What is the stiffness of a 1 m steel bar with 0.2 m² cross-section?
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. See also our Buckling Calculator.
Is the stiffness matrix singular?
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.
What is the difference between a beam element and a bar (truss) element?
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.
What are the important properties of a stiffness 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. You might also find our Young's Modulus Calculator useful.
What does EI/L³ represent in a beam stiffness matrix?
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.
How is the frame element stiffness matrix different from the beam element?
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.
Why do we need to transform the local stiffness matrix to global coordinates?
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.
What inputs do I need to calculate a stiffness matrix?
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.