FoundationalBeamCantileverPoint + UDLSuperposition

Cantilever with a tip load and a UDL

A cantilever turns everything upside-down: the biggest moment is at the support, the whole top face goes into tension, and the tip deflection is the sum of two classical results — for the point load and for the UDL. With one support and no redundancy, statics alone tells the whole story.

Figure 1.The problem: a 3 m cantilever fixed to the wall at A, carrying the tip point load P and the UDL w over its length — the setup only. The fixed-end reactions and the deflected shape come after we solve.

Given

Length: 3 mfixed at A, free tip
: 15 kNpoint load at tip
: 5 kN/mUDL over full length
: 200 GPa
: 80 × 10⁶ mm⁴
: 16 000 kN·m²

Step 1 — Fixed-end reactions

Everything the wall must hold back.

A cantilever has a single support that must react everything — no second support to share the load, and no redundancy to solve for. So statics alone fixes the two reactions at A: vertical equilibrium carries the whole applied load, and moments about the fixed end give the fixing moment.

Vertical equilibrium — the support carries the point load plus the entire UDL:

Moments about the fixed end — the UDL acts at its centroid, from A:

(hogging)

Step 2 — Shear and moment diagrams

The peak moment sits at the support, not midspan.

The shear runs — from 30 kN at the support down to the 15 kN step at the tip load. The moment is hogging everywhere, , largest in magnitude at the support and zero only at the free end.

Nothing sags: the top face is in tension along the whole member, which is why cantilever reinforcement goes in the top — the mirror image of a simply supported beam.

Figure 2.Shear force diagram — from 30 kN at the support down to the 15 kN step at the tip load.

Figure 3.Bending moment diagram — all hogging, peaking at −67.5 kN·m at the fixed end. We follow the US convention (sagging positive, plotted above the axis); the Beam Calculator has a Flip control for the opposite convention.

Step 3 — Tip deflection by superposition

Two classical results, simply added.

The tip deflection is the sum of two textbook cantilever formulas — one for the point load, one for the UDL — added because both act on the same linear-elastic beam:

Figure 4.Deflection diagram — 11.60 mm at the free end.

Figure 5.The solved cantilever — the fixed-end reactions (purple) and the deflected shape (dashed), tip drooping to the computed deflection.

✓

The proof

Hand calculation vs the solver.

Verified — hand calculation vs the solver, to round-off

Quantity	By hand	StructureCalcs
Reaction R_A	30 kN (P + wL)	30 kN
Fixing moment M_A	67.5 kN·m (PL + wL²/2)	67.5 kN·m
Max hogging moment	−67.5 kN·m at x = 0	−67.5 kN·m at x = 0
Tip deflection δ	11.602 mm (PL³/3EI + wL⁴/8EI)	11.602 mm

Every value was worked by hand with the classical method, then checked against this site’s solver — the same engine the Try it button opens. This agreement is re-run automatically on every build.

Now make it yours

Open this exact model in the calculator — then change a load, drag a support, and watch every diagram update in real time. The best way to build intuition is to break it and see what happens.

Take it with you

Export this worked example as a PDF, or download it as a .screport and open it in the Report Builder — the model travels inside the file, so you can reconstruct it, re-solve, and build your own report from it.

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