How the structural analysis is verified
Problems whose exact answers are known in closed form, solved by the program, with the error shown. The same list runs in the test suite on every change, so this page cannot drift from what the program does — the numbers below were computed by it.
This tool analyses 2D structures by the direct stiffness method — linear elastic, small displacements. It does not check buckling, second-order effects or any design code. Results depend on the model you enter and may contain errors, from the model or from the software. They are for education and preliminary checking only and must be independently verified by a qualified engineer before use in design, construction or any decision affecting safety.
Checks on every result
- Overall equilibrium: the applied loads and the reactions sum to zero in both directions and in moment.
- The solution itself: K·D − Q − R is zero over every equation, to within 1e-6 of the largest force in the structure.
- Every joint: the member end forces, the joint loads and the reactions at each joint balance, to the same tolerance.
A result that fails any of these is shown with a warning not to use it, and no report can be printed from it. These checks say the equations were solved; they cannot say the model describes the real structure. That part belongs to the engineer.
Benchmarks
E = 2e+8 kPa, A = 0.01 m², I = 8e-5 m⁴ throughout. Largest relative error over the whole set: < 1e-12.
Simply supported beam under a uniform load
Standard result; Roark 8.1 case 2a
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Reaction at each end | 36 kN | 36 kN | < 1e-12 |
| Moment at mid-span | 54 kN·m | 54 kN·m | < 1e-12 |
| Deflection at mid-span | 0.01265625 m | 0.01265625 m | < 1e-12 |
Simply supported beam under a central point load
Standard result; Roark 8.1 case 1a
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Reaction at each end | 15 kN | 15 kN | < 1e-12 |
| Moment under the load | 45 kN·m | 45 kN·m | < 1e-12 |
| Deflection under the load | 0.0084375 m | 0.0084375 m | < 1e-12 |
Cantilever under a uniform load
Standard result; Roark 8.1 case 2d
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment at the support | 216 kN·m | 216 kN·m | < 1e-12 |
| Reaction | 72 kN | 72 kN | < 1e-12 |
| Deflection at the tip | 0.1215 m | 0.1215 m | < 1e-12 |
Cantilever under a point load at the tip
Standard result; Roark 8.1 case 1d
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment at the support | 180 kN·m | 180 kN·m | < 1e-12 |
| Deflection at the tip | 0.135 m | 0.135 m | < 1e-12 |
Propped cantilever under a uniform load
Standard result; Hibbeler, Structural Analysis, table of fixed-end actions
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment at the fixed end | 54 kN·m | 54 kN·m | < 1e-12 |
| Reaction at the prop | 27 kN | 27 kN | < 1e-12 |
| Reaction at the fixed end | 45 kN | 45 kN | < 1e-12 |
Beam fixed at both ends under a uniform load
Standard result; Roark 8.1 case 2f
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment at each end | 36 kN·m | 36 kN·m | < 1e-12 |
| Moment at mid-span | 18 kN·m | 18 kN·m | < 1e-12 |
| Deflection at mid-span | 0.00253125 m | 0.00253125 m | < 1e-12 |
Beam fixed at both ends under a central point load
Standard result; Roark 8.1 case 1f
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment at each end | 22.5 kN·m | 22.5 kN·m | < 1e-12 |
| Deflection under the load | 0.002109375 m | 0.002109375 m | < 1e-12 |
Two equal spans, continuous, uniform load on both
Three-moment equation; standard continuous-beam coefficients
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Moment over the middle support | 54 kN·m | 54 kN·m | < 1e-12 |
| Reaction at the middle support | 90 kN | 90 kN | < 1e-12 |
| Reaction at each end | 27 kN | 27 kN | < 1e-12 |
Bar in pure tension
δ = PL/EA
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Extension | 9e-5 m | 9e-5 m | < 1e-12 |
| Axial force | 30 kN | 30 kN | < 1e-12 |
| Reaction | -30 kN | -30 kN | < 1e-12 |
Three-member truss, load at the apex
Method of joints
| Quantity | Exact | Program | Relative error |
|---|---|---|---|
| Reaction at each support | 15 kN | 15 kN | < 1e-12 |
| Force in a diagonal | 25 kN | 25 kN | < 1e-12 |
| Force in the bottom tie | 20 kN | 20 kN | < 1e-12 |
| Moment anywhere in a truss member | 0 kN·m | 1.38778e-17 kN·m | < 1e-12 |
The method is written out in full under Learn › Direct stiffness method, and the tool itself is at Structural analysis.