Beam Deflection Calculator
Find the maximum mid-span deflection of a simply supported beam under point, uniform, or triangular loads. Essential for structural engineers verifying serviceability limits and code compliance.
About this calculator
Beam deflection measures how much a structural member bends under load. For a simply supported beam with a central point load, the maximum deflection is δ = PL³ / (48EI). Under a uniformly distributed load it becomes δ = 5wL⁴ / (384EI), and for a triangularly distributed load δ = wL⁴ / (120EI). In all formulas, L is the span length (converted to inches), E is the elastic modulus (stiffness of the material in psi), and I is the second moment of area (in⁴), which captures the beam's cross-sectional resistance to bending. A higher E or I reduces deflection proportionally. Engineers compare the computed deflection against allowable limits — commonly L/360 for live loads — to ensure the beam is serviceable and safe.
How to use
Suppose you have a 10 ft simply supported steel beam (E = 29,000,000 psi, I = 50 in⁴) carrying a central point load of 2,000 lbs. First, convert length: 10 ft × 12 = 120 in. Then apply the point-load formula: δ = PL³ / (48EI) = (2000 × 120³) / (48 × 29,000,000 × 50). Numerator: 2000 × 1,728,000 = 3,456,000,000. Denominator: 48 × 29,000,000 × 50 = 69,600,000,000. δ = 3,456,000,000 / 69,600,000,000 ≈ 0.0497 in. The allowable deflection (L/360) = 120/360 = 0.333 in, so this beam passes comfortably.
Frequently asked questions
What is the formula for maximum deflection of a simply supported beam with a point load?
The classic formula is δ = PL³ / (48EI), where P is the applied point load at mid-span, L is the beam span in inches, E is the material's elastic modulus in psi, and I is the moment of inertia in in⁴. This formula assumes the load is applied exactly at the center of the span and the beam ends are simply supported (pinned-roller). For off-center loads a different, more complex formula applies. The result gives the maximum deflection, which occurs at mid-span.
How does the moment of inertia affect beam deflection?
The moment of inertia I appears in the denominator of all deflection formulas, meaning a larger I directly reduces deflection. I is a geometric property of the beam's cross-section that measures its resistance to bending about the neutral axis. For a rectangular section, I = bh³/12, so doubling the depth h reduces deflection by a factor of eight. This is why I-beams and wide-flange sections are designed with most material in the flanges — far from the neutral axis — to maximize I without adding excessive weight.
When should I be concerned about beam deflection exceeding allowable limits?
Deflection becomes a concern when it exceeds the serviceability limits set by building codes, typically L/360 for live load deflection and L/240 for total load deflection under standards like AISC and IBC. Excessive deflection can crack ceilings or partitions below the beam, cause ponding of water on flat roofs, and create a noticeable sag that is aesthetically unacceptable. Even if a beam is structurally strong enough (stress is within allowable limits), it may still fail the deflection serviceability check and require a deeper or stiffer section. Always verify both strength and serviceability when designing beams.