Cantilever Beam Deflection

A cantilever is held at one end and free at the other: a bracket, a shelf arm, a probe, a tool holder. This calculator takes a point load anywhere along it, a load spread evenly over its length (its own weight, for instance) and a moment at the free end, alone or together, and gives the deflection and slope at the free end, the moment and force the fixed end carries, and the bending stress there.

The first is a point load P at a distance a from the wall (at the free end, a = L and it becomes PL³ ÷ 3EI); the second a load w per unit length over the whole span; the third a moment at the free end. The slopes at the free end are Pa² ÷ 2EI, wL³ ÷ 6EI and M₀L ÷ EI. With more than one load, each is worked alone and the results are added. All loads act the same way: the forces down, the moment bending the free end down.

Section and material. The moment of inertia Icomes from the section's dimensions, about the axis across its width, so the height is the depth in the direction of the load. A rectangular tube is taken with sharp corners; its maker's published I, which counts the corner radii, is a few percent lower and better when you have it. The moduli in the material list are typical values from Machinery's Handbook. A real bar can differ by a few percent, so for a design use the value from the material's specification.

Worked example. A steel bar 1 in wide and 2 in deep (I = 0.6667 in⁴, E = 29 × 10⁶ psi) sticks out 24 in with 500 lbf at the end. EI = 19.33 × 10⁶ lbf·in², so the end drops 500 × 24³ ÷ (3 × 19.33 × 10⁶) = 0.119 in and turns 0.0074 rad (0.43°). The wall carries 500 lbf and 12,000 lbf·in, a bending stress of 18,000 psi. In metric, a 6061-T6 tube 40 mm across with a 3 mm wall, 800 mm long, under 0.5 N/mm (500 N/m): 6.06 mm and 0.58° at the end, 160 N·m at the wall. Load either with the button under the inputs.

Where the formulas stop: they are small-deflection beam theory, for a straight, slender beam of one constant section and one linear-elastic material, held by a wall that does not move or turn. The calculator flags a slope at the free end over 0.1 rad (5.7°), where the curvature the formulas leave out reaches 1.5%, and a span under 10 times the section depth, where deflection from shear, which they also leave out, starts to add (about 1% at that ratio for a solid steel rectangle). Keep the stress at the wall under the material's yield strength, or the beam bends for good and the results no longer hold. A real fixed end gives a little, through bolts, welds or the part it is held in, so a real cantilever deflects somewhat more than this.

Reference: Machinery's Handbook, 27th ed., Table 1, Stresses and Deflections in Beams, pp. 261 to 271: cases 10, 11 and 12 on p. 265; adding loads on p. 272. The slopes are the derivatives of the handbook's general formulas; the end moment is the standard result (Shigley, Table A-9; Roark, Table 8.1). Section formulas: pp. 239 and 241. Moduli: pp. 277, 356 and 420. Reviewer: pending.