Simple mathematical buoyancy functions with application to calculation of shear forces and longitudinal bending moments for ships in arbitrary trimmed conditions

SIMPLE MATHEMATICAL BUOYANCY FUNCTIONS

WITH APPLICATION TO CALCULATION OF

SHEAR FORCES AND LONGITUDINAL BENDING

MOMENTS FOR SHIPS IN ARBITRARY TRIMMED

CONDITIONS

by

ANDERS SVENNERUD

GOTEBORG FEBR. 1964

Technische Hogeschool

Deift

CHALMERS TEKN(SKA HOOSKOLA

WITH APPLICATION TO CALCULATION OF

SHEAR FORCES AND LONGITUDINAL BENDING

MOMENTS FOR SHIPS IN ARBITRARY TRIMMED

CONDITIONS

by

ANDERS SVENNERUD

Report from the Division of Ship Design, Chalmers University of Technology.

Gothenburg Febr. 1964 Anders Sven nerud

The present paper deals with some fundamental, mathematically

describable properties of the hull displacement distribution,

which don't seem to have been treated previously in ship

technical literature

The practical application is chiefly aimed at the calculation

of still water shear forces and bending moments in ships of

full form with long parallel middle body, in arbitrary trim

conditions.

In order that the computing method may be used in the most efficient way, the hydrostatic curvos should be supplemented with the following coefficients:

Prismatic coefficient, _p.

Position o± the half-section, h according to eq. (io).

Shift of the half-section, in terms of the shift of the

centre of buoyancy, at a small change of trim, according to eq. (18 a).

The buoyancy double-moment factor at the half-section, i.e. the minimum value

All these coefficients can easily be computed together with the usual hydrostatic data, and they are sufficient for an exact calculation of the shear forces and bending moments

within the parallel middle body. To make an application of

the method possible also at an early stage of the design, one chapter of the paper deals with calculation of approximate values of the relevant hydrostatic data.

In an appendix (App. iii) the possibilities are discussed of

estimating also the wave induced bending moments and shear forces, in order that the total load distribution on the hull may be determined.

The author wishes to express his sincere gratitude to

Mr. Rutger Bonnet for valuable help with the final compo-sition of the paper.

Buoyancy integral and double-moment functions.

Introduction. Definitions etc0

Appendix II.

Practical applications. Shear force and longitudinal bending moment calculation. Weight distribution

functions

...o..

Appendix III.

Some notes on calculating wave-induced shear forces

and longitudinal bending moments

...lu/i

REFERENCES

J.M. Murray: Longitudinal bending moments. - Trans. lESS, Volume

90,

Febr.

_1947.

A. Mandelli: _{Quick calculation of longitudinal bending} moments. - Shipbuilder, Nov. 1956.

S. Mate0ika: A simplified method of calculating the longi-tudinal still water bending moment at any point in the ship's length. - European Shipbuilding9

No. 3, 1961.

U. Wegner: Uber Näherungsformeln fUr Kurvenblattwerte. - Schiffstechnik9 Band

_7,

Heft

36, 1960.

See also references for appendix III, _{page 111/7.}

11/' 1

Functions for the buoyancy of a parallel middle

body when the ship is on even keel ₇

nctions for the buoyancy of a parallel middle

body when the ship is in a trimmed condition ₁₁ Approximate functions for the buoyancy outside

the parallel middle body 16

Some approximate hydrostatic data to be used in

the preliminary design calculation ₁₉

Notation ₂₄

Appendix I.

Introduction. Definitions etc.

An arbitrary weight or buoyancy distribution _{may be represented} by the o function in figure 1. The area enclosed by the given

function and the x-axis represents the total weight, W, or

buoyancy, L The centre of gravity of the weight or buoyancy

is represented by the centre of gravity of the area mentioned. The statical moment of the weight or buoyancy with respect

to an arbitrary section x-x is represented by the statical

moment of the area with respect to the corresponding axis, _i.e.

the algebraic sum of the moments of the two parts forward and

aft of the section, when the lever is given different signs forward and ai-b. The numerical sum of the moments may be given

a special name and is here called the double-moment and

designated 1V11\II. _{It may also be called the "Murray-Moment" after}

J.Id. Murray, who was the first one to practically use this concept in calculating longitudinal bending moments of ships'

hulls. The double-moment may be given non-dimensionally by the double-moment factor , according to the following equations:

(i)

where L is a representative length. It is suitable _{to use the} length between perpendiculars of the ship for this purpose.

The area aft of the section X-X in figure 1 _{represents the}

weight Wa or buoyancy

a aft of the section, i.e. the weight or buoyancy ci function integrated from aft to the section

mentioned. This weight or buoyancy may be given

W

L

where W and A are the entire weight or buoyancy. 1±' W represents the total weight of the ship9 i.e. both light weight and dead-weight, obviously W = A . I-Iowever9 it may often be preferred

to separate the constant light weight of the ship, W, and

the variable deadweight9 WDW. Obviously the double

moment-factor and the integral moment-factor for the total weight W = A

then may be given by:

WLW _DW = .LLW

;:-

+ DW

T

W W 'DW DW A A where ; and

DW DW are the factors for light weight

and deadweight respectively.

Having thus calculated the integral and double-moment factors for weight and buoyancy separately, the shear force, T, and longitudinal bending moment, M, in an arbitrary section x-x

of the hull girder is given by:

'T

= A

- liA)

M = A L

-Also the shear force and the longitudinal bending moment may

be given non-dimensionally by:

= rW - (4a) (3) (5a) Aa HA A

The non-dimensional calculating system is finally completed by introducing a non-dimensional length coordinate , given

by

X

L

This means, e.g, that the position of the centre of buoyancy,

centre of flotation, half-section (see below) etc. may be given as a non-dimensional coordinate and the real distance is then to be found by multiplying the non-dimensional co-ordinate by the ship's length L.

It may be noticed that the well-known relation between T and.

M, M ₌ $ T dx, can be given in the non-dimensional system by:

M' _{= j T = j}

- rlL) d (7)

Viewed over the ship's length, the i and factors are

continuous functions, characteristic for the actual weight or buoyancy distributions.

It can be mathematically shown, that the following

relation-ship exists between those two functions:

(6)

This means that the double-moment function has a minimum

where rl = , i.e. for the value of F, which corresponds to

that section, where the weight or buoyancy is divided into two parts of the same weight. This section is in the

follow-ing an essential concept. It is designated H and the

non-dimensional coordinate with respect to the half-length L/2 is called h. The section is in the following called

"half-section".

It may be noticed that the moment of' inertia I, and the double-moment IllVI, are analogous, as both have a minimum

valuo. I has its minimum for the axis of centre of gravity,

IM has its minimum for the axis of half-section.

Further it is essential to note, that the double-moment at a section outside the weight distribution area equals the statical moment of the area with respect to the actual

sec-tion. Accordingly the p value at the ends of the area equals

the distance to the centre of gravity in the non-dimensional

system, and the derivative gets the values - i and + i at

the ends.

In the following there is also use for the derivative of the

r function, which is called X.

0O0tO

(9)

Obviously the X function gives a representation of the

original weight or buoyancy distribution function q.

Procedure for calculatin

ï

and functions.

The practical calculation of the 'rl function may be clear by the the above mentioned, especially in according to eq. (2). Thus we have:

i P

j q dx

where W is the entire weight or buoyancy.

The statical moment of the weight aft of section x-x with

respect to that section is

x X

The quantity

f

r d is in the diagram of figure 1 given by

the area, enlosed by the _n

fction, ç-axis and the section

- _{, i.e. the arca A' Q Xt A}

In a similar way it can be shown that the statical moment of the weight forward of section

x-x

with respect to that same axis is

(M

fx

)

= w

L r (i - n) d i

The negative value of the last integral is in the diagram

given by the area F" Q X" F".

Since the double-moment is the numerical sum of

and (Mf) the non-dimensional double-moment factor for

the section X-X may be determined by measuring and summing

the two above mentioned areas in the diagram, thus:

= Area Q X' A' Q ± Area Q X" F!! Q

In accordance with the above the minimum value at the half-section is given by:

= Area H H' A' H + Area H H" F" H

It can easily be shown that having thus found the minimum

value the value at an arbitrary section may be given

by adding to the value twice the area H Q Q' H. In the

same way the p value at a section aft of the half-section

is found by adding twice the area H F F' H. Generally we can write:

L

+2A

X O

X'

where A is the area enclosed by the n function, the

Pinally it may bc mentioned that the contre of' gravity distances from the ends in non-dimensional form, p and are given by:

- Area A' FU AU J?

= Area A' F" F' A'

The various "non-dimensional" areas9 mentioned above, are

found 'ay dividing the actual measured area by the whole area A' A" F" F' A'. If different scales are usad for

and

î,

this arca is a rectangle instead of the square shown

Functions for the buoyancy of a parallel middle

body when the shifl is on even keel.

Figure 2 shows a sectionaa area curve of a ship with a

parallel middle body. The prisma-tic coefficient of the fore body is and that of the after body is defined

by: V V F A cpp - A

-to2JJ

A2L

where VF and are the displacements forward and aft of

the L/2-section respectively. A is the midship sectional area and L is the ship's length between perpendiculars. Since VF and VA includc the displacements outside the per-pendiculars, CPp and _CPA represent the entire displacements

of the two parts. The total prismatic coefficient for the ship is

= _{( + CPA)}

The exact position of the half-section H of the buoyancy can easily be shoa to be:

h _{- (WT1}

-'-

o,.... ...

where h is the non-dimensional coordinate with respect to

L/2. (Positive forward and negative aft of 112.)

In the - r system the non-dimensional integral function is shom and obviously it intersects the line î = at the

distance h from L/2. (Point H).

Within the parallel middle body the differential dn is

given by:

dV A dx

CPLA

Since dx = L d, the derivative 0±' the -r function within

the parallel middle body is:

Accordingly it is very simple to draw the buoyancy integral

function for the parallel middle body in the non-dimensional - n system, because it is a straight line with the constant slope i/ and goes through the half point H. The distances Ht P and H" Q are both cqual to -- p.

