EMNA102: 2023 Semester 2
Stability
Department of Marine Engineering
North Campus, Nelson Mandela University
Gqeberha
Compiled by Aidan August
Aidan.August@mandela.ac.za
Introduction
➢ Keel (K): The lower most point on the vessel centreline and typically the datum/reference
point
➢ Centre of Bouyancy (B): Vertical Centroid of the submerged cross section of the vessel from
keel
➢ Centre of gravity (G): Vertical Centroid of the entire cross section of the vessel from keel
➢ Transverse Metacentre (MT): Imaginary point through the centre of buoyancy and centre of
gravity which intersects the imaginary vertical line through a new centre of buoyancy
created when the body is displaced, or tipped, in the water
Determining Stability Reference points
➢ Since B is the vertical centroid of the submerged cross-sectional area, the value of KB will
be dependent on the draft of the vessel as well as the shape of the cross-sectional area of
the submerged portion
➢ Since G is the vertical centroid of the entire cross-sectional area, the value of KG will be
dependent on the shape of the cross-sectional area of the vessel
➢ The metacentric radius BM may however be determined as follows:
𝐼𝑇
➢ 𝑩𝑴 =
𝑉
➢ Where IT is the transverse second moment of the waterplane area (m4) and V is the
displaced volume (m3) of the vessel
➢
𝐾𝐺 + 𝐺𝑀 = 𝐾𝐵 + 𝐵𝑀
➢
GM is known as the metacentric height of the vessel and is an indication of the initial intact
stability of the vessel as well as the stiffness of a vessel
The Righting Moment (GZ)
➢ For small angles of heel, typically less than 10 degrees, GZ may be determined as follows:
➢ 𝐺𝑍 = 𝐺𝑀 sin θ
➢ Where θ is the angle of heel of the vessel
➢ GM is the metacentric height
Correcting for a vertical shift in G
➢ G is the assumed position of the centre of gravity
➢ G1 is the actual position of the centre of gravity
➢ If G1 is below G then GZ actual (G1Z) is determined as follows:
➢ 𝐺1 𝑍 = 𝐺𝑍 + 𝐺𝐺1 sin θ
➢ If G1 is above G then GZ actual (G1Z) is determined as follows:
➢ 𝐺1 𝑍 = 𝐺𝑍 − 𝐺𝐺1 sin θ
➢ Where GG1 is the difference between the Actual position of G and the assumed position of
G
Range of Stability (ROS)
➢ The range of stability of the vessel may be determined by computing G1Z for various angles
➢ It is important to note that this is an initial intact stability analysis of a vessel and is not a true
indication of what the vessel can do when inclined to a certain angle. This analysis is purely
for comparison
Dynamical Stability
➢ The dynamical stability of the vessel may be determined by calculating the work done over
the ROS as follows:
➢
➢
𝑊𝐷 = 𝐴𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑡ℎ𝑒 𝑅𝑂𝑆 curve × 𝑊 (J)
Where W is the weight of the vessel (N)
➢ When calculating the area under the curve the common interval angle must be in radians
Example
1. A vessel displaces 10 000 tonnes in seawater. The assumed KG is 7.2m and the actual KG is
found to be 7.8m. The following are the GZ values for different angles of inclination at the
assumed KG:
θ (°)
0
15
30
45
60
75
90
GZ (m)
0
0.43
0.93
1.21
1.15
0.85
0.42
a) Determine the range of stability of the vessel
b) Calculate the dynamical stability of the vessel
THANK YOU