2D Frame Analysis Calculator

Build and solve 2D frames (beams + columns) with the matrix stiffness method. Define nodes, members, supports and loads, then view reactions, internal forces and deflections.

Linear elastic Matrix stiffness Shear & moment

1. Define frame geometry & properties

Use consistent units for all inputs (length, E, I, loads).

You can override E and I per member.

Nodes

ID
X
Y
Support

Supports: Free, Pinned (Ux=Uy=0), Fixed (Ux=Uy=θ=0), Roller-X (Uy=0), Roller-Y (Ux=0).

Members

ID
i-node
j-node
E
I
A

E: Young’s modulus, I: second moment of area, A: cross-sectional area.

Loads

Type
Target
Fx / wx
Fy / wy
Start
End

Types: Nodal load (Fx,Fy), Member UDL (wy), Member point load (Fy at relative position).

2. Run analysis & view results

Frame diagram

Support reactions

Node Rx Ry Mz

Member end forces

Local axes: i → j
Member Ni Vi Mi Nj Vj Mj

Nodal displacements

Ux, Uy, θ (global)
Node Ux Uy θ

How this 2D frame analysis calculator works

This tool performs a linear elastic analysis of 2D frames using the matrix stiffness method. Each member is a prismatic beam-column element with 6 degrees of freedom (3 at each node: horizontal displacement Ux, vertical displacement Uy, and rotation θ).

Element stiffness matrix

For a member of length \(L\), axial rigidity \(EA\) and flexural rigidity \(EI\), the local stiffness matrix in the member coordinate system (i → j) is:

\[ k_\text{local} = \begin{bmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 \\ 0 & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & -\frac{12EI}{L^3} & \frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{4EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{2EI}{L} \\ -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 \\ 0 & -\frac{12EI}{L^3} & -\frac{6EI}{L^2} & 0 & \frac{12EI}{L^3} & -\frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{2EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{4EI}{L} \end{bmatrix} \]

The matrix is then rotated into the global X–Y system using a transformation matrix based on the member angle \( \theta = \arctan\frac{\Delta y}{\Delta x} \).

Boundary conditions and solution

  1. Assemble the global stiffness matrix \(K\) from all members.
  2. Apply nodal loads and equivalent fixed-end forces from member loads.
  3. Impose support conditions by constraining the relevant DOFs.
  4. Solve the reduced system \(K_{ff} \, d_f = F_f\) for free DOF displacements.
  5. Back-calculate reactions and member end forces.

Assumptions & limitations

  • Linear elastic material behaviour (no yielding or cracking).
  • Small displacements (no P–Δ or geometric nonlinearity).
  • Members are prismatic with constant E, I and A.
  • Static loading only (no dynamics, vibration or buckling checks).

Step-by-step: using the 2D frame analysis tool

1. Define nodes and supports

  • Add nodes with X and Y coordinates (in m or ft).
  • Assign support type at each node (Pinned, Fixed, Roller, Free).

2. Add members

  • Connect nodes with members (i-node and j-node).
  • Specify E, I and A. Use consistent units (e.g. kN, m, m², m⁴).

3. Apply loads

  • Nodal loads: horizontal Fx and vertical Fy at a node.
  • Member UDL: uniform load along the member (global Y).
  • Member point load: concentrated load at a relative position 0–1.

4. Run analysis and interpret results

  • Click Run 2D frame analysis to solve.
  • Check support reactions and nodal displacements.
  • Review member end forces (axial N, shear V, moment M).
  • Use the diagram toggle to view undeformed vs. deformed shape.

FAQ

Can I use this for design to a specific code?

No. This calculator provides analysis results only. You must perform separate design checks (strength, serviceability, stability) according to the relevant code (e.g. AISC, Eurocode, etc.).

What if my frame is unstable?

If the structure is a mechanism (insufficient supports or bracing), the global stiffness matrix becomes singular and the solver will report an instability. Add supports or members to provide adequate restraint.

How can I validate the results?

Start with simple benchmark problems (e.g. a single-span beam with a UDL) and compare reactions and moments with hand calculations or textbook examples. For more complex frames, compare with trusted commercial software when possible.