Beam Deflection Calculator | Simply Supported & Cantilever

Unit system

Support and load case

Selected case: Simply Supported — Center Point Load Deflected shape is exaggerated and not to scale.

Beam properties and loading

Beam material

Elastic modulus E GPa

Beam length L m

Point load P kN

Second moment of area I

Unit for I

Use the second moment of area about the bending axis. This is a geometric property, not the mass moment of inertia.

How beam deflection is calculated

The calculator uses closed-form Euler–Bernoulli beam equations for prismatic beams with constant elastic modulus and second moment of area.

Flexural rigidity

Beam stiffness depends on both the material stiffness E and the section stiffness I. Increasing either value reduces elastic deflection.

E = Young’s modulus of the material.
I = second moment of area about the bending axis.
L = beam span or cantilever length.
P = concentrated point load.
w = uniformly distributed load per unit length.

Supported equations

Beam and load case	Maximum deflection	Maximum moment
Simply supported, center point load	PL³ / 48EI	PL / 4
Simply supported, uniform load	5wL⁴ / 384EI	wL² / 8
Cantilever, end point load	PL³ / 3EI	PL
Cantilever, uniform load	wL⁴ / 8EI	wL² / 2

How to obtain the second moment of area

Use the value about the actual bending axis. Standard section tables often list two values because a section can have a strong and a weak axis.

Rectangle: I = bh³ / 12

The dimension parallel to bending is cubed.

Circle: I = πd⁴ / 64

Use the outside-minus-inside form for hollow sections.

Worked example

A simply supported structural-steel beam has a 4 m span, carries a 10 kN point load at midspan, and has I = 8,000 cm⁴.

Convert I: 8,000 cm⁴ = 8 × 10⁻⁵ m⁴.
Use E = 200 GPa and δmax = PL³ / 48EI.
δmax = 10,000 × 4³ / (48 × 200 × 10⁹ × 8 × 10⁻⁵).

Maximum deflection ≈ 0.833 mm

Model assumptions

The beam is straight, slender and prismatic.
The material remains linear elastic.
Deflections and rotations are small.
E and I remain constant over the full length.
Shear deformation is neglected.
Supports and load positions match the selected ideal case.

Common mistakes

Using the mass moment of inertia instead of area moment I.
Using I about the wrong bending axis.
Mixing mm⁴, cm⁴, m⁴ and in⁴.
Entering total uniform load instead of load per unit length.
Applying a small-deflection equation to a flexible or deep beam.
Ignoring self-weight when it contributes to the distributed load.

Frequently asked questions

Where does maximum deflection occur?

For the two simply supported cases it occurs at midspan. For the two cantilever cases it occurs at the free end.

What is the difference between E and I?

E is a material property. I is a geometric property of the cross-section about a selected axis. Their product EI is the beam’s flexural rigidity.

Does the calculator determine whether the beam is safe?

No. It calculates elastic response for idealized cases. Strength, stability, fatigue, local effects, connections and code-specific serviceability limits require separate checks.

Why can a real beam deflect differently?

Real supports, load distribution, shear deformation, varying section properties, residual stress, material variation and connection flexibility can change the result.

Accuracy and responsible use

Compare deflection and span-to-deflection ratio with the requirements that apply to the specific structure, machine or component. Do not treat this result as a complete design verification.

Read the Methodology & Accuracy page and the Terms & Disclaimer .