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# Simple Question About Shell constituitve ( ABD) matrix

## Simple Question About Shell constituitve ( ABD) matrix

(OP)
Ok , as I am reading here , on page 15 I see

but the matrix is defined 6 * 6 and I think in shell with one layer we shoul not talk about ABD so ?

but here in this paper

http://www.enu.kz/repository/2010/AIAA-2010-2706.p...

on page 4 , the ABD matrix is 8 * 8

what is taken into consideration and why is it called ABD here then ??!!

### RE: Simple Question About Shell constituitve ( ABD) matrix

An ABD matrix can be used to describe, in a rather limited way, any material whether it has layers or not. One uniform ply of a fibrous composite will have a B-matrix which is all zero but should have useful A and D matrixes. In A these include interactions between X and Y extensions (Poisson's effects) and between extension and shear, and the equivalent bending behaviors in D. However, with a single ply there is no difference between any two positions at different thicknesses so there can't be any bending as a result of stretching, so B11, and all of B in fact, is zero.

However, the matrix on page 32-15 is not an ABD matrix. It appears to be a matrix similar to an A matrix with terms in it for through-thickness shears, sigmayz and sigmazx. There are also extra terms for sigmaxy involving a '4' subscript. I'm not entirely sure how to interpret this; it's quite a complicated document. However, it is clear that the through-thickness shears (yz and zx) are regarded as not interacting with the in-plane stresses, though they can interact with each other via the E56 term.

One of the simplifications in the usual ABD matrix is that is misses out references to through-thickness stresses, which also makes it impossible to describe modification of behavior due to any curvature. This is presumably partly why, in the Kazakhstan ref., the 8x8 matrix has been extended by two rows and columns to include through-thickness shears Qx and y represented by A44 and A55 (this still won't account for any curvature because there is no representation of through-thickness direct stress). In this case the through-thickness shears do not interact with each other or the in-plane stresses.

I've previously seen through-thickness shears handled by a separate shear matrix, as per ESDU 89013 ("Transverse (through-the-thickness) shear stiffnesses of fibre reinforced composite laminated plates"). Apart from the additional Q shears which are assumed not to interact with other directions, their 8x8 matrix is composed of classical A, B and D matrixes. In fact I'm not sure why the Qs are included in the same matrix; you might as well put them in a separate 4x4 matrix as the ESDU data item does.

I hope this helps in spite of the 'holes' in my understanding of what you are doing.

### RE: Simple Question About Shell constituitve ( ABD) matrix

(OP)
This was a great answer , thank you so much !

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