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1-1 INTRODUCTION Ships are built for a wide variety of purposes, but all must meet certain fundamental requirements. They must have reserve buoyancy to enable them to carry their designed loads and resist damage, stability to resist environmental forces or damage, and strength to withstand the stresses imposed on their structure by their own weight, cargo, stores, and the sea. The following discussion provides the salvage engineer with the basics of surface ship construction, stability, and strength. Submarine construction and stability are discussed in the U.S. Navy Ship Salvage Manual, Volume 4 (S0300-MAN-A6-040). Vessels are built to construction specifications based on stability and strength requirements, that are, in turn, based on intended service. Publicly owned vessels (Navy, Coast Guard, etc.) are built to government specifications. Most Navy ships are built to the General Specifications for Ships (GENSPECs), published by the Naval Sea Systems Command (NAVSEA), although some auxiliaries are built to commercial specifications. Stability standards for Navy ships are established by Design Data Sheet (DDS) 079 issued by the Naval Ship Engineering Center. Construction rules and stability standards for commercial vessels are established by classification societies, the International Maritime Organization (IMO), and government regulations for the country of registry; the American Bureau of Shipping (ABS) and United States Coast Guard (USCG) establish and enforce construction rules and stability standards for U.S. vessels. The U.S. rules are often based on IMO standards. The U.S. Maritime Administration (MARAD) may place additional requirements on ships built with Federal financial assistance. MARAD also produces standard designs for certain types of merchant ships. Stability and construction standards are discussed in Appendix C. There is a basic difference in the way naval architects and salvage engineers approach the problems of ship stability and strength. Naval architects, as designers, divide the subject into examinations of intact and damage conditions. The stability and strength of a proposed design is examined in normal operating, or intact, conditions, which must, as matter of course, include free liquid surfaces in tanks. Damage stability analysis examines a ship design in various hypothetical conditions of damage that include breaches in the immersed hull. The salvage engineer on the other hand, deals with damaged stability and strength, i.e., ships in conditions of known or identifiable damage, that may or may not include breaches in the immersed hull. There is a subtle distinction between damage and damaged stability. A salvage engineer doesn't really deal with damage stability, or for that matter, with intact stability either. He deals with damaged stability, and conditions that can reasonably be attained from the initial damaged condition. While the salvage engineer also examines hypothetical conditions, those conditions usually have as a point of departure an initial damaged condition. This chapter discusses ship stability in light of those factors that provide and enhance stability, and those that impair or degrade. Those familiar with standard naval architecture texts may feel that this handbook's treatment of the subject glosses over the distinction between intact and damage stability. This is true to some extent, because in the main, the distinction just doesn't matter to salvage engineers; they deal with stability--good, bad, or indifferent--as they find it. The fact that free surface occurs in intact ships does not obscure the fact that it always impairs stability. 1-2 HULL FORM A ship's hull is a complex geometric form that can be defined accurately by mapping its surface in a three-dimensional orthogonal coordinate system. If a Cartesian coordinate system is used, conventions usually set the Z-axis vertical, the X-axis longitudinal and the Y-axis athwartships. Principal dimensions are measured along these axes. The hull form can be shown in two dimensions by a series of curves formed by the intersection of the hull surface with planes parallel to these axes. The hull form, chosen by the designer, controls the stability and performance characteristics of the ship in its normal environments. 1-2.1 Location of Points Within a Ship. Because a ship is a three-dimensional mobile object, references within the ship itself must be established for locating points in, on, and about the ship. The position of any point in the ship can be described by measuring its position from reference planes or lines. The following planes are most commonly used:

· · · ·

Centerplane ­ A vertical plane passing fore and aft down the center of a ship; the plane of symmetry for most hull forms. Design Waterplane ­ A horizontal plane at which the hull is designed to float. Midship Plane ­ A transverse, vertical plane perpendicular to both the centerplane and the design waterplane, located at the midpoint of the molded hull length between perpendiculars on the design waterplane. Baseplane ­ A horizontal plane passing through the intersection of the centerplane and the midships plane, or through the lowest point of the molded hull.



The intersections of the reference planes with specified locations on the hull create additional reference lines and points:

· · · · ·

Forward Perpendicular (FP) ­ A vertical line through the intersection of the stem and the design or load waterline (DWL, LWL). After Perpendicular (AP) ­ A vertical line at or near the stern of the ship. In naval practice, the after perpendicular passes through the after extremity of the design waterline; in commercial practice, the after perpendicular usually passes through the rudder post, or the centerline of the rudder stock if there is no rudder post. Midship Section ( or MS) ­ An intersection of the midship plane with the molded hull.

Centerline (C or CL) ­ The projection of the centerplane in plan or end views of the hull. L Baseline (B or BL) ­ The projection of the baseplane in the side or end views of the hull. In ships with design drag where the L baseline passes through the intersection of the midships section and the keel, parts of the hull will be below the baseline. For ships with flat-plate keels that float on an even keel, the baseline, bottom of the molded surface, and top of the keel plate coincide; if the keel plate is an outside strake (lapped over the adjacent strakes rather than butt-welded to them), the top of the flat-plate keel is below the bottom of the molded surface by the thickness of the strakes on each side of it (the garboard strakes). In vessels with hanging bar keels, the top of the keel coincides with the bottom of the molded surface.

1-2.2 Location of Points. The position of any point in the ship can be described by its:

· · ·

Height above the baseplane or keel. Athwartships position relative to the centerplane. Longitudinal position relative to the midship section or to one of the perpendiculars.

1-2.3 Ship Dimensions. Molded dimensions, lines, etc., describe the fair surface defined by the framing and are principally of use to the shipbuilder. Displacement dimensions and lines describe the surfaces wetted by the sea and are of principal interest to the naval architect and salvage engineer in determining stability and performance characteristics. Extreme dimensions, such as extreme breadth, account for projections such as overhanging decks, fender rails, etc. Molded dimensions differ from displacement dimensions by the plating, planking, or sheathing thickness. In steel ships, this difference usually amounts to less than one percent of the total displacement. Displacement dimensions are not usually tabulated as such; if desired, they are deduced by adding plating thickness to molded dimensions, or deducting appendage measurements from extreme dimensions. The principal dimensions of a ship are length, beam, and depth. Two other important dimensions are draft and freeboard. Figure 1-1 shows the principal dimensions of a ship.


Length between perpendiculars (L, LBP or Lpp), is used for the calculation of hydrostatic properties. Length overall (LOA) is the maximum length of the vessel, including any extensions beyond the perpendiculars, such as overhanging sterns, raked stems, bulbous bows, etc. Length on the waterline (LWL or LWL) may or may not be the same as LBP, depending on the location of the perpendiculars; tabulated LWL is usually taken on the design waterline. Beam or breadth (B) is the width of the ship. Molded beam is measured amidships or at the widest section from the inside surface of the shell plating. Maximum beam or extreme breadth is the breadth at the widest part of the ship, and is equal to the molded breadth plus twice the plating thickness plus the width of fenders, overhanging decks, or other solid projections. Draft (T) is the vertical distance between the waterline and the deepest part of the ship at any point along the length. Drafts are usually measured to the keel and are given as draft forward (Tf), draft aft (Ta) and mean draft (T or Tm). A ship's forward and after draft marks are seldom at the perpendiculars and mean draft is not necessarily amidships; the slight errors introduced by using drafts at these points can be discounted if trim is not extreme. Molded drafts are measured from the molded baseline, while keel drafts are measured from a horizontal line though the lowest point on the bottom of the keel extended to intersect the forward and after perpendiculars. Navigational or extreme drafts indicate the extreme depth of sonar domes, propellers, pit swords, or other appendages which extend below the keel, and are therefore not used to calculate hydrostatic properties. Draft scales for keel drafts are usually placed on both sides of the ship at each end as near as practical to the respective perpendiculars. The external draft marks are generally Arabic numerals, with height and spacing arranged so that the vertical projection on the vessel of the numeral heights and vertical spacing between numerals are both six inches. The draft figures are placed so that the bottom of the figure indicates the keel draft. Drafts can thus be read to the nearest quarter-foot (3 inches) in relatively calm waters. Freeboard (F) is the vertical distance between the waterline and the uppermost watertight deck. Depth (D) is the vertical distance between the baseline and the uppermost watertight deck and is the sum of freeboard and draft. Molded depth is measured from the top of the outer keel to the underside of the main or freeboard deck at the side. Depending on hull form and ship's attitude, both freeboard and depth can vary along the length of the ship. Unless otherwise specified, tabulated values for depth and freeboard are usually taken at midships or at the point of minimum freeboard.

· ·

· ·



1-2.4 Lines. The shape of a ship is B developed to meet specific requirements of speed, seakeeping ability, and capacity for the intended use of the vessel. The shape MIDSHIPS of the hull is defined by the plan shapes SECTION produced by the intersection of three D families of orthogonal planes and the hull surface. Most hulls are symmetrical about the vertical plane of the centerline. The T intersection of the ship's molded hull surface with this and parallel planes is called a buttock, or buttock line. The term C L buttock was formerly applied only to the AP FP portions of these lines aft of midships; the DWL forward portions were called bow lines. A plane parallel to the baseplane and LBP perpendicular to the centerline plane is a LOA waterplane. The intersection of waterplanes and the molded hull are called Figure 1-1. Principal Dimensions. waterlines (WL). The intersection of transverse planes perpendicular to both waterplanes and buttocks are termed sections. The superimposed sections (body plan), waterplanes (halfbreadth plan), and buttocks (sheer plan) form the lines plan or lines drawing for the ship. Like other engineering drawings, the lines plan is composed of views from ahead or astern, from above, and from the starboard side. Figure FO-1 is the lines plan for an FFG-7 Class ship. The lines plans for steel ships usually show the molded surface. For surface ships, the molded surface is the inside of the shell plating, while the molded surface for submarines is the outside of the hull plating. For vessels with hanging bar keels, the line of the bottom of the keel is shown on the sheer plan to complete the lower contour of the vessel; the keel line is not usually shown for vessels with flat-plate keels because it lies so near the line of the bottom of the molded surface. Because of the greater hull thickness, wooden ships may have separate molded and displacement lines drawings. 1-2.4.1 The Body Plan. The body plan shows the outline of the transverse sections of a ship at equally spaced stations or ordinates along the length of the ship. The distance between perpendiculars is commonly divided into 10 or 20 equal spaces by 11 or 21 stations, including the forward and after perpendiculars. More or fewer stations may be used depending on the complexity of the hull shape. Half-spaced stations may be used when the shape of the hull form changes rapidly, such as near the bow and stern. As the transverse sections are normally symmetrical about the centerline, it is conventional to show only half sections with the forward stations on the right and after stations on the left. Stations are numbered from forward aft, with the forward perpendicular as station zero on U.S. Navy ships. Stations forward of the forward perpendicular (if any) may be designated by negative numbers or letters. Commercial vessels, particularly foreign-built vessels, commonly number stations from aft forward, with the after perpendicular as zero. 1-2.4.2 Halfbreadth Plan. Due to symmetry, it is conventional to show only half of the waterplanes in a halfbreadth plan. Waterlines are designated by their height above the baseline. The waterlines define the shape and area of the waterplane and are spaced closely enough to accurately define the waterplane at any draft. 1-2.4.3 Sheer Plan. Superimposed buttocks form the sheer plan. They are spaced as necessary to adequately define the ship's form.



1-2.4.4 Descriptive Terms. Certain other geometric concepts are useful in describing a ship's form. Figure 1-2 illustrates some of the following definitions:



Parallel midbody ­ In many modern ships, the form of the hull's transverse section in the midships region extends without change for some distance fore and aft. This is called parallel midbody and may be described as extensive or short, or expressed as a fraction of the ship's length. Even in ships without parallel midbody, the form of the fullest transverse section changes only slightly for small distances forward or aft. Forebody ­ The portion of the hull forward of the midship section. After body ­ The portion of the hull abaft the midship section. Entrance ­ The immersed portion of the hull forward of the section of greatest immersed area (not necessarily amidships) or forward of the parallel midbody.





· · ·





Figure 1-2. Hull Form Nomenclature.

· ·

Run ­ The immersed portion of the hull aft of the section of greatest immersed area or aft of the parallel midbody. Deadrise ­ The departure of the bottom from a transverse horizontal line measured from the baseline at the molded breadth line as shown in Figure 1-2. Deadrise is also called rise of floor or rise of bottom. Deadrise is an indicator of the ship's form; fullbodied ships, such as cargo ships and tankers, have little or no deadrise, while fine-lined ships have much greater deadrise along with a large bilge radius. Where there is rise of floor, the line of the bottom commonly intersects the baseline some distance from the centerline, producing a small horizontal portion of the bottom on each side of the keel. The horizontal region of the bottom is called flat of keel, or flat of bottom. While any section of the ship can have deadrise, tabulated deadrise is normally taken at the midships section. Knuckle ­ An abrupt change in the direction of plating or other structure. Chine ­ The line or knuckle formed by the intersection of two relatively flat hull surfaces, continuous over a significant length of the hull. In hard chines, the intersection forms a sharp angle; in soft chines, the connection is rounded. Bilge radius ­ The outline of the midships section of very full ships is very nearly a rectangle with its lower corners rounded. The lower corners are called the bilges and the shape is often circular. The radius of the circular arc is called the bilge radius or turn of the bilge. The turn of the bilge may be described as hard or easy depending on the radius of curvature. If the shape of the bilge follows some curve other than a circle, the radius of curvature of the bilge will increase as it approaches the straight plating of the side and bottom. Small, high-speed or planing hulls often do not have a rounded bilge. In these craft, the side and bottom are joined in a chine. Tumblehome ­ The inward fall of side plating from the vertical as it extends upward towards the deck edge. Tumblehome is measured horizontally from the molded breadth line at the deck edge as shown in Figure 1-2. Tumblehome was a usual feature in sailing ships and many ships built before 1940. Because it is more expensive to construct a hull with tumblehome, this feature is not usually incorporated in modern merchant ship design, unless required by operating conditions or service (tugs and icebreaking vessels, for example). Destroyers and other high-speed combatants are often built with some tumblehome in their mid and after sections to save topside weight.

· · ·





Flare ­ The outward curvature of the hull surface above the waterline, i.e., the opposite of tumblehome. Flared sections cause a commensurately larger increase in local buoyancy than unflared sections when immersed. Flaring bows are often fitted to help keep the forward decks dry and to prevent "nose-diving" in head seas. Camber ­ The convex upwards curve of a deck. Also called round up, round down, or round of beam. In section, the camber shape may be parabolic or consist of several straight line segments. Camber is usually given as the height of the deck on the centerline amidships above a horizontal line connecting port and starboard deck edges. Standard camber is about one-fiftieth of the beam. Camber diminishes towards the ends of the ship as the beam decreases. The principal use of camber is to ensure good drainage in calm seas or in port, although camber does slightly increase righting arms at large angles of inclination (after the deck edge is immersed). Not all ships have cambered decks; ships with cambered weather decks and flat internal decks are not uncommon. Sheer ­ The rise of a deck above the horizontal measured as the height of the deck above a line parallel to the baseline tangent to the deck at its lowest point. In older ships, the deck side line often followed a parabolic profile and sheer was given as its value at the forward and after perpendiculars. Standard sheer was given by: sheer forward = 0.2L + 20 sheer aft = 0.1L + 10



where sheer is measured in inches and L is the length between perpendiculars in feet. Actual sheer often varied considerably from these standard values; the deck side profile was not always parabolic, the lowest point of the upper deck was usually at about 0.6L, and the values of sheer forward and aft were varied to suit the particular design. Many modern ships are built without sheer; in some, the decks are flat for some distance fore and aft of midships and then rise in a straight line towards the ends. Sheer increases the height of the weather decks above water, particularly at the bow, and helps keep the vessel from shipping water as she moves through rough seas. Some small craft and racing yachts are given a reverse or hogged sheer to give headroom amidships without excessive depth at bow and stern.


Rake ­ A departure from the vertical or horizontal of any conspicuous line in profile, defined by a rake angle or by the distance between the profile line and a reference line at a convenient point. Rake of stem, for example, can be expressed as the angle between the stem bar and a vertical line for ships with straight stems. For curved stems, a number of ordinates measured from the forward perpendicular are required to define the stem shape. Ships designed so that the keel is not parallel to the baseline and DWL when floating at their designed drafts are said to have raked keels, or to have drag by the keel. Cut-up ­ When a keel departs from a straight line at a sharp bend, or knuckle, the sloping portion is called a cut-up. High-speed combatants usually have a long cut-up aft (extending 13 to 17 percent of LWL) to enhance propeller performance and maneuverability. Ice-breaking vessels often have a cut-up forward to allow the ship to ride up on the ice. Deadwood ­ Portions of the immersed hull with significant longitudinal and vertical dimensions, but without appreciable transverse dimensions. Deadwood is included in a hull design principally to increase lateral resistance or enhance directional stability without significantly increasing drag when moving ahead. Sailing craft require deadwood to be able to work to windward efficiently. Skegs or fins are fitted on barges to give directional stability. Deadwood aft is detrimental to speed and quick maneuverability and is minimized by use of cut-up sterns in high-speed combatants and by arched keels or sluice keels (with athwartships apertures) in tugs and workboats. Appendages ­ Portions of the vessel that extend beyond the main hull outline or molded surface. Positive appendages, such as rudders, shafts, bosses, bilge keels, sonar domes, etc., increase the underwater volume, while negative appendages, such as bow thruster tunnels and other recesses, decrease the underwater volume. Shell plating, lying outside the molded surface, is normally the largest single appendage, and often accounts for one-half to two-thirds of the total appendage volume. Appendages generally account for 0.2 to 2 percent of total immersed hull volume, depending on ship size, service, and configuration. Paragraph 1-4.10.2 discusses methods for estimating appendage displacement. Hull Surfaces ­ Hull surfaces are either warped, consisting of smoothly faired, complex three-dimensional curves, developed, consisting of portions of cylinders or cones, or flat. Hydroconic hulls are built up of connected flat plates rather than plates rolled to complex curves. Hydroconic construction lowers production costs and may simplify fitting patches to a casualty.







1-2.5 Coefficients of Form. Coefficients of form are dimensionless numbers that describe hull fineness and overall shape characteristics. The coefficients are ratios of areas or volumes for the actual hull form compared to prisms or rectangles defined by the ship's length, breadth, and draft. Since length and breadth on the waterline as well as draft vary with displacement, coefficients of form also vary with displacement. Tabulated coefficients are usually based on the molded breadth and draft at designed displacement. Length between perpendiculars is most often used, although some designers prefer length on the waterline. Coefficients of form can be used to simplify area and volume calculations for stability or strength analyses. As hull form approaches that of a rectangular barge, the coefficients approach their maximum value of 1.0. The following paragraphs describe the most commonly used coefficients. Table 1-1 gives sample coefficients for different type ships. 1-2.5.1 Block. The block coefficient (CB) is the ratio of the immersed hull volume () at a particular draft to that of a rectangular prism of the same length, breadth, and draft as the ship: CB = BTL where: B T L = = = = immersed volume, [length3] beam, [length] draft, [length] length between perpendiculars, [length]

Table 1-1. Typical Coefficients of Form.

Block Coefficient CB Navy Ships Aircraft Carrier (CV-59 Class) Battleship (BB-61 Class) Cruiser (CGN-38 Class) Destroyer (DD-963 Class) Frigate (FFG-7 Class) Replenishment Ship (AOR-1 Class) Salvage Tug (ARS-50 Class) 0.578 0.594 0.510 0.510 0.470 0.652 0.542 Commercial Vessels General Cargo (slow-speed) General Cargo (medium-speed) General Cargo (high-speed) Tanker (35,000-ton DWT) Large Tanker (76,000-ton DWT) VLCC (250,000-ton DWT) Container Ship RO/RO Ore Carrier Great Lakes Bulk Carrier Passenger Liner Barge Carrier Large Car Ferry Ocean Tug, Trawler Offshore Supply Vessel Harbor Tug Ocean Power Yacht (250 ft LWL) 0.800 0.700 0.576 0.757 0.802 0.842 0.600 0.568 0.808 0.900 0.530 0.570 0.530 0.550 0.660 0.585 0.565 0.992 0.980 0.972 0.978 0.997 0.996 0.970 0.972 0.995 0.995 0.956 0.950 0.910 0.833 0.906 0.892 0.938 0.880 0.810 0.695 0.845 0.874 0.916 0.740 0.671 0.883 0.950 0.690 0.820 0.680 0.850 0.892 0.800 0.724 0.984 1.000 0.810 0.850 0.770 0.981 0.908 0.729 0.694 0.780 0.760 0.750 0.777 0.791 Midship Coefficient CM Waterplane Coefficient CWP

Type Ship

1-2.5.2 Midship Section. The midship section coefficient (CM) is the ratio of the area of the immersed midship section (Am) at a particular draft to that of a rectangle of the same draft and breadth as the ship: CM = where: AM = B T = = area of the immersed portion of the midships section, [length2] beam, usually taken at the waterline, [length] draft, [length] AM BT

If the vessel has bulges or blisters below the waterline, CM may be greater than 1.

Coefficients for commercial vessels are typical values; coefficients for specific ships will vary. Coefficients of form for U.S. Navy ships can be obtained from Naval Sea Systems Command, Code 55W. Coefficients for many merchant vessels are available from the National Cargo Bureau, telephone (212) 571-5000. The builder's hull number or name and type of vessel must be provided to access the data files.

1-2.5.3 Waterplane. The waterplane coefficient (CWP) is the ratio of the area of the waterplane (AWP) to that of a rectangle of the same length and breadth as the ship: CWP = AWP LWL B

where: AWP = B LWL area of the waterplane, [length2] = beam, [length] = length on the waterline, [length]



1-2.5.4 Prismatic. The longitudinal prismatic coefficient (CP) is the ratio of the immersed volume to the volume of a prism with length equal to the ship's and cross-section area identical to the midship section: CP = C = B AM L CM

where: = AM = L = immersed volume, [length3] area of the immersed portion of the midships section, [length2] length between perpendiculars, [length]

If length between perpendiculars and length on the waterline are equal (as they are for Navy ships), the prismatic coefficient is equal to the block coefficient divided by the midships section coefficient. The prismatic coefficient thus indicates the longitudinal distribution of the underwater volume of a ship's hull. For a given length, breadth, draft, and displacement, a low (fine) CP indicates a hull with fine ends. A large (full) value for CP indicates a hull with relatively full ends. For this reason, the prismatic coefficient is sometimes called the longitudinal coefficient. The vertical prismatic coefficient (CVP) is the ratio of the immersed hull volume to the volume of a prism having a length equal to the ship's draft and a cross section identical to that of the waterplane: CVP = where: AWP = T immersed volume, [length3] area of the waterplane, [length2] = draft, [length] = AWP T






The vertical prismatic coefficient is equal to the block coefficient divided by the waterplane coefficient and indicates the vertical distribution of the underwater volume. A full CVP indicates a concentration of volume near the keel and a fine CVP, a concentration nearer the waterline.





1-2.6 Ship Proportions. Throughout this handbook and many naval architecture texts, relationships and approximations for 0 various hydrostatic and stability parameters 0 450 600 750 900 are given as applicable to ships of ordinary, LBP, FT or normal form. With the broad range of FROM ELEMENTS OF SHIP DESIGN, R. MUNRO-SMITH, 1975. ship type, size, and service requirements, normal form is best defined by a range of coefficients and dimension ratios. Table Figure 1-3. Approximate Ship Proportions. 1-1 gives typical coefficients of form and Figure 1-3 shows approximate linear relationships between length, beam, depth, and service draft. The relationships given below, adapted from R. Munro-Smith's Elements of Ship Design, and deadweight coefficients (defined in Paragraph 1-3.3), are used to estimate ship dimensions during preliminary design and can help to determine whether a hull should be considered normal. Dimensional Ratios: Ship type General Cargo Tankers VLCC L/B 6.3 to 6.8 7.1 to 7.25 6.4 to 6.5 B/T 2.1 to 2.8 2.4 to 2.6 2.4 to 2.6 T/D 0.66 to 0.74 0.76 to 0.78 0.75 to 0.78



Maximum block coefficient for service conditions: CB 1.00 0.23 Vk L CB 1.00 0.19 Vk L CB 1.00 where: Vk = L = Beam range:

L 9 L 9 L 9

(general cargo ships)

(tankers, bulk carriers)


Vk L


service speed, knots length between perpendiculars, ft

+ 20 ft B + 15 ft B + 39 ft B B

L 9 L 9 L 9 L 5

+ 25 ft (cargo ships) + 21 ft (tankers, bulk carriers) + 50 ft, or 46 ft (VLCC)

where: B = beam, ft Beam to length relationship: B = Ln where B and L are given in feet and: n 0.61 to 0.64 for general cargo ships 0.66 to 0.68 for VLCC

Length-beam product to deadweight relationship: 0.0093LB = where: L B DWT T C = = = = = = = length between perpendiculars, ft beam, ft deadweight, lton draft, ft 0.85 to 2.0 for general cargo ships 0.525 to 0.590 for tankers 0.446 to 0.459 for VLCC DWT C T

1-2.7 Offsets. The hull form can be described in tabular format by a set of measurements known as offsets. Offsets are distances measured from the centerline to the side of the ship at each station and waterline. Molded offsets are measured to the molded surface (inside of shell plating for steel surface ships); displacement offsets are measured to the outer hull surface. Offsets define the hull proper, without appendages. Supplementary appendage offset tables are sometimes available. Molded or displacement offsets are usually presented in a table in the form feet-inches-eighths. The table of offsets for an FFG-7 Class ship shown in Figure FO-1 is typical. The waterline halfbreadth entry for station 4 at the 8' 0" waterline reads 10 - 2 - 3 indicating 10 feet, 23/ 8 inches. Since the station spacing is given as 20.4 feet on the plan (LBP = 408 feet, 408/20 stations = 20.4), this offset precisely locates the point on the skin of the ship 81.6 feet from the forward perpendicular (4 × 20.4), eight feet above the baseline and 10 feet 23/ 8 inches from the centerline. Lines drawings can be constructed from tables of offsets. Of more use to the salvor is the fact that offsets can be obtained from body or halfbreadth plans and used to determine ship volumes and areas by numerical integration (described in Paragraph 1-4). Offsets can be scaled from arrangement drawings, or in the worst case, measured on site.



1-2.8 Wetted Surface. The area of all or part of a ship's hull's wetted surface is important to hydrodynamic resistance and pressure force calculations. Wetted surface multiplied by average shell thickness calculates shell volume to be added to the molded volume to determine total displacement. The area of complex hull surfaces can be calculated by numerical integration from offsets or the shell expansion plan, but this is a tedious and time-consuming task. Wetted surface can be estimated by one of the following empirical relationships: Denny-Mumford Formula: AS = 1.7L T Taylor's formula: A S = C D L Haslar formula for fine-lined ships: AS = 3.3


= 1.7L T T

Table 1-2. Taylor's Coefficient.



3.5 4.0 5.0 8.0 9.0 10.0


16.0 16.5 17.5 20.5 21.3 22.2 23.0 23.8 24.5 25.1 26.3 27.2

L 2.09 1/3

11.0 12.0 13.0 14.0

where: AS L T B CB D C = = = = = = = =

16.0 18.0

wetted surface, ft2, at mean draft T, ft length between perpendiculars, ft (immersed length) displacement volume, ft3 = CB LBT mean draft, ft molded beam, ft block coefficient displacement, ltons a coefficient, ranging from 15.2 to 16.0 for vessels with 0.8 Cm 0.98 and 2.5 B/T 3.5. For shallow draft vessels, C is expressed as a function of B/T in Table 1-2.

1-3 DISPLACEMENT AND BUOYANCY A body immersed in a fluid will experience an upward force equal to the weight of the volume of fluid displaced. This force of buoyancy is the resultant of the normal pressures exerted by the fluid on each element of the immersed body's surface. Buoyancy is opposed by the downward force of gravity, or the object's weight. In order for equilibrium to exist, the two forces must be balanced. An object heavier than an equivalent volume of water has negative buoyancy and will sink until it encounters a solid object or denser liquid, where its apparent weight is decreased by the buoyant force acting on it. Similarly, an object less dense than water will exhibit positive buoyancy and will float with an immersed volume such that the weight of the displaced water exactly equals the object's weight. Deeper immersion requires the application of force. An object whose density equals that of the surrounding water is said to have neutral buoyancy and will float at whatever depth it is placed. A ship floats by enclosing large volumes of less dense material, principally air, in a watertight skin so that its average density is less than that of the surrounding water. To be useful, a ship's effective density must be much less than that of the surrounding water to allow the ship to support not only its own weight, but also that of crew, cargo, stores, etc. 1-3.1 Ship's Weight, Displacement and Capacity. An object's displacement is the weight of the water it displaces; displacement represents the force of buoyancy (B) acting on the object. For a ship in static equilibrium, floating free of any solid support, displacement (D) is equal to the weight of the ship and everything in it (W), measured in long tons of 2,240 pounds. Displacement is usually given for either the lightship--the weight of the ship without cargo or stores--or full-load conditions. A ship's displacement is related to the volume of displaced water, called the displacement volume or volume of displacement ( or V), by the weight density of water (g/gc). D = g gc = W

If mass density is given in slugs per cubic foot, and g in feet per second per second (ft/sec2), g/gc gives weight density in pounds-force per cubic foot. In a standard gravitational field (g = 32.174 ft/sec2) pounds-mass and pounds-force are numerically equal. Since the worldwide variation of gravitational acceleration is slight, weight density in pounds-force per cubic foot () can be taken as numerically equal to mass density, in pounds-mass per cubic foot without significant error.



With weight held constant, the product of displacement volume and water density must also be constant. For a given weight, displacement volume varies inversely with the density of the surrounding water--displacement volume in water of known density can be related to displacement volume water of any density: 1 1 = 2 2 2 = 1 1 2

The density of seawater varies with salinity and temperature, but is approximately 64 pounds per cubic foot; the density of fresh water is about 62.4 pounds per cubic foot. It is sometimes more convenient to use the inverse density, or specific volume (), of 35 cubic feet per ton of seawater. The equivalent figure for fresh water is 35.9, commonly rounded to 36. W = sw 35 = fw 36

W =

36 fw = sw 35 Care must be exercised not to confuse displacement, measured in long tons, with gross, net, or register tonnage. Tonnage is a measurement of the enclosed volume of a ship used to describe her cargo capacity and does not indicate displacement. Register tonnage (gross and net) is measured according to the rules of the country of registry or international rules, and is used as a basis for port fees, canal tolls, and similar charges. Measurement tons were formerly equal to 100 cubic feet, but the more recent international rules determine tonnage by formulas that do not relate volume to tonnage directly. Gross tonnage is a measure of the internal volume of the entire ship--the hull plus enclosed spaces above the main deck. Net tonnage is derived from a formula based on the molded volume of cargo spaces, the number of passengers carried, molded depth, and service draft; net tonnage gives an indication of the ship's earning capacity. Commercial vessels engaged in international voyages are issued a Tonnage Certificate by the country of registry. Certain special tonnages, such as Suez or Panama Canal tonnages, are calculated by somewhat different formulae and recorded on separate certificates. Cargo capacity may also be given in conventional volumetric units. Tank capacities are usually specified in barrels, gallons, or cubic meters. For petroleum products and other liquids subject to thermal expansion, practical capacity is less than net capacity, to ensure that a tank "filled" with cold oil will not overflow as the oil warms. U.S. Navy practice sets oil tank operating capacity at 95 percent of net capacity; U.S. Merchant Marine practice at 98 percent. Dry cargo capacity is specified in cubic feet or cubic meters. Bale capacity is the volume below deck beams and inboard of cargo battens, that is free for the stowage of bags, barrels, crates, bales, pallets, etc. Grain capacity is the net molded underdeck volume, after deductions for the volume of frames, floors, and other structure, that is available for the stowage of granular bulk cargo. Capacity of container ships is expressed as the number of standard 8-foot-wide by 8-foot-high containers of specified length that can be carried, often converted to 20-foot equivalent units (TEU), or 40 foot equivalent units (FEU). Capacity for roll-on/roll-off (RO/RO) cargo and vehicle carriers may be expressed as the number of units that can be carried or as the area of the cargo decks, in square feet or square meters. 1-3.2 Standard Loading Conditions. Displacement and stability characteristics are often referenced to certain standard conditions of loading. 1-3.2.1 U.S. Navy Ships. Characteristics are usually tabulated for the following standard conditions of loading (from NSTM Chapter 096):


Condition A - Lightship ­ The ship complete, ready for service in every respect, including permanent ballast (solid and liquid), onboard repair parts, aviation mobile support equipment as assigned, and liquids in machinery at operating levels, without any items of variable load (provisions, stores, ammunition, crew and effects, cargo, aircraft and aviation stores, passengers, saltwater ballast, fuel and other liquids in storage tanks). Formerly Condition II. Condition A-1 - Lightship ­ Condition A without permanent ballast. Formerly condition II-A. Condition B - Minimum Operating Condition ­ A condition of minimum stability likely to exist in normal operation (following the ship's liquid loading instructions). For warships, Condition B approximates the ship's condition toward the end of a hostile engagement following a long period at sea. Liquids are included in amounts and locations that will provide satisfactory stability, trim, and limitation of list in case of underwater damage. Formerly Condition V. Condition C - Optimum Battle Condition ­ As formerly applied to minor combatants, the ship loaded with full ammunition allowance and two-thirds provisions, fuel, lube oil, etc. Fuel distribution and seawater ballast are in accordance with liquid loading instructions, except that service tanks are assumed half-full and one pair of storage tanks per machinery box are assumed empty. Formerly Condition LS. In current practice, this condition applies only to ships with extensive underwater defense systems, such as aircraft carriers and battleships. Liquids are carried in the amounts and locations that provide the optimum resistance to underwater damage.

· ·





Condition D - Full Load ­ Two different full-load conditions are defined: (1) Full load (contractual) ­ The ship complete, ready for service in every respect; Condition A plus authorized complement of personnel and passengers and their effects, full allowance of ammunition in magazines and ready service spaces, full allowance of aircraft and vehicles with repair parts and stores, provisions and stores for the periods specified in design specifications, sufficient fuel to meet endurance specifications, anti-roll tank liquid, liquids in tanks to required capacity in accordance with liquid load instructions, and cargo in the amounts normally carried or a specified portion of full capacity. This condition is used for weight estimates and reporting. (2) Full load (departure) ­ Same as full load (contractual) except that fuel and lube oil tanks are 95-percent full, potable and feed water tanks 100-percent full. Formerly Condition VI. This condition is used in inclining experiment reports.


Condition E - Capacity Load ­ The ship complete, ready for service in every respect; Condition A plus the maximum number of crew and passengers that can be accommodated, with their effects, maximum stowage of ammunition in magazines and ready service spaces, full allowance of aircraft and vehicles with repair parts and stores, maximum amount of provisions and stores that can be carried in assigned spaces, tanks filled to maximum capacity (95 percent for oil tanks, 100 percent for fresh water), maximum amounts of cargo and supplies, with the provision that the limiting drafts not be exceeded.

Data is sometimes tabulated for special or unusual loading conditions, such as special ballast conditions for amphibious warfare ships. Details for each condition of loading are found in the ship's damage control book. Standard displacement is a condition defined by the Washington Naval Conference of 1923 as "The displacement of the ship, fully manned, engined, and equipped ready for sea, including all armament and ammunition, equipment, outfit, provisions and fresh water for the crew, miscellaneous stores and implements of every description that are intended to be carried in war, but without fuel or reserve feed water on board." Standard displacement was defined primarily as an aid to ensuring compliance to restriction on warship size and total naval tonnage under international treaties, but provides a convenient means of comparing warships and is commonly given in published summaries of naval strength, such as Jane's Fighting Ships. Characteristics for standard displacement are not normally tabulated in damage control books or similar documents. 1-3.2.2 Commercial Vessels. Two major conditions of loading are referenced in dealing with commercial vessels:

· ·

Lightship, Lightweight, or Light Displacement ­ The ship with all items of outfit, equipment, and machinery, including boiler water and lubricating oil in sumps, but without cargo, provisions, stores, crew, or fuel. Fully Loaded ­ Lightship plus cargo, fuel, stores, etc., to settle the ship to her load line. Also loaded, load, or full-load displacement. For ships designed to carry different classes of cargo, full-load conditions may be tabulated for each type of cargo.

The trim and stability booklet will normally tabulate stability data for ballasted and partly loaded conditions, and for end of voyage and intermediate conditions with varying amounts of fuel and stores consumed. 1-3.2.3 Loading Instructions. Specific loading instructions are provided to help operating personnel avoid loading the ship so that her stability is dangerously low or the hull girder is overstressed. The most basic instruction is that ships shall not be loaded so heavily that their load line (merchant) or limiting draft marks (naval) are submerged. Detailed loading instructions are given in the trim and stability booklet for merchant ships or the damage control book for Navy ships. In certain types of ships, such as container ships, RO/RO ships, barge carriers, and ferries, improper loading can easily reduce stability to dangerously low levels. In other ships, such as tankers and ore carriers, improper loading can seriously overstress the hull. Transient conditions created while loading or unloading can also degrade stability or overstress the hull. Load and stability computers supplement or replace loading instructions on many tankers, bulk carriers, and other large ships or ships with unusual stability problems. Load computers are briefly described in Paragraph 4-2.5.3. 1-3.3 Deadweight. Deadweight (DWT) is the load carried by a ship. It is the difference between the lightship displacement and total displacement of the ship at any time. Maximum or load deadweight is the carrying capacity of a ship measured in 2,240-pound long tons, and is the difference between the lightweight and fully loaded displacements. Deadweight includes fuel, provisions, munitions, crew and effects, cargo, or any other weight carried. For a merchant ship, cargo deadweight, paying deadweight, or payload is the part of the deadweight that is cargo and therefore earning income. It is not uncommon for the deadweight of a merchant ship to be given, but not its full-load displacement. A deadweight coefficient (CDWT) can be defined as the ratio of full-load displacement to total deadweight: CDWT = where: CDWT FL DWT = = = deadweight coefficient full-load displacement total deadweight FL DWT FL = DWT CDWT



Typical ranges for deadweight coefficient are given by R. Munro-Smith (Elements of Ship Design, 1979): General cargo ship Ore carrier Bulk carrier Oil tanker Very large tanker, VLCC 1.39 1.30 1.19 1.16 1.28 ­ ­ ­ ­ ­ 1.61 1.39 1.28 1.25 1.32

1-3.4 Change in Draft. Draft is significant as the only principal dimension that varies routinely, while length and beam remain essentially constant. Volume of displacement, and therefore draft, will change as a ship's displacement changes due to loading or discharging cargo, consuming or loading fuel or stores, or flooding. The new volumes and mean drafts can be computed by using the relationships shown. For example: a box-shaped lighter 100 feet long, 30 feet wide, and 10 feet in depth, displacing 429 tons of seawater with zero trim. Because waterplane area is constant at any draft, drafts can be found by: = W = 35 (429) = 15,015 ft3 = L B T = 100 (30) T = 15,015 ft3 T = LB = 15,015 = 5 ft 100 (30)

where: W L B T = = = = = = displacement volume, ft3 = LBTCB; for box-shaped lighter CB = 1.0 total weight of the barge, lton specific volume of seawater = 35 ft3/lton length between perpendiculars, ft beam, ft draft, ft

If weight (displacement) is decreased to 350 tons, the new mean draft is given by: = 35W = 35(350) = 12,250 ft3 T = 12,250 = 4.08 ft = 4 ft 1 in. 3,000

For a complex ship shape, drafts cannot be calculated directly. The change in draft (T) can be determined if certain assumptions are made. The increase in volume can be considered to be a prism of uniform thickness with vertical sides and horizontal section with area equal to the waterplane area. For a wall-sided vessel (one with vertical sides, like the box-shaped lighter), this is mathematically exact; it is sufficiently accurate for most ships for small changes in draft. The thickness of the prism is determined by dividing its volume by the area of the waterplane: T = (15,015) 12,250) = = AWP L B CWP (100)(30)(1.0)

= 0.92 ft = 11 in. where: T AWP = CWP = = change in draft, ft = change in displacement volume, ft3 waterplane area, ft2 waterplane coefficient



The salvor may encounter ships in water of varying densities. The waters of harbors and estuaries might be salty, fresh or brackish; the salinity and density of the water may depend on the state of the tide. The equalities shown can be used to relate displacement volume, draft and displacement of any ship in water of any known density. Recalling that: 1 1 = 2 2 1 1 L B T CB 1 where: L B T CB = = = = = = = displacement volume, ft3 water density, lb/ft3 inverse density or specific volume, ft3/lton length between perpendiculars, ft beam, ft draft, ft block coefficient



2 2 L B T CB 2



With length and breadth constant, and CB assumed constant for a small change in draft, T1 1 T1 T2 = T2 2 1 2


and: T2 = SW FW T12 1 = 35 36

For saltwater and fresh water:

and: 36 TFW = TSW 35 The difference between fresh water and seawater drafts may range from 6 inches for an FFG-7 to 1.2 feet for a large aircraft carrier, or more on a large crude carrier. Differences encountered when dealing with brackish water will be correspondingly less, and may be dealt with by using values for fresh water and saltwater as upper and lower boundaries if the water density is unknown or variable.



1-3.5 Tons per Inch Immersion (TPI). The foregoing analysis can be carried a step further to determine the change in displacement (D) required to cause a change in draft of one inch. For seawater: T = ; AWP Substituting 1 inch = 1/ 12-foot for T: D = AWP (35)(12) = AWP 420 = TPI = 35D T = 35 D AWP

W =

T AWP 35

where: D 35 AWP = TPI = displacement, lton = displacement volume, ft3 = water density, lb/ft3 = specific volume, ft3/lton waterplane area, ft2 = tons per inch immersion, lton/in.

Tons per inch immersion for water of any density can be obtained by a similar calculation. 1-3.6 Reserve Buoyancy. The watertight volume between the waterline and the uppermost continuous watertight deck provides the reserve buoyancy to the ship. Although this volume does not actually provide any buoyancy, it is available to enable the ship to take on additional weight. Freeboard is an indication of the reserve buoyancy remaining. Freeboard and draft can be considered opposite ends of a sliding scale, with draft representing the buoyancy in use and freeboard the buoyancy remaining. 1-3.7 Center of Gravity. A homogeneous body's center of gravity is located at its center of volume, or centroid. The center of gravity of a ship is not so easily definable, but can be assumed to be located on the centerline near the midship plane in a ship floating without list or trim. The center of gravity of a ship is a function of weight distribution; its position varies with loading. With all weights stationary, the center of gravity remains fixed regardless of the movement of the ship. Its position relative to any of the three reference planes along a perpendicular axis (n) is given mathematically by: n dw = G = W where: G n W = = = position of the center of gravity along any axis distance from the origin to an incremental weight dw, or to an individual weight w total weight = w nw w

The location of the center of gravity greatly influences the stability characteristics of a vessel: the vertical location (VCG, or KG) influences a vessel's ability to resist heeling forces; the longitudinal location (LCG) relative to the longitudinal location of the center of buoyancy determines trim; and a transverse location (TCG) off the centerline results in a list.



1-3.8 Center of Buoyancy. The force of buoyancy, like gravity, can be resolved to act upwards through a single point. The center of buoyancy (B) is located at the centroid of the submerged hull form. As the ship inclines, the shape of the underwater volume changes and the center of buoyancy moves to the new geometric center. When a ship is at rest without list, the center of buoyancy is on the centerline directly below the center of gravity. The location of the center of buoyancy responds directly to draft changes. As the ship's displacement is increased or decreased with a corresponding change in draft, the center of buoyancy will move to the new centroid of the redefined submerged hull form. 1-3.9 Metacenter. As shown in Figure 14, vertical lines drawn through successive centers of buoyancy (B1, B2, and B3) as the ship inclines slightly intersect at an imaginary point on the centerline called the metacenter (M). In a stable vessel, M is located above the center of gravity. The vertical location of M is one of the most critical parameters affecting a ship's initial stability. 1-3.10 Center of Flotation. The center of flotation is the point about which the ship trims and heels, and is at the geometric center of the ship's floating waterplane. It is usually located aft of midships, although it may be forward of midships in fullbodied ships.


WL2 WL WL 1 G WL WL1 B2 B1 B WL2

1-3.11 Bonjean's Curves. Bonjean's Curves or Curves of Sectional Areas are a Figure 1-4. Relative Positions of M, B, and G During Small Inclinations. collection of curves plotting sectional area along the X-axis against draft on the Y-axis. The curves are usually presented in one of the two formats shown in Figure FO-3. The section area curve may show area for either the whole section, or for one side only, as noted on the drawing. The areas generally do not account for appendages, but may include shell plating, as noted on the drawing. Section areas can be taken from the curves for any draft and any condition of trim or hull deflection. Section area is converted to unit buoyancy by dividing by the specific volume of water (35 cubic feet per long ton per foot of length for seawater). Volume of displacement and other hydrostatic properties can be determined by integration of section area or derived unit buoyancy ordinates by the numerical methods described in Paragraph 1-4. The rosette arrangement (Figure FO-3A), with all the curves drawn to a single set of axes, produces a more compact drawing and is favored by some designers because lack of fairness in the hull will show itself with the curves lying side by side. Section areas are read from the intersection of a horizontal line through the station draft on the center scale with the appropriate curve. When calculating buoyancies for varying waterlines or wave profiles, it is sometimes more convenient to arrange the curves along the ship's profile, with a vertical axis at each station as shown in Figure FO-3B. With the section area curves arranged in this format, a trimmed waterline can be plotted as a straight line passing through the forward draft at station zero, and the after draft at the after perpendicular, eliminating the need to determine draft at each station. Section areas can be picked off by drawing a horizontal line from the intersection of the waterline with each vertical station marker to the appropriate curves. If the Bonjean's Curves are not available in this format, the curves and area scale can be traced from the rosette onto a hull profile drawn on tracing paper. The horizontal length scale for the hull profile is not critical, but should be consistent throughout its length if buoyancy is to be calculated on waterlines that are not horizontal.



1-4 APPROXIMATE INTEGRATION TECHNIQUES AND APPLICATIONS The salvage engineer may be required to calculate hydrostatic data for a casualty when curves of form or other documents are not available; for a casualty in an unusual condition, such as a ship floated upside down or on its side; or for portions of a ship that has been cut into sections. A ship's form consists of a number of intersecting surfaces, usually of nonmathematical form. Areas and volumes enclosed by these surfaces, as well as moments of areas and volumes, and second moments of area, must be determined to calculate hull hydrostatic characteristics. For a curve plotted on an xy coordinate system, the area under the curve and moments, second moments (moments of inertia), and location of the centroid can be expressed as simple integrals. Since hull forms are seldom definable by mathematical equations, areas, moments, and volumes are calculated by manual integration methods rather than by direct integration. Manual integration methods are also used to evaluate any parameter that can be expressed as a curve of a function of some variable. For example, the total force, location of the center of effort, and force moment of an unevenly distributed force (such as current forces) can be determined from a curve showing the force distribution. Graphical and numerical manual integration methods are described in the following paragraphs. 1-4.1 Graphical Integration. An obvious way to calculate the area under a curve (or within a shape) is to plot the curve to scale on graph paper and count the squares under the curve. This method can be extended to calculate the first moment of area, My = xy dx, by multiplying the height (number of squares, y) in each column by its distance from the origin (x), and summing all such products. In the same way, the second moment is calculated by multiplying the height of each column by x2. By adopting sign conventions and adjusting the location of the origin, moments can be calculated about any desired axis. Graphical integration of large, complex areas is very tedious, but can be very accurate for even the most complex or discontinuous curves. 1-4.2 Numerical Integration. Numerical integration methods, or rules, are based on the same premise as graphical integration; that the area under a curve can be closely approximated by breaking the area up into smaller shapes whose areas can be calculated or estimated easily, and summing the areas of these shapes. Most rules depend upon the substitution of a simple mathematical form for the actual curve to be integrated. The accuracy of the result depends upon the accuracy of the fit between the real and assumed curves. 1-4.3 Trapezoidal Rule. The trapezoidal rule substitutes a series of straight lines for a complex curve to allow integration of the curve in a simple tabular format. Conceptually, the trapezoidal rule is the simplest of the numerical integration rules. A curvilinear shape can be approximated by a series of n trapezoids bounded by n + 1 equally spaced ordinates, y0, y1, y2, y3, ..., yn, (at stations x0, x1, x2, x3, ..., xn) as shown in Figure 1-5. If the station spacing is h, the area (a0,1) of the first trapezoid is: a0,1 = y0 + y1 2 h





y3 ......................... h


yn x




x3 .........................

x n-1


Figure 1-5. Curvilinear Figure Approximated by Series of Trapezoids.

The total area of the shape (A) is approximately equal to the sum of the areas of the trapezoids: A = a0, 1 = = y0 y1 2 h y 2 0 a1, 2 h 2y1 a2, 3 y1 y2 2 h ... an 2 2 y3 . . .

1, n

y2 y3

h yn








y = h 0 2





yn 2

This expression is called the trapezoidal rule, and can be used to calculate areas of any shape bounded by a continuous curve, simply by dividing the shape into a number of equal sections and substituting the ordinate values and the station spacing, or common interval, into the rule. The common multiplier for the trapezoidal rule is the common interval (h). If the common interval and common multiplier (CM) are separated into two factors, the common multiplier for the trapezoidal rule is 1. The factors by which each ordinate is multiplied (1/ 2, 1, 1, 1, ..., 1/ 2) are the individual multipliers (m). The products of the individual multipliers and ordinates are called functions of area, (A). The area under the curve is thus expressed as: A = y dx = h f (A)



Because the trapezoidal rule substitutes a series of straight lines for the curve to be integrated, it is best suited for use with smooth, long-radius curves such as the waterlines of a ship. The rule underestimates the area under convex curves, and overestimates the area under concave curves. Accuracy increases as station spacing is decreased. If greater accuracy is required in regions of considerable curvature, e.g. at the ends of the ship, stations are taken at half-divisions. When half-spaced stations are used, the individual multipliers for the half-stations and adjacent stations must be adjusted. If, for example, a half-station is inserted between ordinates 1 and 2: A = y0 y1 2 h y1 y1.5 h 2 2 y1.5 y2 h 2 2 y2 y3 2 h ... yn






h y 1.5 y1 y1.5 1.5 y2 2y3 ... yn 2 0

3 1 3 1 1 y y y y ... y = h y0 2 4 1 2 1.5 4 2 3 2 n The individual multiplier for the half-station is 1/ 2, and 3/ 4 for the station on either side of it. A similar analysis will show that if several sequential half-stations are inserted (i.e., 21/ 2, 31/ 2, 41/ 2, etc.) the multipliers for all stations and half-stations between the first and last half-stations is 1/ 2, and the multiplier for the two outlying whole stations is 3/ 4. It may be more convenient to use the first form of the rule, to avoid divisors greater than 2, in which case all the individual multipliers are doubled. 1-4.4 Simpson's Rules. The replacement of a complex or small radius curve by a series of straight lines limits the accuracy of calculations, unless a large number of ordinates are used. Integration rules that replace the actual curve with a mathematical curve of higher order are more accurate. Simpson's rules assume that the actual curve can be replaced by a second-order curve (parabola). Figures 1-6 through 1-8 demonstrate the derivations of Simpson's rules. 1-4.4.1 Simpson's First Rule. Figure 1-6 shows a curve of the form y = ax2 + bx + c. It is expressed by three evenly spaced ordinates y0, y1 and y2, at x = 0, 1, and 2 (station spacing = 1). The values of the ordinates are: y0 = a (0)2 y1 = a (1)2 y2 = a (2)2 The area under the curve is:

2 A = (ax 2 0


Y = ax


+ bx + c




x=0 h

x=1 h


X h AREA = __ (y0 + 4y1 + y2 ) 3

Figure 1-6. Simpson's Three-Ordinate Rule.

b (0) b (1) b (2)

c = c c = a c = 4a b 2b c c

for x = 1 for x = 1 for x = 2


c) dx =

ax 3 3

bx 2 2


2 0


8 a 3



Now c = y0 and y1 = y0 + a + b, and y2 = y0 + 4a + 2b. Substituting and solving for a and b: y2 2 y1 = y0 a = ( y2 2b 2 y1 2 y0 a = y1 y0 (y2 2y1 2 y0) = 3 y 2 0 y2 2 2 y1 4a y0 ) 2 y0 2b 2a = y0 2a

b = y1



Area (A) is expressed as: A = 8 a 3 2b 2c = 8 y2 3 4 y 3 2 2 y1 2 y0 3 y0 2 2 4 y = 3 0 1 y 3 0 y2 2 2 y1 4 y 3 1 2 y0 1 y 3 2

= 2 y0 = 1 y 3 0

3 y0 4 y1

y2 y2

4 y1

8 y 3 1

For an ordinate spacing of h rather than unity: A = This relationship is Simpson's first rule, or 3-ordinate rule, commonly called Simpson's rule. The rule calculates correctly the area under a second order curve and will approximate the area under any curve that passes through the same three points. The accuracy depends on how closely the actual curve approaches the parabolic form assumed by the rule. Simpson's Rule is the numerical integration rule used most widely for ship calculations. h (y 3 0 4 y1 y2 )

6 1 1

5 4 4

4 1 1 2

3 4 4

2 1 1 2

1 4 4

0 1 1


Figure 1-7. Simpson's Multipliers for Long Curve.

The rule can be extended to calculate the area under a long nonparabolic curve such as a ship's waterline. If the length of the curve is divided into enough equal parts, as shown in Figure 1-7, it can be reasonably approximated by a series of parabolic segments. For a curve divided into n equal parts, the area between the first (0) and third (2) ordinates would be given by: A0 where: A0-2 h L n = = = = area under the curve between the first and third ordinates distance between ordinates = L/n length of the curve number of sections between ordinates = number of ordinates - 1



h (y + 4y1 + y2) 3 0

Similarly, the area between the third (2) and fifth (4) ordinates would be: A2 The area between the fifth (4) and seventh (6) ordinates: A4 and so on. The total area is the sum of all the two section areas: A = A0 =

2 6 4


h (y + 4y3 + y4) 3 2


h (y + 4y5 + y6) 3 4





... An

2 n

h y 4y1 2y2 4y3 2y4 4y5 2y6 ... yn 3 0

This is the general form of Simpson's rule. Since the rule consists of a summation of areas over two sections of a curve divided into a number of equal sections, the curve must be divided into an even number of sections (by an odd number of stations) to apply the rule. The common multiplier (CM) is 1/ 3; the individual multipliers are 1, 4, 2, 4, 2, 4,..., 2, 4, 1. The derivation of the individual multipliers as a tabular summation of the 3-ordinate rule multipliers for each two adjacent sections is shown in Figure 1-7.



In regions where the slope of the curve changes rapidly, the accuracy of the rule can be increased by inserting intermediate (half-spaced) stations. When half-spaced stations are used, the individual multipliers are modified. For example, a half-station could be inserted at 21/ 2 were there a rapid change in form between the third and fourth stations of the curve in Figure 1-7. The area between the first and second stations is calculated as before: A0



h (y + 4y1 + y2) 3 0

With the insertion of the half-station (21/ 2), the 3-ordinate rule can be applied to the area between the third and fourth ordinates (A2-3), with an ordinate spacing of h/2: h y h y 2 = y2 4y2.5 y3 = 2 2y2.5 3 32 2 3



The area between the fourth and sixth stations (A3-4) is now: A3 and so on. The total area is: A = A0 = =

2 4


h (y + 4y4 + y5) 3 3





... An

1 n

y2 2y3 h 2 y2.5 y y3 4y4 y5 ... yn y0 4y1 y2 3 2 2 h 1 1 y 4y1 1 y2 2y2.5 1 y3 4y4 2y5 ... yn 3 0 2 2

Note that unless another half-spaced station is inserted, the number of sections (n) will be even, and the rule unworkable. Intermediate stations can be inserted at any equal division of the station spacing (third-stations, quarter-stations, etc.) and multipliers deduced in a similar manner. Intermediate stations can be inserted anywhere along the length of the curve so long as two rules are followed:

· ·

An even number of intermediate stations must be inserted, so that the total number of segments remains even (total number of ordinates is odd). Intermediate stations must be inserted so there are an even number of segments in each group of consecutive whole or partial segments (each group of whole or partial segments includes an odd number of ordinates).

Intermediate stations are commonly used near the ends of waterlines where the hull form changes rapidly with respect to length. The individual multipliers can be quickly determined by tabulating and summing the appropriate 3-ordinate rule multipliers as shown in Figure 1-8. 1-4.4.2 Simpson's Second Rule. Rules can be deduced, in a similar manner, for areas bounded by different numbers of evenly spaced ordinates, or by unevenly spaced ordinates. For four evenly spaced ordinates:

6 1/2 1/2

5-1/2 2 2

5 1 1/2 1-1/2

4 4 4

3 1 1/2 1-1/2

2-1/2 2 2

2 1 1/2 1-1/2

1 4 4





Figure 1-8. Simpson's Multipliers with Half-Spaced Stations.

A =

3h (y0 + 3y1 + 3y2 + y3) 8

This is Simpson's second or three-eighths Rule. The general form is: A = 3h (y0 + 3y1 + 3y2 + 2y3 + 3y4 + 3y5 + 2y6 + ... + yn) 8

Simpson's second rule can be used with 4 + 3i ordinates, where i is a positive integer (i.e., 4, 7, 10, 13, etc.). 1-4.5 Applications. The derivations of Simpson's rules and the trapezoidal rule were demonstrated with area computations to aid conceptualization, but the rules can integrate any function that can be plotted on Cartesian coordinates. If, for example, the ordinates represent sectional areas along a ship's length for a given waterline, the products of the multipliers and ordinates are functions of volume, (V), and their summation (integral of the curve) is the volume of displacement. Calculation of areas, moments, centroids, and second moments of areas by the are described in the following paragraphs.



1-4.5.1 Moments and Centroids. As shown in Figure 1-9, the moment of an elemental strip of area about some vertical axis YY is xydx. To determine the moment of a larger area about the axis, the integral M = xy dx must be evaluated. Instead of multiplying the value of y at each station by the appropriate multiplier, the value xy is multiplied, where x is the distance from the station to the reference axis, and dx is the width of each strip, or the common interval h. The value y dx = hyn is the area of the strip an; the first moment of this area about some reference axis YY is: MYY = xnhyn = xnan The total moment is the sum of the moments of all the strips, that is, the integral of the incremental moments along the length:


AREA a = ydx

x 1/2 yn xx dx FOR SHADED STRIP: a = ydx ay2 y3dx i = ___ = ____ 12 12 Myy = xa = x(ydx) Iyy = x 2a = x2(ydx) 3 y2 y dx y2 Ixx = __ a + i = __ ( ydx) + _____ 2 2 12


Figure 1-9. Variables for Moment and Second-Moment Calculations.

L MYY = xn an dx 0

The integral can be evaluated numerically: x a dx = n n where: CM (A) mn = = = common multiplier for the appropriate integration rule function of area = mnyn common multiplier for the appropriate rule and station xn CMf (A) = CM xn f (A)

If the reference axis is chosen to fall on an ordinate station, then the moment arms have the common interval (h) as a common factor, i.e., xn = snh, where xn is the moment arm and sn is the number of stations from the reference axis to station n. The factor h can be brought outside the summation: MYY = CMh sn(A) The products of the number of stations from the reference axis and the functions of area, sn(A), are the functions of moment (M): MYY = CMh (M) The distance from the centroid of the shape to the reference axis (x) is the moment divided by the area: x = MYY A = CM h f (M) = CM f (A) f (M) h f (A)



The centroid of a symmetrical shape lies on the axis of symmetry, and its location can be defined by summing moments about a single axis perpendicular to the axis of symmetry. To precisely locate the centroid of an asymmetrical shape, moments must be summed about another, perpendicular, axis. The calculation can be performed by taking ordinates perpendicular to the first set and integrating with respect to y rather than x. Moments about an axis XX can also be determined using y ordinates, but with slightly less accuracy. Referring again to Figure 1-9, the moment about axis XX of the elemental strip dx is: y y2 y MX X = a = y dx = dx 2 2 2 where y is the height of the strip, and a its area. The total moment is the integral of the incremental moments along the length, and the integral can be evaluated numerically: MX X Ly n an dx = = 0 2 yn 2 CM f (A)n = CM yn f (A)n 2

The product of the y ordinate and the function of area for each segment can be defined as the function of moment about x, (MXX): f MX X = y f (A) = y 2 mn MX X = CM 2 f MX X

where mn is the individual multiplier for the nth ordinate. The distance from the centroid of the shape to the axis XX (y') is the moment divided by the area: MX X A CM f MXX f MXX 2 = CM f (A) 2 f (A)




Moments can be summed about any axis, although it is simplest to sum them about an axis through x0 so that the number of stations from the reference axis is simply the station number. For ship calculations, moments are often summed about the midships section to reduce the size of the products and sums for manual calculation, and because the centers of flotation, buoyancy, and gravity normally lie near midships. When moments are summed about a station other than an end station, a sign convention must be adopted so that distances to one side of the reference axis (and therefore moments and functions of moments) are negative. 1-4.5.2 Second Moments of Area. The second moment of area (moment of inertia, I) of a plane shape about an axis YY parallel to the vertical ordinates is given by:

L IYY = 0 x2y dx

where: IYY = x = L = second moment of area about some axis YY distance from axis YY to elemental vertical strip of height y and width dx length of the area whose second moment is desired, measured along an axis perpendicular to YY

An analysis similar to that taken for the calculation of first moments will show that the second moment of the area under a curve is calculated by: IYY = CMh2 (IYY) where: CM h (IYY) sn mn yn = = = = = = common multiplier common interval function of second moment about axis YY = sn2mnyn number of stations from axis YY to station n individual multiplier for station n height of the ordinate at station n



The second moment of an area (moment of inertia) is always smallest about an axis through its centroid, (the neutral axis in bending stress analysis). If moment of inertia about some axis YY, parallel to the neutral axis is known, the moment of inertia about the neutral axis (INA) is found by the parallel axis theorem: INA = IYY - Ad 2 where d is the distance from axis YY to the neutral axis, and A is the total area of the section. The second moment of area about an axis XX perpendicular to axis YY can be calculated by taking ordinates perpendicular to the first set and integrating twice with respect to y rather than x. To determine the second moment about a horizontal axis of symmetry, such as the moment of inertia of a waterplane about its centerline, the integration can also be performed using the original set of ordinates. In Figure 1-9 (Page 120), y is the half-ordinate of an incremental strip of a waterplane measured from the centerline. The second moment of area of the incremental strip about the centerline is: y 2 ixx = a 2 where: ixx a i0 dx = = = = = second moment of area of incremental strip about the centerline area of the incremental strip second moment of area of the incremental strip about a horizontal centroidal axis (1/ 12)y3dx if strip is assumed to be rectangular width of the incremental strip y 2 i0 = y dx 2 1 3 1 3 y dx = y dx 3 12

The total second moment of half-waterplane area is:

L 1 1 L IXX, half = y 3dx = y 3dx 3 0 0 3 The second moment of the total area is twice this amount, and this will be the second moment about the centerline, since the waterplane is symmetrical about the centerline. The integration y3dx can be performed numerically:

CM h IX X = 2 3 where: CM h (IXX) = = = common multiplier common interval function of second moment about axis XX = mnyn3 individual multiplier for station n height of the half-ordinate at station n

f IX X


mn yn

= =

1-4.5.3 Volumes and Centroids of Volume. Volumes are calculated by integrating a curve of sectional areas. To calculate the volume of the tank shown in Figure 1-10, the shape is first cut at several stations to form section outlines. The area of each section is calculated, and the areas taken as ordinates along the length of the tank. Integrating the area ordinates by the trapezoidal rule: V = a dx = h (V) where: (V) mn an = = =





Figure 1-10. Determination of Volume by Numerical Integration.

function of volume = mnan individual multiplier for station n area of section at station n



The moment of volume about some axis YY is: MYY = h2 (M) where: (M) sn = = function of moment of volume about axis YY = snmnan number of stations from axis YY to station n

The distance of the centroid from axis YY: d = h 2 f (M) = h f (V) f (M) h f (V)

These forms are exactly the same as those used to calculate areas and moments and centroids of areas; the only difference is that ordinate values represent areas rather than linear distances. Integrations can be performed along additional axes to precisely locate the centroid of the shape. 1-4.5.4 General Forms for Area and Moment Calculations. Calculation of areas, moments, centroids, and second moments of area by Simpson's first and second rules can be expressed in general forms: A = (CM) h f (A) MYY = (CM) h f (M) CM MXX = f MXX 2 x = where: A MYY MXX = x y IYY IXX CM h (A) (M) (MXX) (IYY) (IXX) s m yn = area under a curve between selected stations = first moment of area about axis YY first moment of area about axis XX = distance from centroid of area to axis YY = distance from centroid of area to axis XX = second moment of area about axis YY = second moment of area about centerline axis XX = common multiplier for the appropriate rule (1, 1/3, 3/8, etc) = common interval = function of area = mnyn = function of moment about YY = snmnyn = sn(A) = function of moment about XX = mnyn2 = yn(A) = function of second moment about YY = sn2mnyn = sn(M) = sn2(A) 3 = function of second moment about XX = mnyn = number of stations from axis YY (or integration start point) to station n = individual multiplier for station n for the appropriate rule = height of the ordinate at station n (half-ordinate for IXX) (CM) h f (M) = (CM) f (A) f (M) h f (A)

Examples 1-1 and 1-2 demonstrate the use of the trapezoidal rule and Simpson's rule to calculate waterplane functions for an FFG-7 Class ship.




Using 11- and 21-ordinate trapezoidal rules, calculate the waterplane area (AWP), location of the center of flotation (LCF), moment of inertia of the waterplane about the centerline (ICL) and a transverse axis through the LCF (ICF), tons per inch immersion in saltwater (TPI), and waterplane coefficient (CWP) for the 16-foot waterline of an FFG-7 Class ship. Compare these values with actual data. Actual Properties:


= = = =

408 ft 45.6 ft 13,860 ft2 24.1 ft aft of midships = 228.1 ft from forward perpendicular


= 135,888,480 ft4 = 1,664,145 ft4 = 33 tons/in 0.745

Since the waterplane is symmetrical about its centerline, areas and moments can be found by integrating one side of the waterplane along the centerline with half-ordinates (halfbreadths) measured from the centerline, and doubling the results. Halfbreadths for the 16-foot waterline, in feet, inches, and eighths, are taken from Figure FO-1. The integrations are best performed in a tabular format. To integrate on 11 ordinates, halfbreadths for stations 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 are used. Integration on 11 ordinates: Station Ordinate, y ft-in-1/8 ft 0-4-5 0.39 6 -10 - 5 12-11 - 0 17- 9 - 2 20-11 - 5 22- 7 - 1 22- 8 - 3 21- 8 - 4 19- 7 - 1 16- 8 - 6 12- 7 - 0 6.89 12.92 17.77 20.97 22.59 22.70 21.71 19.59 16.73 12.58 Multiplier m


Integration on 21 ordinates: (A) m×y ft2 0.19 6.89 12.92 17.77 20.97 22.59 22.70 21.71 19.59 16.73 6.29 168.34 Lever (M) (IYY) s s × (A) s × (M) ft ft3 ft4 0 0.0 0.0 1 2 3 4 5 6 7 8 9 10 6.89 25.84 53.31 83.88 112.95 136.20 151.97 156.72 150.57 62.90 941.23 6.89 51.68 159.93 335.52 564.75 817.20 1063.37 1253.76 1355.13 629.00 6237.65 (IXX) m × y3 ft4 0.03 327.1 2156.7 5611.3 9221.4 11527.9 11697.1 10232.4 7518.0 4682.6 995.4 63969.9 Station Ordinate, Multiplier y m ft-in-1/8 ft 0 - 4 - 5 0.39 1/2 3 - 7 - 6 3.65 1 6 -10 - 5 6.89 1 10- 0 - 2 10.02 1 12-11 - 0 12.92 1 15- 6 - 1 15.51 1 17- 9 - 2 17.77 1 19- 6 - 7 19.57 1 20-11 - 5 20.97 1 21-11 - 5 21.97 1 22- 7 - 1 22.59 1 22- 9 - 4 22.79 1 22- 8 - 3 22.70 1 22- 3 - 7 22.32 1 21- 8 - 4 21.71 1 20- 9 - 5 20.80 1 19- 7 - 1 19.59 1 18- 2 - 1 18.18 1 16- 8 - 6 16.73 1 15- 1 - 0 15.01 1 12- 7 - 0 12.58 1/2 (A) Lever (M) (IYY) s ft2 ft ft3 ft4 0.19 0 0.0 0.0 3.65 1 3.65 3.65 6.89 2 13.78 27.56 10.02 3 30.06 90.18 12.92 4 51.68 206.72 15.51 5 77.55 387.75 17.77 6 106.62 639.72 19.57 7 136.99 958.93 20.97 8 167.76 1342.08 21.97 9 197.73 1779.57 22.59 10 225.90 2259.00 22.79 11 250.69 2757.59 22.70 12 272.40 3268.80 22.32 13 290.16 3772.08 21.71 14 303.94 4255.16 20.80 15 312.00 4680.00 19.59 16 313.44 5015.04 18.18 17 309.06 5254.02 16.73 18 301.14 5420.52 15.01 19 285.19 5418.61 6.29 20 125.80 2516.00 338.18 3775.54 50052.98 (IXX ) ft4 0.03 48.6 327.1 1006.0 2156.7 3731.1 5611.3 7495.0 9221.4 10604.5 11527.9 11836.8 11697.1 11119.4 10232.4 8998.9 7518.0 6008.7 4682.6 3381.8 995.4 128200.7

0 2 4 6 8 10 12 14 16 18 20


1 1 1 1 1 1 1 1 1



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

h = 408/10 AWP = 2h (A) MFP = 2h 2 (M) x

(M) = -------- h (A) = 2h 3 (IYY) = IFP - Ad 2

= = =

40.8 ft 2(40.8)(168.34) 2(40.8)2(941.23) 941.23 -------- (40.8) 168.34

= 13,736.5 ft2 = 3,133,618 ft3

h = 408/20 AWP = 2h (A) MFP = 2h2 (M) x

(M) = -------- h (A) = 2h 3 (IYY) = IFP - Ad 2

= 20.4 ft = 2(20.4)(338.18) = 2(20.4)2(3775.54) 3775.54 = -------- (20.4) 338.18

= 13,797.5 ft2 = 3,142,457 ft3


= 228.1 ft from FP = LCF

= 227.8 ft from FP = LCF


= =

2(40.8)3(6237.65) = 847,288,842 ft4 847,288,842 - 13,736.5(228.1)2 = 132,516,043 ft4 2(40.8/3)(63,969.9) 13,736.5/420 13,736.5/(408 × 45.6) = 1,739,981 ft4 = 32.7 tons = 0.738


= 2(20.4)3(50,052.98) = 849,865,964 ft4 = 849,865,964 - 13,797.6(227.8)2 = 134,155,856 ft4 = 1,743,529 ft4 = 32.9 tons = 0.742

ICL = 2(h / 3) (IXX) = TPI = AWP / 420 = CWP = AWP / (LB) =


ICL = 2(h / 3) (IXX) = 2(20.4 / 3)(128,200.7) TPI = AWP / 420 = 13,797.6 / 420 CWP = AWP / (LB) = 13,797.6 / (408 × 45.6)


AWP, ft2 LCF, ft fm FP ICF, ft4 ICL, ft4 TPI, tons/in CWP

13,860.0 228.1 135,888,480 1,664,145 33 0.745

11 Ordinate Value Error, % 13,737.8 0.88 228.1 0.00 132,502,924 2.49 1,739,981 4.56 32.7 0.91 0.738 0.94

Value 13,797.500 227.800 134,155,856.000 1,743,529.000 32.900 0.742

21 Ordinate Error, % 0.45 0.13 1.28 4.77 0.30 0.40




Use Simpson's first rule with 11 ordinates to calculate the waterplane properties that were calculated in Example 1-1. Compare the results with actual data and the results by trapezoidal rule. Ship dimensions and actual waterplane properties are the same as for Example 1-1. Halfbreadths for stations 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 from Figure FO-1 are used to integrate on 11 stations. Integration: Station Ordinate, Multiplier (A7) m×y ft2 1 4 2 4 2 4 2 4 2 4 1 0.39 27.56 25.84 71.08 41.94 90.36 45.40 86.84 39.18 66.92 12.58 508.09 Lever (M) s × (A) ft 0.0 27.56 51.68 213.24 167.76 451.80 272.40 607.88 313.44 602.28 125.80 2,833.84 13,820.1 ft2 3,144,882 ft3 (IYY) s × (M) ft4 0.0 27.56 103.36 639.72 671.04 2259.00 1634.40 4255.16 2507.52 5420.52 1258.00 18,776.28 (IXX) m × y3 ft4 0.06 1308.3 4313.4 22445.1 18442.7 46111.4 23394.2 40929.8 15036.0 18730.4 1990.9 192,702.4


ft-in-1/8 0 2 4 6 8 10 12 14 16 18 20 0-4-5 6 -10 - 5 12-11 - 0 17- 9 - 2 20-11 - 5 22- 7 - 1 22- 8 - 3 21- 8 - 4 19- 7 - 1 16- 8 - 6 12- 7 - 0 ft 0.39 6.89 12.92 17.77 20.97 22.59 22.70 21.71 19.59 16.73 12.58



ft 0 1 2 3 4 5 6 7 8 9 10




408/10 / 3 h (A) 2 / 3 h2 (M)


= = =

40.8 ft / 3 (40.8)(508.09) 2 / 3 (40.8)2(2833.84)


= =


(M) -------- h (A) / 3 h 3 (IYY) IFP - Ad2



2833.84 -------- (40.8) 508.09

2 / 3 (40.8)3(18,776.28) 850,156,311 - 13,820.1(227.6)2 2 / 3 (40.8/3)(192,702.4) 13,820.1/420 13,820.1/(408 × 45.6)


227.6 ft from FP 850,156,311 ft4 134,508,685 ft4 1,747,168 ft4 32.9 tons 0.743



= = = =

= = = = =

= = = = =

/ 3 (h/3) (IXX) AWP/420 AWP/(LB)


Actual Value

11 Ordinate Simpson's Rule Value Error, % 0.29 0.22 1.02 4.99 0.30 0.27 13,820.1 227.6 134,508,685 1,747,168 32.9 0.743

Trapezoidal Rule Error, % 11 Ordinate 0.88 0.00 2.49 4.56 0.91 0.92 21 Ordinate 0.45 0.13 1.28 4.77 0.30 0.40

AWP, ft2 LCF, ft fm FP ICF, ft4 ICL, ft4 TPI, tons/in CWP

13,860 228.1 135,888,480 1,664,145 33 0.745

The accuracy of an 11-ordinate Simpson's rule compares favorably with that of a 21-ordinate trapezoidal rule. Simpson's rule with 21 ordinates is only marginally more accurate than with 11 ordinates for this waterplane shape. Note that Simpson's rule calculates the moment of inertia about the centerline with slightly less accuracy than the trapezoidal rule. The derivation of the form: ICL = (CM)(h/3) (IXX) assumes a constant ordinate over the entire section (see Paragraph 1-4.3.3). The Simpson's multipliers do not correct for this assumption. The constant-ordinate assumption is essentially correct for very full ships and barges with extensive parallel midbody, and will yield very accurate values for ICL. Accuracy of ICL calculations for fine-lined ships can be increased only by using very close station spacing or integrating along an axis perpendicular to the centerline. The ± 5 percent accuracy shown here should be sufficiently accurate for most salvage work.



1-4.6 Other Simpson's Rule Forms. Simpson's rules can be derived for numbers of ordinates for which the first two rules do not apply, and to determine areas of "left over" segments at the ends of curves. 1-4.6.1 5, 8, Minus One and 3, 10, Minus One Rules. An additional Simpson's rule, known as the 5, 8, minus one rule, is used to determine the area between two ordinates when three consecutive ordinates are known. For ordinates y0, y1, and y2, the area between the first and second ordinates is given by: 1 h (5y0 + 8y1 - y2) A0-1 = 12 The area between the second and third ordinates can be found by applying the rule backwards: 1 h (-y0 + 8y1 + 5y2) A1-2 = 12 The validity of the 5, 8, minus one rule can be verified by observing that the sum of the expressions for the two sectional areas is the 3-ordinate rule: 1 h 5y0 8y1 y2 A = A0 1 A1 2 = y0 8y1 5y2 12 1 h y0 4y1 y2 = 3 The 5, 8, minus one rule cannot be used for moments. The first moment of the area between the first and second ordinates (A1-2) about the first ordinate is given by the 3, 10, minus one rule: 1 2 h (3y0 + 10y1 - y2) M1 = 24 These two Simpson's rules are at times convenient, but are less accurate than the first and second rules. 1-4.6.2 Simpson's Rules for Any Number of Ordinates. Simpson's rules can be combined one with another to derive rules for numbers of ordinates for which the first two rules do not apply. For example, the first rule can be used for 3, 5, 7, 9, ... ordinates, and the second rule for 4, 7, 10, .... ordinates. A rule can be deduced for six ordinates as shown below: 3 h y0 3y1 3y2 y3 A0 3 = 8 1 h y3 4y4 y5 A3 5 = 3 3 9 9 3 1 4 1 y1 y2 y3 y3 y4 y5 A = A0 3 A3 5 = h y0 8 8 8 3 3 3 8 1 h 9y0 27y1 27y2 17y3 32y4 8y5 = 24 This is not the only rule suitable for six ordinates. By skillful use of the 5, 8, minus one rule, a rule with less awkward multipliers can be deduced: 1 h 5y0 8y1 y2 A0 3 = 12 3 h y1 3y2 3y3 y4 A1 4 = 8 1 h y3 8y4 5y5 A4 5 = 12 A = A0






5 25 25 25 25 5 y0 y1 y2 y3 y4 y5 = h 24 24 24 24 15 12 25 h 0.4y0 y1 y2 y3 y4 0.4 y5 = 24 Substituting the same values for ordinates y0 through y5 in each rule will verify that they are equivalent. Rules deduced in this manner can be used in the general forms described in Paragraph 1-4.4.4. 1-4.7 Other Integration Rules. Simpson's rules and the trapezoidal rule are satisfactory for most manual calculations. The Newton-Cotes', Tchebycheff's, and Gauss' rules are more accurate, but require more tedious manual calculations. These rules are described in most general naval architecture texts, such as Basic Ship Theory by K.J. Rawson and E.C. Tupper, or Muckle's Naval Architecture by W. Muckle and D.A. Taylor.



1-4.8 General Notes For Numerical Integration. The numerical integration rules presented have relative advantages and disadvantages. When time and/or access to high-speed computers permits, the salvage engineer may select the optimum integration rule for a well-defined curve. For curves where ordinates are tabulated for only certain stations, a rule appropriate to that number and spacing of stations must be adopted. Some generalizations about the applicability of integration rules are listed below:


The trapezoidal rule uses constant ordinate spacing and simpler multipliers than the other rules. Any number of ordinates can be used. The rule can accommodate half-stations at any point, and the multipliers for half-stations are easily derived. For a single integration (area calculation) of a gentle curve, the trapezoidal rule is nearly as accurate as the Simpson's rules, but progressively greater errors are introduced on successive integrations (for moments and moments of inertia). Simpson's rules and the trapezoidal rule include the common interval as part of the common multiplier and can therefore calculate areas or volumes, moments, centroids, and second moments of area (single, double, and triple integrations) directly. Simpson's rules are the most commonly used integration rules because they are more accurate than the trapezoidal rule, but simpler to use than the more accurate Newton-Cotes', Tchebycheff's, and Gauss' rules. Simpson's rules exactly integrate first-, second-, and third-order curves. Successive integrations produce progressively higher order curves: the curve of area under a second-order curve is a third order curve, and the curve of the moment of areas is then a fourthorder curve. Simpson's rules will therefore exactly calculate the first moment of a second-order curve, or the second moment of a first-order curve. Calculating the second moment of a second-order or higher curve involves integrating a fourth-order equation, so some error is introduced even for a parabolic curve. Additional error may arise for an arbitrary curve. Experience has shown that Simpson's rule calculates moments and second moments of relatively smooth, continuous curves--such as those describing ship forms--accurately if a sufficiently close station spacing is used. An even-ordinate Simpson rule is only marginally more accurate than the next lower odd-ordinate rule; odd-ordinate Simpson rules are therefore preferred, and almost universally used in salvage.

· · ·


1-4.9 Integration of Discontinuous Curves. The integration rules discussed are applicable to continuous curves. The area under a discontinuous curve can be obtained by applying appropriate rules to the portions of the curve between discontinuities and summing the areas. For curves with large numbers of closely spaced discontinuities, it is simpler to divide the curve into segments at the discontinuities, approximate each segment by a rectangle, triangle, or trapezoid, calculate the area of each segment, and sum the areas to find the total area. The centroid of each segment can be calculated or estimated. Moments, second moments, and the centroid of the entire area can be calculated by summing the products of each area and the lever arm from its centroid to a selected axis in a tabular format. Replacing a segment of the curve between discontinuities (stations) with a horizontal line at a value equal to the average ordinate creates a rectangle with area equal to the area under the curve between the two stations. If the curve between stations can be reasonably approximated by a straight line, a horizontal line intersecting the curve midway between stations has a y value equal to the average ordinate. Repeating this process along the length of the curve creates a stepped curve. If the discontinuities, and subsequent stations, are evenly spaced, the curve can be integrated by a modification of the trapezoidal rule: A = y dx = h

n 1 n


MYY = xy dz = h 2 IYY = x 2y dx = h 3 where: A MYY IYY h sn yn = = = = = =

n 1

n 1

sn 1/2 yn

sn 1/2 2 yn

area under a curve between stations 0 and n first moment of area about axis YY second moment of area about axis YY common interval number of stations from axis YY (or integration start point) to station n height of the mid-ordinate between stations n and n-1

Weight distribution curves for ships are usually drawn assuming a constant weight distribution between stations as stepped curves. The addition of the continuous buoyancy curve and stepped weight curve creates a discontinuous load curve. The load curve is usually stepped as described above to facilitate integration along its length to define the shear curve. Alternatively, the buoyancy curve can be stepped before summing with the weight curve. A stepped 10-segment (11-ordinate) buoyancy curve can be constructed from standard Navy 21-station Bonjean's Curves by taking unit buoyancy calculated from section areas for odd station as the average unit buoyancy for segments bounded by even stations--unit buoyancy for segment 0­2 is based on section area for station 1, that for segment 2­4 on the area for station 3, etc. Example 1-4 includes an integration of this type.



1-4.10 Calculation of Hull Properties. Various integrations of a ship's hull form are used to determine properties such as displacement, locations of centers, tons per inch immersion, etc., known collectively as functions of form, hydrostatic functions, or hydrostatic data. Waterlines, buttocks, and stations of lines drawings are spaced to support numerical integration, usually by Simpson's or the trapezoidal rules. Halfbreadths (offsets) taken along the length of a waterline provide ordinate values to define the waterplane shape; halfbreadths taken at different waterlines at the same station provide ordinate values to define the station shape. Because ships are symmetrical about the centerline, integrations are customarily performed for one side of the section or waterplane only, and doubled to give the total area or moment. When working from offsets, sectional areas are usually calculated by vertical integration on horizontal ordinates from the centerline. An integration up to a waterline gives section area corresponding to that waterline. Integrating the curve of areas along the ship's length gives volume of displacement; the centroid of the volume is the center of buoyancy. Waterlines are integrated along the ship's length to determine area of the waterplane, location of the centroid of the waterplane (center of flotation), and moment of inertia of the waterplane about the centerline and about a transverse axis through the center of flotation. From these properties, tons per inch immersion, location of the metacenter, etc., can be calculated. Displacement volume can be calculated by taking waterplane areas as ordinates and integrating vertically. Longitudinal position of the center of buoyancy (LCB) is obtained by longitudinal integration of the sectional areas. Height of the center of buoyancy (KB) can be obtained by vertical integration of waterplane areas, or by calculating a vertical moment of area for each section. The sum of all the vertical area moments divided by the sum of the sectional areas gives KB. Integrations of this form are included in Example 1-4 and Appendix F. 1-4.10.1 Functions of Form. Functions of hull form are usually calculated for each waterline so they can be plotted as a function of draft as the ship's Curves of Form, also called Hydrostatic Curves, or Displacement and Other Curves (D & O Curves). Figure FO-2 is a reproduction of the curves of form for an FFG-7 Class ship. Hydrostatic data is also recorded in the Functions of Form Diagram (Figure B-1) for Navy ships and Hydrostatic Tables (Figure B-2) for commercial vessels. The salvage engineer may be required to calculate hydrostatic data when curves of form or other documents are not available or for a casualty in an unusual condition. Whether functions of form are calculated for a complete range of drafts or for only a few selected drafts depends on the form of the ship and the nature of information required by salvors. Manual calculations are best performed on organized tabular forms called displacement sheets.

Table 1-3. Appendage Allowances.

Ship Type Appendage allowance: APP/FL . . . . . . . . . . . . . . . . . . . . . 0.0167 0.0200 0.0049 0.0060 0.0106 0.0057 0.0049 0.0024 0.0014 0.0075

1-4.10.2 Appendage Displacement. Volumes and dis. . . 0.0060 . . . 0.0015 placements (buoyancies) based on section areas taken from Bonjean's Curves do not include appendage volume/ dis. . . 0.0050 placement, although sectional areas from some Bonjean's . . . 0.0040 Curves include shell plating. If known, appendage dis. . . 0.0010 placements can be added to the integrated displacement; effect on LCB can be determined by moment balance. When . . . 0.0025 appendage buoyancy is unknown, appendage displacement . . . 0.0081 can be estimated as a fraction of full load displacement, . . . 0.0035 . . . 0.0046 called an appendage allowance. Appendage allowances . . . 0.0015 vary with ship size, type, and configuration. Warships generally have more and larger appendages than auxiliaries or commercial vessels. Vessels with high power-to-size Source: 1Jamestown Marine Services, 1990, unpublished; based on data from 22 ratios have larger screws and rudders than lower powered hull types entered into ship data files for the NAVSEA POSSE Program vessels; appendage allowance increases with the number of screws. Large bow sonar domes on combatants are faired into the hull, and are included in Bonjean's Curves and offsets; keel-mounted domes are appendages. For a given ship type and configuration, appendage allowance generally increases as size decreases. Approximate appendage allowances for different ship types are given in Table 1-3. Appendage displacement is essentially constant with draft, as most appendages (except shell plating) are low on the hull and will be emerged only by extremely low drafts. Once determined, appendage displacement can be added to the integrated displacement for any draft that covers the appendages to determine total displacement. Shell plating displacement can be adjusted for drafts less than full load by assuming that onehalf of the shell plating volume is concentrated in the bottom third of the draft range, and the remaining volume is evenly distributed over the upper two-thirds of the draft range. It is usually safe to assume that LCB for the displacement with appendages is virtually the same as that for the integrated (without appendages) displacement. 1-4.10.3 Station Spacing. In full-bodied ships (low-speed general cargo, large tankers, bulk carriers, etc.) the lengths of the waterlines between stations in the midbody are nearly straight lines. In many modern full-bodied ships, the waterlines over the midbody are, in fact, straight lines, forming a parallel midbody. Integration on 10 equal divisions of length (11 stations, 0-10) is sufficiently accurate for most purposes. If the curvature of the waterlines increases sharply near the ends of the ship, half-spaced stations can be inserted to increase accuracy, for example, at stations 1/ 2, 11/ 2, 81/ 2 and 91/ 2.

Single-screw, small combatant with keel sonar dome1 . . . . . . Twin-screw, small combatant with keel sonar dome1 . . . . . . . Single-screw, small combatant with bow sonar dome1 . . . . . . Twin-screw, small combatant with bow sonar dome1 . . . . . . . Twin-screw amphibious warfare ships with well decks1 . . . . . . shell plating only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . all other appendages . . . . . . . . . . . . . . . . . . . . . . . . . . Twin-screw LST1 without bow thruster . . . . . . . . . . . . . . . . . . . . . . . . . . with tunnel bow thruster (negative appendage) . . . . . . . Single-screw merchant ships and auxiliaries of ordinary form, less than 5,000 tons full load displacement . . . . . . . . . . . . . . shell plating only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . all other appendages . . . . . . . . . . . . . . . . . . . . . . . . . . Single-screw merchant ships and auxiliaries of ordinary form, 5,000 to 15,000 tons full load displacement . . . . . . . . . . . . . shell plating only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . all other appendages . . . . . . . . . . . . . . . . . . . . . . . . . . Single-screw merchant ships and auxiliaries of ordinary form, greater than 15,000 tons full load displacement . . . . . . . . . . . Twin-screw merchant ships and auxiliaries of ordinary form . . shell plating only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . all other appendages . . . . . . . . . . . . . . . . . . . . . . . . . . VLCC, ULCC, very large bulk carriers . . . . . . . . . . . . . . . . . .

... ... ...



Accuracy can be increased by reducing the station spacing throughout the length of the curve. This increases the number of calculations to be performed, but avoids determining additional multipliers and may be simpler to program for computer calculation. For ship calculations, offsets are usually tabulated for either 11 or 21 basic stations (10 or 20 equal divisions), with half-stations as necessary. Offsets for Navy ships are normally tabulated for 21 basic stations, although additional tables may be prepared for very close station spacing. Offset tables for 2-foot station spacing are available for the FFG-7, for example. Even when 21-station offset tables or Bonjean's Curves are available, integration on 11 stations is sufficiently accurate for most hull volume calculations on any smooth hull form, including fine-lined warships. 1-4.10.4 Full Sections. In full, relatively flat-bottomed sections, special care must be taken in calculating the area from the base to the lowest waterline to avoid error. Figure 1-11 shows a section near midships where the turn of the bilge fairs into a straight line (the rise of floor line) at point A. If the entire area below CD is calculated using horizontal ordinates from the centerline, very close ordinate spacing must be used to avoid error because of the rapid change of form in the shell line. The area below CD can be calculated accurately using vertical ordinates from CD, with halfspaced ordinates inserted near the outboard end, or by dividing the area into two segments, as shown. The area KABC is a trapezoid whose area can be calculated accurately when the position of A and rise of floor can be determined. The area ADB can be obtained by using Simpson's rule, either with horizontal ordinates measured from AB, or with vertical ordinates measured from BD.






Figure 1-11. Calculating Sectional Area Below the Lowest Waterline.

1-4.10.5 Lowest Waterlines. When displacement volume is calculated by vertical integration of waterplane areas, the volume under the lowest one or two waterlines is calculated separately. Since the form of the ship changes so rapidly near the keel, the volume under the lowest one or two waterlines is calculated by integrating sectional areas along the ship's length. This volume is added to the volume determined by integrating waterplane areas from the lower waterlines upward to obtain the total volume of displacement. 1-4.10.6 Ends of Full Hull Forms. On SIMPSON'S RULE very full hulls, such as spoon-bowed ASSUMED PARABOLIC barges, large tankers (VLCC, ULCC), and FORM bulk carriers, the parallel midbody extends WATERPLANE nearly to the ends of the ship, where it OUTLINE joins to a short forebody or afterbody with steep or sharply curving lines. The aft ends of the lower waterlines of many fine-lined ships also curve sharply. If the ordinate adjacent to the end ordinate is some 2 1 FP STATIONS distance away from the end of the parallel midbody, the curve from this ordinate to TRAPEZOIDAL RULE ASSUMED STRAIGHT LINE the end ordinate (which is 0 or very small) assumed by Simpson's rules or the trapezoidal rule will fall well inside the Figure 1-12. Inherent Integration Error in Full Waterlines. actual waterline as shown in Figure 1-12. This will cause a serious underestimation of area for the end sections that will lead to even greater errors in calculations of moments and second moments about axes near midships because of the long lever arms. Intermediate stations should be inserted so that there are ordinates near the ends of the parallel midbody and at least one or two ordinates in the forebody and afterbody. Alternatively, waterplane areas for the midbody, forebody, and afterbody can be calculated separately and summed. The midbody area can be treated as a rectangle or integrated by a 3-ordinate Simpson or trapezoidal rule; the midbody and forebody areas can be calculated by any convenient rule with appropriate ordinates. 1-4.10.7 Tank and Compartment Volumes. A compartment's molded volume is greater than its floodable volume (the volume of liquid that can be contained), because of the volume occupied by fittings and structure. Floodable volumes of filled holds, machinery spaces, living spaces, etc., are estimated from molded volumes by use of permeability factors, as explained in Paragraph 1-9.1.1. Framing, sounding tubes, sea chests and similar structures in ordinary skin tanks typically occupy about 21/ 4 to 21/ 2 percent of the molded volume in double-bottom tanks, about 1 percent in cargo tanks (i.e., permeability of empty tanks is 971/ 2 to 973/ 4 percent, and 99 percent, respectively). Heating coils, if fitted, usually occupy an additional 1/ 4 percent of the molded volume. Flush tanks lie entirely within the ship's framing and are externally stiffened, so floodable volume, or capacity, is essentially equal to molded volume. To calculate volumes and centroids of flush tanks, offsets are taken to the inner surface of the tank, rather than the hull molded surface. Bale capacity of holds is calculated from offsets taken from sections showing the line of cargo battens, line of the bottoms of deck beams, and the top of the hold ceiling (above the inner bottom) including any gratings, with deductions for stanchions and other obstructions. Grain capacity is the molded volume, less the volume of structure, hold ceiling, and shifting boards.





Transverse stability is the measure of a ship's ability to resist rotation about its longitudinal axis and return to an upright position after being disturbed by an upsetting force. The following paragraphs define the elements of transverse stability and provide methods to calculate the transverse stability characteristics of a vessel. 1-5.1 Equilibrium and Stability. A ship floating at rest, with or without list and trim, is in static equilibrium; that is, the forces of gravity and buoyancy are equal and acting in opposite directions in line with one another. Stability is the tendency of a ship to return to its original position when disturbed after the disturbing force is removed. Stability can be described as positive, negative, or neutral. 1-5.2 Internal Forces. The internal forces affecting floating bodies are the forces of gravity and buoyancy. Both of these forces act at all times on wholly or partially submerged bodies. Figure 1-13 illustrates the relationship between the forces of buoyancy and gravity. Assuming the prism floats with half its volume submerged, and with the center of gravity located as shown, the prism can come to rest in either position (a), with the center of gravity directly above the center of buoyancy, or (c), with the center of buoyancy above the center of gravity. In either position, the forces of buoyancy and gravity act along the same vertical line. If the prism is inclined from (a) to (b), or from (c) to (d), a couple, or righting moment, is developed between the lines of action of buoyancy and gravity that tends to move the body back to its original position, i.e., the body floats with positive stability in either position. In position (a), with the center of gravity above the center of buoyancy, stability is provided by the body's shape, or form, and is termed form stability. If the width of the prism is reduced while the center of gravity remains on the centerline at the same location, a situation arises in which the center of buoyancy does not move far enough to be to the right of the center of gravity as the body is inclined from (a) to (b). The body can then attain positive stability only in position (c), with the center of buoyancy above the center of gravity. Bodies floating with the center of buoyancy above the center of gravity develop positive initial righting moments regardless of shape. This mode of stability is called weight stability. Sailing yachts with deep weighted keels, spar buoys, conventional ships with very low centers of gravity, and submarines all exhibit weight stability. Capsized ships floating upside down very often have their centers of gravity below the center of buoyancy, and operate in a weight stability mode.



W (c)



(d) Figure 1-13. Stability of a Floating Object.






37° G B Z

45° G Z





61° G







Figure 1-14. Development and Loss of Righting Arm.



The center of buoyancy of a ship moves as the ship is inclined, in a manner that depends on the shape of the hull near the waterline. The center of buoyancy initially moves away from the centerline as the ship is inclined, as shown in Figure 1-14. At some angle of inclination, the center of buoyancy begins to move back towards a vertical reference line drawn through the original position of the center of buoyancy. The vertical line of action of the center of gravity continues to move outward as the ship is inclined. At some angle of inclination, the line of action of gravity moves outboard of the line of action of buoyancy, creating an upsetting moment. Ships that have slowly heeled through progressively greater angles of inclination will suddenly capsize when this angle of zero righting moment (angle of vanishing stability) is passed. In Figure 1-15, the prism is assumed to be neutrally buoyant so that it is wholly submerged but clear of the bottom. An inclination from (a) produces an upsetting moment that tends to rotate the prism away from its initial position. Conversely, a inclination from (c) produces a righting moment. A submerged object clear of the bottom or other restraints can therefore have positive stability in only one position, that is, with the center of buoyancy above the center of gravity. Submerged objects therefore operate in a weight stability mode. The difference in behavior of floating and submerged objects is due to the fact that the center of buoyancy of a submerged object is fixed at the center of volume of the object, while the center of buoyancy of a floating object will generally shift when the object is inclined. Because the center of buoyancy of a submerged object is fixed, the righting moment cannot change to an upsetting moment as the object inclines unless the position of the center of gravity shifts. Stability of submarines and other submerged objects is discussed more completely in the U.S. Navy Ship Salvage Manual, Volume 4 (S0300A6-MAN-040). Figure 1-16 shows how a stable ship subjected to normal disturbances will develop moments tending to return the ship to its original position. A couple is formed as the lines of action of the opposing forces of gravity and buoyancy are separated. The arm of this couple, called the righting arm, is the lever to which the ship's weight is applied to right the ship. Figure 1-17 shows the upsetting arm developed when unstable ships are disturbed.



B (a) B (b)



W (c)

W (d)

Figure 1-15. Stability of a Submerged Object.



Figure 1-16. Righting Arm (GZ).



W1 W B1 M L B L1


Figure 1-17. Upsetting Arm.



1-5.3 External Forces. Ships are inclined by various external forces:

· · · · · ·

Wave action, Wind, Collision, Grounding, Shifting of onboard weights, and Addition or removal of weight.

Any inclination of a ship can be termed heel, but inclinations are broadly defined as heel, list, or roll depending on the duration and nature of the forces causing the inclination.

· · ·

Heel ­ The term heel is specifically applied to noncyclic, transient inclinations caused by forces that may be removed or reversed quickly. Such forces include wind pressure, centrifugal force in high-speed turns, large movable weights, etc. List ­ A list is a permanent, or long-term inclination, caused by forces such as grounding or offcenter weight that are not likely to be removed suddenly. Roll ­ When an inclining force is suddenly removed, a ship does not simply return to its upright position, but inclines to the opposite side and oscillates, or rolls, about its equilibrium position for some time before coming to rest. The natural rolling period (period of roll assumed by a ship free of restraints and exciting forces) is a function of weight and buoyancy distribution. Rolling is cyclic in nature and is induced or aggravated by short duration, repetitive or cyclic forces, such as wave forces.

1-5.4 Heights of Centers. The relative heights of the centers of gravity and buoyancy and the metacenter govern the magnitude and sense of the moment arms developed as the ship inclines. They are, therefore, the primary indicators of a ship's initial stability. Nominally, the symbols KG, KB, and KM indicate the heights of the centers of gravity and buoyancy and the metacenter above the bottom of the keel, while the symbols VCG and VCB indicate the vertical positions of the centers of gravity and buoyancy, measured from the baseline. In practice, KG/KB and VCG/VCB are used almost interchangeably; in steel ships with flat plate keels, the difference in height above baseline and keel for any point is generally less than two inches and is not significant. 1-5.4.1 Height of the Center of Gravity. The height or vertical position of the center of gravity above the keel (KG or VCG) is defined by weight distribution. KG can be varied considerably without change of displacement by shifting weight up or down in the ship. Conversely, it is possible to add or remove weight without altering KG. In most ships, the center of gravity lies between six-tenths of the depth above the keel and the main deck: 0.6D < KG < D where: D = hull depth, keel to main deck For barges with raked or ship-shaped bows and cut-up sterns, lightship KG can be estimated as 0.53D. For tank barges, KG for full load varies little from the lightship value. Table 1-4 gives very approximate values for the height of the center of gravity for several types of merchant ships at lightship, and for some naval ship types at full load. Calculation of KG can be a laborious and time-consuming process, but ignorance of the height of a ship's center of gravity invites disaster. If the height of the ship's center of gravity is known for any condition of loading (lightship, for example), and the location of added or removed weights is known, the new height of the center of gravity can be calculated`: KGnew = where: KG W w kg = = = = height of the ship's center of gravity, G, above the keel total weight of the ship and contents individual weights added (+) or removed (-) height above keel of centers of gravity of added or removed weights, w Wold KGold Wold w (kg) w

Table 1-4. Approximate KG.

KG (D = depth at midships)

Ship Type

Merchantmen (KG at lightship)1: Dry Cargo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passenger/Cargo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulated Cargo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oil Tanker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Naval ships (KG at full load)2: Cruiser/Destroyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frigate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amphibious Warfare without well decks (LST/LKA/LPH) . . . . Amphibious Warfare with well decks (LSD/LPD/LHA/LHD) . . Fleet replenishment (AE/AOE/AOR/AFS/AO) . . . . . . . . . . . . Tender/Repair Ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.55D 0.61D 0.63D 0.72D 0.62D 0.5D 0.68D 0.75D 0.72D 0.68D 0.69D

Source: 1 Applied Naval Architecture, R. Munro-Smith, 1967 2 Jamestown Marine Services, 1990



Height of the center of gravity of cargo can generally be obtained from the ship's officers, usually the chief mate. In the absence of better information, the design estimations proposed by R. Munro-Smith (Applied Naval Architecture, 1967) shown in Table 1-5 may be helpful. 1-5.4.2 Height of the Center of Buoyancy. The height of the center of buoyancy above the keel (KB) is solely a function of the shape of the underwater volume. As the centroid of the underwater hull, the center of buoyancy is lower in flat-bottomed, full-bodied ships, such as tankers and ore carriers, than in finer lined ships like destroyers or frigates. Disregarding changes in the shape of the immersed hull due to trim and heel, KB of any ship is a function of displacement, and therefore of draft. The height of the center of buoyancy can be calculated by summing incremental waterplane areas (aWP) multiplied by their heights above the keel (z) and dividing the result by the displacement volume (): KB = 1 a z dz wp

Table 1-5. Approximate KG of Cargo in Full Holds.

Hold/Space No. 1 No. 2 No. 3 No. 4 No. 5 'tween decks

KG of Cargo (D = depth of hold)

0.7D + depth of double bottom 0.7D + depth of double bottom 0.7D + depth of double bottom 0.7D + depth of double bottom 0.7D + depth of double bottom height above keel to half depth of 'tween deck at mid length of the space

Based on full holds (homogeneous cargo) in general cargo ship with machinery amidships, three holds forward and two aft. In ships with extensive parallel midbody, it may be more appropriate to apply the expression for hold No. 3 to all holds in the parallel midbody, with the expression for No. 1 or No. 2 (depending on fineness of forebody) applied to the forward most hold. A similar analysis should be applied to holds aft of the machinery space, if any.

This expression can be evaluated by numerical integration methods if accurate drawings or offsets are available. In practice, KB can be approximated with sufficient accuracy for salvage work as 0.52T for full-bodied ships and 0.58T for fine-lined ships. At very light drafts, KB is closer to the given waterline because the lower waterlines are usually much finer than the waterlines in the normal draft range. As a vessel's underwater hull form approaches a rectangular prism (CB = 1.0), KB approaches 0.5T. The following empirical relationships give estimates for KB that are very close to calculated values for merchant vessels of ordinary form at normal drafts: KB = 1 5T 3 2 AWP AWP AWP where: Tm AWP = = mean draft, [length] = displacement volume, [length3] waterplane area, [length2] Tm (Morrish's Formula)

KB = Tm

(Posdunine's Formula)

1-5.4.3 Metacentric Height. The transverse metacentric height (GMT), commonly called the metacentric height, of a ship is the vertical separation of the center of gravity and the transverse metacenter (see Figure 1-4) and is a primary indicator of initial stability. A ship with a positive metacentric height (G below M) will tend to right itself by developing righting arms as soon as an inclining force is applied. A ship with a negative metacentric height (G above M) will list to either port or starboard with equal facility until the centers of buoyancy and gravity are on the same vertical line, and thereafter develop positive righting arms. This condition, known as lolling, is a serious symptom of impaired initial stability. Metacentric height is calculated by subtracting the height of the center of gravity from the height of the metacenter above the keel: GMT = KMT - KG Transverse Metacentric Radius. The transverse metacentric radius (BMT) is the vertical distance between the center of buoyancy and the metacenter. This distance is termed a radius because for small heel angles, the locus of successive centers of buoyancy approximates a circular arc, with the transverse metacenter as its center. Metacentric radius is equal to the moment of inertia of the waterplane about its longitudinal centerline (transverse moment of inertia, IT) divided by the underwater volume of the hull (): BMT = IT



For a rectangular waterplane, IT = LB3/12, = LBT and: IT LB 3 12 LBT B2 12T



= where: L B T = = = length between perpendiculars, [length] beam, [length] mean draft, [length]

If the waterplane shape can be accurately defined, the moment of inertia can be determined by numerical integration. If not, the transverse moment of inertia of most ships' waterplanes can be approximated by: IT CIT LB3 where CIT is the transverse inertia coefficient and is approximated by CWP2/11.7 or 0.125CWP - 0.045. These expressions for transverse inertia coefficient are derived from the analysis of numerous ships, and are reasonable approximations for use in salvage for ships with CWP < 0.9. For ships with CWP > 0.9, LB3/12 is a closer approximation of the transverse moment of inertia of the waterplane. Height of the Metacenter. The height of the metacenter above the keel is calculated by adding the metacentric radius to the height of the center of buoyancy above the keel: KM = KB + BM GM = KB + BM KG

When denoting transverse metacenter, BM, KM, and GM, the subscript "T" is often omitted as understood. Ships with large GM develop large initial righting arms and therefore respond to moderate disturbing forces with sharp, short-period rolling. These ships are said to be stiff. Ships with smaller metacentric heights develop smaller initial righting arms and roll more gently in a seaway. Ships with small metacentric heights are said to be tender. Insufficient initial stability results in constant rolling in even gentle seas, making work difficult, and may allow extreme rolling in heavier seas, perhaps causing the ship to take on water or capsize. Excessive initial stability, or stiffness, is also undesirable because it produces an uncomfortable ride, reduces personnel effectiveness, increases requirements on weapons stabilization systems, increases lateral acceleration loads on topside cargo and equipment, and increases hull stresses. These matters usually do not concern the salvage engineer, but very stiff rolling of a casualty under tow may damage sensitive equipment, loosen patches, or place excessive loads on damaged structure. The term seakindly is used to describe a ship whose metacentric height is great enough to give adequate stability, but not large enough to cause excessive stiffness. The natural rolling period is a function of weight and buoyancy distribution and can be expressed as a function of GM and transverse radius of gyration (k): TR = where: TR k = = = GM = g = natural rolling period, seconds transverse radius of gyration of the ship mass, [length] 0.4 to 0.5 times the beam, depending on depth and transverse weight distribution transverse metacentric height, [length] acceleration due to gravity, [length/sec2] 2k g GM



If GM and k are expressed in feet, and g is taken as 32.174 ft/sec2, the rolling period formula reduces to: TR = and:

2 k GM = 1.108 TR

1.108 k GM

If the natural rolling period is known, GM can be estimated. Taking radius of gyration k as beam (B) multiplied by a coefficient (C), a conservative estimate of GM can be made: GM 2 CB T R

The coefficient C can be taken as 0.4 to 0.5 for naval surface ships (0.44 average), 0.4 to 0.45 for submarine hulls based on bodies of revolution, and 0.32 to 0.37 for other submarines. Ships and Marine Engines, Volume IV, The Design of Merchant Ships (Schokker et al, 1953) gives some experimentally derived values for commercial vessels: 0.425 for large cargo and passenger liners, 0.385 for smaller passenger liners, 0.390 for a loaded passenger liner, and 0.405 for an ore ship in ballast. This same text references Laursen's possibly more correct approach of expressing radius of gyration as a function of both beam and depth: k = C B2 + D2 where the constant C ranges from 0.35 to 0.39 for cargo ships of ordinary form. The rolling period formula will not give an accurate estimate of GM for a ship rolling in a seaway because the rolling period is modified by wave and wind forces. Significant changes in GM will be reflected by marked changes in rolling period; increased rolling period is a sign of deteriorating stability. An empirically derived relationship holds that stability is adequate when: TR 2 B where: B = beam, ft 1-5.5 Righting Arm. At equilibrium, the forces of gravity and buoyancy act equally in opposition along the vertical centerline. As the center of buoyancy shifts with a heel, the two opposing forces act along separate and parallel lines. The forces establish the couple which tends to return a stable ship to the upright position. The distance GZ between the lines of action of the center of gravity and the center of buoyancy, as shown in Figure 1-16, is the righting arm. The sine of the angle of inclination () is the ratio of GZ to GM. sin = GZ GM

GZ = GM sin

This relationship applies for heel angles so small that the waterplane shape is not appreciably changed, usually taken as less than 10 degrees for wall-sided ships and 7 degrees for fine-lined ships. At greater angles of heel, the metacenter moves away from the centerline and the relationship between GZ and GM no longer applies. 1-5.6 Righting Moment. The force applied to a righting arm (GZ) is the ship's weight. The righting moment (RM) developed at any angle of heel is given by: RM = W × GZ At any angle of heel, the stability of the ship is measured by the righting moment developed. Since the righting moment is equal to the righting arm times displacement and displacement normally remains constant as the ship heels, the righting arm may also be used to measure stability for a given condition of loading. This assumption lends itself to the use of the cross curves of stability as discussed in Paragraph 1-5.9.



1-5.7 Change of Displacement. Any change of displacement will affect the righting moments developed by the ship. An increased displacement increases W in the expression RM = W × GZ, but also affects GZ by:

· ·

Increasing draft and thereby KB. Increasing , thereby reducing BM as I will not change significantly (BM = I/).

The height of the metacenter is normally reduced as displacement increases because the increase in KB is usually less than the reduction in BM. The opposite effects will be noted when displacement is decreased. Additionally, the location of the added weight will affect the location of the center of gravity and therefore GM and GZ. These effects are simultaneous but not normally compensatory. The net effect of a change in displacement may be either an increase or a decrease in righting moments. In general, the addition of low weight or removal of high weight will increase stability, but each change of displacement must be carefully analyzed to determine its exact effect. 1-5.8 List. List, a long-term inclination of the ship to one side or the other, is caused by:

8 7



· · ·

Offcenter weight. Negative GM. A combination of offcenter weight and negative GM.

6 5 4 3 2 1


Before attempting to correct a list on a ship, the cause must be determined. Inappropriate corrective measures will only aggravate the situation. A list caused by offcenter weight is identified by the ship's tendency to return to its listing condition when an external force is applied temporarily and then removed. A list caused by negative GM is identified by the ship's tendency to loll, or list to either side with equal facility, when disturbed. A list caused by a combination of offcenter weight and negative GM is identified by the ship's tendency to list with equal facility to either side, but with a greater degree of list to one side. Negative GM is the most serious condition that causes a list and should be corrected first. Paragraph 1-9.4 discusses the effects of negative GM in greater detail.

8 7 6 5 4 3 2 1



8 7 6 5 4 3



1-5.9 The Stability Curve. The righting 2 arm GZ is the distance between the lines of 1 action of buoyancy and gravity at any 0 angle of heel. Since the expression GZ = 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 GM sin cannot be used at larger angles of INCLINATION, DEGREES heel, the righting arm for a given heel angle is determined by accurately locating Figure 1-18. Typical Stability Curves. the centers of gravity and buoyancy, and measuring the separation between their lines of action. If movable weights within the ship can be neglected, the center of gravity can be assumed to be fixed. As the ship heels, the center of buoyancy will move to the new center of the underwater volume, which can be determined by numerical integration or graphical means. As a ship heels, it also changes its trim to some extent to maintain constant displacement. This small change in trim can usually be disregarded when calculating righting arms. Centers of buoyancy for various inclinations, and the resulting righting arms are determined by numerical integration. These computations can be shortened somewhat by the methods described in Paragraph 1-5.11. A plot of righting arm against heel angle is variously called a curve of statical stability, stability curve, righting arm curve, or GZ curve. Figure 1-18 shows typical stability curves for various ship types.



1-5.9.1 Cross Curves of Stability. As a ship's displacement is variable, the designers prepare stability curves for a range of displacements. It is customary to plot righting arm values against displacement for each of a number of angles of inclination to create a group of curves known as cross curves of stability. By entering the cross curves with the displacement of the ship and reading the righting arms for each angle of heel, a stability curve for any displacement can be developed. Since height of the center of gravity varies with loading, an assumed position of the center of gravity was used by the designer to develop the cross curves of stability. Once the stability curve has been corrected for the true location of the center of gravity, the following stability data can be obtained:



70° 2 30°


20° 1 10°





· · · · · ·

Range of stability.

Figure 1-19. FFG-7 Class Cross Curves of Stability.

Righting arm and moment at any angle of inclination. Maximum righting arm and moment. Angle of the maximum righting arm and moment. Metacentric height. Angle of deck edge immersion.





The following examples use the FFG-7 Class cross curves of stability from Figure 1-19 to develop the initial and corrected stability curves. Figure 1-20 is the stability curve as taken from the cross curves for a displacement of 3,200 tons.













Figure 1-20. Statical Stability Curve.



1-5.9.2 Correction for Actual KG. If the actual center of gravity lies above the assumed center of gravity, the metacentric height is decreased and the ship is less stable; conversely, if the actual center of gravity is below the assumed center, the metacentric height is increased and the ship is more stable. Figure 1-21 shows that the actual righting arm, GnZn is equal to the assumed righting arm plus or minus the vertical distance between the actual and assumed KG, multiplied by the sine of the angle of heel: GnZn = GZ ± GGn sin The actual, or corrected, stability curve can be constructed graphically as a sine curve correction. The GGn sin curve is plotted to the same scale as the curve of statical stability as shown in Figure 1-22. The ordinates of the corrected curve are the differences between the ordinates of the two curves and can be picked off and plotted using dividers, as shown, or determined by tabular calculation. If the actual height of the center of gravity is less than the assumed height, the correction curve is plotted below the horizontal axis. The assumed KG is sometimes called pole height. It is a common practice, especially with European designers, to develop cross curves based on an assumed pole height of zero. Since the assumed position of the center of gravity coincides with the keel, the resulting cross curves are termed KN curves. 1-5.9.3 Range of Stability. The range of stability--the range of inclinations through The new stability curve is again the difference between the two curves.which the ship develops positive righting arms--is indicated by the intersections of the stability curve with the horizontal axis. For the corrected stability curve in Figure 1-22, the range of stability is from 0 to 75 degrees.

G1 Z1 = GZ - G G1SIN0 G2 Z2 = GZ + G G2SIN0 0 Z1 Z W1 W Z2 0 0


G1 G G2

L L1



Figure 1-21. Assumed KG for Stability Curve.




GZ for KG = 191, = 3200 TONS


GG1sin = 2sin




10 20 30 40 50 60 DEGREES OF INCLINATION 70 80 90







G1Z1 = GZ - GG1 sin















Figure 1-22. Correction to Stability Curve, G Two Feet Higher Than Assumed.



1-5.9.4 Righting Arm and Righting Moment. The righting arm at any inclination is read directly from the curve. Because each stability curve applies only to a specific displacement and KG, the righting moment can be obtained directly for any angle by multiplying the righting arm by the displacement. Maximum righting arm, maximum righting moment, and angle of maximum righting moment can be determined by inspection of the stability curve. From the corrected stability curve in Figure 1-22, maximum righting arm is approximately 1.1 feet at 51 degrees of inclination, giving a maximum righting moment of 3,520 foot-tons (1.1 ft × 3,200 tons). Maximum righting arm and the angle at which it occurs are important parameters when an upsetting moment is applied gradually or statically. Once the upsetting moment exceeds the maximum righting moment, the ship will list past the angle of maximum righting arm. If the upsetting moment is not immediately removed, the ship will capsize, because as the ship heels to progressively greater angles, righting moment, already less than the upsetting moment, will steadily decrease. However, ships can, and do, safely roll past their angle of maximum righting arm in response to short-term or cyclic upsetting forces. 1-5.9.5 Metacentric Height. GM is the measure of the slope of the GZ curve at the origin. The metacentric height is equal to the height of the intersection of a tangent to the statical stability curve at the origin with a perpendicular to the horizontal axis at 57.3 degrees (one radian). Although metacentric height can be approximated from a stability curve by this means, it is more common that GM is known and the intercept is sketched to help draw the initial part of the stability curve. The corrected stability curve in Figure 1-22 indicates a GM of approximately 1.2 feet. 1-5.9.6 Angle of Deck Edge Immersion. For most hull forms, an inflection point in the curve corresponds roughly to the angle of deck edge immersion. This point is not necessarily at or near the angle of maximum righting arm. The inflection results from the abrupt change in the shapes of the waterplane and underwater volume as the deck edge is immersed. The rate of increase in righting arm has changed from positive to negative--i.e., righting arms are still increasing, but at a slower rate. The angle of deck edge immersion varies along the length of the ship, but lies within a relatively narrow range for the large midbody sections that have the greatest influence on the stability curve. The stability curve in Figure 1-22 shows the angle of deck edge immersion to be about 38 degrees. 1-5.9.7 Righting Energy. The area under the stability curve, (foot-degrees, meterradians), is a measure of the ship's dynamic stability--its ability to absorb energy imparted by winds, waves or other external forces. A ship with very little area (righting energy) under its stability curve could be rolled past its range of stability and capsized by even a momentary disturbance. 1-5.10 Effects of Hull Form on the Stability Curve. While initial stability (righting arms at small angles of heel) depends almost entirely on metacentric height, the overall shape of the stability curve is governed by hull form. Figure 123 shows how changing hull form increases or decreases righting arm by altering the position and movement of the center of buoyancy. Figure 1-24 (Page 1-40) illustrates how altering hull form affects the stability curve as described in the following paragraphs.



TUMBLEHOME AND FLARE FINING THE BILGES 1-5.10.1 Beam. Of all the hull dimensions that can be varied by the designer, beam LOCAL INCREASE IN IMMERSED VOLUME has the greatest influence on transverse stability. Metacentric radius (BM) was LOCAL LOSS IN IMMERSED VOLUME shown to be proportional to the ratio B2/T in Paragraph 1-5.4.3. BM, and therefore Figure 1-23. Effects of Changing Hull Form. KM, will increase if beam is increased while draft is held constant. If freeboard is held constant while beam is increased, the angle of deck edge immersion is decreased; righting arms at larger angles and the range of stability are reduced. If the depth remains constant, overall stability will be reduced because KB decreases, increasing BG, although this will be offset at small angles by the increase in BM.



1-5.10.2 Length. If length is increased proportionally to displacement, with beam and draft held constant, KB and BM are unchanged. In practice, increasing length usually causes an increase in KG, reducing initial stability. If length is increased at the expense of beam, righting arms are reduced over the full range of stability. If length is increased at the expense of draft, righting arms will be increased at small angles, but decreased at large angles. 1-5.10.3 Freeboard. Increasing freeboard increases the angle of deck edge immersion, increasing righting arms at larger angles and extending the range of stability. If draft is held constant, increasing freeboard causes a rise in the center of gravity, mitigating the benefits of increased freeboard to some extent. 1-5.10.4 Draft. Reduced draft proportional to reduced displacement increases initial righting arms and the angle of deck edge immersion but decreases righting arms at large angles. 1-5.10.5 Displacement. If length, beam, and draft are held constant, displacement can be increased only by making the ship fuller. The filling out of the waterline will usually compensate for the increased volume of displacement, and BM, as a function of I/, will increase. Height of the center of gravity will also be decreased by filling out the ship's form below the waterline. These changes will enhance stability at all angles.







Figure 1-24. Influence of Hull Form on Stability.

1 S 1-5.10.6 Side and Bottom Profile. As M can be seen in Figures 1-13 and 1-25, the increase in waterplane breadth and area L W caused by inclining a wall-sided ship can Z G W1 be calculated by simple geometry. The stability curve develops good early righting B1 B arms and range of stability. Extreme deadGZ = MS+GMsin rise (fining the bilges) or tumblehome in the vicinity of the inclined waterline reC duces the increase in waterplane area and L outward shift of the center of buoyancy, resulting in a shallow stability curve. Ships Figure 1-25. Residuary Righting Arm. with flaring sides develop large righting arms because of the rapid increase in waterplane area and large shift of the center of buoyancy as the ship is inclined. A round-bottomed ship with vertical sides beginning somewhat above the water line, such as a tug or icebreaker, will roll easily to small angles of inclination but develop strong righting moments at large angles. In the same way, flare or watertight sponsons some distance above the water line will have no effect on initial stability, but will cause a sharp upward turn in the stability curve at larger angles of heel.




1-5.11 Prohaska's Method. As shown in Figure 1-25, the righting arm at large heel angles can be thought of as consisting of two parts: GZ = MS + GMsin The distance from the upright metacenter to the line of action of buoyancy (MS) is called the residuary stability lever. The GMsin term depends principally on KG, while MS is essentially a function of hull form. For inclinations up to about 30 degrees in merchant hulls of ordinary beam to draft ratio, MS can be approximated as: MS = where: BM = metacentric radius of the upright ship A more accurate approach is to define a residuary stability coefficient (CRS): CRS = where : BM = metacentric radius of the upright ship, [length] GZ can now be defined in terms of GM, BM, and CRS: GZ = (BM)CRS + GMsin Using this basic approach, a regression analysis was performed using data from 31 warship hulls to obtain expressions for CRS in terms of other hull parameters. The following expressions give reasonable estimates for CRS at 30 degrees of heel for fine-lined ships: CRS = 0.8566 1.2262 KB T B T 0.035 B T MS BM BM 2 tan sin 2

CRS = 0.1859 KB = 0.8109 T where: KB T B CM = = = = height of the center of buoyancy above the keel, ft mean draft, ft beam, ft midships section coefficient


0.03526 CM

0.2536 CM



1-6 LONGITUDINAL STABILITY Longitudinal stability is the measure of a ship's ability to resist rotation about a transverse axis and to return to its original position. Longitudinal stability is particularly important when refloating stranded ships. The effects of weight shifts, additions, and removal may not be apparent since a grounded ship is restrained from responding as a floating ship would. The effects must be calculated to ensure that the salvor can accurately predict trim and longitudinal stability of the vessel when afloat. 1-6.1 Trim. Because the angles of inclination about transverse axes are quite small compared to typical angles of heel about a longitudinal axis, trim is defined as the difference between the forward and after drafts: t = Taft where: t = trim Regardless of the difference between forward and after drafts, if a ship's waterline is parallel to the design waterline, it has zero trim. Most ships are designed with equal forward and after drafts. Some ships are designed with a deeper draft aft, called keel drag, to keep the propellers adequately submerged in all operating conditions, or with a slightly deeper forward draft. Drag or other designed differences in fore and aft draft should not be confused with trim. For ships with drag, trim is defined as: t = Taft Tfwd drag Tfwd

Trim greater than one percent of the ship's length is usually considered excessive. Excessive trim significantly alters the shape of the underwater volume and can adversely affect transverse stability. 1-6.2 Longitudinal Stability Parameters. The longitudinal positions of centers of buoyancy, gravity, and flotation and their movements influence the longitudinal stability characteristics of a ship. The height of the longitudinal metacenter, similar in concept to the transverse metacenter, is the other major parameter of longitudinal stability. 1-6.2.1 Longitudinal Position of the Center of Gravity. The longitudinal position of the center of gravity (LCG) is determined by summing weight moments about a vertical transverse reference plane, normally through one of the perpendiculars or the midship section. 1-6.2.2 Longitudinal Position of the Center of Buoyancy. The longitudinal position of the center of buoyancy (LCB) is the longitudinal location of the centroid of the underwater hull. For most hull forms, LCB lies near the midships section. For low-speed, full-bodied cargo vessels, the optimum position of the center of buoyancy (from a hull resistance standpoint) is about 0.02LWL forward of midships. As speed increases, the optimum position moves aft. At a speed-to-length ratio (Vk/L) of 1.0 the optimum position is 1 to 2 percent of LWL aft of midships and about 4 percent aft of midships for Vk/L = 2. Table 1-6 gives approximate ranges for the longitudinal position of the center of buoyancy as a function of the block coefficient. In a ship at rest, the longitudinal positions of the centers of gravity and buoyancy lie on the same vertical line. LCB and LCG are therefore the same distance from the midship section in a ship floating on an even keel. In a ship with trim, there is a small difference in the distances of B and G from midships due to their vertical separation, but this difference is so small that it can usually be ignored.

