Ice Actions on Sloping-Sided Structures

Sveinung Løset1,2,3

1Professor, Norwegian University of Science and Technology, Trondheim2Adjunct professor, University Centre in Svalbard, Longyearbyen

3Professor honoris causa, St. Petersburg State Polytechnical University, St. Petersburg

Failure modes

ICE ACTIONS

Interaction geometryIce features Ice properties Design philosophy

Crystallography

Rubble

Temperature

Salinity

Porosity

Surface tension

Limit stress

Ridge

Rafted

Level

Limit momentum

Limit force

Single

Multileg

Out- of-plane shape

Water depth

Waterline shape

Creep

Bending

Buckling

Splitting

Crushing

SpallingDimensions

Concentration

Velocity

Strength

Adhesion

Compressive

Flexure

Tensile

Shearing

Material

Roughness

Iceberg

Friction

Splitting

Sloping structuresSloping structures

FaceFace ofof structurestructure: plane, : plane, conecone or or facetfacet -- slopeslope angle angle αα

The slope changes the failure mode The slope changes the failure mode --> the ice loads are > the ice loads are less than on vertical ones (less than on vertical ones (σσf f < σσcc)

Influence of ice strength, ice thickness, slope and friction Influence of ice strength, ice thickness, slope and friction on the ice loadon the ice load

The slope affects the characteristic breaking frequencies The slope affects the characteristic breaking frequencies and thus reduces potential resonance problemsand thus reduces potential resonance problems

TheThe advantageadvantage ofof slopingsloping structuresstructures maymay be be reducedreduced by:by:–– rubblerubble accumulationaccumulation at at thethe structurestructure–– highhigh velocityvelocity ofof thethe advancingadvancing iceice sheetsheet

Downward breaking cone

Windmill foundation, Denmark

Upward breaking cone

Confederation Bridge, Canada

Different stages

Upward/downward breaking

Same treatment? (Weight Buoyancy)Effects on vertical load and overturning moment

Load prediction models

Ice loads induced by horizontal and vertical componentsIce loads induced by horizontal and vertical components

Limited by: Limited by: –– bending strength, shear stress capacity and thickness of icebending strength, shear stress capacity and thickness of ice–– friction and sloping of the structurefriction and sloping of the structure

Models:Models:–– Croasdale (1980), 2D beam theoryCroasdale (1980), 2D beam theory–– Ralston (1977), 3D plate theoryRalston (1977), 3D plate theory–– FEM simulations (FEM simulations (MMäääättttäänennen et al.)et al.)–– + several other models, see Chao (1992)+ several other models, see Chao (1992)

Forces on structure

α

μNcos α

μNsin αμN

Ncos α

Nsin α

sin cos

cos sinx

y

H F N N

V F N N

α μ α

α μ α

= = +

= = −∑∑

x

y

N

Simple 2D theoryCroasdale (1980)Croasdale (1980)

sin coscos sin

sin coscos sin

H N NV N N

H V V

α μ αα μ α

α μ α ξα μ α

= += −

⎛ ⎞+= = ⋅⎜ ⎟−⎝ ⎠

2D beam on elastic foundation

Note: Only valid for wide structures

Beam

II = second moment of area= second moment of areab b = beam width, = beam width, hh = ice thickness= ice thickness

2

/ 2 32

/ 2

,

:12

u u o o

zA

h

zh

M yIM Mh hI I

I I y dA

bhRectangular cross section I y bdy

σ

σ σ

−

=

= = −

= =

= =

∫

∫

y

x

hu

ho

σo

σu

Simple 2D theorySimple 2D theorycontinuecontinue

Vertical load Vertical load VV limited by the bending strength of icelimited by the bending strength of iceIce sheet assumed as a beam on elastic foundationIce sheet assumed as a beam on elastic foundationStrength limited by the bending moment as:Strength limited by the bending moment as:

,max 3 2

6/ 2/12f

M Mhbh bh

σ = =

Simple 2D theorycontinue

The maximum bending moment capacity for a semiThe maximum bending moment capacity for a semi--infinite beam oninfinite beam onelastic foundation (elastic foundation (HetenyiHetenyi, 1946):, 1946):

where where 1/1/ββ is characteristic length defined byis characteristic length defined by

sin( / 4)exp( / 4)

VM πβ π

=

1/ 4

4KEI

β ⎛ ⎞= ⎜ ⎟⎝ ⎠

3

'( /12)

w

w

K gb foundation constantdensity of water

g acceleration due to gravityE Young s modulusI second moment of area of the cross section bh

ρρ====

=

Simple 2D theorycontinue

By combining the previous equations, the limits of theBy combining the previous equations, the limits of thevertical and horizontal loads read:vertical and horizontal loads read:

1/ 45

1/ 45

0.68

hence

sin cos0.68cos sin

wf

wf

ghV WE

ghH WE

ρσ

ρ α μ ασα μ α

⎛ ⎞= ⎜ ⎟

⎝ ⎠

⎛ ⎞ += ⎜ ⎟ −⎝ ⎠

wherewhereWW = = bb is beam width (breath along the water line on the sloping face)is beam width (breath along the water line on the sloping face)

Simple 2D theorycontinue

Force needed to push ice blocks up the slope:Force needed to push ice blocks up the slope:

i

(sin cos )sin

wheredensity of iceheigth reached by the ice on the slope

iZP hW g

Z

ρ α μ αα

ρ

= +

==

( ) ( )1/ 4 25

1/5

sin cos( sin ) coscos sin

substituting for and

sin cos sin cossin cos0.68cos sin cos sin tan

simplified

wf i

wf

H V P P

V P

ghH W W Zh gE

ghH WE

α μ αα αα μ α

α μ α α μ αρ α μ ασ ρα μ α α μ α α

ρσ

⎛ ⎞+= + +⎜ ⎟−⎝ ⎠

⎛ ⎞+ +⎛ ⎞ ⎛ ⎞+= ⋅ + ⋅ +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠

4

1 2iC W Zh g Cρ⋅ + ⋅ ⋅

Simple 2D theorycontinuecontinue

Constants C1 and C2 predicted vs α and μ

Simple 2D theorycontinue

1/ 45

1 2/ wf i

ghH W C Zh g CE

ρσ ρ⎛ ⎞

= ⋅ + ⋅⎜ ⎟⎝ ⎠

Breaking force Ride-up force

(1)

Example 1: Effects of thickness

Eq. (1), conical structureParameters:

Ride-up force dominates for h > 1 mForceAverage failure pressure almost independent of h

for h = 1-3 m,

45 , 0.30.7 , 5

1 / 144 /3 / 462 /

f MPa Z m

h m H W kN mh m H W kN m

α μσ= == =

= → == → =

h∝p

0.15p MPa≈

Example 2: Effects of αα and and μμ

Eq. (1), conical structureParameters:

Friction effects significant for slopes steeper than 45Friction effects significant for slopes steeper than 45˚̊Steeper angles Steeper angles →→ more crushing more crushing →→ higher loadshigher loadsImportant to maintain smooth surfaces for sloping Important to maintain smooth surfaces for sloping structures to minimize ice loadsstructures to minimize ice loads

0.7 , 5 , 1

45 , 0.1, 0.5 : / 95 / , 235 /55 , 0.1, 0.5 : / 125 / , 430 /

f MPa E GPa h m

H W kN m kN mH W kN m kN m

σ

α μ

α μ

= = =

= = =

= = =

Effects of ice strength

Affects the breaking partAffects the breaking part

Wide structures:Wide structures:–– rideride--up part > breaking part (2D situation)up part > breaking part (2D situation)

Narrow structures:Narrow structures:–– 3D effects, ice strength important3D effects, ice strength important

Effects of ice thickness

Breaking:Breaking:

RideRide--up:up:

Ice thickness is the most important parameter for load Ice thickness is the most important parameter for load estimation on all sloping structures estimation on all sloping structures

1.25breaking ( )F h

ride up ( )F h

Effects of velocity – upward cone

1 0.5 /1 0.5( 0.5) 0.5 /

if V m sV if V m s

η<⎧

= ⎨ + − >⎩

Influence of velocity only if V > 0.5 m/s (F0.5 is

the load at 0.5 m/s)

0.5/VF Fη =

2D vs 3D model

Wide structures:Wide structures:–– 2D assumption valid2D assumption valid–– simple 2D beam on simple 2D beam on

elastic foundation may elastic foundation may be assumedbe assumed

Narrow structures:Narrow structures:–– 3D effects will dominate3D effects will dominate–– failure zone wider than failure zone wider than

structurestructure–– plate theory more valid plate theory more valid

than beam theorythan beam theory

3D modelRalston (1977)Ralston (1977)

3D plate model based on plastic limit analysis (ice as a3D plate model based on plastic limit analysis (ice as aductile material)ductile material)

2 2 24 1 2 3

2 21 2

( )

( )f w w T

w T

H A A h A ghD A gh D D

V B H B gh D D

σ ρ ρ

ρ

⎡ ⎤= + + −⎣ ⎦= + −

DT - top diameter

D – waterline diameter

A, B coefficients

Adfreeze on sloping structuresCroasdale (1980)

Fadfr. - horizontal ice load due to adfreezing (MN)h - ice thickness (m)q - adfreeze bond strength (0.3-1 MPa)W - width of struture (m)

adfreeze tanhqWF πα

=

Kulluk Full-scale - Kulluk

Beaufort Sea

Resistance of a ship

Friction, Buoyancy

Breaking

Inertia ForceHydrodynamic ForceOpen Water Resistance

Res

ista

nce

Speed

IceIce barriersbarriers –– CaspianCaspian SeaSea

IceIce barriersbarriers –– CaspianCaspian SeaSea