Engineering · Mechanics/Statics

Cantilever tip deflection under a uniform load

A cantilever under a uniformly distributed load reaches maximum deflection at the free end.

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Formula

δ_{ma x} = \frac{q L ^{4}}{8 E I}

δ,L in m; q in N/m; E in Pa; I in m⁴

Variables & units

Variables & units
Symbol	Meaning	Unit
$δmax$	Free-end deflection	m
$q$	Uniform line load	N/m
$L$	Beam length	m
$E$	Young modulus	Pa
$I$	Area moment of inertia	m⁴

LaTeX

\delta_{max}=\frac{qL^4}{8EI}

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Derivation & background

The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deformation, constant EI and ideal fixed support. The line load acts over the full length. Shear deformation, separate self-weight and real support compliance are excluded.

Exam blueprint

Validity range

Derivation steps

A cantilever under a uniformly distributed load reaches maximum deflection at the free end.

1State the model assumptions and list quantities with units.
2Substitute into δmax = qL⁴/(8EI); convert quantities to SI first.
3For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

Rearrangements

Base equation for the target quantity

\delta_{max}=\frac{qL^4}{8EI}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Cantilever tip deflection under a uniform load” and check its unit and magnitude.

For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: δ,L in m; q in N/m; E in Pa; I in m⁴.

Exam context

Estimate cantilever deflection under a uniform line load.

Formula cluster

Mechanics/Statics

A cantilever under a uniformly distributed load reaches maximum deflection at the free end.

δ = FL³/(3EI)

Cantilever tip deflection under an end force

M = EIκ ≈ EI·w″

Euler–Bernoulli beam moment–curvature relation

V′ = −q; M′ = V

Distributed load, shear-force and bending-moment relations

Worked example

For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

Applications

Estimate cantilever deflection under a uniform line load.

How do you calculate with Cantilever tip deflection under a uniform load?

Here is how to work through a typical Cantilever tip deflection under a uniform load (δmax = qL⁴/(8EI)) task step by step:

1
Task
Work through the numerical example for “Cantilever tip deflection under a uniform load” and check its unit and magnitude.
Solution path
For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

Frequently asked questions about Cantilever tip deflection under a uniform load

Why does L have a fourth power?+

Uniform loading creates a bending moment varying quadratically along the cantilever. Integrating the beam equation twice makes tip displacement proportional to qL⁴/EI, so length strongly affects stiffness. Assume constant line load and ideal fixed end in small-deflection Euler–Bernoulli theory. Application: Estimate cantilever deflection under a uniform line load. The numerical example For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm anchors the idea in a concrete measurement case. The corresponding units are δ,L in m; q in N/m; E in Pa; I in m⁴. For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm. This checks that the reference quantities are compatible.

Where is the maximum?+

Deflection is zero at the ideal fixed support and grows toward its maximum at the free end. Tip rotation is also nonzero. Different supports or loading change the curve. Assume constant line load and ideal fixed end in small-deflection Euler–Bernoulli theory. The corresponding units are δ,L in m; q in N/m; E in Pa; I in m⁴. For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm. This checks that the reference quantities are compatible. Scope: The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deformation, constant EI and ideal fixed support. The line load acts over the full length. Shear deformation, separate self-weight and real support compliance are excluded. A practical application is Estimate cantilever deflection under a uniform line load.

What does q in N/m mean?+

q is force per beam length, not area pressure in N/m². Convert an area load through the loaded width to line load. The equation assumes constant loading across the full length. Assume constant line load and ideal fixed end in small-deflection Euler–Bernoulli theory. Scope: The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deformation, constant EI and ideal fixed support. The line load acts over the full length. Shear deformation, separate self-weight and real support compliance are excluded. A practical application is Estimate cantilever deflection under a uniform line load.

How do I check the result and units for Cantilever tip deflection under a uniform load?+

Write the quantity and unit from the variable list next to every value: δ,L in m; q in N/m; E in Pa; I in m⁴. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Assume constant line load and ideal fixed end in small-deflection Euler–Bernoulli theory.

Which model limit matters for Cantilever tip deflection under a uniform load?+

The background states this starting condition: The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deformation, constant EI and ideal fixed support. The. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Assume constant line load and ideal fixed end in small-deflection Euler–Bernoulli theory.

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Which formula describes Cantilever tip deflection under a uniform load?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare, Unified Engineering, beam deflection relations.

More formulas: Mechanics, strength and materials

E = \frac{\sigma}{\varepsilon} = \frac{F L_0}{A_0 \Delta L}

Young's modulus

\Delta\vec p=\int_{t_1}^{t_2}\vec F_{\mathrm{ext}}dt, \qquad \Delta\vec L=\int_{t_1}^{t_2}\vec\tau_{\mathrm{ext}}dt

Linear and angular impulse–momentum balance

\sum_i \vec F_i=\vec 0,\quad \sum_i M_{i,O}=0

Static equilibrium conditions

\sigma_\mathrm{m} = \frac{F}{A}

Average axial normal stress

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Quanta Study editorial team

Symbol

Meaning

Unit

δmax

Free-end deflection

q

Uniform line load

N/m

L

Beam length

E

Young modulus

I

Area moment of inertia

m⁴

Exam blueprint

Validity range

Derivation steps

A cantilever under a uniformly distributed load reaches maximum deflection at the free end.

1State the model assumptions and list quantities with units.
2Substitute into δmax = qL⁴/(8EI); convert quantities to SI first.
3For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

How do you calculate with Cantilever tip deflection under a uniform load?

Here is how to work through a typical Cantilever tip deflection under a uniform load (δmax = qL⁴/(8EI)) task step by step:

Task

Work through the numerical example for “Cantilever tip deflection under a uniform load” and check its unit and magnitude.

Solution path

For q=500 N/m, L=1 m, E=200 GPa and I=10⁻⁶ m⁴, δmax=0.3125 mm.

Frequently asked questions about Cantilever tip deflection under a uniform load

Why does L have a fourth power?+

Where is the maximum?+

What does q in N/m mean?+

How do I check the result and units for Cantilever tip deflection under a uniform load?+

Which model limit matters for Cantilever tip deflection under a uniform load?+

Retain Cantilever tip deflection under a uniform load for exams

Create a curated FSRS exam set for δmax = qL⁴/(8EI): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential