EngineeringFormulaCivil & Structural
How to Calculate Beam Deflection
Beam deflection tells you how far a beam bends under load. It is a serviceability check: a beam can be strong enough in bending stress but still sag too much for floors, finishes, equipment alignment, or user comfort.
Final answer from the example
Maximum deflection
1.04 mm
At midspan for this uniform-load case
L/360 limit
11.1 mm
Example serviceability comparison
Status
Passes
1.04 mm is below 11.1 mm
Key formulas
Simply supported beam, center point load
δ = P L³ / (48 E I)
Use when one concentrated load acts at midspan. Maximum deflection occurs at midspan.
Simply supported beam, full uniform load
δ = 5 w L⁴ / (384 E I)
Use when the load is spread evenly across the full span. Maximum deflection occurs at midspan.
Cantilever beam, end point load
δ = P L³ / (3 E I)
Use for a fixed-free beam with a point load at the free end. Maximum deflection occurs at the free end.
Cantilever beam, full uniform load
δ = w L⁴ / (8 E I)
Use for a fixed-free beam with load spread across the full cantilever length.
Example serviceability check
Allowable deflection = L / ratio
Common ratios include L/360, L/240, or project-specific limits depending on use, finish, code, and engineer requirements.
Variables and units
| Symbol | Meaning | Typical units |
|---|---|---|
| δ | Maximum deflection | mm, in, m |
| P | Concentrated point load | N, kN, lbf |
| w | Uniformly distributed load | N/m, kN/m, lbf/ft |
| L | Beam span or cantilever length | m, mm, ft, in |
| E | Modulus of elasticity / Young's modulus | Pa, GPa, psi, ksi |
| I | Second moment of area / area moment of inertia | m⁴, mm⁴, in⁴ |
Quick reference conversions
Steel E
≈ 200 GPa
Typical elastic modulus for structural steel.
Aluminum E
≈ 69 GPa
Lower E means more deflection for the same I.
Wood E
varies widely
Use species, grade, and code-specific design values.
Point load span effect
L³
Doubling span increases point-load deflection by 8×.
Uniform load span effect
L⁴
Doubling span increases uniform-load deflection by 16× if w stays the same.
Flexural rigidity
E × I
Higher EI means lower deflection.
Step-by-step solved example
Example problem
Estimate maximum deflection for a simply supported steel beam with a 4 m span, uniform load w = 5 kN/m, modulus E = 200 GPa, and moment of inertia I = 8×10⁻⁵ m⁴.
1. Pick the matching load case
The beam is simply supported and the load is spread uniformly across the full span, so use δ = 5wL⁴ / (384EI).
2. Convert values to base SI units
w = 5 kN/m = 5,000 N/m. E = 200 GPa = 200×10⁹ Pa. L = 4 m and I = 8×10⁻⁵ m⁴ are already in compatible SI units.
3. Substitute into the formula
δ = (5 × 5,000 × 4⁴) ÷ (384 × 200×10⁹ × 8×10⁻⁵).
4. Calculate the deflection
δ = 0.0010417 m = 1.0417 mm. This is the maximum midspan deflection for the stated beam and load case.
5. Compare with an L/360 limit
For a 4 m span, L/360 = 4,000 mm ÷ 360 = 11.1 mm. The calculated 1.04 mm deflection is below 11.1 mm, so it passes this example L/360 check.
6. Interpret the design meaning
The result is small because the EI stiffness is high relative to the load and span. If deflection were too high, increasing I with a deeper section usually helps more efficiently than changing material E.
Practical field notes
Deflection is not the same as strength
Strength checks ask whether the beam resists bending and shear safely. Deflection checks ask whether movement is acceptable for finishes, comfort, drainage, equipment alignment, or appearance.
Span length dominates the answer
Because L is cubed or raised to the fourth power, small span increases can create large deflection increases. Shortening the span or adding support is often more powerful than small material changes.
Use the exact support and load case
A simply supported beam, cantilever, fixed-fixed beam, off-center point load, and partial uniform load all use different formulas. Matching the physical case matters before the arithmetic starts.
Serviceability limits depend on the project
L/360 is a common starting point for some floor-live-load checks, but roofs, plaster ceilings, brittle finishes, glass, equipment, and local code provisions may require different limits.
Common mistakes to avoid
- Using the point-load formula for a uniform load, or the uniform-load formula for one concentrated load.
- Mixing units, such as using E in GPa while I is entered in mm⁴ without converting consistently.
- Using total load where the formula expects distributed load intensity w in force per length.
- Assuming a beam that passes bending stress automatically passes deflection.
- Comparing calculated deflection to L/360 without checking whether that ratio is actually the governing project limit.
When to use the calculator instead
Use the calculator when you want to switch support/load cases, compare point vs uniform loads, convert units, or quickly test different E and I values. It keeps each formula tied to the correct beam configuration.
Calculation FAQs
What is beam deflection?
Beam deflection is the displacement of a beam from its unloaded position. For common single-span cases, maximum deflection usually occurs at midspan for simply supported beams or at the free end for cantilevers.
Why does span length affect deflection so much?
Span appears as L³ or L⁴ in the standard formulas. That means a longer span increases deflection much faster than load or stiffness changes of the same percentage.
What does EI mean?
EI is flexural rigidity. E is material stiffness and I is cross-section bending stiffness. Deflection is inversely proportional to EI, so increasing either reduces deflection.
Is L/360 always the right deflection limit?
No. L/360 is a common reference ratio, but the correct limit depends on the structure, finish, occupancy, loading type, code, and project specification.
Can this example be used for final structural design?
No. It is a calculation walkthrough for a standard elastic beam case. Final structural design must also check loads, combinations, strength, lateral stability, connections, code provisions, and professional engineering requirements.
References
- Beams Supported at Both Ends with Continuous and Point Loads
Engineering ToolBox
Reference for simply supported beam stress and deflection formulas, including maximum deflection under uniform load.
- Beam Deflection Formulas
Reuven Engineering Tools
Summarizes the four common Euler-Bernoulli closed-form deflection cases used in this example family.