Ship Structural Design Notes

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1. Hull Girder Response Analysis – Prismatic Beam

1.1. Requirements of structural design

Structural design is an iterative process through which the layout and scantlings for a structure are determined, such that it meets all the requirements of structural adequacy. The overall configuration, which are in general dictated by non-structural consideration, such as volume and space requirements, global stability, safety, etc. are required to be achieved in the design.

In general terms the major steps that are involved can be summarized as follows:

a) Identify load and load combinations acting on the structure as a whole, or on its main sub-components.

b) Select initial structural layout and scantlings. In general this is based on past experience with similar structures.

c) Identify structure’s main components, and determine through structural analysis the loads and load combinations acting on each component.

d) Identify relevant limit states and associated factors of safety.

e) Check structural adequacy. If any limit state is violated, adjust scantlings and repeat the analysis and the structural checks. Perform the iterations required to converge to a structurally adequate design.

f) Check other limit state, such as fatigue, which requires the selection of main structural detail configurations. Also check the adequacy of the design against accidental loads. If the structure is found to be inadequate, then new design iterations have to be conducted. g) ‘Optimize’ structural design. Once an adequate design has been achieved it is in general

possible to ‘optimize’ it for a given objective. The objective depends on the structure’s intended use, and can be, for example, the structural weight or the cost of fabrication and installation. Thus, once a new configuration and set of scantlings are derived, structural adequacy (Step d) has to be checked again, in an iterative fashion (Figure 1).

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Figure 1 Strength based design procedure

1.2. Classification of structural analysis

The ship structure can be classified into primary, secondary and tertiary elements as shown in Figure 2. The first level of structure usually considered is the complete hull as a beam; this is called the primary structure or hull girder as shown in Figure 3. In this level the ship is idealized as a simple beam – a floating box girder that is internally stiffened and subdivided – and in which the decks and bottom structure are flanges and the side shell and any longitudinal bulkheads are the webs. Superstructure may also be considered depending on its effectiveness. A part of the overall structure is cut out to show the different forces and moments to be dealt with in beam theory. That part is called a hull module as shown in Figure

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3. Secondary structure consists of stiffened panels and grillages bounded by the decks, bulkheads and the shell. Tertiary structure may be panels of plates bounded by stiffeners or elements of stiffeners themselves (Figure 2).

The forces and moments to be considered are:

1- Vertical longitudinal bending moment (most significant). 2- Horizontal longitudinal bending moment .

3- Longitudinal twisting moment . 4- Vertical shear force .

Stress can be classified in a similar way to the structure in which the stresses are occurring and to the loads, which cause the stresses:

Primary – stresses due to bending, shear and torsion in the main hull girder. Secondary – stresses in a stiffened grillage due to bending and membrane effects. Tertiary – membrane stresses in panels between stiffeners.

The following assumptions must be taken: a) Plane cross sections remain plane.

b) Prismatic beam (no openings or discontinuities).

c) Deflection and distortion caused by shear and torsion do not affect hull girder bending.

d) Material is homogeneous and elastic.

Figure 2 Primary, Secondary and Tertiary structure

Tertiary

Secondary Primary

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1.3. Ship loads and Stresses

Loads can also be classified according to how they vary with time. They are either static, slowly-varying or rapidly-varying. The principal loads on ships are:

1.3.1. Static loads

 Vertical shear and longitudinal bending in still water: A ship floating in still water has unevenly distributed weight owing to both cargo distribution and structural distribution. The buoyancy distribution is also non-uniform since the underwater sectional area is not constant along the length. Total weight and total buoyancy are of course balanced. But at each section there will be a resultant force or load, either an excess of buoyancy or excess of load. Since the vessel remains intact there are vertical upward and downward forces tending to distort the vessel, which are referred to as vertical shearing forces. The variation in the vertical loading will tend to bend the vessel either to sagging or to hogging condition depending on the relative weight and buoyancy forces (Figure 4).

Figure 4 Vertical shear and longitudinal bending in still water

 Longitudinal shear in still water: When the vessel hogs and sags in still water and at sea, shear forces similar to the vertical shear forces will be present in the longitudinal plane. Vertical and longitudinal shear stresses are complementary and exist in conjunction with a change of bending moment between adjacent sections of the hull girder. The magnitude of the longitudinal shear force is greater at the neutral axis and decreases towards the top and bottom of the hull girder (Figure 5).

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Figure 5 longitudinal shear forces

 Dry docking loads: docking a ship on blocks imposes very high vertical loads on ship’s bottom. As all ships may be expected to be docked at some time, it is necessary to design for the docking condition.

 Thermal loads: stresses in the ship structure can be caused by different temperatures in one part with respect to another part, e.g. high air temperature in addition to solar radiation may lead to the upper deck being as much as 40oC hotter than the hull below water causing thermal stresses in the ship’s hull girder.

 Grounding loads: for most ships, grounding is an accident condition and does not directly affect the design of the structure. For vessels expected to ground, such as landing craft and small boats, bending stresses on the hull resulting from grounding must be calculated as well as local loads.

 Lifting loads: some small vessels may be designed to be lifted in slings or from lifting eyes. This kind of loading can be calculated using simple beam theory for the light displacement condition.

1.3.2. Slowly varying loads

 Vertical shear, longitudinal shear and longitudinal bending in seaway: When a ship is in a seaway the waves with their troughs and crests produce a greater variation in the buoyancy forces and therefore can increase the bending moment, vertical and longitudinal shear forces. Classically the extreme effects can be illustrated with the vessel balanced on a wave of length equal to that of the ship. If the crest of the wave is amidships the buoyancy forces will tend to hog the vessel. If the trough is amidships the buoyancy forces will tend to sag the vessel (Figure 6).

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Figure 6 Wave bending moments

 Horizontal bending and torsion: these are caused by wave action. A ship heading obliquely (45o) to a wave will be subjected to righting moments of opposite direction at its ends twisting the hull and putting it in torsion. In most ships, horizontal bending and torsional moments are much lower than bending in the vertical plane and can usually be ignored. However, torsional moments in ship with extremely wide and long deck openings (such as container ships) are significant (Figure 7).

Figure 7 Torsion

 Racking: When a ship is rolling, the deck tends to move laterally relative to the bottom structure and the shell on one side to move vertically relative to the other side. This type of deformation is referred to as ‘racking’. Transverse bulkheads primarily resist such transverse deformation (Figure 8).

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8 Figure 8 Racking

 Sloshing of liquid cargo: specially significant in tankers.

 Shipping of green water on deck: if a ship proceeds at speed even into moderate seas, green water is thrown onto the deck. The forecastle and bridge front will be the worst affected parts.

 Wave slap on sides and foredecks: this is due to the action of the waves as they hit the ship.

 Panting: Panting refers to tendency for the shell plating to work in and out in a bellow like fashion, and is caused by the fluctuating pressures on the hull at the ends when the ship is amongst waves. These forces are most severe when the vessel is running into waves and is pitching heavily.

 Inertial loads: they are caused by the motion of heavy masses such as masts, containers and other heavy objects following the motion of the hull. Provided that the local accelerators are known, the estimation of inertial loads is straightforward

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 Berthing loads: they are extremely variable. They depend on the officer’s skill, the weather conditions and the structure to which the ship is berthed.

 Launching loads: these loads should be checked by the shipbuilder. The bending stresses in the hull girder are moderate. The fore poppet should be carefully designed and the fore end of the hull structure may be temporarily stiffened if necessary.  Ice loads: these are localized loads. Some ships are strengthened to have ice breaking

capability.

 Wheel loads: they result from vehicles, for example on RO/RO vessels. They consist of dead weight and inertia loads.

Distortion of structure Rolling of ship accelerates structure, tending to distort it

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9 1.3.3. Rapidly varying loads

 Slamming: this is impact between the ship’s hull and the water surface. It occurs when the vessel is driven into head seas where some part of the bottom of the forward end of the ship comes out of the water and then re-enters. These slamming stresses are likely to be most severe in a lightly ballasted condition, and occur over an area of the bottom shell aft of the collision bulkhead (Figure 9).

Figure 9 Slamming

 Vibration: there are particular locations on a ship where vibration response may be important. For example, in way of weapons, machinery, and in the stern region.

 Collision loads: this is an accident condition. It is not usual to design a structure to withstand collisions. However, there are some exceptions such as nuclear powered ships and tankers.

 Loads due to underwater explosion: these are taken into consideration only for navy ships.

 Impact loads: due to weapons effects in naval ships.

 Springing loads: are sea excitation forces. Springing is a continuous and steady vibration, it occurs when the natural frequency of the hull and the wave frequency coincide (resonant response).

0.25L or 0.30L 0.05L Slamming region Pitching Summer load waterline Ship forced down

on to water wave profile

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1.4. Failure modes

During structural design and analysis, care must be taken to ensure that all possible failure modes are considered. The possible failure modes are:

1.4.1. Fatigue

The majority of loads on ships are cyclic. Indeed, most structural failures that occur in service are the result of fatigue damage. Generally, fatigue is not included in the main design process, but later in detail design after the final construction drawings are being produced.

1.4.2. Brittle fracture

It depends on the material from which the ship is constructed. The risk is increased by the presence of stress concentrations, notch like defects, exposure to low temperature, impact or high loading rates. Welding procedures are an important factor too.

1.4.3. Yielding or plastic collapse

When yielding occurs, very small increases in load cause large increases in deformation. The two most critical types of load causing yielding in ship structures are lateral load and inplane load. Under lateral loading, a mechanism is formed by the formation of plastic hinges. Under inplane loads, yielding occurs when the combined stresses reach the yield point of the material. Usually, buckling will occur before pure yielding in the case of inplane compressive loads.

1.4.4. Buckling

According to the structural configuration and the loading conditions, it takes many forms. It can occur in the plating between stiffeners, in the stiffeners’ webs or flanges (by tripping or flexure) or in an entire stiffened panel or grillage. All form of buckling may result in complete collapse of the structure. The initial deformations and residual stresses that occur during fabrication almost always lead to some loss of buckling strength.

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11 1.4.5. Excessive deflection

In way of machinery, there are limits on allowable deflections. Large deflections may interfere with the performance of the equipment nearby.

1.5. Function of ship’s structure

A ship is capable of bending in a longitudinal vertical plane and hence there must be material in its structure which will resist this bending.

Any material distributed over a considerable portion of the length of the ship will contribute to its longitudinal strength. Example of such items:

1- Side and bottom shell plating 2- Inner bottom plating

3- Decks

4- Deck and bottom longitudinals 5- Side longitudinals

Items which contribute to transverse strength: 1- Floors

2- Side frames 3- Beams

4- Transverse watertight bulkheads No. 1,2 and 3 form transverse rings.

1.6. Review of load, shear and moment relationships

Consider the beam shown in Figure 10 is subjected to a distributed transverse load of varying intensity .

Consider an element of the beam ∆ in length.

Let = average value of and the resultant load . ∆ acts at a distance of . ∆ from the right-hand face of the element.

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Figure 10 simple beam subjected to an arbitrary distributed load

 

f  f x QQ x M M x (1.1)

From vertical equilibrium:





. 0 a a Q Q f x Q Q f x           (1.2)

The slope of the shear force diagram at any point is equal to the load intensity at that point:

0 lim x dQ Q f dx   x     (1.3)

Taking moments about :





. _a. . . 0 M _a. . M Q x f x x M M Q f x x                  (1.4)

The slope of the moment diagram at any point is equal to the shear force at that point: 2 2 0 lim x dM M d M Q f dx   x dx       (1.5)

Hence, the relationships between the load intensity and the shear force and between the shear force and the bending moment for the beam will be given by:

2 1 1 2 1 . . . x x dQ f dx  Q



f dx C  Q Q 



f dx (1.6) L X1 X2 x x 1 2 f(x) A _B x x O M M+ Q Q+Q f f+f F .xa _{x (01)}

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The change of shear force between two points is equal to the area under the load intensity diagram between these two points.

2 1 2 2 1 . . . x x dM Q dx  M 



Q dx C  M M 



Q dx (1.7) The change in bending moment between two points is equal to the area under the shear force diagram between these two points.

1.7. Basic relationships for ship hull girder

Overall static equilibrium requires that the total upwards buoyancy force equals the weight of the ship and that these two vertical forces coincides; that is, the longitudinal center of buoyancy (LCB) must coincide with the longitudinal center of gravity (LCG).

g = gravitational acceleration

 = displacement

W = weight

B = buoyancy

Similarly, equilibrium of moments requires that:

 

0 0 L L G g a x xdx g m x xdx g l 

_



_

  (1.9) Or B LCB. W LCG.  LCBLCG

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14 Where, l = distance from origin to LCG. _G

However, over any given unit length of the hull the forces will not balance out. At any point x :

Buoyancy per unit length = b x( )ga x

 

The weight per unit length = w x( )m x g

 

Hence, the net force (load) per unit length = f x( )b x( )w x( )

If this net loading is integrated along the length there will be, for any point, a force tending to shear the structure such that:

 

0 0

,

x x

Shear force Q x 



f x dx



_ga x m x g dx_ (1.10) the integration being from one end to the point concerned.

Integrating a second time gives the longitudinal bending moment. That is:

 

0 0 0

,

x x x

 (1.12)

 

   

0 and . x L v x 



 x dx  x v x  x (1.13) Where:  = slope v = vertical displacement  = deflection

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Figure 11 summary of hull girder bending

1.8. Characteristics of shear force and bending moment curves

For any given loading of the ship, the draughts at which it floats can be calculated. Knowing the weight distribution, and finding the buoyancy distribution from the Bonjean curves, gives the net load per unit length.

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Certain approximations are needed to deal with distributed loads such as shell plating. Also the point at which the net force acts may not be in the center of the length of the increment used. However, these approximations are not usually of great significance and certain checks can be placed upon the results (Figure 12):

 First the shear force and bending moment must be zero at the ends of the ship. If after integration there is a residual force or moment this is usually corrected arbitrarily by assuming the difference can be spread along the ship length.

 The shearing force is approximately asymmetric, has a maximum or minimum value at points about a quarter of the length from the ends and is zero near amidships.

 From the relationships deduced above when the net load is zero the shear force will have a local maximum or minimum value and the moment curve will show a point of inflection.

 Where net load is a maximum the shear force curve has a point of inflection.  Where shear force is zero, the bending moment is a local maximum or minimum.  Bending moment will have zero slopes at both ends with small values forward and aft

of the quarter points.

Figure 12 shearing force and bending moment

When the ship is distorted so as to be concave up it is said to sag. The deck is then in compression with the keel in tension. When the ship is convex up it is said to hog. The deck is then in tension and the keel in compression.

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1.9. Weight distribution

The calculation of the longitudinal distribution of weight or mass m x is a difficult process,

 

partly because m x is made up of discrete items rather than being a continuous and regular

 

curve, and partly because at the design stage many of the individual weights are known only approximately.

The weights in a ship fall into two main categories: those which are relatively unchanged, such as the ship’s own structural weight; and those which do change, such as cargo, fuel, stores, and ballast. The first group constitutes the “lightweights” of a ship, that is, the weight when it is without cargo, fuel, and so on (this condition is referred to as the “lightship” condition). The second group is called the “deadweight“. The deadweight changes with each different cargo loading, and hence there are usually several loading conditions which need to be investigated. The two most common conditions are “full load” and “ballast”.

In most cases the following information should be specified for each distinct weight item: 1. Total weight.

2. Vertical and longitudinal center of gravity lcg. 3. Longitudinal extend.

4. The type of distribution over this extend.

In specifying the extend and distribution of individual weights, it is helpful and even necessary to use some approximations and idealizations. Nearly all items can be represented in terms of one or more of three basic types of distribution: point, uniform distribution, and trapezoidal distribution.

Typical examples of point loads are machinery (one point load at each foundation point), masts, winches, and transverse bulkheads.

Examples of uniform loads are: hull steel within the parallel midbody, and cargo, fuel, ballast, and other homogenous weights within prismatic spaces. Outside the parallel midbody and particularly toward the ends of the ship a trapezoidal distribution is appropriate, although even here some items can be accurately represented as uniform loads, such as superstructure. For a trapezoid, say of length l , the relevant information may be specified in two different ways: either as total mass, M , with a specified position of center of gravity within this _o length (say a distance x from the center; see Figure 13) or in terms of mass per unit length at

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the forward and after ends: m and _f m . The formulas for converting from one form to _a another are:

Figure 13 trapezoidal representation of a weight





6 2 f a f a f a o m m l x m m l m m M       _       _{ }      (1.14) and 2 2 6 6 o o a o o f M M x m l l M M x m l l     (1.15)

1.9.1. Hull weight distribution

Hull weight is traditionally defined as lightship minus the weight of the anchor, chain, anchor handling gear, steering gear and main propulsion machinery. Numerous approximation methods for distributing hull weight have been proposed in the past. These approximations are general and appropriate only for initial stage design due to their low fidelity.

A useful first approximation to the hull weight distribution is obtained by assuming that two-thirds of its weight follows the still water buoyancy curve and the remaining one-third is distributed in the form of a trapezoid, with end ordinates such that the center of gravity of the entire hull is in the desired position (Figure 14).

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Figure 14 approximation for hull weight distribution

Trapezoidal approximation is useful for ships with parallel midbody. This approximation uses a uniform weight distribution over the parallel midbody portion and two trapezoids for the end portions, with end ordinates again chosen such that the LCG of the hull is in the desired position as shown in Figure 15. The ordinates indicated in the figure are given by:

Figure 15 Trapezoidal approximation

Hull Weight Ordinate . Length H W Coeff L   (1.16)

Where the coefficient is as indicated in Table 1:

Table 1 coefficients of the trapezoidal approximation

1 2 3 K 0.333 0.333 0.250 a 0.567 0.596 0.572 b 1.195 1.174 1.125 c 0.653 0.706 0.676 C.G. Aft 0.0052L 0.0017L 0.0054L

1 Fine ships – Merchant type 2 Full ships – Merchant type 3 Great lakes Bulk freighters

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Biles presented the hull weight by a trapezoid, frequently called the ‘coffin diagram’, and

gave the following values of the ordinates for passenger and cargo ships (Figure 16):

At F.P. 0.566 over 3amidships 1.195 at A.P. 0.653 H H H W L L W L W L (1.17)

Where W_H = hull weight, L = length of ship

Figure 16 coffin diagram

The centroid of the diagram as given is at 0.0056L abaft amidships. It is permissible to make small adjustments to the end ordinates in order to ensure that the centroid of the diagram corresponds to the longitudinal center of gravity of the hull.

The desired shift of the centroid can be secured by transferring a triangle from one trapezium to the other as indicated by the dotted lines. The shift of centroid of the triangle is

 

7 9 L. Thus, if x is the end ordinate of the triangle to be shifted:

Area of triangle = 1

 

3 2x L Moment of shift = 7 7 2 6 9 54 L x  L xL Shift of centroid = 7 2 54xL W H Thus 54 shift of centroid

7 H W x L L 

Prohaska has given detailed consideration to the diagram (Figure 17) and suggested values

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Figure 17 Prohaska’s approximation

Table 2 Prohaska’s values

Comstock representation is typically used to approximate the hull weight. In this

approximation, 50% of the hull weight is distributed as a rectangular in the middle 0.4 length, and 50% in two trapezoids so as to give the required LCG. If W_H is the total weight to be distributed and d is the LCG of the weight from amidships, then (Figure 18):

Figure 18 Comstock’s representation

1.25 1 20 3 1 20 3 H W h L h d x L h d y L     _  _      _  _   (1.18)

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Cole has proposed a parabolic rule. This method is useful in ships without parallel middle

body. The hull weight is presented by a rectangle and a superimposed parabola (Figure 19). Obviously the centroid of this diagram is at amidships. The centroid can be shifted to a desired position by swinging the parabola as follow (Figure 19, Figure 20):

1) Through the centroid of the parabola draw a line parallel to the base and in length equal to twice the shift desired forward or aft.

2) Through the point so obtained draw a line to the base of the parabola at amidships. 3) The intersection of this line with the horizontal drawn from the intersection of the

midship ordinate with the original parabola determines the position of one point on the new curve.

4) Parallel lines are drawn at other ordinates as shown and the new curve determined.

Figure 19 Cole’s proposal

Figure 20

Small errors in the area and the centroid of the diagram can be corrected by adjustments of the base line. An error in weight can be corrected by raising or lowering the base line. The position of the centroid can be adjusted by tilting the base line as shown in the Biles method.

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23 1.9.2. Total lightweight distribution

When the hull weight distribution has been obtained, the other items of the lightweight (weight of the anchor, chain, anchor handling gear, steering gear and main propulsion machinery) can be added at their centers of gravity. The resulting curve for the lightship weight can be obtained as shown in Figure 21.

Figure 21 lightship weight distribution

1.9.3. Deadweight distribution

For cargo and ballast, the weight per unit length is related to the cross-sectional area of the relevant cargo or ballast space, and their weight distribution may be taken as the product of the sectional area curve of the relevant space times the mass density of the cargo or ballast. If the total volume of cargo spaces and the cargo deadweight of the ship being known, then:

3 volume of cargo spaces

stowage rate in

cargo deadweight  ft ton (1.19)

Because the cargo is the largest item of weight and because there are so many possible variations in its distribution, there are often some distributions and combinations that would cause excessive values of bending moment and that therefore must be avoided. It is more efficient to have the cargo holds or tanks either completely full or completely empty. Given such extreme differences it is important that they be spread out, rather than grouped together, because the latter would give excessive shear force and/or bending moment as shown in Figure 22. Figure 23 shows a typical curve of buoyancy, weight, load, shear force and bending moment for a 30 000 T.D.W. bulk carrier with ore in holds No. 1,3,5 and 7 only.

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Figure 22 effect of cargo distribution

Figure 23 30 000 T.D.W. bulk carrier with ore in holds No. 1,3,5 and 7 only

When the weights per unit length for the deadweight items have been obtained, they are added to the curve of the total lightweight, giving the total weight curve as shown in Figure 24. After the curve is plotted, it should be checked for the total area, giving the weight of the ship in that particular loading condition. Its centroid will give the longitudinal center of gravity of the ship. A sample weight curve is given in Figure 25.

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Figure 24 Total weight curve

Figure 25 typical weight distribution

1.9.4. Modified weight curve

The described weight curve shows many discontinuities. The sudden changes that occur in the weight curve are not at regular intervals in the length direction. This makes some difficulties during integration, particularly by a tabular method. To overcome this difficulty, the length of the ship is divided into a number of equal parts and we assume that the weight per unit length is constant over each division. In this way a stepped weight curve is produced as shown in Figure 26.

