The direct stiffness method
Every commercial structural analysis program — and the analysis tool on this site — solves a frame the same way. The structure is cut into members, each member's resistance to being displaced is written as a matrix, those matrices are added together in one set of axes, and what comes out is a system of linear equations in the displacements of the joints. Solve it, and every reaction, every member force and every deflection follows.
One equation, six pages
The whole method exists to build and then solve this:
| K | The structure stiffness matrix. What the structure does when its joints move. |
|---|---|
| u | The displacements of the joints — two translations and a rotation at each, in a plane frame. |
| F | The loads at the joints, including the loads along members turned into joint loads. |
Nothing in the six guides is harder than that line. They are about where K comes from, what to do about loads that are nowhere near a joint, and how the answer is turned back into the numbers an engineer reads.
The guides
The Direct Stiffness Method: An Introduction
What the direct stiffness method is, why structural analysis is written as matrices, and the six steps that take a drawn frame to its reactions and diagrams.
Member Stiffness in Local Axes
The 6x6 stiffness matrix of a plane frame member, term by term: EA/L for axial action, 12EI/L^3 and 4EI/L for bending, and what each column means.
Changing Axes: Local to Global
How a member stiffness matrix is rotated into global axes with direction cosines, why the transformation is k_global = T-transpose k T, and a worked example.
Loads That Are Not Applied at the Joints
A distributed load acts along a member, not at a joint. Fixed end forces, equivalent joint loads, and how the true member end forces are recovered afterwards.
Assembling the Structure Stiffness Matrix
How member matrices are added into the structure stiffness matrix through the connectivity, why the result is symmetric and sparse, and why it is singular until the supports go on.
The Calculation, Step by Step
Partitioning the equations at the supports, solving for the free displacements, and recovering reactions, member end forces and diagrams — with the checks that prove the answer.
The notation used throughout
Books differ on symbols; these are the ones the six guides keep to. A bar over a symbol means it is written in the member's own axes rather than the structure's.
| E, A, I | Young’s modulus, cross-sectional area, and second moment of area of a member. |
|---|---|
| L | The length of a member, from one joint to the other. |
| k̄ | The 6×6 member stiffness matrix in local axes — along the member and across it. |
| k | The same member, rotated into the structure’s global axes. |
| T | The transformation matrix that does the rotating. |
| K | The structure stiffness matrix, assembled from every member’s k. |
| u | Joint displacements. Global x, global y, and a rotation about z. |
| F | Joint loads, in the same order as u. |
| f̄ | The forces at the two ends of one member, in that member’s own axes: axial, shear, moment. |
| f̄0 | Fixed end forces: what a member carries when both its ends are held still. |
What this method assumes
The same assumptions the analysis tool on this site works under, and they are worth knowing before trusting an answer:
- Linear elastic material. Stress is proportional to strain, and E never changes. No yielding, no cracking, no creep.
- Small displacements. Equilibrium is written on the undeformed shape, so a member's stiffness does not change as it moves. Second-order effects and buckling are outside it.
- Plane behaviour. Everything — the structure, the loads and the displacements — lies in one plane, with three degrees of freedom per joint.
- Bernoulli beam bending. Plane sections stay plane and shear deformation is neglected, which is accurate while members are slender.
- Superposition holds. Because the two above are true, load cases can be scaled and added, which is exactly what a load combination is.
Draw a frame and watch the method run on it.