Numerical Estimation of Ice Forces Acting on Inclined Structures and Ships in Level Ice
Authors: Lindstrom C-A. 
Published: October 2008

ABSTRACT
This paper presents a theoretical method for calculation of dynamic loads and average ice forces on sloping structures and ships in ice of uniform thickness. The method is used for estimation of ice loads on inclined structures, resistance of ice transiting vessels and ice force distribution on ships maneuvering in level ice. The calculations are compared with results from full scale measurements.

INTRODUCTION
The interaction between a sloping structure and a floating ice sheet has been the subject of both theoretical and experimental investigations during recent years. This contact problem is of special interest because the inclination of the structure will considerably reduce the ice forces. This is due to the vertical component of the reaction, which will cause bending failure in the ice sheet at some distance from the structure. Consequently, slopedshapes are commonly used in the waterline area of offshore structures and icebreaking ships.
Based on investigations mentioned above, methods for estimation of ice loads on offshore structures and ice resistance on ships have been developed. Unfortunately, most of the theoretical models are based on the static solution of an elastic plate on an elastic foundation, although the importance of the dynamic behaviour of the ice sheet has been stated.
Varsta1 and Jeberaj et al2 have treated the dynamic interaction between a landing craft bow and an ice sheet by the finite elementmethod. The maximum ice force seems to increase with the relative velocity between the ice and the structure while the impact duration and the size of the broken ice piece decrease. When estimating the average forces acting on ships and offshore structures, all these three parameters are of equal importance.
The use of the finite element method in a general mathematical model is not very convenient. A solution to this problem is given by S?rensen 3 in his work "Interaction betweenFloating Ice Sheets and Sloping Structures", where he treats the problem analytically. The method presented in this paper is partly based on S?rensen's work mentioned above.

THEORY AND DEFINITIONS

Ice Load
In a collision between an inclined structure and an ice sheet, the ice edge breaks down into smaller particles, which flow away from the contact area. The ice pressure integrated over the contact area Ac results in a normal contact force Fn. Because the phenomena in connectionwith the contact pressure are not very well known, an average ice pressure Pav is commonly used:(Available In Full Paper)
In addition to this normal force, a frictional force is assumed to act in the same vertical plane as the relative velocity vector between the structure and the ice sheet.
The contact area in equation (1) depends on the geometry of the structure and the ice edge and on the crushing depth. The structure is usually treated as a rigid body with no deflections. Assuming that the relative motion between the ice and the structure does not change during the impact and that the horizontal deformation of the ice sheet is neglectable, the crushing depth Sc can be expressed: