1975_12_58

Salvage of Heavy Construction Equipment by a Floating Ice Bridge

H. R. KIVISILD Folrndation of Canada Engineering Corporation Limited, 805 8th Avenue, S . W., Calgary, Alberta T2P OM2

G. D. ROSE Forrndarion of Cmrida Engineering Corporation Limited, 9731-51stArte. Ste 210, Edmonton, Alberta T6E4W8

AND

D. M. MASTERSON Forrndation of Cannda Engineering Corporcltion Limited, 805 8th Avenrre, S . W . , Calgary, Albertci 72P OM2

Received May 27, 1974

Accepted July 30, 1974

During the summer of 1972, a barge load of heavy construction equipment under tow to the James Bay Project on the eastern shore of James Bay became grounded on a shoal at the mouth of the Fort George River, only a few miles from its intended destination. Federal Commerce and Navigation Limited retained Foundation of Canada Engineering Corporation Limited (FENCO) to study the feasibility of removing the heavy equipment from the barge by an overice crossing. FENCO personnel visited the site in November and December, 1972, compiled the necessary environmental data and designed a crossing consisting of ice built up by flooding. Following the completion of the 100 ft wide (30.48 m) and 74 in. thick (1.88 m) bridge by Sainte-Marie Construction, the ice bridge was instrumented and tested prior to and during the unloading process. Parameters measured were thickness, width, temperature, ice soundness, and de- flections. Tide readings were also taken.

Loads of 70 t and heavy trucks were removed with no problem. Deflections were very small and cracking was confined to the tidal zone.

Durant I'Cte 1972, une barge chargee d'equipement lourd de construction destine au projet de la Baie James sur la rive est de la Baie, s'est CchouCe sur un banc de sable B I'embouchure de la rivikre Fort George, B seulement quelques milles de sa destination. Federal Commerce and Navigation Limited a retenu les services de FENCO pour etudier la possibilite de decharger I'equipement lourd de la barge par I'intermediaire d'un pont de glace. Le personnel de FENCO a inspect6 le site en novembre et dicembre 1972, a rassemble les donnees nicessaires sur I'environnement et a projete un passage constitue de glace accumulCe par inondation. A la suite de I'achkvement d'un pont de 100 pi. de large (30.48 m) et 74 po. d'kpaisseur (1.88 m) par Sainte-Marie Construction, le pont de glace a ete instrumente et teste avant et pendant les operations de dkchargement de la barge. Les parametres mesures ont Bte I'epaisseur, la largeur, la tempkrature, la qualit6 de la glace et les diflexions. Les niveaux de maree ont 6galement ete relevks.

Des charges de 70 tonnes et des camions lourds ont ete d6charges sans problkme. Les deflexions ont Bte trks petites et la fissuration a CtC limitee ii la zone de maree.

[Traduit par la Revue]

Introduction During the first half of the month of January,

1973, an ice bridge was constructed on the east coast of James Bav from Governors Island in Riviere La Grande 'to a barge grounded on a shoal 4400 f t ( 1341 m) from the shore (see Fig. I ) . FENCO was retained by Federal Commerce to advise on the feasibility of the project and to prepare a report on the design criteria and on the construction procedures re- quired to build a bridge across which heavy construction equipment could be transported from the barge to James Bay construction sites using recommended unloading procedures.

Studies of the climatic conditions of the James Bay estuary indicated that if certain recommended construction procedures were followed, enough ice could be built in the time available to provide a bridge of adequate strength, in spite of the 7 ft (2.1 m) high tides. Design calculations using both elastic plate and beam theory, along with previously gathered field data on ice strengths, made possible the tabulation of optimum road thicknesses and widths for various load conditions.