Figure 26 Stepped weight curve

Indicates deadweight items

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To produce this stepped weight curve, the total weight in each division is calculated and then is divided by the length of the division. This will give the mean weight per unit length for that division. Having obtained the stepped weight curve in this way, the total area and position of its centroid should be checked so as to give the correct weight and center of gravity of the ship.

The steel weight of the far portions in the forward and aft must also be included in the weight curve, thus, the weight curve must be corrected to enclose these weights between the perpendiculars. This can be done through transferring the end weights to the nearest two intervals to compensate for the moment of shifted weight, as shown in Figure 27, such that:

1 2 1 2 3 2 1 2 P P P X P P S X P P S     _  _     _  _   (1.20)

Figure 27 inclusion of end weight

1.10. Buoyancy distribution in still water

The still water buoyancy is a static quantity and depends on the geometry of the underwater portion of the hull. The buoyancy due to waves is both dynamic and probabilistic. It is assumed that all the usual hydrostatic information is available for the ship and that Bonjean curves of area are also available. The problem is then to find the distribution of buoyancy that will give these values of displacement and center of buoyancy so that the ship shall be in static equilibrium either in still water or on a wave.

P P P X S S 0 1 2 1 2

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For the still water condition, the mean draft T is determined from the hydrostatic curves _m according to the loading condition, i.e. at the magnitude of displacement, as shown in Figure 28. If LCB (corresponding to T ) and LCG are not equal, then the total trim _m T caused by this _t difference is determined according to the following equation:





1 t cm LCB LCG T MCT    (1.21)

Figure 28 Hydrostatic curves

The magnitudes of forward and aft trims are determined based on the location of the center of flotation LCF (+ve Fwd) as shown in Figure 29, such that:

0.5 0.5 F t A t L LCF t T L L LCF t T L       (1.22)

Figure 29 Trim calculation

The end drafts are then determined by adding trim to, or subtracting trim from, the mean draft according to the condition of trim. Afterwards, the end drafts should be drawn on a profile of

TPC MCT LCB LCF Tm T   Fwd +ve Aft -ve L LCF tF t_A Tt T_m

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the ship in the normal way to obtain the waterline at which the ship floats as shown in Figure 30.

Figure 30 Bonjean curves

If the Bonjean curves of area are also shown on this profile, it is a simple matter to lift off the immersed areas where the waterline intersects the various sections. The buoyancy per unit length at any section is then simply the area of the section multiplied by the density of water. It must be checked that the areas lifted from the Bonjean curves for the obtained trim line give the correct displacement  and LCB. If the trim is large, some discrepancy may exist so that: ' ' LCB LCB     (1.23)

The position of the waterline must be corrected by moving it a distance ( '  ) TPC and tilting it an amount '(LCB'LCB MCT) ₁_cm, where:

& LCB

 are the required displacement and LCB, '&LCB'

 are those obtained from Bonjean curves’ calculations.

Next areas are lifted and displacement and center of buoyancy calculations are repeated. This second approximation is usually sufficient.

1.11. Buoyancy distribution in seaway

The mass distribution is the same in waves as in still water assuming the same loading condition. The differences in the forces acting are the buoyancy forces and the inertia forces on the masses arising from the motion accelerations, mainly those due to pitch and heave. For the present the latter are ignored and the problem is treated as a quasi-static one by considering the ship balanced on a wave.

Waterline

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29

The buoyancy forces vary from those in still water by virtue of the different draughts at each point along the length due to the wave profile and the pressure changes with depth due to the orbital motion of the wave particles. This latter, the Smith effect, is usually ignored in the standard calculation to be described next. Ignoring the dynamic forces and the Smith effect does not matter as the results are used for comparison.

The concept of considering a ship balanced on the crest, or on the trough, of a wave is clearly an artificial approach although one which has served the naval architect well over many years. Nowadays the naval architect can extend the programs for predicting ship motion to give the forces acting on the ship. Such calculations have been compared with data from model experiments and full scale trials and found to correlate quite well.

The strip theory is commonly used for calculating ship motions. The ship is divided into a number of transverse sections, or strips, and the wave, buoyancy and inertia forces acting on each section are assessed allowing for added mass and damping. From the equations so derived the motions of the ship, as a rigid body, can be determined. The same process can be extended to deduce the bending moments and shear forces acting on the ship at any point along its length. This provides the basis for modern treatments of longitudinal strength.

1.11.1. The trochoidal wave

Figure 31 shows the profile of a regular wave which may be considered to be a deep sea wave. Wave of this type is oscillating waves in which the water particles move in closed paths without bodily movement of fluid. The wave form moves over the surface and energy is transmitted. The distance between successive crests is the wave length L. The distance from the trough to the crest is the wave height h .

Figure 31 profile of a regular wave

Observations on ocean waves have shown that the crests are sharper than the troughs, which assumes that the wave profile is a trochoid. The trochoidal theory shows that the paths of the

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30

water particles are circles, whereas the classical theory shows that the paths are ellipses which tend to circles as depth of water increases.

Both theories show that for a deep sea wave, if conditions are considered some distance below the surface, then the radius of the orbit circles of the particles diminishes. If r_o h 2 is the radius of the surface particles and r is their radius at some sub-surface distance y below the free surface then:

2 exp exp o o y y r r r R L       _ _ _ _     (1.24)

Where y is considered positive downwards.

It will be seen then that the disturbance below the surface diminishes rapidly with depth, the situation being as shown in Figure 32, where the sub-surface trochoids as they may be called rapidly flatten out.

In shallow water where the influence of depth is important, the elliptical orbits of the particles flatten out with depth below the free surface and at the bottom the vertical movement is prevented altogether and the particles move horizontally only.

