Cold Regions Science and Technology, 6 (1982) 37-47 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

37

D E T E R M I N A T I O N OF T H E F L E X U R A L S T R E N G T H A N D E L A S T I C M O D U L U S OF ICE F R O M IN S I T U C A N T I L E V E R - B E A M T E S T S

Jean-Claude Tatinclaux and Ken-lchi Hi rayama* u.s. Army Cold Regions Research and Engineering Laboratory, Ice Engineering Research Branch, Hanover, NH 03755 (U.S.A.)

(Received October 23, 1981 ; accepted in revised form December 18, 1981 )

ABSTRACT

From the theory o f cantilever beams on an elastic foundation, it is shown that the strength index and modulus index o f ice can be determined from measurements o f either the failure load or the tip deflection, or both, o f in situ cantilever beams tested over a wide enough range o f ratio o f beam length to beam thickness. Four methods are proposed, two o f which do not require the measurement o f beam deflection during beam loading, an often difficult task to perform with sufficient reliability, especially in the fieM. The methods have been applied to avail- able field and laboratory data. The initial results show reasonably consistent estimates o f the strength index but a large variation in the predicted values o f the modulus index. One preferred method is suggested but its validity and reliability need to be further evaluated by analyzing a sufficient number o f fieM and laboratory tests.

INTRODUCTION

All common methods of determining the bending strength and bulk modulus of elasticity for ice are based on the elastic theory of homogeneous, iso- tropic material. Ice is neither isotropic nor homoge- neous and behaves as an elastic material only when the rate of deformation is sufficiently high. Conse- quently, it has become customary to refer to the

*On leave from Iwate University, Iwate, Japan.

strength index Ob and modulus index E of ice. It is also well known that different testing methods yield different estimates of Ob and, especially, E. Possibly the most common testing method, both in the field and in the laboratory, is the in situ cantilever-beam test, because of its relative simplicity. From the measurement of the applied load P and deflection 6 of the beam tip, estimates of Ob and E are calculated by applying equations derived from the simple elastic cantilever beam, namely

6PfL Ob - bh 2 (1)

where Pf is the load at failure, L is the beam length, h is its thickness, and b is its width. Also,

E = 7 8 (2)

The above equations neglect the effect of buoyancy exerted by the underlying water on the deforming beams. To minimize the effect of buoyancy, the International Association of Hydraulic Research Committee on Ice Problems recommends that, for sea ice, the length to thickness ratio, L/h, of the beams to be tested should be no greater than 10 (Schwarz, 1981). The analytical basis for this recom- mendation has been presented by Schwarz and Kloppenburg (1976) and by Frederking and H~'usler (1978). Other recommendations of the IAHR Working Group on Testing Methods in Ice are that L[h be no less than 7, to minimize shear effects near

0165-232X/82/0000-0000/$02.75 © 1982 Elsevier Scientific Publishing Company

38

the root of the beam, and that the failure load be reached after one to two seconds of load application to ensure that the strain rate is high enough for the ice to deform elastically.

The determination of E requires measurement of the tip deflection with increasing load P. It is also advisable to measure the beam deflection at several locations along the beam length and to determine the value of E such that the measured deflection curve best fits the theoretical equation (Frederking and H~iusler, 1978). Beam deflection measurements are difficult to perform in the laboratory, because of the small deflections to be measured and the high accuracy required, and even more so in the field because of the often adverse environmental condi- tions. The present paper shows that the limitation L/h < 10 for in situ cantilever-beam tests can be re- moved. Measurements of the failure load Pf, for a series of beams of aspect ratio L/h over a range from, say, 5 - 7 to well above 10, possibly 20-25, can be used for estimating both the strength index and the bulk modulus index.

