Pick a scenario; every slider below stays editable.
Gold: peak compressive stress; blue: peak tensile stress; red dashed: bone strength limits (typical cortical values reduced by bone loss); grey dashed: stress if the abductor muscle did not act (the muscle works as a tension band). Green vertical line: neck base.
A cane in the opposite hand cuts the abductor effort because its lever arm is longer than the abductors lever arm; leaning the trunk (green) lowers it further.
The science
One-leg stance statics. The pelvis must stay level, so the moments about the hip centre balance: Fm b = Weff a − C c. Here Weff = 5W/6 is the weight supported by the stance leg (the swing leg is 1/6 of the body), a the horizontal lever arm of that weight, b the moment arm of the abductor muscles (gluteus medius and minimus, line of action 25° from vertical), C a cane force and c its lever arm. Vertical and horizontal balance then give the joint reaction Ry = Weff − C + Fm cosθ, Rx = Fm sinθ. Typical result: R ≈ 2.5-3 W.
Why a cane helps. The cane acts at a lever arm about 6 times longer than the abductors, so 15 % of body weight through it removes about half of the muscle force; leaning over the hip shortens a = a0 − 450 sin(lean) (mm) in the same way.
Internal loads in the bone. The femur is cut at distance s from the head centre; the force and moment transmitted through the cut are the sum of the loads above it: N = F·t, M = ∑ (ri − rcut) × Fi. The abductor force enters only below the trochanter. The cross-section is a hollow circle (outer diameter D, wall t): A = π(D² − d²)/4, I = π(D&sup4; − d&sup4;)/64, and the extreme-fibre stress is σ = N/A ± M (D/2)/I. The safety factor is the lowest ratio of strength to stress (compression 170 MPa, tension 110 MPa, typical cortical values times (1 − loss)², illustrative). Head deflection is the numerical double integral of M/(EI) along the neck treated as a cantilever.
Tension-band effect. Without the muscle the shaft would be loaded as a pure eccentric column and its bending moment would grow all the way down (grey curve); the abductor force cancels most of it. Muscles protect bones.
Honest simplifications. Frontal plane only, static (the load factor stands in for dynamics); one lumped abductor force; femur drawn as a circular hollow tube with the neck as a solid-wall section (no trabecular core, no elliptical neck cross-section, no sagittal-plane loads); cortical strength and the bone-loss laws are typical or illustrative values, and the safety factors are teaching numbers, not clinical risk estimates. Geometry (hip-centre offsets, 440 mm head-knee length, trochanter position) is idealised. The body silhouette is decorative.
Checks. (1) The pelvis free body balances in force and in moment about a different point than the one used to derive Fm; (2) the one-leg stance hip force falls in the literature range and the cane reduction matches a hand calculation; (3) a numerical double integration of the beam equation reproduces PL³/3EI; (4) the formula for I is compared with a brute-force sum of y² dA.