10

CHAPTER I
RESISTANCE TO DAMAGE

11. Foreword. Combat experience has indicated that it is inappropriate to classify ships as "combatant" and "noncombatant" types. Actually, all Naval ships whether large or small are subject to combat in one form or another even though they never may be a part of a "line of battle" or a carrier striking force.
Action taken after severe damage must be considered from the standpoint of the design of the particular ship with emphasis necessarily placed on those characteristics which provide inherent resistance. In the broad sense, Naval ships can be classified into two major groups according to the elements of resistance to damage incorporated in their design.
1. Ships with torpedoprotection systems (and armor):
a. Battleships.
b. Large aircraft carriers (CV's).

2. Ships without torpedo protection.
a. Armored.
Large cruisers (CB's).
Heavy and light cruisers (CA's and CL's).
Light aircraft carriers (CVL's).
b. Unarmored.
Tenders.
Escort aircraft carriers (CVE's).
Destroyers.
Destroyer escorts.
Transports.
Large landing craft.
Minecraft.
Auxiliaries of various types.

In succeeding discussions concerning damage control efforts the foregoing classification should be borne in mind.
12. Causes of loss. The primary causes of warship loss are the following:
1. Flooding. The ship may capsize, plunge by the bow or stern, or sink bodily.
2. Breaking up. Structural damage may result in the loss of a considerable portion of the boss or


stern, which in turn may lead to plunging. In the case of small, lightly constructed vessels the Lull girder in the middle onehalf length may be so weakened by the destruction of structure that the ship breaks in two.
3. Severe fires. These can cause the abandonment of the ship. If this occurs the fires will rage unchecked until damage below the waterline results, or destruction by own forces becomes necessary.
4. Magazine explosions. These may be caused by hits in the magazines. They also may result from fires which reach the magazines as described above.

13. Types of attack. It must be borne in mind that no ship sinks until underwater damage has been incurred. Such damage can be inflicted by missiles, including projectiles, bombs, and rockets, or by weapons designed to travel underwater. Underwater attack usually means attack by weapons carrying large charges, such as torpedoes, mines, and in special cases, near miss bombs.
14. Missile attack. Bombs and projectiles normally are fitted with fuzes which permit penetration prior to detonation. The effects of detonations within the hull are:
1. Blast, which tears, warps, and demolishes.
2. Fragments or splinters, which spread in a shower and travel with velocities higher than those of projectiles.
3. Incendiary effect, which results partly from the globe of incandescent gases created by the detonation and partly from whitehot fragments of the case containing the explosive.
4. Noxious and toxic gases, which result from the detonation of a high explosive.

Resistance to missile attack is obtained by the installation of armor or ballistic plating and to a lesser extent by providing a liquid layer outboard of vital spaces below the waterline.

11

Figure 1A. Damage due to gunfire. Note the fragment effect on the splinter shield.

12

Bombs ordinarily are classified as GP (general purpose), SAP (semiarmorpiercing), and AP (armorpiercing). There are many other types of bombs but the use of these other types is ordinarily not profitable against ships and they are rarely encountered.
GP bombs come in all sizes ranging from 100 to 2,000 pounds. They have a thin case and a chargeweight ratio of approximately 0.50. They have considerable penetrative power (about one and onehalf inches of deck armor is the maximum) and may be fitted either with instantaneous or short delay fuzes. SAP bombs, usually of moderate size, have a heavier case than GP bombs and will penetrate up to about three inches of deck armor. They have a chargeweight ratio of about 0.30. They should be fitted with short delay fuzes in order to take advantage of their ballistic or penetrating characteristics. AP bombs come in large sizes. They have heavy cases and a low chargeweight ratio (rarely over 0.15). The thickness of deck which they will penetrate depends on the altitude from which they are dropped. For example, the U.S. 1,600 pound AP bomb must be dropped from 16,000 feet in order to penetrate seven and onehalf inches of deck armor. They are fitted with fuzes of somewhat longer delay than are the GP and SAP types.
The projectiles commonly used in naval warfare may be classified as follows:
1. High capacity.
2. Common.
3. Armor piercing.

Highcapacity projectiles have the thinnest walls of the three types, and the highest chargeweight ratio. Even so, the chargeweight ratio is low (0.09 to 0.12) compared to that of bombs. Common projectiles are designed for use against light armor and have a somewhat more sturdy case than the highcapacity projectiles. The chargeweight ratio is rarely over 0.05 and sometimes is as low as 0.02. Armorpiercing projectiles have very heavy cases designed for penetrating the thickest armor. As a result, the chargeweight ratio is seldom higher than 0.015. Highcapacity projectiles are fitted with instantaneous fuzes; the common projectile may have an instantaneous fuze but usually is fitted with a very short delay fuze. The armorpiercing projectile invariably is fitted with a delay fuze.
The major portion of weight devoted to protection in armored ships is provided for the purpose of resisting penetration by armorpiercing projectiles and bombs.


15. Armor protection. Armor designated as Class A comprises heavy plates, installed vertically in sidebelt, barbettes, or turrets, primarily for the purpose of stopping projectiles. It is very highgrade steel which has undergone a facehardening process with the result that the plate has a very hard exposed surface and a strong, ductile back. Side armor is attached to the ship's hull with special bolts and the individual plates are keyed together. Barbettes are keyed together, but being circular require little backing structure. Turret plates are secured to supporting structures by special bolts and rivets, and the individual plates also are keyed together. Class A armor is very heavy, and weight is not available for its installation on lighter warships and auxiliaries.
Armor designated as Class B is of highgrade steel which has been given a special heat treatment. It is not face hardened as is class A armor. Class B armor comes in a large variety of thicknesses beginning with threeinch and including five and sixinch thicknesses. The smaller sizes may be rolled and the heavier thicknesses usually are forged. In the smaller thicknesses it can be worked integral with the hull of the ship so that it forms a part of the hull girder; Class A armor cannot be so worked.
Side armor is installed to protect the waterline area and vitals such as magazines and engineering spaces. At the top of the armor belt a protective deck (STS, or specialtreatment steel  the same material as Class B armor) is installed to protect machinery spaces and magazines. Topside protection in the form of STS of varying thicknesses is provided for exposed personnel and important operating stations. Battleships have a large weight allowance for protection. Cruisers have less weight available for armor protection and their armor belt and protective deck are much lighter and of smaller extent than those of battleships Armor protection cannot be installed on small ships such as destroyers. However, the later destroyers have a portion of their hull plating fabricated from STS which gives them a certain minimum amount of protection against smallcaliber bullets and fragments.
Figure 11A indicates a typical arrangement of armor on a cruiser. Figure 11B shows the typical arrangement for a battleship.
In addition to protection of the ship as a whole by armored belts and armored decks, armor protection usually is provided locally to the steering gear, the conning tower and turrets.
All steel ships have a certain minimum amount of resistance to missile attack by virtue of their steel construction.

