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Alberto Montanari

Professor of Hydraulic Works and Hydrology

Department DICAM - University of Bologna

A ship with a displacement of 3350 t, a waterline length (L_{wl}) of 117 m, a draft of 5 m, a beam of 12 m, pivot point located at 55 m from stern, and side projected wind area of 950 m^{2} is moored to a berth with 6 mooring lines displaced as in Figure 1 (red lines; the yellow lines are additional ones that are shown for the sake of illustrating further possible lines). The ship is subjected to an extreme wind velocity of 50 knots (92.6 km/h) with wind direction perpendicular to the ship center line. The ship is moored with mooring lines composed by three ropes of cross sectional area of 30 cm^{2} each. The elastic modulus of each rope is 125 kN/cm^{2} and the breaking tension is 25 kN/cm^{2}.

- Compute the wind force and yaw moment;
- Compute mooring forces and safety factors for mooring lines by assuming that the ship is moored with the 6 mooring lines that are depicted in red in Figure 1, by assuming that the origin of the axes is placed over the pivot point (PP) and assigning the following values to bollards and chocks x,y coordinates, where x is along the ship center line and y is along the transversal direction:
- B1 = (-50; -10);
- B2 = (-35; -10);
- B3 = (-5; -10);
- B4 = (12; -10);
- B5 = (42; -10);
- B6 = (57; -10);
- C1 = (-50; -4);
- C2 = (-42; -5);
- C3 = (-10; -5.5);
- C4 = (17; -5.5);
- C5 = (49; -5);
- C6 = (57; -4).

To make the computation assume that:

- the mooring lines are horizontal;
- C
_{w}=1.3; - Mooring lines are 15% longer - from bitts to bollards - then the distance between bollards and chocks.

Check that safety factors are always greater than 10.

Figure 1. Mooring lines for the ship**Box 0: Displacement of mooring lines**- The least possible number of mooring lines shall be adopted, to simplify mooring maneuvers and tensioning;
- The mooring lines and the fenders should be simmetrically distributed with respect to the center of the ship. The objective is to distribute the loads homogeneously among all the moorings or fenders and reduce the coupling between different movements. For the calculation it is assumed that the mooring remain tensioned at all times;
- The mooring lines shall be laid as horizontally as possible to reduce the force component on the line itself. The maximum vertical angle should be 25-30 degrees in the worst loading condition and water level;
- The mooring lines shall be laid out as aligned as possible with the movement they seek to restrict;
- For longitudinal mooring, and continuous berthing lines, the optimum disposition of the mooring lines shall consist of head, stern, and breast lines drawn from the ship as close to bow and stern as possible, together with springs connected to the ship at bow and stern at distances equivalent to 1/4 of the length;
- The bow and stern lines shall be laid out forming 45 ± 15 degrees with the longitudinal axis of the ship. The breast lines (B1 and B6 in figure 1) shall be perpendicular to this axis, but layouts with angles of 90° ± 30°can be admitted. The spring lines (B2 and B5 in figure 1) should form angles of 5 to 10°. Other configurations, especially those that omit the breast lines are also customary;
- Likewise, for large ships, spring line duplication can occur, located symmetrically with respect to the bow and stern;
- The mooring lines of the same function (breast lines, bow or stern lines or spring lines) should be of the same material and of equal length in order to maintain load symmetry;
- Mooring lines should preferably be long and few and made of a material (natural or synthetic fibers) with high deformation capacity, to transmit lower loads. When significant movements are to be avoided galvanized steel lines may be a more appropriate solution;
- The optimal length varies from 35 to 50 m, according to the type of ship. Lengths of less than 30 m are to be avoided;
- The number and the optimum separation between fender axes should be less than or equal to 0.15L, where L is the shortest length among the design ships, for continuous berth fronts. The fenders should be positioned along the total length of the berth front. In discontinuous berthing configurations two fenders may be sufficient, with a separation of between 0.25L and 0.50 L.

**Solution**A R code for the computation is available here.

**Computation of wind force**Let's take a coordinate system on the horizontal plane such that the x axis coincides with the ship's center line while that y axis is centered on the ship's pivot point. From eq. (1) here we get:

P

_{w}= C_{w}⋅ (A_{w}⋅ sin^{2}φ + B_{w}⋅ cos^{2}φ) × γ_{w}⋅ V_{w}^{2}/2gIn our case C

_{w}= 1.3, A_{w}= 950 m^{2}, φ = 90°, γ_{w}= 0.01225 kN/m^{3}, V_{w}= 92.6 km/h = 25.72 m/s, g = 9.81 m/s^{2}. Also, lets note that φ = 90° implies that P_{w}= P_{y}, where P_{y}is the component of wind force along the y direction. Therefore one obtains:P

_{y}= 1.3 × 950 ⋅ sin^{2}90° ⋅ 0.01225 × 25.72^{2}/(2 x 9.81) = 510.18 kNSuch wind force is applied to the center of gravity of the windage area.

**Computation of rotational moment**The rotational moment results from the wind force be applied to centre of gravity of the windage area, which is shifted with respect to the pivot point. Then, the rotational moment around the pivot point may be computed by applying the following empirical relationship (see Department of the navy naval sea systems command (1987)):

M

_{w}= C_{xyw}⋅ P_{w}⋅ L_{wl},where C

_{xyw}is an eccentricity coefficient that can be estimated from Figure 2.

Figure 2. Estimation of C_{xyw}. Redrawn from Department of the navy naval sea systems command (1987). Positive sign is associated to anti-clockwise rotation.Therefore one obtains:

M

_{w}= -0.04 ⋅ 392.38 ⋅ 117 = -2387.63 kN m (anti-clockwise)**Computation of mooring forces**The theory behind the computation is described here below. For the application please see the R code available here.

Computation of mooring forces is a complex problem as in general the ship is a rigid body with three degrees of freedom which is berthed with elastic mooring lines. We refer here to the computation method proposed by Department of the navy naval sea systems command (1987). We consider the simplified case that considers that the ship has two degrees of freedom only. Calculations are based on the following assumptions:

- The elastic modulus of the mooring lines does not vary with mooring forces.
- Mooring line arrangement is symmetrical with respect to the y axis.
- Yaw moment M
_{w}and lateral force P_{y}are applied at the ship's pivot point. - The mooring lines are assumed to run horizontally. See Department of the navy naval sea systems command (1987) if interested in relaxing this assumption.

The step-by-step procedure for calculating mooring forces is articulated as follows:

- Take x and y coordinates of bitts (end of the mooring line onboard), chocks and bollards (end of the mooring line on berth) for each mooring line.
- Determine horizontal length of each mooring line l
_{i}from bitts to bollards. - Determine total cross sectional area a
_{i}of each mooring line. This is the product of the rope cross section times the number of parts per mooring line (normally three). - Determine elastic modulus E
_{i}for each mooring line from producer. - Determine the angle θ
_{i}between each mooring line and the y axis (see figure 1). - Determine the length of the mooring line between the chock and mooring bitts, as well as the length between the chock and the bollard. See here to see a picture of a chock.
- Determine total breaking strength for each mooring line BS.
- Determine the spring constant for each mooring line, K
_{i}= a_{i}⋅ E_{i}/ l_{i}. - Determine spring component in the y direction, k
_{y,i}= K_{i}⋅ cos(θ_{i}). - Determine k
_{y,i}⋅ X_{i}, where X_{i}= X_{ch,i}, where X_{ch,i}is the x coordinate of chock i. - Determine k
_{y,i}⋅ X_{i}^{2}. - Take the following summations: A = ∑
_{i}k_{y,i}, B = ∑_{i}k_{y,i}⋅ X_{ch,i}, C = ∑_{i}k_{y,i}⋅ X_{ch,i}^{2}. - Apply total external lateral force P
_{y}and yaw moment M_{w}to the ship's pivot point at X=0. - Calculate ship's movement in the y direction, σ
_{y}= (P_{y}⋅ C - (M_{w}+ P_{y}⋅ X_{pp}) ⋅ B) / (A ⋅ C - B^{2}). X_{pp}is the x coordinate of the pivot point. In our case, X_{pp}= 0. - Calculate rotation of ship around the vertical axis, β = (P
_{y}⋅ B - (M_{w}+ P_{y}⋅ X_{pp}) ⋅ A) / (B^{2}- A ⋅ C). - For each line calculate its force component, P
_{y,i}= k_{y,i}(σ_{y}+ X_{ch,i}⋅ β). - Compute forces in the direction of the mooring line, T
_{i}= P_{y,i}/ cos(θ_{i}). - Determine safety factors for each mooring line, FS = BS / T
_{i}. - One may check the computation by verifying that ∑
_{i}P_{y,i}= P_{y}.

**References**Department of the Navy Naval Sea Systems Command, DDS 582-1 Calculations for mooring systems, Washington DC, 1987.

Last updated on June 9, 2020.

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