Question 5 of 6: Cantilever Deflection — Beam Formula vs. FEA vs. Physical Test

Question 5: Cantilever Deflection — Beam Formula vs. FEA vs. Physical Test (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Simple beam-bending formula. The classical Euler–Bernoulli result for a cantilever with an end point load is δ = FL³/(3EI), where E is the material's elastic modulus and I is the cross-section's second moment of area. Pros: instantaneous to evaluate, needs no software licence or meshing effort, and gives immediate insight into which design variables (L, E, I) matter and by how much — useful for first-pass sizing and quick sensitivity/what-if studies. Cons: it assumes a slender, prismatic, linear-elastic beam with small deflections, a perfectly rigid support, a truly point load, and it neglects shear deformation — every one of those idealizations removes it further from the real, finite-stiffness bracket, fillet, and load patch of an actual part, so it systematically under-predicts deflection, and the error grows as the beam gets shorter/deeper (shear deformation becomes a larger fraction of total deflection) or as the support structure itself has any measurable compliance.

Finite-element analysis. Pros: can represent the true 3-D geometry (fillets, changing cross-section, the real support structure's own flexibility) and can include shear deformation, stress concentrations, and non-linear material or large-deflection effects if requested, so it is generally far closer to reality than the hand formula while still being much cheaper and faster than building a physical prototype. Cons: the result is only as good as the model — mesh density and element type (e.g., under-integrated surface/solid elements can be artificially stiff or soft), the idealized boundary condition (a perfectly rigid “fixed” wall is itself an idealization, just as in the hand formula, unless the surrounding structure is modelled too), and the assumed material properties (a single, isotropic E, when the real material may have some anisotropy or manufacturing-induced variability) are all sources of error that are easy to introduce and not always obvious from the coloured contour plot alone.

Physical prototype test. Pros: captures the true behaviour of the actual part — real material properties and their as-built variability, the real support's finite stiffness, real geometric tolerances and surface finish, and any nonlinearity or assembly play the idealized models omit — and is the only one of the three methods that can be treated as a genuine calibration reference for the other two. Cons: the most expensive and slowest of the three (tooling/machining time, instrumentation, and a physical part is fixed after it is built, so redesign means a new build), and it still carries its own measurement uncertainty — dial-indicator/strain-gauge resolution, load-cell calibration, an imperfectly rigid test fixture, and unit-to-unit manufacturing variation all add scatter, so a single test is a sample, not a guaranteed “true” value.

Approach for an illustrative comparison. Given representative values for a rectangular-section steel cantilever, compute the hand-formula deflection, then discuss how the FEA and test values would be expected to compare to it.

Given. L = 0.40 m, F = 60 N, E = 200 GPa = 200×10³ N/mm², rectangular section b = 25 mm, h = 8 mm.

Find. End deflection δ by the simple beam formula, and the expected sign/magnitude of the difference from an FEA model and from a physical test of the same nominal geometry.

1. Second moment of area. For a rectangular section, I = bh³/12 = (25)(8)³/12 = 1067 mm⁴.
2. End deflection from the cantilever point-load formula. δ = FL³/(3EI). With L = 400 mm: F·L³ = (60)(400)³ = (60)(6.4×10⁷) = 3.84×10⁹. 3EI = 3 × (200×10³) × 1066.7 = 6.4×10⁸. δ = 3.84×10⁹ / 6.4×10⁸ = 6.00 mm (simple formula).
3. Expected FEA result. For a beam this slender (L/h = 400/8 = 50), shear deformation adds only δ_s/δ_b = (E/G)(h/L)²/(4κ) ≈ 0.03% (E/G = 2.6, κ = 5/6). A solid-element FEA model with a perfectly fixed end face should therefore reproduce the formula almost exactly: δ_FEA ≈ 6.0–6.1 mm, with the small excess coming from 3-D effects at the root and mesh discretization. If the model also includes the real mounting, with its own flexibility, the root can rotate slightly and the FEA value rises, typically to about 6.2–6.6 mm for a bolted bracket.
4. Expected physical-test result. The physical prototype should land close to the FEA prediction if the FEA boundary conditions were modelled faithfully, but with additional scatter from manufacturing tolerance on b, h and the fixed-end geometry, from the actual (batch-specific) E of the test material, and from any play/compliance in the physical clamp — a plausible single-test reading is δ_test ≈ 6.2–6.9 mm, with repeat tests scattering by roughly ±5% about that mean.

The consistent ranking — hand formula lowest, FEA and test closest to each other and higher — is the expected outcome whenever the hand formula's idealizations (chiefly the rigid support; shear deformation is negligible at L/h = 50) remove real flexibility that both the FEA model and the physical part actually have. In practice the physical test is treated as the ground truth for validating the FEA model (if they disagree by more than the expected measurement/manufacturing scatter, the FEA boundary conditions or mesh are revisited), while the hand formula remains valuable purely as a fast sanity check and for early design-space exploration before either a CAD model or a prototype exists.