13

Figure 11.

As has been pointed out in the case of destroyers, a few types have a liberal quantity of STS worked into the hull as strength members. By use of STS excellent structural characteristics, as well as ballistic properties, are attained. A high strength steel, commonly referred to as HTS (hightensile steel) is frequently used in light ships when greater strength than is provided with medium steel is desired. HTS has somewhat better ballistic properties than medium steel although these ballistic properties are not comparable to those of STS.
Once penetration has occurred and a missile detonates within a ship, subdivision acts to minimize damage. Interior ballistic bulkheads (usually STS) and armored decks act to absorb the blast and stop or slow down fragments. Mediumsteel decks and bulkheads, which form a great majority of interior structures, act similarly to STS, but are less effective.
16. Underwater protection. The majority of underwater explosions which cause serious damage involve torpedoes, mines and close nearmiss bombs. The effects of such underwater explosions are extensive but damage may be limited, absorbed, or restricted by special protective systems. Battleships and large aircraft carriers are fitted with torpedoprotection systems, also referred to as underwaterprotection systems, whose function it is to prevent water from entering vital spaces, such as magazines, handling rooms, engine rooms, and boiler rooms.
Only the largest ships can provide the space and weight allocations for underwaterprotection systems. Cruisers, for example, do not have such systems. Needless to say, destroyers and most auxiliaries likewise do not have underwaterprotection systems.


Torpedoprotection systems are described in FTP170B and also are discussed in Chapter XV of this volume. At this point it is sufficient to note that underwaterprotection systems in general have functioned as expected, and that the general design of these systems used in the U.S. Navy has been vindicated by war experience. The battleships which were sunk at Pearl Harbor did not sink as a result of the failure of their underwaterprotection systems to function as designed.
An underwaterprotection system consists chiefly of a series of longitudinal bulkheads fitted inboard of the shell. Instructions require that the two outboard longitudinal layers so formed be carried full of liquid and that inboard layer (s) be carried void. Cruisers and smaller ships, as noted above, do not have underwaterprotection systems. Thus, the lack of an underwaterprotection system in cruisers and smaller ships requires a different approach to the problem of stability control than in the case of battleships and large carriers. The differences between ships with underwaterprotection systems and ships without underwaterprotection systems will be discussed at greater length in the following Chapters.
Although cruisers and smaller ships do not have underwaterprotection systems, they do have resistance to underwater attack by virtue of their strength, stability, subdivision, and reserve buoyancy. All cruisers have survived one torpedo hit and several have survived two. Many destroyers have survived one torpedo hit. For such smaller ships the designer's efforts follow three general methods for incorporating resistance to extensive underwater damage:

14

Figure 1B. Collision damage. Good damage control brings such vessels back with minimum danger from progressive flooding.

15

Figure 12. Diagram to show effect of subdivision upon flooding.

1. To insure the smallest possible list after damage.
2. To give magazines as much protection as is possible.
3. To arrange the engineering plant so that damage from one torpedo hit, or other corresponding damage, is unlikely to immobilize the ship.
These factors will be discussed later in more detail. 
17. Collision. Damage from collisions is in general minor when compared with the effects of attack by enemy weapons; however, the same characteristics that enable ships to survive enemy attack, that is, strength, stability, subdivision, and reserve buoyancy, will provide the necessary elements of resistance to collision, groundings, and ramming enemy submarines (in the case of small warships).
18. Loss by flooding. It has been noted previously that flooding is one of the primary causes of loss. When flooding is uncontrolled, that is, when interiors subdivision fails for one reason or another, one of three ultimate results or some combination thereof will ensue:
1. Sinking bodily. When the quantity of water taken aboard utilizes all of the reserve buoyancy, a ship will sink approximately on an even keel. Actually, this dire result occurs most infrequently.
2. Capsizing. When transverse stability characteristics are sufficiently impaired a ship will capsize.
3. Plunging. Flooding at one end or the other will reduce longitudinal stability characteristics to the point that the vessel sinks by the bow or by the stern.



Damagecontrol efforts to prevent loss by flooding must employ one of the following methods:
1. Preventing the entry of damage water.
2. Limiting the extent of damage water.
3. Removing the damage water after it has entered a compartment.
4. Counteracting the effects of damage water.

War experience has indicated that the most effective of the above measures is No. 2  limiting the extent of damage water. This follows from the fact that the entry of damage water into the hull can never be prevented if the hit is below the waterline; removal is difficult in many cases and impossible in others; counteracting the effect of damage water may be dangerous. The function of interior subdivision (that is, watertight decks and bulkheads) is to limit the extent of damage water to the immediate vicinity of damage.
Consider a vessel whose hull contains neither decks nor bulkheads, as shown in figure 12A. A single unplugged hole below the waterline will permit the entire vessel to flood and the ship obviously will sink. If, however, the vessel has one transverse midship bulkhead, as shown in figure 12B, a single unplugged hole below the waterline will flood only onehalf of the ship (see fig. 12C). When onehalf of a ship with a single transverse midship bulkhead is flooded, it will plunge either bow first or stern first (see fig. 12D).
If the number of transverse watertight bulkheads be increased and so spaced that the ship will withstand

^{1}All warships and almost all auxiliaries have watertight subdivision for the purpose of confining the spread of water when damaged.