A N A L Y S I S

The problem of cantilever beams on an elastic foundation has been treated by Schwarz and Kloppenburg (1976) and is depicted schematically in Fig. 1. The deflection line w(x) and the bending moment distribution M(x) are given, respectively, by

P _/sinhX(L - x ) ' c o s Lv "cosh LL -

M~ cosh 2 LL + cos 2 LL wOO - 2 . . . . . .

sin X(L -x) -cosh kx "cos XL

cosh 2 XL + cos 2 XL ] (20

and

eL (cosh X(L x .cosh XL + M(x) = - ~ \ cosh 2 M_, + cos: XL

cos h(1 -x ) - s inh Lr .cos LL ] -

cosh 2 M~ + cos z kL ) (4)

I b

P Deflection w(x)~ x !

zli, ., *4

L

o. Side View

b, Plon View

Fig. 1. Definition sketch.

X ~ j

39

where P = load applied at tip of beam b = beam width L = beam length 7w = subgrade reaction (specific weight of

water)

= ~ 33'w ~ '/4

\Eh 3 ]

h = ice thickness E = bulk modulus index

Equations (3) and (4) were derived for a perfectly elastic, homogeneous, isotropic material, assuming zero deflection and rotation at the root of the beam. They are valid as long as the beam tip is not sub- merged in the water (P applied downward) or as long as it does not fully emerge from the water (P applied upward), i.e. for

6 = w(O)< (1 - P i lh Pw ]

(downward P)

t5 = w(0) ~< P__~i h (upward P) Pw

where Pi is the density of ice and Pw is that of water. Furthermore, the expression for ), given above assumes the beam neutral axis to be located at mid- thickness.

Schwarz and Kloppenburg have shown that for values of EL up to rr/2 the maximum moment occurs at the root of the beam. The strength index Ob can then be obtained as

-Mfh ---6 Mf o b - - ( 5 )

2I bh 2

where Mf is the moment at the beam root (x = L) at failure and I = bh3/12 is the moment of inertia of the beam cross-section. Therefore,

6PfL sin EL-cosh EL + sinh hL-cos XL (6)

Ob - bh z XL(cosh 2 EL + cos 2 EL)

valid for EL ~< 7r/2. Since the quantity Osb = 6PfL/(bh 2) represents

the strength index from simple beam theory (eqn. (1)) for a failure load Pf, eqn. (6) can be written as

Osb LL (cosh 2 EL + cos 2 EL) - - = ( 7 ) Ob sin EL-cosh EL + sinh EL-cos EL

The displacement fi of the beam tip can be expressed by

4P 1 L \3 3(sinh EL.cosh EL -s in EL.cos EL)

= ~ I ~ - / / 2(EL)3 (cosh~ EL + cos~ EL) (8)

Here the quantity

4P ( L / 3

can be interpreted as the tip deflection that a simple beam of modulus index E would experience under load P. Equation (8) can then be rewritten as

~sb 2(EL) 3 ( c°sh2 EL + c °s2 EL) - ( 9 )

6 3 (sinh EL-cosh LL - s in EL-cos EL)

It was found that eqns. (7) and (9) can be approx- imated with a maximum error of less than half a percent over the whole range of 0 < EL < 7r/2 by using the following expressions:

Osb/O b = 1 + 0.3533 (EL)4 + 0.0210 (XL) 8 (10)

6sb/6 = 1 + 0.3140 (EL)4 - 0.0025 (EL)8 (11)

In fact the further approximations, linear in (EL)4

Osb/Ob = 1 + 0.367 (EL)4 (12)

~isbfli = 1 + 0.314 (EL)4 (13)

are excellent approximations of Osb/oh over the range 0 < ~L < 1, and of 8sb/6 over the whole range of 0 < EL < ¢r/2, respectively. These latter two expres- sions, eqns. (12) and (13), can indeed be obtained from polynomial expansions of the hyperbolic and circular functions of EL in eqns. (7) and (9), keeping the first two terms in the final expressions.

