Beam Deflection Calculator

A beam that fails to support its load is an obvious structural failure. A beam that sags more than L/360 under service loads — causing cracked plaster, sticking doors, and visible floor bounce — is a serviceability failure that is equally costly and more common. Deflection governs structural design far more often than strength in modern construction, and the calculator for beam deflection computes the critical mid-span displacement from fundamental elastic beam theory.

Deflection Formulas for Common Loading Cases

Maximum deflection for the two most common beam configurations:

Simply supported beam, central point load P:

delta_max = P × L³ / (48 × E × I)

Cantilever beam, point load P at free end:

delta_max = P × L³ / (3 × E × I)

where P is the applied force (N or kips), L is the beam span (m or ft), E is the elastic modulus (Pa or ksi), and I is the second moment of area (m⁴ or in⁴). The cantilever deflects 16× more than the simply supported beam under the same load and span — which is why cantilever structural elements require much deeper cross-sections for equivalent stiffness. Use this online calculator for any beam configuration. The beam load calculator provides the complementary analysis including bending moments and reactions.

Moment of Inertia for Common Cross-Sections

The second moment of area I (moment of inertia) quantifies a cross-section's resistance to bending. For common structural sections:

• Solid rectangular section (b × h): I = b × h³ / 12; height h is the dimension parallel to the bending direction — making a 2×6 oriented with the 6-inch face vertical (I = 2×6³/12 = 36 in⁴) far stiffer than the same lumber flat (I = 6×2³/12 = 4 in⁴), a 9× difference in deflection
• Hollow rectangular tube: I = (b_outer × h_outer³ − b_inner × h_inner³) / 12
• Wide flange steel beam (W-shape): I values tabulated in AISC Steel Construction Manual; a W8×31 has I_xx = 110 in⁴; a W12×50 has I_xx = 394 in⁴
• Solid circular section: I = pi × d⁴ / 64 where d is diameter

Allowable Deflection Limits in Building Codes

Structural codes specify deflection limits as fractions of the span L to control serviceability:

• L/360: standard limit for floor beams supporting plaster ceilings — the most stringent common requirement; a 20-ft span beam can deflect no more than 0.67 inches
• L/240: typical limit for roof beams without plaster or for floors without sensitive finishes
• L/180: minimum for roof members not supporting ceilings; less stringent
• L/400 to L/600: very tight limits for structures supporting sensitive equipment, precision machinery, or long-span floors where bounce perception is critical

The steel beam calculator and beam and column calculators provide complementary structural analysis tools.

Superposition: Adding Multiple Load Cases

For linear elastic beams, deflections from multiple loads can be added directly (principle of superposition). A simply supported beam with both a central point load P and a uniformly distributed load w: total deflection = P×L³/(48EI) + 5×w×L⁴/(384EI). This additive property is valid as long as deflections are small relative to the span (typically delta/L below 1/50) and the beam material remains elastic. Large deflections and material nonlinearity require more advanced analysis methods beyond the scope of simple beam theory.