Structural analysis of frames and structures by the stiffness method

Model frames, continuous beams, gable frames, three-hinged frames, grids and space frames in 2D or 3D. Draw joints and bars, add fixed, pinned or roller supports, internal hinges and every kind of load, and instantly get the reactions, displacements, axial force, shear and bending moment diagrams and the deflected shape. Includes a detailed step-by-step stiffness-method solution with every bar's matrices.

What you can do

Plane frames with 3 degrees of freedom per joint, and space structures with 6.
Point loads, moments and uniform, trapezoidal, triangular, parabolic or partial distributed loads.
Internal hinges; fixed, pinned, roller or custom supports.
N, V and M diagrams (plus torsion and two-plane bending in 3D) and the deflected shape.
Degree of indeterminacy, reactions with an equilibrium check, and bar end forces.
Step-by-step PDF solution: local and global matrices, assembly and solution.

Frequently asked questions

What is the stiffness method?

It is the matrix method of structural analysis: it takes the joint displacements as unknowns, builds the global stiffness matrix by adding each bar's matrix and solves K·d = P. It works for determinate and indeterminate structures alike.

What loads can I apply?

Forces and moments at the joints, and on the bars point loads, moments and uniform, triangular, trapezoidal, parabolic or partial distributed loads, along gravity, a global or a local axis, per true or projected length.

Guide: frame analysis by the stiffness method

Frames, continuous beams and structures of rigidly connected members are usually statically indeterminate: equilibrium alone cannot solve them. The stiffness method (or displacement method) solves all of them the same way, and it is the basis of structural analysis software. This guide explains its steps and checks them on a beam fixed at both ends.

Degree of indeterminacy

In the plane, for a structure with m members, j joints and r reactions:

DI = 3m − c + r − (3j − d)

c is the number of rotations released by hinges and d the joints whose rotation has no stiffness (every member hinged there). With DI = 0 the structure is statically determinate; with DI > 0, indeterminate; with DI < 0, a mechanism.

Steps of the method

Degrees of freedom: each joint of a plane frame can move in x, in y and rotate. Those displacements are the unknowns.
Member stiffness matrix: it relates the forces at a member's ends to its end displacements. In local axes it has terms EA/L (axial), 12EI/L³ and 6EI/L² (shear) and 4EI/L and 2EI/L (moment).
Transformation and assembly: each matrix is rotated to global axes and added into the global matrix K.
Member loads: they are replaced by their fixed-end forces, applied at the joints with the opposite sign.
Support conditions and solution: the restrained degrees of freedom are removed and K·u = F is solved.
Member forces and reactions: from the displacements, each member gives its internal forces (k·u plus the fixed-end forces), and from them the axial, shear and moment diagrams.

Example: fixed–fixed beam

A 6 m beam, steel IPE 300 section (E = 200 GPa, I = 8356 cm⁴, A = 53.8 cm²), fixed at both ends, with a uniform load w = 10 kN/m.

Indeterminacy: m = 1, j = 2, r = 6: DI = 3 − 0 + 6 − 6 = 3, indeterminate to the third degree.
Stiffnesses: EI = 200 × 10⁶ kPa · 8356 × 10⁻⁸ m⁴ = 16,712 kN·m². Matrix terms: EA/L ≈ 179,333 kN/m; 12EI/L³ ≈ 928.4 kN/m; 6EI/L² ≈ 2785.3 kN; 4EI/L ≈ 11,141 kN·m and 2EI/L ≈ 5570.7 kN·m.
Fixed-end forces: shear wL/2 = 30 kN and moment wL²/12 = 30 kN·m at each end.
Solution: since both joints are fixed, every displacement is zero and the reactions are simply the fixed-end forces: 30 kN upward and 30 kN·m moments at each support.
Diagrams: the moment is −30 kN·m at the ends and +wL²/24 = 15 kN·m at midspan; the maximum deflection, at midspan, is wL⁴/(384EI) ≈ 2.0 mm, five times less than if the beam were simply supported.

The tool gives the same degree of indeterminacy, the same stiffness matrix term by term, the same fixed-end forces and the same reactions. In a frame with free joints, such as the example the tool opens with, the solution step is no longer trivial, but the procedure is identical.

How to use the tool

Draw the joints and members (in 2D or 3D), give each member its section and material, and mark the hinges.
Define each joint's supports (which displacements and rotations they prevent) and the loads: at the joints or along the members (uniform, linear and others).
Read the degree of indeterminacy, the reactions, the deflected shape, each member's N, V and M diagrams and the step-by-step stiffness method.

Common mistakes

Forgetting the fixed-end forces of loads on the members.
Mixing local and global axes when assembling inclined members.
Insufficient supports: a structure that can move like a mechanism gives a singular matrix.
Inconsistent units among E, I, A and the lengths.

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