Beam Deflection Calculator

Maximum deflection and bending moment for simply supported, cantilever, and fixed-fixed beams under a point load or a uniform load, with L/360 and L/240 checks.

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Support Type

Load Type

Span L

Beam length between supports, or from the fixed end to the free end of a cantilever.

Span Unit

Unit for the value entered above.

Point Load P

Used when Load Type is a point load. Ignored for a uniform load.

Point Load Unit

Unit for the value entered above.

Uniform Load w

Force per unit length. Used when Load Type is a uniform load. Ignored for a point load.

Uniform Load Unit

Unit for the value entered above.

Material (Elastic Modulus E)

Custom Elastic Modulus E

Used only when the material is set to Custom.

Custom E Unit

Unit for the value entered above.

Second Moment of Area I

About the bending axis. Use the moment of inertia calculator to find I for a rectangle, circle, or I-beam.

I Unit

Unit for the value entered above.

What This Calculator Does

Beam deflection is the second check after bending strength: a beam can carry its load without yielding and still sag enough to crack plaster, jam doors, or feel bouncy underfoot. This calculator applies the closed-form Euler–Bernoulli results for six standard cases (three support conditions, each with a point load or a uniform load) and reports the maximum deflection, where it occurs, the peak bending moment, the span-to-deflection ratio $L/\delta$, and a pass/fail against the common $L/360$ and $L/240$ limits. The second moment of area $I$ is an input, in m⁴, cm⁴, mm⁴, or in⁴. For a rectangle, circle, or I-beam, get $I$ from the moment of inertia calculator first. Inputs accept metric or imperial units, and results are shown in both.

It combines Support Type, Load Type, Span L, Span Unit to estimate Maximum deflection δmax, Location of maximum deflection, Maximum bending moment Mmax.

Formula & Method

Maximum deflection takes the form

\delta_{max} = C\,\frac{F\,L^{n}}{EI}

with

n = 3

for a point load

P

and

n = 4

for a uniform load

w

(force per length). Simply supported, center point load:

\delta_{max} = \frac{PL^3}{48EI}, \quad M_{max} = \frac{PL}{4}

Simply supported, uniform load:

\delta_{max} = \frac{5wL^4}{384EI}, \quad M_{max} = \frac{wL^2}{8}

Cantilever, end load:

\delta_{max} = \frac{PL^3}{3EI}, \quad M_{max} = PL

Cantilever, uniform load:

\delta_{max} = \frac{wL^4}{8EI}, \quad M_{max} = \frac{wL^2}{2}

Fixed-fixed, center point load:

\delta_{max} = \frac{PL^3}{192EI}, \quad M_{max} = \frac{PL}{8}

Fixed-fixed, uniform load:

\delta_{max} = \frac{wL^4}{384EI}, \quad M_{max} = \frac{wL^2}{12}

Here

E

is the elastic modulus,

I

is the second moment of area about the bending axis, and

L

is the span. Serviceability ratio:

L/\delta

, compared with 360 and 240. The uniform load

w

does not include the beam's own weight unless you add it.

Notation used in the formulas: $R$ = Maximum deflection δmax; $x_{1}$ = Support Type; $x_{2}$ = Load Type; $x_{3}$ = Span L; $x_{4}$ = Span Unit; $x_{5}$ = Point Load P; $x_{6}$ = Point Load Unit.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it for a quick serviceability check of a single-span prismatic beam, or to see how much a change in span, section, or material moves the deflection. Doubling the span multiplies point-load deflection by 8 and uniform-load deflection by 16.

Worked Examples

Steel beam, simply supported, center point load (metric)

Given: L = 4 m, P = 10 kN, E = 200 GPa, I = 8.0×10⁻⁶ m⁴ Method: simply supported, center load, so δ = PL³ / (48EI). EI = 200×10⁹ × 8.0×10⁻⁶ = 1.6×10⁶ N·m² δ = 10,000 × 4³ / (48 × 1.6×10⁶) = 640,000 / 76,800,000 = 0.008333 m = 8.333 mm Limit L/360 = 4000 mm / 360 = 11.11 mm, so the beam passes. L/δ = 480. Mmax = PL/4 = 10 kN.m

Aluminum cantilever, uniform load (imperial)

Given: L = 6 ft, w = 40 lbf/ft, E = 10.0×10⁶ psi (aluminum 6061), I = 1.2 in⁴ Method: cantilever with uniform load, so δ = wL⁴ / (8EI). Convert to inches first. L = 72 in, w = 40/12 = 3.333 lbf/in δ = 3.333 × 72⁴ / (8 × 10.0×10⁶ × 1.2) = 3.333 × 26,873,856 / 96,000,000 = 0.933 in L/δ = 72 / 0.933 = 77. That fails L/240 (limit 0.30 in): the free end deflects about three times the allowed amount.