The exact equations (7) and (9), and their approxi- mations, eqns. (10) and (12), and eqns. (11) and (13), respectively, have been plotted versus (EL)4 .'and EL on Figs. 2a and 2b. As was pointed out by Frederking and H/~usler (1978), it can be seen that Osb/O b remains practically equal to 1 for values of EL up to 0.5-0.6. This observation was the basis for recom- mending that cantilever-beam tests be conducted for values of L/h less than 10, on the basis that typical values of E, in the field, are of the order of 1 -2 GPa, and that ice thickness is of the order of 0.5-0.8 m. It would be more accurate to recommend that L/h

40

17sb/0- b

~sb/~

XL 0.2 0.6 0.65

I I I I ' I ' 0.4 0.5

I ' I

( • ) (Tsb/O-b E xoct (Eq. 7) (O)~sb/~ Exoct (Eq. 9)

(o)

1.05

I I I0 0.1 0.2

(XL)4

Fig. 2a. Plot of exact and approximate equations for Osb/O b and ~ sb/a (0 < LL < 0.65).

O-~b/O" b

,~L 0 6 0.8 1,0 1.2 1.4 "/T/2

I I I I J ~ t I t . , ~

( o ) Usb E x o c t (Eq. 7 )

,,b . / i

Eq. EO

f J J I I , if 2 4 6

( hL)4

Fig. 2b. Plot of exact and approximate equations for asb/ab and a sb/a (0 < LL < 7r/2).

available. From the definition of X, eqns. (10) and (12) can be written as

1.06 0.189 ) Osb = Üb 1 + - - r + - - r 2 (14)

E E 2

and

1.1 ) asb = O b 1 + - - r (15)

E

where r = "Ywh(L[h) 4. Equation (13) can be rewritten in the form

4Pf E = ")'wh ( L ) 4 L - - ~ ( ~ ) - 0.942] (16)

where 6f is the tip deflection at failure. From the definition of Osb, eqn. (16) takes the form

be less than 0.4(E/Twh) 1/4 with E being estimated from the type of ice to be tested.

Rather than limiting the aspect ratio L[h of the beams to be tested, the variation of the ratio Osb/ab and ¢5sb/5 with LL can be used to determine both ab and E from in situ cantilever-beam tests when either Pf or ~f are unavailable, or to arrive at several estimates of E and Ob when such data are

Osb E = - - - 0 . 9 4 2 r (17)

A

where A is defined by

1.5 -/-

and r is as defined earlier.

41

Elimination of Osb between eqn. (14) and eqn. (17) yields

ab 1 + 1.06(r/E) + 0.189(r/E) 2 A = - - • (18)

E 1 + 0.942 (r/E)

An excellent approximation to eqn. 18 over the range 0 <<. r/E ~< 2 is given by

A = - - + 0.1473 - - + 0.0136 (19) E E

Comparison of eqns. (14) and (19) shows that the quantity A, which includes the tip deflection 6f, is much less sensitive to variations of r than Osb ' i.e. to variations in the beam aspect ratio L/h.

Once a series of in situ cantilever-beam tests over a wide range of Lib, from 5 to 15-20, has been per- formed, the quantities Ob and E can be calculated according to one or more of the following proce- dures, depending on the available data.

Method 1 The strength index Ob is calculated as the average

of the values of ash obtained from the beams with L/h less than, say, 10. For each of the remaining beams, the value of E which satisfies eqn. (14) is computed. The corresponding average value is calcu- lated. With this average value of E, the maximum value of L[h which satisfies the condition LL < 0.6 is determined, and the procedure repeated if neces- sary.

Method 2 Values of ab and E can be obtained from a linear

regression analysis of the data Osb and r according to eqn. (15), and may be deemed satisfactory if the cor- responding maximum value of LL for all the beams tested is less than 1, or used as initial values in a non- linear analysis of the data according to eqn. (14). It should be noted that, when the beams all have the same thickness, the value of X, the inverse of the char- acteristic length of the parent ice sheet, can be determined from non-linear regression analysis of trsb versus L according to eqn. (10).

Method 3 When both the failure load Pf and tip deflection

~f at failure have been measured, and the corre- sponding parameters 8sb and A are available, then in

addition to estimating Ob and E by either Method 1 or Method 2 above, other estimates of E can be ob- tained from application of eqn. (17).

