Table of Contents
What Is a Stiffness Matrix?
In structural finite element analysis (FEA), the stiffness matrix relates nodal forces to nodal displacements: [F] = [K][u]. For a simple bar (truss) element, the local stiffness matrix is a 2x2 matrix with the fundamental quantity EA/L, where E is Young's modulus, A is cross-sectional area, and L is element length.
When the bar is oriented at an angle to the global coordinate system, the local matrix must be transformed using a rotation matrix to obtain the 4x4 global stiffness matrix. Assembling all element matrices gives the global structure stiffness matrix used to solve for unknown displacements.
Bar Element Formula
For a 2D element at angle θ, the 4x4 global stiffness matrix uses c = cos(θ) and s = sin(θ), with components (EA/L) × [c², cs, -c², -cs; cs, s², -cs, -s²; -c², -cs, c², cs; -cs, -s², cs, s²].
Material Properties
| Material | E (GPa) | Typical Use |
|---|---|---|
| Structural Steel | 200 | Buildings, bridges |
| Aluminum 6061 | 69 | Aerospace, frames |
| Concrete | 30 | Foundations, columns |
| Timber (Pine) | 12 | Residential framing |
| Titanium | 116 | Aerospace, medical |
| CFRP Composite | 150-230 | Aerospace, sports |
Finite Element Context
- Assembly: Element stiffness matrices are assembled into the global matrix using connectivity information.
- Boundary conditions: Fixed supports are applied by modifying rows/columns of the global matrix.
- Solution: [u] = [K]^-1 [F] gives displacements, then strains and stresses are computed.
- Beam elements: Include bending stiffness (EI/L terms) in addition to axial stiffness.
Frequently Asked Questions
Why is the stiffness matrix symmetric?
The stiffness matrix is symmetric because of Maxwell's reciprocal theorem: the displacement at point i due to a force at point j equals the displacement at j due to the same force at i. Mathematically, this follows from the quadratic nature of strain energy.
What does a negative entry in the stiffness matrix mean?
Negative off-diagonal entries indicate that a positive displacement at one node produces a restoring force at the other. For a bar element, pushing node 1 to the right compresses the bar, creating a leftward force at node 2, hence the -EA/L terms.
How does the stiffness matrix change for beam elements?
Beam elements have 6 DOFs (2 translations + 1 rotation per node in 2D), producing a 6x6 stiffness matrix with additional terms involving EI/L, EI/L^2, and EI/L^3 for bending moment and shear force contributions.