Beam Deflection Calculator

When a beam is loaded, it bends — a phenomenon governed by the Euler–Bernoulli beam theory. The amount of bending, called deflection, depends on four key quantities: the applied load (P), the beam span (L), the material's resistance to deformation expressed as the modulus of elasticity (E), and the cross-section's geometric stiffness expressed as the second moment of area (I). The product EI is called the flexural rigidity of the beam and is the single most important indicator of how stiff a beam is. A higher EI means less deflection for any given load.

This calculator supports four standard configurations. For a simply supported beam with a central point load, the maximum deflection at midspan is δ = PL³ / (48EI). For a simply supported beam with a uniform distributed load (UDL), the formula becomes δ = 5wL⁴ / (384EI), where w is the load per unit length — equivalently expressed as δ = 5PL³ / (384EI) when P is the total load. A cantilever with an end point load deflects by δ = PL³ / (3EI) at its free tip, while a cantilever under a UDL deflects by δ = PL³ / (8EI) at the free end (again using total load P). The maximum slope (rotation) at the critical section is also reported, which is important for checking connection rotation limits and serviceability of attached components.

In practice, deflection limits are prescribed by design codes. AISC and Eurocode 3 typically require that floor beams do not exceed L/360 under live load, or L/240 under total load, to prevent cracking of finishes and ensure user comfort. The span-to-deflection ratio output by this calculator lets you check compliance instantly. Beam deflection calculations also appear in machine design to ensure shafts remain aligned, in aerospace for wing and fuselage stiffness analysis, and in geotechnical work when analysing pile and retaining wall behaviour.

This calculator applies Euler–Bernoulli beam theory, which assumes that plane sections remain plane, that deflections are small relative to the beam length (typically less than 1/10 of the span), and that the beam material is linear-elastic and homogeneous. It does not account for shear deformation, which becomes significant for deep beams with span-to-depth ratios below about 10 — in those cases, Timoshenko beam theory should be used instead. The formulas here cover only four standard load cases; eccentric point loads, multiple point loads, partial UDLs, and moment loads require superposition or more advanced analysis. Lateral-torsional buckling, which can reduce the effective stiffness of unrestrained beams, is also not captured. Real structures may involve composite sections, variable cross-sections, or material nonlinearity — all of which require specialist software such as finite element analysis. Always verify results against applicable design codes (e.g., Eurocode 3, AISC 360, AS 4100) and consult a licensed structural engineer for any safety-critical application.