2025-08-21 14 min

Cantilever Beam Calculator

Calculate deflection, slope, moment, and stress for cantilever beams with point load or UDL.

Calculator

Unit System

Load Type

Point Load (P)(N)

Cantilever Length (L)(mm)

Young Modulus (E)(MPa)

MPa

Moment of Inertia (I)(mm^4)

mm^4

Section Width (b)(mm)

Section Height (h)(mm)

Decimal Precision

Results update live as you type

Results

Normal

Max Deflection (free end)

2.67mm

Max Slope (free end)

0.00rad

Max Moment (fixed end)

2000000.00N*mm

Max Stress (fixed end)

20.00MPa

Reaction Force

1000.00N

Visualizations

Results Overview

2.7mmMax Deflection

2,000,000mmMax Moment

20mmMax Stress

1,000mmReaction Force

Formula

Cantilever beam formulas. Max moment/stress at fixed end. Max deflection/slope at free end.

delta = PL^3/(3EI), theta = PL^2/(2EI), M = PL

Alternative Form

UDL: delta = wL^4/(8EI), theta = wL^3/(6EI), M = wL^2/2

Variables

Symbol	Name	Unit	Description
delta	Maximum Deflection	mm	Deflection at free end
theta	Maximum Slope	radians	Slope at free end
M	Maximum Moment	N*mm	Moment at fixed end
sigma	Maximum Stress	MPa	Bending stress at fixed end
P	Point Load	N	Load at free end
w	UDL Intensity	N/mm	Load per unit length
L	Cantilever Length	mm	Length from fixed end
E	Young Modulus	MPa	Material stiffness
I	Moment of Inertia	mm^4	Section property

Explanation

A cantilever is fixed at one end, free at the other. All moment occurs at the fixed support. Deflection is maximum at the free end.

Derivation

From EI*y = M(x). For point load M(x) = P(L-x). Integrating twice with boundary conditions y(0)=0, y(0)=0.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify loading type (point load or UDL)

Result: Loading determined

Step 2

Calculate section properties I and c

Result: Section properties known

Step 3

Compute deflection at free end

Result: Deflection calculated

Step 4

Compute slope at free end

Result: Slope calculated

Step 5

Compute moment and stress at fixed end

Result: Stress and moment calculated

Step 6

Verify against allowable limits

Result: Design check complete

Final Answer

2.67

Worked Examples

See how the calculator works with different scenarios.

Scenario

500 mm bracket (E=200 GPa, I=1E6 mm^4) with 100 N at free end.

Inputs

Load:100 N

Span:500 mm

Calculation

delta = 100*500^3/(3*200000*1000000) = 0.021 mm. M = 50000 N*mm.

Result

delta = 0.021 mm, M = 50,000 N*mm

Interpretation

Negligible deflection. Very stiff bracket.

Understanding Your Result

Your Result: Max Deflection (free end): 2.67 mm

Typical range: Deflection limit: L/150 to L/250. Steel stress: 100-250 MPa. Concrete stress: 5-15 MPa.

What This Means

Cantilever beams extend from a fixed support with no intermediate support, creating high moments at the fixed end.

Why It Matters

Cantilevers are essential for balconies, bridges, aircraft wings, and overhangs. The fixed connection is the critical design element.

Professional Interpretation

Check both stress and deflection. Pay special attention to the fixed connection design.

Recommendations

•Increase depth to reduce deflection and stress
•Design fixed connection carefully
•Check lateral-torsional buckling

Educational Content

Introduction

Cantilever beams are fundamental structural elements used whenever an overhang or unsupported extension is needed.

Concept Explanation

Fixed at one end, free at the other. Maximum moment at support, maximum deflection at free end.

Applications

▸Balconies and overhangs

▸Aircraft wings

▸Crane jibs

▸Bridge cantilevers

▸Diving boards

Benefits

•Provides clear space underneath
•Versatile for many applications
•Can be combined with other support types

Limitations

•Requires strong fixed connection
•High deflection compared to supported beams
•Vulnerable to dynamic loads

Best Practices

•Design the fixed connection for full moment capacity
•Check deflection carefully
•Consider lateral-torsional buckling

Common Misconceptions

Myth: Cantilevers are weak

Truth: Cantilevers are efficient when properly designed with adequate connections and sections.

Practical Tips

•Depth is most effective for reducing deflection
•Fixed connection is the critical design element
•Check lateral-torsional buckling for long cantilevers
•Self-weight adds significant load

Common Mistakes

Inadequate fixed connection design

The connection must resist the full moment and shear.

Ignoring self-weight

Add self-weight deflection to applied load deflection.