16

flooding of one of its main compartments without sinking, plunging, or capsizing, the ship is said to meet a onecompartment standard of subdivision. If the number of transverse bulkheads be increased, and proper spacing provided, it is possible for the ship to meet a twocompartment or a threecompartment standard of subdivision. Such ships will survive flooding of any two adjacent compartments or any three adjacent compartments, respectively. Reference is made in the above discussion to those compartments which, if flooded, will produce the most adverse conditions of stability and buoyancy. Most Naval auxiliaries meet a twocompartment standard and some have an even better standard. It is apparent, therefore, that even the simplest types of ships can be provided with considerable resistance to underwater damage and consequent flooding by the proper arrangement of bulkheads. That this is the case has been demonstrated many times by survival after torpedoing of tankers, cargo ships, and transports.
Once a given space or a group of spaces has been flooded, the characteristic which keeps the ship from plunging by the bow or stern is longitudinal stability; that characteristic which prevents it from sinking bodily is reserve buoyancy; and, finally, that characteristic which prevents capsizing is transverse stability. In the succeeding Chapters of this volume each of these characteristics will be treated in some detail, together with those corrective measures best adapted to preserving and restoring them after damage.
Only the simplest types of transport and cargo vessels have the minimum compartmentation described above. The large auxiliaries, such as destroyer tenders, and even the smaller offensive types such as destroyers, have several watertight decks below the weather deck and thus have a comparatively large number of watertight compartments. Increasing the subdivision greatly improves the ship's resistance to loss following the entry of damage water. This follows from the facts that the extent of flooding is limited and the quantity of water taken aboard as the result of any given underwater hit is decreased. In small vessels, however, it is dangerous to install longitudinal watertight bulkheads, inasmuch as these will permit offcenter flooding and large heeling or listing moments. A large list, of course, is extremely dangerous.
Ships with armor protection but without torpedoprotection systems, such as cruisers, have subdivision substantially better than the ships just discussed, and accordingly are more resistant to the effects of flooding. Armored ships with torpedoprotection systems, such


as battleships and large carriers, have the best standards of subdivision.
By preventing the penetration of projectiles, armor protects the watertight integrity of the ship in the vicinity of the waterline. However, it rarely covers fore than 60 per cent of the length at the waterline, and in certain modern warships does not lie adjacent to the shell.
Watertight integrity and reserve buoyancy are reduced when the hull is pierced at or above the waterline as well as when the hull is pierced below the waterline. Also, when fragments tear holes in bulkheads and decks above the waterline reserve buoyancy is diminished. If flooding occurs it cannot be confined by the boundaries which are damaged. Accordingly, patching holes in shell plating, decks and bulkheads is a most important job for repair parties. It is emphasized that preservation of watertight integrity should not only be concerned with the underwater body but also with the watertight portion of the vessel which normally lies above the waterline.
19. Loss by structural failure. Only the smaller and more lightly constructed ships may be lost due to physical destruction of their strength members. The larger vessels, including even converted merchant ships, have sufficient strength to resist a considerable amount of such damage. Even on the smallest vessels the margin of strength in most cases can be augmented by executing proper measures after damage.
In general, larger ships including cruisers, battleships, and large aircraft carriers, can withstand a comparatively large amount of structural damage. In the case of a cruiser, for example, the inner bottom, heavy shell, and relatively large number of structural members provides so much strength that breaking up in the middle onehalf length rarely occurs. Actually, only one of our cruisers has been lost by breaking up, and in this instance damage was caused by two successive torpedoes hitting in the same location. For obvious reasons, armored ships larger than cruisers are very unlikely to be in difficulties because of structural strength loss.
110. Loss clue to fires. Firefighting methods, firefighting equipment and firefighting systems are discussed in the Fire Fighting Manual issued by the Bureau of Ships. Firefighting schools for the training of personnel have been established at many of the important shore stations. Attendance of every officer and man aboard at one of these schools is recommended most emphatically. It is important to note that early in World War II many ships were abandoned and lost

17

Figure 13. Diagram to show the relative degree of subdivision on several types of Naval ships.

18

Figure 14. Diagram to show arrangement of boiler rooms and engine rooms in a modern destroyer.

as the result of serious fires, but as the war progressed ships with equally serious fires have been saved handily.
Early World War II experiences with fires aboard ships led to a program for the ruthless removal of fire hazards. As a result inflammable material has been reduced to a bare minimum. The importance of this program cannot be overemphasized.
The problem of fire is one that requires constant vigilance, maintenance, and training. The possibility of serious fires is with the Navy at all times. Therefore, it is a matter of concern that every damage control officer have a thorough understanding of fire fighting.
111. Immobllization of propulsive plant. Strictly speaking, immobilization of a ship is not a primary cause of loss. However, it is obvious that a ship which is dead in the water is an easy target for successive attacks. Proper arrangement and segregation of the engineering plant, together with splitplant operation will reduce the chances of immobilization.
The type of ship most vulnerable to immobilization is the singlescrew vessel with the entire machinery plant in one space. A twinscrew ship has less chance of being immobilized from a single hit, but if both engines are in the same engine room, a single hit can still produce this result. Our earlier destroyers have two adjacent engine rooms with two firerooms forward of them. Thus, one hit could, and in many cases did incapacitate both engine rooms or both firerooms. The latest destroyers have the arrangement shown in figure 14, in which immobilization from a single hit is much less probable than in the earlier arrangement.
Our latest cruisers have the fireroomfireroomengine roomfireroomfireroomengine room arrangement. The newest battleships have four independent plants,


each occupying two compartments. The newest large carriers have even better subdivision and arrangement of the engineering spaces.
112. Loss by magazine explosion. The maximum practicable protection for magazines is attained, even on small vessels, by placing them below the waterline and as far inboard of the shell as is possible. In addition, a liquidfilled space between the magazines and the shell of the ship is most desirable and is installed wherever the size of the vessel permits. On armored ships magazines and handling rooms will, of course, be protected.
113. Summary. Each ship acquired and converted or designed and built by the Navy has the maximum resistance to damage which it is practicable to provide. In all ships some effort is made to protect against abovewater attack by projectiles and bombs, and against underwater attack by torpedoes, mines, and bomb near misses.
In all warships protection is provided by the following characteristics, in part or in whole:
1. Watertight subdivision.
2. Adequate transverse stability.
3. Adequate reserve buoyancy.
4. Structural strength.
5. Fire resistance and protection.
6. Segregation and distribution of propelling machinery.
In the case of larger warships further protection is provided by armor and torpedoprotection systems.
In succeeding Chapters the part that each of these characteristics plays in resisting damage will be analyzed. It is emphasized that all ships, whether large or small, have inherent resistance to battle damage to

19

the maximum degree compatible with their size and other military characteristics. It is noted that this resistance is more effective than is generally expected.
War experience has clearly indicated that our ships can absorb a large amount of damage and survive. To achieve their full power of survival, however, it is


necessary that the men who man them have an understanding of those inherent characteristics which provide resistance. Knowledge which results in correct and prompt action has saved many a severely damaged ship. Ignorance which results in improper measures can cause unnecessary loss. KNOW YOUR SHIP.

