Beam Deflection Calculator — Maximum Deflection, Required Moment of Inertia & Deflection Limit Check
Calculate maximum beam deflection, required moment of inertia, or deflection compliance using beam length, load, Young’s modulus, and section inertia.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Calculation mode
Beam type
Load
Beam length
Material properties
Cross-section
Deflection limit
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
maximum deflection formula:
For a simply supported beam with point load:
δmax = P · L³ / (48 · E · I)
For a simply supported beam with uniformly distributed load:
δmax = 5 · w · L⁴ / (384 · E · I)
For a cantilever beam with point load:
δmax = P · L³ / (3 · E · I)
For a cantilever beam with uniformly distributed load:
δmax = w · L⁴ / (8 · E · I)
required moment of inertia formula:
Deflection limit:
δlimit = L / ratio
For a simply supported beam with point load:
Ireq = P · L³ / (48 · E · δlimit)
For a simply supported beam with uniformly distributed load:
Ireq = 5 · w · L⁴ / (384 · E · δlimit)
For a cantilever beam with point load:
Ireq = P · L³ / (3 · E · δlimit)
For a cantilever beam with uniformly distributed load:
Ireq = w · L⁴ / (8 · E · δlimit)
beam deflection check formula:
δ = calculated beam deflection
Utilization = δ / δlimit
where:
- δmax — maximum deflection (m)
- δ — maximum deflection used for beam check (m)
- Ireq — required moment of inertia (m⁴)
- δlimit — allowable deflection limit (m)
- P — point load (N)
- w — uniformly distributed load (N/m)
- L — beam length (m)
- E — Young’s modulus (Pa)
- I — moment of inertia (m⁴)
- ratio — deflection ratio denominator, for example 250 for L / 250 (-)
When to use this calculator
When to use this calculator:
- Calculate maximum deflection for a simply supported beam under a central point load.
- Calculate maximum deflection for a simply supported beam under a uniformly distributed load.
- Calculate maximum deflection for a cantilever beam under an end point load.
- Calculate maximum deflection for a cantilever beam under a uniformly distributed load.
- Determine the required moment of inertia needed to satisfy a selected deflection ratio.
- Check whether calculated beam deflection is below the selected deflection limit.
How to interpret the result
Maximum deflection is defined as the largest calculated beam displacement for the selected support condition and load type.
Required moment of inertia is defined as the section inertia needed to keep beam deflection within the selected deflection ratio.
Beam deflection depends on beam length, load, Young’s modulus, and moment of inertia. Increasing beam length or load increases deflection. Increasing Young’s modulus or moment of inertia decreases deflection.
- Info — the result is returned for maximum deflection or required moment of inertia without a pass/fail comparison.
- Safe — calculated deflection is less than or equal to the deflection limit, because utilization ≤ 1.
- Unsafe — calculated deflection is greater than the deflection limit, because utilization > 1.
- Invalid — required input data is missing, non-finite, or not greater than zero where required by the calculation.
The result is used to evaluate beam stiffness, required section inertia, or compliance with the selected deflection ratio.
Calculation example
Example:
A user wants to check the maximum deflection of a simply supported beam with a point load.
- Support condition: Simply supported
- Load type: Point load
- P — Point load: 1000 N
- L — Beam length: 2 m
- E — Young’s modulus: 210000000000 Pa
- I — Moment of inertia: 0.000008 m⁴
δmax = 0.0000992 m, which is approximately 0.0992 mm.
Assumptions and limitations
- The calculation uses one of four load cases: simply supported point load, simply supported uniformly distributed load, cantilever point load, or cantilever uniformly distributed load.
- The deflection limit is calculated as beam length divided by the selected deflection ratio.
- Beam stiffness is represented by Young’s modulus and moment of inertia.
- Section-based calculations use section properties returned by the section model when custom inertia is not provided.