Figure 32

A trochoid is a curve produced by a point at radius r within a circle of radius R rolling on a flat base. The equation to a trochoid with respect to the axes shown in Figure 33, is:

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0.607 sin 2 2 , and in meters 0.607 1 cos 2 L L x x z L L z        _     _ (1.27)

The 0.607 L wave has the slight disadvantage that it is not non-dimensional, and units must be checked with care when using this wave and the formulae derived from it.

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32

The wave height 1.1 L is another approximation suggested by Lloyd’s Register. To draw a trochoidal wave surface:

1 – Divide selected wave length (L_W  ) by a convenient number of equally spaced points (L S = spacing).

2 – With each point as a center, draw a circle of diameter equal to the selected wave height (e.g. hL 20).

3 – In each of the circles, draw a radius at an angle increase of the fraction of 360o as the spacing of the circles to wave length.

4 – Connect the ends of the radii

2 _W 2 S S S S R L R R  _ _         (1.28)

1.11.2. Pressure in waves (Smith effect)



r_o2r2





. This is shown in Figure 34. Its effect is that pressure is reduced below the static value at the crest of a wave and is increased beyond the static value at the trough, which influences the buoyancy of ship amongst waves (Figure 35). The

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33

influence of dynamic considerations on the pressure in a wave is often referred to as the ‘Smith effect’.

Figure 34

Figure 35 Buoyancy distribution of ship in waves (a) wave crests at ends. (b) troughs at ends

1.11.3. The standard static longitudinal strength approach

Too long a wave length, several times the length of the ship, makes the profile look like a horizontal line on the ship and the buoyancy distribution would be essentially as that in still water. On the other hand, a very short wave length, i.e., small fraction of the length of the ship, makes small undulations in the still water buoyancy curve that would have little effect on the bending moment produced on the ship.

It would appear that there would be some length of wave in between these two extremes that would have the maximum effect on the buoyancy distribution. The standard that has usually been accepted is that the ship is assumed to be poised, in a state of equilibrium, on a trochoidal wave of length equal to that of the ship. This is not easy and can involve a number

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34

of successive approximations to the ship's attitude before the buoyancy force equals the weight and the center of buoyancy is in line with the center of gravity. Clearly this is a situation that can never occur in practice but the results can be used to indicate the maximum bending moments the ship is likely to experience in waves.

Two conditions are considered, one with a wave crest amidships and the other with wave crests at the ends of the ship. In the former the ship will hog and in the latter it will sag (Figure 36). Figure 37 shows the resulting buoyancy distribution. The bending moments obtained include the still water moments. It is useful to separate the two. Whilst the still water bending moment depends upon the mass distribution besides the buoyancy distribution, the bending moment due to the waves themselves depends only on the geometry of the ship and wave.

The influence of the still water bending moment on the total moment is shown in Figure 38. Whether the greater bending moment occurs in sagging or hogging depends on the type of ship depending, mainly, upon the block coefficient. At low block coefficients the sagging bending moment is likely to be greater, the difference reducing as block coefficient increases.

Figure 36 ship on wave

Figure 37 Shape of buoyancy distribution

Wave profile Wave profile Sagging Hogging Buoyancy in still water Buoyancy (Wave crests at amidships) Buoyancy (Wave crests

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35

Figure 38 still water and wave bending moments

1.11.4. W. Muckle’s Method

The position of the still-water level corresponding to the wave form is first calculated. This is a distance r2 2R below the center of the wave, where rh 2 and RL 2 , so that:

Still-water level below wave center = h2 4L

The still-water level of the wave is now placed on the waterline at which the ship would float in still water for the condition of loading for which the calculation is being carried out. If the sagging condition is being considered, then the wave must be raised a certain amount and tilted. Thus, the amount by which the wave must be raised at any position in the length of the ship can be written:

y a bx L (1.30)

where a and b are constants yet to be determined and x is measured from, say, the after perpendicular.

Figure 39 shows the Bonjean curve of area for a section in the length of the ship. The point C is where the wave cuts the section before it has been adjusted by the amount y . The corresponding sectional area is A_o.

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36 Figure 39

The buoyancy curve should satisfy the following two equilibrium conditions.

The total volume after the wave has been shifted must be equal to the required volume for the condition of loading, hence:

0 L x W A dx       

_

(1.31)

The moment of the total displacement after the wave has been shifted must be equal to the moment of the ship weight, hence:

0 L G B B x G B W X X X A xdx X X       



  (1.32)

Where, X and _G X are the LCG and the LCB, respectively, measured from the after _B perpendicular. W is the total weight of the ship.

Let the area at a position m m above the position C be A . An approximation of the Bonjean _m curves will show that the curve between these two positions could be very closely represented by a straight line. If A is the area at any height y y above C, then:

y o m o m o y o A A A A A A A A y m y m   _ _ _         (1.33) From Eqn (1.30): m o m o m o y o o A A A A A A x x A A a b A a b L m m m L             _  _ _  _ _ _ _        (1.34) From Eqn (1.31):

(37)

37 0 0 0 0 L L L L m o m o y o A A A A x W A dx A dx a dx b dx m m L          _ _  _ _       



(1.35) From Eqn (1.32): 2 0 0 0 0 L L L L m o m o y o G B A A A A x x x x A dx A dx a dx b dx L L m L m L X X W L  L          _ _  _ __{ }         



(1.36)

We thus have two equations (1.35) and (1.36) from which the two unknowns a and b can be calculated.

1.11.4.1. Example

A ship 460 ft in length and of 8996 tons displacement with the center of gravity 6.32 ft aft of amidships.

