Free Finite Element Method (FEM) beam analysis — model spans, supports, point loads, and distributed loads, then get instant shear force and bending moment diagrams, deflections, and reactions. No account required. Jump to how it works ↓
This calculator uses the Finite Element Method (FEM) — specifically the direct stiffness method for Euler-Bernoulli beam elements — to analyze beams under combined point loads and uniformly distributed loads (UDLs), with any mix of fixed, pinned, and roller supports.
Each span between two nodes is modeled as a beam element with two degrees of freedom per node: vertical deflection and rotation. The tool assembles a global stiffness matrix from every element, applies your boundary conditions (supports), and solves the resulting system of equations to find nodal displacements. From there, it works back through equilibrium at 21 points along every element to build the shear force and bending moment diagrams you see above.
The calculation itself runs on a protected server — only your model (node positions, section properties, supports, and loads) is sent out, and only the results come back. This keeps the underlying formulas off the page while still giving you instant, accurate results in the browser.
Choosing the right support type changes how many degrees of freedom are restrained at that node, which directly affects reactions, deflections, and the shape of your SFD/BMD:
Restrains vertical displacement and rotation. Produces both a vertical reaction and a moment reaction. Used for beams built into a wall or column, e.g. a cantilever.
Restrains vertical displacement only — the beam is free to rotate at that node. Produces a vertical reaction, no moment reaction. Common at bearing points and simple supports.
Also restrains vertical displacement only, behaving the same as a pinned support in this 2D vertical-load model. Used where a support must allow horizontal movement (thermal expansion, etc.) while still carrying vertical load.
Consider a wide-flange W310×52 steel beam spanning two segments — a 5 m span between a fixed support and a pinned support, then a 2.5 m cantilever beyond the pinned support — carrying a uniformly distributed load of 25 kN/m along its full length.
Each element's local stiffness matrix follows the standard Euler-Bernoulli beam formulation:
\[ K(E,I,L) = \frac{E \cdot I}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix} \]
Assembling both elements' stiffness matrices into the global system, applying the fixed support at node 1 (\(U_{11}=U_{12}=0\)) and the pinned support at node 2 (\(U_{21}=0\)), and solving the reduced system \(K_r \cdot u = F_r\) gives the free-DOF displacements:
\[ u = K_r^{-1} \cdot F_r = \begin{bmatrix} -0.0013887 \\ -0.0086796 \\ -0.0041662 \end{bmatrix} \begin{matrix} rad \\ m \\ rad \end{matrix} \]
Which gives the full result set at every node:
| Node | Position [m] | Deflection [mm] | Rotation [mrad] |
|---|---|---|---|
| 1 | 0.00 | 0.0000 | 0.0000 |
| 2 | 5.00 | 0.0000 | 1.3888 |
| 3 | 7.50 | 8.6797 | 4.1663 |
| Support | Vertical Reaction [kN] | Moment Reaction [kN·m] |
|---|---|---|
| Node 1 (Fixed) | 54.69 | 39.06 |
| Node 2 (Pinned) | 132.81 | N/A |
Total applied load (25 kN/m × 7.5 m) = 187.50 kN, matching the sum of the two vertical reactions exactly — confirming equilibrium.
This exact problem was independently checked against a hand solution (direct stiffness method) and Autodesk Robot Structural Analysis — both agree with this tool's results to within 0.02%. Try it yourself: enter 3 nodes at 0 m, 5 m, and 7.5 m above, set a fixed support at node 1, a pinned support at node 2, a UDL of 25 kN/m on both elements, and compare your results to the table above.