Assumptions

This calculator makes the following assumptions to provide estimates:

Perfect Fixed End: Support prevents all rotation and translation.

Elastic Behavior: Material remains in linear elastic range.

Limitations

Be aware of these limitations when using this calculator:

Connection Flexibility: Real supports are not perfectly rigid.

Buckling: Lateral-torsional buckling may govern for long cantilevers.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a cantilever beam?

A cantilever beam is fixed at one end and free at the other, carrying loads without support at the free end.

What is the deflection formula for a cantilever?

For end point load: delta = PL^3/(3EI). For UDL: delta = wL^4/(8EI). Maximum at free end.

What is the slope formula for a cantilever?

For end point load: theta = PL^2/(2EI). For UDL: theta = wL^3/(6EI). Maximum at free end.

What is the maximum moment in a cantilever?

For point load: M = PL. For UDL: M = wL^2/2. Both occur at the fixed end.

What is the maximum stress in a cantilever?

sigma = M*c/I. Maximum at the fixed end, outermost fiber.

How does cantilever length affect performance?

Deflection scales with L^3 (point) or L^4 (UDL). Moment scales with L or L^2. Length is critical.

What is the reaction at the fixed end?

For point load: R = P, M_reaction = PL. For UDL: R = wL, M = wL^2/2.

How do you design a cantilever beam?

Calculate max moment and stress, check against allowable. Verify deflection limits. Design the fixed connection.

What is a real-world example of a cantilever?

Bridges, aircraft wings, diving boards, balconies, crane jibs, stadium roofs.

How do cantilevers differ from simply supported beams?

Cantilevers have 16x more deflection and 4x more moment than simply supported beams for the same load and span.

What is the stress distribution in a cantilever?

Linear from zero at neutral axis to maximum at outer fiber. Top in tension, bottom in compression for downward loads.

How do you reduce cantilever deflection?

Increase depth, reduce length, increase E or I, add a prop (propped cantilever), or use post-tensioning.

What is a propped cantilever?

A cantilever with a prop at the free end, reducing deflection and moment compared to unconstrained cantilever.

What is the effect of self-weight on a cantilever?

Self-weight acts as UDL: w = rho*g*b*h. Add self-weight deflection to applied load deflection.

What materials are best for cantilevers?

Steel (high E and yield), reinforced concrete (mass and fire resistance), FRP composites (strength-to-weight).

What is the maximum span for a cantilever?

Steel 15-20 m, concrete 10-15 m, timber 3-5 m. Deflection usually governs over strength.

How do you calculate cantilever with intermediate load?

For load P at distance a from fixed end: delta = Pa^2(3L-a)/(6EI).

What is the slope at the fixed end?

Always zero by definition. The support prevents rotation.

What is the moment diagram shape?

Point load: linear from 0 at free end to PL at fixed. UDL: parabolic from 0 to wL^2/2.

How do support conditions affect cantilever design?

The fixed support must resist the full moment. Connection design is critical for safety.

What is the shear force in a cantilever?

V = P (point load) constant, or V = w*x (UDL) linear. Maximum at fixed end.

How does temperature affect a cantilever?

Temperature gradient causes curvature and deflection. Top heating causes downward deflection.

What is the dynamic response of a cantilever?

Natural frequency f = 3.52/(2*pi) * sqrt(EI/(rho*A*L^4)). First mode at free end.

What is the buckling load of a cantilever column?

P_cr = pi^2*EI/(4L^2). Effective length factor is 2.0 for fixed-free.

What is the vibration frequency of a cantilever?

f1 = 0.1615 * sqrt(EI/(m*L^4)) where m is mass per unit length.

How do cantilevers behave under seismic loading?

Vulnerable due to single support. Dynamic amplification increases moment and deflection.

What is the fatigue life of a cantilever?

Fatigue at the fixed end is primary concern. Stress range and connection details determine life.

What are common cantilever failure modes?

Connection failure, excessive deflection, lateral-torsional buckling, and fatigue cracking.

How does load position affect cantilever deflection?

Loads closer to the fixed end cause less deflection. Deflection is proportional to a^2 for intermediate loads.

What is the deflection at the load point for intermediate load?

For load at distance a from fixed end: delta = Pa^3/(3EI). Free end has additional deflection.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Cantilever: A beam fixed at one end and free at the other.
Fixed End: The supported end that prevents rotation and translation.
Free End: The unsupported end where deflection and slope are maximum.
Propped Cantilever: A cantilever with additional support at the free end.

Authoritative References

Trusted sources and standards referenced by this calculator.

Shigleys Mechanical Engineering Design AISC Steel Construction Manual Eurocode 2 - Concrete Design

Applicable Standards

AISC 360Eurocode 3ACI 318

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.