Its a beam with 3 displacement variables.
I use a linear interpolation function for "u" and Hermit cubic interpolation function for "w" and "fi".
When calculating stiffness matrix I achieve 3 matrices with different dimensions.
2 by 2 matrix for u.
2 by 4 matrix for u*w and u*fi.
4 by 4 Matrix for w*w and fi*w and fi*fi.
and now I need to assemble them. Would you please tell me how to do that or introduce a book which I can reach?
Thanks a million.
Dear Arash imani aria, the stiffness matrix for your beam is 6x6, so there is no problem at all.
Do the assembly.
Could you provide more details about the problem, it is really hard to refer to the target issue.
From you problem, I understand you try to assemble a local stiffness matrix with 3 variables(dofs). So if the problem is 1D, you local stiffness matrix should be 3 by 3, if it is 2D, the matrix should be 6*6 (3by2). If it is 3D, the matrix should be 9*9(3by3).
You choice of interpolation should be related to the shape function order. In general, u = N11*u1 +N12*u2 + N13*u3(or similar form). You can rewrite this with the matrix multiplication form: (I do not know the number of nodes in the elements and dofs)
[U .....N11, 0, 0, N12, 0, 0, N13, 0, 0
W, = [ 0 , N21, 0, 0, N22, 0, 0, N23, 0 ] * [u1, w1, fi1, u2, w2,fi2, u3, w3,fi3]T
Fi]....... 0, 0, N31, 0, 0, N32, 0, 0, N33
I can only try to answer your question, however, I need more details.
Dear Mohannad,
Yes it is 6*6, and I know how to assemble 2*2 (rod) with 4*4 (beam), the problem is a 2*4 matrix which is derived by u*w and u*fi.
Dear Arash imani aria,
For an ordinary beam, there is no relation between u & w in local coordinates. So all the elements of u*w equal to zero. But they are not equal to zero in global coordinates (after transformation).
In your beam, they are already nonzero elements.
The same thing for u*f.
No problem.
Dear Nohannad,
My beam is not an ordinary beam, its an FGM sandwich beam and my u*w terms are not zero.
Dear Arash imani aria,
I said that before, your beam is not an ordinary beam and u*w terms and u*f terms are not zero. So where is the problem ?
I wonder if you check your second matrix (2 by 4 matrix for u*w and u*fi.), it should be 6x6.
Dear Mohannad,
If you look at shape functions for "u" its 2by1 and for w and fi its 4by1, so their multiplication will be 2by4.
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