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Bar element stiffness matrix

In the finite element method, a bar is replaced by two nodes and a stiffness matrix that links the nodal displacements to the nodal forces. For a linear 1D bar element the stiffness is EA/LEA/L, just as in ordinary spring theory, but written as a 2×22\times2 matrix that acts on both degrees of freedom.

k=EAL(1−1−11)\mathbf{k} = \dfrac{EA}{L}\begin{pmatrix}1&-1\\-1&1\end{pmatrix}element stiffness matrix, 1D bar element

Symbols

EEmodulus of elasticityPa
AAcross-sectional aream²
LLelement lengthm

Example

E=210E = 210 GPa, A=100A = 100 mm², L=500L = 500 mm:

EA/L=210 000⋅100/500=42 000EA/L = 210\,000\cdot 100/500 = 42\,000 N/mm.

The matrix alone is singular – it only describes stiffness, not how the bar is supported.
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