20

CHAPTER II
BASIC PRINCIPLES OF MATHEMATICS AND MECHANICS

21. Foreword. The purpose of the following discussion is to present certain basic mathematical processes and physical principles which are vital to the study of stability. Understanding of succeeding chapters requires a knowledge of arithmetic, some algebra, some trigonometry, and certain aspects of physics from the field of mechanics. The ensuing presentation has been simplified, and it is not intended to be an allinclusive review of the subjects mentioned.
22. Algebra. Algebra is that part of mathematics in which letters represent numbers. Thus the multiplication 4 X 3= 12 can be expressed as a X b = c or ab = c, where a replaces 4, b replaces 3, and c is 12. Through the use of letters in place of numbers, we are able to develop formulas general in application. In a formula letters are used until the numbers which the letters represent are known. In finding the area of a rectangle, for example, we have the formula Area = b X l, where b is the width and l the length of the rectangle. When we learn the values of b and l in terms of numbers, we can obtain the area of the rectangle numerically.
Figure 21. Diagram to show certain trigonometric relationships.


The other part of algebra to be reviewed is that which is concerned with "exponents." An exponent is a small number placed to the right and above another number to indicate how many times the basic number is multiplied by itself. For example, 4^{3} = 4 X 4 X 4 = 64. In this case, 3 is the exponent. 4^{3} is expressed as "four to the third power" or "four cubed." 5^{2} is 5 X 5, or "five to the second power," or "five squared."
Since letters can be used in place of numbers, a^{2} = a X a.
23. Trigonometry. Trigonometry is the study of triangles, and the interrelationship of the sides and angles of a triangle. In damagecontrol work, only a very small and simple section of trigonometry is used; that part dealing with right triangles. Advantage is taken of the fact that there is a fixed relationship between the angles of a right triangle and the lengths of the sides.
The ratio of the sides of a right triangle, or trigonometric functions as they are known, may be determined from the right triangle shown in figure 21. The ratios required for this study are the sine, cosine, and tangent. Angles are customarily represented by the Greek letter theta, θ. The sine of the angle θ (sin θ) is the ratio of the side of the triangle opposite the angle θ, divided by the hypotenuse.
As the angle increases in size, side a grows larger and the sine increases. Changes of sine value corresponding to changes in the size of the angle are shown on the curve of figure 22. This curve is called a sine curve. The size of the angle is plotted horizontally and the value of the sine vertically.
At any angle the vertical height between the base line and the curve is the value of the sine of the angle. A feature of this curve is that the sine at 30° is half of the sine value at 90°. At 0°, sin θ equals zero. At 90°, sin θ = 1.
The cosine is the ratio of the side adjacent to the angle θ, divided by the hypotenuse.

21

Figure 22. A sine curve.
Figure 23. A cosine curve.

In contrast with the sine, the cosine decreases as the angle θ becomes larger. This relationship is shown on the curve in figure 23. At 0° the cosine equals 1, and at 90° the cosine equals 0. At 60° the cosine is half the value of the cosine at 0°.
The tangent of the angle, θ, is the ratio of the side opposite the angle θ to the side adjacent.
A curve could be drawn for the tangent, but it is not needed in this treatment.
The other term to be discussed is the radian. In a circle of any size, if the radius is laid off along the circumference, the angle of the arc thus laid off is equal to one radian. In figure 24 the radius (r) is measured along the circumference and forms the arc of angle θ. θ = one radian. One radian is always equal to 57.3 degrees.
24. Volume, density, weight, and center of gravity. The volume of any object is determined by the number of cubic feet or cubic inches contained in the object. The underwater volume of a ship is found by


determining the number of cubic feet in the part of the hull below the waterline.
The density of any material, solid or liquid, is obtained by weighing a unit volume of the material. For example, if we take one cubic foot of sea water and weigh it, the weight is 1/35th. of a ton (one ton equals
Figure 24. Diagram to illustrate the principle of a radian.

22

2,240 pounds). Therefore we say that sea water has a density of 1/35th. ton per cubic foot.
If we know the volume of an object and the density of material, the weight of the object is found by multiplying the volume by the density. In a similar manner the volume or density may be found by knowing the weight and the other factor.
W=V X d (Weight volume X density).
When an object floats in a liquid, the weight of the volume of liquid displaced by the object is equal to the weight of the object. Thus if we know the volume of displaced liquid, the weight of the object is obtainable by multiplying the volume by the density of the liquid. For example if a ship displaces 35,000 cubic feet of salt water, the ship weighs 1,000 tons.
35,000 cubic feet X 1/35 th. ton per cubic foot = 1,000 tons.
The center of gravity of an object is the center of the various weights that make it up. It is the point at which all the weight of the unit or system could be concentrated and have the same effect as that of all of the component parts.
25. Force. A force is a push or pull. It tends to produce motion or a change in motion. A force may act on an object without being in direct contact with it. The most common example of this is the phenomenon known as gravitational attraction. Since gravitational forces are allpervading, forces are usually expressed in terms of weight units; i.e., pounds, tons, ounces, and the like.
To indicate the action of force on a body, an arrow pointing in the direction of the force is drawn to represent the force. Thus, the force F in figure 25 indicates a push on the object.
Figure 25. Diagram to show how a force may be indicated.
If a number of parallel forces act on a body, they may be combined for purposes of computation into one force which is equal to the sum of all forces acting in the same direction and so located as to produce the same effect. F_{4} (see fig. 26) is the resultant or net force of F_{1}, F_{2}, and F_{3}, and is equal to F_{1} + F_{2} + F_{3}.
Whether we consider the individual forces F_{1}, F_{2}, and F_{3}, or just F_{4} alone, the action of these forces or force on the object will move the body in the direction