Both the minimum valuo for the double-moment function at

the half-section and the non-dimensional centre of buoyancy

coordinato h can be very accurately calculated by means of the two shaded areas, A and AA, shown in the _{- n diagram.} These areas are enclosed by the n curve, the line PQ and the

two lines n = O and n = 1.

According to the previous chapter the following can easily

be shown to be valid:

i

po = : rp + A _{+ AA}

b = h + A _{- AA}

It is to be noted that since AA and A include the integrated displacements also outside the perpendiculars the above

equations (12) and (13) give exact values. In fact equation

(13) points out a way for a very accurate determination of

the centre of buoyancy position, having in mind that equation

(io) is an exact expression of the half-section _{coordinate h.}

Integrating eq. (ii) and using the half-section as origins

i.e. _{= -} for = 0, we get the following expression for the buoyancy integral function within the parallel midship body:

...(ii)

Integrating this equation and using the relationship between

and given by eq. (8) we get the equation for the

func-tion. Still keeping the origin at the half-section, where

= we get the expression for the buoyancy double-moment

function within the parallel midship body.

= +

i2

a e e eOet

000..G0.t.*...SS a

O

a 0

(15) Note: In eq. (14) and (15) is the non-dimensional coordinate with respect to the half-section H (+ forw.; - aft).

If the double-moment factor is calculated with respect to the L/2 - axis, and that value the minimum value at the half-section can he calculated according to eq. (15), putting = - h; thus

12

= /2 -

h

lic:

a.000 e

e ...oes....

IC)

Provided that we know the corresponding functions for the

ship's total weight, we can obviously carry out an exact

calculation of the shear forces and longitudinal bending

moments within the parallel middle body by means of the above

equations. In most cases the maximum bending moment falls

within the parallel part, and the equations concerning the

double-moment given above may be sufficient.

Outside the parallel part, within the "entrance" and "run"

parts of the ship, the approximate functions given in a

following chapter can be used for determining especially the shear forces in these regions. The approximation f orimlas

are the same for a ship in a trimmed condition as for the

ship on even keel, but the initial data must obviously be different.

and R, (see figure 2) are known, the non-dimensional distances mf and ma (both positive) from those points to the half point may be determined. Then the initial data for the above

men-tioned approximations for even keel conditions are:

A2roximation forward: Approximation aft:

= _{- b} = ± ! (mf)2 = + mf. E cp =

_{+ b}

= R o co a R

__ma

Note: The above expressions for and means the centre of

buoyancy distances from the perpendiculars. If there are

displacement parts outside the perpendiculars these expressions only approximately equal the double-moment factors at the per-pendiculars.

In the following chapter it will be discussed how the buoyancy double-moment function changes when the ship is trimmed. In

appendix I it is shown that the

value is constant for one

section only, the centro of flotation section. That constant value, may be an initial value for the calculations con-cerning longitudinal bending moments in the trimmed condition.

If the centre of flotation has the non-dimensional coordinate c with respect to L/2 (+ forw.; - aft) and the corresponding coordinate for the half-section is h, we can easily get from

the previous equations:

2 1 2

=

-

- e)

= I/2 + - (e - e h) (17)

/

o

AF

5

lo H L

_rr

15 4'

A

Functions for the buoyancY of a arallel middle

body when the shi is in a trimmed condition.

Note: The equations given below, which are deduced in

appendix I, assume the following:

The ordinary laws for trimming are valid.

No part of the bilge in way of the parallel middle

body is above the water line.

The ship's sides are vertical in way of the parallel

middle body.

Figure 3 is intended to show what happens when the ship is

brought into a trimmed condition.

The ship pivots around the centre of flotation axis C, which

has the non-dimensional coordinate o with respect to L/2. Thus the draught at C is unchanged = the mean draught d.

The centre of buoyancy moves from B to B'. The position of B' is the same as the centre of gravity of the actual total weight, which can easily be calculated when the load con-dition is known. The moving distance in non-dimensional form is designated CB* Obviously

E L means the buoyancy trimming

lever, which as well as the total trim t = d _{- dk can} be

calculated if' the corresponding hydrostatic data are known.

The half-section moves in the same direction as the moving

of the centre of buoyancy. The moving distance in

non-dimen-sional form is designated Ç and is usually somewhat greater

than

The exact value of can be calculated according to appendix

I. For practical purpose the following approximate expression means a handy but very accurate value, valid also for great

(SL

t

ç

I d B

t means here the a'osolute value (without regard to the siga) of the total trim dF _- d.. d is the draught in the

correspond-ing even keel condition, i.e. the draught at the centre of

flotation C.

The non-dimensional factor is an essential concept for the purpose of calculating shear forces and. longitudinal bending

moments in a trimmed condition. The factor refers to the

properties of the water-line arca. L is the ship's length, I

is the total moment of inertia of the water-line area with

respect to the centre of flotation (= centre of gravity) axis. S is the statical moment with respect to that same axis of one of the two parts of the area forward or aft of the centre of flotation axis.

