Engineers have checked finite element programs against these problems for decades. Each one has a value from theory or from the literature. Below is what Femra computes for each one, how far it is from the reference, and where the problem comes from.
Every case runs in Femra's test suite before each change is merged, and is in the app under Project → Verification, where you can run all ten or open one as a model. The values on this page come from the run of 11 October 2026.
| Problem | Quantity | Reference | Femra | Deviation |
|---|---|---|---|---|
| Beams and frames | ||||
| Cantilever, tip load | tip deflection, m | 0.00191962 | 0.00191962 | 0.00 % |
| Pinned column, Euler load | critical load factor | 690.164 | 690.167 | 0.00 % |
| Cantilever, first frequency | f₁, Hz | 5.1721 | 5.1643 | −0.15 % |
| Plates | ||||
| Simply supported square plate | centre deflection, m | 2.1124·10⁻⁴ | 2.1157·10⁻⁴ | +0.15 % |
| Clamped square plate | centre deflection, m | 6.5797·10⁻⁵ | 6.5805·10⁻⁵ | +0.01 % |
| Morley's 30° skew plate | centre deflection, m | 2.1216·10⁻⁵ | 2.1361·10⁻⁵ | +0.68 % |
| Membranes | ||||
| Patch test, distorted mesh | end displacement | 4 (exact) | 4.00000 | exact |
| Cook's membrane | displacement at (48, 52) | 23.96 | 23.944 | −0.07 % |
| Shells | ||||
| Scordelis–Lo roof | free-edge deflection | 0.3024 | 0.3040 | +0.53 % |
| Pinched cylinder | radial displacement | 1.8248·10⁻⁵ | 1.790·10⁻⁵ | −1.91 % |
Deviations are at the meshes each case defines. Each case has a tolerance in the tests, from 10⁻⁶ for the cantilever to 15 % for the pinched cylinder; see each case for why.
A steel beam, 100 × 200 mm and 2 m long, fixed at one end and loaded at the other. The exact deflection includes shear: δ = PL³/(3EI) + PL/(κGA), with κ = 5/6. Femra's Timoshenko beam element is exact for this load.
A 100 × 100 mm steel column, 5 m long, pinned at both ends under a unit axial load. Femra's linear buckling analysis finds the critical load factor, which matches Euler's load with Engesser's shear correction, P_cr = P_E / (1 + P_E/κGA).
A steel cantilever, 4 m long and 100 × 100 mm, vibrating under its own mass. Euler–Bernoulli theory gives β₁L = 1.875104. Femra lumps the mass at the nodes and includes shear and rotary inertia, so it lands slightly below the slender-beam value.
A thin square plate under uniform pressure, simply supported on all four edges. Navier's double series gives w = 0.00406235 qa⁴/D at the centre. Femra's shells include shear deformation (Mindlin), which adds a few tenths of a per cent at this slenderness.
The same plate with all edges clamped. The classical centre deflection is w = 0.00126532 qa⁴/D. A clamped plate is stiffer in bending, so shear deformation counts for more; the tolerance allows for that.
A rhombic plate with 30° skew, simply supported, under uniform pressure. Morley's solution gives w = 0.000408 qL⁴/D at the centre. The obtuse corners carry a stress singularity, so element results approach the value slowly and from above.
A 4 × 4 panel pulled by a uniform edge load. Femra's free mesher meshes it with distorted quadrilaterals around an interior point. A correct element must reproduce the constant stress exactly on any mesh. Femra does, at every node, to rounding.
A tapered panel, clamped on its left edge and sheared on its right, in plane stress with E = 1 and ν = 1/3. The value is taken at the middle of the loaded edge. Using the top corner instead gives about 25.2, which is a common mix-up. Femra's own check with constant-strain triangles, extrapolated, gives 23.97.
A cylindrical roof on rigid end diaphragms, loaded by its own weight. The deflection at the middle of the free edge is the classic shell benchmark. Femra models the curved shell with flat facets, which approach the reference as they get finer.
A thin cylinder with rigid diaphragms at its ends, pinched by two opposite point loads. The test is severe because bending and membrane action are coupled at the load. Four-node flat shells converge slowly on it, so Femra reports its value at the mesh the case defines and holds it to a wide tolerance.