Method 4 If only the tip deflections are available, the

modulus index E and the ratio ablE can theoretically be determined from a non-linear regression analysis of the pairs of data (A,r) according to eqn. (19). However, it can be immediately realized that because of the very slow increase of A with r as predicted by eqn. (19) and the difficulty in measuring 6f with great accuracy and reliability, it is likely that only a fair estimate of able can be obtained as the average of the measured values of A, but that the estimate of E predicted by this method may be seriously in error.

EXAMPLES OF APPLICATION

1. Field data

The field data presented by Frederking and H/iusler (1978) are listed in Table 1. Frederking (1981) provided the tip deflections t~f listed in Table 1. Eleven beams were tested, seven with L/h less than 10 and only four with L/h ranging from 17.8-29.5. In the application of Methods 1 and 2, the tip deflection was assumed to be unknown.

Method 1 The average value of Osb for the seven beams with

L/h < 10 was

ab = 0.33 MPa

With this value for trb, the corresponding values of E for each of the four beams with L/h > 10 were calcu- lated from eqn. (14) and are listed in Table 1. They are seen to vary from 0.52 GPa- l . 42 GPa with an average value of 0.95 GPa. The values of X/_, corre- sponding to E = 0.95 GPa have been calculated and are also listed in Table 1. For beam 16, the calculated value of X/, is greater than rr/2, which is in agreement with the observation by Frederking and H~usler that this particular beam had broken not only at the root but also at a distance of 4.75 m from the root. When the data for beam 16 are discarded, the average value E is 0.88 GPa. The corresponding values of XL, listed in Table 1, are seen to never exceed rr/2.

42

TABLE 1

Field data (from Frederking and H/iusler, 1978): (hL) I calculated withE = 0.95 GPa, (hL) 2 calculated with E = 0.88 GPa (beam 16 discarded)

Beam b L h L/h Osb r Results from Method 1 Method 3 (m) (m) (m) (MPa) (MPa)

E (KL) 1 (;kL) 2 8f ~ E (GPa) (mm) (× 10 4) (GPa)

L/h < 10

5 0.67 3.8 0.44 8.64 0.34 24.0 0.49 0.54 5.7 2.60 1.29 9 0.72 3.95 0.45 8.78 0.36 26.2 0.51 0.55 6.9 2.98 1.18 7 0.60 3.98 0.43 9.26 0.45 31.0 0.53 0.57 5.7 2.32 1.92 6 0.64 3.96 0.42 9.43 0.33 32.6 0.54 0.58 5.6 2.25 1.36 8 0.78 4.01 0.435 9.22 0.31 30.8 0.53 0.57 6.4 2.60 1.53

12 0.61 3.57 0.38 9.38 0.27 28.9 0.52 0.56 9.8 4.40 0.59 13 0.65 3.56 0.37 9.62 0.24 31.1 0.53 0.57 5.4 3.24 0.99

L/h > 10

15 0.65 7.65 0.43 17.79 0.44 422.5 1.42 1.08 1.10 21.3 2.35 1.52 11 0.64 7.79 0.42 18.55 0.61 487.9 0.69 1.12 1.14 25.6 2.66 2.07 10 0.63 8.15 0.41 19.88 0.85 628.2 0.52 1.19 1.21 25.0 2.31 3.05 16 0.66 11.50 0.39 29.49 1 . 6 1 2893.6 1.15 1.66 NA 70.0 NA NA

Method 2 A regression analysis of all the data according to

eqn. (15) yielded

crb = 0.34 MPa

E = 0.81 GPa

while the non-linear regression analysis according to eqn. (14) yielded

crb = 0.36 MPa

E = 1.22 Gpa

Here again the value of LL for beam 16 computed with either of the two values of E above is greater than 7r/2. When the data for beam 16 are discarded, eqn. (15) yields Ob = 0.30 MPa and E = 0.48 GPa, while eqn. (14) yields ab = 0.30 MPa and E = 0.54 GPa.

Methods 3 and 4 From the measured values of trsb and 8f the corre-

sponding values of E have been calculated for all beams except beam 16 for which tSf/h = 0.18 greater

than ( 1 - Pi/Pw), and are listed in Table 1. These values are seen to vary from 0.6 G P a - 3 . 0 GPa with an average of 1.55 GPa.

When the calculated values of A for all beams except beam 16 were analyzed according to eqn. (19), the calculated value of E/Ob was found to be equal to 3300 but a negative value of E was obtained. This latter result could have been expected since the values of A for the longer beams are usually smaller than for the shorter beams, contrary to eqn. (19). This result also confirms the difficulty in extracting reliable information from the sole measurement of the tip deflection at failure.

The results of the various analyses above are sum- marized in tabular form in Table 2 and in graphical form in Figs. 3a and 3b. While the values of the strength index Ob obtained from the different methods are quite consistent, the predicted values of the modulus index E vary widely from 0.48 G P a - 1.55 GPa, and are consistently smaller than the initial tangent moduli obtained by Frederking and H/iusler from beam deflection measurements at three points along the length of the seven shorter beams (L/h < 10) which ranged from 1.45 GPa--3.17 GPa with an average of 2.0 GPa.

2. Laboratory data

The model ice used in ice-structure interaction

43

TABLE 2

Field data - summary of results

o b (MPa) E (GPa) E/ob

Method 1 All data 0.33 +- 0.07 0.95 -+ 0.41 Less beam 16 0.33 -+ 0.07 0.88 -+ 0.48

Method 2 Eqn. 14 (all data) 0.36 +_ 0.07 1.22 -+ 0.30 Eqn. 14 (less beam 16) 0.30 +- 0.06 0.54 ± 0.22 Eqn. (15) (all data) 0.34 -+ 0.07 0.81 +- 0.24 Eqn. 15 (less beam 16) 0.30 -+ 0.07 0.48 -+ 0.24

Method 3 Less beam 16

Method 4 Less beam 16

0.33 +- 0.07 1.55 -+ 0.68

3300 -+ 450

(Tsb (MPa)

' I ' I

Eq. 15 / / / ~ - (O'b= 0.:34 MPa ) / ~ / t (E = o.e, G / P o ) / ~

~ q. 14

• / . ~ . ' ~ (O'b = 0.56 Mpa ) j (E=I.22Gpo)

(o)

i I i I J I 2

T(GPo)

Fig. 3a. Results of analysis of field data by Frederking and Hgusler (1978). All data included.

O'sb (MPo)

1.0

0.8

0.6

0.4

0.2

I i I I I ' I i /

(b) . ~ / / - • /

fib = O.30 MPo - - - - - r q . 15 E = O . 4 8 G P a

Orb= 0.30MPQ ~ ~'q. 14 E = O . 5 4 G P a

0 2 0.4 0.6 0.8 'C (GPo)

1.0

Fig. 3b. Results of analysis of field data by Frederking and H~iusler (1978). Beam 16 excluded.

model tests at the Cold Regions Research and Engineering Laboratory is grown from a 0.95% aqueous solution of urea. The model-ice strength is moni tored at various time intervals during growth and warm-up by in situ cantilever-beam tests. The beams are cut with a hand-saw and the load is applied at the beam tip by a hand-held push-pull force gage. Beam deflections were not measured, precluding application o f Methods 3 and 4. After failure, the length of the beam is measured with a ruler to the nearest millimetre, and its width and thickness are measured with a caliper to the nearest tenth of a millimetre. In some cases, the modulus index of the

ice sheet is determined from plate-deflection tests by applying known loads at the center o f the ice sheet and measuring the sheet deflection at the point o f application of the load. The results of cantilever- beam tests with a significant variation in the param- eter L/h are listed in Tables 3 - 5 for three ice sheets.

