2025-08-21 14 min

Beam Deflection Calculator

Calculate beam deflection for cantilever and simply supported beams with point loads or UDL.

Calculator

Unit System

Beam Type

Point Load (P)(N)

Beam Span (L)(mm)

Young Modulus (E)(MPa) Help

MPa

Steel ~200000, Al ~70000

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

0.42mm

Max Slope

0.00rad

Max Moment

5000000.00N*mm

Visualizations

Results Overview

0.4mmMax Deflection

5,000,000mmMax Moment

Formula

Deflection depends on loading type and support conditions. All formulas assume linear elastic behavior.

Cantilever: delta = PL^3/(3EI) | Simply Supported: delta = PL^3/(48EI)

Alternative Form

UDL Cantilever: delta = wL^4/(8EI) | UDL Simply Supported: delta = 5wL^4/(384EI)

Variables

Symbol	Name	Unit	Description
delta	Maximum Deflection	mm	Largest vertical displacement
P	Point Load	N	Concentrated force on the beam
w	UDL Intensity	N/mm	Load per unit length
L	Beam Span	mm	Length between supports
E	Young Modulus	MPa	Material stiffness
I	Moment of Inertia	mm^4	Section property resisting bending

Explanation

Deflection is inversely proportional to EI and strongly dependent on span length. The beam type and loading determine the coefficient.

Derivation

From Euler-Bernoulli beam equation: EI * d^2y/dx^2 = M(x). Integrating twice with boundary conditions gives the deflection curve.

Step-by-Step Calculation

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

Step 1

Identify beam type (cantilever or simply supported) and loading

Result: Configuration determined

Step 2

Calculate EI = E * I for the beam

Result: Flexural rigidity computed

Step 3

Apply the appropriate deflection formula

Result: Deflection calculated

Step 4

Calculate slope and moment if needed

Result: Additional results computed

Step 5

Compare against allowable deflection limits (e.g. L/360)

Result: Serviceability check

Final Answer

0.42

Worked Examples

See how the calculator works with different scenarios.

Scenario

600 mm steel bracket (E=200 GPa, I=0.5E6 mm^4) with 200 N end load.

Inputs

Load:200 N

Span:600 mm

Young Modulus:200000 MPa

Moment of Inertia:500000 mm^4

Calculation

delta = 200*600^3/(3*200000*500000) = 0.432 mm.

Result

delta = 0.432 mm

Interpretation

Very small deflection. Well within typical limits.

Understanding Your Result

Your Result: Max Deflection: 0.42 mm

Typical range: Floors: L/240-L/360. Roofs: L/180-L/240. Machinery: L/480-L/1000.

What This Means

Beam deflection measures how much a beam bends under load, determining serviceability compliance.

Why It Matters

Excessive deflection causes cracking, ponding, misalignment, and occupant discomfort even if structurally adequate.

Professional Interpretation

Compare against code-specified limits. Check both immediate and long-term creep deflection.

Recommendations

•Depth is most effective for reducing deflection
•Check deflection before increasing strength
•Consider long-term creep effects

Educational Content

Introduction

Beam deflection analysis ensures structures meet serviceability requirements.

Concept Explanation

Deflection results from curvature: EI*y = M(x). Integrating gives the deflection curve.

Applications

▸Floor and roof framing

▸Bridge decks

▸Machine tools

▸Aerospace structures

▸Building code compliance

Benefits

•Simple analytical solutions
•Direct relationship to geometry and material
•Foundation for vibration analysis

Limitations

•Assumes linear elastic behavior
•Neglects shear deformation for deep beams
•Does not account for connection flexibility

Best Practices

•Always check deflection alongside stress
•Use consistent N and mm units
•Account for creep effects

Common Misconceptions

Myth: Stronger beams deflect less

Truth: Stiffness (EI) controls deflection, not strength (yield).

Practical Tips

•Deflection scales with L^3 (point) or L^4 (UDL)
•Doubling depth reduces deflection by 8x
•Common limit is L/360
•Check deflection at worst-case location

Common Mistakes

Ignoring deflection when stress passes

Serviceability often governs over strength.

Wrong I axis

Use I about the bending axis, not the other principal axis.

Assumptions

This calculator makes the following assumptions to provide estimates:

Elastic Behavior: Material remains in the linear elastic range.

Small Deflections: Deflections are small relative to beam span.

Limitations

Be aware of these limitations when using this calculator:

Deep Beams: Shear deformation significant for L/h < 4.

Plastic Behavior: Beyond yield, elastic formulas are invalid.

Frequently Asked Questions

Answers to common questions about this calculator.

What is beam deflection?

