METHOD
What this calculator solves
This page uses linear superposition for a prismatic cantilever with constant E and I. Enter all loads as non-negative magnitudes acting in one common bending sense. Set an unused action to zero; the result panel also shows the separate free-end deflection contributions so the combined result can be audited.
|Mfixed| = wL2/2 + Pa + M0
Vfixed = wL + P
δfree = wL4/(8EI) + Pa2(3L − a)/(6EI) + M0L2/(2EI)
θfree = wL3/(6EI) + Pa2/(2EI) + M0L/(EI)
Input convention: w is a full-span UDL in kN/m, P is a point-load magnitude in kN at a measured from the fixed end, and M0 is a free-end moment magnitude in kN·m. E is entered in GPa and I in cm4. The existing calculator treats the three actions as same-sense positive inputs; opposite-sense loading is not represented by this page.
The parameterized form contains several familiar textbook cases. Use UDL only for the full-span UDL solution; point load only for an intermediate point load, with a = L giving the free-end point-load case; and end moment only for the pure applied-couple case. Non-zero combinations give UDL + point load, UDL + end moment, point load + end moment, or all three actions.
For UDL only, the fixed moment is wL2/2 and free-end deflection is wL4/(8EI). For end moment only, fixed shear is zero, free-end deflection is M0L2/(2EI), and free-end rotation is M0L/(EI).
SCOPE
Assumptions and limitations
- Linear-elastic superposition, small deflection, and constant E and I are assumed.
- The root is perfectly fixed; support movement, connection flexibility, and partial restraint are excluded.
- The UDL acts over the full span, only one point load is available, and the applied couple is at the free end.
- All entered actions must act in the same sense. Do not use this page for cancellation between opposite-sense actions.
- Shear deformation, second-order effects, material nonlinearity, large rotation, torsion, dynamic response, and stability are excluded.
- The page reports fixed-end actions, free-end deflection and rotation, and deflection components; it does not provide a full shear/moment/deflection diagram or search every possible maximum.
- This is an analysis aid, not a section-capacity, connection, serviceability, or code-compliance check.
REFERENCES
Further reading
For comparable individual load cases, see StructX: cantilever beam with a uniform distributed load, StructX: cantilever beam with a point load at any point, and StructX: cantilever beam with a free-end point load. For the beam-theory basis and boundary conditions, see MIT OpenCourseWare beam theory notes.
Include self-weight in w only when the full-span UDL idealisation is appropriate, and confirm that the assumed common load sense matches the physical combination before relying on the sum.