About Beam Deflection Calculator
The beam deflection calculator returns the maximum deflection, maximum bending moment, and maximum shear force for four common single-span beam configurations: a simply supported beam under a uniformly distributed load, a simply supported beam under a central point load, a cantilever under a uniformly distributed load, and a cantilever under an end point load.
Enter the modulus of elasticity E, the second moment of area I, the span or cantilever length L, and the applied load, all in a consistent SI unit set (newtons, millimetres, megapascals). The tool evaluates the closed-form elastic solutions and plots the deflected shape so you can verify serviceability against span-over-deflection limits.
How It Works
- Choose one of the four load cases (simply supported or cantilever, with a distributed or point load).
- Enter E (MPa), I (mm^4), L (mm), and the load magnitude (w in N/mm for distributed loads, P in N for point loads).
- The calculator applies the classical Euler-Bernoulli closed-form expressions for maximum deflection, moment, and shear.
- It samples the elastic deflection equation along the member to draw the deflected shape and report the location of the maximum.
Worked Example
A simply supported steel beam with E = 200,000 MPa, I = 1.0e8 mm^4, span L = 5000 mm carries a uniformly distributed load w = 10 N/mm. The maximum deflection is 5 w L^4 / (384 E I) = 5 * 10 * 5000^4 / (384 * 200000 * 1e8) = 4.069 mm at midspan. The maximum moment is w L^2 / 8 = 10 * 5000^2 / 8 = 3.125e7 N.mm, and the maximum shear is w L / 2 = 25,000 N at each support.
Formulas
- Simply supported, UDL - max deflection
d_max = 5 * w * L^4 / (384 * E * I)- Simply supported, central point load - max deflection
d_max = P * L^3 / (48 * E * I)- Cantilever, UDL - max deflection
d_max = w * L^4 / (8 * E * I)- Cantilever, end point load - max deflection
d_max = P * L^3 / (3 * E * I)- Maximum moment and shear (summary)
SS-UDL: M = w L^2 / 8, V = w L / 2 | SS-P: M = P L / 4, V = P / 2 | Cant-UDL: M = w L^2 / 2, V = w L | Cant-P: M = P L, V = P
Standards & References
- Euler-Bernoulli beam theory
- Roark’s Formulas for Stress and Strain
- Gere & Timoshenko, Mechanics of Materials
Frequently Asked Questions
Which load cases does the beam deflection calculator support?
It supports four single-span cases: simply supported with a uniformly distributed load, simply supported with a central point load, cantilever with a uniformly distributed load, and cantilever with an end point load.
What units should I use for the inputs?
Use a consistent SI set: E in MPa (N/mm^2), I in mm^4, L in mm, distributed load w in N/mm, and point load P in N. Deflection then comes out in mm, moment in N.mm, and shear in N.
Why is the cantilever deflection so much larger than the simply supported case?
A cantilever is restrained at only one end, so the same load produces far larger rotations and tip deflection. For an end point load the cantilever deflects P L^3 / 3EI versus P L^3 / 48EI at the midspan of a simply supported beam, sixteen times more.
Does the calculator account for self-weight or shear deformation?
No. It uses the Euler-Bernoulli closed-form solutions for the applied load only and neglects shear deformation, which is accurate for slender beams. Add self-weight to the distributed load if you need to include it.