Beam Deflection Calculator
Find the maximum midpoint deflection of a simply supported or cantilever beam under a central point load. Essential for structural engineers checking serviceability limits and avoiding excessive sag.
About this calculator
Beam deflection quantifies how much a beam bends under an applied load, governed by Euler-Bernoulli beam theory. For a simply supported beam with a central point load, maximum deflection is δ = (F × L³) / (48 × E × I). For a cantilever beam with a tip load, it is δ = (F × L³) / (3 × E × I). Here F is the applied force in Newtons, L is the beam span in metres, E is Young's modulus (elastic modulus) in Pascals, and I is the second moment of area (moment of inertia) in m⁴. The cubic dependence on length means doubling the span increases deflection eightfold. E × I is called flexural rigidity — a higher value means a stiffer beam. Engineers compare the result against serviceability limits, often L/360 for floors or L/180 for roofs, to ensure user comfort and structural integrity.
How to use
A simply supported steel beam spans L = 5 m and carries a central point load of F = 10,000 N. Steel has E = 200 × 10⁹ Pa and the cross-section has I = 8.33 × 10⁻⁶ m⁴. Step 1: Compute the numerator — 10,000 × 5³ = 10,000 × 125 = 1,250,000. Step 2: Compute the denominator — 48 × 200 × 10⁹ × 8.33 × 10⁻⁶ = 48 × 1,666,000 = 79,968,000. Step 3: δ = 1,250,000 / 79,968,000 ≈ 0.01563 m = 15.6 mm. The allowable limit of L/360 = 13.9 mm, so this beam marginally exceeds serviceability — a larger section is needed.
Frequently asked questions
What is the difference between deflection in a simply supported beam versus a cantilever beam?
A simply supported beam is restrained at both ends but free to rotate, so a central load causes the beam to bow downward with maximum deflection at midspan, calculated as FL³/(48EI). A cantilever is fixed at one end and free at the other, causing maximum deflection at the free tip, calculated as FL³/(3EI). For the same load and geometry, a cantilever deflects 16 times more than a simply supported beam, making cantilevers far more sensitive to span length and load magnitude.
How does the moment of inertia affect beam stiffness and deflection?
The second moment of area (I) measures how a cross-section's material is distributed relative to its neutral axis. Placing more material farther from the neutral axis — as in an I-beam — dramatically increases I and thus flexural rigidity EI. Since deflection is inversely proportional to I, doubling I halves the deflection. This is why I-beams and hollow sections are preferred in structural applications: they achieve high stiffness with minimal material weight.
What elastic modulus value should I use for common structural materials?
Young's modulus (E) varies significantly by material: structural steel is approximately 200 GPa, aluminium alloy around 70 GPa, timber roughly 8–15 GPa depending on species and grain direction, and reinforced concrete between 25–35 GPa. Using an incorrect E value will produce proportionally wrong deflection results. Always verify the value for your specific grade and standard (e.g. EN or ASTM) and consider temperature effects for high-temperature applications.