Beam Deflection Calculator
When a load is applied to a structural beam, the beam bends and the midpoint drops below its unloaded position. That drop is called deflection, and controlling it is just as important in structural design as controlling stress. Building codes commonly limit deflection to span divided by 300 or span divided by 500 for serviceability reasons, even when a beam has plenty of strength in reserve. This calculator uses the standard formula for the most common beam configuration: a simply supported beam (resting on two supports at each end, free to rotate) carrying a single point load at the midpoint of the span. The maximum deflection occurs at mid-span and equals P times L cubed divided by 48 times E times I, where P is the applied force, L is the clear span, E is the Young's modulus of the beam material, and I is the second moment of area of the beam cross-section about its bending axis. You enter the span in metres, the point load in kilonewtons, the Young's modulus in gigapascals, and the second moment of area in the box provided using scientific notation if needed (for example 8.33e-6 for a value of 8.33 times 10 to the minus 6 m to the fourth). The calculator returns maximum deflection in millimetres and maximum bending moment in kilonewton-metres. These results are for preliminary design and checking; a qualified engineer should review all structural calculations.
Simply supported beam with central point load only. For other load patterns (UDL, off-centre loads) or other support conditions, a different formula applies. For structural design, consult a qualified engineer.
How it works
For a simply supported beam of span L with a central point load P, the maximum deflection at mid-span is: delta = P * L^3 / (48 * E * I). Units must be consistent: P in newtons, L in metres, E in pascals, I in m to the fourth power. The calculator converts kN to N (multiply by 1000) and GPa to Pa (multiply by 10 to the 9). Maximum bending moment for a central point load is M = P * L / 4, expressed in kN·m. The deflection-to-span ratio L/delta indicates whether the beam meets typical serviceability limits (L/300 or L/500 are common thresholds).
Worked example
A simply supported steel beam has a 3 m span (L = 3 m), carries a central point load of 10 kN (P = 10,000 N), has a Young's modulus of 200 GPa (E = 200 times 10 to the 9 Pa), and a second moment of area I = 8.33 times 10 to the minus 6 m to the fourth. Deflection = 10000 times 3 cubed divided by (48 times 200e9 times 8.33e-6) = 10000 times 27 divided by 79,968,000 = 270,000 divided by 79,968,000 = 0.003376 m = 3.38 mm. Maximum bending moment = 10 times 3 divided by 4 = 7.50 kN·m. Deflection ratio = 3000 / 3.38 = L/888, well within the L/300 serviceability limit.
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