# Stiffness Matrix Calculator

URL: https://speedycalculator.com/calculators/physics/stiffness-matrix/
Last updated: 2026-10-30

> The stiffness matrix calculator finds the coefficients of the local element stiffness matrix for a bar (truss), beam or 2D frame element from E, A, I and L.

## Inputs

- Element type
- Young's modulus E (GPa, MPa, Pa, psi, Mpsi)
- Cross-section area A (mm², cm², m², in²)
- Area moment of inertia I (mm⁴, cm⁴, m⁴, in⁴)
- Element length L (m, cm, mm, km, in, ft, yd, mi)

## Results

- EA/L (axial)
- 12EI/L³
- 6EI/L²
- 4EI/L
- 2EI/L

## 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.
```

## Explanation

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

FieldWhat to enterElement typeChoose one: Bar / truss (2 DOF), Beam (4 DOF), Frame (6 DOF)Young's modulus EUnits: GPa, MPa, Pa, psi, MpsiCross-section area AUnits: mm², cm², m², in²Area moment of inertia IUnits: mm⁴, cm⁴, m⁴, in⁴Element length LUnits: 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)100125,000,000150187,500,000200250,000,000250312,500,000300375,000,000400500,000,000

## 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.

## FAQ

**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.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)