2D Frame Analysis Calculator

What does a 2D frame analysis give?

A frame is a structure in which beams and columns are rigidly connected. Besides axial force, its members carry shear force and bending moment. This calculator takes the joints, members, supports and loads and uses the direct stiffness method to find the end forces of every member (N, V, M), the support reactions and the joint displacements and rotations. It draws the bending moment, shear force and axial force diagrams and a scaled deflected shape. Continuous beams are solved with the same tool.

Method

Each joint has three degrees of freedom: two translations and one rotation. The 6 × 6 member stiffness matrix is built from the axial term E × A / L and the bending terms 12 × E × I / L³, 6 × E × I / L², 4 × E × I / L and 2 × E × I / L, rotated to the global axes by the angle of the member and assembled into the system matrix. Loads on a member are moved to the joints through fixed end forces: the end moment is w × L² / 12 for a uniform load and P × L / 8 for a midspan point load. Then K × u = F is solved. Moment and shear along a member follow from the end forces plus the effect of the member load.

Sign convention

A member load is positive when it acts to the right as you look from the first joint of the member to the second: downward on a beam defined left to right, and to the right on a column defined bottom to top. For joint loads x is to the right, y is up and moment is positive counterclockwise. Axial force is positive in tension. The moment diagram is drawn on the tension side.

Example

For a continuous beam with two equal spans of 4 m under 10 kN/m, the moment over the middle support is w × L² / 8 = 20 kN·m, the end reactions are 3 × w × L / 8 = 15 kN and the middle reaction is 10 × w × L / 8 = 50 kN. The largest span moment is 9 × w × L² / 128 = 11.25 kN·m, located 1.5 m from the end support. The built in example reproduces these values.

Limits

The analysis is first order and linear elastic. Shear deformation, second order effects, buckling, member end hinges, semi rigid connections, inclined supports and support settlement are not included. Member loads act only perpendicular to the member axis. Member self weight has to be entered separately as a load. Section selection and resistance checks follow the relevant standard.