For normal trim, however, the following simple equation may

be used.:

SL

I

The half-section position in the trimmed condition is expressed

by the non-dimensional

coordinate h' with respect to L/2,

thus:

h' = h t O

O ...O0

...(19)

where h is the half-section coordinate in the corresponding

even keel condition. The plus sign refers to trim by stem, i.e. dF > dA and the minus sign refers to trim by stern.

In the following we will use the new half-section H' as origin for the non-dimensional length coordinate ' within

the parallel middle body of the ship in the trimmed condition.

cp

The integral function, which is designated r' in the trimmed

condition9 is no more a straight line in way of the parallel

middle body but a curve of the second power, with the

follow-ing slope equation:

X - = A + B ' (20)

In this equation the coefficients A and B are:

dr_dAl

i +

(h'

- c)

-+

(h'

-

e)

kL

d-c

B=

dF_dA

E

In the above expressions the different terms mean:

= prismatic coefficient in the corresp. even keel conci.

e = centre of flotation coord. with respect to L/2.

h' = half-section coordinate U il fi

=

draught at

the forward perpendicular.

dA = fi ft aft ii

ci =

in

the corresp. even keel condition.

c = ci (i _{- ).}

Note:

c is constant for the ship, see fig.

= mi.dsh, area coeff. in the corresp. even keel cand. k = moment of inertia coeff. for the waterl. area

L

_BL

=

trimming

lever in non-dimensional form. Plus

sign

to be used for trim by stem9 i.e. d > _dA, minus

sign to

be used for trim by stern.

= non-dimensional length coordinate with respect to the

actual half-section in the trimmed conci.

(

forw.; - aft).

The equation for the integral function r' itself is found by integrating the above eq. (20) and putting

r' =

for

= O as a boundary condition. Thus we get:

In an analogous way we get from the above equation and eq. (8) the equation for the double-moment function in the trimmed condition:

= + A

()2

(y)3

(22)

As a boundary condition we have here used the uidcnown minimum value for the actual half-section, whore ' = O. To get

this value it is necessary to know one value at some point within the parallel middle body. In appendix I it will be

shown that the double-moment factor for the section of the centre of flotation is approximately independent of the trim. Thus:

By means of eq. (17)

c can be calculated for the ship in the

corresponding even keel condition, and consequently also

is known. We can now use eq. (22) in order to get the minimum

value p' by putting ' = - (h' - c). Thus:

o

i

(h' c)2 + (h'

-dF

.. .

(2)

Since (h' - c) for normal trim is a rather small quantity,

we can with sufficient accuracy neglect its third power. Thus

we get the more simple expression:

- i _(h' 2 - o)

c _ç

...(23)

(24a)

According to the above equatioc tbe.intgral eaJ

douhl-moment functions may be determinated within the parallel middle body. Outside the parallel part the approximate

func-tions given in the next chapter may be used. The initial data for the approximation functions, can easily be found by

buoyancy position and the non-dimensional distances m' and

m'a from the ends of the parallel part to the actual

half-section H' . Thus we get:

Approximation forward: Approximation aft:

= ' + A (mtf)2 ±

(m')3

= + b'

= + A m'f + (mtf)2 = _{- A m'a +}

(m'a)2

XE = A + B m'f

XE = A - B mf

Note: rn'f and m'a are both positive, b' is the centre of

buoyancy coordinate with respect to

L/2.

E) _D

= + A (m' ) - a (m'

)3

A. P

H

H'

t

-Approximate functions for the- buoyancy outside the parallel middle body.

Then the actual buoyancy distribution outside the parallel middle body is not known, exact calculations of the integral

and double-moment functions can obviously not be carried out.

Referring to the previously mentioned it may, however, be

clear that several boundary conditions can be connected to those functions. In fact, the number of boundary conditions are sufficient for very accurate approximations.

G-enerally the double-moment factor t can be written as a func-tion of the non-dimensional length coordinate . The integral

function and its slope function X are related to the p func-tian according to eq. (8) and (9). Referring to figure 2 the ends of the parallel middle body are designated E and R.

The point E is chosen as origin for the functions within the

uentranceu part of the- buoyancy. The entrance length in the non-dimensional system is designated e- and the non-dimensional length coordinate, which is positive- forward of' E, is

Thus we have, if' any buoyancy forward of the forward per-pendicular is neglected:

The five boundary conditions above can be fulfilled by a

-polynominal of the fourth degree. Thus we get the following

equations for the r and functions forward of the point E:

±A

2+B

e3 e e e e (25)

+ (2E

+ XE 2 A + B _(25a) e 3 e e 2 e e = f 1 1 d jL X

Y1l

= = = E XE for ) for e e = = O e

11_2

_2d

2

_!

e 2 de-2

where the coefficients A and B are: e e e -

C -

_XE e2 + 2

E]

B

=_[_e+4EeXEe2_3(F_E)]

e

For the approximate functions in the in part, i.e. abaft the point R, we choose that point as origin. The length co-ordinate is now positive abaf t the point R. The length of the

"run"

part is desiaated r in the non-dimensional system.

Thus we have, if any buoyancy abaft the aft perpendicular is neglected:

LLR

= = R for r

=0

X=XR

_J i ldp. t

I

-

2d2

=O

j

fori =r

r r

Thus we have for the functions aft of the point R:

_Brr3

(26)

+ (1 _{- 2nR)}

r + XR r2 3 Ar + 2 Br r (26a)

The coefficients Ar B are:

A =

[_2r+3flRr_XRr2+2

(A-R)]

r 3

r

B=

{+3r_4

_{Rr+ XRr _3 (A-R)]}

The above equations, which fulfil all available boundary

conditions, can also be used for ships without a parallel

middle body. The non-dimensional distances e and r then means the non-dimensional distances from the perpendiculars

to the half-section;

e and r means the non-dimensional coordinates (both positive in this case) with respect to the

half-section. All equations concerning the position of the

half-section, both on even keel and in a trimmed condition,

which are deduced on the assumption of a parallel middle

body, e.g. eq. (io) and (18), are then obviously only

approx-imate. Finally is _{riE = =}

E = = and XE = =

Some approximate hydrostatic data to be used in the preliminary dcsign calculation.

In the previous chapters there aro used sorne concepts, which

normally are not considered at the ship design. These are thé

half-section position in the even keel condition, h, and the 'stress" trim factor, SL/I. In the following, approximate methods of dete.inining these factors for use at preliminary

calculations, will be described. The buoyancy double-moment factor for L/2,

L/2' as well as the waterline moment of

inertia coefficient, kL, are nowadays usually available in

sorne approximate form, but also these factors will be further discussed.

The half-section position.

The centre of buoyancy coordinate b with respect to L/2 is assumed to be known in the even keel condition. In order to get an approximate relation between b and the half-section coordinate h, we may proceed according to the following.

In figure 4 the length L* represents the total buoyancy length, i.e. the actual waterline length. The midpoint of that length has the non-dimensional coordinate q with respect to the

ordi-nary L/2, where L is the length between perpendiculars.

Sup-pose that when the centre of buoyancy is located at I*/2 (the

half waterline length) the sectional area curve is

approxi-mately symmetrical with respect to L*/2. Consequently the half-section also is located on L*/2. Now let the sectional area curve be unsymmetrical by moving all points the distance x, lincarely decreasing from 2 (h - q) to O according to the

figure. I-t can easily be shown that the non-dimensional half-section coordinate with respect to the ordinary L/2 for the

unsymmetrical sectional area curve now is h. The centre of buoyancy coordinate with respect to L/2 is now b and the

moving distance oC the centre of buoyancy is b - q. According to the figure we have:

2(h_q)A®

b-q

- z'

where z is the centre of' gravity distance from the baseline of' the area, enclosed by the sectional area curve. An in-vestigation of' different sectional area curves has sho that the following approximate relationship exists:

i + Cr)*

A

L

4 ' S

'L*

where ç fundamentally should be based on the actual buoyancy length, but for our purpose it may be sufficient to use the

ordinary value of ç', based on the length between perpendi-culars. Thus we finally get an approximate relationship bet-ween the half-section coordinate h and the centre of' buoyancy

coordinate b:

hb1

(27)

where q is the non-dimensional coordinate with respect to L/2

of the midpoint of the waterline.

The above expression means a rather good approximation. The difference between the exact value according to eq. (io) and the approximate value is normally less than 1/4 of the

ship's length. Obviously eq. (27) may be used for approximate

determination of centres of gravity for different areas of

similar types, where the exact position of' the half-section can be calculated according to eq. (iO)

Aroximatc roerties of waterlinEs and. sectional area

curves.

The three curves shown in figure 5 may represent different

types of' waterlines or sectional area curves. Curve "E" is an ellipse, "P" is a parabola and "DP" consists of' two parabolas. Thus the three curves have mathematically determinable areas,

21.

moments and moments of inertia with respect to the x and y

axes. At any ratio the moment and moment of inertia ca be calculated and graphically represented versus the three

coefficients of fineness. For intermediate values the moment and the moment of inertia can be obtained by interpolation. The double moment of a sectional area curve and. the moment of inertia of a waterline area can then be approximately

cal-culated with respect -to I/2, assuming that the curves can be

made symmetric according to figure 6. The results of such

calculations are non-dimensionally represented versus the co-efficients of fineness in the figures 7 and 8. Figure 9 shows

the factor SL/I, where S is obtained from figure 7 by putting

S = IVII'iI and c = . I is calculated according to figuro 8.

Obviously the ratio of parallel length to ttal length, has a great influence on the results and means in all cases

an essential parameter.

The double-moment factor with res2ect to L/2.

Approximate values of the buoyancy double-moment, which are dealt with in the literature, aro generally based on the block

coefficient and no consideration io paid to the influence of the parallel length. It may be clear from the above that the

prismatic coefficient ç means an adequato base, and the length cf the parallel middle body is an essential parameter for

approximate determination of the buoyancy double moment factor

Figure 7 can be used for the purpose, but it may be

noted that no real sectional area curves aro used for the

diagram.