of the force. To prevent motion, i.e., to keep the body at rest, another force of equal size in the same line and in opposite direction to F_{4} must be applied. The new force and F_{4} would cancel each other, and there would be no movement.
Figure 26. Diagram to show the resultant of several forces.
26. Moments. In addition to the size of a force and its direction of action, the location of the force is important. Let us take, for example, the simple seesaw. If two persons weighing the same sit on opposite ends equally distant from the support (see fig. 27), the seesaw will balance. However, if one of the persons moves, the seesaw will no longer remain balanced. The person farthest away from the fulcrum will move down because the effect of the force of his weight is greater.
Figure 27.
This effect of the location of a force is known as the moment of a force. It is equal to the force multiplied by the distance from an axis about which we want to find its effect. The moment of force is the tendency of the force to produce rotation. Since the force is expressed in terms of weight units, and moment is force times distance, the units for moment are tonsfeet, poundsinches or inchespounds, etc. The moment of force F in figure 28 about an axis at point a is F X d; d being called the moment arm.
The moment of a force can be measured about any point or axis, but of course the moment will be different in each case wherein the moment arm differs. It may be noted that the moment of a force tends to produce motion, but of a rotary nature. In figure 28,

23

for example, the force F would produce a clockwise rotation. If the moment of existing forces tending to produce clockwise rotation is exactly equal to the moment of those tending to produce counterclockwise rotation, there will be no rotation. The body will be in equilibrium.
Figure 28. Diagram to illustrate moment.
A special case of moments occurs when two equal and opposite forces acting on a body are not in the same line. These forces rotate the body. This system of two forces is termed a couple, as shown in figure 29. The moment of the couple is equal to the product of one of the forces times the perpendicular distance between them. Thus, in the example shown in figure 210:
Moment = F X d = 50 X 12 = 600 foot pounds.
Figure 29. Two equal and opposite forces acting on a body but not in the same line. The arrow indicates direction of rotation.
In addition to moment of force, there can be moment of weight (weight being a force), moment of volume (if there is the volume, there is weight and therefore force), or moment of area. Some examples of moments which are used in connection with damagecontrol studies are inclining moments, trimming moments, and vertical moments. Any athwartship movement of weight on board ship will produce an inclining moment equal to the magnitude of the weight times the distance moved.
A 20 ton weight moved from the centerline 20 feet outboard (see fig. 211) produces an inclining


moment of 400 foot tons. A foreandaft movement of a weight produces a trimming moment equal to the weight multiplied by the distance moved. Thus. 2C tons moved 50 feet forward (see fig. 212) will produce a trimming moment equal to 20 tons X 50 feet, or 1,000 foot tons.
Figure 210. Diagram to illustrate a couple (forces).
Figure 211. Diagram to illustrate inclining moment produced by moving a weight outboard.
Figure 212. Diagram to illustrate trimming moment produced by moving a weight forward.
It is also possible to calculate the vertical moment of any part of a ship's structure or the vertical moment of any weight carried on board. In these cases, the axis about which the moments are taken is the ship's base line, or keel, as shown in figure 213. The vertical moment of a 5inch gun on the main deck of a cruiser is equal to 15 tons X 40 feet 600 foot tons.

24

27. Center of gravity. When a group of objects, such as the various parts that make up a ship, have different densities, the center of gravity of the system is not at the geometric center. The center of gravity is then obtained by taking moments of the weights of the various component parts about any point. The sum of the moments is divided by the total weight to give the distance from the axis point to the system's center of gravity. If moments of the various parts are now taken about the center of gravity just found, the moments tending to produce clockwise motion will exactly equal those tending to rotate counterclockwise.
Figure 213. Diagram to illustrate vertical moment of a weight carried on board.
The foregoing facts are illustrated by the example shown in figure 214:
10 pounds X 10 feet  = 100 foot pounds 
30 X 40  = 1,200 foot pounds 
10 X 60  = 600 foot pounds 
_______  ______ 
50 pounds  1,900 foot pounds. 
CG = 1,900/50 = 38 feet from P. 
Taking moments about CG:
Clockwise
10 pounds X 22 feet  = 220 foot pounds 
30X 2  = 60 foot pounds 
_______  ______ 
280  foot pounds. 


Counterclockwise
10 pounds X 28 feet  = 280 foot pounds. 
Net moment  = 0. 
If any part of a system of weights is relocated, the center of gravity (G) of the system shifts. The shift in G will be proportional to the moment of the shifted weight (w) (i.e., w multiplied by the distance moved), and will also depend upon the weight (W) of the entire system. An example is shown in figure 215. For any given moment of a moved weight, the center of gravity will shift more in smaller systems and less in larger ones. The shift in center of gravity from G to G_{1} equals GG_{1}, equals ws/W (where s equals distance w moved), and will be in a direction parallel to the direction of the weight's (w) motion.
When a weight is added to a system of weights, the new center of gravity of the system may be found by taking moments of the old system plus that of the new weight and dividing this total moment by the total weight. An example is shown in figure 216. Applying the principle just cited:
((W X h)+(w X h_{1})) / (W + w)
= New position of center of gravity relative to a base line.
If:
W  = 100 pounds. 
h  = 10 feet. 
w  = 20 pounds. 
h_{1}  = 25 feet. 
Then: 
100 X 10  = 1,000 foot pounds 
20 X 25  = 500 foot pounds 
______  _____ 
120  1,500 
Hence, new center of gravity (G_{1}) = 1,500/120 = 12.5 feet above base line. Center of gravity has moved upward 2.5 feet.

Figure 214. A system of weights. CG is the center of gravity of the system.

