Beam Deflection (Center Load)
Compute max deflection δ = PL³/(48EI) of a simply-supported beam with center point load.
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Beam deflection: δ = 5·w·L⁴ / (384·E·I)
Take a beam supported at both ends carrying a uniformly distributed load w across the whole span L. Its largest deflection happens at mid-span and comes out to δ = 5·w·L⁴ / (384·E·I), with E being Young's modulus and I the second moment of area. Steel runs around E ≈ 200 GPa. Concrete bends much more easily at E ≈ 25 GPa, and aluminium falls in between near 70 GPa. Watch that L⁴ term closely. Double the span and the deflection jumps by a factor of 16, which is exactly why long spans tend to fail serviceability checks long before they run out of strength. Most codes cap deflection at δ < L/300, loosening to L/250 for roofs and tightening to L/500 where partitions are sensitive. To see it in numbers: L = 4 m, w = 5 kN/m, E = 200 GPa, I = 2·10⁷ mm⁴ gives δ ≈ 8 mm, which works out to L/500 and stays inside the limit.
Applications
It shows up when you size reinforced-concrete beams and slabs (NBR 6118), steel rafters and joists for industrial roofing (NBR 8800), or timber floors (NBR 7190). Engineers also lean on it to check vibration comfort, to size crane runways and bridges, and to verify serviceability limit states (SLS).
FAQ
Why is the limit L/300 and not based on stress? A stress check tells you whether the beam will actually break. A deflection check is about whether people can live with it: cracked plaster, doors that stick, a floor that bounces underfoot. You need to pass both.
What if the load is concentrated, not distributed? For a single point load at mid-span, switch to δ = P·L³/(48·E·I). The exponent on L drops from 4 to 3, though everything else works the same way.
Does the formula handle cantilevers? No. A cantilever loaded at its free end follows δ = P·L³/(3·E·I), which makes it roughly 16 times more flexible than the same beam supported at both ends.
Related Tools
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Computes max deflection of a simply supported rectangular slab under uniform load.
Beam Deflection (Uniform Load)
Max deflection δ = 5wL⁴/(384EI) of simply-supported beam with uniformly distributed load. Useful for basic structural sizing.
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Computes max deflection of a simply supported beam under uniformly distributed load.
Cubic Mean Load (Bearing)
Calculate the equivalent mean load of a bearing under a cycle with two different loads, P_m = ∛(P₁³·U₁ + P₂³·U₂), from the loads P₁ and P₂ (N) and the time (or revolution) fractions during which they act U₁ and U₂ (with U₁ + U₂ = 1). Many bearings do not work under CONSTANT load: the load varies over the operating cycle (a press loading and unloading, a motor accelerating and decelerating, a machine with different work phases). To compute life in this case, the variable cycle is replaced by an equivalent CONSTANT load causing the same fatigue damage — the mean load. But the mean is NOT arithmetic: since fatigue damage is proportional to load CUBED (the life exponent p=3), the mean load is a time-fraction-weighted mean, but with the loads cubed (then cube-rooted) — the so-called cubic mean or 'fatigue-weighted mean'. This makes HIGH loads weigh disproportionately more (a double load causes 8× more damage), so even a small fraction of time at high load dominates the result. This formula (here for two load levels; it generalizes to several) is essential to size bearings in variable-load machines, avoiding underestimating the damage. Enter the two loads and their time fractions.
Linear Explosive Charge
Calculate the linear loading density of a blast hole, q = (π/4) × d² × ρ, from the hole diameter d (mm) and the explosive density ρ (g/cm³). The result, in kg of explosive per meter of hole, is how much explosive fits in each meter of charged column — a central parameter of rock blast design. Multiplied by the hole charge height, it gives the charge per hole; combined with the blasted rock volume, it gives the powder factor. Larger diameters and denser explosives raise the linear charge. Enter the hole diameter and the explosive density.
Eccentricity Ratio
Calculate the eccentricity ratio of a hydrodynamic bearing, ε = e ÷ c, dividing the eccentricity e (shaft centre offset from bearing centre) by the radial clearance c. The result (between 0 and 1) describes the shaft position within the bearing under load: ε = 0 means a centred shaft (no load); ε near 1 means the shaft nearly touches the bearing (heavily loaded, minimum film at the limit). Eccentricity grows with load and decreases with viscosity and speed. The minimum film thickness is h_min = c·(1 − ε). Enter the eccentricity and the radial clearance.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.