For a wave having a length equal to the length of the ship and a height equal to L 20 the still-water level is 2 2 11.5 0.905 2 2 460 2 r R     ft

Below the center of the wave. This still-water level was put on the water line at which the vessel would float in still water, and ordinates were read from the Bonjean curves where the wave cut the various sections.

As the sagging condition was being considered, the final position of the wave would be higher than this, so that ordinates of area at positions 4 ft above these intersections were also lifted as described in the foregoing.

Table 3shows the values of A and _o A read from the curves, and also shows how the various ₄ integrations were made by using Simpson’s first rule.

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38 Table 3 Section A o S.M. Area function Lever Moment function A 4 A4Ao Area function Moment function Moment of inertia function 0 125 1₂ 63 0 - 210 85 43 - - 1 2 465 2 930 1 2 465 615 150 300 150 75 1 770 1 770 1 770 948 178 178 178 178 1 2 1 960 2 1920 1 1₂ 2880 1175 215 430 645 968 2 1005 1 1₂ 1507 2 3014 1235 230 345 690 1380 3 790 4 3160 3 9480 1025 235 940 2820 8460 4 475 2 950 4 3800 718 243 486 1944 7776 5 310 4 1240 5 6200 555 245 980 4900 24500 6 412 2 824 6 4944 655 243 486 2916 17496 7 705 4 2820 7 19740 948 243 972 6804 47628 8 900 1 1₂ 1350 8 10800 1115 215 323 2584 20672 1 2 8 838 2 1676 8 1₂ 14246 1025 187 374 3180 27030 9 625 1 625 9 5625 770 145 145 1305 11745 1 2 9 300 2 600 9 1₂ 5700 375 75 150 1425 13538 10 - 1₂ - 10 - - - - 18435 87664 6152 29541 181446

39 Equations (1.35) and (1.36) may now be written as:

282670 23583 a11324b314860 134418 11324 a6956b153104 That is, 0.4802 1.365 a b 0.6143 1.6501 a b

From these we obtain a0.3441 and b2.126.

The area to be added to each section is now = 4





0.3441 2.126 4 o A A x L  

Table 4shows these areas together with the calculation for the additional displacement and its centroid.

Table 4

Section Additional area S.M. Area function Lever Moment function

0 7.3 1₂ 3.6 5 18.0 1 2 17.0 2 34.0 1 2 4 153.0 1 24.8 1 24.8 4 99.2 1 2 1 35.6 2 71.2 3 1₂ 249.2 2 44.2 1 1₂ 66.3 3 198.9 3 57.7 4 230.8 2 461.6 4 72.6 2 145.2 1 145.2 5 86.2 4 344.8 0 1325.1 6 98.4 2 196.8 1 196.8 7 111.3 4 445.2 2 890.4 8 109.9 1 1₂ 164.8 3 494.4 1 2 8 100.6 2 201.2 3 1₂ 704.2 9 81.8 1 81.8 4 327.2 1 2 9 44.3 2 88.6 4 1₂ 398.7 10 - 1₂ - 5 - 2099.1 3011.7 Additional displacement = 2099.1 46 919 35  3  tons

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40

Centroid from amidships = 3011.7 1325.1 46 36.96 ford.

2099.1 ft

 _ _

Original displacement obtained from 0 282670 8076 35 L o A dx  tons



Corresponding LCB = 134418 0.5 460 11.27 aft. 282670 ft  _ _ _    

Final displacement = Original displacement + Additional displacement = 8076 + 919 = 8995 tons

Final LCB = 8076 11.27 919 36.96 57050 6.34

8076 919 8995 ft

   _ _

 aft of amidships

These results compare very favorably with the required values of 8996 tons displacement with a position of center of buoyancy of 6.32 ft aft of amidships.

It remains now to add the areas in the second column of Table 4 to the values A obtained _o originally from the Bonjean curves. The resulting areas, when divided by 35, give the ordinates of the buoyancy curve in tons/ft.

Should the hogging condition be considered instead of the sagging condition, the wave would be required to be lowered instead of raised. It would be necessary, therefore, to take areas from the Bonjean curves at positions 4 ft below those at which the areas A are read. _o Otherwise, the calculation would be exactly the same as outline here.

1.12. Load distribution

The load on the structure at any point in the length of the ship is the difference between the weight per unit length and the buoyancy per unit length:

p b w (1.37)

The total area enclosed by the load curve should be zero, the area underneath the base being considered negative, i.e. total vertical force = 0 or static equilibrium.

If the load curve is integrated, then the shearing force on the structure is obtained, so that:

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41

By integrating the shearing force curve, the bending moment on the structure can be obtained:

M 



Q dx (1.39)

The shearing force and bending moment curves are shown in Figure 40. Both of these curves should be zero at xL, which is the consequence of the equilibrium condition. The load distribution is what mainly affects shear force and bending moment (maximum values).

Figure 40 Load, shearing force and bending moment curves

1.13. Integration procedure

The integration is performed using one of the following two methods: 1.13.1. First method:

In this method, the integration of the weight and buoyancy curves is performed separately. The first integral of these two curves gives the curves shown in Figure 41 and the shearing force is the difference between the two curves. If these two first integral curves are integrated, then curves of the type shown in Figure 42 are obtained and one again the difference between these two curves gives the bending moment.

Figure 41 Integration of weight and buoyancy curves

Sh_ea rin g fo rce Load Be nd_ing m om en_t

Shearing force Integra l of buo yancy Integ ral o f wei ght

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42

Figure 42 Double integration of weight and buoyancy curves

1.13.2. Second method:

The tabular method of integration is one used very frequently and is particularly suited where the stepped type of weight curve has been employed. Suppose that the length of the ship is divided into a number of equal intervals of length s (usually 40 intervals) and that b and _m

m

w are the mean ordinates of the buoyancy and weight curves for an interval. The integral of the load curve for this division is then given by



b_mw_m



s. Then the shearing force at any point will be:



m m



Qs



b w (1.40)

If Q is the mean value of the shearing force for an interval, then the bending moment is _m given by:

m

M s



Q (1.41)

Table 5 shows how the calculation can be performed using the second method.