Table 1-6. Longitudinal Position of the Center of Buoyancy.


0.60 0.65 0.70 0.75 0.80

LCB Relative to the Midship Section

0.016L aft to 0.002L forward 0.011L aft to 0.009L forward 0.002L aft to 0.020L forward 0.010L forward to 0.027L forward 0.015L forward to 0.030L forward

From Ships and Marine Engines, Volume IV, Design of Merchant Ships, Schokker et al, 1953

1-6.2.3 Longitudinal Position of the Center of Flotation (LCF). The center of flotation is the geometric center of the ship's waterplane. The ship trims about a transverse axis through the LCF. The location of the center of flotation is required to calculate final drafts after a change in trim. This can be calculated if the shape of the waterplane is known. In ships of normal form, the center of flotation may lie either slightly forward or slightly aft of midships. The center of flotation of fine-lined ships is usually about five percent of the ship's length aft of midships. A broad transom increases the relative proportion of waterplane area aft of midships and will tend to shift LCF aft. If unknown, the center of flotation can be assumed to be amidships without introducing significant error to most salvage calculations. 1-6.2.4 Longitudinal Metacenter. The longitudinal positions of the centers of buoyancy and gravity are simply projections of these centers onto the vertical centerplane. The longitudinal metacenter, in contrast, is a point distinct from its transverse counterpart. Its height is an indication of the ship's ability to resist trimming forces. Longitudinal Metacentric Radius. The longitudinal metacentric radius (BML) is the vertical distance between the center of buoyancy and the longitudinal metacenter. The longitudinal metacentric radius is calculated by: BML = IL



If the waterplane shape is defined by ordinate stations, the moment of inertia can be determined numerically. If not, the longitudinal moment of inertia of most ships' waterplanes can be approximated by: IL B L 3 CIL where CIL = tegression analysis derived longitudinal inertia coefficient, approximated by 0.143CWP - 0.0659. For a rectangular barge, IL = B(L3)/12; the value of CIL for a rectangular waterplane (the limiting value) is 1/12 or 0.0833. Because the longitudinal moment of inertia is proportional to the cube of the ship's length rather than beam, the longitudinal moment of inertia and longitudinal metacentric radius are much greater than their transverse counterparts. Height of the Longitudinal Metacenter. The height of the longitudinal metacenter (KML) is given by: KML = KB + BML Longitudinal Metacentric Height. The longitudinal metacentric height (GML) is the distance between the center of gravity and the longitudinal metacenter. GML = KML - KG = KB + BML - KG 1-6.3 Trimming Arms and Moments. If the center of gravity is displaced from its longitudinal position in vertical line with the center of buoyancy, as shown in Figure 1-26, a trimming moment (MT) equal to GG1(W) tends to rotate the ship about a transverse axis through the center of flotation. As the ship inclines, the shape of the underwater volume changes and the center of buoyancy moves until it is again in line with the center of gravity. Simultaneously, the projection of the position of the center of gravity onto a horizontal plane moves towards the high end of the ship. For small trim angles, the horizontal translation of the position of the center of gravity can be neglected. The trim resulting from a known trimming moment could be determined precisely by iterative numerical integration, but this would be a tedious process. Simple methods to estimate trim with reasonable accuracy are described in the following paragraphs. A ship trims about an axis through its center of flotation because LCF lies at the centroid of the waterplane. The moments of volumes of the wedges immersed and emerged as the ship trims are equal, although the volumes are not. Because the volumes are not equal, the ships will settle or rise slightly as it trims to maintain constant displacment. LCF also shifts slifhtly as the ship trims and changes draft.






W LCF W1 B G1 B1

L1 L

Figure 1-26. Trim due to Shift in LCG.



W W1

Figure 1-27. Trimming Moments and Longitudinal Metacenter.

1-6.4 Moment to Change Trim One Inch (MT1). A trimming moment applied to the ship in Figure 1-27 causes a longitudinal inclination or trim angle, . The immersion and emergence of the two wedges of buoyancy causes the center of buoyancy to move forward a distance BB1. A longitudinal righting arm GZL develops. Because the small vertical separation between B and G is much less than the longitudinal metacentric height, GZL and BB1 are approximately equal. The moment arm GZL can be related to the longitudinal metacentric height as in transverse inclinations: GZL sin = , GZL = GML sin Mt = W GML sin GML where: GZL = GML longitudinal righting arm, [length] = longitudinal metacentric height, [length] Mt = trimming moment, [length-force] W = ship's weight, [force]



By similarity of triangles: sin = where: t L = = change of trim, [length] = Tf ± Ta length between perpendiculars, [length] t L

Setting change in trim equal to one inch or 1/ 12-foot: Mt = where: GML Mt W L = = = = longitudinal metacentric height, ft trimming moment, ft-tons ship's weight, lton length between perpendiculars, ft W (GML ) 12 L

This moment is called the moment to change trim one inch (MT1); in metric units, a moment to trim one centimeter (MTCM) is similarly defined. MT1 is useful for evaluating the effect of trimming moments so long as the change in trim is not great enough to change the waterplane area or shape appreciably: Mt t = MT1 If longitudinal metacentric height (GML) is unknown, MT1 can be closely approximated by using metacentric radius (BML), since the difference between GML and BML is small a percentage of their values: IL (BML ) W IL 35 = (seawater) MT1 = 12 L 420 L 12 L This value is known as the approximate moment to trim one inch. MT1 can also be approximated less accurately by an empirical relationship: MT l = where: TPI = B = tons per inch immersion, lton/in ship's beam, ft 30 (TPI)2 B

1-6.5 Drafts After a Change in Trim. As a ship trims about the center of flotation, the change in draft at the bow is proportional to the ratio of the distance between the forward perpendicular and the center of flotation to the length of the ship: Tf = t df

L New Tf = Tf ± Tf Likewise, the change in draft aft: L New Ta = Ta ± Ta where: Tf t df L Ta da = = = = = = change in draft forward change of trim distance from forward perpendicular to LCF length between perpendiculars change in draft aft distance from after perpendicular to LCF Ta = tda

and distance, draft, trim, and length are measured in like units.



1-6.6 Movement of LCB and LCG with Change of Trim. As discussed in Paragraph 1-5.3, movements of LCB and LCG accompany changes of trim. From Figures 1-26 and 1-27: BB1 GG1 t = = tan = BML GML l BB1 = where: BML GML t L = = = = = longitudinal metacentric radius longitudinal metacentric height change of trim length between perpendiculars trim angle BML t L , and GG1 = GML t L

and trim and length are measured in like units. The shift of LCG or LCB with a change in trim can be closely approximated by: t (MT1) BB1 or GG1 = W where: t MT1 W = = = change of trim, in. moment to trim one inch, lton/in. ship's weight, lton

1-7 PARAMETRIC DETERMINATION OF HULL CHARACTERISTICS The hull characteristics of a ship are determined and tabulated when the ship is designed and verified following construction. This information is contained in a number of different documents, described in detail in Appendix B. The two most useful documents are the previously discussed cross curves of stability and curves of form. In the absence of detailed stability information or the precise mapping of the hull form necessary to develop hydrostatic characteristics by numerical integration, hull characteristics must be estimated. Methods of estimating some of the required parameters have been presented in the previous sections. When information is extremely limited, an analytical method, based on a parametric hull model, can be employed. This method has been shown to yield results within 10 percent of rigorously determined values for most ship forms. The parametric method has its inception in a regression analysis of 31 commercial hull types published by Joseph D. Porricelli, J. Huntly Boyd, Jr., and Keith E. Schleiffer in the Society of Naval Architects and Marine Engineers Transactions, Vol.91, pp. 307-327, August 1983. Many of the relationships were subsequently refined though further regression analysis by Herbert Engineering Corporation as part of the NAVSEA Program of Ship Salvage Engineering (POSSE) development work in 1990 (use of POSSE is detailed in Volume 2 of the Salvage Engineer's Handbook). At the same time, relationships for stability parameters and weight distributions applicable to warships and other finelined ships were developed. The parametric factors for warships and naval auxiliaries were derived from analysis of U.S. Navy hulls and may not apply precisely to ships of other navies. This is particularly true of amphibious warfare ships and fleet replenishment auxiliaries. U.S. Navy amphibious warfare ships and replenishment auxiliaries are designed for a 20-knot service speed and are correspondingly finer than slower auxiliaries and bow-door LSTs with typical speeds in the 10- to 16-knot range. 1-7.1 Parametric Model. The method creates a baseline parametric model of the hull, consisting of the following parameters for the full-load condition: Coefficients of form, CB, CWP, CP, CM Displacement and weight, D, W Height of the center of buoyancy, KB Height of the Metacenter, KM Height of the Center of Gravity, KG Metacentric radius, BM Metacentric Height, GM Tons per Inch Immersion, TPI Moment to Trim One Inch, MT1 Longitudinal position of the center of buoyancy, LCB Longitudinal position of the center of flotation, LCF Longitudinal position of the center of gravity, LCG Parameters for other conditions are extrapolated from the baseline, or full-load model.



1-7.1.1 Required Information. This method requires only limited information: Length between perpendiculars, L Breadth, B Depth, D Maximum summer draft amidships, T Design sea speed at normal service draft, Vk This information is available from sources such as the ABS Record, Jane's Shipping Registry, Lloyds Register of Shipping, etc., or may be compiled from other sources, including the ship's crew or agents. 1-7.1.2 Displacement and Coefficients of Form. To determine the necessary hydrostatic characteristics of a ship, the coefficients of form are first estimated, starting with the block coefficient: V CB = f1 1.10736 0.550401 k L where: Vk L f1 = = = design sea speed at service draft, knots length between perpendiculars, ft 1.61 for destroyer type hulls (including cruisers based on destroyer hulls, such as CG-16, CG-26, CG-47, etc.) 1.41 for frigates 1.28 for cruisers 1.08 for bulk carriers 1.06 for liquid petroleum gas (LPG) carriers 1.04 for liquid natural gas (LNG) carriers 1.03 for ore-bulk-oil (OBO) carriers 1.03 for lumber ships 1.025 for product tankers/chemical carriers 1.01 for crude carriers 1.00 for breakbulk freighters and most barges with rake* 0.98 for cargo liners (16-18 kts) 0.97 for container ships 0.96 for Navy replenishment oilers (Vk 20 kts, AO/AOE/AOR) 0.95 for RO/RO ships 0.93 for Navy replenishment vessels other than oilers (Vk 20 kts, AE/AFS) 0.91 for amphibious warfare ships (LSD/LPD/LPH/LKA/LST) 0.89 for barge carriers, Navy repair ships/tenders (AR/AD/AS)


In the context of the following discussions, the phrase "barges with rake" refers to ocean going barges with raked, ship-shaped or spoon-shaped bows, and cut-up sterns, usually with skegs. It does not apply to box-shaped lighters or to barges designed for harbor use with identical flat rake at bow and stern. CWP = k1 0.702 CB

Waterplane coefficient:

where: k1 = 0.360 0.325 0.336 0.339 0.387 0.370 0.316 0.306 for for for for for for for for barge carriers and barges with rake container ships RO/RO ships naval repair ships/tenders destroyers, frigates, and cruisers well deck type amphibious warfare ships (LSD/LPD) Navy replenshishment ships and fast LKA, LST (20 kts) other merchant ship types and slow-speed naval auxiliaries CP = 0.917 CB + k2 where: k2 = 0.073 for merchant ships and naval auxiliaries 0.075 for barges with rake 0.147 for destroyers, frigates, and cruisers

Longitudinal prismatic coefficient:



Midships coefficient: CM = CB CP

With an estimate for block coefficient, displacement volume and displacement can be estimated: = L B T CB = where: = D = displacement volume at full load full-load displacement = W = specific volume of water = 35 ft3/lton for seawater ship's weight at full load L B T CB = W

1-7.1.3 Heights of Centers. Height of the center of buoyancy (KB) is estimated by a form of Posdunine's formula: KB = where: CWP = CB waterplane coefficient = block coefficient T = mean draft CWP CB + CWP T

Metacentric radius is equal to the transverse moment of inertia of the waterplane (IT) divided by the displacement volume (): BM = IT can be expressed as: IT = L B 3 CIT where CIT is the transverse inertia coefficient and is a function of waterplane shape. CIT is determined from the waterplane coefficient (CWP): CIT = 0.125 CWP = 0.125 CWP 0.045 0.043 for ships for barges with rake IT

Transverse metacentric height for the full-load departure condition (corrected for free surface) is correlated to beam, or beam to depth ratio, depending on ship type: B for cargo liners and container ships GM = 2.816 - 1.88 D B = 15.86 - 19.62 D B = 0.714 + 2.2 D = f2 B B2 T = f3 + - 0.53 D 12 T 2 where: f2 = 0.055 for barge carriers and RO/RO ships 0.065 for bulk carriers 0.075 for OBO carriers f3 = = 1.18 for barges with rake 1.00 for barges without rake for tankers in general

for cargo ships in general for other merchant ship types for barges



From the estimates for KB, BM, and GM, KM and KG can be estimated: KM = KB + BM KG = KM - GM Since the estimate for KG is based on the parameterized GM estimate, the value returned is the virtual, or effective KG (corrected for free surface). GM does not parameterize well for U.S. Navy hulls because Navy stability standards (described in Appendix C) do not include minimum GM requirements. Uncorrected full-load KG does parameterize well, as a function of depth: KG = f4D where: f4 = = = = = = 0.55 0.61 0.63 0.72 0.62 0.50 for for for for for for cruisers and destroyers frigates amphibious warfare ships without well decks amphibious warfare ships with well decks (LSD/LPD) fleet replenishment auxiliaries repair ships/tenders (Navy hulls)

For Navy hulls, GM (uncorrected for free surface) is calculated from the estimates for KB and KG. The parametric factors were derived from an analysis of U.S. Navy hulls and may not apply precisely to ships of other navies. 1-7.1.4 Tons Per Inch Immersion. TPI is calculated directly, using the estimated waterplane coefficient to estimate waterplane area: L B CWP TPI = 420 where L and B are measured in feet. 1-7.1.5 Moment to Trim One Inch. A value for MT1 is found using estimates for longitudinal metacentric height or radius: GML W 12 L where BML is given by: BML = IL = IL 35 W MT1 = IL 420 L BML W 12 L

MT 1 =

The longitudinal moment of inertia, IL, of a ship-shaped waterplane can be expressed as: IL = B L 3 CIL where the longitudinal inertia coefficient, CIL, is given by: CIL = 0.143 CWP - k3 where: k3 = 0.0659 0.0664 0.0643 0.0634 for for for for merchant ships and slow-speed auxiliaries replenishment auxiliaries amphibious warfare ships destroyers, frigates, and cruisers



1-7.1.6 Longitudinal Positions of Centers. The distance from the forward perpendicular to LCF, LCB, and LCG can be estimated as follows. LCF is estimated as a function of speed (Vk) and length (L): V LCF = 0.5L k + 0.914 160 V = 0.485 L k 100 V = 0.5 k 135 0.95 = 0.5 L V k V = L 0.5 k 135 V = 0.5 L k 135 where Vk is given in knots and L in feet. LCB at full load and zero trim is approximated as a function of length (L) and prismatic coefficient (CP): LCB = L 0.5 - 0.175CP - k4 where: k4 = 0.125 0.111 0.117 0.126 0.146 for for for for for merchant ships and slow-speed auxiliaries replenishment auxiliaries amphibious warfare ships destroyers, frigates, and cruisers barges with rake 0.9 for tankers

for bulk carriers

0.924 1.03 0.924 0.95 0.23

for single-screw cargo ships and naval auxiliaries

for twin-screw cargo ships with transom sterns

for twin-screw cargo ships with cruiser sterns

for barges with rake

To estimate the longitudinal position of the center of gravity, trim must be known or estimated. If unknown, trim can be estimated from similar ships as a percentage of length. Multiplying trim (t) in inches by MT1 gives the trimming moment Mt: MT 1 (t) = Mt Trimming moment divided by weight (W) gives the trim arm or lever (GZL): = GZL W Since the trim arm is the horizontal separation between LCB and LCG prior to trimming: LCB ± GZL = LCG Upon trimming, LCB will relocate to a position in vertical line with LCG. LCG can be assumed to be directly above the estimated LCB for a ship with zero trim at normal full-load departure condition. 1-7.2 Changes. The values calculated are for the full-load departure conditions, and must be corrected for other conditions. Floating or grounded drafts can be observed on site. New floating displacement, drafts and location of center of gravity are determined by evaluating the effects of all weight changes from the normal full-load departure condition. Hydrostatic properties are assumed to vary linearly with draft according to: TPI2 = TPI1 MT12 = MT11 LCB2 = LCB1 LCF2 = LCF1 TPI1 0.0075 T1 - T2 MT11 0.025 T1 - T2 LCB1 0.002 T1 - T2 LCF1 0.004 T1 - T2 Mt

Where the subscript 1 denotes the full-load condition and the subscript 2 the new condition. The drafts T1 and T2 are taken at the LCF for each condition.







1-7.3 Calculation Hierarchy. Only CB, GM (or KG), and LCF are calculated directly from the basic input data (L, B, T, D, Vk). Because other parameters are successively calculated from previously calculated parameters, basic data, and empirically derived factors, there is a hierarchy of accuracy among the calculated parameters. This hierarchy is shown in the two panels of Figure 1-28. Two panels are used to reduce the complexity of the diagram. The basic input parameters are listed across the top of each of the two panels. 1-7.4 Cautions. The parametric method described in this paragraph was developed through regression analysis of typical, conventional hull forms. The less typical a particular hull, compared to ships of the same type, the greater the error introduced by use of the relationships given. As this method is based primarily on analysis of the speed-to-length ratio, errors will be larger for an underpowered hull--for example, a hull designed for 20 knots but actually powered for only 16 knots.




Figure 1-28. Calculation Hierarchy.

Because of the interdependence among various parameters, changing any parameter (except LCF) creates a ripple effect that necessitates recalculation of other parameters. Mixing bits of actual data with data calculated by the analytical method in a set of salvage calculations without recalculating lower precedence parameters tends to give poorer results than complete sets of either calculated or actual data. Specifically, hydrostatic properties and coefficients of form must be compatible. Within the framework of these limitations, the parametric method yields results sufficiently accurate for salvage work, and provides a means to evaluate a casualty's condition when only limited information is available. 1-7.5 Applications to Salvage Calculations. The nature of the relationships in the analytical method dictates the methodology of their use. From the input data, the method calculates parameters and creates a baseline ship model in the full-load condition. From the base condition, parameters at other conditions are calculated by one of two approaches.


The new condition is defined by drafts (for example, drafts on departure from last port). Change in block coefficient is calculated first. With the new block coefficient, mean draft and trim, a new set of parameters is calculated. The difference between old and new displacements gives the required weight change between the full-load and new condition. If the change in draft results from stranding, the difference between old and new displacements is the ground reaction. This approach can also be used to determine the amount and LCG of weight that must be added or removed to reach a desired draft and trim. The new condition is defined by change in weight (consumption of fuel and consumables, flooding, cargo discharge, etc.). The sum of weight change and old displacement gives the new displacement. Change in draft is calculated from the total weight change and TPI. For large weight changes, the change in draft is calculated incrementally, recalculating TPI for each intermediate draft. Shift of LCG is calculated by moment balance. A new block coefficient is calculated from the new displacement and mean draft. With the new block coefficient and mean draft, a new set of parameters is calculated as for the full-load condition, except that the new LCB is calculated from the new LCG.






Longitudinal locations are referenced to the forward perpendicular. These relationships apply only so long as the change in draft or trim does not cause a significant change in the shape of the underwater hull form. KM does not vary significantly with draft until draft is dramatically decreased, to approximately two-thirds full-load draft, after which it increases.











The salvage engineer must fully appreciate the relationship between weight and ship stability. The addition and removal of weight is the most common evolution affecting a ship's stability and can be the result of onloading and offloading cargo and equipment, refueling, consuming stores or fuel, ballasting, etc. Weight additions and removals have three effects:

· · ·

Change of displacement with attendant change of draft. Movement of the center of gravity. Development of trimming or inclining moments.

Displacement changes cause draft changes and changes in the hull characteristics. The change in the transverse metacentric radius is particularly important because of its potential effect on stability. Both weight additions and removals may change the moment of inertia of the waterplane. Weight additions will increase and weight removals will decrease displacement volume. Table 1-7 illustrates the general effect of weight changes on an intact ship. To evaluate a weight change, it is simplest to assume that the weight is added or removed at the center of gravity (G) for the purpose of calculating the effect on mean draft, and then moved to its final location in a series of steps to account for the effects of its vertical, transverse, and longitudinal moments.

Table 1-7. Effect of Weight Movements. CENTER OF GRAVITY Up Down Port/Starboard No change Up Down No change Down Up CENTER OF BUOYANCY No change No change To low side Up Up Up Down Down Down

ACTION Weight shift up Weight shift down Weight shift transverse Weight added at G Weight added above G Weight added below G Weight removed at G Weight removed above G Weight removed below G

STABILITY Decrease Increase Decrease Decrease Decrease Increase Increase Increase Decrease

METACENTER* No change No change No change Down Down Down Up Up Up

*Relative movement of metacenter is based on the relationship BM = I / and assumption that waterplane shape and area do not change appreciably for moderate changes of draft and displacement. As draft increases with added weight, the reduction in BM [I /] is greater than the rise of B. Conversely, as draft and displacement decrease, the increase of BM is greater than the lowering of B.

1-8.1 Weight Shifts. When weights are moved about the ship, displacement and mean draft remain constant; stability parameters that are functions of displacement or draft, such as height of metacenter, are therefore unaffected. The distance the center of gravity moves when a weight is shifted is the product of the weight (w) times the distance moved (d), divided by the total weight of the ship (W): wd GG1 = W This distance can be resolved into vertical, transverse, or longitudinal components. A single weight shift can cause any combination of transverse, vertical, or longitudinal shifts of the center of gravity with attendant effects on longitudinal and transverse stability. Although they occur simultaneously, each effect can be assumed to occur independently; the effects can be calculated separately as though they were occurring sequentially. Change of KG alters GM and righting arms as discussed in Paragraph 1-5. The effects of longitudinal and transverse weight shifts are discussed in the following paragraphs.



1-8.1.1 Longitudinal Effects of Weight Shifts. When a weight movement has a longitudinal component, LCG shifts and the ship's weight acting through the new center of gravity and buoyancy acting through the old center of buoyancy form a couple, or trimming moment, as shown in Figure 1-26. The magnitude of the trimming moment is: Mt = W GG1 where: W Mt GG1 = = = ship's weight trimming moment longitudinal distance from the old LCG to the new LCG

W1 W



L Z G1 B G L1

The trimming moment is also equal to the product of the weight moved (w) and the longitudinal distance moved (d). Mt = wd 1-8.1.2 Offcenter Weight. The effect of offcenter weight is to create an inclining moment. This effect can be evaluated by calculating the lateral movement of the ship's center of gravity off the centerline. The magnitude of the inclining moment is: MI = W(GG1) where: GG1 MI W = = = lateral (horizontal) shift of center of gravity, [length] inclining moment, [forcelength] ship's weight (including the offcenter weight), [force] wd GG1 = W where: d = lateral (horizontal) distance that the weight w is moved, [length] W GG1 = wd = MI


Figure 1-29. List Due to Transverse Shift of G.




W Z Z1 B T


L G L1





Figure 1-30. Reduced Righting Arm due to Transverse Shift of G.

The inclining moment will cause the ship to list to an angle where the center of buoyancy is again in vertical line with the center of gravity. The angle of list becomes the new equilibrium position; when disturbed, the ship will roll about the angle of list. The effect of a permanent list is to reduce the righting arms and range of stability when the ship rolls towards the list, and increase them when the ship rolls away from the list. For small angles of inclination (less than 7 to 10 degrees), list can be found by reference to the metacentric height. From Figure 1-29, the list due to an offcenter weight can be seen to be: GG1 tan = GM wd = tan 1 W GM



1-8.1.3 Stability Curve Correction for Offcenter Weight. Figure 1-30 shows a ship whose center of gravity has moved from G to G1. When inclined towards G1 to some angle , the righting arm developed is not GZ, but a smaller arm, G1Z1. The reduction in righting arm (GT) is: GT = GG1cos As with the sine correction for actual KG, the offcenter weight correction, as a cosine curve, is plotted to the same scale as the curve of statical stability as shown in Figure 1-31. The corrected stability curve is the difference between the two curves. The angle at which the corrected curve crosses the horizontal axis is the angle of list caused by the offcenter weight. Extending the curve to the left of the origin shows the increased righting arms developed on the side away from the list. In dynamic situations, the increase in righting energy on the side away from the list does not increase stability because the ship will roll about the angle of list. If the ship is subjected to a constant upsetting force, such as a steady beam wind, the increased righting arms provide additional stability if the ship is oriented so that the upsetting force heels the ship away from the list, towards its strong side. The increased righting arms and energy must also be overcome if the salvage plan calls for the ship to be heeled away from the list by external forces. It should also be remembered that if the ship is heeled towards its strong side, the area under the curve from the point where the curve crosses the axis to the angle of heel represents stored energy. If this area is larger than the area under the stability curve on the weak side, the ship could capsize if suddenly released.



2 1 0 1 2 3 90 80 70 60 50 40 30 20 10



















Figure 1-31. Correction to Statical Stability Curve for Transverse Shift of G.

1-8.2 Weight Additions and Removals. Weight addition or removal at the center of gravity changes displacement without introducing trimming or inclining moments. The increase or decrease in mean draft in inches (T) is approximately equal to the weight added or removed (w) in tons divided by the tons per inch immersion (TPI): w T = TPI



1-8.2.1 Weight Changes Away From the Center of Gravity. When weights are added or removed at some distance from the center of gravity, the center of gravity moves toward the added weight, or away from a removed weight, to a new position determined by the size and location of the weight. The weight change can be treated as an addition (or removal) at the center of gravity, followed by a shift to the location where the weight is added: (Gg) (w) GG1 = (W1) where: GG1 Gg W1 w = = = = shift of ship's center of gravity, [length] distance between ship and added weight centers of gravity, [length] = the distance d that the weight is "shifted" new total weight of ship, [force] = W ± w weight added (+) or removed (­)

The new vertical, transverse, and longitudinal positions of the center of gravity can also be calculated directly, by summing moments. Height of the center of gravity is given by: W (KG) ± w (kg) KG1 = W ± w where: KG1 W KG w kg = = = = = height of the ship's center of gravity after weight change, [length] original weight (displacement) of the ship, [force] original height of the ship's center of gravity, [length] weight added (+) or removed (-), [force] height of the center of gravity of the added or removed weight above the keel, [length]

New transverse and longitudinal positions of the center of gravity can be determined by the same method. A longtitudinal moment caused by weight addition or removal will not necessarily trim the ship. Most ships are not symmetrical about a transverse axis; as a ship settles or rises, the change in buoyancy is weighted towards one end, causing LCB to shift towards the fuller end. If the buoyancy moment generated by the shift in LCB equals the trimming moment, the ship will not trim. Conversely, a weight added directly above or below the center of gravity may cause the ship to trim to keep the centers of buoyancy and gravity in vertical line. For any weight addition or removal, a ship will assume the trim that brings the center of buoyancy directly under the new center of gravity. The trim resulting from a weight change can be determined very precisely by calculating LCB for trimmed waterlines at the new displacement until a trim is found that brings LCB under LCG. Simpler approximate methods to determine trim resulting from weight changes can be derived by determining where weights must be added or removed from a ship to change draft without changing trim. These methods are described in the following paragraphs, and are sufficiently accurate for virtually all situations. 1-8.2.2 Weight Changes Without Change of Trim. If a weight is to be added to a ship without changing trim, it must be added at a location that will be in vertical line with the resultant upward force of the added buoyancy. If the rise or sinkage is parallel, the added buoyancy results from the immersion of a layer of uniform thickness between the old and new waterplanes. The center of buoyancy of this layer is very close to the midpoint of a line connecting the centroids (centers of flotation) of the old and new waterplanes. For small draft changes through a ship's normal range of drafts, the old and new waterplanes are very nearly the same size and shape. The line connecting the centroids is therefore essentially vertical and the center of buoyancy of the immersed layer is in line with the centroid of the old waterplane, or center of flotation. For moderate weight changes, causing small changes in draft, at locations other than the center of flotation, trim can be closely approximated by: a. Taking the distance from the added or removed weight to the LCF as the trimming arm, b. Multiplying the trimming arm by the weight to determine trimming moment, and c. Dividing the trimming moment by MT1 to find the resulting trim. For larger weights whose addition or removal causes draft changes large enough to appreciably change hydrostatic functions, the trimming arm is taken as the distance from the new LCG to the LCB at the new waterline. Since TPI varies with draft, an iterative solution is required, as shown in Example 1-3.




This example calculates trim resulting from moderate (causing small changes in draft) and large weight additions at various locations on an FFG-7 Class ship. a. Calculate the change of trim when a 100-ton weight is added to an FFG-7 Class ship at the following locations: (1) Center of Flotation. First estimate of new mean draft: (2) Center of Gravity. (3) 50 feet abaft the forward perpendicular. FFG-7 Curves of Form are given in Figure FO-2. Initial drafts are 14 feet, 6 inches, forward and aft, LBP is 408 feet. From the curves of form: T Tnew = w/TPI = 1,000/32 = 31.25 31 inches Told - T = 14' 6" - 31" = 11 feet 11 inches b. Calculate the location for the center of gravity of 1,000 tons of weight to be removed from an FFG-7 Class ship with initial drafts of 14 feet 6 inches forward and aft without changing trim.


Second estimate of new mean draft:


= = = = =

32 23.4 feet abaft midships LCG = 1.4 feet abaft midships 745 foot-tons 3,495 tons

TPI at 11' 11" TPIavg T Tnew = LCF at 11' 9"

= = = Tnew =

28.5 (32 + 28.5)/2 = 30.25 1,000/30.5 = 33.06 33 inches - T = 14' 6" - 33" = 11 feet 9 inches 14 feet abaft midships

Center of buoyancy of immersed layer (lcb) is approximately midway between the old and new LCF, (23.4 + 14) = 18.7 feet abaft midships lcb = 2 Removing the 1,000 tons so that the center of gravity of the removed weight is approximately 19 feet abaft midships will cause no noticeable trim. c. Calculate the change in trim for an FFG-7 Class ship with initial drafts of 14' 6" forward and aft if 1,000 tons are removed from the following locations: (1) LCF.

Calculate the increase in mean draft: T =


w 100 = 3.125 inches 3 inches = TPI 32 = Tnew T = 14 feet 6 inches 3 inches = 14 feet 9 inches

Calculate the change in trim for 100 tons added at: (1) Center of Flotation

(2) LCG. The change in draft is small, so adding the weight at LCF causes no change of trim. This is verified by observing that the LCF at the new mean draft of 14 feet 9 inches is 23.5 feet. The center of the new waterplane (LCF) is only 0.1 foot from the center of the old waterplane, so the center of buoyancy of the immersed layer is essentially directly over the old LCF. (2) Center of Gravity Trim arm = = = = distance from LCF to added weight 23.4 - 1.4 = 22 feet w(trim arm) = 100(22) = 2,200 foot-tons Mt / MT1 = 2,200/745 = 2.95 3 inches by the bow (3) 100 feet forward of midships.

Tnew LCB at 11' 9" MT1 at 11' 9" MT1avg

= = = =

11' 9" (from part b.) 6 feet forward of midships 565 foot-tons (745 + 565)/2 = 655

(1) 1,000 tons removed at original LCF

Mt t


trim arm

= = = = = = = =

(3) 50 feet abaft the forward perpendicular 50 feet abaft the forward perpendicular is 154 (204 -50) feet forward of midships Trim arm Mt t = = = 23.4 + 154 = 177.4 feet 100(177.4) = 17,740 foot-tons Mt/MT1 = 17,740/745 = 23.81 23 inches by the bow

Mt t

(Gg)(w)/(W + w) 23.4 -1.4 = 22 feet (22)(1,000) / (3,495 - 1,000) = 8.8 feet forward -1.4 feet (aft) + 8.8 feet (forward) = 7.4 feet forward of midships distance from new LCG to new LCB 7.4 - 6 = 1.4 feet (LCG is forward of LCB) 1,000(1.4) = 1,400 foot-tons Mt/MT1 = 1,400/655 2 inches by the bow

(2) 1,000 tons removed at original LCG

GG1 LCG1 trim arm Mt t

= = = = =

0 1.4 feet abaft midships 6 + 1.4 = 7.4 feet (LCG is aft of LCB) 1,000(7.4) = 7,400 foot-tons Mt / MT1 = 7,400/655 11 inches by the stern

(3) 1,000 tons removed 150 feet forward of midships


trim arm Mt t

= = = = = =

150 + 1.4 = 151.4 feet (151.4)(1,000)/(3,495 - 1,000) = 60.7 feet aft 1.4 feet (aft) + 60.7 feet (aft) = 62.1 feet abaft midships 62.1 + 6 = 68.1 feet (LCG is aft of LCB) 1,000(68.1) = 68,100 foot-tons Mt/MT1 = 68,100/655 104 inches by the stern



1-8.2.3 Point of Constant Draft. When a weight is added at some point away from the LCF, the ship trims as it sinks to a new mean draft. Drafts on the opposite side of the LCF are reduced by the effect of trim, but increased by parallel sinkage. At some point the reduction in draft caused by trim equals the increase in draft caused by parallel sinkage: T due to parallel sinkage = T due to change of trim w Tparallel sinkage = TPI wd1 wd1 d2 Ttrim = t = MT 1 MT 1 L wd1d2 w = TPI MT 1 (L) where: t d1 MT1 L TPI d2 w = = = = = = = change of form, in. distance from the LCF to the added or removed weight, ft moment to change trim one inch, ft-ton/in length between perpendiculars, ft tons per inch immersion, lton/in distance from the point of constant draft to the LCF, ft weight added or removed, lton

The relationship can be solved to determine the point of constant draft for weight added or removed at a known location. It is generally more useful to solve for d1 to find the point where weight must be added or removed to keep draft constant at some point: (MT1) (L) d1 = (TPI) (d2) Note that w cancels out of the equation. So long as the weight change is not large enough to significantly alter MT1, TPI, or the position of LCF, the amount of weight added or removed does not affect the location of the point where weight must be added or removed to keep draft constant at another point. 1-8.3 Inclining Experiment. The predictable and measurable effects of offcenter weight are used to determine height of center of gravity in an inclining experiment. By shifting a known weight a specified distance, the movement of the center of gravity can be determined. The resulting inclination (heel) observed and the tangent formula (see Paragraph 1-8.1.2): GG1 wd tan = = GMeff W (GM) is solved for the as inclined, or effective metacentric height, GMeff: GMeff = GG1 tan = wd W tan

Inclining experiment reports are an important source of data for ship characteristics, especially a baseline vertical position for the center of gravity. 1-8.4 Sallying Ship. Sallying ship is a procedure in which the ship is rocked, or sallied, by rapidly shifting weights back and forth, by rhythmically heaving on the deck edge with a crane, or by personnel running back and forth. If, after inducing rolling, all exciting forces are removed, the ship will roll with the time of roll equal to her natural rolling period. It is impossible to remove all exciting forces, but if the ship is sallied in calm water, is clear of the bottom throughout her roll, the number of mooring lines has been reduced to the minimum acceptable and those remaining are slack, and the ship is free of any other significant restraints, her rolling period will closely approximate the natural rolling period, TR. GM can be estimated by means of the rolling period formula: 1.108 k 2 C B 2 0.44B 2 GM = T T T R R R To determine the rolling period accurately, the ship should be timed through several rolls and the result divided by the number of rolls to find the average rolling period. A derivation of the rolling period formula, with constants for various ship types, is given in Paragraph 1-5.4.3. Sallying ship is often performed in conjunction with an inclining experiment as a check on the accuracy of the experiment or to provide a means to calculate an initial estimate of GM. The accuracy of the procedure is degraded by the influences of offcenter weights, free surfaces, and exciting or restraining forces, such as personnel moving about the ship, unslackened crane hoists or mooring lines, hydrodynamic effects of water entrained by the moving hull surface in confined basins, etc.



1-8.5 Ballast. A ship's loading varies considerably during a voyage as fuel and stores are consumed, and for merchant ships and auxiliaries, from one leg of a voyage to another as cargo is taken on and discharged. Ballast, liquid or solid, is carried to maintain stability or seakindliness. As fuel is consumed from double-bottom tanks, the ship's center of gravity rises and metacentric height is reduced. Saltwater ballast taken into low tanks restores metacentric height to a safe value. All ships require certain drafts, displacement, and trim for seakindliness, propulsion efficiency, and steering control. Discharge of cargo from forward holds and tanks trims the ship by the stern. A light draft forward causes pounding and slamming in a seaway, reduces visibility from the bridge, and makes steering difficult in beam winds. Fuel and cargo oil tanks were formerly used alternately as sea water ballast tanks in most ships. Environmental protection standards now prohibit discharge of oily water in most areas, so modern ships are usually designed with dedicated or segregated ballast tanks (SBT). Normal practice is to provide ballast capacity such that the ship's displacement in ballast is 40 to 60 percent of the full-load displacement. Cargo tanks are often piped for ballast; if the tanks have been cleaned prior to taking ballast, the ballast is clean and can be discharged overboard; otherwise the ballast is dirty and is discharged to receiving facilities ashore. Ballast tanks are distributed over the length of the ship to provide flexibility in controlling trim and hull bending moments. In general cargo ships, the combined center of the ballast tanks is usually near or below the combined center of the fuel tanks. Ships designed to carry dense cargo, such as stone and ore carriers, have an excess of volume that is taken up by wing ballast tanks. Some of these vessels are very stiff in light condition, so high ballast tanks are fitted to reduce metacentric height. Fuel tanks are still commonly piped for saltwater ballast for emergency use. Many warships are fitted with compensating fuel tanks that admit seawater through openings in the bottom of the tanks as fuel is drawn off the top, maintaining nearly constant weight and center of gravity in the tank. Solid ballast, usually consisting of loose stone or sand, river mud, or other dredge spoil, is sometimes carried by cargo ships. Decomposing organic material in mud ballast can produce flammable and toxic gases, such as methane or hydrogen sulfide. Solid ballast, carried in holds or 'tween decks, can degrade stability by shifting, as explained in Paragraph 1-9.3. Fixed solid ballast is sometimes fitted, particularly after conversions involving addition of high weight and in submarines. Ordinary concrete or special heavy aggregate concrete is commonly used. The U.S. Navy has used cast iron ingots or lead pigs weighing about 60 pounds each. The cast iron ingots are sometimes covered with a layer of 3 to 4 inches of cement mortar. High density drilling mud stowed in double-bottom tanks is also used as ballast. Ballasting instructions, where applicable, are included in the damage control book for Navy ships, and in the trim and stability booklet or loading instructions for commercial vessels.


A ship's afloat stability can be impaired or otherwise changed by any of the following:

· · · · ·

Addition, removal, or shift of weight, changing KG, Change in the shape of the submerged hull from grounding or battle damage changing KM, Free surface effect of loose liquids (FS), causing a virtual rise of G, Free communication with the sea (FC), causing a virtual rise of G, or Any combination of the above.

The first three conditions affect stability of the intact ship as well. Only free communication with the sea is predicated on damage to the hull. As the primary indicator of initial stability, GM can be expressed as a function of the above effects: GM = KM KG FS FC The following paragraphs demonstrate the methods to calculate and apply the effect of these conditions on stability. 1-9.1 Flooding. Flooding can be caused by breaches in the hull, accumulating firefighting water, damaged saltwater systems, or any other condition that admits uncontrolled amounts of liquid into the watertight envelope of the ship. Seawater flooding increases displacement and reduces reserve buoyancy. Offcenter flooding causes list and reduces transverse stability. Major flooding towards the ends of the ship reduces longitudinal stability, and in extreme cases may result in the loss of the ship by plunging. The effects of added weight on stability and trim are addressed in Paragraph 1-8. In addition to the increased weight, loose water causes other serious consequences discussed in the following paragraphs.



