Beam Deflection Formulas
Beam deflection is a measure of the displacement of a structural element under a load. The calculation varies depending on the beam's supports and the type of load applied.
Frequently Asked Questions
The maximum deflection occurs at the center and is calculated as: delta = (P * L^3) / (48 * E * I).
Beam deflection is the vertical displacement of a structural beam from its unloaded position under applied loads. Excessive deflection causes serviceability issues (cracking in finishes, door binding) even if the beam is structurally adequate. Most codes limit deflection to L/360 for live loads and L/240 for total loads, where L is the span length.
Beam deflection depends on: applied load type and magnitude; span length (deflection increases with the cube of span — doubling span increases deflection 8×); moment of inertia (I) of the cross-section (larger I = less deflection); modulus of elasticity (E) of the material (steel deflects less than timber); and support conditions (cantilever deflects 4× more than simply supported for the same loading).
For a cantilever beam with a point load P at the free end: maximum deflection δ = PL³ / (3EI), occurring at the free end. For a uniformly distributed load w (N/m): δ = wL⁴ / (8EI). Cantilever beams deflect 4 to 5 times more than simply supported beams of the same span and loading — this is a critical design consideration.
Modulus of elasticity (Young's Modulus): Structural Steel — 200 GPa; Aluminium alloy — 69 GPa; Concrete — 20–30 GPa (varies with grade); Timber along grain — 8–15 GPa (species dependent); Carbon fibre composite — 70–300 GPa. Steel is 3–10× stiffer than timber, which is why steel beams deflect far less for the same cross-section.
The second moment of area (I), or moment of inertia, measures a cross-section's resistance to bending. For a rectangular section: I = b×h³/12, where b is width and h is depth. Doubling the depth increases I by 8×, reducing deflection 8×. This is why I-beams (wide flange sections) are efficient — most material is placed far from the neutral axis where it contributes most to stiffness.
Beam Deflection: Euler-Bernoulli Theory in Practice
Calculating how much a loaded beam will sag is one of the foundational problems of structural engineering. The Euler-Bernoulli beam theory, developed in the 1750s and still the workhorse for the great majority of practical beam analyses today, gives closed-form deflection equations for the standard load and support cases. The theory assumes the beam is long compared to its cross-section, the material is linearly elastic, and plane sections remain plane after bending — all reasonable for most timber, steel, and concrete beams in service. For very deep beams or short spans, Timoshenko beam theory (which adds shear deformation) is the next step up.
The Standard Cases — Closed-Form Solutions
Simply supported beam, central point load P, span L: max deflection at midspan δmax = PL³ / (48EI); max bending moment at midspan M = PL/4. Simply supported beam, uniformly distributed load w (force per length), span L: δmax = 5wL⁴ / (384EI); max moment M = wL²/8. Cantilever, tip point load P: δmax = PL³ / (3EI); max moment at fixed end M = PL. Cantilever, UDL w: δmax = wL⁴ / (8EI); max moment M = wL²/2. Fixed-fixed beam, central point load P: δmax = PL³ / (192EI) — quarter the deflection of the simply-supported version because the end moments share the load. The beam calculator above implements all of these and lets you specify cross-section to compute I.
Section Property: I, the Second Moment of Area
The flexural-rigidity term EI in the deflection formulae has two parts: E is the modulus of elasticity of the material (200 GPa for structural steel, 25–33 GPa for normal-weight concrete depending on grade, 10–14 GPa for typical structural timber along the grain), and I is the second moment of area of the cross-section. For a rectangular beam: I = bh³ / 12, where b is width and h is height. The cube on h shows why depth dominates: doubling beam height reduces deflection by a factor of 8, but doubling width only halves it. For a circular section: I = π D⁴ / 64. For a hollow circular: I = π(Do⁴ − Di⁴) / 64. For I-sections (W-shapes), the value is published in steel-section tables.
Worked Example: A Wooden Floor Joist
A 50 mm × 200 mm timber floor joist (deep dimension vertical) spans 4 m, simply supported, carrying a UDL of 2 kN/m. E (Douglas fir) = 12.4 GPa = 12,400 N/mm². I = (50 × 200³) / 12 = 33,333,333 mm⁴ = 3.33 × 10⁻⁵ m⁴. δmax = (5 × 2,000 × 4⁴) / (384 × 12.4 × 10⁹ × 3.33 × 10⁻⁵) = 2,560,000 / 158,720 = 16.1 mm. Span/deflection ratio = 4,000 / 16.1 = 248. For a residential floor (typical L/360 limit for brittle finishes), required deflection limit = 4,000/360 = 11.1 mm. The joist fails serviceability — need a deeper section, e.g. 50 × 250 (which would give δ ~ 8.3 mm and L/δ = 482). The beam calculator above performs this exact analysis and shows the L/δ ratio against typical code limits.
Serviceability Limits in Common Codes
Codes set deflection limits to keep finishes intact and occupants comfortable, not just to prevent structural failure: L/180 for cantilevers carrying ductile finishes; L/240 for floors with no brittle finish; L/360 for floors carrying brittle finishes such as plaster, tile, or stone; L/480 for sensitive equipment (museum cases, precision laboratory benches). The beam calculator displays the L/δ result alongside these reference limits so you can immediately see whether the section passes serviceability for a given application.
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