The double-moment factor with respect to L/2 for full load

displacement may be taken from the following expression, valid for normal ship types

L/2 0,250 - 0,258

/' (i

-

(28)

If an accurate value is known at a fixed draught d, where

the corresponding prismatic coefficient is the value at an

can be calculated according to:

- 0,25 (ç - ₍₂₉₎

This equation may be deduced from the mean slope of the

different curves in figuro 7. It is noted that the parallel length of the buoyancy is independent of the draught.

The following relationships between the different coefficients

may be recollected:

ç = = i - (i _{- (30)}

ç = prismatic coeff.

ô block coeff.

= midship area coeff. d. = draught

Trim calculation.

'L

The longitudinal metacentric radius = may be given in nondimensional form by a coefficient rL, according to:

Li

rL ==kL

T

where k _{is called the moment of inertia coeff.}

L

_LB

Approximate values of kL, acc. to the above mentioned, are

represented in figure 8. For normal waterline forms the following expression may be used, where is the waterline area coefficient.

lcL 0,083 - 0,23 (i - (32)

The buoyancy trimming lever, i.e. the distance between the

±

CB

centres of buoyancy (or the weight's centres of gravity) in the even keel and the trimmed conditions, is previously

designated ±

B in the non-dimensional form. Assuming the longitudinal metacontric height approximately equal to the longitudinal metacentric radius ML, we get from the

com-mon trimming law:

t=dF_dAL

rL

where the plus sign is to be used for trim 'by stem and the minus sign for trim by stern.

Note: If the actual waterline moment of inertia coefficient

is available, it is not necessary to calculate the trim

itself, when the only purpose is a stress calculation. See expressions for the coefficients A and B for use in equations

(20) - (22) and comp. eq.. (18a).

The "stress" trimfactor sii/i.

The non-dimensional trim factor sii/I is, according to the previously mentioned., represented in figure 9, based upon the waterline area coefficient . For normal types of

water-lines the following approximate expression may be used for

the determination of that factor:

1,50 + 1,6 (1 _{- (34)}

y y

¿

LÇ

'ç'

XlIk.+

x4)

4

6

0.2 O.2Z

aDt

O.2o o. 0.1 0.1

7

Buoy

1

4

//

i

H

4 ii#e4

ì'e/1r

h

I-.

. * . . .

o. g

C.70 Ö.7.Ç O.g coe

r

0oZ

0.01

0.0 0.05 0. 0.

o.c

070

0.75 ¿.SS O.9ø _feet

citA ¿.*.//._O

/

1 o.

//

*,.

///

/

-y'

rje hr/.

oo.l,o,

i

. . . . . . F

o L 4 7 SL

I

N I

N

-Li

- &

-

ry-i í r

r

-1 -r T i I I r f i

rr

i r----i -i---i---v-o,

o."-o. O.go

o.o

Ô. E S

Notation

Dimensional

L length between perpendiculars E breadth of ship

d = draught, mean

= at forward perpendicular

T? ?t _aft

t = total trim

e _{small distance; see figuro 1:2, appendix I}

x length coordinate

W = weight

= buoyancy (weight) V = displacement (volume)

= weight pr unit length

M

= moment, statical or longitudinal bending moment = double-moment

T = shear force

S = statical moment of an area I = moment of inertia

A = area

Non-dimensional

= prismatic coefficient

o = block coefficient

= waterline area coefficient sectional T?

= double-moment function

= minimum double-moment factor at the half-section = integral function

X = slope of n-function

= length coordinate

A = area in the - r system h = half-section coordinato

b = centre of buoyancy coordinate

o = u T?

flotation U

= shift of half-section at trim = moment of' inertia coefficient

m = distance from half-section to ends of parallel part

e = length of entrance part

r run part

*

M = bending moment coefficient

= shear force coefficient

Subscri2ts etc.

L/2 = middle point o± length between perpendiculars

F forw. perpendicular or part forw. of L/2

A = aft U U U _aft f!

E = forward point of parallel middle body

R = aft U t!

C = centre of flotation

H = half-section

LW = ship's light weight

DW = " _deadweight.

Trim condition is denoted: ', e.g. H' ' ' etc.

9 -u- 5

APPEI1DIX I

Deducing of' equations.

Relation between the p and functions. E. (8).

See figure 1. According to the definitions the double-moment

at the section x-x is:

MM=Wg+Wfgf

where Wa and Wf are the weights aft and forward of the

sec-tion X-X respectively, and gf' are the centre of gravity distances of the two parts to that section. With respect to an axis at the small distance ix from X-X the double-moment

is:

-MM

=W (g+x)+qxf+Wf(gf_x)-qx

2 x+Ax a a = MM + (W - Wf) x + q

(x)2

x a

The double-moment difference at the two adjacent sections

therefore is: = (Wa - Wf) x + q

(Lx)2

; thus:

-W +qx

a f MM

um

MM

òxx,Ox

a

f'

Thus the derivative of the double-moment at any section equals

the difference between the weights aft and forward of that section.

