Each of the squares or rectangular elements is made of different materials with different Young's Modulus (see the picture below). I can compute the 8 x 8 stiffness matrix for each of the elements. I need an explanation on how to compute the global stiffness matrix in order to be able to calculate the displacements on each node if I apply the force boundary condition. Thank you.
Hi Samuel,
When you develop stiffness matrices for each zone, you are traditionally writing forces on the zone in function of the nodal displacements.
The nodal displacements are shared by adjacent zones, so you have a compatibility requirement there. (displacements being the same)
Furthermore, you have an equilibrium requirement for the nodes. This means that all the different forces acting on the nodes through the different zones should be in equilibrium.
In general, this means you simply add the constituent stiffness matrices when combining them in the global stiffness matrix, while taking into account the node coordinates.
(note that 'simply adding' depends on your sign convention. You should check this by calculating equilibrium equations by hand for a single node).
See a slightly lengthier discussion in section 3.1.2 of my dissertation "Towards reliability-based structural fire safety: development and probabilistic applications of a
direct stiffness method..." uploaded on Researchgate.
It may relate to fire (and my stiffness matrices have only 6x6 components), but every element has a different stiffness as in your problem.
Hope this helps.
Best regards,
Ruben
My thought is that finite element modeling is a method to transfer the actual problem to a mathematical one. So as you assembly the stiffness matrices of each element into the global stiffness matrix, it becomes a group of equations.
Hello Samuel Ayinde
Your question is kind of surprising. What you are investigating is a standard problem in finite element analysis where you have an irregular mesh with variable material properties. What you need is a mesh generator to generate the required mesh. By virtue of the rectangular element shapes mesh generation is an elementary problem. Next your FE software should be able to assign different material properties to the elements. Then the assembly of the global stiffness matrix will proceed as usual with each element stiffness matrix being computed from
Ke = ∫BTDiB d(vol)
where Di is the D-matrix for the ith element. The presence of different material properties does not affect the procedure. But beware! The type of mesh you used with narrow rectangular elements will not give accurate results. For accurate solution the elements should be approximately square near the points of load application. In that case the mesh would have been finer than that the one you used. THe FE analysis of solids with narrow material inclusions can be quite expensive because of this requirement of square elements.
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