by Felipe Ferrari de Oliveira.

felipe.ferrari.212@gmail.com, v0.1, september 2018

felipe.ferrari.212@gmail.com, v0.1, september 2018

Catenary representation of a hanging mooring line between reference
and transiction points. For further explanation regards
hypotheses, assumption, and formulation please consult:
Bernardo, A. (2016). Mooring Lines. In Applied Topics in Marine
Hydrodynamics (Chapter 7). São Paulo, SP: EPUSP.

Mooring line density: [kg/m]

Mooring line length: [m]

Depth: [m]

Horizontal force: [kgf]

**\(w = \) Mooring line density (kg/m):** Weight per unit along the chain.

**\(l = \) Mooring line length (m):** Total length of the chain including hanging line and horizontal line.

**\(z = \) Depth (m):** Vertical distance of hull attachment on the ship from the seabed, also corresponding to sea depth.

**\(H = \) Horizontal force (kgf):** Horizontal component of the tension, which is uniform along the chain.

**\(d = \) Attraction radius (m):** Horizontal distance from the ship to the attraction point. This distace represents the radius which the ship is allowed to discribe for whatever force direction.

**\(w = \) Suspended mooring line length (m):** Lenght of the mooring line between the point of contact of seabed and hull attachment point.

**\(T_{max} = \) Maximum traction force (kgf):** Maximum force in all line, the point where the maximum force occurs is the same of hull attachment point for this model.

**\(V_{max} = \) Maximum vertical force (kgf):** Maximum vertical force applied due gravitational force, the point where the maximum vertical force occurs is the same of hull attachment point for this model.

**\(\theta = \) Angle in ship attraction point (°):** Angle between the tangent to the chain and the x-axis at hull attachment point.

- The upper extremity of the line is attached to a point at the waterline level of a ship platform hull (attachment point);
- The lower extremity of the line is attached to an anchor;
- \(x_{s}\) is horizontal, pointing to the right;
- \(y_{s}\) is vertical, pointing upwards;
- The line is completely contained in the \((x,y)\) plane;
- The orign of the system is at the point where hanging line first achieves zero tangent from the hull to the anchor;
- Only traction forces are applied to the line, whichs act along the tangent. The line does not resist compression nor shear forces or bending moments;
- No elongation of the line;
- Hidrostatics forces on line irrelevant in front of weight and enviromental forces acting on the hull;
- Static equilibrium only.

The \(T_{max}\) can be calculated by the sum of horizontal and vertical forces on the hooking point:

$$T_{max} = H + w.z $$

Let \(a\) be the horizontal force weight ratio in \((m)\):

$$a = {H \over w} $$

$$s = \sqrt{z.(z+2a)} $$

$$V_{max} = w.s $$

$$\theta = \bigg({180 \over \pi}\bigg)\arctan{\bigg({s \over a} \bigg)} $$

Let \((x_s,y_s)\) be the ordinate point of suspended mooring line: $$y_s = a.\bigg(\text{cosh}\bigg({x_s \over a}\bigg)-1\bigg)$$