Cantilever with a point load at any position.

METHOD

What this calculator solves

This is the constant-EI Euler-Bernoulli solution for a cantilever fixed at one end and free at the other. Enter the downward point-load magnitude P and its distance a from the fixed end. The unloaded segment beyond the load moves with the rotation and translation already accumulated at the load.

Input convention: P is a non-negative downward magnitude; a is measured from the fixed end and may range from 0 to L; E is entered in GPa and I in cm⁴. Actions are shown as magnitudes, so the fixed-end moment is labelled by magnitude rather than a signed sagging/hogging algebraic value.

The classic free-end point-load case is recovered at a = L: the fixed-end moment is PL, free-end deflection is PL³/(3EI), and free-end rotation is PL²/(2EI). At a = 0, the entered point load has no lever arm and produces zero transverse response. Intermediate positions cover the familiar “load somewhere along a cantilever” case.

SCOPE

Assumptions and limitations

• Euler-Bernoulli, linear-elastic beam theory with constant E and I.
• The root is perfectly fixed and the free end is unrestrained; connection flexibility and support movement are excluded.
• One transverse point load is included; no distributed load, axial force, torsion, or self-weight is added by this page.
• Shear deformation, second-order effects, material nonlinearity, large rotation, and dynamic response are excluded.
• The page reports deflection at the load and at the free end, plus free-end rotation; it does not search for a separate maximum along the span.
• This is an analysis aid, not a section-capacity, connection, serviceability, or code-compliance check.

REFERENCES

Further reading

Compare the position-dependent load case with StructX: cantilever beam with point load at any point. Its free-end limit is also shown by StructX: cantilever beam with a free-end point load. For beam-theory assumptions and boundary conditions, see MIT OpenCourseWare beam theory notes.