Figure 3 Vertical ice forces acting on an inverted conical structure
As shown above, the actions of vertical ice forces can be differentiated according to the form of a structure and the presence or absence of ice adfrozen to a structure.
Vertical ice forces act differently when the water level is rising and when it is falling. When it is rising, the occurrence of flooding also need to be considered.
The methods to compute ice forces under different conditions are presented in Table1.
Table1 Forms of structures and actions of vertical ice forces
3. Methods to Compute Vertical Ice Forces
3-1. Deflection of a floating ice sheet
The equation to compute the deflection of a floating ice sheet is:
D･∇4W + Kw･W = 0 (1)
where,
W is the deflection in distance r, Kw is the unit weight of water, and D is the flexual rigidity of an ice sheet, given by the following equation:
where,
E is Young's modulus of an ice sheet, h is the thickness of an ice sheet, and v is Poisson's ratio.
3-2. Vertical ice forces with ice adfrozen to a structure
Vertical ice forces are produced by changes in water level when an ice sheet is adfrozen to a circular cross-sectional structure (Figure 1). When the change is at a low speed,the structure is not subject to significant ice forces since an ice sheet exhibits viscoelastic properties. When the change is at a high speed, however, the structure is subject to considerable ice forces since an ice sheet displays elastic characteristics. Kerr1） formulated the following equation to compute the vertical ice force generated by a change in water level from the theory of elasticity of flat plates:
where,
P is the vertical ice force exerted on a structure,
k1 = kei(λa)ker'(λa) - kei'(λa)ker(λa),
λ = (Kw/D)1/4
Kw is the unit weight of water, a is the radius of the structure, and Δ is the change in the water level.
However, as the change in water level increases, either bending failure of an ice sheet is induced by the bending moment of the ice, or adfreeze failure is caused by shear forces acting on the area of contact between the ice sheet and the structure. Thus, both bending and adfreeze failures should be incorporated into the computation of vertical ice forces generated by changes in water level.
3-3. Vertical ice forces when ice is not adfrozen to a structure
Saeki et al. 4) proposed the following equation to compute the vertical ice force when an ice sheet is not adfrozen to a structure:
