Beam Deflection Calculator
Find the maximum mid-span deflection of a simply supported beam under a uniformly distributed load. Use this during structural design to verify serviceability limits before specifying a section.
About this calculator
For a simply supported beam carrying a uniform load w (N/m) over span L (m), the maximum deflection occurs at mid-span and is given by: δ_max = (5 × w × L⁴) / (384 × E × I). Here E is the modulus of elasticity in MPa and I is the second moment of area (moment of inertia) in cm⁴. Because E is in MPa (N/mm²) and I is in cm⁴, a unit conversion factor of 10,000 is applied in the denominator to keep the result in metres, matching the formula used here: δ = (5 × w × L⁴) / (384 × E × I × 10,000). This equation assumes linear elastic behaviour, a prismatic cross-section, and negligible shear deformation. Deflection limits in most codes (e.g., L/360 for live load) must be checked against this value.
How to use
Consider a steel beam with span L = 6 m, uniform load w = 10,000 N/m, E = 200,000 MPa, and I = 8,000 cm⁴. Substituting: δ = (5 × 10,000 × 6⁴) / (384 × 200,000 × 8,000 × 10,000). Numerator: 5 × 10,000 × 1,296 = 64,800,000. Denominator: 384 × 200,000 × 8,000 × 10,000 = 6.144 × 10¹⁵ … wait — computing carefully: 384 × 200,000 = 76,800,000; × 8,000 = 614,400,000,000; × 10,000 = 6.144 × 10¹⁵. δ = 64,800,000 / 6,144,000,000,000 ≈ 0.0000106 m ≈ 10.6 mm. The L/360 limit for a 6 m span is 16.7 mm, so this beam passes serviceability.
Frequently asked questions
What is the formula for maximum deflection of a simply supported beam with a uniform load?
The standard formula is δ_max = (5wL⁴) / (384EI), where w is the load per unit length, L is the span, E is Young's modulus, and I is the second moment of area. This result comes from integrating the bending moment equation twice and applying boundary conditions of zero deflection at both supports. The factor 5/384 is specific to uniform loading; other load patterns (point load at centre, cantilever, etc.) produce different numerical coefficients.
How do I choose the right moment of inertia for my beam section?
The moment of inertia I depends on the cross-sectional shape and the axis of bending. For an I-beam bending about its strong axis, use the value tabulated in steel section tables (e.g., AISC or BS 4). For a rectangular timber section of width b and depth d, I = bd³/12. Always use the axis that corresponds to the direction of loading — weak-axis bending gives a much larger deflection. Getting I wrong is one of the most common errors in preliminary beam sizing.
When does beam deflection become a structural or serviceability problem?
Codes distinguish between strength (stress) and serviceability (deflection) limits. A beam may be structurally safe in bending yet still fail serviceability if it deflects excessively, causing cracked ceilings, jammed doors, or ponding on flat roofs. Typical limits are L/360 under live load and L/240 under total load per IBC/ASCE 7. Long-term creep in timber or composite beams can double the initial elastic deflection, so designers often apply a creep multiplier (e.g., 2.0 for seasoned timber) to the calculated value.