Note that if there is an error in shearing force or bending moment, it is corrected as shown in Figure 43 and Table 5. Since Q and ₄₀ M must be equal to zero, the base lines are corrected. ₄₀

Bending moment b dxdx  w dxdx 

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43

Figure 43 Base line correction for shear force and bending moment

Be_nd ing_m om en_t She arin g fo rce B A

O Original base line

Corrected base line for bending moment Corrected base line for shearing force

(44)

(45)

45

1.14. Wave bending moment for a box-shaped vessel on a sine wave

Figure 44

The wave surface equation is given by:

2 cos 2 h x y L    (1.42)

Where, h = wave height (hL 20 or h1.1 L), and L = length of the vessel.

The wave bending moment is the bending moment resulting from the difference in the buoyancy between the wave surface and the still water surface.

2 ( ) cos 2 h x Load p x By B L       (1.43)

Where,  is the specific weight of water and B is the breadth of the vessel

 

₀

 

₀ 2 2 cos sin 2 2 2 x x h x h L x Shear force Q x p x dx B dx B L L      



 



  (1.44)

The maximum shear force, Q_max occurs when dQ dx p x( )0, at xL 4 and x3L 4









max max 4 2 2 4 3 4 4 h L BhL Q x L B BhL Q x L              (1.45)

 

0 0 2 2 2 2 0 2 sin 2 2 2 2 cos 1 cos 8 8 x x x h L x Bending moment M x Q x dx B dx L BhL x BhL x M x L L                   _ _   _  _    



(1.46) L 0 X Y B h

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46

The maximum bending moment, M_max occurs when dM dx Q 0, hence, sin 2



x L



0 , at x0, xL 2, and xL





2 2 max 2 2 2 2 8 4 BhL BhL M x L           (1.47)

1.15. Wave bending moment for a diamond-shaped vessel on a sine wave

The wave surface equation is given by:

2 cos 2 h x y L    (1.48)

The breadth is a function of x as follows:







max max 1 3 M Diamond  M Box (1.53)

1.16. Wave bending moment and shear force for a ship on a sine wave

1.16.1. Wave bending moment

The magnitude of the wave bending moment for a ship shape will be between the wave bending moments of diamond and box shapes, such that:

Diamond Ship Box

M M M (1.54)

Let, M_Ship .M_Box  



1 



.M_Diamond, where  = factor determined by the following:





0.5

B

C Diamond  & C_B



Box



 1





1. 0.5 1





B

C Ship    , then,  2



C_B0.5



, C_B0.5 For diamond shape: C_B 0.5, and   0

For box shape: C_B 1.0, and   1 Therefore,











 



2 0.5 1 2 1 .

2 0.5 1

Ship B Box B Diamond

Box B B Diamond M C M C M M C C M       _    _ (1.55) and



 







ax 4 1 M B Max M Ship  C  M Diamond (1.56)

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48

1.16.2. Shear force

For a box shape:

 

max max 2 1 cos 2 2 sin M x M x L dM x x Q x M dx L L       _  _     (1.57)

The maximum shear force, Q_max occurs when dQ dx p x

 

0, at xL 4 and x3L 4

max max max max

Q M for box shape

L C

Q M for ship shape

L 

 

(1.58)

For passenger ship: C = 3.5 For cargo ship: C = 3.75 For tankers: C = 4.3

1.17. Examples Ex. 1:

Consider a vessel of constant rectangular cross section, 140m long, 20m beam and 13m deep, with total mass 25 830 tones, 20 830 of which is uniformly spread over the length and the rest distributed uniformly over the central 10m. Calculate the bending moments and shearing forces.

(49)

49 Solution:

The mass distribution will be constant at 20830 148.8

140  t/m over the entire length; plus 5000

500

10  t/m over the central 10m.

The still water buoyancy distribution will be constant at 25830 184.5 140  t/m

The maximum bending moment (BM) will be at amidships. The BM amidships due to buoyancy = 25830 140 9.81 4434

2  4 1000 MNm

The BM amidships due to weight = 20830 140 5000 5 9.81 3637

2 4 2 2 1000

 _ _ _ _ _

 

  MNm

The net BM amidships = 4434 – 3637 = 797 MNm sagging moment

Suppose now that the vessel is poised on a sinusoidal wave equal to its own length and of height 0.607( )L 0.5, that is a wave 140m long and 7.2 m high. The wave height at any point above the still water level, when the wave crests are at the ends, is:

Figure 47 wave load

7.2 2 cos 2 140 x h _  _   (1.59)

The wave buoyancy distribution = 1.025 20 3.6 9.81 10 cos3 2 0.724 cos 2

140 140 x x            _ _ _ _     MN/m

By integration, the wave shearing force is:

0 2 140 2 0.724 cos d 0.724 sin 140 2 140 x x x Q  x        _ _   _ _    



(50)

50 Integrating again the bending moment is:

2 2

0

140 2 140 2 140 2

0.724 sin d 0.724 cos 0.724 359 1 cos

2 140 4 140 4 140 x x x x M  x                _ _    _ _   _  _ __       



Putting x = 70 the wave bending moment at amidships is found to be 718MNm sagging. With the wave crest amidships the wave moment would be of the same magnitude but hogging. The total moments are obtained by adding the still water and wave moments, giving:

Sagging = 797 + 718 = 1515MNm Hogging = 797 – 718 = 79MNm

Had the mass of 5000 tones been distributed uniformly over the whole ship length the still water bending moment would have been zero, giving equal sagging and hogging wave bending moments of 718 MNm.