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Some Observations on Simply Supported Circular Ice Plates Loaded at
the Centre
Gold, L. W.
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DOI ci-dessous.
Publisher’s version / Version de l'éditeur:
http://doi.org/10.4224/20337958
Internal Report (National Research Council Canada. Division of Building
Research), 1957-03-01
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NATIONAL RESEARCH COUNCIL
CANADA
DIVISION OF BUILDING RESEARCH
SOME OBSERVATIONS ON SIMPLY SUPPORTED
CIRCULAR ICE PLATES LOADED AT THE
CENTRE
by
L. W. Gold
Report No. 117 of the
Division of Building
Research
OTTAWA
March
1957
PREFACE
There is a great need for a rational method of
predicting the strength of ice sheets on lakes and
rivers. Aircraft and truck and tractor trains use
frozen rivers and lakes in Northern Canada in winter,
and pUlpwood logging companies establish log dumps on
lakes to which logs are trucked as soon as the ice
cover has attained sufficient strength to carry the
loads. The study of the strength of ice is one of the
major projects of the Snow and Ice Section of this
Division.
Studies of full-scale ice sheets are relatively
costly and control of even the major factors may be
difficult, if not impossible. Most fortunately it
has been possible to study in the laboratory many
features of the larger problem by means of model tests
such as the one now reported. The value of this approach
has already been demonstrated. Further studies involving
both model and large-scale tests are in progress.
Ottawa,
March,
Ｑ Ｙ Ｕ
N. B. Hutcheon 7
Assistant Director.
SOME OBSERVATIONS ON SIMPLY SUPPORTED CIRCULAR ICE PLATES
LOADED AT THE CENTRE
by
L. W. Gold
Before it became too involved with the various
aspects of the problem of the bearing strength of ice, the
Snow and Ice Section of the Division of Building Research,
National Research Council, thought that it would be profitable to gain an understanding of the behaviour of ice
plates under load. Therefore, a series of observations
was undertaken on the deflection of a circular ice plate
simply supported at the edges and loaded at the centre.
From these observations valuable experience was gained on
the elastic behaviour of ice under conditions of short
duration loads.
It was possible also to obtain a value
for the rigidity modulus of ice and a rough estimate of
Young's modulus and Poisson's ratio. Finally, the results
indicated the usefulness of continuing such studies on the
behaviour of ice plates under load, but on a larger scale.
THEORY
Love (1) gives for the deflection of a circular
plate loaded at the centre and simply supported at the edge,
w =
P
a
27TD
r
where
w is the deflection at radius r
P is the total load applied at the centre
D is the rigidity modulus
ｾ
ｩ
the Poisson's ratio
a is the plate radius
Seely and Smith (2) discuss the conditions under which the
assumptions implicit in the derivation of equation(l)are
valid. Experiments have ｳ ｨ ｯ ｾ ｶ that this equation is
(1)
- 2 -
Q
ZO.l and the maximum deflection is less
a
correct for
ｾ
than
where h is the thickness of the plate.
These
conditions ensure that any straight line drawn through the
plate, normal to the middle surface, before the plate is
bent remains reasonably straight and normal to the middle
surface after the ｰ ｬ ｡ ｴ ･
bent, and that no direct tensile
stresses due to large deflections are developed.
If we let a equal 10 inches in ･ ｱ ｵ ｡ ｴ ｩ ｯ ｮ Ｈ ｾ
it is
seen that the last term on the right is of the order of
1 per cent of the second term. Therefore, this term was
neglected in applying the theory to the data.
Roark (3), for the deflection w when load is applied
uniformly over an area of radius r O' gives
w
=L
ｾ
f7r42"*"r802)ln
11
2111)
l'
2 2-r2
3+'\I(a2_r2)+1 1-q- 1'0 (a
8 1+c:r
16 Ｑ Ｋ
a2
)lJ
for 1'0> 0
and
w
=...l....
1
[8
2TTD
for
l'
3+ q- a 2 _ r0
Ｑ Ｋ
8
2
In
ｾ
2
r
_ 1'0
O
32
7+3\OJ
1+ ｾ
J
= O.