The actual construction of the bridge OC-

curred between December 29, 1972, and Janu- ary 16, 1973. Water was applied in layers not

Can. Geotech. J . , 12.58 (1975)

KIVISILD ET AL.: FLOATING ICE BRIDGE

0 I m I

NO ice Build-up Necessary In

/ ~ i d o i F la t Zone

DEBARCADERE

J A M E S B A Y

J I I

-,- - . ... - I I

lOOOOft S C A L E : 9 , FIG. 1 . Location of barge and crossing.

greater than 1.5 in. (38 mm) thick by means of pumps and in the 2 weeks a minimum of 58 in. (1.5 m) of ice was built up.

After completion of the construction, ice strength tests were performed, thickness mea- surements were made and temperature profiles and salinity samples of ice and water were taken by FENCO personnel. The bridge was closely inspected in the tidal crack zone and was deemed safe for the purpose intended. Levels taken at various times during unloading on two days indicated a maximum vertical dis- placement due to tides of about 7 ft (2.1 m) . No serious damage to the bridge approaches or delays in unloading resulted.

Special Problems Tides

This area of James Bay is an estuary and is subject to 7 ft (2.1 m) tidal fluctuations at

certain times of the month. Because of this, it was feared that the ice, as it rose and fell with the tides, would develop large cracks and shear discontinuities near the shore and render ap- proach to the bridge difficult or impossible.

Where tides exist in ice covered waters, there is always a section of grounded ice at the shore adjoined by a zone of tidal cracking (Fig. 2) . As the floating ice sheet responds to the semi- diurnal tidal fluctuations, and the grounded shore ice cannot, flexural cracks develop where the two meet. These cracks are called tidal cracks and can be seen, and heard, by a patient observer, as the tide flows and ebbs. These cracks can completely penetrate the ice sheet and allow flooding of the grounded ice.

It was decided, partially on the basis of past observation and partially through deduction that, while this cracking due to tides should not be taken lightly, there should be no serious

CAN. GEOTECH. J. VOL. 12, 1975

GROUNDED ICE [ TIDAL CRACK ZONE

ICE THICKENED FROM FLOODING

FIG. 2. Ice in the shore zone.

5- DISCONTINUITY

- GROUNDED ICE , . T I D A L CRACK E

FLEXURAL I? I /MECHANISM +i

I

FIG. 3. Tide crack mechanism.

Z O N E m

impediment imposed by it on the use of the ice bridge. If necessary, fill could be placed in the affected area to facilitate travel.

It was reasoned that either a shear discon- tinuity could occur between the floating and grounded ice or that some sort of a flexural mechanism could result (Figs. 3a, b ) . Shear discontinuities as shown in Fig. 3a are highly unlikely since tidal cracking usually occurs over a zone rather than at one location. It was thus concluded that the flexural mechanism of Fig. 3b was more likely to occur and a study of the geometry of a very simplified form of this mechanism serves to illustrate the reasons be- hind the optimism regarding the effects of tidal cracking.

As a worst case, suppose that the floating

ice sheet, attached to the shore ice at low tide level, rises a distance A due to a full tidal surge and causes two tension cracks connected by a flexurally rigid block of ice of length 1 (Fig. 3c).

The crack opening is then

Dl Ah 6 = : -

1 If A = 7f t (2 .1m)

h = 7 St (2.1 m) 1 = 50 St (15.2 m), a reasonable value to

assume as indicated by experience, then the maximum crack width at the ice surface would be about 1 f t (0.3 m) , certainly noth- ing which is beyond repair. In actual fact, the flexural mechanisms are more complex than this, resulting in more hinges and narrower

KIVISILD ET AL.: FLOATING ICE BRIDGE

PROBABIL ITY O F OCCURANCE

1.01 1.1 1.2 1.4 1.7 2 2.5 3 4 5 6 8 10 15 20 30

N

ONE CHANCE I N I N ' T H A T T H E THICKNESS SHOWN WILL B E EQUALLED OR E X C E E D E D I N ANY ONE YEAR.

FIG. 4. Summary of ice growth (values are maximum possible buildup under close control)

cracks. Also, elastic bending of the floating ice has been neglected here and this could result in some stress relief in the tidal zone.

At any rate, it was decided that it should be possible to maintain access to the bridge in spite of the rather severe tides.

Ice Buildup Rates Preliminary calculations indicated that the

naturally occurring ice in the estuary would have to be artificially thickened in order that the heavy loads on the barge could be safely transported. An analysis of the local weather data and previously established ice buildup rates, (Kingery 1963, p. 283) given certain weather conditions, led to the curves shown in Fig. 4. These curves were produced by plot- ting the time-temperature product on prob- ability paper for each 5-day interval of January for each year. From them, it was concluded that most certainly by the middle of January, enough ice could be produced. As will be shown later, actual progress verified this pre- diction.

Salinity Since the area under question is an estuary,

it could possibly have been salty or at least subject to influxes of salt water. Brine pockets will develop in frozen sea water if the ice be-

comes too warm. When rapid artificial buildup is under progress, the lower layers of ice can often approach the melting point. Sea water ice in this temperature range, where brine drainage has not been possible, contains large slushy brine pockets which have no strength. Thus, in the event that the estuary was salty, careful control and inspection was necessary during and after construction to make sure that these pockets were not too abundant.

Stress Analysis and Design Stress analysis of the ice bridge was carried

out on the basis of the loading diagram shown in Fig. 5. It was assumed that a crawler tractor, of weight 1/3 of the total load, PT, was pulling a trailer with its load equally distributed be- tween the front and rear axles. Total loads of 20, 40, 60, and 80 t were considered. Maxi- mum flexural stresses were checked considering the bridge as: (a) an infinite beam on an elastic foundation with both longitudinal and transverse bending; ( b ) an infinite plate on an elastic foundation. The ice was assumed in all cases to be elastic, homogeneous, and isotropic. In the event that the bridge should crack under loading, stress calculations were carried out assuming that complete loss of flexural strength occurred behind the front axle of the float,

CAN. GEOTECH. J. VOL. 12, 1975

CRAWLER P / 3 ON FRONT LOAD P,/Q ~ A ~ X L E

P, = TOTAL LOAD

FIG. 5. Assumed ice loading configuration.

leaving a semi-infinite beam on an elastic [6] T = PZ/2 foundation loaded by 2/3 of the total load.

The formulas used in the calculation can be The maximum torsional stress, assuming

listed as: elastic behavior,

( a ) Infinite beam on an elastic foundation [71 = for w 3 2.5h subjected to concentrated loading (Den Hartog 0.968(h2)w 1952). (e) Stresses in the beam after cracking

(loss of flexural strength) (Den Hartog 1952). In this case

[2] Deflection =

6 =-e PB -ex(cos BX + sin BX) 2k

183 Moment = M = PeCpx sin PX B [3] Moment =

[9] Shear = V = ~ e - * ~ ( c o s PX - sin BX)

where P is assumed equal to 2PY7/3. The maxi- mum moment, and hence maximum flexural stress, occurs when

[lo] V = 0 = P ~ a ~ ~ ( c o s /3X - sin p X )

since P # 0 and dX -+ 0 only as X + a then

P -

M = ~ e pX(cos /3X - sin PX)

It was considered, for the purpose of calcu- lation, that the beam is subjected to two concentrated loads, one equal to 2PT/3 at x = 0 and one equal to P7./3 at 30 ft (9.1 m). The stresses and deflections from the two con- centrated loads were then superimposed to give the maximum values.