1-9.1.1 Permeability. The effects of flooding are mitigated by the contents of the flooded compartment. The space occupied by solid objects or watertight volumes cannot be occupied by floodwater, so total volume and weight of floodwater admitted is reduced. This effect is called permeability, and a permeability factor, or ratio of the volume of floodwater to the total volume of the space, can be defined: available volume µ = total volume The volume of the water entering a flooded space can be determined by calculating the volume of the space and multiplying by an appropriate permeability factor. The permeability of tanks can usually be taken as the percentage of full capacity to which they are filled to calculate the amount of floodwater admitted. Not using a permeability factor will result in overestimating the amount of water a space contains. If the exact quantity of floodwater cannot be determined, it is usually safest to err on the high side by disregarding permeability. Permeability for cargo can be calculated from cargo density or stowage factor, as explained in Appendix E, U.S. Navy Ship Salvage Manual, Volume 1 (S0300-A6-MAN-010); the appendix includes an extensive list of material densities and cargo stowage factors. Permeabilities calculated from cargo stowage factors or cargo densities may not be entirely accurate for breakbulk cargo in rigid watertight packaging (cans, steel boxes, etc.) as water will not be able to enter all void spaces in the cargo. Permeability factors for some typical spaces and cargoes are given in Table 1-8. 1-9.1.2 Downflooding. Downflooding occurs when a ship heels sufficiently to immerse an opening above the normal waterline, such as an open door or holed shell plating. This angle of heel is defined as the downflooding angle. Righting arms are reduced as the water accumulates on the low side, and as an offcenter weight creates an additional upsetting moment. A ship rolling so that it cyclically immerses a shell opening may assume a permanent list or increase the period and angle of roll due to the free surface effect described in the next paragraph. As roll angle and period increase, the time the opening is immersed increases, admitting greater amounts of water. 1-9.1.3 Flooding into Liquid-filled Spaces. Tanks often contain immiscible liquids, such as fuel or cargo oil, with densities different from seawater. If an oil tank is holed, there may be either a net inflow or outflow of liquid. There may be an inflow even if the liquid level in the tank is above sea level. If the density of the oil in the tank is low enough that its head pressure at the hull penetration is less than the seawater head pressure, water will flow into the tank. Head pressure is a function of liquid depth and density: Ph = h = where: Ph h = = = = head pressure liquid weight density liquid specific volume = 1/ liquid depth at point in question h

Table 1-8. Selected Permeability Factors.

Miscellaneous Spaces on Naval and Commercial Ships1: Permeability, µ Space Full Load Minimum Operating Condition

Engine rooms (steam turbine) fully flooded above mid height below mid height lower third Engine rooms (diesel and gas turbine) Firerooms Auxiliary machinery spaces Pump rooms Steering gear rooms Shops Offices, electronics spaces Living spaces General stores Magazines Powder Small arms Small arms ammunition Rocket stowage Torpedo stowage Handling rooms Chain locker

0.85 0.90 0.75 0.70 0.85 0.90 0.85 0.90 0.85-0.90 0.90 0.95 0.95 0.80-0.90 0.60 0.80 0.60 0.80 0.70 0.80 0.65

0.85 0.90 0.75 0.70 0.85 0.90 0.85 0.90 0.85-0.90 0.90 0.95 0.95 0.95 0.95 0.80 0.95 0.95 0.95 0.95 0.65

Cargo Spaces: Space Tanks, empty, on molded volume2 Double-bottom tanks Cargo tanks Tanks of known capacity Empty With liquid contents Bulk and breakbulk cargo (average)3 Container holds3 RO/RO holds (average)4 Liquids in cans or barrels1 Permeability, µ

0.97 0.99 1.00 1 - percent full 0.60-0.80 0.70 0.85 0.40


1 2

From Naval Ship Engineering Center Design Data Sheet, DDS 079-1, Aug 75 See Paragraph 1-4.10.7 for discussion. 3 See Appendix E, U.S. Navy Ship Salvage Manual, Volume 1 (S0300-A6-MAN-010) for discussion of how to calculate permeability/volume of floodwater from cargo stowage factor/density. 4 Permeability of hold around containers; does not include space inside containers/ trailers.



The equilibrium liquid level in the tank is the level that will give the same head pressure as the seawater. When there is an outflow of liquid from the tank, the equilibrium level can be determined simply: 1 h1 = sw hsw h1 = sw hsw i where the subscripts i and sw denote properties of the liquid inside the tank and of the seawater outside the hull, respectively. Since specific gravity is directly related to density , the ratio of seawater to product specific gravities can be substituted for the density ratio. The outflow of liquid lightens the ship, and may trim or heel it, varying hsw, so an iterative solution is required. When there is an inflow of seawater into the tank, a water bottom forms. If the tank is holed at its bottom, hi remains essentially constant, but lies over the water bottom of depth hsw,i. Equilibrium head pressure at the hull penetration is now expressed: i hi + sw hsw, i = sw hsw The inflow of seawater adds weight and may trim or heel the ship. It is possible that the liquid level will reach the tank top before equilibrium is reached; the block of oil is held in place by sea pressure, and there can be no further weight addition, even if the ship continues to settle, unless oil escapes through tank vents or other avenues. Tankers carrying light oils that have suffered severe bottom damage may float in this manner, with much of the ship's weight transmitted from the tank tops to the water through the oil mass, rather than through the sides of the hull to the bottom structure. Since the lower level of the liquid mass is above the hull penetration, and separated from it by a water bottom, there is little leakage in calm seas. If the side of a tank is holed at a height such that the internal head pressure is less than the seawater head pressure, water will flow into the tank. If the hole is low enough that it is covered by the water bottom, the situation is identical to that described above. If the hole is above the top of the initial water bottom, there will be an ongoing oil-seawater exchange until the water bottom covers the opening. The local seawater depth over a hull opening can vary with time as the ship rises, settles, trims, or lists in response to weight changes, or as tide rises and falls around a stranded or sunken ship. Tanks may be subject to either inflow or outflow at different times. Heavily damaged tanks will normally reach equilibrium in 20 minutes or less, although significant leakage will continue from casualties that strand at a tide that is higher than subsequent low tides. It is not always necessary to discharge a damaged tank completely to stop oil or other light liquids from leaking into the sea. The water bottom formed when a tank is damaged near its bottom can prevent further discharge of liquids lighter than water. For example, in a tanker with a 50foot molded depth and a 30-foot draft, there is a 20-foot difference in head between sea level and oil level in full cargo tanks. If a full tank is breached through its bottom plating, oil leaks out until the internal oil head balances the external seawater head. The depth of oil is determined by converting the water head to an oil head. For the tanker described, and an oil specific gravity of 0.8: hi = where: hsw hi g, sw g, i = = = = depth to tank penetration = local draft for bottom rupture = 30 ft oil depth with head equivalent to seawater head, ft seawater specific gravity = 1.025 oil specific gravity = 0.8 g, sw g, i hsw = 1.025 30 = 38.44 ft 0.8

For fresh water, specific gravity is taken as 1.0, and oil depth is found by dividing the draft or penetration depth by specific gravity; for the case described above, the equivalent oil head is 37.5 feet. As a practical matter, the equilibrium oil depth has been reached when the cargo pumps begin to draw water instead of oil. The thickness of the water bottom can be increased by drawing oil from the top of the tanks with portable pumps, allowing water to flow in through the breached plating. In the initial stages of a pollution incident, salvors should attempt to create or increase water bottoms in damaged tanks, especially if pumping or storage capacity is limited and several tanks are leaking. As operations continue, water bottoms can be systematically increased until the tanks are completely discharged. Liquid and solid pollutants can be removed by the methods discussed in Paragraphs 3-3 and 3-4, and the U.S. Navy Ship Salvage Manual, Volume 5 (S0300-A6-MAN-050). The effectiveness of water bottoms is limited for water-soluble liquids or liquids with a specific gravity very near one. Water bottoms cannot be created at all under liquids with specific gravities greater than one. Many bulk chemicals fall into this category, as well as some crude oils and bunker fuels. Many chemicals are also highly soluble in water and cannot be contained by water bottoms.



1-9.2 Loose Water. Liquid in a partially flooded compartment is free to move as the ship inclines. The adverse effects of loose water result from the unrestrained movement of masses of water. The movement of significant weights causes the ship's center of gravity to move off the centerline as the ship inclines. 1-9.2.1 Free Surface Effect. The movement of the ship's center of gravity caused by loose water movement can be related to the width of the free surface and the angle of inclination. The loss of righting arm results from the weight of a wedge of water transferred from the high to the low side, as shown in Figure 1-32. For small angles, the volume of the wedge in a rectangular tank can be calculated: y (y tan) dl = Vwedge = 0 0 2

l l

B M W1 W 0 Gv 0 Z G2 W g 0 G g L L1

y tan dl 2


where: l y = = = length of the tank half-width of the tank (from its centerline) angle of inclination


Figure 1-32. Free Surface Effect.

For a rectangular tank, the centroids of the wedges are at 2/ 3y from the centerline of the tank; the plan area of most tanks approximates a rectangle sufficiently to assume that the centroid of the wedge lies 2/ 3y from the centerline. The centroid of the transferred wedge therefore moves a total distance of 4/ 3y. The moment of volume of the transferred wedge is:

l y 2 tan l 2 4 d l × y = tan y 3 dl moment of volume = 3 0 0 2 3

The integral 0l 2/ 3y3 dl is the second moment of area (moment of inertia), i, of the liquid surface (see Paragraph 1-4.5.2 for a derivation). Substituting: moment of volume = i tan The weight moment of the transferred wedge is: weight moment = f i tan where f is the weight density of the fluid in the tank. The weight shift and accompanying moment will cause a shift of the ship's center of gravity parallel to the inclined liquid surface (and the inclined waterline) to a new position G2: f i tan f i tan = GG2 = W w Righting arms are reduced by the transverse shift of center of gravity; the transverse component of the shift GG2 is found by multiplying by the cosine of the angle of inclination: i tan f i sin GG2 transverse = GG2 cos = f cos = w w The righting arm with free surface is found by subtracting the transverse shift of G from the righting arm without free surface: f i sin GZcorr = GZ GG2 transverse = GZ w where: W w GZcorr = = = = weight of the ship weight density of the water in which the ship is floating volume of displacement righting arm corrected for new position of the center of gravity, G2



The free surface correction is applied to the basic statical stability curve by graphical or tabular means in the same way the sine correction for increased KG is applied (see Paragraph 1-5.10.1). The effect on stability of a free surface can be much greater than the effect of the weight of the floodwater. The total correction is the sum of the corrections for each free liquid surface. The component of the weight moment causing the transverse shift of center of gravity, f isin, is called the moment of transference. For many ships, moments of transference are tabulated for each tank, with f expressed in long tons per cubic foot. Moments of transference are normally calculated for a slack condition (50 percent full) and a full condition (100 percent for water tanks, 95 percent for Navy fuel tanks, 98 percent for commercial vessel cargo tanks) for a series of heel angles. The free surface correction for each tank at each angle is obtained by dividing the moment of transference by the ship's displacement. Tabulated moments of transference are included in the damage control books of newer Navy ships. Approximate moments of transference can be calculated by assuming a rectangular free surface: moment of transference rectangle = f i sin = f where: l b = = compartment length compartment width

W W1

3 lb 3 1 lb sin = 12 sin 12 f

L1 L

For seawater flooding, where f is 35 cubic feet per long ton, the expression reduces to: l b 3 sin moment of transference sw = 420 where l and b are measured in feet. If a tank or flooded space is nearly full or nearly empty, the liquid pockets when the ship heels; that is, the liquid moves to expose the deck or to cover the overhead, as shown in Figure 1-33. Once the liquid begins to pocket, the center of gravity, g, of the liquid mass moves very little as heel angle increases. The reduction in righting arm is simply that of an offcenter weight of known location. Model tests have shown that pocketing normally decreases free surface effect by approximately 25 percent. The angle at which pocketing occurs can be predicted by geometry. As the tank shown in Figure 1-34 is inclined, a wedge of liquid shifts from the high side to the low side. The increase in water level on the low side is equal to the decrease on the high side. This distance (h) can be expressed as a function of the tank breadth (b) and the angle of inclination, : b h = tan 2 Pocketing occurs at angles of inclination where h is equal to or greater than the liquid depth in the tank (d) or the overhead clearance (a) as shown in Figure 1-34. Solving for : 2h p = tan 1 p b where: p hp = = angle of inclination where pocketing begins a or d, whichever is less

L1 W W1 L

Figure 1-33. Pocketing.

C L b

l1 a b tan = h _ 2 y w l

d w1

Figure 1-34. Pocketing Angle.



Tabulated moments of transference account for pocketing and tank shape. When using approximate moments, a statical stability curve can be constructed by applying a free surface correction for angles up to p, and an offcenter weight (cosine) correction for larger angles. Alternatively, the gradual diminishment of the moment of transference can be evaluated by defining the moment of transference as the product of f i and some factor C that is less than sin: moment of transference = f i C where: f i C = = = fluid density (tank contents), lton/ft3 moment of inertia of the free surface, ft4 transference factor from Table 1-9, 1-10, or 1-11

Table 1-9. Transference Factor ­ Tanks 50 Percent Full.

Ratio of depth to breadth 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Angle of inclination, deg 10 0.13 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 20 0.14 0.21 0.27 0.31 0.35 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 30 0.14 0.21 0.27 0.34 0.40 0.50 0.57 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 40 0.12 0.19 0.26 0.33 0.40 0.53 0.65 0.74 0.83 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 50 0.11 0.16 0.23 0.30 0.37 0.51 0.66 0.80 0.94 1.06 1.16 1.24 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 60 0.09 0.14 0.20 0.26 0.33 0.47 0.63 0.79 0.96 1.13 1.30 1.47 1.7 2.0 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 70 0.06 0.10 0.16 0.21 0.27 0.41 0.56 0.74 0.92 1.12 1.34 1.56 2.0 2.7 3.7 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 80 0.04 0.07 0.11 0.16 0.21 0.33 0.47 0.65 0.85 1.06 1.30 1.56 2.1 3.1 5.0 9.3 13.4 16.2 16.8 16.8 16.8 16.8 16.8 90 0.02 0.03 0.06 0.09 0.14 0.24 0.38 0.54 0.74 0.96 1.22 1.50 2.2 3.4 6.0 13.5 24.0 37.0 54.0 73.0 96.0 121.0 150.0

The moment of transference factor C depends on the degree of fullness, ratio of depth to breadth of the compartment, and the angle of inclination. To simplify evaluation of the factor C, tanks or flooded spaces are assumed to be full or empty (no free surface), half-full (worst case) or 95 percent full in naval practice or 98 percent full in merchant practice (normal operating condition). Tables 1-9 through 1-11, reproduced from the Society of Naval Architects and Marine Engineers' Principles of Naval Architecture, give factors for 50, 95, and 98 percent full tanks. These tables have been derived for rectangular tanks but will provide sufficient accuracy for most tanks if certain adjustments are made to the entering parameters of breadth and depth. Tanks with substantial variation in breadth, such as those that are approximately trapezoidal in plan view, usually have a small free surface effect; the breadth at the narrow end should generally be used to determine the depth to breadth ratio. If greater accuracy is required, breadth can be taken as:


Table 1-10. Transference Factor ­ Tanks 95 Percent Full.

Ratio of depth to breadth 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Angle of inclination, deg 10 0.02 0.04 0.05 0.06 0.06 0.08 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 20 0.02 0.04 0.05 0.06 0.07 0.09 0.11 0.13 0.14 0.16 0.18 0.19 0.22 0.25 0.30 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 30 0.02 0.04 0.05 0.06 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.20 0.24 0.28 0.35 0.46 0.53 0.57 0.58 0.58 0.58 0.58 0.58 40 0.02 0.03 0.04 0.06 0.07 0.09 0.11 0.13 0.15 0.17 0.18 0.20 0.24 0.29 0.38 0.52 0.64 0.74 0.80 0.85 0.87 0.87 0.87 50 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.24 0.29 0.38 0.56 0.71 0.85 0.97 1.09 1.16 1.22 1.27 60 0.01 0.02 0.03 0.04 0.05 0.07 0.09 0.11 0.13 0.14 0.16 0.18 0.23 0.29 0.38 0.58 0.78 0.96 1.14 1.30 1.46 1.6 1.7 70 0.01 0.02 0.03 0.03 0.04 0.06 0.08 0.10 0.12 0.13 0.15 0.17 0.22 0.28 0.39 0.62 0.87 1.12 1.36 1.6 1.9 2.1 2.3 80 0.01 0.01 0.02 0.03 0.04 0.05 0.07 0.09 0.11 0.14 0.16 0.18 0.23 0.31 0.45 0.77 1.12 1.5 1.9 2.3 2.7 3.2 3.6 90 0.00 0.01 0.01 0.02 0.03 0.05 0.07 0.10 0.14 0.18 0.23 0.28 0.41 0.64 1.14 2.6 4.6 7.1 10.3 14.0 18.2 23.0 28.5

b =

12 i l

For tanks not rectangular in transverse section, the depth should normally be taken as the greatest depth. Accuracy can be increased by taking depth as n times the distance from the free surface to the tank top, where n is 2 for tanks 50 percent full, 20 for tanks 95 percent full, or 50 for tanks 98 percent full. The tables should be entered with the next larger value for depth to breadth ratio unless interpolations are made. The increase in accuracy gained by interpolation is usually insignificant.



Computing moments of transference may be time-consuming and tedious. Figure 1-32 shows that an equivalent righting arm Gv Z can be developed by extending the line of action of gravity back through the ship's centerline. Raising the ship's center of gravity to Gv has the same effect on stability as shifting it to G2. The virtual rise in the center of gravity can be related to the actual transverse shift: GG2 = GGv sin At small angles (less than 7 to 10 degrees), GZ = GMsin; the reduction in righting arm is approximately GGvsin: GZcorr = GM sin GGv sin Setting the two expressions for GZcorr equal: f i sin GM sin GGv sin = GZ w Noting that GMsin = GZ and canceling common terms:

Table 1-11. Transference Factor ­ 98 Percent Full.

Ratio of depth to breadth 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Angle of inclination, deg 10 0.01 0.02 0.02 0.03 0.03 0.04 0.05 0.05 0.06 0.07 0.08 0.08 0.09 0.11 0.13 0.16 0.17 0.18 0.18 0.18 0.18 0.18 0.18 20 0.01 0.02 0.02 0.03 0.03 0.04 0.05 0.06 0.07 0.07 0.08 0.09 0.11 0.13 0.16 0.22 0.27 0.30 0.33 0.35 0.36 0.36 0.36 30 0.01 0.02 0.02 0.02 0.03 0.04 0.05 0.06 0.07 0.07 0.08 0.09 0.11 0.13 0.17 0.24 0.30 0.35 0.40 0.44 0.48 0.51 0.54 40 0.01 0.01 0.02 0.02 0.03 0.04 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.13 0.17 0.24 0.31 0.38 0.44 0.49 0.55 0.60 0.64 50 0.01 0.01 0.02 0.02 0.02 0.03 0.04 0.05 0.06 0.06 0.07 0.08 0.10 0.12 0.16 0.24 0.31 0.38 0.46 0.52 0.59 0.65 0.71 60 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.05 0.06 0.06 0.07 0.09 0.11 0.15 0.22 0.30 0.38 0.46 0.54 0.62 0.70 0.78 70 0.01 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.05 0.05 0.06 0.08 0.10 0.13 0.22 0.30 0.38 0.48 0.58 0.67 0.77 0.87 80 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.03 0.04 0.04 0.05 0.05 0.07 0.09 0.14 0.23 0.34 0.45 0.58 0.70 0.84 0.98 1.12 90 0.00 0.00 0.01 0.01 0.01 0.02 0.03 0.04 0.06 0.08 0.10 0.12 0.17 0.27 0.47 1.06 1.9 2.9 4.2 5.8 7.5 9.5 11.8

Virtual rise of G = F S = GGv =

f i w f i =


w i f i f

= For flooding from the sea, the density ratio becomes one, and: GGv = where: GGv i = = = virtual rise of the center of gravity from free surface effect transverse moment of inertia of the free surface volume of displacement i

If free surface exists in several tanks or compartments, the virtual rise of G is calculated separately for each compartment and the results summed to determine the total virtual rise. The virtual position of the center of gravity is then used to develop a corrected stability curve, as described in Paragraph 1-5.9.1. Treating free surface effect as a virtual rise of the center of gravity provides a relatively quick and easy estimate of the reduction in initial stability. The method overestimates the reduction in righting arm at larger angles because it does not account for pocketing or the reduction in lever arms of the transferred wedge as heel angle increases, but is acceptably accurate if the sum of i for all slack tanks in ft4 is less than twenty times the displacement in long tons. When virtually all free liquid surfaces are subject to pocketing at small angles, as in ships with nearly full fuel load or cargo tanks, it is common practice to determine the reduction in righting arm (by transference) at an arbitrarily selected angle of 5 or 10 degrees, and translate the reduction in righting arm into loss of metacentric height by dividing by the sine of the angle. Equipment, cargo, or stores that pierce the floodwater surface reduce the area and effect of the free surface; this effect is called surface permeability. The surface permeability factor is the moment of inertia of the actual free surface divided by the moment of inertia of an unpierced plane surface with the same outer perimeter. Surface permeability is very difficult to estimate accurately. An error in estimation can cause the salvor to believe the ship is more stable than it actually is. If, on the other hand, surface permeability is neglected, the calculations will indicate less stability than the ship actually possesses, erring on the safe side for the salvor.



1-9.2.2 Cross-flooding. Situations exist where, by design or damage, liquids can freely transfer, or cross-flood, between athwartships tanks:

· · · ·

Damaged longitudinal bulkheads. Cross-flooding ducts fitted between shaft alleys, voids, and similar spaces in small ships to prevent the large offcenter weight moments that would result if only one side flooded. Faulty or inadvertently opened valves or valve manifolds, especially those connecting deep tanks where the liquid surface is above the level of the valve. Anti-roll tanks consisting of two tanks, normally carried about half-full, on opposite sides of the ship connected by relatively smalldiameter sluice pipes.

The shift of liquid from one space to another is treated as a moment of transference between the two tanks to determine reduction in righting arm. The effect on initial stability, as a loss of metacentric height, is calculated for each tank separately. 1-9.2.3 Liquids of Different Densities. A tank may contain two different liquids--one of them is usually seawater. Examples include ruptured cargo or fuel tanks and compensating tanks with water bottoms. Even if the tank is filled with liquid, there is a free surface at the interface between the two liquids that will remain parallel to the inclined waterline. There will be a wedge of volume on the low side where the denser liquid displaces the less dense, and a corresponding wedge on the high side where the less dense liquid displaces the denser, causing the center of gravity of the tank to shift. This effect can be evaluated by using the difference in densities for the value f in the expressions for moment of transference and virtual rise of G. 1-9.2.4 Bulk Cargoes. Bulk cargoes, such as grain and ore, and loose solid ballast, can produce an effect similar to that of free surface, but the effect is modified by friction and inertia of the individual particles. In general, bulk cargo will begin to shift when the angle of inclination is approximately equal to the angle of repose of the cargo. This is the angle between the horizontal and the slope of a granular bulk material that is freely poured onto a horizontal surface. However, violent or cyclic ship motions or vibration can cause the cargo to shift at smaller angles. A cargo that shifts during a heavy roll to one side will not necessarily shift back when the ship rolls to the opposite side. The tendency to roll to greater angles on the low side can cause progressive cargo shifting that can lead to capsize. Some cargoes, especially certain ores, may act like semi-liquid slurries in the presence of even a small amount of moisture, and shift readily when inclined. Ships designed to carry bulk cargo, such as grain, are fitted with permanent or temporary longitudinal bulkheads in their holds that may be supplemented with shifting boards to limit cargo movement. The cargo is normally pressed up to the tops of the holds and between the overhead deck beams. If the cargo is not large enough to fill the hold, a portion of the grain is bagged and laid over the bulk grain to prevent shifting. The cargo may also be tommed down by placing tomming boards, held in place by shores extending to the deck above, over the leveled cargo. 1-9.2.5 Free Communication Effect. A partially flooded, noncenterline space open to the sea introduces the effects of both offcenter weight and free surface. In addition, floodwater is free to enter or leave the space as the ship inclines. The distribution and weight of floodwater varies with time as the ship inclines. This creates virtual rise in the center of gravity, in addition to that caused by free surface: Virtual rise of G = F C = GGc = where: A = y = 1 = plan area of the flooded compartment transverse distance from the center of the flooded compartment to the ship's centerline volume of displacement to the after flooding to the waterline Ay2 1

Free communication exists only when the water level inside the damaged compartment remains the same as the sea level outside the hull. This occurs only when the hull opening is relatively large compared to the volume of the space, and the compartment is vented. 1-9.3 Icing. Ice accumulation in freezing weather steadily adds high weight, increasing displacement and raising center of gravity. In severe conditions, ice thicknesses of six inches or more can collect on weather decks in a short time. Ice builds up as spray or precipitation freeze onto above-water structures. The rate of accumulation is therefore influenced by relative direction of winds and seas, and is seldom uniform on both sides of the ship. The offcenter weight of accumulated ice will cause list that may cause increased ice accumulation on the low side, especially if the primary source of ice is wind-driven spray. High winds often accompany icing conditions; ice loading can severely degrade the ship's ability to withstand heeling moments from beam winds. As an example, a destroyer that has adequate stability for a 100-knot beam wind without ice meets the wind heel criterion (see Appendix D) for only 80 knots with 200 tons of accumulated ice. The 200-ton ice accumulation corresponds to an average ice thickness of 5 to 6 inches over those areas subject to icing. The effect is more severe on smaller vessels; 50 tons of topside ice on a 140-foot minesweeper reduces maximum righting arm from 1.2 feet to 0.7 feet, and reduces maximum allowable beam wind from 85 to 40 knots.



Once ice has started to form, it will continue to form as long as conditions favor icing. The only recourse is to remove the ice or leave the area where ice formation is likely. Frequent heading changes can help prevent the accumulations of large weights of offcenter ice. Icing presents particular difficulties to ships that are not free to maneuver, such as strandings and vessels under tow. The effects of accumulated weights of ice (and snow) must be evaluated before refloating a heavily coated stranding. Removing ice from an unmanned vessel under tow may be difficult or impossible; conditions favorable to icing are often also unfavorable for at-sea personnel transfers. At slow towing speeds, the time needed to reach an area where conditions are significantly less favorable to icing may be considerable. Offcenter ice accumulation is likely on towed vessels because tows follow a relatively steady course. It is important to ensure that a casualty has adequate stability under icing conditions, or that heaters or other means to prevent icing be installed, if the casualty is to be towed through areas where icing is likely. The U. S. Department of Commerce Publication Climatological and Oceanographic Atlas for Mariners provides guidance for expected winds and icing conditions. In general, heavy to severe icing will occur when wind speed is greater than 30 knots and air temperature less than 28 degrees Fahrenheit. Icing predictions can also be provided by Fleet Weather Centers and the National Oceanic and Atmospheric Administration (NOAA). Damage control books for some Navy ships include icing studies and limiting wind velocity curves for various thicknesses of accumulated ice. Figure 1-35 is the limiting wind curve for an FFG-7 Class ship with 9 inches of ice on the foredeck; there are also curves for 6 inches and 12 inches of ice. The fuel-ballast sequence numbers refer to steps in the prescribed tank emptying and ballasting sequence. The plot is entered by reading vertically from the appropriate fuel-ballast sequence number to the solid wind heel curve, and then horizontally to the maximum wind speed for which the ship meets the Navy wind heel criteria. The dashed lines show the increase in allowable wind that can be gained by ballasting the indicated tanks. For example, at fuel sequence 6, the ship has adequate stability to withstand 58-knot beam winds with 9 inches of ice on the foredeck. Continuing vertically along the sequence 6 line shows that the limiting wind can be increased to 62 knots by ballasting tank 5-32-0-W, or 72 knots by ballasting 5-32-0-W, 5-116-0-W, and 5326-1 and 2-W. If necessary, fuel tanks 5250-1 and 2-F, which are emptied by sequence 4, can be ballasted to increase limiting wind to 83 knots. Limiting wind curves from damage control books are based on specific loading conditions, and the assumption that the prescribed tank emptying/ballasting sequence has been followed. They are not valid for conditions that differ significantly from these assumptions. 1-9.4 Added Weight Versus Lost Buoyancy. The foregoing discussions have assumed that flooding, with or without free communication, increases the weight of the ship by the weight of the floodwater. This method, called the added weight method, assumes that none of the hull surface exposed to the buoyant force of the water is lost.


90 85 80


5-100-3&4-F 5-250-1&2-F

75 70 65 60 55 50 45

5-116-0-W 5-326-1&2-W 5-32-0-W

40 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18


Figure 1-35. FFG-7 Class Ship Limiting Winds for Icing Conditions.

Table 1-12. Added Weight Versus Lost Buoyancy.

Item Change in displacement Change in volume of displacement Change in draft, trim, and list Shift of center of gravity Shift of center of buoyancy Shift of metacenter Free surface correction required Free communication correction required

Added Weight yes yes yes yes yes yes yes yes

Lost Buoyancy no no yes no yes yes no no

An alternative method, called the lost buoyancy method, can be used where floodwater in free communication with the sea is assumed to remain part of the sea, and the flooded portion of the ship no longer contributes buoyancy. The vertical pressure forces about the flooded compartment are assumed to act on the sea rather than on the ship. Flooding in free communication with the sea can be assessed by either method, but the two methods cannot be mixed during calculations. Table 1-12 itemizes the important points of the two methods. The method used is a matter of personal preference, although the added weight method is more commonly used. Unless otherwise specified, hydrostatic and stability calculations in this book are made by the added weight method. A more complete discussion of the lost buoyancy calculation method can be found in the Society of Naval Architects and Marine Engineers' Principles of Naval Architecture.



1-9.5 Loss of GM. Loss of GM can result from added high weight (raised G) or increased displacement (lowered M), or both. New initial righting arms are calculated using the new value for GM. The stability curve can be corrected for the new KG with a sine curve correction as described in Paragraph 1-5.9. A ship with a very low metacentric height will roll sluggishly. If GM is negative, the ship is initially unstable and will loll to some angle where the center of buoyancy has moved sufficiently to begin to develop positive righting arms. The ship will settle with equal facility to the same angle of loll on either side. The angle of loll may be estimated by: = tan






Figure 1-36. Stability Curve Showing Range of Instability (Lolling).

Where GM is the absolute value of GM. When GM is negative, the corrected stability curve will indicate the list or angle of loll and a measure of the stability remaining beyond the angle of loll as shown in Figure 1-36. A warship or laden merchant vessel with negative metacentric height is in a very dangerous condition. A positive metacentric height should be restored immediately. In general, negative metacentric height is dealt with by one of three methods:

Recovering lost waterplane to increase the transverse metacentric radius. Free surface is suppressed by pumping from slack tanks directly overboard or by consolidating the contents of slack tanks to press up as many tanks as possible. Partially flooded spaces should be dewatered if they can be made tight and pumped, or allowed to flood to the overhead. When there are several slack tanks or partially flooded spaces, judicious selection of spaces to be pumped down can result in a simultaneous suppression of free surface and a lowering of G. The effects of both the reduction of free surface and loss of low weight should be calculated before emptying low tanks or spaces. In some cases, the net effect of pumping out is to raise the center of gravity unacceptably--flooding the space from the sea would be more effective. The dewatering sequence should be arranged to avoid reducing GM dangerously while pumping out. In ships with marginal stability, the transient free surface created while pumping down solid flooded spaces can cause loss of GM. Shifting weights transversely to correct a list caused by negative GM will only aggravate an already dangerous situation. If enough weight is shifted or added to bring the ship upright, it will list to the opposite side to an angle approximately twice that of the original list; the loll angle is now added to the list due to offcenter weight. 1-9.6 Drydocking. A ship being drydocked is subject to an unusual loading situation; part of the ship's weight is supported by keel blocks, part by the surrounding water. This condition is complicated by changes in the size and shape of the submerged hull form as draft changes while the dock is pumped out. This situation is analogous in many ways to that of a grounded ship, where part of the ship's weight is supported by the ground and part by water, and hull form changes with the state of the tide or passage of waves. The fundamental stability problem is to determine whether the ship will remain stable from the time it first touches the blocks until it has completely settled, or landed, on them. On undocking, the problem is whether the ship will be stable from the time it begins to leave the blocks until it is completely afloat. Positive GM is taken as the indicator of adequate stability. The following discussion of docking stability is summarized from NAVSHIPS Technical Manual (NSTM) 997, Docking Instructions and Routine Work in Drydock. 1-9.6.1 Block Reaction and Residual Buoyancy. When the keel of a ship begins to land on the blocks in a drydock, it pushes down with an initial force w, causing a block reaction, P. A ship with trim, t, by the stern, will contact the aftermost keel block first. This block is called the knuckle block because the ship pivots on it. Strictly speaking, the knuckle rer1 action is not the entire block reaction, but r can be assumed to be in most cases. The Gv block reaction has two effects: a virtual W L weight removal at the keel and a longG B1 W1 L1 itudinal trimming moment. As the ship setP tles on the blocks, P increases from zero and is distributed over all the blocks. As the water level falls, the distributed block reaction increases until it equals the ship's weight, W. The actual or residual buoyancy, B, is equal to W - P. It is the residual buoyancy that determines the ship's hydrostatic characteristics. Figure 1Figure 1-37. Drydocking Forces. 37 diagrams the forces on a ship during drydocking.

· · ·

Suppressing free surface to lower virtual height of the center of gravity, Shifting weight downward in the ship, removing high weight or adding low weight to lower the center of gravity, or




1-9.6.2 Docking Stability. Stability while docking is analyzed either by evaluating the effect of weight removal at the keel, or by balancing moments about the point of first contact. Draft at landing and draft at instability (GM = 0) are determined and compared. Figure 1-38 shows sample plots for an FFG-7 Class ship. Draft at Landing. Summing longitudinal moments about the knuckle block: ML = Wr B1r1 where: W r B1 r1 = = = = ship's weight distance from knuckle block to LCG, as shown in Figure 1-38 residual buoyancy of the ship at current draft distance from knuckle block to LCB, as shown in Figure 1-38

17 16 15 14 13 12 11 2.5 WEIGHT MOMENT, Wr = 3,769(118.4) 446,250 FOOT-TONS 3.0 3.5 4.0 4.5 5.0 RESIDUAL BUOYANCY MOMENT B1 r1 DRAFT AT LANDING 14.9 FEET




16 15 14 13 12 11 5

W(KG)= =3,769(18) =67,842 FOOT-TONS

KM1 B 1

The weight moment (Wr) is constant while the residual displacement and LCB vary with draft. The draft at landing is the draft where ML is zero with the keel parallel to the tops of the keel blocks; that is, where the weight and buoyancy moments are equal, with B1 and r1 determined for the ship with her keel parallel to the keel blocks. Buoyancy moments can be calculated for a range of drafts and plotted as shown in Figure 1-38. The draft at landing is indicated by the intersection of the weight moment and buoyancy moment curves. Draft at landing can be estimated by:

13.25 FEET

6 7 8 9






Figure 1-38. Drydocking Plots.

T1 = Tm where: Tl Tm = = draft at landing, ft mean draft on entering the dock, ft

P 12(TPI) TPI = tons per inch immersion, lton/in

The block reaction at landing, PL, is given by: PL = where: t MT1 = = trim on entering the dock, in moment to trim one inch, ft-lton/in h = distance from application of P (knuckle block) to LCF, ft t (MT1) h

A rule of thumb for estimating draft at landing is: T1 = Tmax 2 (t) 3

where Tmax is the deepest draft on entering the dock, and Tmax and t are given in consistent units. Draft at Instability. After touching the keel blocks, GM is given by: GM1 = KM1 - KGv where: GM1 KM1 = = metacentric height after touching blocks height of the metacenter after touching blocks KGv = virtual height of the center of gravity



The center of gravity undergoes a virtual rise due to the addition of negative weight at the keel. The height of the virtual center of gravity is: KGv = w(kg) W(KG) ( P) (0) W(KG) = = W P B1 w

It is useful to plot GM1 for various drafts to visualize the relationship between the metacentric height and draft while the ship is on the blocks. The draft at instability is found by setting GM1 equal to zero: 0 = KM1 KGv = KM1 KM1 = W(KG) B1 W(KG) B1

KM1 (B1) = W(KG) By considering the products as moments and plotting moments against drafts as shown in Figure 1-38, the draft at instability is shown by the intersection of the two curves. If this draft is less than the draft at landing by a comfortable margin, the ship should remain stable until firmly supported by the keel blocks, or when it begins to leave the blocks on refloating. Example 1-4 illustrates the stability calculations for an FFG-7 Class ship entering drydock.


An FFG-7 Class ship with initial conditions as shown is to be drydocked. Determine draft at landing and whether the ship will remain stable throughout the docking. Initial conditions:

B1r1 as a function of draft: r1 Tm ft

= 330 - [408/2 - LCB*]

B1 ltons

3,769 3,660 3,290 2,910 2,550 2,210

LCB* ft

-3.18 -2.6 -0.08 2.6 5.32 8.1

r1 ft

122.82 123.42 125.92 128.6 131.32 134.1

B 1r 1 ft-tons

462,909 451,644 414,277 374,226 334,866 296,361


= = = = = = =

408 ft 14 ft 3 in 16 ft 1 in 15 ft 2 in 3,769 tons 7.6 ft abaft midships 18 ft

15.17 15.0 14.0 13.0 12.0 11.0

The knuckle block will contact the keel at a point 330 feet abaft the forward perpendicular. From the Curves of Form (FO-2):

* from midships, negative values aft and positive forward

Wr and B1r1 are plotted as functions of draft in Figure 1-38, showing a draft at landing of approximately 14.9 feet.

c. Draft at instability:



= = = =

3.2 ft abaft midships 23.8 ft abaft midships 773 ft-tons 32.5 tons

KM1(B1) = W(KG) =

W(KG) 3769(18) = 67,842 foot-tons

Initial estimates for draft at landing: 408 h = 330 23.8 = 102.2ft 2 t (MT1) 22 (773) P = = = 166.4 102.2 h P 166.4 T1 = Tm = 14.74 ft = 15.17 12 (32.5) 12 (TPI) or

KM1(B1) as a function of draft: Tm ft

15.17 15.0 14.0 13.0 12.0 11.0

B1 ltons

3,769 3,660 3,290 2,910 2,550 2,210

KM1 ft

122.82 123.42 125.92 128.6 131.32 134.1

KM1B1 ft-tons

84,049 81,764 73,992 65,882 58,089 50,609

Tl t Tl


= = =

Tmax - 2/3(t) 22 in = 22/12 ft 16.08 - [2/3(22/12)] = 14.86 ft

W(KG) and KM1(B1) are plotted as functions of draft in Figure 1-38, showing a draft at instability of approximately 13.25 feet.

d. Margin between draft at landing and draft at instability: Draft at landing - Draft at instability = 14.9 - 13.25 = 1.65 feet Draft at landing exceeds draft at instability by 1.65 feet; the ship will be completely settled on the docking blocks well before the residual buoyancy ceases to provide adequate stability.

Draft at landing by plotting:

Ml = r = Wr =

Wr B 1r 1 330 - [408/2 - (-7.6)] = 118.4 3769(118.4) = 446,249.6 446,250 foot-tons




Vessels are built to construction specifications based on intended service. Publicly owned vessels (Navy, Coast Guard, etc.) are built to government specifications. Most Navy ships are built to General Specifications for Ships (GENSPECs), although some auxiliaries are built to commercial specifications. Construction rules for commercial vessels are established by classification societies and government regulations for the country of registry; the American Bureau of Shipping (ABS) and United States Coast Guard (USCG) establish construction rules for the United States. The hull structure consists of a watertight grillage of stiffened plates supported by a framework of mutually supporting longitudinal and transverse members. The framework and shell plating work together to carry imposed loads. The framework carries imposed loads and stiffens the shell plating to allow it to function effectively as a strength member under edge and lateral loading. The arrangement of the structural members is dictated by the framing system. Structural members, with the exception of shell plating and stanchions, are categorized as either longitudinal, with their long axes approximately parallel to the ships centerline, or transverse, with their long axes athwartships or vertical, approximately perpendicular to the longitudinal members. In a general context, any structural stiffener can be called a frame, although the term is usually reserved for the transverse frames described in Paragraph 1-10.3.1. 1-10.1 Framing Systems. While ships vary considerably in the details of their construction, most conform to one of two basic framing systems. Some reflect a combination of the two systems. With longitudinal and transverse structural members crossing at right angles, only one can be continuous. In the longitudinal system, shown in Figure 1-39, this conflict is resolved by the use of closely spaced continuous longitudinal members with intercostal transverses. The transverse system, shown in Figure 1-40 (Page 1-70), uses closely spaced continuous transverse members with intercostal longitudinals.













Figure 1-39. Longitudinal Framing.



In wooden ships and riveted steel construction, continuity of the intercostal members depends on the strength of the joining connections; the intercostal members contribute less direct strength to the framing grillage and serve primarily to stiffen the longitudinal members and shell plating. With good alignment and modern welding practices, full strength can be maintained, regardless of the previous assembly continuity of members. In modern, welded-construction ships, framing systems are distinguished by the relative size, number, and spacing of transverse and longitudinal members. Longitudinally framed ships have many small, closely spaced longitudinals, with fewer, larger, and more widely spaced transverses; transversely framed ships have many small, closely spaced transverses, with fewer, larger, and more widely spaced longitudinals. For average merchant ships, typical close spacing is 2 to 4 feet, typical wide spacing is 10 to 15 feet. Merchant ships and naval auxiliaries may use either longitudinal or transverse framing, depending on the service of the ship. Generally, the same system is used throughout the ship. Most naval combatants (except submarines) are longitudinally framed, with transverse framing near the bow and stern. Because naval ships require a greater reserve of strength to provide damage resistance, their frame members are generally deeper and/or more closely spaced than those of similarly sized merchant vessels. Appendix B describes the construction and characteristics of different types of ships.
















