Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 152

R

Load y

y

x

h

Pressure
y

Fig. 5.8 Schematic picture of the bending of ice sheet under load; h is ice thickness and R is the
characteristic length (Goldstein et al. 2014)

drilling rigs have risen again in connection with the planned development of oil and gas
resources on the shelf of Russian Arctic.
The physical basis of the bearing capacity problem was presented by Hertz (1884).
Among the

first practical results was a special issue of the Proceedings of the Scienti c

and Technical Committee of the People s Commissariat of Transport of the USSR (see

Goldstein et al. 2014), where there were also links to the papers by Sergeev (1929) and
Bernshteyn (1929). The formulation of the theory by Wyman (1950) has been widely used
in the western literature. For a point load, the ice is forced down, while it is supported by
water pressure (Fig. 5.8 ). The supporting pressure is

'

ection, and
thus a liquid water body acts mathematically in the same way as an elastic foundation (i.e.,
the response of the foundation is proportional to the de

ˁ w gw, where w is the de

ection). The general theory gives

the equation of de ection as (e.g., Michel 1978):

r 4 w ¼ q w gw

Þ
The load acts in the origin and is accounted for in the boundary conditions. Under a
static point load, the solution for the de

ð 5 : 25

ection shows an exponentially damping flexural

sine wave with maximum under the load. The wavelength is 2 ˀʻ , where

1 = 4

Eh 3
12 q w g ð 1 l 2 Þ

k ¼ ð 5 : 26 Þ

is the characteristic length of ice plate on water foundation. 5 Equation ( 5.25 ) can be
reduced to the Bessel equation (see Goldstein 2014). For an in nite plate, centrally

symmetric solution has the form

P
4 q w gL 2 ReH 0

r
L

f ð r Þ ¼ ð 5 : 27 Þ

5 Note that, in one-dimensional case, the characteristic length is defined for beams and that is larger
than the plate value by a factor of p , see Eqs. ( 5.22d ) and ( 5.26 ).