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A slender cantilever with an end force deflects in proportion to force and the cube of its length.
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Formula
δ and L in m; F in N; E in Pa; I in m4
| Symbol | Meaning | Unit |
|---|---|---|
| Vertical tip deflection | m | |
| Force at the free end | N | |
| Beam length | m | |
| Young modulus | Pa | |
| Area moment of inertia | m4 |
\delta=\frac{FL^3}{3EI}The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deflection, constant EI and ideal fixed support. Shear deformation, self-weight, local indentation and plasticity are excluded. Doubling length increases deflection eightfold with other values fixed.
The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deflection, constant EI and ideal fixed support. Shear deformation, self-weight, local indentation and plasticity are excluded. Doubling length increases deflection eightfold with other values fixed.
A slender cantilever with an end force deflects in proportion to force and the cube of its length.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Cantilever tip deflection under an end force” and check its unit and magnitude.
For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: δ and L in m; F in N; E in Pa; I in m4.
Formula cluster
A slender cantilever with an end force deflects in proportion to force and the cube of its length.
For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm.
Estimate stiffness of a slender cantilever loaded at its free end.
Here is how to work through a typical Cantilever tip deflection under an end force (δ = FL³/(3EI)) task step by step:
Task
Work through the numerical example for “Cantilever tip deflection under an end force” and check its unit and magnitude.
Solution path
For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm.
Curvature scales with bending moment divided by EI; for a tip force, moment grows along the beam. Integrating the beam equation twice yields the L3 tip-deflection dependence. Long cantilevers are especially sensitive. Assume small elastic deflection, constant EI and an ideal fixed support. Application: Estimate stiffness of a slender cantilever loaded at its free end. The numerical example For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm anchors the idea in a concrete measurement case. The corresponding units are δ and L in m; F in N; E in Pa; I in m4. For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm. This checks that the reference quantities are compatible.
Use the area moment about the bending axis corresponding to the deflection direction. For unsymmetric sections, the weak principal axis may govern. I is not mass moment of inertia and has units m4. Assume small elastic deflection, constant EI and an ideal fixed support. The corresponding units are δ and L in m; F in N; E in Pa; I in m4. For F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 mm. This checks that the reference quantities are compatible.
The ideal boundary fixes translation and rotation. A real support can rotate or move, adding compliance. Accurate design must model support stiffness in its boundary conditions. Assume small elastic deflection, constant EI and an ideal fixed support. Scope: The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deflection, constant EI and ideal fixed support. Shear deformation, self-weight, local indentation and plasticity are excluded. Doubling length increases deflection eightfold with other values fixed. A practical application is Estimate stiffness of a slender cantilever loaded at its free end.
Write the quantity and unit from the variable list next to every value: δ and L in m; F in N; 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 F=100 N, L=1 m, E=200 GPa and I=10−6 m4, δ≈0.167 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 small elastic deflection, constant EI and an ideal fixed support.
The background states this starting condition: The Euler–Bernoulli solution assumes a slender prismatic beam, small elastic deflection, constant EI and ideal fixed support. Shear. 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 small elastic deflection, constant EI and an ideal fixed support.
Quanta exam set
Curated exam set for "Cantilever tip deflection under an end force":
Question (front)
Which formula describes Cantilever tip deflection under an end force?
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How do you rearrange Cantilever tip deflection under an end force for Base equation for the target quantity?
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Which common mistake happens with Cantilever tip deflection under an end force?
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