Beam Deflection Calculator (Max δ)
Choose a beam and load case, enter inputs with units, and calculate the maximum deflection.
- Euler–Bernoulli beam theory for small deflections: deflection varies inversely with E·I.
- Standard closed-form max-deflection cases are used for simply supported and cantilever beams.
- The off-center simply supported point-load case evaluates the valid stationary point to get the true maximum deflection.
Use this Beam Deflection Calculator to estimate the maximum deflection of simply supported and cantilever beams under common point loads, uniform loads, and end moments.
Beam Deflection Formulas
Beam deflection depends mainly on the applied load, span length, Young’s modulus E, and area moment of inertia I.
Simply supported — center point load:
δmax = PL3 ÷ (48EI)
Simply supported — uniform load:
δmax = 5wL4 ÷ (384EI)
Cantilever — end point load:
δmax = PL3 ÷ (3EI)
Cantilever — uniform load:
δmax = wL4 ÷ (8EI)
Cantilever — end moment:
δmax = ML2 ÷ (2EI)
For a simply supported beam with an off-center point load, the maximum deflection location depends on the load position. The calculator evaluates the beam deflection curve to determine the maximum value and its location.
Where:
- P = point load
- w = uniform load per unit length
- L = beam span or cantilever length
- E = Young’s modulus
- I = area moment of inertia
- M = applied end moment
Example
Suppose a simply supported beam has a 3.5 m span, a 25 kN point load at midspan, Young’s modulus of 200 GPa, and an area moment of inertia of 8,000 cm4.
Using:
δmax = PL3 ÷ (48EI)
Convert the values to compatible SI units:
P = 25,000 N
E = 200,000,000,000 Pa
I = 0.00008 m4
δmax = (25,000 × 3.53) ÷ (48 × 200,000,000,000 × 0.00008)
δmax ≈ 0.001396 m = 1.40 mm
The estimated maximum deflection is approximately 1.40 mm at midspan.
Important Notes
- Select the beam support and load case that matches the actual situation.
- Use compatible units for length, load, Young’s modulus, and moment of inertia.
- The formulas assume elastic, small-deflection beam behaviour with constant E and I.
- They do not account for yielding, cracking, shear deformation, creep, vibration, buckling, connection flexibility, or complex load combinations.
- A low calculated deflection does not by itself prove that a beam is structurally safe.
For structural beams in buildings or other safety-critical applications, deflection must be checked together with strength, stability, load combinations, serviceability requirements, and the applicable design code.