You went ice fishing with Grandpa on a lake in rural Minnesota. You were concern

Question

You went ice fishing with Grandpa on a lake in rural Minnesota. You were concerned the ice would crack and you would fall with Grandpa onto the lake. The temperature right above the lake was 50C, Your Grandpa said the lake wont crack, it is 65 feet deep and the temperature at the bottom of the lake is 200C and he wanted to walk on the lake and start ice fishing. You didnt know if you should listen to Grandpa so you needed to calculate the thickness of the ice to see if it can hold. Your mass is 80 kg and your grandpas mass is 54 kg. Was Grandpa right?

Explanation / Answer

There are several formulae, but the simplest and most widely used one is Gold's Formula, as utilised in [1]:

P=Ah2 where:

P: the allowed load (kg)
h: the effective thickness of the ice sheet (cm)
A: a parameter that depends on the strength of the ice and includes a safety factor (between 3.5 ~ 7)

As far as load bearing capacity is concerned, there are 2 kinds of ice:
(a) Blue ice (clear ice) - formed by the freezing of water. All calculations are based on the strength of an ice sheet consisting of blue ice.
(b) White ice (snow ice) - formed when water-saturated snow freezes on top of ice, making an opaque white ice which is not as strong as clear ice. Its thickness counts ashalf.
Also:

Effective ice thickness (h) is defined as the good quality, well-bonded white and blue ice that is measured in an ice cover. Poor quality or poorly bonded ice should not be included in the measurement of ice thickness. The following are examples of ice that should be excluded from the measurements if they are encountered:

Ice layer with water lens (>5 mm diameter) with a cumulative volume greater than 10% of the total volume.

Ice layer with visible incompletely frozen frazil (slush) ice.

Ice layer that is poorly bonded to the adjoining layer.

Ice layer that has been found to have a strength less than 50% of good quality blue ice (a number of specialized methods are available for determining ice strength).

Ice that has wet cracks.

The contact pressure is not critical because [3]:

The bending of the ice under the load causes exural stress to be imposed on the ice cross section. If the maximum exural stress does not exceed the ice strength, the load will be supported. Ice is a material weak in tension and relatively strong in compression. Thus the critical stress is the maximum tensile stress at the bottom of the ice directly under the load.

The maximum flexural stress, according to Westergaard [3]:

max=0.275(1+)Ph2log10(Eh3kb4)

where:
max: maximum exural stress in the ice sheet
: Poisson's ratio ( = 0.33 for ice)
P: magnitude of load
E: elastic modulus ( = 5*10^6 kPa for ice)
h: ice thickness
k: subgrade reaction ( = 9.81 kN/m^3 for water )

b=1.6c2+h20.675h , for c1.724h
b=c , for c1.724h

c: radius of loaded area.

A value of 3.5h2 to 5h2 corresponds to a simply loaded ice sheet operating exural stress level of 500 kPa to 550 kPa.

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