Beam Deflection Calculator
Calculate beam deflection, maximum displacement, and flexural rigidity for simply-supported and cantilever beams under point or uniformly distributed loads. Free online structural engineering tool.
About This Calculator
The Beam Deflection Calculator helps structural engineers, architects, civil engineering students, and construction professionals determine the maximum deflection of simply-supported and cantilever beams under common load configurations. Understanding beam deflection is critical for ensuring structural safety, serviceability, and building code compliance in bridges, buildings, platforms, and other load-bearing structures.
The calculator uses the standard beam deflection formulas from Euler-Bernoulli beam theory. For a simply-supported beam with a point load at the center: δ = PL³ / (48EI). For a cantilever beam with a point load at the free end: δ = PL³ / (3EI). For uniformly distributed loads, the formulas are δ = 5wL⁴ / (384EI) for simply-supported and δ = wL⁴ / (8EI) for cantilever beams. The flexural rigidity EI combines the material's modulus of elasticity with the cross-section's area moment of inertia.
The modulus of elasticity (E) depends on the beam material — steel (~200 GPa), aluminum (~69 GPa), concrete (15-50 GPa), and various wood species (5-15 GPa). The moment of inertia (I) depends on the cross-sectional shape — for rectangular beams: I = (width × height³) / 12, oriented about the bending axis.
Regional Notes
India: Structural design follows IS 456:2000 for concrete and IS 800:2007 for steel. Common beam spans range from 3-8 meters for residential buildings. Steel reinforcement bars (rebars) follow IS 1786 standards with Fe415 and Fe500 grades having E ≈ 200 GPa.
United States: Building design follows AISC 360 for steel and ACI 318 for concrete structures. Typical steel wide-flange beams (W-shapes) have published moment of inertia values in the AISC manual. Allowable deflection limits are typically L/360 for floors and L/240 for roofs under live loads.
United Kingdom: Structural design follows Eurocode 3 (BS EN 1993) for steel and Eurocode 2 (BS EN 1992) for concrete. Serviceability deflection limits are typically span/250 for total deflection and span/300 for live load deflection under BS EN 1990.
Frequently Asked Questions
What is beam deflection?
Beam deflection is the vertical displacement of a point along the centroid of a beam when it is subjected to a load. It measures how much a beam bends under the applied force or weight.
How do I use this beam deflection calculator?
Select the beam type (simply-supported or cantilever) and load type (point load at center or uniformly distributed load). Enter the span length in meters, the load value, modulus of elasticity in GPa, and moment of inertia in cm⁴. Click Calculate to see the maximum deflection and flexural rigidity.
What is the formula for beam deflection?
For a simply-supported beam with a point load at the center: δ = PL³/(48EI). For a cantilever beam with a point load at the free end: δ = PL³/(3EI). For uniformly distributed loads, use δ = 5wL⁴/(384EI) for simply-supported and δ = wL⁴/(8EI) for cantilever beams.
What is flexural rigidity?
Flexural rigidity (EI) is the product of the modulus of elasticity (E) and the area moment of inertia (I) of a beam's cross-section. It represents the beam's resistance to bending — the higher the flexural rigidity, the smaller the deflection under load.
What is a typical modulus of elasticity for steel?
Structural steel has a modulus of elasticity of approximately 200 GPa (29,000 ksi). Aluminum is about 69 GPa (10,000 ksi), concrete ranges from 15-50 GPa depending on the mix, and wood (e.g., eastern white pine) is around 6.8 GPa.
How is moment of inertia calculated for a rectangular beam?
For a rectangular cross-section, the area moment of inertia about the horizontal axis is I = (width × height³) / 12. For example, a 20 cm wide by 30 cm deep beam gives I = 20 × 30³ / 12 = 45,000 cm⁴.
What is the difference between simply-supported and cantilever beams?
A simply-supported beam rests on supports at both ends and deflects downward in the middle under load. A cantilever beam is fixed at one end and free at the other, with maximum deflection occurring at the free end. Cantilever beams deflect significantly more than simply-supported beams under the same load.
How does beam length affect deflection?
Beam deflection increases with the cube (for point loads) or the fourth power (for UDL) of the span length. Doubling the span length increases deflection by 8 times for point loads or 16 times for distributed loads. This makes span length the most critical factor in beam design.