In most of the load tests that were made
r
O
l'
a
= 0.5inches
=
=
5 inches
10 inches
Assuming u-= 0.3 and considering the terms in equation (2)
1'2 In
l'
= 4.33
r0 2 In §;
0r
= 0.02
'4
1
8
1
16
§;
3+ \J (a2_r2)
1+\r
Ｑ ｟ ｾ ｲ Ｐ
---Ｑ Ｋ
a2
= 23.8
(a 2_r2)
= 0.01
(2)
- 3 -
Considering the terms in equation (3)
3+'1 a 2 = 31 • 7
1
8
1+'1
In
r 2
--!L
= 0.09
7+3 V- = 0.05
1 + 'f
32
Therefore it is evident that all the terms involving r O
are less than 1 per cent of the significant terms and were,
therefore, neglected in analysing the load test results.
The equations considered valid are, therefore,
w =
U
p
27TD
2
- p
o - '2D
W
Wo
P
ｾ
a
- 1
r
8
Ｓ Ｋ
l+q-
(a 2 - r2)]
0
ｲ
r
In ｾ
r
t
1 3+\la 2J
ｬ Ｋ
8
=0
:;: _1.
Ｓ Ｋ
8 1 +<::J
0:;
=k
0 ｾ
21fD
Substituting in equation (4)
Therefore,
ra
D = -----.....,..--..,r---a 2-r2
r
2
s rrt
For r
In
k-k O
a2
)
= 5 inches
(6)
- 4 From equation (5)
r = _ •50
3
1 + .50 kO D
kO D +
Because of the sign of w, k O and k are negative numbers.
5
From the definition of D
D=
Eh3
12 (
I-V2 )
(8)
where E is Young's modulus
(9)
APPARATUS
A loading frame (Fig. 1) was constructed of steel
angle and clamped to a metal-topped rigid bench. The ice
plate to be loaded was supported on a steel ring 20 inches
in diameter which contained sixteen i-inch screws uniformly
spaced around the ring circumference. These could be adjusted to ensure that the ring had uniform support. Dial
gauges with an accuracy of 10- 4 inches per division were
mounted on a cross-piece to measure along a diameter of the
ice plate the deflections at the centre and at 5 inches on
either side. The load was applied to the ice plate over a
circular area loaded through a ball by a lever.
EXPERIMENTAL PROCEDURE
The ice plate was made artificially in a large tank,
cut to a diameter of 20 inches with a band saw, and planed
to uniform thickness. It was then placed on the steel ring
of the loading frame and preloaded by a small weight to
ensure firm contact. The screws on the ring were adjusted
until they just conta.cted the ice as indica:ted by the gauges
mounted for measuring deflections.
Initial observations had indicated that if a loading
cycle was completed in 10 seconds or less, the behaviour of
the ice was essentially elastic. A loading procedure was
adopted therefore, in which the load was applied, the gauges
read, and the load removed, all in approximately 5 seconds.
A range of loads was used in which the magnitude was not
increased or decreased in a regular way but chosen at
random. On completion of the test, the deflection at the
centre and the mean of the deflections at r = 5 inches were
- Splotted against the applied load and the values of k and
O
k S obtained.
DATA
Observations were carried out on three plates whose
thicknesses were 0.44 inches, 0.S4 inches and 0.66 inches.
More than one series of load cycles was carried out on each
plate. The dates on which the tests were made, along with
the measured values of k and k s and the calculated values
for D and Dk O' are givenOin Table 1. All tests were made
at -9°C.
Figure 2 shows a typical set of load deflection
curves. In all cases k O and k were constant over the
S
loading range.
From Table 1 it is seen that the accuracy in the
calculation of individual values of D was of the order of
+ 10 per cent. The source of this error was not so much in
the actual measurement of load and resulting deflection as
in factors which affected each reading in any given test in
such a way as to alter the slope of ｾ ｨ
load-deflection
curve and thus, k O and kS. Possible sources of this error
are non-isotropic behaviour of the plate, the load not being
applied exactly in the centre, and deformation of the plate
at the ring support due to the compressive stresses existing
there. A rough estimate of the edge deformation indicated
that it should be less than 2 per cent of the central
deflection. This source of error is largely eliminated in
the calculation of D, as it is the differences k S - ｾ kO' that
occur in equation (64
From
･ ｱ ｵ ｡ ｴ ｩ ｯ ｮ Ｈ
it is seen that
(10) .
where C is a constant.