Results of the analyses of the data by Method 1 are listed in Tables 3 - 5 and are summarized in Table 6, where the results given by Method 2 are also presented. The value of E obtained from the plate method for ice sheet 3 is given, and is larger than that obtained from beam tests, a common observation. The overall results are presented graphically in Figs.

44

TABLE 3

Laboratory data - ice sheet 1: (kL), calculated with E = 65.3 MPa, (kL): calculated with E = 52.6 MPa (beam 17 discarded)

Beam b L h L/h asb r Results from Method 1 (cm) (cm) (cm) (kPa) (MPa)

E (hL) 1 (hL) 2 (MPa)

L/h < 10

1 9.70 14.8 3.99 3.73 54.7 0.076 0.24 0.25 2 9.84 18.9 4.10 4.6 54.9 0.18 0.30 0.31 3 10.16 19.3 4.07 4.74 55.3 0.22 0.31 0.32 4 10.69 19.8 4.10 4.8 50.1 0.22 0.31 0.33 5 9.95 21.8 4.03 5.4 47.9 0.34 0.35 0.37 6 9.36 24.8 4.06 6.11 49.1 0.55 0.40 0.41 7 10.46 26.7 4.10 6.5 47.9 0.72 0.43 0.44 8 10.16 29.1 4.10 7.1 45.1 1.02 0.47 0.48 9 9.90 33.3 4.06 8.2 39.0 1.80 0.54 0.56

10 10.81 34.2 4.11 8.3 49.6 1.94 0.54 0.56 11 11.93 36.2 4.10 8.8 45.1 2.44 0.58 0.60 12 10.27 36.5 4.10 8.9 70.9 2.52 0.58 0.61 13 10.56 36.8 4.10 9.0 51.2 2.61 0.59 0.61

L/h > 10

14 9.48 40.6 4.00 10.1 54.4 4.16 63.6 0.66 0.68 15 11.12 44.8 4.12 10.87 55.9 5.65 61.1 0.71 0.74 16 11.24 48.5 4,13 11.74 57.5 7.71 63.7 0.77 0.80 17 10.65 48.7 4.10 11.90 53.66 8.01 154.2 0.78 0.81 18 10.06 50.2 4.10 12.20 68.47 9.04 29.2 0.80 0.83 19 11.25 60.8 4.10 14.8 70.93 19.44 55.4 0.97 1.01 20 11.61 61.1 4.10 14.9 71.91 19.83 54.0 0.98 1.01 21 11.84 63.9 4.10 15.6 85.05 23.72 41.2 1.02 1.06

4 - 6 . As expected, the values of Ob obtained by the various methods are very close to one another and, since the values of kL are seldom greater than one, the values of the modulus index predicted by either eqn. (14) or eqn. (15) are practically identical. It can also be noted that in the application of Method 1, when no data discrimination is attempted, the average value of E is significantly greater than that predicted by Method 2. However, when those data for which the calculated values of E are significantly different

from the average (i.e. beam 17 of ice sheet 1, beam 16 of ice sheet 2, and beams 8 and 12 of ice sheet 3) are discarded, the average value of E from Method 1 becomes in reasonable agreement with the values predicted by Method 2. Nevertheless, such a discrimi- nation is always subjective and therefore question- able, and for that reason Method 2 is considered to be the better method to apply. Furthermore, Method 2

requires less computation and can be performed with- out prior ordering of the data with respect to L/h.

I00

~ b (kPo)

50

I ' I l

Eq. 15 • / (O'b= 493 kP°) (E = 42.1MPa) ~ ' ~

~ P a ) elE"-~ e (E= 42.9MPa) - -

I I I I I 0 I 0 20 30

"/" (MPa}

Fig. 4. Results of analysis of laboratory data - ice sheet 1.

