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OVERALL STABILITY 4.1 External Forces Acting on a Vessel

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OVERALL STABILITY 4.1 External Forces Acting on a Vessel OVERALL STABILITY 4.1 External Forces Acting on a Vessel In Chapter 4 we will study five areas: 1. The concept of a ship’s Righting Moment (RM), the chief measure of stability. 2. KG and TCG changes and their effects on RM. 3. How Stability is effected by Damage to the Hull using the “Added Weight” method. 4. Effects of a “Free Surface”. 5. Effects of Negative GM on ship stability. 4.2 Internal Righting Moment EXTERNAL FORCES cause a vessel to heel. Recall Force x Distance = Moment – External Moment can be caused by wind pushing on one side of the vessel and water resisting the motion on the other side. – Each distributed force can be resolved into a resultant force vector. The wind acts above the waterline and the water resistance acts below the waterline. Internal Righting Moment MT External upsetting force f Ds Righting Arm G Z WLf F f B FB Water resistance CL Internal Righting Moment The two forces create a couple because they are equal in magnitude, opposite in direction, and not aligned. The couple causes rotation or heeling. The vessel will continue to rotate until it returns to Static Equilibrium (i.e. an Internal Moment is created which is equal in magnitude and opposite in direction). Giving M=0. Internal Righting Moment Internal Forces create a Righting Moment to counter the Upsetting Moment of the External Forces. The two internal forces are the weight of the vessel (s) and the resultant buoyant force (FB). Internal Righting Moment The perpendicular distance between the Weight and the Buoyancy Force vectors is defined as the RIGHTING ARM (GZ). The moment created by the resultant Weight and the resultant Force of Buoyancy is defined as the RIGHTING MOMENT (RM). It may be calculated by: = = RM GZ s GZ FB Internal Righting Moment Where: RM is the internal righting moment of the ship in ft-LT. s is displacement of the ship in LT. FB is the magnitude of the resultant buoyant force in LT. GZ is the righting arm in feet. A ship in static equilibrium is affected by outside forces that will alter its state of equilibrium. MT Wind Water Resistance The forces of wind- and the opposing force of the water below the waterline- will cause an external moment couple about the ship’s center of flotation. The ship reacts to this external moment couple by pivoting about F, causing a shift in the center of buoyancy. MT Wind B Water Resistance The center of buoyancy will shift because the submerged volume will change. Note that there is no change in weight or it’s distribution so there is NO change in the location of G! Because the location of B changes, the location of where the FB is applied also changes. Because G does not move, the location of the Δs force does not change. MT s B FB The displacement force and the buoyant for are no longer aligned. The heeling over causes the creation of an internal moment couple. The external moment couple causes the creation of the internal moment couple to oppose it. MT s Wind F B Water Resistance FB As a result, the ship is now back into equilibrium, even as it heels over due to the wind force. We are concerned with the created internal moment caused by the offsetting of the ship’s weight and the buoyant force. Ds MT f Z B FB The offset distance of the applied forces, GZ, is called the MOMENT ARM. The length of this moment arm is a function of the heeling angle, φ. Remember that a moment is created when a force acts at a distance from a given point. In the case of the created internal moment couple, we have the two force, Ds and FB, acting over the distance GZ. The created moment is called the internal RM = GZDs = GZFB This illustrates just one potential moment arm based upon one particular angle of φ. There are an infinite number of angles possible, therefore, an infinite number of moment arms that vary with the degree of heel, φ. Ds MT f Z B FB If we can plot the heeling angle f versus the created moment arm GZ, we can create the Intact Statical Stability Curve. 4.3 Curve of Intact Statical Stability “Curve of Intact Statical Stability” or “The Righting Arm Curve” – Shows the Heeling Angle () versus the righting arm (GZ). – Assumes the vessel is heeled over quasi- statically in calm water (i.e. external moments are applied in infinitely small steps). This is a typical curve. Notice that it plots the angle of heel on the x-axis and the righting arm on the y-axis. The curve is in both the 1st and 3rd quadrants (the 3rd shows a heel to port). Typically only the curve showing a heel to starboard is shown as it is symmetrical. Measure of Overall Stability Curve of Statical Stability Range of Stability Slope is a measure of tenderness or stiffness. Righting Arm - Dynamical Stability GZ -(feet) Maximum Righting Arm Angle of Maximum Righting Arm Angles of Inclination: (Degrees) Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle The above chart plots the data presented in the text on p. 4-6 an 4-7. Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle With φ at 0 degrees, the moment arm is also is 0. The buoyant force and the ship’s weight are aligned. No moment is created. Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle As the angle of heel increases, the moment arm also increases. At 25 degrees, shown here, GZ is 2.5ft. Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle As the angle increases, the moment arm increases to a maximum… here it is 4ft. As φ increases beyond this point the moment arm begins to decrease and the ship becomes in danger of capsizing… Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle ...Remember, the internal moment couple created here is in response to the external couple created by outside forces. At GZ max the ship is creating its maximum internal moment. If the external moment is greater than the internal moment, then the ship will continue to heel over until capsized. Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle The angle of heel continues to increase, but the moment arm GZ, and thus the internal moment couple, decreases. Intact Statical Stability 4.5 4 3.5 3 2.5 2 1.5 Moment Arm GZ Arm Moment 1 0.5 0 0 25 50 75 85 Heeling Angle The angle has now increased to the point that G and B are now aligned again, but not in a good way. GZ is now at 0 and no internal moment couple is present. Beyond this point the ship is officially capsized, unable to right itself. Curve of Intact Statical Stability Caveats! Predictions made by the Curves of Intact Statical Stability are not accurate for dynamic seaways because additional external forces and momentum are not included in the analysis. ”Added Mass” However, it is a simple, useful tool for comparison and has been used to develop both intact and damaged stability criterion for the US Navy. Curve of Intact Statical Stability Typical Curve of Intact Statical Stability – Vessel is upright when no external forces are applied and the Center of Gravity is assumed on the centerline. (Hydrostatics) – As an external force is applied, the vessel heels over causing the Center of Buoyancy to move off the centerline. The Righting Arm (GZ) is no longer zero. Curve of Intact Statical Stability Typical Curve of Intact Statical Stability (cont.) – As the angle of heel increases, the Center of Buoyancy moves farther and farther outboard (increasing the Righting Arm). – The max Righting Arm will happen when the Center of Buoyancy is the furthest from the CG. This is max stability. – If the vessel continues to heel, the Center of Buoyancy will move back towards the CG and the Righting Arm will decrease. Curve of Intact Statical Stability Typical Curve of Intact Statical Stability (cont.) – Since stability is a function of displacement, there is a different curve for each displacement and KG. These are called the Cross Curves. For all ships, there exists the CROSS CURVES OF STABILITY. Like the Curves of Form, they are a series of curves presented on a common axis. • The x-axis is the ship’s displacement, Δs, in LT • The y-axis is the righting arm, GZ, in ft • A series of curves are presented, each representing a different angle of heel f By plotting the data from the Cross Curves of Stability for a given displacement, you can create an Intact Statical Stability Curve. In the Cross Curves of Stability, the data is presented assuming that: KG = 0 (on the keel) This is, of course, not realistic.
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