Some observations on simply supported circular ice plates loaded at the centre

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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

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, ＱＹＵｾ

N. B. Hutcheon7

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 profi-table 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 27TD

a

r (1)

where

w is the deflection at radius r

P is the total load applied at the centre D is the rigidity modulus

ｾｩｳ 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 ｳｨｯｾｶｮ that this equation is

2

-correct for

Q

ZO.l and the maximum deflection is less a

than ｾ 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･ｾ 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Ｌ 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

f7r42"*"r802)ln

ｾｬ

3+'\I(a2_r2)+1 1-q- 1'02(a2-r2

)l

(2) 2111)

11

l ' 8 1+c:r 16

ＱＫｾ

a 2

J

for 1'0> 0 and w

=...l....

[81 3+ q- a 2 _ r02 In

ｾ

_ 1'0 2 7+3\OJ

J

2TTD ＱＫｾ 8 r O 32 1+ ｾ for l '

=

O.

In most of the load tests that were made r

O = 0.5inches

l '

=

5

inches

a

=

10 inches

Assuming

u-=

0.3 and considering the terms in equation (2) 1'2 In §;

=

_4.33

'4

l ' r02 In §;

=

0.02 0 - r 1 _{3+\J (a2_r2)}

₌

_23.8 8 1+\r 1

_{- - - -}

Ｑ｟ｾｲＰＲ _{(a2_r2)}

₌

_0.01 16 ＱＫｾ a 2

3

-Considering the terms in equation (3)

1 8 3+'1 _{a 2 = 31 7} • 1+'1

In

= 0.09 r 2

--!L

7+3 V-= 0.05 32 1 + 'f

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 = p

U

_r2

In

ｾ

-

1

ＳＫｾ

(a2 - r2)] 27TD r 8 l+q-ｲｾ 0 W

o

-

- '2D

p

t

1₈ 3+\l_ｬＫｾ a 2J

r

=

0

Wo

ｾ

:;:

_1.

ＳＫｾ

=

k 21fD P a0:; 8 1+<::J 0 ｾ Substituting in equation (4) Therefore, ) D

=

---...,..--..,r----2 a

r

In

r

a2-r2

s rrt

_{k-kO a2} For r

=

5

inches (6)

4

-From equation

(5)

r

= _ •

50 k O D + 3 1 + .50 kO D

Because of the sign of w, kO and k5 are negative numbers. From the definition of D

D

=

Eh3

12 (

I-V2 )

where E is Young's modulus

(8)

(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 ad-justed 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

S

-plotted against the applied load and the values of k

O and 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

O and k

s

and the calculated values for D and DkO' are given in Table 1. All tests were made at -9°C.

Figure 2 shows a typical set of load deflection curves. _{In all cases kO and kS were constant over the} 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ｾ it is seen that

where C is a constant. (10) . Therefore

wo

p

= _{k O} = 1 21TCh3 3+l+T<:Ta

2J .

(11) In Figure 3 the logarithm of the average value for kO 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

TABLE I

Date h rO kO k₅ TI kOTI

inches inches inches/pound inches/pound pound-inches

22/11/ 0.55 0.5 2.12 x 10- 4 1.30 x 10- 4 2.38 x 10 4 5.04

24/111

0.55 0.5 2.14 1.28 2.15 4.61

13/12/

PLATE

BROKE

DUHING

TEST

Average 2.13 1.29 2.27 4.83 23/11/ 0.66 0.5 1.19 x 10-4 0.74 x 10- 4 4.60 x 10 4

5.48

24/11/

0.56 0.5 1.22 0.74 -.:. R'7 _4.68 ＯＧｾｊ

13/12/

0.56 0.5 1.26 0.80 4.60 5.80

9/1/

0.66 0.5 1.23 0.76 4.06 5.00 11/1/ 0.66 0.5 1.30 0.83 4.93 6.40 11/1/ 0.66 0.5 1.32 0.81 3.84 5.06