1-10.1.1 Longitudinal Framing. LongiFigure 1-40. Transverse Framing. tudinal framing systems (Figures 1-39A and 1-39B) are more efficient structurally, providing greater strength for the same weight; they are, however, less efficient in the use of internal space because of the deep web frames supporting the longitudinals. Longitudinal framing has been widely used in tankers and bulk carriers where the disruption of internal spaces caused by the web frames is unimportant. Modern practice tends increasingly towards longitudinal framing, or a combination system, in most types of ships. 1-10.1.2 Transverse Framing. Transverse framing (Figure 1-40) is most often found in dry cargo vessels where deep web frames would interfere with cargo stowage. Wooden ships are transversely framed. Given the load-carrying capacity of wood, the lack of longitudinal strength of this system limits the maximum length of wooden vessels. Conversely, this system provides good resistance to racking stresses caused by lateral forces that tend to distort a vessel's cross section. 1-10.1.3 Combination Systems. There are framing systems that combine elements of both longitudinal and transverse framing. Figure 1-41 shows two common combination framing systems. The combination framing system was introduced to overcome the disadvantages of longitudinal framing for dry cargo vessels. Longitudinal strength is provided by longitudinal framing in the double bottom and under the strength deck; transverse framing is used along the side plating where longitudinal bending stresses are smaller. Plate floors and heavy transverse beams are fitted at intervals to support the main deck and bottom longitudinals and increase transverse strength. Cantilever framing is a modification of the combination framing system with some special features. It was developed to facilitate the building of ships with very long and wide hatchways where the remaining deck structure provides insufficient transverse and longitudinal strength. Transverse strength is maintained by the use of special web frames, or cantilevers, at frequent intervals abreast the hatchways. The ship is strengthened longitudinally by heavier than normal sheerstrakes and deck stringer plates. The side plating may be extended upward at the sheerstrake as a heavy bulwark, in place of the usual light bulwark or rails. Hatch side coamings are deep and may be continuous through the length of the hatch deck. If the ship has two hatches abreast, a deck girder or longitudinal bulkhead is fitted on the centerline.



1-10.1.4 Connections. In riveted construction, a variety of plates, angles, and scarfs were used to create strong and rigid joints between structural members. In welded construction, most connections between plates and shapes are made directly through butt or fillet welds, although brackets and angle bars are used in some joints for extra stiffness. 1-10.2 Longitudinal Members. Longitudinal structural members resist bending about athwartships axes.






1-10.2.1 Keel. The keel is a major longitudinal member that runs the length of the ship's bottom along the centerline. In CANTILEVER FRAMING COMBINATION SYSTEM large ships, the keel normally consists of an outer flat keel, the inner (plate) keel, a vertical keel (sometimes called the center Figure 1-41. Combination Framing Systems. vertical keel, or CVK), and a horizontal top flange called the keel rider plate. In small vessels, the outer keel, vertical keel, and rider plates may consist of an I- or H-beam, while in large vessels, the keel is a built-up section. Duct keels are flat-plate keels with two center girders, instead of one, on either side of the keel plates. Duct keels are commonly used forward of propulsion machinery spaces to provide a pipe tunnel. The keel usually varies in cross section along the length of the ship. Some newer vessels have no distinct keel. Instead, there is a cellular double bottom consisting of a grillage of heavy stiffeners plated over top and bottom. In this system, the center girder is generally distinguishable from the side girders only by location. In very large, broad vessels, specially strengthened longitudinals, called docking keels, are fitted at some distance to either side of the center keel. The docking keels help distribute docking loads as the ship rests on three rows of keel blocks. In smaller vessels and some older merchant vessels, an outer vertical keel or bar keel is fitted. In wooden vessels, the keel is usually a large timber, or series of timbers scarfed together. A timber keelson may be fixed atop the keel to increase strength. In glass-reinforced plastic (GRP) vessels, the keel may be a wooden or metal member firmly bonded to the GRP skin, or may consist of a multiple-fiber layup. 1-10.2.2 Other Longitudinal Members. Structural members that run the length of the vessel along shell plating or decks are variously termed stringers, girders, or longitudinals. These members stiffen the entire structure against longitudinal bending loads, and reinforce shell and deck plating against local loads. They may be built-up sections or standard structural sections. In the U.S. Navy, longitudinal members along the side plating are called stringers; those along the bottom plating, longitudinals; and those under decks, girders. In large ships, heavy, deep, bottom longitudinals may be fitted at some distance to either side of the keel. These members are often sized and located to carry the vertical loads imposed by side blocks when dry docking. The heavy longitudinals are variously called sidegirders, keelsons, or docking keels. Bilge keels may be fitted externally at the turn of the bilge to improve seakeeping by resisting rolling. Bilge keels are not usually structural members; if they are attached by load carrying connections and extend for a significant length of the ship, they may contribute to the ship's longitudinal strength. 1-10.3 Transverse Structural Members. Transverse members are fitted primarily to stiffen the hull and enable it to resist shear and torsional loads. 1-10.3.1 Frames. Transverse frames are analogous to ribs extending from the backbone of the keel inside the shell plating. They may continue to the upper decks in their full cross section or be reduced in size at some height above the keel. Frame spacing and dimensions often vary throughout the length of the ship to compensate for variations in loading. Intermediate partial frames may be added for local strengthening. Web frames--deeper-than-normal frames with heavy flanges--are often placed at intervals of several frame spaces, to stiffen and strengthen the hull. Frames connect the longitudinal members and maintain spatial relationships in the face of shear and torsion. They also strengthen the plating against bending under hydrostatic and dynamic loads or buckling under hull shear and bending, and act as ring stiffeners. U.S. Navy practice is to number frames from the forward perpendicular (frame 0) aft; most foreign and many U.S. commercial vessels number frames from aft forward. Frames forward of the forward perpendicular are designated by letters or negative numbers. 1-10.3.2 Floors. The portion of the frame from the keel to the turn of the bilge is a floor. Floors that do not continue into frames are sometimes used for local strengthening or machinery foundations. Deep floors--deeper than the standard floors--are used at the ends of the ship and in high-load areas.



1-10.3.3 Beams. Athwartships deck stiffeners are called beams. They strengthen the deck against local loads, including hydrostatic loads for weather decks, and contribute to overall ship strength by increasing rigidity. Deck beams normally join directly to frames at their outboard ends, forming a continuous frame ring. Triangular brackets, called beam brackets or beam knees, are fitted to stiffen the joint, or the beam is faired in to the frame in a smooth arc to form a continuous structure, as shown in Figures 1-39, 1-40, and 1-41. 1-10.4 Shell Plating. Shell plating is the side and bottom plating; i.e., those portions of the ship's skin that hold back the sea. Bottom plating extends from the keel to the turn of the bilge, side plating from the turn of the bilge to or slightly beyond the upper or main deck edge. Shell and deck plating is arrayed in longitudinal strips called strakes. The strake adjacent to the keel is called the garboard strake. The outer keel may be incorporated into a keel strake. Strakes are lettered from the keel outboard, starting with the garboard strake as A. The strake at the turn of the bilge is the bilge strake. The uppermost strake, which joins to the strength deck plating, is the sheer strake. The keel, garboard, bilge, and sheer strakes contribute significantly to longitudinal strength, and are usually constructed of heavier or stronger plate.














1-10.5 Decks. Decks subdivide the vessel into horizontal levels; weather decks also close the top of the hull and maintain the ship's watertight integrity. Decks add Figure 1-42. Stems significant strength and rigidity to the structure as a whole and limit the extent of flooding after damage, provided they are or can be made watertight. Decks may be steel or aluminum plating or wooden planking, and may be covered or sheathed with wood, tile, linoleum, or other materials. The main deck is the highest continuous watertight deck and is usually the strength deck or upper flange of the hull girder. Because of the main deck's significance to hull strength and watertight integrity, it is used as the reference for numbering other decks. The outboard strake of main deck plating is normally designated the main deck stringer and is either heavier or reinforced to provide longitudinal strength. The connection of the deck to the sheer strake is critical to hull strength. Deck to sheer strake connections are often made by means of a welded T-joint which may be backed up with an angle called the deck stringer angle or gunwale bar. Alternatively, the connection may be made by means of a riveted gunwale bar, or the sheer strake may be rounded and butt-welded to the deck stringer. The U.S. Navy uses the following definitions:

· · · · · ·


Platform or Platform Deck ­ Deck extending less than the full length of the ship below the lowest complete deck; sometimes called an orlop deck. Flats ­ Noncontinuous platforms between deck levels. Half-Deck ­ A partial deck above the lowest complete deck and below the main deck. Forecastle Deck ­ A partial deck above the main deck at the bow. Poop Deck ­ A partial deck above the main deck at the stern. Upper Deck ­ A partial deck above the main deck in the midships region, or one extending from the waists to either bow or stern.


Decks above the main deck are called superstructure decks and may be referred to as levels. The term level also refers to nonwatertight horizontal subdivision, usually by gratings of very deep compartments; for example, the upper level of a machinery space. In merchant ships and auxiliaries, 'tween decks are often fitted to provide one or two levels above the hold bottom to allow cargo to be subdivided or carried high to prevent stiff rolling. 1-10.6 Bulkheads. Bulkheads further subdivide levels or decks into compartments of varying size. Bulkheads may extend through one or several decks and may be classed as structural, watertight, or joiner (also called partition or screen) bulkheads. Structural bulkheads are those that, by design, contribute significantly to the ship's strength. They stiffen the hull by resisting racking and torsional stresses and distribute vertical loads. Watertight bulkheads are designed to withstand significant hydrostatic loads and are installed to increase the ship's resistance to damage by containing flooding. Transverse watertight bulkheads extend upward to a specified deck called the bulkhead deck. Bulkheads are strengthened by angle or bar stiffeners where necessary, or are constructed of corrugated plate. Joiner or partition bulkheads separate and subdivide living, working, storage or other spaces, but impart no watertight integrity or significant strength to the ship's structure. Bulkheads often fit into more than one class, although all bulkheads act as partitions. In practice, watertight bulkheads are almost always structural, while structural bulkheads are often watertight. 1-10.7 Other Structural Members. The Stem Assembly (Figure 1-42) forms the bow of the ship. In its original and simplest form, still used in wooden ships and boats, the stem or stem post consisted of a heavy, rectangular timber which is, in essence, an upward continuation of the keel to which the side planking was attached. In ships of iron or steel construction, the stem was a rectangular forged bar attached at its base to the keel, usually through a forefoot casting. This type of bar stem has been largely superseded by the plate stem, built up of curved wrapper plates, although bar or heavy pipe stems are still commonly used on Great Lakes bulk carriers. The sharper portions of the stem are formed by welding the side plates to an ordinary stem bar or length of round bar or tube, or by butt-welding the plates together. The entire assembly is reinforced by a closely spaced network of deep floors, frames, stringers, and horizontal plate breasthooks. Vertical centerline stiffeners are fitted in stems of large radius and bulbous bows. Stern Assemblies, seen in Figure 1-43, close the aft end of the hull and must accommodate propeller shafts and rudder assemblies, as well as resist the dynamic loads imposed by the rudders. In singlerudder ships, a stern post or frame is fitted at the aft end of the keel. It is generally constructed of castings and forgings arranged to allow for the propeller shaft and rudder stock bosses. The upper part of the stern which extends past the rudder post is supported by a special arrangement of framing. This framing is carried by the transom consisting of a deep, heavy transom floor in conjunction with a transverse transom frame and beam. In counter sterns (also called ordinary, overhanging, or elliptical sterns), which may be found in older merchant vessels, a system of cant framing radiates from the center of the transom like the spokes of a wheel. Cruiser sterns have a system of transverse frames and longitudinal girders with a number of cant frames fitted abaft the aftermost transverse frame. Transom sterns are similar to cruiser sterns, but end in a flat plate, called the transom, and have no cant frames. In twin-rudder vessels, the stern post is omitted and the reinforced stern structure extends forward of the rudder posts.







A double bottom may be fitted to increase FLOOR strength and resistance to underwater damage. The inner bottom plating is laid over the grillage of floors and longitudinals, forming spaces often used as tankage for TRANSOM STERN bunker fuel or other liquids. The outer strake of the inner bottom is called the margin plate, which may extend in a Figure 1-43. Stern Assemblies. horizontal line to the side plating, or be inclined downward near the turn of the bilge to form the side of the double bottom. The double bottom may or may not be continuous over the length of the ship. Large combatants such as aircraft carriers and battleships may have more than one inner bottom. Stanchions or pillars are used to support decks, distribute vertical loads, and stiffen the hull structure between bulkheads.



1-10.8 Superstructures and Deckhouses. The term superstructure is applied to a portion of a ship's structure above the main or upper deck extending the width of the ship and forming an integral part of the main hull. A deckhouse is a lighter structure, usually not extending the width of the ship, that is placed on the hull rather than forming a part of it. In practice, the two terms are often confused or used interchangeably. In naval combatants and passenger liners, deckhouses or superstructures may extend for most of the vessel's length; in most other types, they occupy a small portion of the ship's length. These structures generally house accommodation, communications, navigational, or control spaces. They may house workshops or specialized machinery; in warships, weapons control spaces and weapons mounts are often located on or in the superstructure or deckhouse. Deckhouses are not normally designed to contribute to overall hull girder strength, but being rigidly attached to the hull, they carry some stresses. Superstructures, as an integral part of the hull, are normally designed to carry hull stresses. 1-10.9 Damage-resistant Features of Ships. While the entire structure of a ship is designed to resist some damage, certain features are incorporated into ships specifically to prevent loss of the ship when damaged. Loss may result from flooding or structural failure of the hull girder. Features enhancing a ship's ability to resist damage are described in the following paragraphs. 1-10.9.1 Subdivision. Subdivision, or compartmentation, is a ship's primary means of resisting damage. A system of watertight decks, bulkheads, and an inner bottom limits the spread of flooding, fire, blast effects, weapon fragments, and fumes or gases. Extensive subdivision is an inconvenience to everyone; production cost is increased, cargo storage is complicated, access and movement around the ship is hampered. The degree of subdivision is therefore a compromise between safety and other requirements. Factors considered include the following:

· · · · · · · · ·

Ability to resist battle damage. Ability to survive underwater damage. Ability to resist bow collision damage. Ability to resist damage from stranding. Protection of vital spaces against flooding. Ability to resist spread of fire, smoke, and airborne contaminants. Interference of subdivision with arrangements. Interference of subdivision with access and systems. Provisions for carrying liquids.

Tankers between 492 and 738 ft in length Tankers less than 492 ft in length

Table 1-13. Standards of Subdivision.

Type Ship Standard of Subdivision

Navy Ships (without side protection systems)1 Seagoing craft under 100 ft in length Ships 100-300 ft in length Ships over 300 ft in length: Combatants and Personnel Carriers, such as Hospital Ships and Troop Transports All other ships Withstand rapid flooding from a shell opening equal to 15% of length between perpendiculars at any point fore or aft Withstand flooding from an opening equal to 12.5% of the length between perpendiculars Coast Guard Standards for Commercial Vessels2 Tankers over 738 ft in length Withstand solid flooding from a shell opening with length equal to the lesser of 0.495L2/3 or 47.6 ft, width equal to the lesser of B/5 or 37.74 ft, from the keel upwards without limit, at any point between perpendiculars Withstand flooding from damage described above at any point except at an aft machinery room bulkhead Withstand flooding from damage described above at any point between main transverse bulkheads, except to an aft machinery room Withstand solid flooding from a shell opening with length equal to the lesser of 0.495L2/3 or 47.6 ft, width of 4.2 ft, from the keel upwards without limit, between any two main transverse bulkheads Withstand flooding from damage with length of 6 ft, width of 30 in, from the keel upwards without limit, at any point, including the intersection of a transverse and longitudinal bulkhead 1 compartment 2 compartments

1-10.9.2 Flooding. A principal concern in many casualty situations is limiting flooding. Floodwater may be admitted to the ship by collision, grounding, weapons strike, firefighting, or other means. However flooding occurs, it is necessary to limit its extent to minimize the following:

· · ·

Loss of transverse and longitudinal stability. Loss of reserve buoyancy.

Great Lakes dry bulk carriers

Barges carrying very hazardous materials

Damage to cargo and ship systems.

Ideally, a ship should be able to sustain increasing amounts of flooding until it founders from loss of reserve buoyancy. Barges carrying moderately hazardous Withstand flooding from damage described above at materials any point, except on a transverse watertight bulkhead Loss of transverse or longitudinal stability can cause a ship 1 to capsize or plunge, even when a sizable reserve buoyancy Naval Ship Engineering Center Design Data Sheet, DDS079-1, Stability and Buoyancy of U.S. Naval Surface Ships, 1 Aug 75 remains. Offcenter flooding and its serious effects on 2 Title 46, US Code of Federal Regulations (46 CFR), Subchapter S. Requirements have transverse stability can be avoided by using transverse been simplified. Additional definitions and exceptions apply. Subdivision requirements subdivision only. Complete avoidance of longitudinal for passenger ships are especially diverse. watertight boundaries is not always possible or advisable, but most modern ships follow a general pattern of transverse watertight subdivision, at the expense of admitting a larger volume of floodwater. Some longitudinal subdivision is necessary to reduce free surface effect, especially in tanks. This subdivision normally takes the form of a centerline bulkhead dividing the inner bottom into port and starboard tanks, or use of wing tanks smaller than the adjacent centerline tanks. Sills, seen in Figure 1-44, or baffle plates are sometimes used to reduce the free surface effects of rolling or shallow flooding but are ineffective against unchecked flooding. Transverse watertight bulkheads near the extremities of the ship limit flooding, and prevent the large and dangerous trims that large amounts of floodwater at the ends of the ship would produce. Additional transverse watertight bulkheads are spaced to permit the ship to remain afloat after a specific number of adjacent compartments, usually 1, 2, or 3, are flooded. The number of compartments that can be flooded without causing foundering is the ship's standard of subdivision or standard of flooding. For example, the FFG-7 Class frigate shown in Figure 1-45 can remain afloat if any 3 of its 13 major watertight compartments are flooded--it is said to be a 3-compartment ship. Progressive flooding is defeated by carrying each watertight bulkhead intact from the bottom plating to a height above the expected flooding water level. Watertight bulkheads are normally carried watertight to a specified deck, called the bulkhead deck. The bulkhead deck on most designs is the main or weather deck and may be either a continuous or stepped deck. For the FFG-7 Class ship shown in Figure 1-45, the main deck is the bulkhead deck. Standards of subdivision for Navy and commercial ships are given in Table 1-13.



Ships are assigned a minimum freeboard based on the reserve buoyancy required to sustain flooding to their standard of subdivision without foundering. This freeboard is measured from a margin line that represents the highest allowable waterline in a damaged condition. The margin line is usually established near the bulkhead deck or a designated freeboard deck. Load lines for cargo ships and tankers or limiting draft marks for warships are marked at a distance below the margin line corresponding to the required freeboard. If the load line or limiting draft mark is not immersed before damage, and flooding is equal to or less than the standard of subdivision, the ship will remain afloat at a waterline at or below the margin line after damage. Salvors may not be able to restore a ship's required minimum freeboard; reduced freeboard must be recognized as a loss of reserve buoyancy and damage resistance. This is particularly important if the casualty is to be towed some distance to safe haven. In such a case, a salvage engineer may wish to calculate the standard of subdivision for the ship in its actual condition. 1-10.9.3 Likely Damage. Certain features are incorporated into ships to isolate common or likely forms of damage. Because the ends of the ship are more vulnerable to damage from collision or grounding, a collision bulkhead is required at about five percent of the ship's length from the bow, along with an afterpeak bulkhead near the stern, enclosing the propeller shaft penetration into the hull. A second collision bulkhead may be required in large ships. Watertight double bottoms are required in some classes of vessels to provide protection against grounding and limited protection against underwater weapons. Machinery spaces are segregated from the rest of the ship by watertight bulkheads that (1) protect the ship from intense machinery space fires, and (2) protect vital equipment located in the machinery spaces from flooding in other parts of the ship. Sheer can prevent or delay progressive flooding through deck openings when trim is extreme, as shown in Figure 1-46. Wing tanks, common in tankers, ore carriers, and large combatants, limit flooding from damage to the sides. The effect of offcenter flooding can be mitigated by constructing the wing tanks with volumes that are small compared to the center tanks or holds, or by keeping wing tanks filled at least to the waterline. A system of wing tanks combined with a double bottom produces, in effect, a double hull. 1-10.9.4 Structural Damage. Structural failure is resisted by the use of materials of consistent and known strength, and by building in reserve strength. Ships' scantlings are selected to result in bending stresses on the order of 15,000 to 22,000 pounds per square inch, considerably less than the yield stress of shipbuilding steels (32,000 psi or greater). This stress level is often contingent on specified loading sequences and conditions, particularly in very large tankers or bulk carriers. Hull strength is addressed in greater detail in Paragraph 1-11.

Figure 1-44. Effects of a Sill.







250 212 180


100 84 64

32 20


Figure 1-45. FFG-7 Transverse Subdivision.



Figure 1-46. Sheer Defeating Progressive Flooding.

1-10.9.5 Additional Features of Naval Ships. Both naval and merchant ships use the damage-resistant features previously described. Naval ships, intended to go "in harm's way," incorporate additional damage-resistant features in their construction. Naval ships will usually have more extensive subdivision than merchant vessels, although some naval auxiliaries are built to classification society standards. Combatants are built with a much greater degree of subdivision and greater reserve of strength than auxiliaries or merchant ships of the same size. Naval vessels often have multiple machinery spaces segregated by watertight bulkheads, as well as auxiliary machinery spaces located remotely from the main machinery rooms. Additional vital spaces, such as ship control stations or weapons spaces, are designated and protected by watertight subdivision. In all commissioned vessels of the U.S. Navy, a damage control (DC) deck is designated. The DC deck, on which damage control equipment and stations are located, is considered a vital space and is made watertight where feasible. Remote operators for certain vital piping and electrical systems are located on the DC deck. The damage control deck is located high in the ship and is usually covered; fore and aft access is provided through watertight openings in the main transverse bulkheads. Doors and nonwatertight fittings in main transverse bulkheads are not permitted below the DC deck. Doors through transverse bulkheads into shaft alleys are not allowed; no penetrations are allowed through the collision bulkhead. In addition to armored decks and side armor, large combatants, such as aircraft carriers and battleships, are fitted with underwater defense systems (also called side protective or torpedo protection systems) consisting of layered wing and bottom tanks. These are alternately empty or liquid-filled to absorb the shock of underwater explosions. The tank boundaries form a series of barriers that must be breached before major spaces are flooded.





1-11.1 Stresses in Ships. Ships, like all structures, are subject to load-induced stress and the resulting strains. Simple beam theory is employed to predict ship responses to various conditions of loading by treating assuming the ship's structure as a built-up box girder bearing an distributed load (weight of the ship and contents) and supported by a distributed reaction (buoyancy). Of principal concern are the compound bending and shear stresses resulting from the ship's loading and wave action. Torsional stresses are also important, and can be severely aggravated by grounding in large ships. Stresses may be divided into three groups:



· ·

Primary or Structural Affecting the hull girder.



Secondary or Local ­ Affecting major substructures or definable areas of the hull, such as a hold or bulkhead. Tertiary ­ Very localized, affecting small areas of plating or single stiffeners.

Figure 1-47. Deflections from Primary, Secondary, and Tertiary Stresses.



The distinctions among primary, secondary, and tertiary stresses are illustrated by the character of the accompanying structural deflections, as shown in Figure 1-47. The total stress on any portion of structure is the sum of primary, secondary and tertiary stresses that may tend to either reinforce or cancel one another. 1-11.1.1 Structural Stresses. The principal structural stresses are caused by the following conditions:




WEIGHT Weight and Buoyancy Distribution. Differences in buoyBUOYANCY ancy and weight distribution cause longitudinal bending stresses and accompanying shear stresses. An excess of buoyancy in the midships region with an excess of weight near the ends of the ship places the deck in tension and the keel in compression. The resulting convex deflection is called hog or SAGGING hogging. An excess of weight in the midships region and excess buoyancy near the Figure 1-48. Hull Girder Bending. ends places the deck in compression and the keel in tension. The concave deflection is called sag, or sagging. Long waves can impose hogging or sagging conditions as shown in Figure 1-48. Bending stresses are resisted by the longitudinal strength members, particularly those of the strength deck, sheer strake and bottom. Bending stresses are normally greatest in the midships region of an intact ship, while maximum shear stresses occur in the quarter length regions.




Water Pressure. The distributed force of buoyancy, as water pressure, is resisted by the side and bottom plating stiffened by a network of frames, floors, longitudinals, etc. All weight loads are ultimately transmitted through the ship structure to be borne by water pressure. The differences in weight and water pressure distribution produce varying loads as shown in Figure 1-49. Racking. Transverse waves alter the water pressure distribution around the ship, as shown in Figure 1-50. The unequal pressure distribution tends to bend side plating and transverse frames about a horizontal longitudinal axis. The transverse distortion is called racking and is resisted by shear stresses in the ship's structure. Racking stresses are highest on the corners of a ship's cross section. Racking is resisted by transverse bulkheads and frame ring, particularly the corner brackets. Drydocking. Ships supported by a single line of drydock keel blocks will hog transversely. A cellular double bottom stiffens the hull against such hog, but additional lines of side blocks are more effective. Stranding. Stranding changes the bending stress distribution in the hull girder by altering the buoyancy distribution and introducing concentrated loading along the bottom. Point loads similar to those caused by docking blocks, but naturally much less predictable, result if the ship strands on uneven or rocky ground. Large ships may sag transversely if stranded over a narrow width near the centerline.






Figure 1-49. Water Pressure.



Figure 1-50. Racking.



1-11.1.2 Local Stresses. Secondary and tertiary stresses result from localized loads such as the following:

· · · ·

Panting. Panting is an oscillatory motion of the shell plating, principally near the bow and stern of a ship, caused by uneven water pressure as the ship passes through waves. The fore-end (and sometimes the after) structure is reinforced with a system of panting beams, panting stringers, panting frames, breasthooks, and deep floors to withstand panting loads. Pounding or slamming. Pounding occurs when the bows of a pitching ship clear the water and come down heavily. Pounding is most severe in full-bowed ships in the bottom structure in the forward quarter length of the ship. In this pounding region, plating and bottom stiffeners are often heavier and/or more closely spaced than in the rest of the ship. Local Loads. Local strengthening enables the ship structure to carry loads caused by large local weights, such as machinery or cargo. Similar measures are used to strengthen structure in way of fittings that transmit high loads, such as padeyes, winch mounts, and kingpost foundations. The geometry of portions of the hull or fittings may cause stress raisers, requiring local reinforcement to increase load-carrying capacity. Figure 1-51 shows some forms of local reinforcement. Vibration. Vibration from engines, propellers, etc., causes stresses in various parts of the ship. Vibration-induced stresses are resisted by local stiffening of areas in way of vibration sources.

1-11.1.3 Weapons Effects. Impact and shock effects of airborne, underwater, and contact explosions can cause severe and not wholly predictable loads on ship structure. Warships are constructed with this kind of loading in mind, and are therefore strengthened to withstand blast and impact loads over much of their structure. The exact nature of this strengthening varies from ship to ship but generally consists of closer stiffener and bulkhead spacing than would be found in an equivalent-sized merchant ship or auxiliary. Weapons effects are discussed in greater detail in the U.S. Navy Ship Salvage Manual, Volume 3 (S0300A6-MAN-030). 1-11.2 Longitudinal Bending Stress. The magnitude of the longitudinal bending stresses in the hull girder is a function of the total bending moment, cross-sectional area distribution. The bending moment is a function of the shear force distribution along the ship's length, which is in turn a function of the ship's load distribution. The hull is assumed to be a statically loaded beam that behaves in accordance with the theory of flexure (see Paragraph 2-3). The downward loads on the beam are the weights of the component parts of the ship and any weights carried on the ship. Upward loads are the forces of buoyancy (and ground reaction or block reaction for stranded, beached, or dry docked ships). Bending moment is calculated by a double integration of the static load curve. The steps in the longitudinal stress calculation are:







· · · · · · · ·


Determine longitudinal weight and buoyancy distributions. Statically balance the ship on still water or a wave. Develop the longitudinal load distribution or curve. Integrate the load curve to give shear forces. Integrate the shear curve to give bending moments. Determine which structure in sections of interest is effective. Determine moment of inertia, section modulus and location of the neutral axis for sections of interest. Calculate bending and shear stresses in sections of interest.

Figure 1-51. Local Strengthening.


These steps are examined separately in the following paragraphs. Amplifying information can be found in the Naval Ship Engineering Center Design Data Sheet DDS 100-6, or any good naval architecture text. Examples 1-5, F-3, and F-5 demonstrate longitudinal strength calculations. 1-11.2.1 Load Curve. The load on the hull girder at any point is the difference between the buoyant force and weight at that point. This is graphically represented by superimposing buoyancy and weight curves. The areas under the curves represent total buoyancy and total weight. For a floating ship, the two areas must be equal, with their geometric centers in vertical line. Figure 1-61 shows the load curve developed for Example 1-5. For the shear and bending moment integrations to close properly, the ship must be statically balanced; that is, weight and buoyancy, as calculated by integration of the respective curves, should be within 0.5 percent, and LCB and LCG should be within one foot of each other. It is important to adopt sign conventions for the directions of forces and distances, and carry them through subsequent calculations. The calculations in this handbook follow the intuitive convention that downward forces (weight) are negative and upward forces (buoyancy) are positive, resulting in load curves that are predominantly positive over the middle portion for hogging hulls, and predominantly positive at the ends for sagging hulls. 1-11.2.2 Buoyancy Curve. The magnitude of the buoyant force at any point is a function of the cross-sectional area below the water line and the water density. The buoyancy curve will therefore follow the curve of areas. Areas of sections are most easily obtained from Bonjean's Curves, shown in Figure FO-3 and described in Paragraph 1-3.11. Lines drawings, offsets, or general plans can also be used to determine sectional areas by numerical integration. The still water buoyancy curve is developed by dividing sectional areas by 35 (cubic feet per long ton of seawater) to convert to unit buoyancy (tons per foot) and plotting these values as ordinates. A buoyancy curve based on ordinates taken from Bonjean's Curves will not include appendage buoyancy. If known, appendage buoyancies can be added to the basic curve as rectangles or trapezoids. When appendage buoyancy is unknown, a simpler and generally adequate solution is to assume that an appropriate appendage allowance (a fraction of full-load displacement) is distributed over the length of the ship. Final buoyancy ordinates are determined by an appendage allowance adjusted for the ship's condition, i.e., the appendage allowance divided by actual displacement. Buoyancy ordinates multiplied by the adjusted appendage allowance plus one give adjusted buoyancy ordinates. Integrating the adjusted buoyancy ordinates should give a correct total buoyancy equal to total weight. Appendage allowances are discussed in Paragraph 1-4.10.2. As part of the regression analysis described in Paragraph 1-7, Porricelli, Boyd, and Schlieffer developed a method of approximating the buoyancy curve for merchant and auxiliary hulls with a series of trapezoids. The method is reasonably accurate for full-bodied ships (CB > 0.6). The ship is first divided into three segments: the parallel midbody (pmb), the forebody (fb), and the afterbody (ab). The forebody and afterbody are then divided into two sections each. A uniform buoyancy distribution is assumed for the parallel midbody and represented by a rectangle. Ordinates are plotted at the forward and after perpendiculars and the boundaries of the sections of the hull and connected by straight lines to form the buoyancy curve. Buoyancy of the parallel midbody (Bpmb), lengths of sections (Ln, bn) and heights of ordinates (yn) are calculated as shown in Figure 1-52.

y3 y4 y5


y2 y1 b5 b4 Lab b3 L pmb b1 = (0.61 - 0.615 CB)L b2 = Lfb - b1 b3 = Lpmb b4 = Lab - b5 b = 0.2L


b2 Lfb

b1 FP


L pmb = (1.74CB - 1.002)L Lfb Lab = (1.186 - 1.17CB)L

y1 = 0.04y3 y2 = CB y3 y3 = Bpmb /Lpmb y4 = CB y3 y5 = 0.08y3

= L - Lpmb - Lfb Lpmb B Tm Cm B pmb = 35

To facilitate summing weight and buoyancy Figure 1-52. Approximate Buoyancy Curve for Full-Bodied Ship. curves to develop the load curve, the buoyancy curve is often stepped, that is, approximated by a series of horizontal segments at a height corresponding to the mean buoyancy ordinate for that segment. The procedure for stepping a curve is described in Paragraph 1-4.9. It is not necessary for the buoyancy curve to have the same number of segments as the weight curve, although it is convenient for all of the bounding stations for the curve with fewer segments to coincide with stations on the other curve. The load curve resulting when the two curves are summed will have the same number of segments as the curve with the most segments. 1-11.2.3 Weight Curve. Weight distribution tables or curves are often difficult to obtain, even though they are developed during the design of the hull girder. For U.S. Navy ships, a Longitudinal Strength and Inertia Sections drawing is prepared, showing weight distribution, usually for full load. A portion of the longitudinal strength drawing for FFG-7 Class ships is reproduced in Figure FO-4. The complete drawing includes section scantlings, similar to Figure 1-58, for a number of stations along the ship's length. Format and content of longitudinal strength drawings for Navy ships are more completely described in Appendix B.



Weight distributions for Navy ships are tabulated or drawn for 20 standard ship segments between perpendiculars, plus one segment forward of the forward perpendicular and one aft of the after perpendicular (22 segments). The segments forward and aft of the perpendiculars extend from the perpendiculars to the ends of the ship and are not necessarily the same length as the segments between perpendiculars. Segments are identified by the stations that bound them, numbered from 0 at the forward perpendicular to 20 at the after perpendicular. Weight distribution is assumed to be uniform within each segment, producing a stepped curve. For cargo ships, tankers, etc., where loading may vary by compartment, it may be more convenient to segment the ship by compartments. Weight distributions for a number of Navy ships are given in Appendix B. The weight curve from a longitudinal strength drawing or other source must be corrected for the ship's actual weight distribution, including any major alterations (SHIPALTS). Often this information is not available and weight change estimates must be made until the weight distribution sums to the known ship displacement. If detailed weight curves are not available, weight distribution can be estimated by one of the methods described in Paragraph 1-11.13. 1-11.2.4 Shear and Bending Moment Curves. A fundamental principle of beam theory is that at any point in an elastic beam: P = dS dx = d 2M dx 2

S = Pdx and M = Sdx = Pdx where: P S M = = = load shear bending moment


Vertical shear at any section is the sum of the vertical forces to one side of the section; the shear curve is therefore developed by integrating the load curve (the sum of the weight and buoyancy curves) along its length, starting from either end of the ship. The total positive area under the shear curve should equal the total negative area for static equilibrium. Shear is zero at the ends of the ship; for most ships, shear will be maximum near the quarter-lengths and change signs near midships. Bending moment at any section is the sum of force moments about the section. The bending moment curve is developed by integrating the shear curve along its length. Bending moment is zero at the ends of the ship, and is maximum where shear changes sign. The load and shear curves cannot be defined mathematically, so graphical or numerical methods are used to perform the integrations, as shown in Paragraph 1-4 and Appendix F. Several important relationships between the load, shear and bending moment curves, illustrated in Figure 1-53, act as checks on the completed curves:







B >W

· · ·


When P is 0, S is a maximum or minimum and M is at an inflection point. When P is a maximum, S is at an inflection point.

Figure 1-53. Load, Shear, Bending Moment Curve Relations and Conventions.

When S is 0, M is a maximum or minimum.


The load and shear curves can be integrated from either end. Each integration should close to zero at the end opposite the beginning. Small errors in closing are unavoidable if the areas under the weight and buoyancy curve are not precisely equal, and LCG and LCB are not coincident. It is sometimes useful to integrate each curve twice, once in each direction, and compare the results. If the integrations close to zero, integrating in the opposite direction will reverse the sign of the ordinate at each station, but will not change the magnitude. If the integrations do not close precisely, integrating in the opposite direction will change the magnitude of the shear and moment ordinates at each station, and is a means of estimating the error range of the calculated values. If the shear curve does not close, the sections of maximum shear and bending moment will also shift somewhat when integrating in the opposite direction. For small errors in closing, the magnitude of the shear and bending moment ordinates in the middle portion of the curve will be fairly reliable, but the ordinates near the ends of the ship should not be trusted. A useful convention is to integrate the load curve from left to right (from aft forward) to develop the shear curve, and the shear curve from right to left (from forward aft) to develop the moment curve. Following this convention, along with taking downward forces as negative, will result in shear and moment curves with the features shown in Figure 1-53:


For sagging hulls: (1) Positive shear on the left side of the plot (aft). (2) Negative shear on the right side of the plot (forward). (3) Negative (convex downwards) bending moment.


For hogging hulls: (1) Negative shear on the left side of the plot (aft). (2) Positive shear on the right side of the plot (forward). (3) Positive (convex upwards) bending moment.

This convention is useful because the bending moment curves superficially resemble a sagging or hogging hull, as appropriate. Other conventions may be encountered in ship design data. Shear curves that are the mirror image of the convention described above are common and result when both shear and moment integrations are run in the same direction. U.S. Navy longitudinal strength drawings disregard the sign of bending moments and shear forces and show all curves above the axis to save space. Example 1-5 calculates still water bending moment and shear curves for an FFG-7 hull; the curves are illustrated in Figures 1-62. 1-11.3 Variations in Loading. Any change in weight or buoyancy distribution will alter the load curve. 1-11.3.1 Changes in Weight Distribution. Changes in weight distribution generally result from deliberate actions, such as taking on or discharging cargo, ballasting, launching or recovering aircraft and boats, use of fuels or other consumables, or shifting weights. Weight distribution can also be changed in a casualty by:

· · · ·

Flooding. Major fires which consume flammable materials. Spilled cargo. Loss of structure or fittings.

Weight additions or removals change total weight, and therefore affect total buoyancy and buoyancy distribution. Weight shifts that significantly alter trim also affect the buoyancy distribution. Buoyancy distribution can change without an accompanying change in weight distribution. Such changes result from:

· · ·

Waves. Grounding. Drydocking.



1-11.3.2 Wave-induced Buoyancy Distribution. In all but the stillest water, buoyancy distribution changes constantly in proportion to the variations in draft along the ship's length as successive wave trains pass. A wave-induced buoyancy curve is developed by superimposing a wave profile, or series of profiles, on the ship profile, instead of using a horizontal waterline, to determine drafts at stations. An infinite number of waves are possible; in practice, it is usually sufficient to examine only worst-case situations. Maximum midships bending moments result from the two situations shown in Figure 1-48. Ships are designed to carry the stresses imposed by these conditions, based on a trochoidal or sinusoidal wave form with length equal to the ship's length (L). Standard wave heights were formerly taken as L/20, and then 1.1L as ship size increased, but with steady increases in ship length, these formulae yield unrealistically large waves. More recent ABS construction rules specify different formulae for different ranges of length, although Navy design practice still uses the 1.1L wave. Although artificial, these assumed conditions have proven adequate for design work; they are used here to illustrate the procedures for analyzing wave-induced stresses in ships. The salvage engineer who finds it necessary to evaluate the strength of a casualty exposed to wave action should base his worst cases on observed or expected waves and the actual loading and structural condition of the casualty. Total bending moment is sometimes spoken of as the sum of a still water bending moment and a wave-induced bending moment. The total bending moment is simply the bending moment resulting from the load distribution at that instant. The bending moment can be evaluated by adding to or subtracting from the still water buoyancy curve or by starting from scratch by superimposing a wave profile over the Bonjean's curves to develop the buoyancy curve, as shown in Figure 1-54. As before, the area under the buoyancy curve must equal the area under the weight curve.






Figure 1-54. Wave-Induced Buoyancy.

R h r




Figure 1-55. Trochoidial Wave Form.

To ensure that shear and bending moment integrations close, the ship must be statically balanced on the wave; that is, the waterline must be adjusted until weight equals buoyancy and the center of buoyancy is in vertical line with the center of gravity. When using Bonjean's Curves in the profile format, this is most easily accomplished by plotting the wave profile to the same vertical and horizontal scales as the Bonjean's Curves on a piece of tracing paper. The wave profile is laid over the Bonjean's Curves, with either the crest or trough at the midship station, as appropriate. Section areas are picked off as ordinates to a trial buoyancy curve, which is integrated to determine buoyancy and LCB. If the first guess does not match buoyancy and weight within limits, successive calculations are made, moving the wave up and down and trimming it until a position is found where buoyancy is within one percent of weight, and LCB is within one foot of LCG. When the final position of the ship on the wave is determined, the section areas are converted to unit buoyancies to plot a precise buoyancy curve that is used to determine the mean unit buoyancy over each segment of the ship's length. Buoyancy and weight curves are then summed to calculate the load curve; shear and bending moment integrations are conducted as for the still water condition. When the rosette format Bonjean's Curves are used, drafts at each station must be determined by interpolation so the section areas can be read from the curves. Alternatively, rosette format curves can be traced onto a profile of the ship. The horizontal scale of the ship profile (not the same as the Bonjean's Curve area scale) is not critical, but should not be more than twice the vertical scale; if the horizontal scale is too great, portions of the wave profile will be steep enough that small errors in plotting will cause significant errors in reading sectional areas.