45

TABLE 4

Laboratory data - ice sheet 2: (KL) 1 calculated with E = 111.1 MPa, (KL) 2 calculated with E = 78.8 MPa (beam 16 discarded)

Beam b L h L/h asb r Results from Method 1 (cm) (cm) (cm) (kPa) (MPa)

E (X/-')l ( ~ ) 2

L/h < 10

1 10.9 18.9 4.15 4.6 65.3 0.17 0.27 0.29 2 10.7 20.2 4.32 4.68 46.4 0.20 0.27 0.30 3 11,0 20.4 4.20 4.9 66.6 0.23 0.28 0.31 4 10.9 22.2 4.27 5.2 59.4 0.31 0.30 0.33 5 9.4 27.9 4.17 6.7 52.0 0.82 0.39 0.42 6 10.8 29.8 4.21 7.1 58.7 1.03 0.41 0.45 7 10.8 29.8 4.22 7.1 48.8 1.04 0.42 0.45 8 10.1 31.2 4.29 7.3 61.1 1.18 0.42 0.46 9 10.8 40.1 4.25 9.4 54.5 3.30 0.54 0.59

10 8.8 41.2 4.24 9.7 59.4 3.71 0.56 0.61 11 10.9 41.7 4.17 10.0 54.6 4.09 0.58 0.63

L/h > 10

12 10.8 48.6 4.20 11.6 60.0 7.38 149.0 0.67 0.73 13 10.4 49.1 4.10 11.9 64.7 8.10 64.8 0.68 0.74 14 10.9 50.5 4.22 12.0 67.8 8.49 48.9 0.69 0.76 15 10.9 53.1 4.22 12.6 72.1 10.38 43.2 0.73 0.79 16 10,7 55.3 4.23 13.1 59.4 12.12 304.9 0.76 0.83 17 11.0 57.5 4.17 13.8 65.2 14.79 111.3 0.80 0.87 18 10.5 58.8 4.22 13.9 74.7 15.61 55.9 0.80 0.88

% (kPa)

50

I001 I I I

I -" (fib = 56kPa)

(E = 71MPa)

, 1 J 0 IO 20

2-(MPa)

Fig. 5. Results o f analysis o f laboratory data - ice sheet 2.

I I I . I 14 and 15 v ~ Eq.

_ _ (O"b=BZkPo) , / __ I00 ; . ( E ~ : ~ o u P ~ i •

e l

O-~b (WPo)

5 0 - -

O IO 20 T (MPa)

Fig. 6. Results of analysis of laboratory data - ice sheet 3.

46

TABLE 5

Laboratory data - ice sheet 3: (kL) t calculated with E = 189 MPa, (kL) 2 calculated with E = 58.8 MPa (beams 8 and 12 dis- carded)

Beam b L h L[h Osb r Results from Method 1 (cm) (cm) (cm) (kPa) (MPa)

E (M_,) 2 (X.L)= (MPa)

L/h < 10

1 6.82 29.3 4.57 6.41 85.0 0.76 0.32 0.44 2 6.53 29.2 4.53 6.45 79.3 0.77 0.33 0.45 3 6.41 32.3 4.59 7.03 90.1 1.11 0.36 0.43 4 7.14 32.6 4.55 7.16 93.2 1.17 0.37 0.49 5 6.63 35.3 4.53 7.79 78.8 1.66 0.40 0.54 6 6.91 38.9 4.60 8.46 77.7 2.31 0.44 0.59 7 7.23 38.8 4.58 8.47 91.7 2.31 0.44 0.59

L/h > 10

8 7.11 46.5 4.57 10.2 86.4 4.81 338.4 0.53 0.71 9 5.65 47.0 4.58 10.3 91.5 4.99 71.4 0.53 0.71

10 6.84 49.4 4.56 10.8 82.4 6.16 - 0.56 0.75 11 6.75 53.3 4.51 11.8 101.7 8.63 48.4 0.61 0.81 12 7.44 58.7 4.56 12.8 87.1 12.28 560.1 0.66 0.88 13 5.55 62.4 4.55 13.7 115.1 15.79 50.2 0.71 0.95 14 6.83 63.0 4.59 13.7 108.2 15.98 65.2 0.71 0.95

TABLE 6

Laboratory data - summary of results

Ice sheet Data

Method 1

Ob (kPa) E (MPa)

Method 2

Eqn. (14) Eqn. (15) Eqn. (10) Eplat e

trb(kPa) E(MPa) crb(kPa) E(MPa) ob(kPa) 1/X(em) (MPa)

1 All data 51 ± 7.5 Less beam 17 51 -+ 7.5

2 All data 57 ± 6.4 Less beam 17 57 ± 6.4

3 All data 85 ± 6.6 Less beams 8 and 12 85 26.6

6 5 ± 3 8 4 9 ± 3 4 3 ± 1 3 4 9 ± 3 42± 13 4 9 ± 3 5 7 ± 4 5 3 ± 1 3 5 0 ± 3 4 4 ± 1 3 4 9 ± 3 4 2 ± 3 5 0 ± 3 5 7 ± 4

111 ±94 5 6 ± 4 7 1 ± 4 2 5 6 ± 4 7 1 ± 4 4 5 6 ± 4 6 5 ± 1 0 7 9 ± 4 2 5 6 ± 4 6 2 ± 3 4 5 6 ± 4 6 2 ± 3 5 56-+3.5 63± 10

1 8 9 ± 2 1 4 8 2 ± 5 5 9 ± 2 7 8 2 ± 5 5 9 ± 2 9 82_+5 6 6 ± 8 5 9 2 1 1 8 2 2 5 5 1 2 1 8 8 2 2 5 5 1 2 1 8 8 2 2 5 64-+6

74

SUMMARY AND CONCLUSIONS

F o u r m e t h o d s have b e e n p r o p o s e d for the es t ima-

t i on o f the s t r eng th i ndex and m o d u l u s i n d e x o f ice

f rom the resul ts o f in si tu can t i l eve r -beam tes ts per-

f o r m e d over a wide range o f the aspect ra t io o f b e a m

l eng th to b e a m th ickness f r o m a p p r o x i m a t e l y 5 - 7

to 1 5 - 2 0 . In two o f these m e t h o d s , on ly the fai lure

load needs to be measured , in add i t i on to the b e a m

d imens ions , b u t the m e a s u r e m e n t o f the b e a m deflec-

t i on is n o t requi red . The t r ade -o f f b e t w e e n a large

n u m b e r o f b e a m tes ts and n o t ip de f l ec t ion measure-

m e n t m a y be advan tageous in the field as well as in

the l abo ra to ry . The second o f these two m e t h o d s ,

47

which consists of a non-linear regression analysis of the data according to eqn. (14), is considered prefer- able since it requires less manipulation and no sub- jective discrimination of the data.

Method 4 can theoretically yield estimates of both Ob and E when only the tip deflection at failure is available; however, while it may lead to acceptable values of E/eb, values of E predicted by this method may be seriously in error, because of the lack of sensitivity of ~f with L.

It is emphasized that the values of (r b and especial- ly E obtained by any of the proposed methods are only indices, which ought to be used only for com- parison purposes between different types of ice with the condition that the same test method and data analysis method are used in all cases.

The proposed methods are currently used at the Cold Regions Research and Engineering Laboratory, in addition to other conventional methods. It is

hoped that other workers in the field of ice engineer- ing will find them worth trying and will report both field and laboratory results to confirm or refute their validity.

REFERENCES

Frederking, R. and H/iusler, F.U. (1978), The flexural behav- iour of ice from in situ cantilever-beam tests, Proe. IAHR Symp. on Ice Problems, Lule~', Sweden, Part I, pp. 197- 215.

Frederking, R. ( 1981), Personal communication. Schwarz, J. and Kloppenburg, M. (1976), Untersuchung fiber

das Widerstands-verh~iltnis zwisehen Model und Grossaus- fiJhrung eisbrechender Sehiffe-Sehlussbericht, Ham- burgische Schiffbau-Versuchsanstalt, Berieht No. E82/76.

Sehwarz, J. (1981), Standardized testing methods for measuring mechanical properties of ice, Cold Regions Sci. and Technol., 4" 245-253.

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