Stiffness Matrix Calculator for 2D Beams

What Is a Beam Stiffness Matrix?

In finite element analysis, each beam element has a local stiffness matrix that relates nodal forces and moments to displacements and rotations. For a 2D Euler-Bernoulli beam in bending, the 4×4 matrix couples vertical deflections and end rotations at both nodes.

Local Bending Stiffness Matrix

For a prismatic beam of length $L$, Young's modulus $E$, and second moment of area $I$, the bending stiffness matrix with DOFs $[v_1, \theta_1, v_2, \theta_2]$ is:

$$[k] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}$$

Axial stiffness for the same element is $EA/L$, where $A$ is the cross-sectional area. Related tools: Rotational Stiffness Calculator and Torque Calculator.

Frequently Asked Questions

What DOFs does the 4×4 matrix use?

The matrix uses vertical deflection and rotation at each end: v₁, θ₁, v₂, and θ₂. Axial extension uses a separate EA/L term.

Why is the matrix symmetric?

Linear elastic structures obey Maxwell-Betti reciprocity, so the stiffness matrix is symmetric and positive semi-definite.

What units appear in the matrix?

Rows and columns mix force per length (N/m), moment per rotation (N·m/rad), and coupled terms with units of N and N·m/rad depending on the DOF pair.

Can I use this for a 3D frame?

This tool covers the 2D bending element. A full 3D frame element uses a 6×6 or 12×12 matrix with additional axial and torsional terms.

How do I assemble multiple elements?

Map each local DOF to global DOFs, then add element matrices into the global stiffness matrix at matching indices.

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Stiffness Matrix Calculator

What Is a Beam Stiffness Matrix?

Local Bending Stiffness Matrix

Frequently Asked Questions

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