Beam Deflection Calculator
Determine the maximum deflection of a simply supported beam under a concentrated point load. Use this when sizing structural beams to ensure they stay within allowable deflection limits.
About this calculator
For a simply supported beam carrying a single point load P at midspan, the maximum deflection δ is given by δ = P·L³ / (48·E·I), where L is the beam length, E is the elastic modulus, and I is the second moment of area (moment of inertia). This formula comes from integrating the bending moment equation twice and applying boundary conditions at the supports. When the load is off-center, a correction factor is applied: roughly 0.74× for a load at the third-point and 0.42× for a load near the quarter-point. Stiffer materials (higher E) and deeper cross-sections (higher I) dramatically reduce deflection because both appear in the denominator. Engineers use deflection checks alongside stress checks to ensure serviceability under design loads.
How to use
Suppose a steel beam (E = 200,000 MPa, I = 8,500,000 mm⁴) spans L = 3,000 mm with a midspan load P = 5,000 N. Using the midspan formula: δ = P·L³ / (48·E·I) = 5,000 × 3,000³ / (48 × 200,000 × 8,500,000). Numerator: 5,000 × 27,000,000,000 = 1.35 × 10¹³. Denominator: 48 × 200,000 × 8,500,000 = 8.16 × 10¹³. δ = 1.35 × 10¹³ / 8.16 × 10¹³ ≈ 0.165 mm. This is well within the typical L/300 limit of 10 mm for a 3 m span.
Frequently asked questions
What is the formula for maximum deflection of a simply supported beam with a midspan point load?
The standard formula is δ = P·L³ / (48·E·I), where P is the applied load in Newtons, L is the span in mm, E is the elastic modulus of the material, and I is the second moment of area of the cross-section. This result is derived by double-integrating the bending moment diagram and applying zero-deflection boundary conditions at both pin supports. It gives the deflection at the exact midpoint, which is also the maximum for a symmetric load case.
How does load position affect beam deflection in a simply supported beam?
A load at midspan produces the greatest deflection because the bending moment is maximized at the center of the span. Moving the load toward a support reduces deflection significantly — a load at the third-point yields roughly 74% of the midspan deflection, and a load at the quarter-point yields around 42%. These correction factors arise from the different moment distributions along the beam. Engineers must check the actual load position when the load cannot be guaranteed to sit at midspan.
Why does the moment of inertia have such a large effect on beam deflection?
The moment of inertia I represents how efficiently the cross-section's area is distributed away from the neutral axis. Because I appears in the denominator of the deflection formula, doubling I halves the deflection. Increasing the beam depth is particularly effective because I scales with the cube of depth for rectangular sections. This is why I-beams and hollow sections are preferred in structural design — they place material far from the neutral axis, maximizing I for a given weight of steel.