A trochoid is the curve traced by a point inside a circle as the circle rolls along a horizontal line, as shown in Figure 1-55. Coordinates for the trochoidal wave form are developed from the relationships: x = L sin + h 360 2 1 cos 2

y = h

The relationships are not linear, so there is no fixed interval that will match the x interval to station spacing; x and y coordinates are determined for values of from 0 to 360 at convenient increments, such as 30 degrees. Because the ordinates to the trochoidal wave do not fall on Bonjean's stations, it is important to plot the curve carefully to minimize error. The area under a sagging trochoid is less than the length multiplied by half the height, so the line of centers (see Figure 1-55) must be placed above the still water line for buoyancy to equal ship's weight (for a hogging wave, the line is placed below the still waterline). The area under a trochoid is equal to that of a rectangle with the same length and an upper boundary formed by a line r2/2R below the line of centers. Since the circle describing the trochoid makes one revolution in the ship's length, L = 2R, and 2R = L/. For an L/20 wave, r = L/40, and: L 2 40 L L2 1,600 L L 1,600 L wave 20

r 2R





= 0.00196 L

As an initial estimate, the line of centers of the trochoidal wave should be placed 0.00196L above the still waterline. If r is expressed as 0.55L, L will cancel out of the ratio, giving no solution. For a 1.1L wave, r is expressed as h/2, and:


r 2R


h 2 2 L


h 2 4L


0.785 h 2 L

1.1 L wave

For manual calculations, it is often simpler to use sinusoidal waves (y = Lsin), as they are not horizontal-scale dependent. The full wave form is developed in 180 degrees, and ordinates calculated at even increments of are plotted at evenly spaced stations. If increments of are set equal to 180 divided by the number of segments, the wave ordinate stations correspond to the Bonjean's curve stations, simplifying determination of section areas. Sinusoidal waves are somewhat steeper than trochoidal waves. For fine-lined ships, maximum hogging moments will be lower and maximum sagging moments higher than moments based on trochoidal waves of the same length and height. For full-bodied ships, both hogging and sagging moments will be higher when based on sinusoidal waves. For a ship with block coefficient of 0.46, the standard 1.1L sine wave bending moment is 6 percent less than trochoidal for hogging and 2 percent higher for sagging. For a block coefficient of 1.0, the standard sine wave bending moment is 11 percent higher for hogging and 9 percent higher for sagging. 1-11.4 Curve Scales. It is sometimes convenient to draw the load, shear, and bending moment curves on the same plan. To standardize drawing size and simplify manual integration, the U. S. Navy has adopted the following scaling criteria for longitudinal strength drawings like that shown in Figure FO-4.

· · · · · ·

Base length for all curves is 20 units. Base length corresponds to the length between perpendiculars, so the horizontal scale is one unit = L/20 feet. The mean heights of the weight and buoyancy curves are three units for the full load condition. Vertical scale for weight, buoyancy, and load curves is one unit = W/3L tons per foot of length. One square unit of area under the weight, buoyancy, or load curves represents L/20 × W/3L = W/60 tons. The shear curve is drawn so that one unit of ordinate represents two square units of area under the load curve; the vertical shear scale is one unit = W/30 tons. One square unit of area under the shear curve represents L/20 × W/30 = WL/600 foot-tons. The bending moment curve is drawn so that one unit of ordinate represents three square units under the shear curve; the vertical moment scale is one unit = WL/200 foot-tons.



Navy drawings use one inch as the base unit, but any convenient unit or multiple can be used. When there is no requirement to plot curves on the same plan, it is more convenient to make all the integration calculations in the base units without scale conversions. 1-11.5 Section Modulus. From beam theory, the bending stress () at any point is given by: My = I where:





M y

= =



bending moment at the section in question vertical distance from the neutral axis to the fiber (element) in question moment of inertia of the section in question about the neutral axis


This relationship shows that the maximum tensile and compressive stresses will occur in the beam elements furthest from the neutral axis. The distance from the neutral axis to the outer fibers is designated c. The term I/c is sometimes calculated separately and called the section modulus (Z or SM). Substituting: max = Mc M = I Z







If, as is common, bending moment is exSECTION A-A pressed in foot-tons, moment of inertia in in2-ft2, and distances from the neutral axis Figure 1-56. Ineffective Shadow Zones at Discontinuities. in feet, the calculation yields bending stress in long tons per square inch. It is best to convert tons per square inch to pounds per square inch for comparison with material strengths (normally tabulated in psi) and to avoid confusion between long, short, and metric tons. 1-11.5.1 Effective Structure. Calculating the moment of inertia for a simple girder is straightforward; the relatively complex cross section of a ship is another matter. Judgement must be used to determine which elements of the ship's structure effectively contribute to longitudinal strength. Elements that are subject to buckling, tripping and other forms of load shirking, or that are inadequately joined to the overall structure, cannot be assumed to contribute to longitudinal strength. As load shirking by panels with a width-to-thickness ratio greater than 70 is likely, contribution of unsupported plating panels should be limited to 70 times the thickness. Material not structurally continuous for at least 40 percent of the length of the ship about the section being examined is assumed to be ineffective. Only the net cross-sectional area of longitudinally continuous components of longitudinal strength members, excluding openings and ineffective shadow areas forward and aft of openings or other discontinuities, are included when calculating the moment of inertia. The shadow area of an opening is the area forward and aft of the opening between converging lines drawn tangent to the radiused corners at a slope of one transverse unit to four longitudinal units, as shown in Figure 1-56. All structures, including longitudinal framing and other connected structures within this area, are considered ineffective. For openings caused by damage or with sharp corners, lines bounding shadow areas should be drawn tangent to points outside the area of wrinkled or upset plating, or at a distance equal to 30 times the plating thickness from the edge of the opening, whichever is greater. Shadow areas adjacent to discontinuities such as the ends of longitudinal bulkheads, strength decks, and inner bottoms, are bounded by lines with a 1:4 slope, as shown in Figure 1-56.



1-11.5.2 Calculating Section Modulus. After the elements to be included have been selected, moment of inertia, I, is calculated by summing second moments of area (ay2) of individual elements about an arbitrary axis. It is most convenient to sum moments about the keel (some authorities prefer to use an assumed neutral axis). Moments of inertia (i) of elements with significant vertical dimensions are added to the summed second moments of elemental areas. Moment of inertia about the keel (IK) is then: IK = where: IK a y ay2 i = = = = = moment of inertia of section about the keel, in2-ft2 area of individual section element, in2 height of centroid of section element above the keel, ft second moment of area of individual section element, in2-ft moment of inertia of individual section elements, in2-ft2 (ay 2) (i)

Measuring areas in square inches and vertical distances from the axis in feet gives second moments of area (moments of inertia) in in2-ft2, rather than the in4, ft4, cm4, etc., customarily used in other branches of engineering. Moment of inertia of a rectangle is equal to bh3/12, where h is the height and b the breadth of the rectangle: i = If area is given in square inches, and height in feet, the units of moments of inertia of individual elements are consistent with the units of ay2. Individual moments of inertia for inclined or curved plates with significant vertical dimensions are determined by calculating the square of the radius of gyration (k) as shown in Figure 1-57. Moment of inertia can then be calculated from the definition of radius of gyration. i = ak2

y y h g

bh 3 (bh) h 2 = 12 12


ah 2 12


To obtain i in in2-ft2, a must be given in square inches, and k in feet. If the inclined flat-plate section shown in Figure 1-58 is 5/ 8-inch thick, 54 inches wide, and inclined so that h is 40 inches, then: k2 = = h2 12 = 402 12 = 133.33 in2

h k2 = 12



i = ak2

Figure 1-57. Moment of Inertia for Inclined Plates.

133.33 144

= 0.926 ft2

5 i = a k 2 = 54 × (0.926) = 31.25 in2 ft2 8 Since the neutral axis of the ship's section passes through the centroid of the section, height of the neutral axis above the keel is found by dividing the first moment of areas by the sum of areas of the section. The moment of inertia about the neutral axis is found by the parallel axis theorem: INA = IK Ad 2 where: INA IK A d = = = = moment of inertia about the neutral axis, in2-ft2 moment of inertia about the keel, in2-ft2 = (ay2) + (i) total area of individual section elements, in2 = (a) height of the neutral axis above the keel, ft = (ay2)/(a)



Once INA and height of the neutral axis are known, section modulus (INA/c) is easily calculated. The neutral axis is not usually equidistant from the top and bottom flanges of the hull girder (strength deck and keel), so each flange has its own value for c and therefore Z. The summations required to find height of the neutral axis and moment of inertia can be methodically performed in a tabular format. Table 1-14 is a sample section modulus calculation for the ship section shown in Figure 1-58. In an intact ship of uniform cross section, maximum bending stress occurs at the location of maximum bending moment. A vessel's cross section is not normally uniform throughout its length, but the scantlings at each section are selected by the designer to keep bending stresses within acceptable limits based on the anticipated bending moment.

5 x 4 x 6.00#T 5 x 5 3/4 x 13.0#T 20.5' 5.75' 15.3# 30 SHADOW 25.5# 30.0' ABV BL 6 x 6 1/2 x 13.0#T 25 5 x 4 x 6.00#T 4 x 4 x 5.00#T 20 21.0' ABV BL 6 x 4 x 7#T




2' 7-1/2" x .500 P HY-80 L SHELL DOUBLER

2' 6"x0.75" P HY-80 L SHELL DOUBLER



L 20 L 19 L 18 L 17 "E"-20.4 P L HY-80#




10.2# P L

L 16 L 15 L 14


6 x 4 x 8.00#T 6 x 6 1/2 x 13.0#T

L 13 L 12 L 11

"D"-12.75# P L


7 x 6 3/4 x 15#T L 10 18 x 7 1/2 x 50#I-T 8 x 7 x 22.5#T L9 L8 L7 L6 L5 L3 L1 35.7# L2 "A"-38.25# P HY80 L NOTE: I - T SHAPES ARE FORMED FROM W SHAPES BY CUTTING LOWER FLANGE FROM WEB, USUALLY WITH TWO VERTICAL CUTS L4 "B"-20.4# P L 2' 9" x 0.75 P M.S. L SHELL DOUBLER "C"-15.3# P HY-80 L

5 9 x 7 1/2 x 25#T 25 x 13 x 162# I-T CVK 0


P F.K. L


Figure 1-58. Frigate Hull Section at Station 10.



Table 1-14. Section Modulus for FFG-7, Station 10.



a (in2)

12.39 15.24 14.80 64.13 52.50 25.31 12.75 29.06 50.63 31.50 46.63 72.00 15.75 22.50 24.75 3.82 3.82 1.77 1.77 2.08 2.08 2.36 2.36 3.82 3.82 3.82 3.82 3.82 4.42 4.42 10.60 6.63 6.63 7.33 7.33 16.38 6.13 598.95

y (ft)

29.613 26.613 20.746 30.000 30.000 21.000 21.000 17.875 16.500 7.000 3.000 0.875 28.000 26.500 0.500 28.000 26.500 24.500 22.750 19.250 17.500 16.000 14.750 12.500 11.750 9.000 7.500 6.250 5.500 4.500 4.25 2.750 2.000 1.500 1.000 1.500 0.073

ay (in2 ft)

366.91 405.58 307.04 1923.75 1575.00 531.56 267.75 519.49 835.31 220.50 139.88 63.00 441.00 596.25 12.38 106.96 101.25 43.37 40.27 40.04 36.40 37.76 34.81 47.75 44.89 34.38 28.65 23.88 24.31 19.89 45.05 18.23 13.26 11.00 7.33 24.57 0.45 8989.85

ay2 (in2 ft2)

10865.16 10793.76 6369.87 57712.50 47250.00 11162.81 5622.00 9285.92 13782.66 1543.50 419.63 55.13 12348.00 15800.63 6.19 2994.88 2682.60 1062.44 916.09 790.77 637.00 604.16 513.45 596.88 527.40 309.42 214.88 149.22 133.70 89.51 191.46 50.14 26.52 16.49 7.33 36.85 0.03 215549.69

h or k* (ft)

i = ah2/12 or ak2* (in2 ft2)

Mn Dk Girders, Inbd (7) - T Mn Dk Grdrs, Outbd (4) - T 2nd Dk Girders, (10) - T Mn Dk Plating, Inbd, less shadow zones Mn Dk Plating, Outbd 2nd Dk Plating, Inbd, less shadow zones 2nd Dk Pltg, Outbd "E" Strake "D" Strake "C" Strake "B" Strake "A" Strake "E" Doubler, upper "E" Doubler, lower "A" Doubler Side Stringers L20 - T L19 - T L18 - T L17 - T L16 - T L15 - T L14 - T L13 - T L12 - T L11 - T L10 - T L9 - T L8 - T Bottom Longitudinals L7 - T L6 - T L5 - I - T L4 - T L3 - T L2 - T L1 - T CVK (1/2) I - T Flat Keel (1/2) Totals

5 × 4 × 6# 5 × 5.75 × 13# 4 × 4 × 5# (246 - 75) × 0.375 84 × .625 (225 - 90) × .25 51 × .25 93 × .3125 162 × .3125 84 × .375 93.25 × .5 96 × .75 31.5 × .5 30 × .75 33 × .75 6 6 5 5 6 6 6 6 6 6 6 6 6 × × × × × × × × × × × × × 6 × 13# 6 × 13# 4 × 6# 4 × 6# 4 × 7# 4 × 7# 4 × 8# 4 × 8# 6.5 × 13# 6.5 × 13# 6.5 × 13# 6.5 × 13# 6.5 × 13#

7.75 12.50 1.88* 0.42* 0.26* 2.63 2.50

145.45 659.18 111.33* 8.23* 4.87* 9.04 11.72

7 × 6.75 × 15# 7 × 6.75 × 15# 18 × 7.5 × 50# 8 × 7 × 22.5# 8 × 7 × 22.5# 9 × 7.5 × 25# 9 × 7.5 × 25# 25 × 13 × 162# 14 × .875




3.18 971.75



(ay)/a (ay 2) + i IK - Ad 2 2INA for half-section Depth - d INA/c t d INA/c b

= = = = = = = =



15.01 ft 216,521.44 in2 ft2 81,577.95 in2 ft2 163,155.90 in2 ft2 14.99 ft 10,884.32 in2 ft 10,869.81 in2 ft

IK for half-section = INA for half-section = INA for full section = cDK ZDK cK ZK

= = = =

215,549.69 + 971.75 = 216,521.44 - (598.95 × 15.012) = 2(81,577.95) 30 - 15.01 163,155.90/14.99 15.01 ft 163,155.90/15.01 = = = =

Notes: Areas and centroids for T-shapes taken from AISC Manual for Steel Construction, 8th Edition. i of vertical web only












Figure 1-59. Shear Stress in the Hull Girder.

1-11.6 Shear Stress. Shear stresses result from vertical shear, caused by the uneven force distribution along the ship's length, and horizontal shear, caused by longitudinal bending and racking, as shown in Figure 1-59. The shear force is distributed over the section, each element contributing to the total. Shear stress distribution can be modeled by the theory of thin-walled sections, as explained in the Society of Naval Architects and Marine Engineers' Principles of Naval Architecture, but this method requires the evaluation of indefinite line integrals, and may be too tedious for field calculations. For salvage calculations, shear stress, , along any horizontal axis BB can be adequately approximated by the expression: SQ = INA b where: S Q a y INA b = = = = = = = = shear stress shear at the section in question first moment of area about the neutral axis of the area of effective structure above axis BB ay area of individual structural element vertical distance of individual structural elements from neutral axis moment of inertia of the section about the neutral axis total width of material resisting shear along axis BB, in

Moment of inertia is obtained as part of the section modulus calculation. The first moment of area, Q, is determined by summing the products of areas and their distances from the neutral axis in the same manner that ay about the keel is determined in the section modulus calculation. The material width, b, is normally twice the shell-plating thickness (to account for both sides), plus the thickness of effective longitudinal bulkheads, i.e., those that extend from the strength deck to the bottom of the ship and are firmly anchored at both top and bottom. Consistent units must be used, along with appropriate conversion factors. If moment of inertia and first moment of area are in the customary units of in2-ft2 and in2-ft, a conversion factor of 12 must be applied to obtain stress in units of force per square inch: = SQ 12 INA b









Figure 1-60. Shear Stress.

Shear is normally determined in long tons, giving shear stress in long tons per square inch; shear stress, like bending stress, is converted to pounds per square inch by multiplying by 2,240 pounds per long ton. Shear stresses act in pairs, are equal on all four faces of a plane element, and are maximum on planes parallel and perpendicular to the shear force, as shown in Figure 1-60. Because the paired stresses tend to change the angle between faces of an element and lengthen the diagonal, shear yield in plating panels is evidenced by diagonal wrinkles. The form of the expression implies that shear stress in any section is zero at the deck and keel and maximum at the neutral axis, where Q is maximum: SQmax max = 12INAb where: Qmax = first moment of the area above neutral axis about the neutral axis Although shear stress in the deck is very low, and may approach zero near the centerline, shear stress is not usually zero at the deck edge; the expression does estimate shear stress in the middle portion of the side shell (where it is normally of greatest concern) accurately.




This example illustrates the detailed still water strength calculations for an FFG-7 Class ship, including steps to reconcile inconsistent data, and to balance weight and buoyancy. Examples 4-5 through 4-12 in the U.S. Navy Ship Salvage Manual, Volume 1 (S0300-A6-MAN-010) illustrate simplified calculations for a simple barge. For an FFG-7 Class ship in the 1/3 Consumed Stores loading condition, calculate: Deck and keel bending stresses for stations 3 through 17 Maximum shear stress From the Damage Control Book (DC Book) loading summary (Appendix F): 1/3 Consumed Stores, Sequence 6 Fuel/Ballast:


Weight tons

lcg fm Comments midships ft

161.8 80.0 -141.1 -141.1 Saltwater ballast tanks listed as empty for full load

Clean Ballast: 5-34-0-W 5-116-0-W 5-328-1-W 5-328-2-W

32.04 53.56 19.62 19.62

Oily Ballast: 5-100-3-F 5-100-4-F 5-250-1-F 5-250-2-F

9.47 9.47 9.9 9.9

92.3 92.3 -59.8 -59.8

Fuel/ballast tanks, filled with fuel for departure full load. Listed weights are differences between weights of equal volumes of fuel and seawater


= = = = = = =

14' 8" 15' 8" 15.23' (LCF 23.79 ft abaft midships) 3748.15 tons 5.53 ft abaft midships 3.06 ft abaft midships 769.01 ft-tons

Miscellaneous Holding Tanks: 5-132-0-F 5-164-0-F 5-170-0-F 4-170-0-W 19.21 44.00 16.31 11.84 68.4 36.9 29.0 29.7 Contaminated Oil Settling Tank Waste Oil Retention Tank Oily Waste Water Holding Tank Sewage Collection, Holding and Transfer (CHT) Tank

Full-load Displacement = 3,951.79 tons From Curves of Form (FO-2) for TLCF = 15.23':


254.94 tons

W = 4,224.83 - 254.94 = 3,969.89

The difference between the corrected longitudinal strength drawing displacement and the full-load departure displacement from the DC Book is: 3,969.89 - 3,951.79 = 18.1 tons or 4.6 percent. The discrepancy cannot be resolved further without additional data. It is not necessary to constuct a corrected full-load curve that would then be corrected for the actual loading condition. The two corrections can be made simultaneously. b. Initial Weight Curve for 1/3 Consumed Stores condition (3,748.15 tons)


= = = =

3,750 tons 3.1 ft abaft midships 23.8 ft abaft midships 770 ft-tons

From Longitudinal Strength and Inertia Sections Drawing (FO-4):

W = 4,224.83 tons

Scale Factors: Length 1 in. Weight Ordinates 1 in. Weight Area 1 in2 Shear Ordinates 1 in. Shear Area 1 in2 Moment Ordinates 1 in. a. = = = = = = 408/20 4,224.83/3L 4,224.83/60 4,224.83/30 4,224.83(408)/600 4,224.83(408)/200 = = = = = = 20.4 ft 3.45 tons/ft 70.41 tons 140.83 tons 2,872.88 ft-tons 8,618.65 ft-tons

The weight curve is created by deducting the weight differences between the full-load condition and the actual condition from the full-load curve at their locations. The corrections to the full-load curve described in Paragraph a. above are deducted at the same time. Examination of the DC Book loading summaries for the full load and 1/3 consumed stores conditions reveals the following weight differences: Item Full Load Weight tons 1/3 Consumed Weight tons Difference tons lcg from Midships ft

Resolution of discrepancies in raw data

The data from the DC Book and Curves of Form are in good agreement. However, at equilibrium, LCB and LCG must be aligned vertically. The Curves of Form give LCB for the ship with 0 trim. Assuming the same to be true for the DC Book, the initial trim arm (BGL) is 2.47 feet (5.53 - 3.06). The resulting trim would be:

Provisions and Stores Dry provisions Frozen Chill Clothing, Small Stores Ship Stores General Stores Deck Gear Flammable Liq & Paints Bosun Storeroom Medical Stores Misc Storerooms Potable Water 5-292-3-W 5-308-1-W 5-308-2-W 8.73 7.88 7.88 8.71 2.37 2.37 0.02 5.51 5.51 -94.4 aft 115.8 fwd 115.8 fwd 13.95 4.84 4.79 0.31 3.49 9.29 3.23 3.19 0.21 2.33 4.66 1.61 1.60 0.10 1.16 9.0 fwd 20.0 fwd 20.0 fwd 145.5 fwd 4.0 fwd

t = W(BGL)/MT1 = 3,748.15(2.47)/769.01 = 12.04 in by the stern

This is consistent with the tabulated drafts. In constructing the weight and buoyancy curves, it will be assumed that the actual centers of gravity and buoyancy are on a vertical line 5.53 feet aft of midships. There is a discrepancy of 273 tons between the full-load weights as given by the DC Book (3,951.79 tons) and the longitudinal strength drawing (4,224.83 tons). This discrepancy must be resolved as completely as possible before proceeding. The longitudinal strength drawing is prepared for the most extreme loading conditions. It is therefore likely that items of weight were included that are not included in the operating full-load departure condition described in the DC Book. The most probable items that would be included for the longitudinal strength drawing but deleted from the operational full load are saltwater ballast and waste-holding tanks that would be presumed empty for the departure condition. An examination of the full-load condition and tank capacity tables from the DC Book reveals the following potential weights.

2.37 3.77 4.13 1.00 7.46

1.58 2.51 2.75 0.67 4.98

0.79 1.26 1.38 0.33 2.48

81.3 fwd 115.5 fwd 137.1 fwd -176.0 aft -68.5 aft



Segment Item Full Load Weight tons 1/3 Consumed Difference Weight tons tons lcg from Midships ft -1.4-0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-20.6 Totals

Ordinate y in. 0.15 0.62 1.37 1.59 2.93 3.12 3.11 3.56 2.86 2.01 3.76 3.76 3.49 1.99 4.21 3.55 2.57 2.40 1.88 2.35 1.95 0.49

Length l in. 1.40 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60

Area y×l in2 0.21 0.62 1.37 1.59 2.93 3.12 3.11 3.56 2.86 2.01 3.76 3.76 3.49 1.99 4.21 3.55 2.57 2.40 1.88 2.35 1.95 0.29 53.58

lcg from FP Moment at midordinate lcg × area in. in2

-0.70 0.50 1.50 2.50 3.50 4.50 5.50 6.50 7.50 8.50 9.50 10.50 11.50 12.50 13.50 14.50 15.50 16.50 17.50 18.50 19.50 20.30 -0.15 0.31 2.06 3.98 10.26 14.04 17.11 23.14 21.45 17.08 35.72 39.48 40.14 24.88 56.84 51.48 39.84 39.60 32.90 43.48 38.03 5.97 557.59

Lubricating Oil 3-272-2-F 3-278-2-F 3-286-2-F 3-208-4-F 3-236-1-F 3-236-2-F 3-292-8-F Fuel Oil, Storage 5-100-3-F 5-100-4-F 5-116-1-F 5-140-1-F 5-250-1-F 5-250-1-F Fuel Oil, Service 5-204-2-F 3-240-2-F 3-292-6-F 5-201-3-F JP-5 5-344-0-J 29.81 8.54 21.27 -150.9 aft 46.47 2.54 1.21 1.33 23.18 0.53 0.25 0.28 23.29 2.01 0.96 1.05 -4.0 aft -40.9 aft -89.3 aft 1.7 fwd 32.12 32.12 65.69 28.43 33.60 33.60 0.00 0.00 63.60 22.21 0.00 0.00 32.12 32.12 2.09 6.22 33.60 33.60 92.3 fwd 92.3 fwd 75.5 fwd 51.8 fwd -59.8 aft -59.8 aft 3.50 4.00 2.75 0.95 1.05 1.05 0.92 2.35 2.68 1.84 0.63 0.70 0.70 0.61 1.15 1.32 0.91 0.32 0.35 0.35 0.31 -70.7 aft -77.9 aft -85.0 aft -6.0 aft -33.9 aft -33.9 aft -89.3 aft

W = area × scale factor = 53.58(70.41) = 3772.85 tons centroid = moment/area = 557.59/53.58 = 10.41 in fm FP LCG = centroid × scale factor = 10.41(20.4) = 212.28 ft fm FP = 212.28 - 204 = 8.28 ft aft of midships LCG of the curve is more than one foot from the known LCG (5.53 ft aft of midships), so the curve must be adjusted to move the LCG forward. The initial buoyancy curve is developed for comparison before correcting the weight curve.

c. Initial Buoyancy Curve for 1/3 Consumed Stores condition (3,748.15 tons) The buoyancy curve ordinates are calculated by determining section areas for each station from the Bonjean's Curves (FO-3), dividing the area by 35 to convert to unit buoyancy (tons per foot), and dividing the unit buoyancy by the scale factor (3.45). Drafts at each station are calculated assuming no hog or sag. Before calculating ordinates from the section areas, the area curve is integrated to compare total buoyancy and LCB with total weight and LCG from the weight curve. The integration is performed by Simpson's rule on 21 stations: Station Draft T ft 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sums 14.67 14.72 14.77 14.82 14.87 14.92 14.97 15.02 15.07 15.12 15.17 15.22 15.27 15.32 15.37 15.42 15.47 15.52 15.57 15.62 15.67 Ordinate Multiplier (Section Area) y m ft2 2 55 131 205 270 326 379 428 471 499 515 519 500 470 418 357 285 215 153 95 41 1 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 1 18,963

Miscellaneous Tanks 5-132-0-F 5-170-0-F 5-164-0-F Total: 0.00 0.00 0.00 203.64 9.61 4.08 2.12 -9.61 -4.08 -2.12 68.4 fwd 29.0 fwd 37.0 fwd

W1/3 = 3,951.79 - 203.64 = 3,748.15

The ordinates for the weight curve are calculated by consolidating the differences by weight segments, distributing the weight difference over the length of the segment, and dividing the distributed weight difference by the scale factor (3.45). The new weight curve ordinates are calculated in the following table: Segment Old Weight Dist Load Ordinate New Ordinate Ordinate Difference wt diff/20.4 Difference Old ord - diff dl/3.45 in. tons tons/ft in. in. 0.15 0.62 1.37 2.05 2.95 3.29 4.29 4.50 2.95 2.95 3.90 4.10 3.50 3.25 4.28 3.58 2.57 2.96 2.18 2.35 1.95 0.49 0.00 0.00 0.00 -32.14 -1.38 -12.28 -83.18 -66.04 -6.22 -65.95 -10.08 -23.61 -0.70 -89.01 -4.95 -2.20 0.00 -39.24 -21.27 -0.33 0.00 0.00 0.00 0.00 0.00 -1.58 -0.07 -0.60 -4.08 -3.24 -0.30 -3.23 -0.49 -1.16 -0.03 -4.36 -0.24 -0.11 0.00 -1.92 -1.04 -0.02 0.00 0.00 0.00 0.00 0.00 -0.46 -0.02 -0.17 -1.18 -0.94 -0.09 -0.94 -0.14 -0.34 -0.01 -1.26 -0.07 -0.03 0.00 -0.56 -0.30 0.00 0.00 0.00 0.15 0.62 1.37 1.59 2.93 3.12 3.11 3.56 2.86 2.01 3.76 3.76 3.49 1.99 4.21 3.55 2.57 2.40 1.88 2.35 1.95 0.49

f(V) y×m ft3

2 220 262 820 540 1,304 758 1,712 942 1,996 1,030 2,076 1,000 1,880 836 1,428 570 860 306 380 41


f(M) s × (V) ft4

0 220 524 2,460 2,160 6,520 4,548 1,1984 7,536 1,7964 10,300 22,836 12,000 24,440 11,704 21,420 9,120 14,620 5,508 7,220 820 193,904

s ft

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-1.4-0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-20.6

The weight curve is integrated on these ordinates to determine total area (weight) and longitudinal position of the centroid (center of gravity). The integration is carried out in a tabular format:


= = = = =

20.4 (h/3) × (V) = (20.4/3)(18,963) = 128,948.4 ft3 V/35 = 128,948.4/35 = 3684.24 tons h(M)/(V) = 20.4(193,904)/(18,963) = 208.6 ft fm FP 208.6 - 204 = 4.6 ft aft of midships



d. Adjusting Weight and Buoyancy Curves The weight and buoyancy curves disagree by 88.61 tons on total area. This error is undesirable, but probably tolerable. The 3.68-foot separation between the centers of gravity and buoyancy is excessive and must be corrected. The ordinates of both curves must be adjusted to bring the centers of gravity and buoyancy to within one foot of each other and within one foot of the point 5.53 feet abaft midships. Total buoyancy is corrected first by gradually increasing the area curve ordinates until the buoyancy (area under the curve divided by 35) equals total weight. There is a greater probability of error in reading the section areas for the middle stations because the Bonjean's Curves for the middle stations slope more gently than those near the ends. The corrections are therefore weighted towards the center of the curve. LCB is then moved aft by transferring a strip of uniform thickness from the forward half of the curve to the aft half. The thickness of the strip is determined by trial and error. After several iterations, the following section areas were determined: Station Ordinate Multiplier (Section Area) y m ft2 0 54 134 204 274 329 379 430 475 510 524 524 509 479 429 369 299 229 167 109 59 1 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 1

LCG of the initial weight curve is moved forward by transferring strips of uniform thickness from segments in the after half of the curve to the corresponding segments in the forward half, and by reducing some ordinates in the after half to lower total weight slightly. The thickness of the strips are determined by trial and error. After several iterations, ordinates were determined and integrated as follows:

Segment Ordinate y in. -1.4-0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-20.6 Totals 0.15 0.66 1.41 1.63 2.97 3.16 3.15 3.60 2.90 2.05 3.80 3.72 3.45 1.95 4.17 3.51 2.53 2.36 1.82 2.29 1.86 0.49 Length l in. 1.40 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60 Area y×l in2 0.21 0.66 1.41 1.63 2.97 3.16 3.15 3.60 2.90 2.05 3.80 3.72 3.45 1.95 4.17 3.51 2.53 2.36 1.82 2.29 1.86 0.29 53.49 lcg from FP Moment at lcg × Area Midordinate in. in2 -0.70 0.50 1.50 2.50 3.50 4.50 5.50 6.50 7.50 8.50 9.50 10.50 11.50 12.50 13.50 14.50 15.50 16.50 17.50 18.50 19.50 20.30 -0.15 0.33 2.12 4.08 10.40 14.22 17.33 23.40 21.75 17.42 36.10 39.06 39.68 24.38 56.30 50.90 39.22 38.94 31.85 42.37 36.27 5.97 551.90

f(V) y×m ft3

0 216 268 816 548 1,316 758 1,720 950 2,040 1,048 2,096 1,018 1,916 858 1,476 598 916 334 436 59 19,387


f(M) s × (V) ft4

0 216 536 2,448 2,192 6,580 4,548 12,040 7,600 18,360 10,480 23,056 12,216 24,908 12,012 22,140 9,568 15,572 6,012 8,284 1,180 199,948

s ft

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sums

W = area × scale factor = 53.49(70.41) = 3766.51 tons centroid = moment/area = 551.90/53.49 = 10.32 in fm FP LCG = centroid × scale factor = 10.32(20.4) = 210.53 ft fm FP = 210.53 - 204 = 6.53 ft aft of midships

The adjusted weight and buoyancy curves are shown in Figure 1-61. e. Shear and Bending Moment Curves Ordinates to the load shear and bending moment curves are determined by a continuous tabular calculation. Curve segments are identified by the bounding stations in the first column. The weight ordinates are written in the second column. The mean buoyancy ordinates for each segment are written in the third column. The load ordinate in the fourth column is found by subtracting the weight ordinate (column 2) from the mean buoyancy ordinate (column 3). The load curve is integrated along its length by keeping a running total of the area under the load curve in the fifth column. In keeping with the convention of integrating the load curve from left to right, the area total is run from bottom to top in this table. The area for each segment is the ordinate multiplied by the segment length (1 inch for all but the two end segments). The area total is the area up to the forward station of the segment. The shear ordinates in the sixth column are determined by dividing the areas in column 5 by two. The shear curve defined by these ordinates is shown in Figure 1-62. The shear ordinates are carried into the following table and written in the second column, next to the appropriate station (column 1). It is necessary to interpolate the x intercept (station 10.41) to properly integrate the curve and to determine the section of maximum bending moment. The mean shear ordinate for each segment is written in the third column. The shear curve is integrated along its length from forward aft (top to bottom); the running total is written in the fourth column. The shear areas are divided by 3 and written in the fifth column as the moment ordinates. The resulting bending moment curve is shown in Figure 1-62. Bending moments for use in the bending stress calculations are determined by multiplying the moment ordinate by the scale factor, 8,618.65 ft-tons/in.


= = = = =

20.4 (h/3) (V) = (20.4/3)(19,387) = 131,831.6 ft3 V/35 = 131,831.6/35 = 3,766.62 tons h(M)/(V) = 20.4(199,948)/(19,387) = 210.4 ft fm FP 210.4 - 204 = 6.4 ft abaft midships

Now that the total buoyancy and location of LCB are both acceptably near the known values, the buoyancy curve ordinates are calculated: Station 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Section Area ft2 0 54 134 204 274 329 379 430 475 510 524 524 509 479 429 369 299 229 167 109 59 Unit Buoyancy B = A/35 tons/ft 0.00 1.54 3.83 5.83 7.83 9.40 10.83 12.29 13.57 14.57 14.97 14.97 14.54 13.69 12.26 10.54 8.54 6.54 4.77 3.11 1.69 Ordinate B/3.45 in. 0.00 0.45 1.11 1.69 2.27 2.72 3.14 3.56 3.93 4.22 4.34 4.34 4.22 3.97 3.55 3.06 2.48 1.90 1.38 0.90 0.49









Segment Weight Mean Load Cum. Area Shear Ordinate Buoyancy Ordinate under Ordinate w Ordinate b - w Load Area/2 in. b in. Curve in. in. in2 -1.4-0 0.15 0.00 -0.15 -0.02 -0.012 0-1 0.66 0.23 -0.43 0.19 0.093 1-2 1.41 0.78 -0.63 0.62 0.308 2-3 1.63 1.40 -0.23 1.25 0.623 3-4 2.97 1.98 -0.99 1.48 0.738 4-5 3.16 2.49 -0.67 2.47 1.233 5-6 3.15 2.93 -0.22 3.14 1.568 6-7 3.60 3.35 -0.25 3.36 1.678 7-8 2.90 3.75 0.85 3.61 1.803 8-9 2.05 4.07 2.02 2.76 1.378 9-10 3.80 4.28 0.48 0.74 0.368 10-11 3.72 4.34 0.62 0.26 0.128 11-12 3.45 4.28 0.83 -0.36 -0.182 12-13 1.95 4.09 2.14 -1.19 -0.597 13-14 4.17 3.76 -0.41 -3.33 -1.667 14-15 3.51 3.31 -0.20 -2.92 -1.462 15-16 2.53 2.77 0.24 -2.72 -1.362 16-17 2.36 2.19 -0.17 -2.96 -1.482 17-18 1.82 1.64 -0.18 -2.79 -1.397 18-19 2.29 1.14 -1.15 -2.61 -1.307 19-20 1.86 0.69 -1.17 -1.46 -0.732 20-20.6 0.49 0.00 -0.49 -0.29 -0.147 2 3 4 Shear Mean Area Station Ordinate Shear under Ordinate Shear Curve in. in. in2 -1.4 -0.120 0 0.41 0 0.093 0.57 0.62 1 0.308 1.19 0.47 2 0.623 1.65 0.68 3 0.738 2.33 0.99 4 1.233 3.32 1.40 5 1.568 4.72 1.62 6 1.678 6.34 1.74 7 1.803 8.08 1.59 8 1.378 9.67 0.87 9 0.368 10.55 0.18 10 0.128 10.73 0.06 10.4 0.000 10.76 -0.09 11 -0.182 10.70 -0.39 12 -0.597 10.31 -1.13 13 -1.667 9.18 -1.56 14 -1.462 7.62 -1.41 15 -1.362 6.20 -1.42 16 -1.482 4.78 -1.44 17 -1.397 3.34 -1.35 18 -1.307 1.99 -1.02 19 -0.732 0.97 -0.44 20 -0.147 0.53 -0.07 20.6 0.000 0.46 f. Bending Stresses 1 5 6 Moment Moment Ordinate Mom. Ord Shear x 8618.65 Area/3 in. ft-tons 0.00 0 0.19 0.40 0.55 0.78 1.11 1.57 2.11 2.69 3.22 3.52 3.58 3.59 3.57 3.44 3.06 2.54 2.07 1.59 1.11 0.66 0.32 0.18 0.15 1,629 3,407 4,745 6,700 9,531 13,554 18,217 23,217 27,787 30,295 30,823 30,899 30,744 29,625 26,373 21,879 17,822 13,737 9,601 5,717 2,788 1,526 1,314

5 4 3



5 4 3 2

2 1 0 WEIGHT

1 0

2 1 0 -1 AP 19 18 17 16 15 14 13 12 11 10 9 STATIONS 8 7


2 1 0 -1






1 FP

Figure 1-61. Buoyancy, Weight, and Load Curves for FFG-7.

10.4 5 4 3



5 4 3 2 1 0 LOAD -1 -2 -3 -4 -5

2 1 0 -1 -2 -3 -4 -5 AP 19 18 17 16 15 14 13 12 11 10 9 STATIONS 8 7 6 5 4 3


1 FP

Figure 1-62. Still Water Load, Shear, and Bending Moment Curves for FFG-7.

Bending stresses are calculated using the tabulated moments of inertia from the Longitudinal Strength and Inertia Sections Drawing (FO-4): Station Moment M ft-tons 3 4 5 6 7 8 9 10 10.4 11 12 13 14 15 16 17 6,700 9,531 13,554 18,217 23,217 27,787 30,295 30,823 30,899 30,744 29,625 26,373 21,879 17,822 13,737 9,601


in -ft

2 2


ft 15.09 15.68 14.18 15.32 15.37 15.53 15.36 14.62 14.90 14.16 14.27 15.00 12.87 11.70 10.12 9.46

deck Mc/I tons/in2 0.91 1.32 1.88 2.04 2.74 3.12 2.92 2.64 2.85 2.60 2.70 2.92 2.56 2.33 2.01 1.59


ft 20.58 18.84 19.27 17.23 16.38 15.59 15.21 15.45 15.02 15.51 15.10 14.27 15.29 14.83 14.32 13.07

keel Mc/I tons/in2 1.25 1.59 2.55 2.29 2.92 3.13 2.89 2.79 2.88 2.85 2.86 2.78 3.04 2.95 2.85 2.19

Since the ship is hogging, the deck is in tension and the keel in compression. All weight and buoyancy forces were given in long tons, so the stresses are in long tons per square inch. Stresses are converted to psi by multiplying by 2,240. Deck and keel bending stresses are plotted in Figure 1-63 (Page 1-94). Note that the maximum bending stresses do not occur at the section of maximum bending moment. g. Maximum Shear Stress

110,681 112,994 102,384 136,770 130,123 138,267 159,477 170,416 161,280 167,165 156,553 135,444 110,066 89,467 69,084 57,188


S(Q) 12 INA b

Shear stress is a function of shear force (S), moment of inertia (I), and plating thickness (b), and is maximum at the neutral axis for any section. Maximum shear occurs at station 7. Moments of inertia for adjacent stations and other stations of high shear are equal to or greater than that for station 7. Side-plating thickness at the neutral axis is constant between stations 3 and 17 (information taken from the section drawings of the Longitudinal Strength and Inertia Sections drawing - not reproduced in this handbook). Maximum shear stress can therefore be assumed to occur at or near station 7 at the neutral axis. The first moment of area about the neutral axis and shear stress for station 7 are calculated in a tabular format as shown on the following page.




Dimensions in.

a in2

y ft

ay in2ft

Mn Dk Girders 5 x 4 x 6# (12) 2nd Dk 4 x 4 x 5# Girders (11) Mn Dk Plating, 192 x .25 Inbd Mn Dk Plating, 84 x .375 Outbd 2nd Dk 202.5 x .1875 Plating, Inbd 2nd Dk Pltg, 52.5 x .25 Outbd "D" Strake 105 x .375 "C" Strake above N.A. 84 x .3125 (16.33') "D" Doubler 30 x .75 Side Stringers L20 6 x 4 x 7# L19 6 x 4 x 7# L18 5 x 4 x 6# L17 5 x 4 x 6# L16 6 x 4 x 7# L15 6 x 4 x 7# L14 6 x 4 x 8# Totals

15.93 15.100 240.54 10.66 6.062 64.61 48.00 15.370 737.76 31.50 15.370 484.16 37.97 6.312 239.66 13.13 6.312 82.85 39.38 10.995 432.93 26.25 3.500 91.88 22.50 11.750 264.38 1.67 1.67 1.33 1.33 1.67 1.67 1.96 270.5 13.625 11.750 9.875 8.125 4.625 2.875 1.250 22.77 19.64 13.11 10.79 7.73 4.80 2.45 2856.2

9 8 7 6 5 4 3 2 1 17 16 15 14 13 12


9 8 7 6 5 4 3 2 1 8 7 6 5 4 3



9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

9 8 7 6 5 4 3 2 1 8 7 6 5 4 3


Qhalf-section = Qwhole section = INA b S

ay = 2,856.2 in2-ft 2(2,856.2) = 5,712.4 in2-ft = 130,123 (from Longitudinal Strength and Inertia Sections drawing) = 2 × plate thickness @ NA = 0.625 in. = 1.803 x 140.83 = 255.61 tons = S(Q)/12INAb = 1.5 tons/in2 = 1.5 × 2,240 = 3,360 psi


Figure 1-63. Still Water Bending Stresses for FFG-7.

1-11.7 Bending Stress in Inclined Ships. If a ship is inclined, as shown in Figure 164, the depth of sections is increased and bending stresses at the "corners" may be increased. For a ship heeled to an angle , the new axis of bending is parallel to the water line. The bending moment, M, can be resolved into Mcos about the old (horizontal) neutral axis and Msin about the centerline of the ship. Each component produces stress as if it acted independently, and the total stress at some point P, with coordinates (x,y), is: t = where: t y x = = = total bending stress at (x,y) distance from the old neutral axis to the point in question distance from the centerline to the point in question moment of inertia about the old neutral axis moment of inertia about the centerline My cos INA Mx sin ICL






O 0




Figure 1-64. Stresses in Inclined Ships.




Since the section is not symmetrical about its new bending axis, the neutral axis is not parallel to the waterline (horizontal) but is inclined to it by some angle , as shown in Figure 1-64. The angle, , between the new neutral axis and the horizontal can be found from: INA ICL tan = tan

If the point farthest from the neutral axis has coordinates (x1, y1) referenced to the centerline and the old neutral (horizontal) axis, maximum bending stress in the section is: max = My1 cos INA Mx1 sin ICL

Tabulated section moments of inertia about the centerline are not normally available to the salvage engineer, and must be calculated. Calculating ICL is somewhat simpler and shorter than calculating INA because the incremental second moments are taken about a known axis (the centerline). There is therefore no need to sum first moments about an arbitrary axis to locate the neutral axis. For intact sections, only the incremental second moments of area for one side need be summed; the moment of inertia is twice the sum for one side. Distances from the centerline are scaled from section drawings. Maximum bending stresses in an inclined ship may be 20 percent greater than when the ship is upright. 1-11.8 Combined Stresses. The bending (tensile or compressive) and shear stresses in a ship or other beam combine to form the principal stress at any point. It can be shown that: s (s ) = 2 where: s = = = principal stress at any point simple tensile or compressive stress at the point in question shear stress at the point in question

This relationship does not solve for s so iterative or trial and error methods are used to determine principal stress. The presence of shear in the hull girder distorts the sections so that the conditions on which simple beam theory are based are not strictly fulfilled (see Chapter 2 for an explanation of basic beam theory). This alters bending stress distribution across the section from that predicted by beam theory. Analysis of this problem is beyond the scope of this book, but the general effect is to increase bending stress at the corners of the section, i.e., the deck edges and the bilge, and reduce bending stresses at the center of the deck and bottom. This effect is appreciable only when the ratio of length to depth is small. 1-11.9 Acceptable Stress Levels. The stress that any material can withstand without failure is a function of the properties of that material and the definition of failure. Fracture is an obvious and final form of failure. Permanent or plastic deformation, or unacceptable extents of deflection or elastic deformation can also be considered failure. 1-11.9.1 Failure Definition. In many engineered systems, deflection or deformation of a component in excess of certain limits interferes with the operation of the mechanism and is considered failure. Plastic deformation is often considered failure because of the discontinuous behavior of the material as it yields. Plastic behavior may be acceptable in components subjected to in-line, tensile loading where elongation will not cause interference with any other components. The deformation may render the component unsuitable for continued use, but many salvage evolutions are one-time events. Plastic behavior or excessive deflection/deformation should be carefully examined, as such deformation in components can alter stress levels in other components in unforeseen or unpredictable ways. Plastic failure in ship hulls is unacceptable because it unpredictably alters load responses. Failure of a given component must be defined accurately, so that limiting stress values for that component can be set. The limiting stress values define limiting loads for components; the degree of load sharing among components will define system load limits. 1-11.9.2 Factors of Safety. Use of an appropriate factor of safety keeps stresses well below the failure point and allows for manufacturing defects and inconsistencies in loading. Safety factors are specified by various regulatory agencies, depending on intended use of systems and components. In salvage it is not always possible to use a standard safety factor, so reduced factors of safety must often be accepted. This does not mean that salvors can disregard safety factors. Each situation must be examined to determine acceptable stresses and loads. A reduced safety factor represents an increased chance of failure. The consequences of failure must be considered and precautions taken.



1-11.9.3 Common Materials. The most commonly used shipbuilding materials are:

· · · · ·

Steel and Iron. Aluminum. Wood. Glass Reinforced Plastic (GRP). Copper and Copper Alloys.

In addition to encountering these as components of a ship, the salvor may use any of them in on-site repairs or fabrication of salvage systems, along with concrete or other materials. The ultimate or yield stresses of many materials vary depending on whether tensile, compressive or shear stress is experienced. This is an important factor in salvage operations, where components may be loaded in ways other than those anticipated by the designer. The mechanical properties of commonly used materials are given in Appendix E. Steel, in the form of rolled plate, rolled or forged structural shapes, or complex castings, is the most commonly used shipbuilding material. Shipbuilding steel meeting ABS and Navy specifications has a yield stress of not less than 32,000 psi and an ultimate stress of 58,000 -70,000 psi. In the United States, structural shapes and plates for general use are usually manufactured to American Society for Testing of Materials (ASTM) Standard A36, requiring a tensile yield strength of not less than 36,000 psi. Unless otherwise specified, mild steel can be assumed to have a yield strength of about 30,000 psi, although some alloys have yield strengths as low as 20,000 psi. Plating thickness is often specified by weight per square foot. Steel weighs approximately 490 pounds per cubic foot, so a 40.8-pound plate is approximately 1-inch thick. Iron weighs 480 pounds per cubic foot, so 1-inch iron plate weighs exactly 40 pounds per square foot. In common usage, the decimal fraction is often dropped when naming steel plate; 1-inch steel plate is called 40-pound plate, quarter-inch steel plate is called 10-pound plate, etc. This practice can sometimes lead to confusion--steel plate and shapes are sometimes fabricated to dimensions specified by weight per area or linear dimension. The thickness of plate so manufactured will be slightly less than assumed by dividing the weight by 40. Table E-15 correlates steel-plate thickness to weight per square foot. Major load-bearing members, such as sheer and garboard strakes, main deck stringers and bottom girders, etc., and submarine pressure hulls are frequently fabricated of high-stength steels. High-strength steels are designated by an "HY" (high yield), "HSLA" (high-strength, low-alloy) or number, i.e., HY80, HSLA80, HY100, HY140, etc.; the number specifying the nominal yield stress in thousands of pounds per square inch. High-strength steels are difficult to weld and cut. Intermediate-strength steels, with yield stresses in the 35,000 - 45,000 psi range, are often used for the major strength members of larger merchant hulls to provide the required strength with lighter scantlings. These steels have been called high-tensile (HTS) or higher strength steels by classification societies to avoid confusion with truly high-strength steels. Corrosion-resistant steels (CRES), sometimes called stainless steels, are used extensively where corrosion or appearance are important factors. Strength and other properties vary widely, depending on composition. Because of their resistance to oxidation, corrosion-resistant steels are considered nonferrous metals, and are difficult to cut with oxygen-fuel or oxygen-arc cutting equipment. Low magnetic signature alloys are sometimes used on mine countermeasures ships. Cast iron is used occasionally for complex shapes not subject to tensile loads. Wrought iron is more malleable and corrosion-resistant than mild steel, and nearly as strong. Wrought iron is no longer produced in the United States, but was formerly used in place of steel in ship construction, and may be encountered in older ships. Wrought iron stud-link chain is found occasionally. Aluminum is used extensively in small ships, boats, and landing craft. The yield stress of pure aluminum is about 5,000 psi, but some alloys have yield stresses as high as 78,000 psi. Aluminum alloys used in shipbuilding have yield stresses in the range of 12,000 - 20,000 psi. Because of aluminum's low density, aluminum alloy members are lighter, but bulkier, than steel members of the same strength; aluminum is often used in superstructures to reduce topside weight. Wood is used in the construction of mine countermeasures ships and small craft. The hardness and density of wood vary with species and water content. Green wood contains varying amounts of water as sap; wood absorbs water in humid climates or when immersed. The strength characteristics of wood vary with species and type of stress; all species are much stronger against normal stresses than against shear; most are stronger in tension than in compression. Glass Reinforced Plastic is used in the hulls of small craft and some mine countermeasures ships, in piping systems, as sheathing over wooden hulls and in joiner bulkheads. It is also frequently used as a patching material for other materials. Strength varies depending on the orientation of the glass fibers and plastic resins used. Copper and its alloys, such as brass, bronze, monel, and copper-nickels, are used in piping systems, propellers, and fittings where corrosion resistance or low magnetic signature are required. Although certain copper alloys are very strong, they are seldom used as structural members or fittings, except on mine countermeasures ships, because of their high cost.



1-11.10 Hull Girder Deflection. Hull girder deflection is a function of the fourth integral of the load curve with respect to ship length, and girder stiffness, indicated by the product of moment of inertia (I) and modulus of elasticity (E). Deflection is determined by double integration of the curve of bending moment divided by EI. Since I, and sometimes E vary along a ship's length, M/EI is calculated at several stations to construct an M/EI curve. The curve is integrated from left to right to determine the ordinates to the first integral curve, which is again integrated from left to right to determine ordinates to the second integral curve. A straight line is drawn between the ends of the second integral curve, as shown in Figure 1-65. The vertical separation between the straight line and the second integral curve at any station is the deflection at that station. As shown in Figure 1-65, the straight line in the deflection plot corresponds to a straight line connecting forward and after drafts in a floating ship, i.e., deflection is assumed zero at the fore and after perpendiculars. The hull deflection of a stranded or damaged casualty is readily observable; a salvage engineer does not usually calculate hull deflection unless unusually extreme loadings are contemplated and the degree of hull deflection may affect salvage work or conditions. Observed deflection is a rough indicator of hull stress; a first estimate of stress can be obtained by comparing a casualty's deflection with the stress corresponding to similar deflections in ships of similar form and size. Table 1-15 gives stresses and deflections calculated for four different ships in various conditions. 1-11.11 Approximate Strength Calculations. Lack of detailed ship data or time for rigorous calculations may necessitate the approximation of all or part of the strength calculations. The following paragraphs describe methods to estimate weight distribution, section properties, and still water or wave bending moment. 1-11.12 Weight Curve Approximations. There are a number of empirically derived approximations for weight distribution, none of which is equally applicable to all ship types. The station coefficient method, presented below, is probably the most accurate, but is applicable to only three ship types at present. Less accurate, but more generally applicable methods are presented in the following paragraphs.






__ M EI



Figure 1-65. Hull Girder Deflection Determination.

Table 1-15. Hull Deflection.


Ship Type:


T-AO 187




LBP, ft Beam, ft Depth, ft Deflection conditions: Full Load Maximum stress, ksi Maximum deflection, in. Ballast Maximum stress, ksi Maximum deflection, in. Full load w/hogging wave Maximum stress, ksi Maximum deflection, in. Stranded on one pinnacle (hogging) ­ deflection at 34 ksi Stranded on two pinnacles (sagging) ­ deflection at 34 ksi

046 408 47 30

0.56 650 97.5 50

0.87 1050 175.9 90.5

0.49 673 105.7 66.5

Stresses and Deflections:

5.6 2.4

-3.4 2.5

15.6 6.2

-4.3 -1.1

5.5 2.3

-11.1 5.9

13.0 5.3

10.3 8.1

17.8 7.2 10.6 -20.3

-18.7 10.6 14.0 -27.2

29.3 17.7 17.7 -42.3

23.4 11.2 11.0 -20.4

Note: Positive stresses indicate tension, negative stresses compression From Hull Deflection Versus Bending Moment Study for Supervisor of Salvage, U.S. Navy, Herbert Engineering Corporation, 5 March 1991



1-11.12.1 Station Coefficient Method. This method was developed as part of the Pouricelli-Boyd-Schleiffer regression analysis discussed in Paragraph 1-7 and provides a means to approximate lightship weight distribution of three types of merchant hulls:

Table 1-16. Station Coefficients, CSN.

Station A AFT 20.5-21 20-20.615 20-20.5 19.5-20 19-19.5 18.5-19 18-18.5 17.5-18 17-17.5 16.5-17 16-16.5 15.5-16 15-15.5 14.5-15 14-14.5 13.5-14 13-13.5 12.5-13 12-12.5 11.5-12 11-11.5 10.5-11 0.006303 0.015807



Station A


B 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.022542 0.021834 0.021352 0.022349 0.021834 0.021352 0.020387 0.017493 0.016496 C 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.024942 0.023199 0.022668 0.021606 0.021114 0.020052 0.019522 0.018991 0.01793 0.017399 0.016338


Breakbulk (general) cargo ship with engine room and accommodations three-quarters aft of the forward perpendicular. Container ship with forward and aft accommodations. Tanker with engine room and accommodations aft.

· ·

The length between perpendiculars (LBP) is divided into 20 basic segments. The breakbulk cargo ship has a segment forward of the forward perpendicular and the tanker and container ship each have segments aft of the aft perpendicular. Station coefficients (CSN) from Table 1-16 for the appropriate ship type are used to determine the weight ordinate (OSN) for each half segment: OSN = CSN W1s where: Wls = lightship weight The weight ordinates are plotted as shown in Figure 1-66 to develop the lightship weight curve. Variable weights (cargo, flooding, etc.) are added as rectangles or trapezoids at the appropriate station for the ship's actual load condition. 1-11.12.2 Bare Hull Estimates. For ship types other than the three mentioned above, the lightship weight curve is approximated in three steps:

10-10.5 0.023068 9.5-10 0.023068 0.012377 0.010676 9-9.5 0.023068 0.014333 0.015049 0.01793 8.5-9 0.023068 0.022157 0.017975 0.021114 8-8.5 0.023068 0.020875 0.020387 0.034267 7.5-8 0.023068 0.020875 0.022831 0.038513 7-7.5 0.023068 0.020875 0.02476 0.039536 6.5-7 0.023068 0.021516 0.028169 0.034267 6-6.5 0.022157 0.022157 0.029616 0.025321 5.5-6 0.021516 0.023472 0.025243 0.025321 5-5.5 0.020875 0.032612 0.025243 0.025321 4.5-5 0.020201 0.033252 0.038845 0.024942 4-4.5 0.019560 0.041076 0.038845 0.024942 3.5-4 0.018919 0.053453 0.040774 0.024942 3-3.5 0.018245 0.055409 0.042736 0.024942 2.5-3 0.017604 0.052172 0.043701 0.024942 2-2.5 0.016963 0.028025 0.022542 0.024942 1.5-2 0.016289 0.023068 0.022542 0.024942 1-1.5 0.015142 0.023068 0.022542 0.024942 0.5-1 0.014333 0.023068 0.022542 0.024942 0-0.5 0.013692 0.023068 0.022542 0.024942 -0.55-0 0.013051 FWD FWD Ship A ­ Breakbulk cargo ship - engine room and accommodations three-quarters aft from Ship B ­ Container ship with forward and aft accommodations Ship C ­ Tanker with engineroom aft







The hull steel weight is calculated or estimated by deducting weights of machinery, propellers, and superstructure from the lightship weight, or by the methods described below. The bare hull weight distribution is estimated by one of the methods described in the following paragraphs. The deducted items are added at their locations to complete the lightship weight curve.










TANKER WITH AFT ENGINEROOM Figure 1-66. Station Coefficient Weight Curves.



After the distribution of the hull weight of the ship has been estimated, the variable weights of fuel, stores, cargo, boats, aircraft, ballast, ammunition, crew and effects, etc., are added by superimposing rectangles or trapezoids on the curve at their locations. Hull steel weight for commercial vessels can be estimated by the two relationships shown below: W H L B D k 1, WH L (B where: WH = L = B D k1 = = = = = hull weight, ltons length between perpendiculars, feet molded beam, feet molded depth, feet weight coefficient 0.0027 for welded construction 0.0030 for riveted construction weight coefficient 0.0433 for welded construction 0.0558 for riveted construction or,



2D) k2


= = =

20 4 6 8 10 12 14 18 22 26 30 34 38 42 SHP RATING OF PROPULSION PLANT (THOUSANDS)


Weights of machinery and outfits can sometimes be obtained from the ship's information book (SIB), operating and Figure 1-67. Machinery Weight. technical manuals, or manufacturers' data. Machinery weight for commercial vessels can be estimated very approximately by use of the "power density" factors taken from Figure 1-67.

There is no standard definition of what is included in the term machinery weight, so figures given in ship's data must be investigated to determine what items are included. Values taken from the curves in Figure 1-67 include the weight of main propulsion units, shafting, bearings, propellers, boilers, stacks, condensers, generators, switchboards, and pumps; all piping, floors, ladders and gratings in the machinery spaces; water in boilers, engines, and piping; and refrigerating and steam heating systems for a normal vessel. Weights of steering gear, deck machinery, and piping outside the machinery spaces are not included. Machinery weights are subject to variation, depending on the ship type and service. In ship types that require particularly rugged or reliable machinery, machinery weight will be about 10 percent higher than the values from Figure 1-67. Different makes of diesel engine of the same horsepower will vary in weight by as much as 50 percent. Total machinery weight in Table 1-17. Machinery Weights for Combatants. specialized vessels will include items not fitted on ordinary ships, or larger numbers of common items. Examples are the refrigeration plant on a refrigerated cargo ship, additional pumps and generators on salvage and service vessels, dredge machinery, etc. BB, CV 50-60 pounds/SHP Because of their high speed and correspondingly powerful machinery, the weight of machinery of naval combatants is a large portion of the total weight of the ship. Emphasis on machinery weight savings during design results in lower weight per horsepower than in the average commercial vessel. Machinery dry weight for different types of combatants can be taken from Table 1-17.

CG, CL, CA DD, FF DD, FF, CG (gas turbine) 35-40 pounds/SHP 27-30 pounds/SHP 20-25 pounds/SHP



Table 1-18 gives weights of bronze propellers as a function of shaft horsepower and rpm. Table 1-19 gives summarized weight lists for different types of ships to illustrate general trends in weight distribution. Additional weight summaries are included in Appendix B.

Table 1-18. Weights of Bronze Propellers (lbs).

Shaft RPM 160 2,030 3,880 7,140 10,360 14,545 24,245 32,400 40,115

SHP 500 1,000 2,000 3,000 5,000 10,000 15,000 20,000

100 3,415 6,545 12,080 17,410 24,905 55,100 62,155

120 2,775 5,270 9,830 14,105 20,495 35,705 50,030

140 2,315 4,475 8,265 11,680 17,190 29,315 40,335 50,910

180 1,785 3,460 6,350

200 1,585 3,150 5,730

250 1,255 2,445 4,630

300 970 1,915 3,670

350 750 1,520 2,975

From Ships and Marine Engineers, Volume IV, The Design of Merchant Ships, Schokker, Newerburg, Bossnack, and Burghgracf, The Technical Publishing Company H. Stam, 1953

Table 1-19. Lightship Weight Summaries.

Ship Type Mariner With Added General Features, Cargo Ship1 1962 5,115 2,586 1,039 --8,746 5,011 2,230 867 --8,108 Combination Passenger/ Reefer Container Ship2 5,482 3,959 982 --10,4235 BargeBargecarrying carrying ship Ship (LASH)4 (SEABEE)5 9,588 2,937 1,105 --13,630 12,983 2,979 1,421 --17,383

Item Steel Outfit Machinery10 Fixed Ballast Lightship

Container Ship3 10,282 2,525 1,911 --14,718

Tanker6 11,519 1,844 831 --14,194

Ore Carrier7 12,137 1,600 980 --14,717

Small Freighter8 2,248 574 398 --3,220

Passenger Container Ship Vessel9 11,850 6,875 2,525 --21,250 4,557 1,739 837 3,329 10,452

Weights in Long Tons

From Ship Design and Construction, Amelio M. D'Arcangelo; Society of Naval Architects and Marine Engineers, 1969 and Princples of Naval Architecture, Society of Naval Architects and Marine Engineers, Second Edition, 1967 and Third Edition, 1988 Notes: 1 573' LOA, machinery and house 3/ 4 aft, 6 holds, 2 'tween decks, 24,000 SHP, 23 Kts. 2 574' LOA, machinery and house midships, 19,800 SHP, 20 Kts. 3 752' LOA, machinery 3/ 4 aft, house forward, 1,920 TEU, 60,000 SHP, twin screw, 27 Kts. 4 820' LOA, machinery 3/ 4 aft, house forward, 79 LASH barges, 32,000 SHP, 27.5 Kts. 5 824' LOA, machinery 3/ 4 aft, house forward, 38 SEABEE barges, 36,000 SHP, 20 Kts. 6 810' LOA, machinery and house aft, single bottom, 5 center and 8 wing tanks, 19,000 SHP, 17 Kts. 7 765' LOA, machinery and house aft, 7 holds, 19,000 SHP, 16.5 Kts. 8 390' LPB, two deck, three-island design, 3,150 SHP, 13 Kts. 9 661' LPB, ten deck, 1,200 passenger, 650 crew, 30,000 SHP, 20 Kts. 10 Steam turbine plants in all cases, single screw unless otherwise noted.

Table 1-20. Prohasha's Ordinates for the Coffin Diagram.

Type of Ship Tanker Full-bodied cargo ships w/o erections Fine-lined cargo ships w/o erections Full-bodied cargo ships with erections

Prohaska's ordinates a&c b 0.75WH / L 1.125WH / L 0.65WH / L 1.175WH / L 0.60WH / L 1.20WH / L 0.55WH / L 1.225WH / L

Type of Ship Fine-lined cargo ships with erections Small passenger ships Large passenger ships

Prohaska's ordinates a&c b 0.45WH / L 1.275WH / L 0.40WH / L 1.30WH / L 0.30WH / L 1.35WH / L

where: WH = Hull weight, ltons (less propelling machinery) L = Length overall, ft

Reproduced from Applied Naval Architecture, R. Munro, 1967



1-11.12.3 Coffin Diagram. Bare hull weight distribution for ships with parallel midbody can be approximated by a line diagram, commonly called a coffin diagram, consisting of a rectangle over the length of the midbody and trapezoids at the bow and stern. Three hull weight distribution methods are based on the coffin diagram. The Biles and Prohaska methods each divide the length overall into three equal segments as shown in Figure 1-68. A third method, that may be termed the general parallel midbody method, divides the length into three segments based on the observed length of the parallel midbody. Biles method ordinates for ordinary cargo and passenger vessels are shown in the Figure 1-68, Prohaska method ordinates for different ship types are given in Table 1-20. The centroid of the Biles diagram is 0.0056L abaft midships. Small adjustments can be made to the end ordinates so that the centroid of the diagram corresponds to the longitudinal position of the center of gravity of the hull. LCG of the bare hull is not at the same location as the light ship LCG. The position of the centroid of the coffin diagram must be chosen so that LCG will shift to a known or estimated position as weights are added, corresponding to the condition where LCG is known. By shortening one end ordinate and lengthening the other by an equal amount, a triangle is transferred from one trapezoid to the other, as shown by the dotted lines in Figure 1-69. The centroid of each triangle lies one-third of its length from its base: 1 L = 3 3 L 9





L/3 AP

L/3 BILES METHOD ORDINATES WH a = 0.566 __ L WH b = 1.195 __ L

L/3 FP



Figure 1-68. Coffin Diagram.

__ L 7 9

x G G1


L/3 AP

L/3 54(WH )GG1 x = ____________ 7L2

L/3 FP

where L is the length of the diagram, corresponding to length overall (LOA) of the ship. The shift of the centroid of the total area is therefore (7/9)L. If the base of the triangle is taken as x, and its height as L/3, then,

Figure 1-69. Adjusting LCG of the Coffin Diagram.

1 L Area of triangle = x = 2 3 xL 7L Moment of the shift = = 6 9 The shift of the centroid of the diagram, representing the LCG of the hull is thus: L2 7 Shift of LCG = ( x ) WH 54

xL 6 7xL 2 54

where WH is the bare hull weight. The triangle base, x, required to give the desired shift of LCG is: x = 54 (WH) (desired shift of LCG) 7L 2



In the general parallel midbody method, the beginning and end points and length of the parallel midbody are determined by inspection. The middle ordinate (b) is defined as shown in Figure 1-70. The end ordinates are chosen so that the centroid of the entire diagram corresponds to the bare hull LCG. Figure 1-71 shows how to select end ordinates for a trapezoid to place the center of the trapezoid in a desired location. 1-11.12.4 Ships Without Parallel Midbody. An approximate weight curve for ships without parallel midbody can be constructed as a parabola over a rectangle, with the area under each representing half the bare hull weight (Cole, reproduced in Applied Naval Architecture, R. MunroSmith, 1967). The ordinate for the rectangle is WH/2L; the maximum (midships) ordinate for the parabola is 3WH/4L, as shown in Figure 1-72. LCG of this figure is amidships. Correction for LCG lying forward or aft of midships is made by swinging the parabola. A line parallel to the base is drawn through the centroid of the area under the parabolic curve. A second line is drawn from the base of the parabola at its midlength to intersect the first line at a distance from the midships ordinate equal to twice the desired shift in LCG. This line is extended beyond the contour of the parabola. The intersection of this line with a horizontal line drawn from the center of the parabolic curve defines one point on the new curve. Parallel lines drawn at other ordinates define other points on the new curve, as shown in Figure 1-72. For ships without parallel midbody, a bare hull weight curve can also be generated by assuming that two-thirds of the hull weight follows the still water buoyancy curve and distributing the remaining one-third in the form of a trapezoid so arranged that the center of gravity lies above the center of buoyancy, as shown in Figure 1-73. This method has been found to yield close approximations to the hull weight distribution for large warships.

1.4 b l 1.3 WH __ L W = Hull Weight b = b1 x l = LENGTH OF TANK SECTIONS OR PARALLEL MIDDLE BODY





1.0 0.2 0.3 0.4 l __ L





Figure 1-70. General Parallel Midbody Weight Curve.






1-11.13 Wave Bending Moment with Nonstandard Waves. The salvage engineer must often assess the ability of a damaged casualty to withstand wave l bending loads, either during the salvage 2a 3x operation or during transit to a repair __ __ b1 = 2a (2 - 3x ); b2= __ ( __ -1) l l l l facility. Because of the tedious nature of the calculations, the usual first task is to determine the stresses imposed by a Figure 1-71. Centroid of a Trapezoid. standard L/20 or 1.1L wave with length equal to ship's length. If the ship can carry loads imposed by a standard wave, no further calculations need be performed in most cases. If, however, the stresses imposed by the standard wave are excessive, calculations must be performed for trial wave heights and lengths until the maximum acceptable wave is determined, unless bending moment caused by waves with differing length and height can be correlated to those caused by the standard wave.



A 1991 analysis by Herbert Engineering Corporation of five hull forms with block coefficients ranging from 0.46 to 1.0 developed factors that relate nonstandard wave bending moments to normalized standard bending moment. The factors are functions of block coefficient, wavelength, and wave height. The analysis revealed that for fine-lined ships, maximum wave bending moment occurs at wavelengths slightly less than the ship's length (approximately 0.75L), and may be as much as 15 percent higher than bending moment for the standard wave. Figure 1-74 shows the relationship between wavelength and bending moment for an FFG-7 Class ship (CB = 0.46) for a 1.1L wave height. Figure 1-75 (Page 1-104) shows the relationship between standard wave bending moment and maximum wave bending moment as a function of block coefficient. Figure 1-76 (Page 1-104) shows normalized maximum and standard hogging and sagging moments as a function of block coefficient. All curves are based on 1.1L trochoidal waves. The normalized bending moment is given by: NBM = WBM 35 L 2Bh



a + b = 1.25

WH L b= 3WH 4L







Figure 1-72. Parabolic Weight Curve.


1/ W 3 H AP FP


Figure 1-73. Alternate Weight Distribution for Ships Without Parallel Midbody.


0.015 0.01


where: NBM = normalized wave bending moment, dimensionless wave bending moment, ft-lton standard seawater specific gravity, ft3/lton length between perpendiculars, ft beam, ft wave height, ft = 1.1L

0.005 0 -0.005

WBM = 35 L B h = = = =


-0.01 -0.015 -0.02 0.9 0.8 0.7 0.6 0.5 0.4 LOCATION FROM FP (X/LBP) 0.3 0.2 0.1

Figure 1-76 (Page 1-104) can be entered with block coefficient to get an estimate of the standard bending moment (waveheight = 1.1L, wavelength = L).

L = 1.0 LBP L = .75 LBP

L = .50 LBP L = .25 LBP

L = 1.0 LBP L = .75 LBP

L = .50 LBP L = .25 LBP



Figure 1-74. FFG-7 Bending Moment with Varying Wavelength.



The plots in Figures 1-77 and 1-78 are entered with wavelength expressed as a function of ship length to determine the ratio between wave bending moment for the wavelength and the standard wave bending moment. The ratio is then applied to wave bending moment determined from Figure 176 or by rigorous calculation to estimate wave bending moment for the nonstandard wavelength. Figure 1-79 (Page 1-106) gives normalized bending moments for wavelengths equal to L with nonstandard waveheight. 1-11.14 Murray's Method for Approximating Maximum Bending Moment. An approximation of determining maximum bending moment has been developed by J. M. Murray, former Chief Ship Surveyor to Lloyd's Register of Shipping. Murray's method computes still water bending moment by taking moments of weight and buoyancy about midships. Wave bending moment is calculated by use of empirical coefficients. The sum of the two gives total bending moment at midships, which can be taken as the maximum bending moment in most cases. The method is reasonably accurate for ships floating at a trim of less than one percent of their length. 1-11.14.1 Still Water Bending Moment. Still water bending moment (SWBM) is given by: SWBM = MW where: MW = MB

1.16 1.14 1.12


1.1 1.08 1.06 1.04 1.02 1 0.4 HOG 0.5 0.6 0.7 BLOCK COEFFICIENT 0.8 0.9 1 SAG


Figure 1-75. Ratio of Maximum to Standard Wave Bending Moment as a Function of Block Coefficient.

0.025 0.02


0.015 0.01 0.005 0 -0.005 -0.01 -0.015

mean moment of weight -0.02 Mwf + Mwa -0.025 = __________ 0.8 0.9 1 0.4 0.5 0.6 0.7 2 BLOCK COEFFICIENT MAX. HOG MAX. SAG Mwf = moment of weight STD. HOG STD. SAG forward of midships, ft-lton or m-tonne FROM WAVEHEIGHT AND WAVELENGTH VERSUS BENDING MOMENT STUDY FOR SUPERVISOR OF SALVAGE U.S. NAVY, HERBERT ENGINEERING CORP., 20 FEBRUARY 1991 = Wf(LCGf) Mwa = moment of weight aft of midships, ft-lton or mFigure 1-76. Normalized Wave Bending Moment as a Function of Block Coefficient. tonne = Wf, a(LCGfa) Wf, a = weight of the forebody or afterbody, lton or m-tonne LCGf, a = LCG of the forebody or afterbody, measured from midships, ft or m Mbf + Mba MB = mean moment of buoyancy = _________ 2 Mbf = moment of buoyancy forward of midships, ft-lton or m-tonne = B f (LCBf ) Mba = moment of buoyancy aft of midships, ft-lton or m-tonne = Ba(LCBa) Bf,a = buoyancy of the forebody or afterbody, lton or m-tonne LCBf, a = LCB of the forebody or afterbody, measured from midships, ft or m




1.25 1 0.75 0.5 0.25 0 0.25 SAG 0.5 0.75 1 1.25 1.5 2 1.5 1 0.5 0 0.5 1 1.5 2 WAVELENGTH / LBP HOG


Figure 1-77. Ratio of Wave Bending Moment to Standard Bending Moment, CB = 0.46.


1 0.8 0.6 0.4 HOG 0.2 0 0.2 SAG 0.4 0.6 0.8 1 2 1.5 1 0.5 0 0.5 1 1.5 2 WAVELENGTH / LBP


Figure 1-78. Ratio of Wave Bending Moment to Standard Bending Moment, CB = 1.0.



Since total weight and buoyancy moments are mean moments, they are numerically equal to the product of the mean weight or buoyancy and the mean lever arm: MW = Mwf + Mwa 2 Mbf + Mba 2 W + Wa = f LCGm 2 B + Ba = f LCBm 2

MB = where: LCGm = mean distance from midships of the centers of gravity of the fore and after bodies = mean distance from midships of the centers of buoyancy of the fore and after bodies

0.025 0.02


0.015 0.01 0.005 HOG 0 SAG -0.005 -0.01 -0.015 -0.02 -0.025 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 WAVE HEIGHT / STANDARD WAVE HEIGHT Cb = 0.46 Cb = 0.58 Cb = 0.78 Cb = 0.84 Cb = 1.0


Since the sum of the weights of the fore and after bodies is equal to the total weight, which is equal to displacement, which is similarly equal to the sum of the buoyancies of the fore and after bodies, still water bending moment can be expressed: SWBM = (LCGm 2 LCBm)

If the mean of the centers of gravity is greater than the mean of the centers of buoyancy, the weight levers are longer than the buoyancy levers, and the net moment is hogging, as shown in Figure 1-80. If the mean of buoyancy centers is greater, the net moment is negative, and sagging. Forward and after weight moments are determined by summing the moments of individual weights. Weights and centers of variable weights can be obtained from ship's officers or estimated with reasonable accuracy. Machinery weight can be approximated from the factors given in Paragraph 1-11.12.2; machinery lcg is determined by inspection. Hull weight can be estimated as described in Paragraph 1-11.12.2. The mean distance from midships of the centers of gravity of the forward and after bodies of the hull can be expressed as a portion of length between perpendiculars: mean lcg = aL where: a = = an empirical coefficient 0.223 for a cargo ship with forecastle and poop; deckhouse and machinery amidships 0.24 for a tanker with forecastle, bridge, and poop 0.233 for a cargo ship with machinery aft


Figure 1-79. Normalized Wave Bending Moment as a Function of Wave Height.











MEAN DISTANCE TO FORE AND AFT LCBs GREATER THAN DISTANCE TO LCGs - SAGGING Figure 1-80. Determination of Still Water Bending Moment by Murray's Method.

= =

Values of a for different configurations can be estimated from those given above. For example, 0.225 might be used for a cargo ship with machinery slightly aft of midships.



Mean buoyancy moment can be estimated as: MB = where: cL L c = = = = total buoyancy (displacement), lton or tonne mean position of LCB, ft or m length between perpendiculars, ft or m empirical coefficient based on block coefficient and draft from Table 1-21 cL 2

Table 1-21. Coefficient c for Mean LCB in Murray's Method.

Draft 0.06L 0.05L 0.04L 0.03L

c 0.179CB + 0.063 0.189CB + 0.052 0.199CB + 0.041 0.209CB + 0.030

L = length between perpendiculars, block coefficient, CB is taken at draft equal to 0.06L


Calculate the still water bending moment for a cargo ship with machinery and accommodations three-quarters aft with the following characteristics: length between perpendiculars 570 feet beam 80 feet molded depth 55 feet full load draft 35 feet block coefficient 0.71 displacement 32,400 lton deadweight 23,800 lton hull weight 6,250 lton weight of propulsion machinery 1,200 lton center of machinery room 145 ft aft of midships Variable Weight Distribution: item Cargo: Hold 1 Hold 2 Hold 3 Hold 4 Hold 5 Oil fuel in deep tank Oil fuel in double bottom tanks Feed water Potable water Crew & effects, stores Calculation: Mean distance from midships of centers of buoyancy The load draft of 35 ft is approximately 0.06L, CB = 0.71, Weight lton 3000 4200 6100 6800 3700 370 435 20 250 75 lcg from midships ft 231 F 142 F 60 F 95 A 250 A 200 F 85 A 170 A 122 A 165 A

Weight moments, after body: item Weight lton Hold 4 Hold 5 O.F. (double bottom) Feed water Potable water Machinery Crew & effects, stores Total: 6,800 3,700 435 20 250 1,200 75 12,480

lcg from midships ft 95 A 250 A 85 A 170 A 122 A 147 A 165 A

Moment ft-lton 646,000 925,000 36,975 3,400 30,500 176,400 12,370 1,830,645

Weight moments, fore body: item weight lton Hold 1 3,000 Hold 2 4,200 Hold 3 6,100 O.F. (deep tank) 370 Total: 13,670 Total weight moments: item hull after body fore body Total:

lcg from midships ft 231 F 142 F 60 F 200 F

moment ft-lton 693,000 596,400 366,000 74,000 1,729,400

weight lton 6,250 12,480 13,670 32,400

moment ft-lton 819,375 1,830,645 1,729,400 4,379,420

Mean distance from midships of centers of gravity:


= Total moment/total weight = 4,379,420/32,400 = 135.2 ft


= 0.179CB + 0.063 = 0.190 Still water bending moment:

cL = 0.19(570) = 108.3 ft = LCBm

Hull weight moment = WHaL (take a to be 0.23) = 6,250(0.23)(570) = 819,375 ft-lton

SWBM = /2 (LCGm - LCBm)

= (32,400/2)(135.2 - 108.3) = 435,780 ft-lton

LCGm is greater than LCBm; the net moment is positive, or hogging



1-11.14.2 Wave Bending Moment. Wave bending moment, for a standard wave with length equal to the ship's length, can be estimated as: WBM = bL 3B 1,000,000 2.2 b L 2.5 B 100,000 for wave height = L 20

Table 1-22. Wave Bending Coefficient for Murray's Method.

Wave bending coefficient b Block Coefficient CB Hogging (wave crest at midships) Sagging (wave trough at midships)

= where: WBM L B b

for wave height = 1.1 L

= = = =

wave bending moment, ft-lton length between perpendiculars, ft beam, ft empirical coefficient based on block coefficient and wave position, from Table 1-22

1-11.15 Section Property Design Rules. In the absence of better information, empirical relationships and construction standards can be used to estimate section modulus or moment of inertia. The following design rules are taken from Applied Naval Architecture, R. Munro-Smith, 1967. A first approximation of the midships section moment of inertia can be made from: I = where: I B D c = = = = moment of inertia, ft4 or m4 molded beam, ft or m depth to strength deck, ft or m empirical coefficient, ranging from 0.14 to 0.16 0.18 for cargo ships 0.22 for large tankers 0.175 to 0.21 for small tankers cBD3

0.80 0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60

25.00 24.25 23.55 22.85 22.10 21.35 20.65 19.90 19.20 18.45 17.75

28.00 27.25 26.50 25.70 24.90 24.10 23.35 22.60 21.80 21.05 20.30

CB taken at draft = 0.06L

An estimate for section modulus and/or moment of inertia can be made by reference to preliminary design expressions for maximum shear force and bending moment, and assuming the ship was built to withstand that force and moment. 12 Smax 9








where: Smax = maximum shear, lton = displacement, lton Mmax = maximum bending moment, ft-lton L = length between perpendiculars, ft = block coefficient CB C = a constant, generally ranging from 20 to 40 35 for most auxiliaries, merchant ships, and vessels with large longitudinal prismatic coefficient Mmax = LBT/1600 (CB taken as 0.75) 20 for destroyers, and vessels with small longitudinal prismatic coefficient Mmax = LBT/1490 (CB taken as 0.47) These relationships give a good approximation for the full-load condition on a standard hogging wave. For most merchant ships, hogging moments are greater than sagging moments.



1-11.16 By Rule Section Modulus. Classification society rules set minimum standards for midships section modulus. Midships section modulus of an in class ship will not be lower than the minimum standard, and is unlikely to be much higher. Bending stresses in the midships region can be roughly estimated without determining section modulus rigorously, provided the following are true:

· · ·

The ship was built to classification society standards or other specifications requiring minimum section modulus, and is currently in class. The minimum section modulus standards are known. The ship has not suffered damage that will reduce section modulus in the sections where stresses are to be determined.

A summary of section modulus requirements established by the American Bureau of Shipping (ABS) is given in Appendic C. 1-11.17 Strength Considerations in Salvage Operations. A ship is designed and constructed to withstand expected shear forces and bending moments. In an intact floating ship, maximum bending moment occurs in the midships region and maximum shear near the quarter-length points. These sections are designed to ensure that stresses remain below acceptable limits. Three conditions common to salvage operations may require that the stress levels be examined at other points:

· · ·

The ship may be loaded in ways not foreseen by the designer. Because of flooding, grounding or other unusual conditions of loading, maximum bending moment can occur at some section other than midships. Similarly, maximum shear may be at some point other than at the quarters. Damage can alter the stress distribution at a section so that maximum stress can occur in some section other than where maximum bending moment or shear occurs. Damage, even over a short distance, disrupts the continuity of longitudinal members and reduces the section modulus for some distance on either side of the damaged section. Local damage or distortion can render plating and stiffeners more susceptible to tripping, buckling, or other forms of load shirking, thereby reducing effective moment of inertia.

The load, shear, and bending moment curves of a casualty must be carefully examined:



Stresses should be determined wherever shear or bending moment are maximum or the effective moment of inertia is reduced. The effects of salvage actions on load, shear and bending moment should be examined before taking the action. Accesses should not be cut in locations that will reduce the section modulus or strength member continuity.








A useful salvage technique is to calculate and plot the maximum acceptable shear and bending moments along the length of the ship. The bending moments and shear resulting from planned actions can be compared with the allowable limits to determine if the planned action is safe. Figure 1-81 shows maximum acceptable bending moments for an FFG-7 Class ship.


















Figure 1-81. Maximum Bending Moment for FFG-7.

1-109 (1-110 blank)


109 pages

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