( b ) Maximum tensile bending stress under a concentrated load for an infinite plate on an elastic foundation (Timoshenko 1940).

cos j?X = sin BX

7T and /?X = -

4

[4] Max. Stress =

Substituting this into the equation for the bend- ing moments, we get (c) Transverse bending of the ice bridge

under a line load acting along the bridge center line (Hetenyi 195 8). C11-I

P M,,, = -- enI4 sin (n/4)

B P, cosh /3w - cos Pw

[5] Max. Moment = - - 48 sinh pw + sin /3w

or M,,, = 0.3224 P/B

Now (d) Eccentric loading (Timoshenko 195 1 ) . Calculations were carried out assuming that the bridge was loaded 20 ft (6.1 m) off center to try and ascertain whether torsional stresses could be critical.

Under a concentrated load P, the torque is

P [12] Thus a,,, = 1.9344 -T

Bwh

= maximum stress

KIVISILD ET AL.: FLOATING ICE BRIDGE

IOm 2 0 m 3 0 m

WIDTH OF ICE BRIDGE ( ft )

CURVES ARE FOR FRESH WATER ICE, 2 0 0 PS.1. BENDING STRENGTH AND A FACTOR OF SAFETY OF 3 . 0 CRACKED CONDITION F S 1.0

FIG. 6 . Bearing capacity of ice bridge.

Calculations of maximum tensile and tor- sional stresses were carried out on the com- puter using the above formulas and imposing a factor of safety of 3.0 against tension failure in the ice. The design curves of Fig. 6 were compiled. Transverse beam or torsional stresses were always well below the critical limit.

The two governing formulas were those de- scribing the stress in an infinite beam and an infinite plate. The curves slope downward to the right, indicating, as would be expected, that less ice thickness is required for wide beams. The curves eventually became horizontal, indi- cating that as the beam width increases in- definitely it begins to act as a plate and a further increase of flooded area is of no use. Thus, making use of both beam and plate theory, it was possible to arrive at the optimum width of structure. The actual width of the bridge was 100 ft (30.5 m) .

It must be admitted that calculations based on extreme fibre elastic stresses yield a lower- bound solution to the problem of the strength of structures such as this, especially since ulti- mate strength behavior and compressive mem- brane action result in actual load capacities considerably higher than elastic analysis pre- dicts. However, because safety for personnel and cargo was considered of prime importance, elastic analysis was considered the wisest ap- proach to the problem.

Rate of Travel and Spacing of Vehicles In the interest of safety it was recommended

that vehicle spacing as outlined by the U.S. Navy Cold Region Engineering Department in manual NAVFAC Dm 9 be followed. Accord- ing to this, 180 ft (54.9 m) clearance should be maintained between two trailing vehicles and 290 ft (88.4 m) between vehicles in a train.

In addition, vehicles travelling over the bridge were cautioned to move according to Fig. 7 (after Nevel 1970). Vehicle speeds in excess of 10 mph were discouraged and, to minimize creep effects, it was recommended that parking of heavy loads on the floating ice bridge be avoided. Comments on the necessity of these precautions will be made later.

Construction of the Ice Bridge Construction of the bridge was carried out

by Sainte-Marie Construction between Decem- ber 29, 1972 and January 16, 1973. Water was flooded in layers not greater than 1.5 in. (38 mm) by means of pumps and was then allowed to freeze before another layer was applied. Work proceeded in shifts on a round the clock basis. In the time period, an average of 58 in. (1.5 m) of ice was frozen. Two lengths of bridge were built (Fig. I ) , one 4400 ft (1 341 m) in length and the other 620 ft (189 m) long. The ice on the tidal flats was

CAN. GEOTECH. J. VOL. 12, 1975

0 I

20 40 60 80 100

ICE THICKNESS (Inches)

FIG. 7. Recommended maximum speed of vehicle travel.

grounded and needed no thickening. The port- able nature of the pumps used for the flooding made them ideal for the continuous flooding operation, as they could be moved up and down the ice bridge as required. Spare pumps were kept in a heated building so that no time need be lost from freezing and breakdown. The bridge was kept lighted by sodium vapor lights mounted on poles cut from the surround- ing bush. These lights were operated by a generator placed at one end of the bridge. On the whole the operation was very efficient.

Results of the Field Investigations Salinity Measurements

Ice from cores taken at two stations was analyzed chemically and the total salinity was determined. At each station, one sample of the natural ice and one of built-up ice were analyzed. A sample of water taken directly from one of the pumps was also tested. The results are listed in Table 1 and indicate that the water and ice were fresh and that there was no danger of forming large brine pockets within the bridge.

Ice Strength Tests Unconfined and confined compressive

strengths and elastic modulus of the ice, basic engineering parameters, were determined for the built-up ice by means of in situ plate jack- ing tests performed near the island end of the bridge. These tests have been developed for arctic use and have proved to give reliable results and are relatively easy to execute.

The average unconfined compressive strength

TABLE 1. Results of salinity tests -.

Salinity (%,)

Natural Built-up Station* ice ice Pump water

2 + 66 0.018 0.122 0.333 18 + 23 0.009 0.444 0.333

'All chainages are given in feet using standard surveying notation. Origin o f chainage, 0 + 00, is shown in Fig. 1.

of the ice was found to be 624 p.s.i. (4.3 X lo6 N/m2).

The confined compressive strength of the ice as given here is determined by loading a circu- lar area of the ice sheet to failure using a hydraulic jack, similarly as is commonly done in soil and rock tests. The strength obtained was 1920 p.s.i. (1 3.2 x lo6 N/m2). Deformations were measured during the confined compressive strength tests (Fig. 8 ) and, using theory de- veloped for the load deformation relationship during such tests in rock, the elastic modulus of the built-up ice was determined to be about 1.2 x 106 p.s.i. (8.3 x lo9 N/m2), a value comparing very well with those determined for naturally formed fresh water ice.

Visual Examinations Ice cores were taken at various stations along

the bridge using a SIPRE ice core auger. In all cases the built-up ice was sound and of good quality. From examination of the cores it was easy to determine the natural-built-up ice in- terface as the natural ice was clear while the built-up ice was cloudy because of sediments trapped during freezing. As a result, it was

KIVISILD ET AL.: FLOATING ICE BRIDGE 65

FIG. 8. Results of 5 in. (127 mm) diameter plate bearing tests.

possible to determine, since core recovery was good, the exact thicknesses of both natural and built-up ice. The results are listed in Table 2.

The ice cores showed clearly visible layers with no evidence of brine pockets and full co- hesion between layers. Individual layers were an average of 1.5 in. (38 mm) thick. The absence of brine pockets was encouraging and, since the ice was fairly warm, indicated that the estuary was fresh at the time of flooding.

Bridge Level Profiles Levels were taken along the full length of the

bridge and also across the bridge out onto the parent ice at several stations. In all cases the results indicated that the bridge was floating

TABLE 2. Results of ice coring

Natural ice Total ice thickness thickness

Chainane in. m in. mm

isostatically (Fig. 9 ) and was not supported in any way by the shore or the surrounding natural ice. Since no large cracks or discontinuities of any sort were noticed at the edges of the bridge, it was concluded that the surcharging of the parent ice had been gradual enough to allow the large plastic deformations imposed by the settling of the bridge to isostatic level.

Tide Measurements Tide measurements were recorded at inter-

vals of 0.5 to 1 h for January 21, 1973 by taking levels from a bench mark established on shore. Levels were taken at stations 1 + 21,l 2 + 66, and 4 + 05, shots being taken both on top of the ice and on top of the water through a hole in the ice at 2 + 66. The results are plotted in Fig. 10. Comparing the shots on the ice and the water at 2 + 66, we are once again reassured that the bridge was floating and responding exactly to tidal fluctuations. The bridge at station 1 + 21 seems to have been somewhat out of phase with locations further out, possibly because of interference from shore

'All chainages are given in feet using standard surveying notation. Origin of chainage, 0 + 00, is shown in Fig. 1.

66 CAN. GEOTECH. J. VOL. 12, 1975

(a ) SECTION

BUILT-UP ICE

( b ) P R O F I L E

FIG. 9. Final section and profile of floating ice bridge.

T I M E ( h )

FIG. 10. Tidal readings for January 21, 1973.

ice & this point. At any rate, it was concluded Temperature Profiles from theseAmeasureme&s that the 7 f t (2.1 m) her mist or probes were inserted at station maximum predicted tide was reasonable. 18 + 23 at various depths in the ice as shown

While these readings were being taken, a in Fig. 11. A hole was drilled in the ice at this large tidal crack opened up near station 1 + 31. station, the probes were attached to a weighted This crack was the widest one noticed, about cord, let down into the hole and then the hole 4 in. (10.2 cm) wide at the ice surface, but was filled with water. A period of 2 or 3 days did not interfere with unloading operations. was allowed to elapse so that the hole tempera-

KIVISILD ET AL.: FLOATING ICE BRIDGE

(OP O F I C E A F T E R THEORETICAL FLOODING ON J A N . 1 7 / 7 3 f (Carslaw 8 Jaeger 1959)

I I I I I I ! - 2 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

T E M P E R A T U R E ( O C )

FIG. 11. M e a s u r e d temperature p r o f i l e s at s t a t i o n 18 + 23.

ture could reach equilibrium with the surround- ing ice and then temperature readings were commenced. Readings were taken from January 17 to January 19 when the probe leads were damaged beyond repair by construction work. The results are plotted in Fig. 11.

Three sets of readings were taken on January 17, one in the morning before the last flooding and two in the afternoon after flooding was completed. The high reading for probe number two can only be explained on the basis of a pocket of still unfrozen water in that particular area. It apparently rectified itself on following days, as the profiles show.

As can be seen from the profiles, the flooding on January 17 had a profound effect on the

temperature of the ice in the foot below the top of the ice.

It is unfortunate that we did not have more probes because the temperature profile before and after flooding in the top foot would most surely have looked more like the theoretical curve. It is doubtful if the temperature dis- turbances, at least so soon after flooding, would extend down farther than a few inches. The cdd ambient then tends to bring the temper- ature curve back towards its original position. The destruction of the probe leads made it impossible to determine whether the original profile was eventually restored.

What is of importance here is the fact that the ice just below the surface layer was quite

68 CAN. GEOTECH. J. VOL. 12, 1975

FIG. 12. Ice crusher crossing.

warm and obviously, from the temperaturc pro- files taken, the rapid flooding and buildup rate was having considerable effect on the mean tempcrature of the ice mass. Newly applied laycrs of ice plus heat influx from the warm water beneath prevent the majority of the ice mass from attaining temperatures much below the freezing point. This can be critical if the water used for flooding is sea water as the ice formed therefrom has a high salinity, as much as 20 ; i r , and when warm has a very high brine content. Since this brine is detrimental to ice strengths, it is very important to keep a close watch on temperatures as flooding and freezing with sea water proceeds, especially if the result- ing structure is to be of a permanent nature.

Unloading the Barge The equipment was unloaded from the barge

between January 15 and 27, 1973. The heaviest load to cross at one time was the rock crusher pulled by a grader; the combined weight of the two being approximately 70 t (Fig. 12) . This equipment crossed the bridge when the tidal fluctuations were at the highest and no difficul- ties were experienced in the tidal crack zone. Attempts were made, while the equipment was crossing, to measure deflections using a sur- veyor's level but they were too small to be detected in this manner.

Type 769 B 35-t trucks with a tare weight of approximately 29 t were used to remove drums of fuel and other miscellaneous items from the barge. The trucks were often loaded with 30 drums of fuel (for a total weight 33.5 t ) and, after a few crossings at low speeds, were ob- served finally to be crossing the bridge at speeds up to 20 m.p.h. (32 km/h) and on occasion pass other trucks returning to the barge. These rates of travel were not observed to have any detrimental cffect on the bridge.

Conclusions Elastic platc and beam theory with a factor

of safety of 3.0 in the uncracked condition and 1.0 in thc cracked condition were used to de- sign the bridge. A safe structure 100 ft wide (30.5 m) and 74 in. thick (1.9 m) which sup- ported well the loads of up to 70 t imposed upon it was built. Principle flexural stress, as opposed to transverse beam or torsional stresses, werc found to govern the design.

Actual construction of the bridge proceeded quickly and 58 in. (1.5 m) of built-up ice was produced in a period of two weeks. The ice was built-up by the flooding and freezing in layers not greater than 1.5 in. (38 mm). Strength tests and visual examination of cores showed the ice to be of good quality with an average

KIVISILD ET AL.: FLOATING ICE BRIDGE 69

unconfined compressive strength of 624 p.s.i. (4.3 X 10W/m" and an elastic modulus of 1.2 x 10".s.i. (8.9 X loD N/m2).

The water in the estuary where the bridge was located was found to be of very low salinity and thus no brine problems were en- countered in the natural or built-up ice.

In spite of 7 ft (2.1 m) of tide, the bridge behaved well in the tidal crack zone and there was no impediment to the unloading although it took place during the time of maximum fluctua- tions at both high and low tide.

Profile and section levels showed the bridge to be completely isostatic with no discontinuities noticeable at its junction with the parent ice.

Limitations imposed on the rate of travel for the 33.5 t trucks were found to be unnecessary at the stress levels encountered as they were observed to cross the bridge at speeds of 20 m.p.h. (32 km/h) and also met other trucks while crossing.

Acknowledgments The authors are very pleased to have had the

opportunity to work on this project with Fed- eral Commerce and Navigation Limited. They also appreciate the initiative shown by Eastern Canada Towing Limited and the aid given by F. S. Minniken of the Salvage Association. Special thanks is extended to Ste. Marie Con- struction for the help and cooperation shown FENCO personnel during the inspection of the bridge and the gathering of field data.

DEN HARTOG, J. P. 1952. Advanced strength of materials. McGraw-Hill, Toronto, Ont.

HETENYI, M. 1958. Beams on elastic foundation. Univ. Mich. Press, Ann Arbor, Mich.

NEVEL, DONALD E: 1970. Moving loads on a floating ice sheet. U.S. Army, Cold Reg. Res. Eng. Lab., Hanover, New Hampshire, Res. Rep. 261.

TIMOSHENKO, S. 1940. Theory of plates and shells. McGraw-Hill, Toronto, Ont.

1951. Theory of elasticity. McGraw-Hill, Toronto, Ont.

CARSLAW, H. S., and JAEGAR, J. C. 1959. Conduction of heat in solids. Oxford Univ. Press, Oxford, Engl.

KINGERY, W. D. 1963. Ice and snow. M.I.T. Press, Cam- bridge, Mass.

Appendix - List of Symbols h = ice thickness ft (m) E = elastic modulus of ice

= lo6 p.s.i. (6.9 x lo9 N/m2) p = Poisson's ratio of ice

= 0.3 a = stress p.s.i. (N/m2) a, = ultimate tensile strength of ice

= 200 p.s.i. (1.4 x lo6 N/m2) k = subgrade modulus

= 62.4 lb/ft3 (2450 N/m3) I = moment of inertia ft4 (m4)

P , = total load lb (N) P = value of concentrated load Ib (N)

X = distance from load along beam ft (m) b = Jm - 0.675 h for c < 1.724 h c = radius of loaded area ft (m)

rr x tire pressure P, = load per unit width of beam lb/ft (N/m)

= PT/40 T = torque ft-lb (N-m) Z = eccentricity of loading ft (m) w = width of beam ft (m)

M = bending momentlunit width ft-lb (N-m) V = shear lb (N)