Therefore
wo
p
1
= kO =
21TCh3
3+ <:T a
l+T
2J .
(11)
In Figure 3 the logarithm of the average value for k O for
each plate is plotted against the logarithm of h. A line
with a slope of -3 is dravVll through the plotted points.
The logarithm of k S is also plotted on the same figure.
It is seen that the results for the O.SS-inch and the 0.66-inch
plates are consistent with the theory, but the results for
the 0.44-inch plate deviate markedly. There was no apparent
TABLE I
Date
22/11/
24/111
13/12/
Average
23/11/
24/11/
13/12/
9/1/
11/1/
11/1/
11!11
11/11
Average
4/11
9/1/
11/1/
11/1/
11/1/
11/1/
Average
h
inches
rO
inches
kO
inches/pound
k
5
inches/pound
pound-inches
0.55
0.55
0.5
0.5
2.12 x 10- 4
2.14
1.30 x 10- 4
1.28
2.38 x 10 4
2.15
5.04
4.61
2.13
1.29
2.27
4.83
4.60 x 10 4
PLATE
BROKE
DUHING
TEST
TI
0.66
0.56
0.56
0.66
0.66
0.66
0.66
0.66
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1.19 x 10 -4
1.22
1.26
1.23
1.30
1.32
1.23
1.26
1.25
0.74 x 10- 4
0.74
0.80
0.76
0.83
0.81
0.77
0.77
0.78
4.60
4.06
4.93
3.84
4.31
4.06
4.28
0.44
0.4·4
0.44
0.44
0.44
0.44
0.5
0.5
0.75
0.25
1.0
0.5
5.16 x 10- 4
5.17
5.13
5.28
5.56
5.18
5.25
3.09 x 10- 4
3.05
3.17
3.24
3.13
3.21
3.15
0.88 x 10 4
0.83
1.01
0.96
1.11
1.01
0.97
-.:.
Ｏ
Ｇ
ｾ
R'7
kOTI
5.48
4.68
5.80
5.00
6.40
5.06
5.30
5.12
5.35
4.56
4.30
5.20
5.06
5.56
5.26
4.99
m
- 7 -
reason for this deviation; it was probably due to the
difficulty of establisl1ing the ｰ ｬ ｾ ｴ
thicbless accurately.
In Fig. 4 the average ｶ ｾ ｬ ｵ
each plate is plotted against h).
found to be equal to 1.47 x 104
for D calculated for
From this Figure, C is
Therefore,
D ::;: 1.47
x
105 h 3 .
(12 )
From Fig. 3 it is found that
k
o
::;:-
0.3Lt
x
10 4
(13 )
h3
Utilising these values for k O and D,
kOD ::;: 5.0,
which agrees favourably v,rith the values given in Table I,
particularly the over-all average. Substitutinp' this value
for kOD in equation (7) gives
0.33.
ｾ Ｚ Ｚ ［
Using this value for Poisson's ratio, and sUbstituting
equation (12) for D in equation (9) and solving for Young's
modulus gives
E ::;: 1.57 x 10 6
ｕ ｌ ｔ ｉ ｔ ｨ ｾ ｔ
STRENGTH OF ICE IN TillJSION
For the radial and tangential stress at the centre
of a simply supported plate loaded at the centre, Roark
gives
Sr ::;: St ::;: - 2 3P
h2
Letting T=
a
r O ::;:
Sr
[ｬ Ｋ Ｈ ｬ Ｋ ｾ
In ｾ
]
(14)
0.33
inches
0.5 inches
= 10
= St
- 2.4P
- hT
Since the stress-deflection curve is linear to failure for
ice loaded dynamically, it will be assumed that equation
(15) is valid for the calculation of ultimate strengths.
(15 )
- 8 -
Table 2 gives the results of a number of tests on plates
which had undergone previous loading tests. The approximate total number of load cycles to which the plate had
been subjected before failure occurred is given.
Figure 5a is a photograph of one of the plates after
failure. Most of the non-radial fractures occurred when
pieces of the plate fell from the loading ring. Figures
5b and c are sections of the same plate photographed with
polarized light. Some failures which coincided with grain
boundaries are indicated.
