Rear axle types: semi-, three-quarter and full-floating

Why it matters

In a live rear axle the half-shafts drive the wheels, but something must also carry the vehicle's weight, the cornering side thrust and the end thrust at each wheel. Semi-, three-quarter- and full-floating axles differ only in where the wheel bearings sit, and that decides which loads the half-shaft must carry, how thick it must be, and whether the wheel stays on if the shaft breaks. It is a standard theory question and a good exercise in combined bending and torsion.

Key ideas

• Loads at a driving wheel. (1) Driving (and braking) torque – twists the half-shaft. (2) Vertical load (share of vehicle weight) – acting at the tyre centre plane, it causes shear and, if the wheel overhangs its bearing, a bending moment. (3) Side thrust in cornering – acting at the tyre contact patch, a radius r below the shaft axis, it causes a bending moment F_s · r and an axial end thrust. (4) Driving and braking thrust (fore-aft) and their bending.
• Semi-floating axle.
  • One bearing sits between the outer end of the half-shaft and the inside of the axle casing; the wheel hub is fixed directly on the overhanging shaft end.
  • The shaft carries all loads: torque, shear and bending from the vehicle weight (because the wheel overhangs the bearing), bending and end thrust from side forces.
  • Simple, light and cheap – the usual choice for passenger cars and light commercial vehicles with live axles. The shaft must be larger in diameter than in the other types, and if it breaks outboard of the bearing, the wheel can come off. The shaft cannot be removed without removing the wheel.
• Three-quarter-floating axle.
  • A single bearing is mounted on the outside of the axle casing, inside the wheel hub, so the vehicle weight goes from hub through the bearing directly to the casing; the hub is still rigidly keyed or splined to the shaft end.
  • The shaft is relieved of the shear and bending due to the vehicle weight, but because one bearing cannot resist the tilting of the wheel, it still carries bending and end thrust from side forces, plus the driving torque.
  • A compromise of cost and strength; used on some older cars and light vehicles.
• Full-floating axle.
  • The hub runs on two widely spaced bearings (usually taper-roller) mounted on the outside of the casing. The half-shaft ends in a flange bolted to the hub.
  • The bearings and casing carry weight, side thrust and end thrust; the shaft carries only the driving torque.
  • The shaft can be withdrawn without removing the wheel or jacking the vehicle, and the wheel stays on its bearings if the shaft breaks (the vehicle can even be towed with shafts removed). Heavier and more expensive; standard on trucks, buses and heavy 4×4s.
• Design consequences. Moving from semi- to full-floating reduces the shaft's job from combined bending and torsion to pure torsion, so a smaller shaft suffices – but the casing ends and bearings become heavier.

Formulas

M_w = W · a

• M_w: bending moment at the bearing due to wheel load (N·m); W: vertical load on the wheel (N); a: overhang from the wheel centre plane to the bearing centre (m). Applies to semi-floating axles.

M_s = F_s · r

• M_s: bending moment from side thrust (N·m); F_s: lateral force at the tyre contact patch (N); r: tyre rolling radius (m). Applies to semi- and three-quarter-floating axles.

T_e = √(M² + T²), M_e = ½ · (M + √(M² + T²))

• T_e: equivalent twisting moment (N·m), used with the maximum-shear-stress theory; M_e: equivalent bending moment (N·m), used with the maximum-normal-stress theory; M: total bending moment at the critical section (N·m); T: driving torque on the shaft (N·m).

τ_max = 16 · T_e / (π · d³), σ_max = 32 · M_e / (π · d³)

• Maximum shear and normal stresses in a solid shaft of diameter d (m), in Pa.

τ = 16 · T / (π · d³) (full-floating, torque only)

Worked examples

Example 1 (standard). A semi-floating half-shaft carries a wheel load of 4000 N with the wheel centre plane 0.10 m outboard of the bearing, and a driving torque of 1200 N·m. The shaft diameter at the bearing is 45 mm. Find the maximum shear stress and maximum normal stress (straight-line driving, no side thrust).

1. Bending moment: M = W · a = 4000 × 0.10 = 400 N·m.
2. T_e = √(400² + 1200²) = 1264.9 N·m.
3. M_e = ½ (400 + 1264.9) = 832.5 N·m.
4. π d³ = π × 0.045³ = 2.863 × 10⁻⁴ m³.
5. τ_max = 16 × 1264.9 / 2.863 × 10⁻⁴ = 70.7 MPa.
6. σ_max = 32 × 832.5 / 2.863 × 10⁻⁴ = 93.1 MPa.

Answer: τ_max ≈ 70.7 MPa; σ_max ≈ 93.1 MPa.

Example 2 (GATE level). For the same wheel, a cornering side force of 0.6 × wheel load acts at the tyre contact patch (rolling radius 0.30 m) on the wheel where its moment adds to that of the wheel load. Take the same torque 1200 N·m and an allowable shear stress of 70 MPa. Find the minimum shaft diameter for (a) a semi-floating axle and (b) a full-floating axle.

1. Side-thrust moment: M_s = 0.6 × 4000 × 0.30 = 720 N·m; total M = 400 + 720 = 1120 N·m.
2. (a) T_e = √(1120² + 1200²) = 1641.5 N·m; d = (16 T_e / (π τ))^(1/3) = (16 × 1641.5 / (π × 70 × 10⁶))^(1/3) = 0.04925 m.
3. (b) Full-floating carries torque only: d = (16 × 1200 / (π × 70 × 10⁶))^(1/3) = 0.04436 m.

Answer: semi-floating ≈ 49.3 mm; full-floating ≈ 44.4 mm (choose the next standard sizes). The difference grows with heavier axle loads, which is why trucks use full-floating axles.

Common mistakes

• Saying the three-quarter-floating shaft still carries vehicle weight; it carries side-thrust bending and end thrust, while the weight goes through the bearing on the casing.
• Putting two bearings in a three-quarter-floating hub; two widely spaced hub bearings define the full-floating axle.
• Using τ = T / (π r²) or T / A for torsion; for a solid shaft τ = 16T/(πd³) = 2T/(πr³).
• Forgetting that side thrust acts at the tyre contact patch, so its moment arm is the rolling radius.
• Mixing up the equivalent-moment formulas: T_e for shear (Guest/Tresca), M_e for normal stress (Rankine).

For GATE ME

Descriptive questions ask which loads each axle type's half-shaft carries and why heavy vehicles use full-floating axles. Numerical questions are really machine-design shaft problems: combined bending and torsion with equivalent moments, diameter from allowable stress, and pure-torsion sizing for a full-floating shaft. Practise the equivalent-moment formulas with consistent units.

Quick check

1. Which axle type's shaft carries only torque?
2. Where is the bearing in a three-quarter-floating axle?
3. A full-floating shaft of 50 mm diameter carries 1000 N·m. What is the shear stress?
4. In a semi-floating axle, which loads cause bending at the bearing?
5. Why can a truck with full-floating axles be towed with its half-shafts removed?

Answers: 1. Full-floating. 2. On the outside of the axle casing, inside the hub (a single bearing). 3. 16 × 1000 / (π × 0.05³) = 40.7 MPa. 4. The wheel load acting at an overhang and the side thrust acting at the rolling radius (plus fore-aft thrust). 5. The wheels run on hub bearings carried by the casing, independent of the shafts.