Finite Element Method for Structural Analysis

The local stiffness matrix $k_{local}$ for an element of length $L$ and area $A$ and Young’s Modulus is given by:

[F_{1} F_{2}] = [k - k - k k] [δ_{1} δ_{2}], k = \frac{E A}{L}

Each element matrix is assembled into the global matrix. Note that if you have 3 elements, there will be 4 nodes so the matrix will be $4 \times 4$ !

For example, if we have 3 elements:

Then we can add the local matrices to the global matrix like this:

For Element 1:

k_{1} - k_{1} 00 - k_{1} k_{1} 00 00000000

For Element 2:

k_{1} - k_{1} 00 - k_{1} k_{1} + k_{2} - k_{2} 0 0 - k_{2} k_{2} 0 0000

For Element 3:

k_{1} - k_{1} 00 - k_{1} k_{1} + k_{2} - k_{2} 0 0 - k_{2} k_{2} + k_{3} - k_{3} 00 - k_{3} k_{3}

If an element is positioned at angle, such as many truss members, we have to use a transformation matrix.

T = cos θ - sin θ 00 sin θ cos θ 00 00 cos θ - sin θ 00 sin θ cos θ

Note that this matrix is $4 \times 4$ as truss members have $4 \times 4$ local matrices (2 degrees of freedom at each node)

A local element stiffness matrix can be transformed to global coordinates can be found by:

T^{T} \cdot [k] \cdot T

This can also be directly done with:

[k] = \frac{E A}{L} cos (θ)^{2} cos (θ) sin (θ) - cos (θ)^{2} - cos (θ) sin (θ) cos (θ) sin (θ) sin (θ)^{2} - cos (θ) sin (θ) - sin (θ)^{2} - cos (θ)^{2} - cos (θ) sin (θ) cos (θ)^{2} cos (θ) sin (θ) - cos (θ) sin (θ) - sin (θ)^{2} cos (θ) sin (θ) sin (θ)^{2}

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