Validation / 5 of 10

Cantilever beam from a STEP file

The cantilever again, but built the way you build a model in the app: import a STEP block, mesh it, fix one end face and load the other. The mesh is an unstructured Netgen mesh of tetrahedra rather than a hand-made grid. The model opens ready to solve, and after the solve the Results panel compares the tip deflection with the hand formula.

Tip deflection δ; reference δ = P·L³ / (3·E·I)
Reference	KoFEM	Difference
−1.9048 mm	−1.9040 mm	−0.04 %

Open in KoFEMOpens the model in the app, ready to solve.

The problem

Block of length L along x, b × h section, I = b·h³/12. Sketch, not to scale.

$L$	1000 mm
$b \times h$	100 × 100 mm
$E$	210 GPa
$ν$	0.3
$P$	10 kN

Supports	x = 0 face fixed (Ux, Uy, Uz), picked as one CAD face.
Load	P in −y as a force on the x = L face, picked as one CAD face.
Measured	Mean y-displacement of the x = L face.

Analytical solution

Euler–Bernoulli beam theory assumes that cross-sections stay plane and perpendicular to the deformed axis, which neglects shear deformation. Take $w (x)$ as the deflection, positive upwards, with the load $P$ acting downwards at the tip. Equilibrium of the part beyond a section $x$ gives the bending moment

M (x) = - P (L - x),

and the curvature is proportional to it:

E I \frac{d^{2} w}{d x^{2}} = M (x), I = \frac{b h^{3}}{12} .

Integrating twice with the clamp conditions $w (0) = 0$ and $\frac{d w}{d x} (0) = 0$ gives the deflection curve

w (x) = - \frac{P x^{2}}{6 E I} (3 L - x),

and at the tip

δ = w (L) = - \frac{P L^{3}}{3 E I} .

For this beam $I = 8.3333 \times 10^{6} {mm}^{4}$ and $δ = - 1.9048 mm$ .

Timoshenko beam theory adds the shear deformation that Euler–Bernoulli leaves out. For a cantilever with a tip load it adds a second term,

δ_{s} = - \frac{P L}{κ G A}, κ = \frac{5}{6}, G = \frac{E}{2 (1 + ν)},

which is 0.78 % of the bending term for this beam (span ten times the depth). A converged 3-D solid model should therefore deflect slightly more than $P L^{3} / (3 E I)$ .

Finite element model

Mesh	Netgen, 25 mm maximum element size: 2,991 quadratic 10-node tetrahedra (P2)
Unknowns	5,532 nodes, 16,596 degrees of freedom
Material	Linear elastic, isotropic
Solver	KoFEM engine (MFEM), the WebAssembly build the app runs: engine-5ceb743373dd

Netgen meshes the block with a 25 mm maximum element size, about 4 elements across the section. The elements are quadratic: linear tetrahedra lock in bending even more than the linear hexahedra do. The support and the load are each one CAD face, picked with one click, so they stay on the model when you change the element size and mesh it again. The deflection is the mean $u_{y}$ of the loaded face, which is what the app's hand check reads.

Results

Quantity	tip deflection δ
Reference	δ = P·L³ / (3·E·I) = −1.9048 mm (Euler–Bernoulli beam theory)
KoFEM	−1.9040 mm
Difference	−0.04 %

Discussion

KoFEM's tip deflection differs from $P L^{3} / (3 E I)$ by −0.04 %. The beam-theory value is not the whole story: the shear deformation it leaves out adds +0.78 %, so a converged solid model would deflect slightly more. Against the Timoshenko value the result differs by −0.81 %, which is what the mesh still leaves out at this element size.

The quadratic hexahedra on a coarse grid come out +1.84 %. Both show the same thing: in bending, second-order elements are accurate on meshes coarse enough to solve in a second.

Reproduce

In the app: open the model, then Studies → Static 1 → Run static solve, select the result plot and set Field to Uy. The Result summary shows Min Uy -1.904e+0 mm.

From a clone (Node.js 20+, bash, curl), this rebuilds the model and its figure:

git clone https://github.com/mkofler96/KoFEM.git && cd KoFEM
bash scripts/fetch-wasm-engine.sh
cd web && bun install && bun run examples:generate-cantilever

References

J. M. Gere and B. J. Goodno, Mechanics of Materials, Cengage Learning: axially loaded members, torsion, deflections of beams.
S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars”, Philosophical Magazine 41 (1921) 744–746.
J. Schöberl, “NETGEN: An advancing front 2D/3D-mesh generator based on abstract rules”, Computing and Visualization in Science 1 (1997) 41–52.
R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, 2002.

KoFEM is research software and is not certified for safety-critical design. It solves linear-static, linear-elastic problems. Check results that matter against your own references.