Initial Transverse Metacentre Is defined as the point of intersection of successive lines of action of buoyancy force (B_{f}) when the ship is in the initial upright condition and subsequently heeled conditions.

It is assumed to be at a fixed position when the ship is heeled within small angles only.

**Consider the ship shown.**

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**When the ship heels beyond small angles the point of intersection has to move, hence the term Initial Transverse Metacentre.**

**It’s position is expressed as a height above the keel in metres and is termed KM. The value of KM is tabulated in ship’s hydrostatic data and its position varies with draught.**

**Calculation of KM for Box Shaped Vessel**

**It is convenient to consider the KM for a box-shaped vessel because the maths is simple, however, the same principles will apply for ship shapes. KM is calculated by the formula:**

**For a box-shaped vessel on an even keel:**

**BM is calculated by:**

**where I is the moment of inertia of the water-plane area (WPA) about a longitudinal axis of rotation passing through the centre of the water-plane area; and V is the volume of displacement of the vessel.**

**Moment of inertia (second moment of area) of the water-plane area**

**Consider the water-plane area of a box-shaped vessel shown.**

** **

**C is the geometric centre of the water-plane area and XX is the longitudinal axis about which it is to be rotated.**

* **(In reality, it is the ship that will rotate about this axis as it heels, however it is convenient to think of the water-plane area rotating about the same axis instead!)*

**For a box-shaped vessel:**

**where L and B are the length and breadth of the water-plane area respectively. I is in units of metres4 (m4).**

*A simplistic, but convenient, way of considering the effect of the moment of inertia of the **Water-plane area is to consider that it gives a ship resistance to heeling!*

** Therefore, the larger the water-plane area, the less easily a ship will heel.**

**If the formula for BM is considered: It is the breadth of the water-plane area that is most influential. It is generally accepted that the broader a ship is, the more stable it will be. **

**To summarize:**

**and for a box-shaped vessel:**

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