How Thick Must the Ice Be: A Bearing Bound for Crossing a Frozen Lake on Foot, by Sledge, and by Loaded Vehicle

Every winter, people judge lake ice by eye, by habit, or by thicknesses remembered from other winters, and every winter some of them are wrong. This paper replaces the guess with a bound computed from stated assumptions. The ice sheet is modelled as a floating elastic plate whose breakthrough load scales with the square of its thickness, with a dimensionless plate coefficient of 0.9, a flexural strength for clear freshwater ice stated as a range of 0.4–1.0 MPa with a design value of 0.5 MPa, a safety factor of 3.0 for loads that keep moving, and a safety factor of 6.0 for loads that park, because sustained loading lets ice creep. For wheeled vehicles the governing load is the heaviest axle rather than the gross weight. The resulting minimum clear-ice thicknesses for moving loads are 8.5 cm by formula for a person with a pack — raised to a mandatory floor of 10.0 cm, for reasons stated in the text — 16.2 cm for a loaded sledge of 400 kg, 25.6 cm for an animal cart of 1,000 kg, 37.1 cm for a light truck of 3,500 kg, and 57.2 cm for a loaded truck of 10,000 kg. Parking doubles the required load capacity and so multiplies every thickness by 1.414: a loaded truck that may cross at 57.2 cm needs 80.9 cm to park. White ice formed from frozen snow and slush is charged at half the strength of clear ice, which also multiplies thickness by 1.414, so the two discounts are numerically equal and compound to a factor of 2 in thickness when both apply. At 30 cm of clear ice the allowable moving governing load is 1,376 kg against a breakthrough load of 4,128 kg. The bound is deliberately conservative, and its most important output is negative: ice that carries a walking person many times over can still be less than halfway, in the load it may bear, to carrying the vehicle that follows the person onto the lake.