Considering a 2-nodes beam element, having 3 dofs per node, making it 6 per element, this element will be studied according to a M+N model, where both bending and traction would be taken into consideration.
As mentioned, each node has 3 dofs :
- u : x-axis translation
- v : y-axis translation
- theta : section rotation
For this 2 nodes element, I found the representation of the stiffness matrix for both bending (Km) and traction (Kn).
The element's stiffness matrix would be a sum of the two.
My question is related to the 3-nodes version of the element, being a plane shell, having the same dofs, 3 per node, 9 per element.
What would be the representation of the element's stiffness matrix in this case ?
It's very exciting that you are going to build an FEM Model from scratch! The stiffness matrix in your case is simply: $$ K_m+K_n $$ But this stiffness matrix only applies to each edge's local coordinate system respectively, while the variables shown in the triangle are inevitable in a global coordinate system.
A conventional routine would be:
(1) first construct the local stiffness matrix (6 DOF per node) matrix per edge：
But don't forget to transform it into a global coordinate system by $$ K=\mathbf{v}^T k' \mathbf{v} $$ where the $12\times 12 $ matrix $\mathbf{v}$ represents the transformation according to the edge's direction.
(2) Virtually construct a full matrix $A$ whose dimension is $3N\times 3N$ (truss) or $6N \times 6N$ (frame), where $N$ denotes the number of nodes. Then add every edge's local matrix onto $A$ (the nodal index in the full matrix should be consistent with that in local matrix).
(3) distinguish the free nodal displacements $\mathbf{x}_a$ from the constrained ones $\mathbf{x}_c$ according to boundary conditions. Then $A$ can be partitioned into: $$ \begin{bmatrix} A_{aa} & A_{ac} \\ A_{ac} & A_{cc} \end{bmatrix} \begin{bmatrix} \mathbf{x}_{a} \\ \mathbf{x}_{c} \end{bmatrix} = \begin{bmatrix} F_{a} \\ F_{c} \end{bmatrix} $$
It's a magic if you get a full rank square matrix $A_{aa}$. Then solve
$$ A_{aa} \mathbf{x}_{a} = F_{a} $$
to find out the unknown displacements $\mathbf{x}_{a}$.
(4) The displacement $\mathbf{x}_{a}$ in the last step is represented in a global coordinate system. One need to go back to each edge's local coordinate system to compute its axis forces and bending moments.