# Mooring Graphic

by Felipe Ferrari de Oliveira.
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.

## Insert data:

Mooring line density: [kg/m]

Mooring line length: [m]

Depth: [m]

Horizontal force: [kgf]

## Results

### General Inputs

$$w =$$ 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.

### Results

$$d =$$ 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.

### Assumption

• 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.

### Formulation

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: $$y_s = a.\bigg(\text{cosh}\bigg({x_s \over a}\bigg)-1\bigg)$$