11!11

0.66 0.5 1.23 0.77 4.31 5.30 m

11/11

0.66 0.5 1.26 0.77 4.06 5.12 Average 1.25 0.78 4.28 5.35

4/11

0.44 0.5 5.16 x 10- 4 3.09 x 10- 4 0.88 x 10 4 4.56 9/1/ 0.4·4 0.5 5.17 3.05 0.83 4.30

11/1/

0.44 0.75 5.13 3.17 1.01 5.20 11/1/ 0.44 0.25 5.28 3.24 0.96 5.06 11/1/ 0.44 1.0 5.56 3.13 1.11 5.56 11/1/ 0.44 0.5 5.18 3.21 1.01 5.26 Average 5.25 3.15 0.97 4.99

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･ for D calculated for each plate is plotted against h). From this Figure, C is found to be equal to

1.47

x

104

Therefore,

D ::;: 1.47

x

_105h

3 .

From Fig. 3 it is found that

(12 )

k

::;:-o

0.3Lt

_hx₃

104

(13 )

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

ｾＺＺ［Ｚ 0.33.

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 106

ｕｌｔｉｔｨｾｔｅ 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

3P

[

]

Sr ::;: St ::;: - 2 h 2 ｬＫＨｬＫｾＩ In ｾｏ Letting T= 0.33 a

=

10

inches r _O ::;:

0.5

inches S_r

=

S_{t -}- 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.

(14)

8

-Table 2 gives the results of a number of tests on plates which had undergone previous loading tests. The approxi-mate 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 No. of load Pat ｔ･ｭｾ･ｲ｡ｴｵｲ･ sr

=

ｓｾ

h inches cycles failure Ibs. C. Ib/in

0.46 8 56 -10 640 0.62 20 118 -10 746 0.75 76 124 -10 532 0.65 13 90.1 -10 515 0.68 20 135.3 -10 704 0.81 34 236.7 -10 860 0.54 31 78.8 -10 648 0.44 37 50.5 -10 626

Average ultimate strength 660 Minimum ultimate strength 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

contri-bution 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 depen-dence 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 re-designing 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.

BR

6313

Figure 1 Photograph of Apparatus

160 140 120 60 80 100 DEFLECTION

x

104 (INCHES) 40 20 00 80 I I I I I " I I I - I 20 I " ｾ I I I I I I I 100I i i , i , i i i i i i i i i i •

ar

60 I I I / ' 1/ I 'c Iｾ I I I .J

-

a, c

«

o

...J 40 I I /1 ｾ I I I I I

FIGURE 2

TYPICAL LOAD-DEFLECTION CURVE FOR

r

=0

AND r=5 INCHES

10·0 9-0 8-0 7·0 4'0 _ 2-5 m ...J ... 3-0 Z

--2

1-0 -9 C Z ·8

« .

7

-2

-6 -5 ·3 -IS \ \

\

\

,

•

SLOPE=-3

ｾ

ｾ

':\

ｾ

'.

...

\

ｾ X

1\

-,

1\

\

X

,

LEGEND _!

_1\

-.

•

Ko

X

K5

-4 -5 -6 -7 '8 -9 1-0 h (INCHES)

FIGURE 3

LOGARITHM OF AVERAGE

Ko

a

K5

PLOTTED AGAINST THE LOGARITHM

OF

·5

·4

.,

/

/

/

,/r

/0

V

i

o

o

2 6 )C.

o

(/)4

3

;:)

o

o

ｾ ｾ 3

-o

(,!)

a::

V '0

-N Z ...

ｾＵ

-FIGURE 4

RIGIDITY MODULUS D PLOTTED AGAINST

h

3

BR 6312 Figure 5a Photograph of one of the ice

plates after failure

Figure 5b Section of same ice plate photo-graphed using polarized light

BR 6310

Figure 5c Another section of ice plate photographed using polarized light

BR 6311 DBR Internal Report No.117