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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.

🧊Open the interactive 3D frame calculator→

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: ux, uy, uz, θx, θy, θz. 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 Iz and about local y using Iy, 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.

Frequently asked questions

A 2D frame has 3 degrees of freedom per node (two translations and one in-plane rotation). A 3D space frame has 6 DOF per node - three translations and three rotations - so each beam element is a 12×12 stiffness matrix that carries axial force, torsion and bending about two perpendicular axes simultaneously.

Beyond area A and Young’s modulus E, a 3D beam needs the shear modulus G, the torsion constant J, and the two second moments of area Iy and Iz (bending about each local axis). G is derived from E and Poisson’s ratio when not given: G = E / (2(1+ν)).

The local x-axis runs from the start node to the end node. The local y and z axes (the cross-section’s principal axes) are fixed by a reference “up” direction plus an optional roll angle about the member axis. For a non-vertical member the local z lies in the vertical plane containing the member; vertical members fall back to a global-X reference.

A space truss is a 3D structure of axial-only bars (pin-connected, carrying no bending or torsion). In the solver a truss member keeps only its axial stiffness; a joint connected solely to truss members has its three rotational DOFs suppressed, because pin joints have no rotational stiffness.

This tool performs a first-order linear-elastic static analysis by the direct stiffness method - it does not include P-delta (geometric) or material non-linearity. Results are valid for small displacements within the elastic range.

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

🧊Open the interactive 3D frame calculator→
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