Mechanics · Beams and deflection

Beam deflection

Choose a common beam case, select a profile and its strong or weak axis, then calculate maximum deflection from L, E, I and the applied load.

Created Jul 13, 2026v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Beams and deflection

Calculation inputs

Configuration

Unit system

Supports and loading

Geometry and material

Span or length L

Young’s modulus E

Profile and second moment

Second-moment definition IThe library automatically fills I from the profile nominal geometry.

Profile family

Profile designation

Section orientationThe strong axis minimises deflection; the weak axis is the more flexible orientation.

Property definitionChoose the published catalog value or the Formulaxis analytical value.

Second moment used I

Value calculated from the selected profile or entered manually, depending on the selected mode.

Point load F

Verified profile

Profile used in this calculation

IPE 200IPE sectionsMajor axisAnalytical value

Value used in the calculation: I1,845.5902 cm⁴
Provenance: Value recalculated by Formulaxis from the nominal geometry.

Major axis — catalog y-y — Formulaxis x-x

IPE 200. Major axis. Analytical value. Value used in the calculation: I = 1,845.5902 cm⁴

View this profile’s properties and provenance

Next stepCheck bending stress

Results are up to date.

Results

Maximum deflection

1.451336mm

Maximum transverse displacement.

Maximum slope

1.451336mrad

Maximum rotation for the selected case.

Characteristic reaction

5kN

Reaction per support or fixed-end reaction, depending on the case.

Maximum bending moment

7.5kN·m

Maximum absolute bending-moment value.

Same mechanical case

Profile comparison

Profile AIPE 200

Profile BChange profile B

Both profiles use the same calculation data.

Axis: Major axis
Basis: Analytical values

Values recalculated by Formulaxis from the nominal geometry.

Comparison of IPE 200 and IPE 100 for the same beam-deflection case
Measure	Profile AIPE 200	Profile BIPE 100	Δ B vs A
Linear mass	—	—	—
Section propertySecond moment of area I	—	—	—
Maximum deflection	—	—	—

View calculation detailsFormulas, SI conversion, numerical substitution and resultsOpen ↓

Calculation note

Full calculation details

Straight prismatic beam with constant E and I. Linear-elastic behaviour and small deflection. Shear deformation neglected. Idealized static loads and perfect supports.

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IPE beam: a complete worked example

Major axis: 22.8 MPa and 1.81 mm. Minor axis: 156 MPa and 24.8 mm. These are model results, not an approval of the beam.

01Formulas and symbols

Formulas used

δFL³/(48EI)Simply supported, centre point load

δ5qL⁴/(384EI)Simply supported, uniform load

δFL³/(3EI)Cantilever, end point load

δqL⁴/(8EI)Cantilever, uniform load

02Assumptions and limits

Scope of validity

Straight prismatic beam with constant E and I.
Linear-elastic behaviour and small deflection.
Shear deformation neglected.
Idealized static loads and perfect supports.

03Validation example

Reference numerical case

Simply supported, F = 10 kN at centre, L = 3 m, E = 210 GPa and I = 8,000,000 mm⁴.
Maximum deflection = 3.3482 mm.
Support slope = 3.3482 mrad.
Reaction per support = 5 kN and Mmax = 7.5 kN·m.

04Frequently asked questions

Questions about the calculation

Which I value should be entered?

Use the second moment about the bending axis. A vertical load with section height along y generally uses Ix.

Is shear deflection included?

No. The formulas use Euler–Bernoulli theory and are most suitable for slender beams.

Can several loads be combined?

The first version handles one load case at a time. Linear superposition is possible when support conditions and conventions are consistent.

05References

Technical references

Roark’s Formulas for Stress and Strain, 9th edition, beam tables.
Gere & Goodno, Mechanics of Materials, elastic curve equation.