25

Another method of accomplishing the same result is to assume that the weight is first placed at the original center of gravity of the system, and then moved to its ultimate location. When the weight is placed at the center of gravity, G remains unchanged. When the weight is moved, however, the following change occurs:
GG_{1} = (ws)/(W + w) = (20 X 15)/(100 + 20) = 2.5 feet.
Figure 215. Diagram to illustrate effect of weight shift upon G. G_{1} is the new center of gravity; s equals distance w is moved.
28. Work. From the standpoint of physics, work is done only when a force moves through a distance (a force that does not move does no work). Work, therefore, is measured in pounds or tons of force multiplied by distance in feet or inches through which the force moves. In lifting a weight of 10 pounds 5 feet vertically, 50 footpounds of work has been done.
Similarly, when a body is rotated, work is done and is measured by the moment causing the rotation multiplied by the angle through which the moment acts. (The angle is measured in radians which may be


obtained by dividing the angle in degrees by 57.3°). Units of work are always force times distance, whether a body is in translation or rotation.
A couple whose moment equals 500 footpounds, moves a body through an angle of 13 degrees. The work done equals:
500 foot pounds X (13° / 57.3°) = 113 foot pounds of work.
Figure 216.
Figure 217.
29. Moment of inertia. Moment of inertia, designated by the letter I, is defined as the moment of a moment. A moment is a force multiplied by a distance; moment of inertia is force times distance times distance, or F X d X d, or F X d^{2}. It will be remembered that in considering moments it was possible to find a moment of a volume or a moment of an area. Likewise there can be a moment of inertia of a volume or area. In our study, the only need will be for moment of inertia of area. The unit of moment of inertia of area is feet^{4}.

26

The value of the moment of inertia of a rectangular area about an axis passing through its center of gravity
Figure 218.


is obtained by dividing the rectangle into a large number of small areas. The moments of inertia of these small areas are equal to their areas times the square of the distance to the axis. An example is diagrammed in figure 217. (i=ay^{2}, where i is moment of inertia of the small area, a the small area, and y the distance from the center of the small area to the axis). This summation is made by the use of the calculus. The moment of inertia of a rectangular area calculated by adding all she small i's is equal to:
I = b^{3}l/12.
The moment of inertia of the rectangle about axis AA shown in figure 218 is equal to:
I = b^{3}l/12 = ((20)^{3} X 6) / 12 = 4,000 feet^{4}
About Axis BB, I = ((6)^{3} X 20) / 12 = 360 feet^{4}.
Moments of inertia for different axes would have different values. Also, different sized or different shaped areas have different moments of inertia. Each can be derived in a manner similar to that indicated above for the rectangle. Most mathematical or engineering handbooks give these valises for various shapes of areas and axes.

27

CHAPTER III
FUNDAMENTALS OF BUOYANCY AND TRANSVERSE STABILITY

31. Foreword. In this Chapter and in the Chapters to follow, two basic properties of a ship are to be studied in detail. One of these properties is buoyancy. The other is stability. Buoyancy may be defined as the
Figure 31. A steel cube, and a box made from the same volume of steel.


ability of a ship to float. Stability is the ability of a ship to stay rightsideup. Separate treatments are to be given transverse and longitudinal stability.
32. Why ships float. Assume that an object of given volume is placed under water. If the weight of this object is greater than the weight of an equal volume of water, the object will sink. It sinks because the force which buoys it up is less than its own weight. However, if the weight of the object is less than the weight of an equal volume of water, the object will rise. It rises because the force which buoys it up is greater than its own weight, and it will continue rising until part of it is above the surface of the water. Here it floats at such a depth that the submerged part of the object displaces a volume of water whose weight is equal to the weight of the object.
For example, the cube of steel in figure 31A is solid, and measures 1 foot by 1 foot by 1 foot. If dropped in water the steel will sink. But hammer it out into a flat plate 8 feet by 8 feet, bend the edges up one foot all around, and the 6 by 6 by 1 box (fig. 31B) that you have formed will float. In fact, it will not only float but will, in calm water, carry an additional 1,800 pounds of weight.

Figure 32. Diagram to indicate how displacement varies with changes in draft.

28

Figure 33. Displacement curve of a cruiser.

It is obviously the submerged part of a floating ship which provides the buoyancy. If the ship is at rest the buoyancy, which is the weight of the displaced water, must be just equal to the weight of the ship. For this reason, the weight of a ship is generally referred to as her displacement.
33. Displacement. Since weight (W) is equal to displacement, it is possible to measure the volume of the underwater body (V) in cubic feet and multiply this volume by the weight of a cubic foot of sea water, in order to find what a ship weighs. This relationship may be written as:
V = 35W.
V = volume of displacement in cubic feet.
W = weight in tons.
35 = cubic feet of sea water per ton.

In dealing with ships it is customary to use the long ton of 2,240 pounds; in this text the word "ton" refers to the long ton exclusively.
Displacement varies with draft. As the draft increases, the displacement increases. This is indicated in figure 32 by the series of displacements shown for successive draft lines on the midship section of a cruiser. The volume of the underwater body for a given draft line can be measured in the drafting room, using graphic or mathematical means. This is done for a series of drafts throughout the probable range of displacements in which a ship is likely to operate. The


values obtained are plotted on a grid, on which feet of draft is measured vertically, and tons displacement horizontally. A smooth line is faired through the points plotted, providing a curve of displacement versus draft, or a displacement curve, as it is generally called. The result is shown in figure 33 for a typical cruiser. It is one of the curves of form which are supplied to every Naval ship under the title displacement and other curves (see Chapt. XI).
To find the displacement when the draft is given, locate the value of mean draft on the draft scale at the left of figure 33. Then proceed horizontally across the diagram to the curve. From this point proceed vertically downward to the scale of tons, where the displacement can be read from the scale. For example, given a mean draft of 22 feet, the displacement found from the curve is approximately 13,400 tons. The curve also can be used in reverse. If given a displacement of 10,000 tons, the draft would be approximately 17.5 feet.
34. Reserve buoyancy. The volume of the watertight portion of the ship above the waterline is the ship's reserve buoyancy. Freeboard is a rough measure of reserve buoyancy, being the distance in feet from the waterline to the weather deck edge. Unless otherwise stated references normally are to mean or midship freeboard. As shown in figure 34, freeboard plus draft always equals the depth of the hull in feet.

29

Figure 34. Diagram to show physical relationships of reserve buoyancy, freeboard, draft, and depth of hull.

