The finite element method
Then, axial stiffness matrix is given as when applied equilibrium equation at node: 14
𝐸𝐴 𝐿
𝑥 1 𝑥 2
𝐹 𝑥1 𝐹 𝑥2
1 −1
{
}=
[
]{
}
−1 1
And rotational stiffness matrix is given as: 15
𝐹 𝑦1 𝑀 1 𝐹 𝑦2 𝑀 2
𝑦 1 𝜃 1 𝑦 2 𝜃 2
12 6𝐿 6𝐿 4𝐿 2
−12 6𝐿 −6𝐿 2𝐿 2
𝐸 𝐿 3
{
}=
[
]{
}
−12 −6𝐿 6𝐿 2𝐿 2
12 −6𝐿 −6𝐿 4𝐿 2
Combining the previous matrices gives us the frame matrix of the beam: 16
𝐴 0 0 0 12𝐼 𝐿 2 6𝐼 𝐿 0 6𝐼 𝐿 4𝐼
−𝐴 0 0 0 −12𝐼 𝐿 2 6𝐼 𝐿 0 −6𝐼 𝐿 2𝐼 𝐴 0 0 0 12𝐼 𝐿 2 −6𝐼 𝐿 0 −6𝐼 𝐿
𝐹 𝑥1 𝐹 𝑦1 𝑀 1
𝑥 1 𝑦 1 𝜃 1 𝑥 2 𝑦 2 𝜃 2 }
𝐸 𝐿
=
𝐹 𝑥2 𝐹 𝑦2 𝑀 2 }
−𝐴 0 0 0 −12𝐼 𝐿 2 −6𝐼 𝐿 0 6𝐼 𝐿 2𝐼
{
{
4𝐼 ]
[
These frame matrices of the individual element will then be inputted into a global matrix. However, the process of inputting each stiffness matrix into the global system can be intricate. This is different from the combination of axial and rotational matrix as each beam will be connected to make up the whole mesh. 17 The global matrix is determined according to the total number of degrees of freedom at each node. But as beams are connected at nodes, continuity states that as two (or more) beams share a same node, the displacement for both nodes would be the same. 18 Hence, some elements in the global matrix that refer to the individual nodal values will be affected by the frame matrices of several elements in the discrete model. After finishing the assembling of the global matrix from the frame matrix of each individual node, there is still one last step before passing the problem onto computation to solve the system of equations: setting up boundary conditions. Usually, there are two types of boundary condition in FEM: Dirichlet and Neumann. In static conditions, displacement of given points is dependent on Dirichlet boundary conditions while stresses and strains of the structure is guided by Neumann boundary conditions. 19 14 Sevier, M. Stiffness matrix for a frame element in finite element analysis . https://www.youtube.com/watch?v=jij28UFNfGw. Consulted: 10/8/2024. See also Abbasi, N. Derivation Of The Beam Sti ff ness Matrix. https://www.12000.org/my_notes/stiffness_matrix/stiffness_matrix_report.htm. Consulted: 10/8/2024. 15 Juricek, L. & J. Kveton. Rotational stiffness – the direct stiffness approach. https://www.ideastatica.com/blog/rotational-stiffness-the-direct-stiffness- approach#:~:text=The%20stiffness%20matrix%20governs%20the,matrix%20with%20dimensions%20of%206 x6. Consulted: 10/8/2024. See also Sevier (n. 14) and Abbasi (n. 14).
16 Sevier (n. 14). 17 Dean (n. 12).
18 The Efficient Engineer (n. 7). 19 What Are Boundary Conditions?
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