Commonly Used Deflection Limits

Limits are set by the governing code, the material standard, or the project specification. The ratios below are widely quoted starting points, not code values. Check the code that applies to your project.

Limit	Typical application
L/180	Roof members with no ceiling or finish to protect, under live load
L/240	Roof or floor members, total load; members carrying non-brittle finishes
L/360	Floor members under live load; members supporting plaster or other brittle finishes
L/480	Floors where vibration or sensitive equipment is a concern; stricter project specs

For a cantilever, many codes apply the ratio to twice the cantilever length. This calculator uses the physical span $L$ you enter, so double-check the convention you need.

Reference

The six cases are standard results of Euler–Bernoulli beam theory, tabulated in Roark's Formulas for Stress and Strain (Young, Budynas, and Sadegh) and in the beam diagrams and formulas of the AISC Steel Construction Manual.

Common Mistakes

Mixing units: entering $I$ in cm⁴ but $E$ in GPa and $L$ in meters without converting. The unit selectors on each field convert for you, so set each one to match the number you typed.
Using $I$ about the wrong axis. A standard I-beam has a much larger $I_x$ (strong axis) than $I_y$. Use the axis the load actually bends the beam about.
Treating a total load as a per-length load. A 20 kN load spread along a 4 m beam is $w = 5$ kN/m, not 20 kN/m.
Forgetting self-weight. For long steel or timber spans, add the beam's own weight per length to $w$.
Applying a deflection limit to the wrong load combination. $L/360$ is normally checked against live load only, not total load.
Assuming a fixed end is perfectly rigid. A connection that rotates even slightly makes the beam behave between the fixed-fixed and simply supported results, and the fixed-fixed answer then underestimates deflection.

When Not To Use This Calculator

Large deflections. The formulas assume small, linear-elastic bending, so they lose accuracy once deflection exceeds a few percent of the span.
Plastic behavior. If the stress reaches yield, the load-deflection relationship is no longer linear.
Composite, tapered, or non-prismatic beams. $EI$ must be constant along the span.
Short, deep beams where shear deformation matters. Euler–Bernoulli theory ignores it. Timoshenko beam theory includes it.
Multiple point loads, partial loads, or continuous beams over several supports. Use superposition of the cases here or a structural analysis program.

Frequently Asked Questions

What deflection limit should I use, L/360 or L/240?: It depends on what the beam carries and which code governs. L/360 is the common limit for floor members under live load and for members that support brittle finishes such as plaster. L/240 is a common limit for roof members or for total load on members with flexible finishes. Project specifications and local building codes override these defaults, so confirm the limit before relying on a pass.
Why does beam deflection scale with L³ or L⁴?: Bending moment grows with span, and the curvature that moment produces accumulates over the length. For a point load the result is proportional to PL³, and for a uniform load, where the total load also grows with span, it is proportional to wL⁴. Doubling the span multiplies point-load deflection by 8 and uniform-load deflection by 16.
Where do I get I for an I-beam?: Take Ix from the manufacturer's section table (AISC shape tables list Ix for every W shape) or compute it with the moment of inertia calculator. Use Ix, the strong-axis value, when the load acts in the plane of the web.
How does the end support change the deflection?: Fixed ends stiffen a beam a great deal. For the same span, load, and section, a fixed-fixed beam with a center point load deflects one quarter as much as a simply supported beam, and a fixed-fixed beam with a uniform load deflects one fifth as much. A cantilever is much more flexible: its free-end deflection under an end load is 16 times that of a simply supported beam of the same span.
Does this include the weight of the beam?: No. The uniform load w is whatever you enter. To include self-weight, compute the beam's weight per unit length and add it to w.

Inputs Used

Support Type: Used directly in the calculation.
Load Type: Used directly in the calculation.
Span L: Beam length between supports, or from the fixed end to the free end of a cantilever.
Span Unit: Unit for the value entered above.
Point Load P: Used when Load Type is a point load. Ignored for a uniform load.
Point Load Unit: Unit for the value entered above.
Uniform Load w: Force per unit length. Used when Load Type is a uniform load. Ignored for a point load.
Uniform Load Unit: Unit for the value entered above.
Material (Elastic Modulus E): Used directly in the calculation.
Custom Elastic Modulus E: Used only when the material is set to Custom.
Custom E Unit: Unit for the value entered above.
Second Moment of Area I: About the bending axis. Use the moment of inertia calculator to find I for a rectangle, circle, or I-beam.
I Unit: Unit for the value entered above.

See Also

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