A simply supported 1,000 mm beam carries a 1,000 N point load 200 mm from the left support. Is the deflection under the load also the maximum deflection?
Under the load isn't always the maximum
For a simply supported beam with a point load F at distance a from the left support (b = L − a), the deflection under the load is δ(a) = F·a²·b² / (3·E·I·L). When the load isn't centred (a ≠ b), the true maximum deflection sits elsewhere on the beam, closer to midspan from the short side. Only with a centred load (a = b = L/2) do 'under the load' and 'maximum' coincide.
Worked example: load at 200 mm of 1,000 mm
With F = 1,000 N, L = 1,000 mm, a = 200 mm, E = 210,000 N/mm² and I = 1,000,000 mm⁴, the deflection under the load is ≈ 0.0406 mm. The true maximum deflection is larger: ≈ 0.0575 mm, at x ≈ 434.3 mm from the support nearest the load. The maximum is about 42% higher than the under-load deflection; that local value is about 29% below the true maximum.
A cantilever makes the gap even bigger
On a cantilever (fixed at x = 0) with a load F at distance a from the fixed end, the segment beyond the load carries no further moment and stays straight at the slope reached at x = a — so the tip translates further than the point under the load itself. With F = 500 N, L = 800 mm, a = 300 mm, E = 210,000 N/mm² and I = 500,000 mm⁴, the deflection under the load is ≈ 0.0429 mm, but at the free tip ≈ 0.150 mm — 3.5 times as much. The maximum moment (at the fixed end, M = F·a) stays the same either way; only the deflection differs.
Assumptions and applicability limits
The model calculates small elastic deformation per Euler-Bernoulli, a single point load or a single uniformly distributed load per calculation — not combined. Shear deformation, dynamics/vibration and non-ideal supports fall outside it. Whether a result is acceptable depends on the application: the tool gives generic guideline ratios (L/180 to L/750) as orientation, not a replacement for the applicable standard (the applicable structural standard and functional requirements for the machine or crane).