Beam deflection is the vertical displacement of a beam from its original position under applied loads, measuring the beam stiffness and serviceability.

What is the deflection formula for a cantilever beam?

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

What is the deflection formula for a simply supported beam?

For central point load: delta = PL^3/(48EI). For UDL: delta = 5wL^4/(384EI). Maximum deflection occurs at midspan.

What is EI in beam deflection?

EI is flexural rigidity, the product of Young modulus E and moment of inertia I. Higher EI means less deflection.

How does beam length affect deflection?

Deflection is proportional to L^3 for point loads and L^4 for UDL. Doubling length increases deflection by 8x or 16x respectively.

How does depth affect deflection?

Deflection is inversely proportional to I, which scales with h^3 for rectangles. Doubling depth reduces deflection by 8x.

What is the allowable deflection?

Common limits: L/240 for floors, L/360 for roofs, L/480 for finishes. Serviceability limits often govern over strength.

What is the slope of a cantilever beam?

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

How does Young modulus affect deflection?

Deflection is inversely proportional to E. Aluminum (E=70 GPa) deflects 2.86x more than steel (E=200 GPa).

How do you calculate deflection for multiple loads?

Use superposition: calculate deflection for each load separately and sum them algebraically.

How does temperature affect deflection?

Temperature gradients cause thermal bowing. Differential temperature across the depth causes curvature and deflection.

How do you reduce beam deflection?

Increase depth (most effective), increase width, use higher E material, reduce span, or add intermediate supports.

What is the relationship between deflection and stress?

Stress depends on M/I while deflection depends on L^3/(EI). A beam can pass stress check but fail deflection limits.

What is deflection due to self-weight?

For simply supported beam: delta = 5wL^4/(384EI) where w is the self-weight per unit length.

How do you measure beam deflection?

Using dial indicators, LVDTs, laser sensors, or optical methods. Digital image correlation works for large deflections.

What is creep deflection?

Creep is time-dependent deflection under sustained load. Concrete and wood creep can be 2-3x the initial elastic deflection.

How do support conditions affect deflection?

Cantilever deflects most (PL^3/3EI), simply supported is intermediate (PL^3/48EI), fixed-fixed least (PL^3/192EI).

What is the deflection formula for fixed-fixed beam?

Central point load: delta = PL^3/(192EI). UDL: delta = wL^4/(384EI). Fixed ends significantly reduce deflection.

What is the deflection of a composite beam?

Transform the section using modular ratio n = E1/E2, then apply standard formulas with the transformed moment of inertia.

What is the deflection of a tapered beam?

Requires integration with variable I(x). For linear taper, use average I or numerical integration methods.

What is the effect of shear deformation?

Shear deformation adds to bending deflection, especially for deep beams. Total deflection includes both bending and shear components.

What is the deflection of a beam on elastic foundation?

Depends on foundation modulus k. Characteristic length beta = (k/(4EI))^(1/4) governs the deflection shape.

How do holes affect beam deflection?

Holes reduce I, increasing deflection. Recalculate I for the net section when holes are significant.

What is dynamic deflection?

Dynamic deflection includes static deflection plus amplification from resonance. The amplification factor depends on frequency ratio and damping.

How does pre-stressing affect deflection?

Pre-stressing creates upward camber counteracting gravity deflection. Net deflection is the sum of all load effects including prestress.

What is the effective length for deflection?

Depends on end conditions. Simply supported uses actual span, cantilever uses free length, continuous beams use span between inflection points.

What is the maximum deflection in a cantilever?

Always at the free end. For point load: PL^3/(3EI), which is 16x larger than simply supported with same load.

What is the difference between deflection and deformation?

Deflection is displacement under loads. Deformation is any change in shape from any cause including thermal and residual effects.

What is the deflection of a continuous beam?

Analyzed using moment distribution, slope-deflection, or three-moment equation methods for multi-span configurations.

How does the modular ratio affect composite beam deflection?

The modular ratio n = E1/E2 transforms one material to equivalent width of the other. The deflection uses the transformed I with standard formulas.

What is the deflection formula for an eccentric load?

For simply supported beam with load at distance a from left support: deflection at load point = Pa^2*b^2/(3EIL) where b = L-a.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Deflection: Vertical displacement of a beam under load.
Flexural Rigidity: EI, product of Young modulus and moment of inertia.
Serviceability: Performance criteria including deflection, vibration, and crack control.
Creep: Time-dependent deflection under sustained load.

Authoritative References

Trusted sources and standards referenced by this calculator.

Shigleys Mechanical Engineering Design AISC Steel Construction Manual Eurocode 3 - Design of Steel Structures

Applicable Standards

AISC 360Eurocode 3ACI 318NDS for Wood

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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.