2 A®

We have: V

= F A® and VA = A A® thus:

h

= F - CPA)

Trim half-section position. Eq. (18) and (18a).

Eq. (10)

See figure 1:10 The position of the half-section is known for the ship on even keel, The umknown shift is called L.

According to the elementary theories concerning ships' trim the draught is unchanged at the centre of flotation section

C The volume of each of the wedges C A A and C F P! is:

'0 =

_àx

Wa W

SinceWf=W-W

wehave-1=1-a _W Therefore:

w -w

a - 2 - 1 Eq0 (8) X

Even keel half-section position. E. (io).

See figure 2. The displacement aft of L/2 is VA and that

for-ward of L/2 is VF. The sectional area A® is constant over the

parallel middle body. If VF > VA we will find the half-sec-tion forward of L/2 at the distance h L from L/2. The

dis-placement aft of the half-section is now VA + A® h L and that

forward of L/2 is VF - A® h L. These two displacements are equal because the half-section divides the entire displace-ment in to equal parts. Thus we have:

VA+A®hL=VF_A®hL

- V

f

xydx=S

C C

where S is the statical moment of each part of the waterplane area forward or aft of the centre of flotation axis with

re-spect to that axis, t is the total trim = d

- d.

The volume of the small wedge C P !

where S( is the statical moment of the waterplane area between C and Hwìth respect to the centre of flotation axis.

Obvious-ly

S is a function of the unknown

The ship's total displacement is V and consequently the

dis-placement aft of the original half-section H in the even keel

condition is

Assuming the ship with a trim by stem we obtain for the dis-placement aft of the new half-section H:

V =+iLA®_V+vç

_{a 2}

Since H' divides the displacement in two equal parts, _{Va =} = . Thus we have

HLA®VVÇ

(1:3)

In the above A® represents the sectional _{area for the mean} draught d, i.e. the section at the centre of flotation.

Equations (1:1), (1:2) and (1:3) give:

ss

(1W, H

AL

HT -t;

ç xydx=b

(1:2) C

The shift of the centre of buoyancy is

B L, which is

calcul-ated by means of the statical moments with respect to C ol' the two displacement wedges C A A' and C F F' , thus:

-t

CB L V = mom, of wedges

-I

t CB ₂

AL

p

Thus we have from eq. (I:'l-) and (1:5):

(s - L I CH _SL

DT2

₂ S ₌

_{(h_c+CH)}

2

From eq. (1:6) and (1:7) we have:

5.i=

BL3

i

(h-c+ç

21

From eq. (1:5) and A = E d we have:

F A

2 2

J x y dx + $ x y dx

C C

where I = moment of inertia of the entire waterplane area

with respect to C. Thus we get the well-known expression:

CB L and with V= A® L

When the trim is small S may be neglected; therefore:

t

(1:5)

(1:6)

Eq. (18a)

Then the trim is great SC must not be neglected. Within the

parallel part of ship we have:

(1:7)

2

-

-The above equation (1:9) means the exact expression for the

half-section position at trim. However, this equation is not

very convenient for practical use, but vie can get an

approxi-mate solution in the following maimer.

The original even keel half-section H and the centre of

flota-tion C are in practice rather near to each other and.

may be

assumed to coincide. Thus h - c

= O and we get

t

Figure (1:3) gives

versus the factor

_B

for sonic

cliffe-rent values of

c.

Besides the relation ô =

the following approximate

rela-tionships between the different coefficients

are hereby used:

= 0,93 + 0,07 cp

;

2 1

= 1,50 + 1,6 ce

(i

- e

It is seen from the figure that the difference

_{curves can be}

approximately replaced by straight lines with the slope

=

= - cp. Thus eci.

(1:10) can be replaced. by the following

simple

expression

CH _SL

t

_SL

t

B

Eq. (18)

SL

it

h-c

H2

(I:D)

SL i

H2t

(1:10)

d.

-c

A =A

z

d-c

Thus we have from eq

(I:3) with V

_{= A®} L cp:

The slope equation of the buoyancy integral curve. Eq. (20).

Figaro 1:2 shows the midship section integral curve (Bon Jean

curve)0 For a ship with vertical sides this _{curve is a straight}

line above the range of the bilge. The continuation of the

straight line intersects the vertical axis at the distance

from the horisontal baseline0 Within the parallel middle body

the sectional area at an arbitrary draught _{d may be written}

. . . (1:11)

The half-section at trim FU is assumed to be kno _{and is}

chosen as origin for the --coordinate system.

The draught at an arbitrary section within the parallel middle body for a trimmed ship is then

= d ± [(h' -- o) + ] (d-r _{- dA) (1:12)}

where d is the draught at the centre of flotation, which _is

assumed to be constant and dF - dA is the total trim. Prom the above equations (1:11) and (1:12) _{we get the}

ex-pression for the sectional area at the section :

= A® i + [(h' - o) +

J

d

}

(I: 13)

The slope of the buoyancy integral curve _{may be written:}

um

L>0

= =

{ 1 + [(h' - c) ±

d

.... (1:14)

Substituting '

for and

î'

for

î,

it is seen that this equation is the same as Eq. (20).