When weight is added to a ship, draft and displacement increase in the same amount that freeboard and reserve buoyancy decrease. A substantial amount of reserve buoyancy is, of course, essential to the seaworthiness of a ship. Some approximate values of reserve buoyancy in Naval ships of various types are given below:
BB (new)  55% of displacement. 
CV (large)  130% of displacement. 
CA, CL  100% of displacement. 
DD (2,100 ton)  100% of displacement. 
DD (1,630 ton)  75% of displacement. 
DE  more than 100% of displacement. 
35. Forces at rest. A ship floating at rest in calm water is acted upon by only two sets of forces:
1. The downward force of gravity.
2. The upward force of buoyancy.
The force of gravity is a resultant or composite force, involving the weights of all portions of a ship's structure, and every item of equipment, cargo, and personnel in her. The force of gravity, or the weight of the ship, may be regarded as a single force acting vertically downward through the ship's center of gravity (G). The force of buoyancy is also a resultant or composite force. It results from the pressure of the water on the ship's hull. Horizontal pressures against one side of the ship cancel the horizontal pressure against the opposite side, and only the vertical pressures are effective. The vertical pressures may be regarded as resulting in a single force, the force of buoyancy, acting vertically upward through the center of buoyancy (B). When the ship is floating at rest in


calm water, the forces of buoyancy and gravity are equal and lie in the same vertical line, as indicated in figure 35.
36. Center of buoyancy. The point B (fig. 35) , through which the force of buoyancy always acts, is known as the center of buoyancy. It is the geometric
Figure 35. Forces of buoyancy and gravity. G = center of gravity; B = center of buoyancy.
center of the ship's underwater body. When the ship is upright and at rest, the center of buoyancy lies in the centerline plane and usually is near the midship section. Its vertical height above the keel usually is a little more than half of the draft. As the draft increases, B rises with respect to the keel. Figure 36 shows how different drafts result in different values of KB, the height of the center of buoyancy above the keel.

30

Figure 36. Successive centers of buoyancy (B) for different drafts. K = keel.

The vertical height of B above the keel is determined by graphic or mathematical means. A series of values of KB for successive drafts is obtained, and a curve plotted to show KB versus draft. Sometimes, in plotting this curve of center of buoyancy above base an auxiliary line called a diagonal is used. Figure 37 depicts a KB curve for a typical cruiser, together with the diagonal.
To read KB when the draft is known, enter the scale on the left (fig. 37) at the proper value of draft and proceed horizontally to the diagonal. Then drop vertically downward to the curve. Then proceed horizontally hack to the left, reading KB on the draft scale. Thus, if the ship were floating at a mean draft of 19feet, the KB found from the chart would be 11 feet. The KB curve is one of the curves of form; it is included in the displacement and other curves previously mentioned. The diagonal, which is merely an aid in reading the curve, is supplied wherever required.
37. Ship inclined. Whenever a disturbing force exerts an inclining moment on a ship, causing it to heel over to some angle, the inclined waterline plane changes the shape of the ship's underwater body. In fact, it relocates the underwater volume, shifting its bulk in the direction of the heel. This causes the center of buoyancy (B) to leave the ship's centerline and shift in the direction of the heel. As a result, the lines of action of the forces of buoyancy and gravity separate, and in so doing, exert a moment on the ship, which normally tends to restore the ship toward an even keel. This righting moment, as shown in figure


38, consists of two equal and opposite forces, each of W tons magnitude, separated by a distance GZ, which constitutes the lever arm of the moment. The
Figure 37. Curve of center of buoyancy above base, with diagonal.
ship in figure 38 is stable because B has shifted far enough to place the buoyant force where it exerts a restoring effect. However, it is possible for conditions

31

Figure 38. Diagram to illustrate development of righting moment when a stable ship inclines.
Figure 39. Diagram to illustrate development of an upsetting moment when an unstable ship inclines.

to exist which do not permit B to move far enough to place the buoyant force outboard of the force of gravity. The moment produced in such a case will tend to upset the ship, rendering it unstable. Figure 39 shows an


unstable ship in which the relative positions of B and G have produced an upsetting moment.
A moment is the product of a force times a distance. In the case of two equal and opposite forces separated

32

Figure 310. Diagram to indicate initial location of the metacenter.

by a distance, the moment produced is a couple, measured by one of the forces times the distance which separates them. Righting moment in a ship is therefore the product of the force of buoyancy (displacement) times the distance, GZ, which separates the forces of buoyancy and gravity. Or it might be expressed as the force of gravity (weight) times GZ. The distance GZ is known as the ship's righting arm. In mathematical terms:
Righting moment  = W X GZ (expressed in foottons). 
Where: W  = displacement in tons. 
GZ  = righting arm in feet. 
38. Initial stability. The term. "initial," as used in the discussions which follow, refers to the first few degrees of heel from the upright position. Specifically, initial stability refers to the tendency to return to the upright, which the vessel develops as it rolls from 0° up to about 10°. (Note that the word "initial" in this text does not refer either to time or to condition, but only to angle of heel.)


Figure 310 shows two lines of buoyant force. One of these represents the ship on an even keel. For angles up to about 10° they intersect to establish the original location of the metacenter (M). By definition, a ship's metacenter is the intersection of two successive lines of action of the force of buoyancy as the ship heels through a very small angle. When angles of heel are larger than about 10°, M moves, the path of its movement being a curve. However, it is the initial position of the metacenter that is most useful in studying stability; in succeeding discussions "metacenter" will invariably refer to initial metacenter. The distance from the center of buoyancy (B) to M when the ship is on an even keel is BM, the metacentric radius, which is used in stability investigations.
39. Metacentric height. The distance from the center of gravity (G) to the metacenter (M) is known as the ship's metacentric height. The symbol for metacentric height is GM. In figure 311, a ship is heeled over through a small angle (this has been exaggerated in the drawing) , establishing a metacenter at M. Here, the ship's righting arm is GZ, which forms a triangle

33

Figure 311. Diagram to illustrate a stable condition. G is below M.

