EngineeringFormulaCivil & Structural Engineering
Beam Deflection Calculator
Calculate maximum beam deflection for common support and load cases.
Beam deflection is the displacement of a beam under load, measured perpendicular to its unloaded axis. Select a support condition and load case, enter the beam's span, material and cross-section properties, and this calculator returns the maximum deflection using standard Euler-Bernoulli beam theory.
L · Length of the beam between supports, or from support to free end.
P · Concentrated force applied to the beam.
E · Material stiffness (Young's modulus).
I · Cross-sectional resistance to bending (second moment of area).
Solution
Enter the required values to calculate deflection.
δ = P L³ / (48 E I)
Formula Sheet
- δDeflection
- PPoint Load
- wDistributed Load
- LSpan Length
- EModulus of Elasticity
- IMoment of Inertia
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| δ | Deflection | Maximum displacement of the beam under load. | mm, in |
| P | Point Load | Concentrated force applied to the beam. | N, kN, lbf |
| w | Distributed Load | Load intensity spread evenly along the span. | N/m, kN/m, lbf/ft |
| L | Span Length | Length of the beam between supports, or from support to free end. | mm, m, ft |
| E | Modulus of Elasticity | Material stiffness (Young's modulus). | GPa, psi, ksi |
| I | Moment of Inertia | Cross-sectional resistance to bending (second moment of area). | mm⁴, cm⁴, in⁴ |
How to Use This Calculator
- 01Select a beam support condition (Simply Supported or Cantilever) and a load type (Point Load or Uniform Load) using the configuration selectors.
- 02Enter the span Length, the load magnitude, the material's Modulus of Elasticity, and the cross-section's Moment of Inertia.
- 03Select Calculate to see the maximum deflection for that configuration, alongside the matching formula substitution.
- 04Compare the result with the deflection limit that applies to your job, such as L/360 for many floor-serviceability checks or L/240 for less sensitive applications. The calculator gives the deflection; the governing allowable limit comes from your code, project spec, or engineer of record.
- 05Compare configurations by switching the beam type or load case - each combination uses its own formula, shown in the Formula Sheet.
How the Formula Works
Beam deflection theory predicts how much a beam bends under load, based on standard Euler-Bernoulli beam theory. Each combination of support condition and load type has its own closed-form deflection formula, but all four share the same underlying structure: deflection increases with load and span length, and decreases with the material's stiffness (E) and the cross-section's resistance to bending (I).
Span length has an outsized effect because it appears raised to the third or fourth power in every formula above - doubling the span multiplies deflection by 8× (point-load cases) or 16× (uniform-load cases) if nothing else changes. A cantilever, supported at only one end, deflects several times more than a simply supported beam under the same span and load, because the full bending moment builds up over the entire unsupported length with no second support to share it.
Worked Example 01
Simply supported beam with a central point load
Known
- Span Length (L): 4 m
- Point Load (P): 10 kN
- Modulus of Elasticity (E): 200 GPa
- Moment of Inertia (I): 8 × 10⁻⁵ m⁴
Formula
δ = P L³ / (48 E I)
Substitution
δ = (10,000 × 4³) / (48 × 200×10⁹ × 8×10⁻⁵)
Result
δ = 0.8333 mm
A 4 m steel beam under a 10 kN central point load deflects about 0.83 mm at midspan - well within typical serviceability limits.
Worked Example 02
Simply supported beam with a uniform load
Known
- Span Length (L): 4 m
- Distributed Load (w): 5 kN/m
- Modulus of Elasticity (E): 200 GPa
- Moment of Inertia (I): 8 × 10⁻⁵ m⁴
Formula
δ = 5 w L⁴ / (384 E I)
Substitution
δ = (5 × 5,000 × 4⁴) / (384 × 200×10⁹ × 8×10⁻⁵)
Result
δ = 1.0417 mm
The same 4 m steel beam, now under a 5 kN/m load spread evenly across the full span, deflects about 1.04 mm - slightly more than the equivalent point load at midspan, since a uniform load applies force along the entire length.
