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1/2
, and r is the distance from
where H
0
(r/L) is the Hankel function of zero order, L =
ʻʼ
the point of the load application. The maximum de
ection occurs at the point
r
=0:
P
8
q
w
g
k
2
f
ð
0
Þ
¼
ð
5
:
28
Þ
In the case of a centrally loaded ice plate, the characteristic length is taken as the size of
the de
ection bowl or action radius. The stresses at
r
→
0 are equal to
c
r
r
¼
r
u
¼
31
þ
l
ð
Þ
P
h
2
k
r
log
2
ð
5
:
29
Þ
2
p
where c is Euler
'
s constant (c = 0.577216).
The de
ection of the plate and its elastic surface curvature changes sign, and at
r
> two
to three times
ʻ
it is practically invisible. At greater distances an ice plate can be con-
sidered in
nite with respect to the vertical load. The maximum tensile stress acts on the
lower ice plate surface under the point of the force application. Within the region
r
<
ʻ
the
upper surface of the ice is compressed. For practical evaluation of the bowl de
ection for
freshwater ice at short-term loading, it is convenient to use a simple relation (Gold 1971):
ʻ ≈
3/4
m
1/4
. The analytical solution shows that the maximum concentrated load at the
edge of the half-in
16
h
nite ice plate or a semi-in
nite crack, is half of that for an in
nite plate,
while the de
ection is increased by four times.
Equation (
5.25
) admits centrally symmetric solutions, also with several axes of sym-
metry. Radial cracks occur initially at the lower surface directly under the ice load.
Solutions for vertical loads of other distributions may be obtained by integration of the
problem on the action of the concentrated load. Calculations show a small effect of a
distribution of uniform loads on the maximum values of the ice plate deformation
parameters. Increasing the radius of the loaded area to 10 m reduces the de
ection at the
center on 12 %.
The limit load can be written as
1
r
f
h
2
k
P
cr
¼ 31
þ l
ð
Þ
C
1
ð
5
:
30a
Þ
where b is the characteristic size of the loaded area, and C
1
(b/
ʻ
) is a coef
cient (Fig.
5.9
).
cients determined
by complicated functions, while obtainable dependences of the limit load until crack
formation, in the most sought interval of the parameter (b/
Existing solutions for the evaluation of the limit load comprise coef
ʻ
) from 0.07 to 1.0, can be
represented more simply as (Pan
lov 1960)
r
t
h
2
P
cr
¼ A 1
þ
B
k
ð
5
:
30b
Þ
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