Stiffness Matrix Calculator

For a steel bar with E = 200 GPa, A = 0.2 m², and L = 1 m, the axial stiffness is k = AE/L = 0.2 × 200×10⁹ / 1 = 40,000,000 kN/m (40 GN/m). This represents how much force is required per unit displacement along the bar's axis.

Yes, the element stiffness matrix is always singular because it has no boundary conditions applied — rigid body motion is still possible. This means its determinant is zero and it cannot be inverted directly. Boundary conditions (supports/constraints) must be applied to the global stiffness matrix to make it non-singular and solvable.

A bar (truss) element carries only axial forces along its length, giving it a 2×2 local stiffness matrix based on AE/L. A beam element carries transverse (shear) forces and bending moments, resulting in a 4×4 stiffness matrix involving EI/L³, EI/L², and EI/L terms. Frame elements combine both behaviors into a 6×6 matrix.

The stiffness matrix is always symmetric (Kij = Kji), which follows from the principle of reciprocal work. It is positive semi-definite before boundary conditions are applied, becoming positive definite afterward. It is also sparse (mostly zeros for large structures) and banded, which numerical solvers exploit for efficiency.

EI/L³ represents the lateral (transverse) stiffness term of a beam element — it relates the transverse force required to produce a unit transverse displacement at a node. It appears with coefficients of 12 and -12 in the standard Euler-Bernoulli beam stiffness matrix.

A frame element can resist axial forces, shear forces, and bending moments, giving it a 6×6 local stiffness matrix (3 DOF per node: axial, transverse, rotation). A beam element only resists shear and bending (4×4 matrix, 2 DOF per node). The frame stiffness matrix is essentially a combination of the truss and beam stiffness contributions.

Each element has its own local coordinate system aligned with its axis. When assembling the global stiffness matrix for a structure with elements at different orientations, the local stiffness matrices must be transformed using a rotation matrix based on the element's angle φ. This ensures all element contributions are expressed in a common global reference frame before assembly.

For a truss element you need the elastic modulus (E), cross-sectional area (A), and length (L). For beam elements, you additionally need the moment of inertia (I). For frame elements, all four parameters are required, plus the element orientation angle (φ) for global coordinate transformation.