3-4. Vertical ice forces under flooding conditions
As Figure 4 demonstrates, when the water level rises and the flooding conditions are present, the deflect ions of the ice sheet in flooded areas (r｡蚌0, W1) and in non-flooded areas (r > r0, W2) are described by:
Figure 4 A conceptual figure to compute vertical ice forces when the water level is rising
When the water level is falling and flooding is present, general solutions for the deflection of the floating ice sheet in the flooded areas (r≦r0, W1) can be obtained from the following equation (for r > r0, W2， use equation (9)):
W1(r) = A1 + A2log(r) + A3r2log(r) + A4r2 (10)
(1)When ice is adfrozen to a structure The following equations give the boundary conditions when an ice sheet is adfrozen to the structure, where a and r0 are the radius of the structure and the distance to the boundary between the flooded and non-flooded areas, respectively:
W1(a) = Δ (11)
(when r = a, the deflection is equal to the change in the water level)
W′1(a) =0 (12)
(when r = a, the inclination is equal to zero)
W1(r0) = nh (13)
(when r = r0， the deflection is constant)
W2(r0) = nh (14)
(when r = r0， the deflection is constant)
W′1(r0) = W′2(r) (15)
(when r = r0， the inclination is constant)
(2)When ice is not adfrozen to a structure When an ice sheet is not adfrozen to a structure, the following equation and equations (12)-(17) describe the boundary conditions:
(when r = a, the bending moment is equal to zero)
The seven unknown variables, A1 - A4′B3′B4′and r0， can be solved by the above equations for the boundary conditions. However, because r0 is a variable of Kelvin's function, it seems rational to take a limit by assigning an approximate value of r0． In this method, if seven unknown quantities are known, the shear force (Q) and the vertical ice force (P) under flooded conditions can be computed from the following equations:
