Simple-span center-load deflection
δ_max = PL³/(48EI)
Maximum deflection occurs at midspan for a centered point load.
Beam Deflection Calculator
Calculate elastic deflection for two standard Euler–Bernoulli beam cases: a simply supported beam with a center point load, or a cantilever with a point load at the free end.
Access: Free to use, no installation, and No account required.
Inputs
This is linear-elastic small-deflection beam theory with constant E and I. It does not check yielding, lateral-torsional buckling, shear deformation, connections, local buckling, vibration, fatigue, or code compliance.
Results
Standard formulas
For a simply supported beam with a center point load, δmax = PL³/(48EI). For a cantilever with an end point load, δmax = PL³/(3EI).
Section property warning
The second moment of area must use the bending axis that matches the actual beam orientation. A section rotated 90° can have a substantially different I.
Engineering reference
Beam Deflection Calculator: background and worked detail
Use the Beam Deflection Calculator to estimate maximum elastic deflection, bending moment, reactions, and slope for a simply supported beam with a center point load or a cantilever with an end point load.
Deflection under standard load cases
Closed-form beam-deflection formulas come from Euler-Bernoulli beam theory under small-deflection linear-elastic assumptions. Bending stiffness is the product EI, so both material modulus and the section's second moment of area strongly influence deflection.
The orientation of the section matters because I must be taken about the actual bending axis. The focused calculator covers two standard load cases; general loading and code design require the full structural or strength workbench.
Equations used by this calculator
Cantilever end-load deflection
δ_max = PL³/(3EI)
Maximum deflection occurs at the free end.
Bending stiffness
EI
Elastic modulus times second moment of area controls flexural rigidity.
2 m steel simple span with 1 kN center load
Use P = 1,000 N, L = 2 m, E = 200 GPa, and I = 1,000 cm⁴.
- Convert I = 1,000 cm⁴ to 1.0×10⁻⁵ m⁴.
- δ_max = PL³/(48EI) ≈ 0.0833 mm.
- Maximum bending moment is PL/4 = 500 N·m.
Result: The small deflection is consistent with the high EI value in this example.
Validation checks
Load scaling
Case: Double P while holding E, I, and L fixed.
Expected: Deflection and bending moment should double.
Length scaling
Case: Double L for the same center-load simple span.
Expected: Deflection should increase by a factor of eight because of the L³ dependence.
Elastic, small-deflection limits
Assumptions
- Small deflection and linear-elastic Euler-Bernoulli beam behavior.
- Constant E and I over the span.
- Loads are static and match one of the two supported standard cases.
Limitations
- Does not check stress capacity, shear deformation, lateral-torsional buckling, vibration, local buckling, connections, or code serviceability limits.
- Large-deflection, tapered, composite, cracked, or nonlinear members require a different model.
Beam Deflection Calculator FAQ
Why does beam orientation matter?
The second moment of area depends on the bending axis. Rotating a rectangular section can change I by a large factor.
Does this calculator check whether the beam is strong enough?
No. Deflection is a serviceability response. Strength, stability, connections, and applicable code limits must be checked separately.
Why does deflection grow so quickly with span?
For these point-load cases, elastic deflection is proportional to L³, so span is a dominant design variable.
Where this calculation comes from
Shared with the Statics and Strength Workbench, which adds combined loading, shear diagrams, and section design.
Open the source workbench →Read calculation and source methodology →