3D Space-Frame Analysis - The Stiffness Method Explained

The theory behind this 3D structural analysis calculator: how the direct stiffness method extends to six degrees of freedom per node, the 12×12 beam element that couples axial force, torsion and bending about two axes, how members are oriented in space, and how reactions, member forces and the deflected shape are obtained.

Run this method in the free calculator and export the result to Excel (.xlsx) or PDF.

Six degrees of freedom per node

In three dimensions every joint can translate along three axes and rotate about three axes, giving 6 DOF per node: u_x, u_y, u_z, θ_x, θ_y, θ_z. The whole structure is assembled into a global stiffness matrix K and solved for the unknown displacements u from the equilibrium system K·u = F, where F is the vector of applied nodal forces and moments. After solving, support reactions follow from R = K·u − Fat the restrained DOFs, and each member’s end forces from its own stiffness.

The 12×12 beam element

A two-node 3D beam has 12 DOF, so its local stiffness matrix is 12×12. It superimposes four uncoupled actions in the member’s local axes: axial (EA/L), torsion (GJ/L), and bending in two planes - about local z using I_z and about local y using I_y, each the familiar 4×4 Euler–Bernoulli bending block. The local matrix is rotated into global axes by a transformation built from the member’s direction cosines.

Member orientation in space

Unlike a 2D member (defined by a single inclination angle), a 3D member needs a full local triad. The local x-axis runs start→end; the local y and z axes are set by a reference “up” direction (so local z stays as vertical as possible) plus an optional roll angle about the member axis. Getting this orientation right is what makes biaxial bending and torsion come out correctly.

Supports, loads and space trusses

Supports restrain any subset of the six DOF - a fixed support locks all six, a pinned support locks the three translations, and a roller locks one translation. Loads are forces and moments in the three global axes. A space truss is modelled with axial-only members; joints connected only to truss bars have their rotational DOFs suppressed, since pin joints carry no moment.

What the analysis gives you - and its limits

From the solved displacements the tool recovers everything an engineer needs to size the structure: the support reactions, each member’s end forces (axial, two shears, torsion and two bending moments), the internal force diagrams along every member, and the deflected shape. This is a first-order, linear-elasticanalysis - equilibrium is written on the undeformed geometry and the material obeys Hooke’s law. That is the right model for the vast majority of building frames, but it does not capture second-order (P-delta) effects on very slender sway frames, member or global buckling, plastic hinges, or large deflections; those need a dedicated nonlinear or buckling analysis. Use this tool to size members and to see how a space frame carries its load, and confirm stability and any second-order sensitivity separately where the structure is slender.

Frequently asked questions

Ready to analyse your own structure? Build a 3D frame or truss, add supports and loads, and solve for reactions, member forces and deflection.