between the points G, M, and Z. In triangle GZM, the angle of heel is at M (the sides GM and ZM are respectively perpendicular to the waterline upright and the waterline inclined). It is evident that for any given angle of heel, there will be a definite relation between GM and GZ; viz., GZ = GM sin θ. Double GM, and GZ doubles. Thus, GM acts as a measure of GZ, the righting arm. For the first few degrees, GM does not vary with angle of heel, and it is this value of GM which is used as a measure of initial stability.
GM is not only a measure of the size of GZ, but is also an indicator of whether the ship is stable or unstable. If M is above G, the metacentric height is positive, the moments which develop are righting moments, and the ship is stable. But if M is below G, the metacentric height is negative, the moments which develop are upsetting moments, and the ship is unstable. An example of the latter circumstance is shown in figure 312.
310. Influence of metacentric height. If the metacentric height of a given ship is large, large righting arms develop at small angles of heel. Such a ship is stiff, and will tend to roll with a "snap." But if GM is small, the righting arms that develop will be smaller for any given angle. Such a ship is tender and will


roll slowly. A high value of GM is desirable in a Naval ship because it bespeaks ability to sustain damage without fatal loss of stability. However, a lower GM is desirable because a slow easy roll makes for accurate gunnery. Consequently, the value of GM in a Naval ship is a matter of compromise.
The following are some typical values of transverse metacentric height for Naval ships of various types in operating conditions:
BB  8 to 10 feet.  CVE  3 to 6 feet. 
CV  5 to 11 feet.  DD (new)  3 to 4 feet. 
CVL  4 to 6 feet.  DD (old)  1 to 3 feet. 
CA  4 to 6 feet.  DE  4 to 5 feet. 
CL  3 to 6 feet.  AK  1 to 6 feet. 
311. Relationships of metacentric height (GM). Inasmuch as GM is a measure of initial stability, it is desirable that damage control officers understand GM and the factors which produce and vary it. Metacentric height is the distance between two points whose locations are established by independent causes. These two points are M, the metacenter, and G, the center of gravity. The relation between M and G may be expressed by a formula:
GM = KM  KG.
KG is the height of the ship's center of gravity

34

Figure 312. Diagram to illustrate an unstable condition. G is above M.

above the keel (base line) (see fig. 313). KG is determined by the amounts and vertical heights of all the weights which go to make up the ship's structure, machinery, and loading. If topside weights are predominant, KG will be large; however, if low weights are predominant, KG will be relatively small.
KM is a property of the underwater body, and depends upon the movement of B, which shifts the line of action of the buoyant force. The location of B for a given angle of heel depends upon the size and shape of the underwater body, and these in turn depend upon draft, as previously has been indicated. The development of KM may be expressed as:
KM= KB + BM.
It was shown in Article 36 how KB is developed and furnished as a curve of KB versus draft. It remains now to find BM, the metacentric radius.
312. Computation of metacentric radius. The metacentric radius, BM, may be found for any given draft by calculation from the formula:
BM = i/V.
In this expression, I is the moment of inertia of the ship's waterline plane area about its own foreandaft centerline. V is the ship's volume of displacement in cubic feet. (V = 35W.) In Article 29 it was indicated that the moment of inertia of any rectangular


area about its own centerline may be calculated from:
i = b^{3}l / 12.
Figure 313. Diagram to show relative positions of K and G, and initial positions of B and M.
This expression may be used in finding the moment of inertia of an irregular area like the ship's waterplane, provided the area is symmetrical about its centerline, and can be divided up into a series of small rectangles, as in figure 314.
Thus, athwartship lines are used to divide the ship's waterline plane area into a series of adjacent rectangles (fig. 314). The breadth (b) of each of these rectangles is the average beam of the ship at that location,

35

Figure 314.

and the length (l) of each rectangle is the foreandaft spacing of the athwartship lines. The smaller this spacing, the more accurate the result when the formula above is used to find i for each rectangle, and all the i's are added to give I for the waterplane. In other words:
I = i_{1} + i_{2} + i_{3} + i_{4}... etc.
Since b is in feet and occurs three times, l is in feet and occurs once, and the factor 12 is a pure number, moment of inertia of an area is in units of feet^{4}. The value of BM in feet is the result of dividing feet^{4} by feet^{3}. Both I and V change with a change of draft. As draft increases, the size of the waterline plane will either increase or decrease, and this results in a new I. A greater draft produces a greater displacement, and V gets larger. The net effect on BM of an increased draft usually is a reduction at normal waterlines, because I changes little compared to V in the denominator of the expression i/V. Making use of the principles explained in this Article, the design activity calculates a series of values of BM for successive drafts.
313. Curve of transverse metacenter above keel. Having obtained BM as explained in the preceding Article, its value is added to the corresponding KB for a given draft. This is done for a series of drafts throughout the probable range of displacements.
KM = KB + BM.
A curve is then plotted of KM versus draft. It is one of the curves of form as supplied to the ship in the sheet of displacement and other curves. This KM curve usually is labelled transverse metacenter above


base line, and is most frequently so plotted as to utilize the same diagonal as the KB curve, although it may be plotted separately with its own scale. Figure 315 is a typical KM curve using a diagonal. The corresponding KB curve is shown as a broken line.
Figure 315. KM curve and diagonal, with corresponding KB curve.

36

To find KM on this curve, when the mean draft is given, enter the draft scale at the left at the given value, proceed horizontally to intersect the diagonal, then vertically upward to intersect the curve. From this point proceed horizontally back to the left and read KM on the draft scale. For example, given a mean draft of 23 feet, KM is 27.1 feet (see fig. 315).
When the KM and KB curves are plotted with the same diagonal, the value of BM is the distance between the two curves. Thus, for a draft of 23 feet, EM is found to be 14.0 feet in figure 315.
314. Summary. In the foregoing Articles of this Chapter it has been shown that ships float because they displace a certain amount of water which buoys them lip, and that they have a substantial reserve of buoyancy. It has also been shown that ships tend to remain upright because in heeling they develop a righting


moment. The stability of a ship can be measured by the righting moment which it generates at any given angle. Metacentric height is a measure of stability as long as two conditions obtain:
1. The angle of heel is less than about 10°.
2. The displacement of the ship remains constant.
In the chapters which follow, the uses and limitations of GM, BM, KM, as well as other properties of stability, will be developed and analyzed. Bear in mind that whenever comparisons are made, the intention is to compare the effect of different conditions on any one given ship, and not to compare one ship with another especially not to compare ships of different types. For example, a destroyer cannot be changed into an aircraft carrier, but there is much that can be done to render it either a seaworthy or an unseaworthy destroyer.

