Stiffness Matrix Calculator
Calculate stiffness matrices for truss, beam, and frame elements in structural analysis
Calculate Stiffness Matrix
Choose the type of structural element for stiffness matrix calculation
Elastic modulus of the material
Length of the structural element
Cross-sectional area for axial stiffness
Angle with respect to global x-axis
Stiffness Matrix Results
Enter all required parameters to calculate the stiffness matrix
Example Calculation
Steel Beam Example
Element: Beam
Young's modulus: 200 GPa
Moment of inertia: 8.33×10⁴ mm⁴
Length: 3 m
Key Results
EI = 200×10⁹ × 8.33×10⁻⁸ = 16.66 N⋅m²
k₁₁ = 12EI/L³ = 7.40×10⁻² N/m
k₁₂ = 6EI/L² = 1.11×10⁻¹ N
k₂₂ = 4EI/L = 2.22×10¹ N⋅m
Element Types
Bar/Truss
Axial forces only
4×4 matrix with angle transformation
Beam
Bending and shear
4×4 matrix with transverse DOF
Frame
Combined axial and bending
6×6 matrix with all DOF
Key Formulas
Bar Stiffness
k = AE/L
Flexural Rigidity
EI
Beam Terms
12EI/L³ (force/displacement)
6EI/L² (force/rotation)
4EI/L (moment/rotation)
Understanding Stiffness Matrices
What is a Stiffness Matrix?
A stiffness matrix relates forces to displacements in structural elements. It's fundamental to finite element analysis, representing how an element resists deformation under applied forces.
Matrix Properties
- •Always symmetric (K = K^T)
- •Singular without boundary conditions
- •Positive semi-definite
- •Units depend on force-displacement relationships
Fundamental Equation
[K]{u} = {F}
- [K]: Stiffness matrix
- {u}: Displacement vector
- {F}: Force vector
Engineering Application: Stiffness matrices are assembled from individual elements to analyze entire structures, enabling prediction of deformations and stresses.
Element Type Comparison
| Element | DOF per Node | Matrix Size | Primary Forces | Key Parameters |
|---|---|---|---|---|
| Bar/Truss | 2 (u, v) | 4×4 | Axial | A, E, L, φ |
| Beam | 2 (v, θ) | 4×4 | Bending, Shear | E, I, L |
| Frame | 3 (u, v, θ) | 6×6 | Axial, Bending, Shear | A, E, I, L |
Applications
Structural Analysis
- • Building frame analysis
- • Bridge design
- • Truss optimization
Finite Element Method
- • Element assembly
- • Global stiffness matrix
- • Solution of K{u} = {F}
Design Verification
- • Deflection limits
- • Dynamic analysis
- • Stability checks