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Unformatted text preview: ma: lo_ 120
_ (1.40)
MB 6Mox
mf— 1 + 12 2.5.1. 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 onethird 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 26). . " F.. — 1 _ __I Figure 26 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 27. The ordinates indicated in the ﬁgure are given by: Hull Weight WH
Length L Ordinate = Coeﬁ‘. x (1.41) Where the coefﬁcient is as indicated in Table 1: Table l 1 2 3
0.333 0.333 0.250
0.567 0.596 0.572
1.195 1.174 1.125
0.653 0.706 0.676 0.0052L 0.0017L 0.0054L 1 Fine ships — Merchant type
2 Full ships — Merchant type
3 Great lakes Bulk freighters 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 28): AtF.P. 0.566wH /L
over L/ 3 amidships 1.195WH / L (1.42)
at A.P. 0.653wH /L Where WH = hull weight, L = length of ship Figure 28 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. 29 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: 1 Area of triangle = E x(L/3)
Moment of shift = XEX l L = l XL2
6 9 54 Shift of centroid : l xL2 /WH
54 Thus x = EEW
7 L L Prohaska has given detailed consideration to the diagram (Figure 29) and suggested values for a number of different types of ships as given in Table 2. Figure 29 Table 2
a“ Prohaska’s values a/(s) Type of ship Tankers 075 1125
Full cargo ships without erections 0‘65 H75
Fine cargo ships without erections 060 120
Full cargo ships with erections 055 l225
Fine cargo ships with erections 0~45 1275
Small passenger ships 040 130
Large passenger ships 030 135 —_*_—_—__ 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, 30 and 50% in two trapezoids so as to give the required LCG. If WH is the total weight to be distributed and d is the LCG of the weight from amidships, then (Figure 30): h=1.25E
L
“Eb—203) (1.43)
3 L
h d
=—1+20—
y 3[ Ll 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 31). 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 31, Figure 32): 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. 31 Note:— I = desired Shlfl Corrected Curve X Desired Shift 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. 2.5.2. 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 33. Figure 33 lightship weight distribution 32 2.5.3. Deadweight distribution For cargo and ballast, the weight per unit length is related to the crosssectional 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: W = stowage rate in 1? /ton (144) cargo deadweight 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 34. Figure 35 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. fl
“1
l J Figure 34 effect of cargo distribution 33 Figure 35 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
36. 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 37. m Indicates deadweight items Figure 36 Total weight curve Figure 37 typical weight distribution 34 2.5.4. 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 38. Figure 38 Stepped weight curve 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 39, such that: P=H—g
Pl 2(3+§]P (1.45)
2 S
2 S 35 Figure 39 inclusion of end weight 2.6. Buoyancy distribution 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. For the still water condition, the mean draft Trrl is determined from the hydrostatic curves
according to the loading condition, i.e. at the magnitude of displacement, as shown in Figure
40. If LCB (corresponding to Tm) and LCG are not equal, then the total trim Tt caused by this
difference is determined according to the following equation: T _ A(LCBiLCG)
‘_ MCT 1cm (1.46) LCB LCF l l , Aft ve <— CX)—— Fwd +ve Figure 40 Hydrostatic curves 36 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 41, such that: : 0.5LLLCF th
0.5L+LCF
tA=Tth F (1.47) Tt
tA /
 a? J m Figure 41 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 the ship in the normal way to obtain the waterline at which the ship ﬂoats as shown in Figure 42. Waterline
g Bonjean curve of area Figure 42 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 A and LCB. If the trim is large, some discrepancy may exist so that: 37 A' ¢A
LCBilﬂG (1.48) The position of the waterline must be corrected by moving it a distance (A —A)/TPC and tilting it an amount A'(LCB' — LCB)/MCT1 where: cm’ A & LCB are the required displacement and LCB, A' & IJCB are those obtained from Bonj ean curves’ calculations. Next areas are lifted and displacement and center of buoyancy calculations are repeated. This second approximation is usually sufficient. 2.7. Forces on a ship in a 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 quasistatic one by considering the ship balanced on a wave. 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 in 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 38 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. 2.8. The wave Figure 43 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 43 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
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 r0 = h/ 2 is the radius of the surface particles and r is their radius at some subsurface distance y below the free surface then: r = r0 exp[—%] = r0 exp (1.49) 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 44, where the subsurface trochoids as they may be called rapidly ﬂatten out. 39 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. 'u'v'uvelengw L Figure 44 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 45, is: Figure 45 construction of a trochoid R6—rsin6
r(1—cos¢9) X (1.50) Z One accepted standard wave is that having a height from trough to crest of one twentieth of its length from crest to crest. In this case, L = 27rR and r = h/2 = L/40 and the equation to the wave is: 40 xziﬂ—isinﬁ 2L” 40 (1.51)
22— l—cos0
40( ) Research has shown that the L/ 20 wave is somewhat optimistic for wave lengths from 90m up to about 150m in length. Above 150m, the L/20 wave becomes progressively more unsatisfactory and at 300m is probably so exaggerated in height that it is no longer a satisfactory criterion of comparison. This has resulted in the adoption of a trochoidal wave of height 0.607x/E as a standard wave in the comparative longitudinal strength calculation. This wave has the equation: XZL9_M,M
2” 2 x, zand Lin meters (1.52)
0.607ﬁ z 2 TO —cos 6') The 0.607x/E wave has the slight disadvantage that it is not nondimensional, and units must be checked with care when using this wave and the formulae derived from it. To draw a trochoidal wave surface: 1 — Divide selected wave length (LW 2 L) by a convenient number of equally spaced points (
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 360° as the spacing of the circles to wave length. 4 — Connect the ends of the radii 2 s 2 6R (1.53) 2.9. The standard static longitudinal strength approach The ship is assumed to be poised, in a state of equilibrium, on a trochoidal wave of length equal to that of the ship. Clearly this is a situation that can never occur in practice but the 41 ...
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This note was uploaded on 03/19/2012 for the course ECON 256 taught by Professor Lopez during the Spring '10 term at École Normale Supérieure.
 Spring '10
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