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Visco-elastic buckling analysis of floating ice sheets
CoM Regions Science and Technology, 11 (1985) 2 4 1 - 2 4 6
241
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
VISCO-ELASTIC BUCKLING ANALYSIS OF FLOATING ICE SHEETS Stig-G6ran Sj61ind Department of Mechanical Engineering, University of Oulu, Linnanmaa 90570, Oulu 57 (Finland)
(Received November 5, 1984; accepted in revised form May 1,1985)
ABSTRACT Buckling o f ice sheets has been analysed by several authors. In all o f these analyses it was assumed that the ice sheet and the foundation were elastic, lce, however, is a viscoelastic material and the viscous properties will have some effect on the buckling phenomenon in many cases. In this paper it is shown that there is a limit wave length above which buckling cannot occur at a certain load level. It is also shown that the viscoelastic buckling phenomenon is selective with respect to critical buckling modes and the actual critical buckling time can be obtained if the assumed initial imperfection contains components also from the critical mode. Calculated results "show that the wave length in viscoelastic buckling is always shorter than wave lengths in elastic buckling. The loads at which buckling can occur can be very small if there is enough time for the buckling process.
INTRODUCTION When a floating ice sheet impinges on a vertical, rigid structure the ice sheet can fail either by crushing or by buckling. Many theoretical and experimental investigations have been performed for creating methods for estimation of forces which moving ice sheets exert on rigid structures. Most of the results obtained are for cases where crushing of the ice sheet is the dominating mode of failure, i.e., for cases where the aspect ratio (ratio of structures width to ice thickness) is small and the velocity of the ice sheet is relatively high. Observations in nature and 0165-232X/85/$03.30
© 1985 Elsevier Science Publishers B.V.
results from model testing in basins, however, have shown that buckling of ice sheets is possible even when the thickness of the ice sheet is very large (Kato and Sodhi, 1983; Kovacs and Sodhi, 1979; Sodhi et al., 1983). Theoretical calculations for the description of the buckling phenomenon have given satisfactory results only when the aspect ratio is relatively large and the velocities of the ice sheet are high. Theoretical calculations for the determination of buckling loads for sheets have mostly been based upon assumptions of elastic behaviour of the ice sheet and the supporting foundation (Kerr, 1978; Nevel, 1979; Sodhi and Nevel, 1980). Ice, however, is a nonlinear viscoelastic material for which the behaviour during stationary viscous deformation has been described using Norton's creep law or Odqvist's generalization of Norton's creep law in several dimensional stress conditions. The elastic model can be considered satisfactory only in cases when the velocity of the ice sheet is so high that there is no time for any stress relaxations to take place. In such cases the buckling problem is a dynamic one, so the dynamical behaviour of the ice sheet and the foundation (added mass of foundation, travelling surface waves etc.) has to be considered (Sodhi, 1983). For lower velocities, when there is time for some stress relaxation to take place, the elastic material model gives too high buckling loads and the observed buckling phenomenon of thick ice sheets against relatively narrow structures cannot be explained.
BUCKLING OF ICE BEAMS As the most trivial case we shall first consider an ice sheet loaded uniaxially with a constant, dis-
242
~2~b
B/~B
p
/~--x
(/ I 0
/,h- . . . . . .
/
_L,~,"
(4)
A limit for i~creasing q(x,t) can be obtained from the condition {l(X,t) = 0 i.e. ~2ff N~x 2 +Bc~v=O (5)
H/
Fig. 1. Beam on elastic foundation, tributed pressure load p as shown in Fig. 1. The loaded edges are simply supported and the unloaded edges are free to move vertically. This case can be assumed to describe the situation in the middle of an ice sheet moving against a wide, vertical structure. We assume also that the foundation can be described accurately enough with Winkler's foundation model. The elastic buckling load for the structure in Fig. 1 is (El Naschie, 1974) E1 c L2 Ncr = Pcr B = n27r2--~ + B n2 rr2
(1)
where n is the number of buckling half-waves in the critical buckling mode, E1 is the elastic bending stiffness of the beam and c is the foundation coefficient. The solution Ncr can be obtained by solving the homogeneous, linearized transverse equilibrium equation ~2M ~2w bx 2 + N ~ x 2 + B c w = O
(2)
where M is the bending moment in the beam and w(x) is the transverse deflection. At the load Ncr the eleastic bifurcation points of the beam are reached and the deflection amplitude becomes indefinite. If the material in the beam is viscoelastic the variables in eqn. (2) are time dependent functions and the corresponding quasistatic equilibrium equation for constant N is 0x 2 - - - N ~
~ 2 +Bc~b >0 {l(X,t) = - - N Ox
-Bcff
(3)
This homogeneous equation can be solved by separating variables and by taking account of the boundary conditions w(O,t) = w(L,t) = 0. The solution is w(x,t) = A(t) sin (/3x)
(6)
where j3 = (B c/N) v2 and A(t) is a time dependent amplitude. This solution describes the physical situation when all bending stresses have relaxed and the beam is free from bending moments. The length Lv of the longest buckling half-wave which has any possibility to increase in amplitude with time is Lv = rr
v?
(7)
Half-waves longer than Lv are stable at the load level N and their amplitudes cannot increase. Amplitudes of half-waves shorter than Lv can however increase with time and initial imperfections with these wavelengths are starting to grow. The elastic foundation has a destabilizing effect on the bifurcation points of the beam (El Naschie, 1974; Sj61ind, 1984a) and the bifurcation points for beams on elastic foundation are almost always unstable. The locus of limit points on the equilibrium curves of the beam can therefore often be reached after a certain time tcr (Sj61ind, 1984a) at which elastic snap-buckling can occur. The critical time tcr can be calculated starting either from eqn. ( 3 ) b y taking account of the viscoelastic material properties, initial and boundary conditions or from the corresponding three dimensional, quasistatic nonlinear initial-boundary value problem using the finite element method as described in Sj6lind (1984a,
b). where time differentiated quantities are denoted with dots. The expression on the right hand side of eqn. (3) can be considered a time dependent transverse load gl(X,t) which can increase with time only if
APPLICATIONS Figures 2 and 3 show results obtained from calculations of a beam with L = 5 n Lb, Lb = (EI/cB) 'A ,
243
•cr
selected as a c o m b i n a t i o n o f the ten lowest buckling modes o f the beam
5
10
120
Wo(X) = ~
Ai sin (izrx/L)
(8)
i=1 80
t.o
0 m
Fig. 2. Critical buckling times tcr versus mode number m for a viscoelastic beam on elastic foundation N = 0.9 Ner.
where all IAi(0+)I = 0.001 m and the sign (-+) was selected randomly. Figure 4 shows a typical initial deflection Wo(X) o b t a i n e d from eqn. (8). The initial deflection in Fig. 4 was used in calculating the critical time when N = 0.9 Ncr. The analysis was performed with 30 isoparametric three dimensional 16-node elements using a time step At = 0.35 s. Figure 5 shows the d e f o r m a t i o n w(x) just before t = tcr. The buckling mode m = 5 is as one can see the dominating mode at t = tcr.
~¢r
5
w.
120C
crn 1. i
800
,,
i
I
J
t+O0
'Z,
5
6
7
8
9
1"0
m
Fig. 3. Critical buckling times tcr versus mode number m for a viscoelastic beam on elastic foundation N = 0.5 Ncr.
. . . . . . . . . . . . . . . .
-3. 0
i ~ x
0.5
1.0
L
Fig. 4. Initial deflection w o of the beam at time t = 0 ÷. H --- 0.2 m, B = 1.0 m, N1 = 0.5 Ncr, N = 0.9 Ncr and with initial imperfections Ao = 0.001 m for t = 0 +, i.e., after application o f the load. In each solved case buckling is o n l y possible in one mode. The material properties are E = 7.0 GPa, v = 0.35 and the creep law el/" = 6.842 X 10 -4 (Oe) 2 Si], where o e is V o n Mises's equivalent stress and Si! is the stress deviator. The vertical lines in Figs. 2 and 3 show the value o f m at which the buckling waves become stable and tcr ~ o~. In the analysed cases the shortest buckling time was o b t a i n e d for half-waves with m ~ 1.4 mcr. The same buckling p r o b l e m has also been solved b y the finite element m e t h o d using the program which has been described b y Sj61ind (1984a, b). In these cases the initial imperfections Wo(X)were
w crn 30.
15.
-15. "30. E . . . . . . . .
o
o.s
....
Fig. 5. Deflection w(x) at time t ~ tcr, N = 0.9 Ncr.
___, ×
~.o
244 w
--
Cffl
Or2
30.
(rMr) + 2 0(
=---
15.
0w
r 00
•
0
Fig. 6. Deflection
*
A
i
J
i
J
a
0.5 w(x)
1•0
L
at time t ~ tcr, N = 0.5 Ncr.
Figure 6 shows corresponding results for N = 0.5 Ncr which has been calculated with At = 16 s. The results in Figs. 4 - 6 show that the buckling process for a viscoelastic beam on an elastic foundation is selective with respect to the critical buckling mode. The real buckling time tcr can be obtained if the assumed (or measured) initial deflection contains components also from the final critical buckling mode. The buckling mode in Fig. 6 contains components from both m = 6 and m = 7. Buckling mode m = 6 is somewhat more dominating than the results in Fig. 3 would suggest. This is probably due to the inability of the finite element model to describe the higher buckling mode with so few elements.
BUCKLING OF ICE SHEETS
When the structure is relatively narrow the ice sheet has to be analysed as a viscoelastic plate on an elastic foundation. Of some interest in connection with floating ice sheets is the buckling force an ice sheet can exert on a cylindrical structure• The elastic buckling of ice sheets against cylindrical structures has been analysed by Sodhi and Nevel (1980) and Wang (1978). The transverse equilibrium equation for a plate on elastic foundation in polar co-ordinates (r, 0) can be written as
002
Or
0w) (9)
Or +NrO " ~
0(
-15.
+
r N r --
Or
Owl
Ow No - - + - ~ 00 NrO Or r 00
O.
-30.
\ ~rO0
+rcw
The right hand side of eqn. (9) can be considered as an equivalent transverse load q(r,O), which is time dependent when the material in the plate is viscoelastic. This transverse load can increase only when ?l(r,O,t) > O. A limit for increasing q(x,O) can be calculated by setting O(x,O,t) = 0. If stress redistribution in the plate is neglected we obtain
_
0 (rNr Ofv 0~b\+ 0 [ O~v No O~v ) Or Or +Nro " ~ ) - ~ (Nro Or + -r - ~ _
_
_
(lo)
-rc~v=O
The solution to this equation gives the wave lengths of the buckling waves, which become stable at the load level, Nr, No. We shall now consider the most trivial case when the membrane stress has components only in the radial direction. We assume that 2Ph
Nr=---
cos0
(11)
fir
No =Nro = 0
(12) These equations describe the distribution of membrane stresses in an elastic half-plane with a total load P applied at r = R, where R is the radius of the cylindrical structure, and can be considered to approximately describe the radial membrane stress distribution in a moving ice sheet in front of a cylindrical structure also in the viscoelastic case (Wang and Ralston, 1983). Substitution into eqn. (10) gives
a 2Cv Or2
lr c r fv +
2Ph cos 0
-0
(13)
where h is the thickness of the ice sheet. The solution to this equation with the boundary condition w(R,O) = 0 is
245 W
w(r,O) = A(t) [cos (fir) - cot (fi0R) sin (fir)]
cm
where fl = (rrcr/(2Ph cos 0)) ½ and flo = (rrcR/2Ph cos 0) '/2. The shortest stable half-wave length Lv = zr/fl is in this case dependent on the distance r and Lv decreases with r. The half-wave length in elastic buckling has been calculated only numerically b y Wang (1978). Figure 7 shows a structure which has been analysed using the finite element program described by Sj61ind (1984a). The ice sheet is moving with a velocity which is constant v = 0.375 cm/s in the negative y-direction at the outer edge o f the sheet. The ice sheet is fixed to the cylindrical structure and its boundary y = 0 is free and x = 0 is free to move in the z-direction. The dimensions o f the ice sheet and the structure are L = 33 m , H = 0.2 m and D =
10¸
5
10
15
Fig. 9. Maximum deflection of the ice sheet as a function of time. F
MN 20
15
S
10 C~
m
H
W
.xq--z l. Fig. 7. Ice sheet moving against a cylindrical structure.
2~..._
I ~ L - - - - - - - - " ~ 2b
~)
m~ Y
30
m
W.
1"o
Fig. 8. Initial deflection w o and final deflection w of the ice sheet.
5
10
15
cm
Fig. 10. Total force exerted on the cylindrical structure as a function of maximum deflection w. 4 m. The material properties o f the ice sheet are E = 7.0 GPa, v = 0.3 and the foundation modulus is c = 10,000 N/m 3. The initial deflection Wo and the final deflection w just before the ice sheet looses its stability are shown in Fig. 8. The largest amplitude as a function o f time is shown in Fig. 9 and the total force F exerted on the structure as a function o f the maximum deflection is shown in Fig. 10.
CONCLUSIONS
The results obtained show that the buckling behaviour o f an ice sheet can be substantially affected b y the viscous behaviour o f the ice. The viscous behaviour is especially important when the velocity
246
o f the ice sheet is so low that there is e n o u g h time for stress relaxations to take place. In such cases the actual buckling load can be calculated only by m e t h o d s which can take account o f the viscous properties o f the ice. It has also been shown that the viscoelastic buckling process is selective w i t h respect to the critical m o d e shape and that it is therefore not very sensitive to the selection o f initial d e f o r m a t i o n shape.
REFERENCES E1 Naschie, M.S. (1974). Exact asymptotic solution for the initial post-buckling of a strut on linear elastic foundation. ZAMM, 54: 677-683. Kato, K. and Sodhi, D.S. (1983). Ice Action on Pairs of Cylindrical and Conical Structures. CRREL Report 83-25, 35 p. Kerr, A.D. (1978). On the Determination of Horizont:d Forces a Floating Ice Plate Exerts on a Structure. CRREL Report 78-15, 9 p. Kovacs, A. and Sodhi, D.S. (1979). Ice pile-up and ride-up on arctic and subarctic beaches. Proc. Fifth International Conference on Port and Ocean Engineering under Arctic Conditions (POAC 79), Trondheim, Norway. Vol. 1, pp. 127-146.
Nevel, D.E. (1979). Bending and buckling of wedge on an elastic foundation. IUTAM Symp. on the Physics and Mechanics of Ice, Technical University of Denmark, Copenhagen, 6 - 1 0 August, 1979. Sj61ind, S.-G. (1984a). Analys av stabilitet hos olinjffra viskoelastiska balkar och plattor p~ elastiskt underlag (Stability analysis of nonlinear viscoelastic beams and plates on elastic foundations), (in Swedish). Licentiate Th., University of Oulu, Finland, 87 p. Sj/51ind, S.-G. (1984b). Viscoelastic buckling of beams and plates on elastic foundation. Proc. IAHR Ice Symp. 1984, Hamburg, Vol 1, pp. 63-72. Sodhi, D.S. (1983). Dynamic buckling analysis of floating ice sheets. Proc. of the Seventh International Conference on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland. Vol. 2, pp. 822-833. Sodhi, D.S., Haynes, F.D., Kato, K. and Hirayama, K. (1983). Experimental determination of the buckling loads of floating ice sheets, Ann. Glaciol., 4: 260-265. Sodhi, D.S. and Nevel, D.E. (1980). A Review of Buckling Analyses of Ice Sheets, CRREL Special Report 80-26, pp. 131-146. Wang, Y.S. (1978). Buckling of a half ice sheet against a cylinder. J. Eng. Mech. Div., EM5: 1131-1145. Wang, Y.S. and Ralston, T.D. (1983). Elastic plastic stress and strain distributions in an ice sheet moving against a circular structure. Proc. Seventh International Conference on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland, Vol. 2, pp. 940-951.
241
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
VISCO-ELASTIC BUCKLING ANALYSIS OF FLOATING ICE SHEETS Stig-G6ran Sj61ind Department of Mechanical Engineering, University of Oulu, Linnanmaa 90570, Oulu 57 (Finland)
(Received November 5, 1984; accepted in revised form May 1,1985)
ABSTRACT Buckling o f ice sheets has been analysed by several authors. In all o f these analyses it was assumed that the ice sheet and the foundation were elastic, lce, however, is a viscoelastic material and the viscous properties will have some effect on the buckling phenomenon in many cases. In this paper it is shown that there is a limit wave length above which buckling cannot occur at a certain load level. It is also shown that the viscoelastic buckling phenomenon is selective with respect to critical buckling modes and the actual critical buckling time can be obtained if the assumed initial imperfection contains components also from the critical mode. Calculated results "show that the wave length in viscoelastic buckling is always shorter than wave lengths in elastic buckling. The loads at which buckling can occur can be very small if there is enough time for the buckling process.
INTRODUCTION When a floating ice sheet impinges on a vertical, rigid structure the ice sheet can fail either by crushing or by buckling. Many theoretical and experimental investigations have been performed for creating methods for estimation of forces which moving ice sheets exert on rigid structures. Most of the results obtained are for cases where crushing of the ice sheet is the dominating mode of failure, i.e., for cases where the aspect ratio (ratio of structures width to ice thickness) is small and the velocity of the ice sheet is relatively high. Observations in nature and 0165-232X/85/$03.30
© 1985 Elsevier Science Publishers B.V.
results from model testing in basins, however, have shown that buckling of ice sheets is possible even when the thickness of the ice sheet is very large (Kato and Sodhi, 1983; Kovacs and Sodhi, 1979; Sodhi et al., 1983). Theoretical calculations for the description of the buckling phenomenon have given satisfactory results only when the aspect ratio is relatively large and the velocities of the ice sheet are high. Theoretical calculations for the determination of buckling loads for sheets have mostly been based upon assumptions of elastic behaviour of the ice sheet and the supporting foundation (Kerr, 1978; Nevel, 1979; Sodhi and Nevel, 1980). Ice, however, is a nonlinear viscoelastic material for which the behaviour during stationary viscous deformation has been described using Norton's creep law or Odqvist's generalization of Norton's creep law in several dimensional stress conditions. The elastic model can be considered satisfactory only in cases when the velocity of the ice sheet is so high that there is no time for any stress relaxations to take place. In such cases the buckling problem is a dynamic one, so the dynamical behaviour of the ice sheet and the foundation (added mass of foundation, travelling surface waves etc.) has to be considered (Sodhi, 1983). For lower velocities, when there is time for some stress relaxation to take place, the elastic material model gives too high buckling loads and the observed buckling phenomenon of thick ice sheets against relatively narrow structures cannot be explained.
BUCKLING OF ICE BEAMS As the most trivial case we shall first consider an ice sheet loaded uniaxially with a constant, dis-
242
~2~b
B/~B
p
/~--x
(/ I 0
/,h- . . . . . .
