The finite element method builds a structure’s behaviour from small elements, each described by a stiffness matrix linking nodal forces to displacements. The matrices for bars and beams follow directly from E, A, I and L.
How to use the Stiffness Matrix Calculator
- Element type
- Young’s modulus E
- Cross-section area A
- Area moment of inertia I
- Element length L
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the element type and enter its properties. The calculator lists every distinct coefficient; place them into the matrix layout shown in the formula.
Inputs explained
| Field | What to enter |
|---|---|
| Element type | Choose one: Bar / truss (2 DOF), Beam (4 DOF), Frame (6 DOF) |
| Young’s modulus E | Units: GPa, MPa, Pa, psi, Mpsi |
| Cross-section area A | Units: mm², cm², m², in² |
| Area moment of inertia I | Units: mm⁴, cm⁴, m⁴, in⁴ |
| Element length L | Units: m, cm, mm, km, in, ft … |
What each result means
- EA/L (axial) = [Young’s modulus E]×[Cross-section area A]÷[Element length L]
- 12EI/L³ = 12×[Young’s modulus E]×[Area moment of inertia I]÷[Element length L]^3
- 6EI/L² = 6×[Young’s modulus E]×[Area moment of inertia I]÷[Element length L]^2
- 4EI/L = 4×[Young’s modulus E]×[Area moment of inertia I]÷[Element length L]
- 2EI/L = 2×[Young’s modulus E]×[Area moment of inertia I]÷[Element length L]
Example values
How the main result changes when Young’s modulus E changes and everything else stays at the example values above:
| Young’s modulus E (GPa) | EA/L (axial) (N/m) |
|---|---|
| 100 | 125,000,000 |
| 150 | 187,500,000 |
| 200 | 250,000,000 |
| 250 | 312,500,000 |
| 300 | 375,000,000 |
| 400 | 500,000,000 |
Formula
Bar: k = (EA/L)·[[1, −1], [−1, 1]]
Beam: k = (EI/L³)·[[12, 6L, −12, 6L], [6L, 4L², −6L, 2L²], [−12, −6L, 12, −6L], [6L, 2L², −6L, 4L²]]
Frame: the bar terms on axial DOFs plus the beam terms on transverse and rotational DOFs.
Worked example
Inputs — Element type: Frame (6 DOF); Young's modulus E: 200 GPa; Cross-section area A: 5,000 mm²; Area moment of inertia I: 50,000,000 mm⁴; Element length L: 4 m.
Results — EA/L (axial) = 250,000,000 N/m; 12EI/L³ = 1,875,000 N/m; 6EI/L² = 3,750,000 N; 4EI/L = 10,000,000 N·m; 2EI/L = 5,000,000 N·m.
Key terms
- Degree of freedom
- An independent displacement or rotation at a node.
Frequently asked questions
What is a stiffness matrix?
It relates nodal forces to nodal displacements: F = k·u.
Why is it symmetric?
Elastic structures obey Maxwell–Betti reciprocity.
Why is the bar matrix singular?
A free element can move as a rigid body; supports remove this.
How is the global matrix built?
By transforming each element to global axes and adding them at shared nodes.