirchoff's theory) shear is not taken into account, as it does not have impact on displacement.
However there are shear forces in the slab, which I am trying to get.
I am looking at bending of very thin slabs, so CKZ triangles are enough for me. My plate is loaded perpendiculary to the surface, so there are no membrane forces.
I have displacements, and moments which are correct, but I do not know how to get shear forces.
Could you advise how can I get them using displacements, moments?
My idea was to get shear forces from formulas (which I found in internet)
Vx=(dmx/dx)+(dmxy/dy)
Vy=(dmy/dy)+(dmxy/dx)
dm/dx, dm/dy,dmxy/dy, dmxy/dx I calculated doing derivation of matrix B (strain displacement matrix)
But I have wrong results, so I am doing something wrong.
If you know any book, publication, with explanation, how to get shear forces, I would be grateful.
Regards
Kris

### RE: Shear internal forces in Kirchoff element plate

#### Quote:

Could you advise how can I get them [i.e. shear forces] using displacements, moments?

Simple statics. With one piece at a time. For a beam element for example, you can take any segment, sum forces about one end, and it's simple algebra to arrive at the shear. (If you have the correct moments.)

EDIT: Another approach might be formula:

qx= K * ((1-v)/2) * λ2 * (θx + wx)

where:

qx= shear force for a particular axis (x in this case)

K= E*h3/(12*(1-v2))

E= Modulus of Elasticity

h= slab thickness

v= Poisson's ratio

λ2= 10/(h2)

θx= rotation about a particular axis (at a point)

wx= deflection

I found this formula in 'Structural Analysis with Finite Elements', by: Hartmann, et al. (2004), p.344

Seems to be based on the Reissner-Mindlin plate theory.

### RE: Shear internal forces in Kirchoff element plate

I edited my last post to add a formula. Hopefully it will be useful.

Be careful though: I seem to recall Reissner-Mindlin plate theory giving spurious shears at certain thicknesses.

### RE: Shear internal forces in Kirchoff element plate

(OP)
Thank you WARrose,
I found this book very useful.
There are formulas on p.327 for Kirchoff plates, and they seem to be the same as those I used.
I will then try to search for mistakes I made during calculations of derivation of matrix B.
Regards,
kris

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