/
_L,~,"
(4)
A limit for i~creasing q(x,t) can be obtained from the condition {l(X,t) = 0 i.e. ~2ff N~x 2 +Bc~v=O (5)
H/
Fig. 1. Beam on elastic foundation, tributed pressure load p as shown in Fig. 1. The loaded edges are simply supported and the unloaded edges are free to move vertically. This case can be assumed to describe the situation in the middle of an ice sheet moving against a wide, vertical structure. We assume also that the foundation can be described accurately enough with Winkler's foundation model. The elastic buckling load for the structure in Fig. 1 is (El Naschie, 1974) E1 c L2 Ncr = Pcr B = n27r2--~ + B n2 rr2
(1)
where n is the number of buckling half-waves in the critical buckling mode, E1 is the elastic bending stiffness of the beam and c is the foundation coefficient. The solution Ncr can be obtained by solving the homogeneous, linearized transverse equilibrium equation ~2M ~2w bx 2 + N ~ x 2 + B c w = O
(2)
where M is the bending moment in the beam and w(x) is the transverse deflection. At the load Ncr the eleastic bifurcation points of the beam are reached and the deflection amplitude becomes indefinite. If the material in the beam is viscoelastic the variables in eqn. (2) are time dependent functions and the corresponding quasistatic equilibrium equation for constant N is 0x 2 - - - N ~
~ 2 +Bc~b >0 {l(X,t) = - - N Ox
-Bcff
(3)
This homogeneous equation can be solved by separating variables and by taking account of the boundary conditions w(O,t) = w(L,t) = 0. The solution is w(x,t) = A(t) sin (/3x)
(6)
where j3 = (B c/N) v2 and A(t) is a time dependent amplitude. This solution describes the physical situation when all bending stresses have relaxed and the beam is free from bending moments. The length Lv of the longest buckling half-wave which has any possibility to increase in amplitude with time is Lv = rr
v?
(7)
Half-waves longer than Lv are stable at the load level N and their amplitudes cannot increase. Amplitudes of half-waves shorter than Lv can however increase with time and initial imperfections with these wavelengths are starting to grow. The elastic foundation has a destabilizing effect on the bifurcation points of the beam (El Naschie, 1974; Sj61ind, 1984a) and the bifurcation points for beams on elastic foundation are almost always unstable. The locus of limit points on the equilibrium curves of the beam can therefore often be reached after a certain time tcr (Sj61ind, 1984a) at which elastic snap-buckling can occur. The critical time tcr can be calculated starting either from eqn. ( 3 ) b y taking account of the viscoelastic material properties, initial and boundary conditions or from the corresponding three dimensional, quasistatic nonlinear initial-boundary value problem using the finite element method as described in Sj6lind (1984a,
b). where time differentiated quantities are denoted with dots. The expression on the right hand side of eqn. (3) can be considered a time dependent transverse load gl(X,t) which can increase with time only if
APPLICATIONS Figures 2 and 3 show results obtained from calculations of a beam with L = 5 n Lb, Lb = (EI/cB) 'A ,
243
•cr
selected as a c o m b i n a t i o n o f the ten lowest buckling modes o f the beam
5
10
120
Wo(X) = ~
Ai sin (izrx/L)
(8)
i=1 80
t.o
0 m
Fig. 2. Critical buckling times tcr versus mode number m for a viscoelastic beam on elastic foundation N = 0.9 Ner.
where all IAi(0+)I = 0.001 m and the sign (-+) was selected randomly. Figure 4 shows a typical initial deflection Wo(X) o b t a i n e d from eqn. (8). The initial deflection in Fig. 4 was used in calculating the critical time when N = 0.9 Ncr. The analysis was performed with 30 isoparametric three dimensional 16-node elements using a time step At = 0.35 s. Figure 5 shows the d e f o r m a t i o n w(x) just before t = tcr. The buckling mode m = 5 is as one can see the dominating mode at t = tcr.
~¢r
5
w.
120C
crn 1. i
800
,,
i
I
J
t+O0
'Z,
5
6
7
8
9
1"0
m
Fig. 3. Critical buckling times tcr versus mode number m for a viscoelastic beam on elastic foundation N = 0.5 Ncr.
. . . . . . . . . . . . . . . .
-3. 0
i ~ x
0.5
1.0
L
Fig. 4. Initial deflection w o of the beam at time t = 0 ÷. H --- 0.2 m, B = 1.0 m, N1 = 0.5 Ncr, N = 0.9 Ncr and with initial imperfections Ao = 0.001 m for t = 0 +, i.e., after application o f the load. In each solved case buckling is o n l y possible in one mode. The material properties are E = 7.0 GPa, v = 0.35 and the creep law el/" = 6.842 X 10 -4 (Oe) 2 Si], where o e is V o n Mises's equivalent stress and Si! is the stress deviator. The vertical lines in Figs. 2 and 3 show the value o f m at which the buckling waves become stable and tcr ~ o~. In the analysed cases the shortest buckling time was o b t a i n e d for half-waves with m ~ 1.4 mcr. The same buckling p r o b l e m has also been solved b y the finite element m e t h o d using the program which has been described b y Sj61ind (1984a, b). In these cases the initial imperfections Wo(X)were
w crn 30.
