Exercise - Computation of mooring forces

A ship with a displacement of 3350 t, a waterline length (Lwl) 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 m2 is moored to a berth with 10 mooring lines displaced as in Figure 1. 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 cm2 each. The elastic modulus of each rope is 125 kN/cm2 and the breaking tension is 25 kN/cm2.
  • 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;
    • Cw=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


    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:

    Pw= Cw ⋅ (Aw ⋅ sin2 φ + Bw ⋅ cos2 φ) × γw ⋅ Vw2/2g

    In our case Cw = 1.3, Aw = 950 m2, φ = 90°, γw = 0.01225 kN/m3, Vw = 92.6 km/h = 25.72 m/s, g = 9.81 m/s2. Also, lets note that φ = 90° implies that Pw = Py, where Py is the component of wind force along the y direction. Therefore one obtains:

    Py= 1.3 × 950 ⋅ sin2 90° ⋅ 0.01225 × 25.722/(2 x 9.81) = 510.18 kN

    Such 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)):

    Mw = Cxyw ⋅ Pw ⋅ Lwl,

    where Cxyw is an eccentricity coefficient that can be estimated from Figure 2.

    Figure 2. Estimation of Cxyw. Redrawn from Department of the navy naval sea systems command (1987). Positive sign is associated to anti-clockwise rotation.

    Therefore one obtains:

    Mw = -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 Mw and lateral force Py 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 li from bitts to bollards.
    • Determine total cross sectional area ai 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 Ei for each mooring line from producer.
    • Determine the angle θi 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, Ki = ai ⋅ Ei / li.
    • Determine spring component in the y direction, ky,i = Ki ⋅ cos(θi).
    • Determine ky,i ⋅ Xi, where Xi = Xch,i, where Xch,i is the x coordinate of chock i.
    • Determine ky,i ⋅ Xi2.
    • Take the following summations: A = ∑iky,i, B = ∑iky,i ⋅ Xch,i, C = ∑iky,i ⋅ Xch,i2.
    • Apply total external lateral force Py and yaw moment Mw to the ship's pivot point at X=0.
    • Calculate ship's translation in the y direction, σy = (Py ⋅ C - (Mw + Py ⋅ Xpp) ⋅ B) / (A ⋅ C - B2). Xpp is the x coordinate of the pivot point. In our case, Xpp = 0.
    • Calculate rotation of ship around the vertical axis, β = (Py ⋅ B - (Mw + Py ⋅ Xpp) ⋅ A) / (B2 - A ⋅ C).
    • For each line calculate its force component, Py,i = ky,iy + Xch,i ⋅ β).
    • Compute forces in the direction of the mooring line, Ti = Py,i / cos(θi).
    • Determine safety factors for each mooring line, FS = BS / Ti.
    • One may check the computation by verifying that ∑iPy,i = Py.


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

    Last updated on April 29, 2020.