Stiffness Matrix Calculator — 2-Node Bar/Truss Element
Enter the material stiffness (E), cross-section area (A) and element length (L) to compute the 2×2 local stiffness matrix for a 2-node bar element — the fundamental building block of truss and frame finite element analysis.
GPa
mm²
m
20,000,000 N/m — diagonal entry of the 2×2 element stiffness matrix
Convert inputs to SI base units
Compute axial stiffness k = AE / L
2×2 element stiffness matrix K = k·[1 −1; −1 1]
- 1
Young's modulus in Pa
200 × 10⁹ = 200,000,000,000 - 2
Cross-section area in m²
100 × 10⁻⁶ = 0.0001 - 3
Axial stiffness k = AE ÷ L
0.0001 × 200,000,000,000 ÷ 1 = 20,000,000 - 4
k in kN/m (÷ 1000)
20,000,000 ÷ 1000 = 20,000
How does this calculator work?
For a 2-node bar element, axial stiffness k = AE/L (N/m). The 2×2 stiffness matrix is K = k·[[1,−1],[−1,1]]. The matrix is symmetric and singular (det = 0 due to one rigid-body mode). Fix at least one node to solve for displacements under applied load.
Formula
How this is calculated
In the direct stiffness method (finite element analysis), a 1D bar or truss element with two nodes has two degrees of freedom (axial displacement at each node). The relationship between nodal forces {F} and displacements {u} is {F} = [K]{u}, where [K] is the 2×2 element stiffness matrix. The axial stiffness k = AE/L, so [K] = k·[[1,−1],[−1,1]].
The off-diagonal terms −k enforce compatibility: applying a displacement at node 2 with node 1 fixed induces an equal-and-opposite reaction force at node 1. The matrix is singular (determinant = 0) because the element has one rigid-body mode (pure translation without deformation). In a complete structural model, boundary conditions remove this singularity before solving.
This calculator computes the local stiffness matrix in the element's axial direction. For 2D or 3D truss assemblies you would transform this into global coordinates using the element orientation angle, then assemble all elements into the global stiffness matrix. This simplification assumes linear elastic material behaviour and small deformations (no geometric or material nonlinearity).
Frequently asked questions
A free bar element can translate rigidly without any deformation — this rigid-body mode makes the matrix singular. The physical meaning is that the system has no unique solution until you fix at least one degree of freedom (impose a boundary condition). Assembly into a global model with proper supports removes the singularity.
Structural steel: ~200 GPa, aluminium alloys: ~70 GPa, titanium: ~110 GPa, concrete: ~25–35 GPa, wood (along grain): ~10–15 GPa, rubber: ~0.01–0.1 GPa. Stiffer materials (higher E) resist deformation more and produce larger stiffness values for the same cross-section and length.
Each element contributes its stiffness to the global matrix by adding each K[i,j] entry to the corresponding global degree-of-freedom pair. For a two-element truss with three nodes the global matrix is 3×3: each 2×2 element matrix overlaps at the shared interior node. After assembly, apply boundary conditions, then solve {F} = [K]{u} for unknown displacements.
Also known as
TG we-Calculate Editorial Team. (2026). Stiffness Matrix Calculator — 2-Node Bar/Truss Element [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/stiffness-matrix-calculator
TG we-Calculate Editorial Team. "Stiffness Matrix Calculator — 2-Node Bar/Truss Element." TG we-Calculate. 2026. https://we-calculate.com/calculator/stiffness-matrix-calculator.
TG we-Calculate Editorial Team, "Stiffness Matrix Calculator — 2-Node Bar/Truss Element," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/stiffness-matrix-calculator
@misc{wecalculate_stiffness_matrix_calculator, title = {Stiffness Matrix Calculator — 2-Node Bar/Truss Element}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/stiffness-matrix-calculator}}, year = {2026}, note = {TG we-Calculate} }
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