15.
-15. "30. E . . . . . . . .
o
o.s
....
Fig. 5. Deflection w(x) at time t ~ tcr, N = 0.9 Ncr.
___, ×
~.o
244 w
--
Cffl
Or2
30.
(rMr) + 2 0(
=---
15.
0w
r 00
•
0
Fig. 6. Deflection
*
A
i
J
i
J
a
0.5 w(x)
1•0
L
at time t ~ tcr, N = 0.5 Ncr.
Figure 6 shows corresponding results for N = 0.5 Ncr which has been calculated with At = 16 s. The results in Figs. 4 - 6 show that the buckling process for a viscoelastic beam on an elastic foundation is selective with respect to the critical buckling mode. The real buckling time tcr can be obtained if the assumed (or measured) initial deflection contains components also from the final critical buckling mode. The buckling mode in Fig. 6 contains components from both m = 6 and m = 7. Buckling mode m = 6 is somewhat more dominating than the results in Fig. 3 would suggest. This is probably due to the inability of the finite element model to describe the higher buckling mode with so few elements.
BUCKLING OF ICE SHEETS
When the structure is relatively narrow the ice sheet has to be analysed as a viscoelastic plate on an elastic foundation. Of some interest in connection with floating ice sheets is the buckling force an ice sheet can exert on a cylindrical structure• The elastic buckling of ice sheets against cylindrical structures has been analysed by Sodhi and Nevel (1980) and Wang (1978). The transverse equilibrium equation for a plate on elastic foundation in polar co-ordinates (r, 0) can be written as
002
Or
0w) (9)
Or +NrO " ~
0(
-15.
+
r N r --
Or
Owl
Ow No - - + - ~ 00 NrO Or r 00
O.
-30.
\ ~rO0
+rcw
The right hand side of eqn. (9) can be considered as an equivalent transverse load q(r,O), which is time dependent when the material in the plate is viscoelastic. This transverse load can increase only when ?l(r,O,t) > O. A limit for increasing q(x,O) can be calculated by setting O(x,O,t) = 0. If stress redistribution in the plate is neglected we obtain
_
0 (rNr Ofv 0~b\+ 0 [ O~v No O~v ) Or Or +Nro " ~ ) - ~ (Nro Or + -r - ~ _
_
_
(lo)
-rc~v=O
The solution to this equation gives the wave lengths of the buckling waves, which become stable at the load level, Nr, No. We shall now consider the most trivial case when the membrane stress has components only in the radial direction. We assume that 2Ph
Nr=---
cos0
(11)
fir
No =Nro = 0
(12) These equations describe the distribution of membrane stresses in an elastic half-plane with a total load P applied at r = R, where R is the radius of the cylindrical structure, and can be considered to approximately describe the radial membrane stress distribution in a moving ice sheet in front of a cylindrical structure also in the viscoelastic case (Wang and Ralston, 1983). Substitution into eqn. (10) gives
a 2Cv Or2
lr c r fv +
2Ph cos 0
-0
(13)
where h is the thickness of the ice sheet. The solution to this equation with the boundary condition w(R,O) = 0 is
245 W
w(r,O) = A(t) [cos (fir) - cot (fi0R) sin (fir)]
cm
where fl = (rrcr/(2Ph cos 0)) ½ and flo = (rrcR/2Ph cos 0) '/2. The shortest stable half-wave length Lv = zr/fl is in this case dependent on the distance r and Lv decreases with r. The half-wave length in elastic buckling has been calculated only numerically b y Wang (1978). Figure 7 shows a structure which has been analysed using the finite element program described by Sj61ind (1984a). The ice sheet is moving with a velocity which is constant v = 0.375 cm/s in the negative y-direction at the outer edge o f the sheet. The ice sheet is fixed to the cylindrical structure and its boundary y = 0 is free and x = 0 is free to move in the z-direction. The dimensions o f the ice sheet and the structure are L = 33 m , H = 0.2 m and D =
10¸
5
10
15
Fig. 9. Maximum deflection of the ice sheet as a function of time. F
MN 20
15
S
10 C~
m
H
W
.xq--z l. Fig. 7. Ice sheet moving against a cylindrical structure.
2~..._
I ~ L - - - - - - - - " ~ 2b
~)
m~ Y
30
m
W.
1"o
Fig. 8. Initial deflection w o and final deflection w of the ice sheet.
5
10
15
cm
Fig. 10. Total force exerted on the cylindrical structure as a function of maximum deflection w. 4 m. The material properties o f the ice sheet are E = 7.0 GPa, v = 0.3 and the foundation modulus is c = 10,000 N/m 3. The initial deflection Wo and the final deflection w just before the ice sheet looses its stability are shown in Fig. 8. The largest amplitude as a function o f time is shown in Fig. 9 and the total force F exerted on the structure as a function o f the maximum deflection is shown in Fig. 10.
CONCLUSIONS
The results obtained show that the buckling behaviour o f an ice sheet can be substantially affected b y the viscous behaviour o f the ice. The viscous behaviour is especially important when the velocity
246
o f the ice sheet is so low that there is e n o u g h time for stress relaxations to take place. In such cases the actual buckling load can be calculated only by m e t h o d s which can take account o f the viscous properties o f the ice. It has also been shown that the viscoelastic buckling process is selective w i t h respect to the critical m o d e shape and that it is therefore not very sensitive to the selection o f initial d e f o r m a t i o n shape.
REFERENCES E1 Naschie, M.S. (1974). Exact asymptotic solution for the initial post-buckling of a strut on linear elastic foundation. ZAMM, 54: 677-683. Kato, K. and Sodhi, D.S. (1983). Ice Action on Pairs of Cylindrical and Conical Structures. CRREL Report 83-25, 35 p. Kerr, A.D. (1978). On the Determination of Horizont:d Forces a Floating Ice Plate Exerts on a Structure. CRREL Report 78-15, 9 p. Kovacs, A. and Sodhi, D.S. (1979). Ice pile-up and ride-up on arctic and subarctic beaches. Proc. Fifth International Conference on Port and Ocean Engineering under Arctic Conditions (POAC 79), Trondheim, Norway. Vol. 1, pp. 127-146.
Nevel, D.E. (1979). Bending and buckling of wedge on an elastic foundation. IUTAM Symp. on the Physics and Mechanics of Ice, Technical University of Denmark, Copenhagen, 6 - 1 0 August, 1979. Sj61ind, S.-G. (1984a). Analys av stabilitet hos olinjffra viskoelastiska balkar och plattor p~ elastiskt underlag (Stability analysis of nonlinear viscoelastic beams and plates on elastic foundations), (in Swedish). Licentiate Th., University of Oulu, Finland, 87 p. Sj/51ind, S.-G. (1984b). Viscoelastic buckling of beams and plates on elastic foundation. Proc. IAHR Ice Symp. 1984, Hamburg, Vol 1, pp. 63-72. Sodhi, D.S. (1983). Dynamic buckling analysis of floating ice sheets. Proc. of the Seventh International Conference on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland. Vol. 2, pp. 822-833. Sodhi, D.S., Haynes, F.D., Kato, K. and Hirayama, K. (1983). Experimental determination of the buckling loads of floating ice sheets, Ann. Glaciol., 4: 260-265. Sodhi, D.S. and Nevel, D.E. (1980). A Review of Buckling Analyses of Ice Sheets, CRREL Special Report 80-26, pp. 131-146. Wang, Y.S. (1978). Buckling of a half ice sheet against a cylinder. J. Eng. Mech. Div., EM5: 1131-1145. Wang, Y.S. and Ralston, T.D. (1983). Elastic plastic stress and strain distributions in an ice sheet moving against a circular structure. Proc. Seventh International Conference on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland, Vol. 2, pp. 940-951.