TABLE 2
Plate thickness
h inches
No. of load
cycles
0.46
0.62
0.75
0.65
0.68
0.81
0.54
0.44
8
20
76
13
20
34
31
37
Pat
failure Ibs.
56
118
124
90.1
135.3
236.7
78.8
50.5
Average ultimate strength
Minimum ultimate strength
ｔ ･ ｭ ｾ ･ ｲ ｡ ｴ ｵ ｲ
C.
-10
-10
-10
-10
-10
-10
-10
-10
sr = ｓ
Ib/in
640
746
532
515
704
860
648
626
660
515
DISCUSSION
One interesting result of these tests was that the values
for Young's modulus and the ultimate strength of ice were
significantly larger than the values normally reported in
the literature. This is not surprising in the light of the
symmetrical stress distribution which exists under the type
of loading used. The calculated values for Young's modulus
and Poisson's ratio are in good agreement with observea
values which have been obtained in the laboratory for single
ice crystals. This tends to confirm the conclusion that
when shear stresses exist, there is an additional contribution to the strain which likely arises at the grain boundaries.
Though this method of testing does not lend itself to
accurate determinations of Young's modulus and Poisson's
ratio on the basis of tests ona single plate, from tests
on a number of plates of different thicknesses, the dependence of D and k on h3 can be determined and a much hieher
accuracy achieveg. The apparatus used in the present tests
limits the maximum plate thickness to about 1 inch. The
success achieved to date indicates the advantages of redesigning the apparatus so that plates 2 to 3 inches thick
can be tested.
- 9 -
The tests to date have been carried out in such a way
as to ensure that the ice behaves elastically. Information
is now required on the behaviour of ice plates when the
load is essentially static. An apparatus capable of loading
plates up to 3 inches thick would be very useful for this
study.
REFERENCES
1.
Love, A.E.H., A treatise on the mathematical theory of
elasticity, 4th edition. Cambridge University
Press, 1952, 643 p.
2.
Seely, F.B. and J.O.Smith, Advanced mechanics of materials,
2nd. edition. Wiley and Sons, New York, 1952, 680 p.
3.
Roark, R.J., Formulas for stress and strain, 3rd edition.
McGraw-Hill, New York, 1954, 381 p.
BR 6313
Figure 1
Photograph of Apparatus
DBR Internal Report No. 117
100 I
i i
,
i
80 I
I
I
ar 60 I
I
I
I
/1
-
,
i i i
/'
i i i
i
i i i
I
I"
I
I
I
1/
I
Iｾ
I
I
I
I
I
I
I
I
I
I
I
I
I
120
140
160
.J
'c
-
•
I
a,
c
«
o
...J
40 I
20 I
00
"
ｾ
I
20
40
ｾ
I
60
80
100
4
DEFLECTION x 10 (INCHES)
FIGURE 2
TYPICAL LOAD-DEFLECTION CURVE FOR
AND r=5 INCHES
r =0
10·0
\
9-0
\
8-0
\
7·0
4'0
_
\
SLOPE=-3
ｾ
,
ｾ
•
':\
ｾ
2-5
m
...J
....... 3-0
-
'....
\
Z
X
ｾ
1\-,
-2 1--90
C
Z
·8
7
« .
-2
-6
-5
1\
X
LEGEND
•
Ko
X
K5
!
,
\
1\-.
·3
-IS
-4
-5
-6 -7 '8 -9 1-0
h (INCHES)
FIGURE 3
LOGARITHM OF AVERAGE Ko a K5
PLOTTED AGAINST THE LOGARITHM
OF h-
INT. REPT. 1/7
6
-
N
Z
.......
ｾ
-
/
/
V
'0
)C.
o
(/)4
3
;:)
/
o
o
ｾ
-oｾ
,/r
3
(,!)
a::
/0
2
i
o
o
V
.,
·4
·5
FIGURE 4
RIGIDITY MODULUS D PLOTTED AGAINST h3
INT. REPT. 117
BR 6312
Figure 5a
Photograph of one of the ice
plates after failure
DBR Internal Report No.117
Figure 5b Section of same
ice plate photographed using
polarized light
BR 6310
Figure 5c Another section of
ice plate photographed
using polarized light
BR 6311
DBR Internal Report No.117