Worked Example 03
Cantilever beam with an end point load
Known
- Span Length (L): 4 m
- Point Load (P): 10 kN
- Modulus of Elasticity (E): 200 GPa
- Moment of Inertia (I): 8 × 10⁻⁵ m⁴
Formula
δ = P L³ / (3 E I)
Substitution
δ = (10,000 × 4³) / (3 × 200×10⁹ × 8×10⁻⁵)
Result
δ = 13.333 mm
The same beam and load, now cantilevered (fixed at one end, free at the other) instead of simply supported, deflects about 13.3 mm - sixteen times more than the simply supported case, since only one end resists the bending moment.
Worked Example 04
Cantilever beam with a uniform load
Known
- Span Length (L): 4 m
- Distributed Load (w): 5 kN/m
- Modulus of Elasticity (E): 200 GPa
- Moment of Inertia (I): 8 × 10⁻⁵ m⁴
Formula
δ = w L⁴ / (8 E I)
Substitution
δ = (5,000 × 4⁴) / (8 × 200×10⁹ × 8×10⁻⁵)
Result
δ = 10 mm
A cantilevered version of the uniform-load case deflects about 10 mm - again far more than the simply supported equivalent, for the same reason: no second support to share the bending moment.
Applications
- 01Preliminary sizing of beams, joists, and shelving
- 02Checking serviceability (deflection) limits during design
- 03Comparing deflection across different materials or cross-sections
Typical Modulus of Elasticity by Material
| Material | E (GPa) | E (psi) |
|---|---|---|
| Steel | 200 | ≈ 29,000,000 |
| Aluminum | 69 | ≈ 10,000,000 |
| Concrete (typical) | 30 | ≈ 4,350,000 |
| Wood (structural lumber) | 11 | ≈ 1,600,000 |
Assumptions
- 01The beam is linearly elastic, homogeneous, and prismatic (constant cross-section).
- 02Deflections are small relative to the span length.
- 03For point-load cases, the load is applied at midspan (simply supported) or at the free end (cantilever).
- 04For uniform-load cases, the load is spread evenly across the full span.
- 05Shear deformation is neglected (standard Euler-Bernoulli beam theory).
Where This Model Stops
- 01This calculator provides preliminary reference values only - final structural designs must be checked against applicable codes, standards, and qualified professional engineering judgment.
- 02Does not account for shear deflection, which can be significant for short, deep beams.
- 03Only supports the four configurations listed - off-center point loads, partial-span distributed loads, and multi-span or continuous beams require a different analysis.
- 04Does not check bending stress, shear stress, vibration, lateral-torsional buckling, connection capacity, or code-required load combinations.
- 05Serviceability limits such as L/240, L/360, or L/480 depend on occupancy, finish sensitivity, material, and jurisdiction; treat them as a separate pass/fail check, not a universal calculator output.
References
- [1]
Euler-Bernoulli beam theory - standard deflection formulas
Standard structural/mechanical engineering fundamentals
Maximum deflection formulas for simply supported and cantilever beams under point or uniformly distributed loads.
Frequently Asked Questions
Why does a cantilever deflect so much more than a simply supported beam?
A cantilever is only supported at one end, so the full bending moment builds up over the entire span with no second support to share the load - for the same load and span, a cantilever deflects several times more than a simply supported beam.
What if my load isn't centered or evenly distributed?
This calculator covers the four standard textbook cases (centered point load or full-span uniform load, for simply supported or cantilever beams). Off-center or partial loads follow different deflection equations and aren't covered here.
Does a larger moment of inertia always mean less deflection?
Yes - deflection is inversely proportional to moment of inertia in every case above. Increasing I (a deeper or differently-shaped cross-section) reduces deflection for the same load and material.
Where do I get E and I for a real beam?
Use the material specification or code reference for E, and use a section table, manufacturer data sheet, or this site's Area Moment of Inertia Calculator for I. Guessing either value can move the deflection result by a large amount.
How is this different from the Beam Reaction or Bending Stress Calculator?
Beam reaction finds support forces, bending stress checks strength from moment and section modulus, and this calculator checks serviceability deflection from load, span, E, and I. A real beam can pass one check and fail another.