The slope of the straight part of the midship section inte-gral curve is:

B òd g-a--- a

From figure 1:2 we also have:

-

; where A may be written:

A =Bd A

_e

z z

e ==d

(1-B Z Z

The expression d - e in eq. (20) thus may be replaced by:

d - e = d _(1:15)

The centre of flotation doublemoment factor. _{Eci. (23).}

IVB1' MM - (M _{) + (M}

)

e e

wa

wf

where A =E d

z z z

See figuro 1:1. The displacement doublemoment with respect to the centre of flotation axìs is assumed to be Imown for the ship in the even keel condition and denoted MM0. Then the ship is in the trimmed condition the doublemoment with

respect to that same axis is:

(Mw)a and. (M)±' are the moments of the two displacement wedges C A A' and C F F' with respect to the centre of flotation axis C. According to the figure and the elementary trim theories we have: P

(M)

_wf

$ yx dx=I±'

2 t C (I: 17) À

war

5

yx2dx=I

a C

where I and I are the moments of inertia of the two

water-f a

plane parts forward and. aft of the centre of flotation axis with respect to that axis.

The non-dimensional double-moment factors at the centre of flotation in the even keel and trimmed condition respectively

may be written:

c , o

L=-

; =

IJL

VL

where V is the ship's displacement.

Thus we have from the above with t = d, _{- dA.}

,

d -d

I

-I

F A i' ' L + C C L

VL

(I: 18)

The above equation moans the exact expression for the dis-placement or buoyancy double-moment factor with respect to

the centre of flotation axis in the trimmed condition.

The well-known expression for the ship's longitudinal

mcta-centric radius can be written

Since is of the same order as the ship's length L it may be seen that the expression

- I

I a

V L

has a very small value. Since also the trim dF _{- dA is rather}

small compared with the ship's length L it is seen that the

last term in equation (1:18) can be neglected. Thus we finally

have:

1)4

-o

liz

-o

1,1 1)0 C 0

0,05

_0»o

Q,j t" 1'

/,jure I:Z

W

e

'\'\\

s.L.

_Vz

- L

/

+ i

'&L

I

1:3

d

Practical ap'lioation.

The practical application of the previously mentioned theories

can be made in different ways. The following intends to give sorne illustrations.

Shear force calculations.

Pig. 11:1 shows a simple longitudinal view of a ship. The ship's light weight distribution is assumed to be known. At each bulkhead, which bounds important holds or tanks, the

light weight aft of the corresponding section can be given

and is here designated

a' b' c etc. Thus Pg in this case

means the entire light weight. In an analogous manner the

distribution of the deadweight can be given as

a' b' 0 etc.

and consequently Q means the entire deadweight. Thus the

total weight distribution is given by W

= + which

means the total weight aft of an arbitrary section x. Wg means the total weight W, which also is the ship's buoyancy A.

The nondimensional weight integral factor at an arbitrary section x is now easily found by:

2 +0

X 'X

nW =

A

and the weight integral curve can be drawn in the corresponding - r system. The curve is here shown as the broken line r.

This means that both light weight and deadweight _distribution

is considered constant between the bulkheads. If the holds

are very long or the actual load cannot be considered to 'oc evenly distributed over the hold's length it may be necessary

to use ordinates also between the bulkheads.

The buoyancy integral curve _{riA can be calculated according to} the previous theorieo. The calculation is of _{course very}

The shear force at an arbitrary section can then be read as

the difference between the two integral curves, multiplied 'by

the actual buoyancy thus

=

; T =

In the example shown, there are seven values of interest, i.e. the four maximum points '1-4 and the three zero points

I-III,

where the bending moment has its maxima. The two absolute

shear force maxima both fall outside the parallel middle body9 but it may be noted that the difference between 'the actual

buoyancy integral curve and the continuation of the middle body curve generally is very small even rather far from the

end points of the parallel iart. This means that in many cases

it is not necessary to use the more accurate, but a little

laborious approximate formulas for the "entrance' and "run" parts, which are given in the foregoing.

Bonding moment calculations.

For estimating ships' longitudinal strength the longitudinal

benclìng moment ìs nowadays mostly calculated with respect to

the L/2 - section only according to the method given 'by

LIurray. In most cases this simple calculation method is fully sufficient from practical point of view, duc to the fact that the niaxinmm value appears very near amidships at normal load distributions.

In many cases however, it is of interest to know how the

ben-ding moment varies over a longer part of the ship. This is

especially the case when the load is very unevenly distributed, e.g. according to figure 11:1, where the bending moment curve will have both maxima and minima rather far from amidships.

In such cases the longitudinal bending moment at the

charac-teristic points may be calculated according to the following.

The longitudinal bending moment is first calculated with

respect to L,/2, thus we have the "non-dimensional" moment amidships

Seo equations (3), (5) and (5a).

The "non-dimensional bending moment at an arbitrary section

may now be calculated by means of cq. (7) and by use of the fl-curves in figure II:L

Thus we have for the moment at section I:

M1

_{= ML/2}

+

L2

(flr - fl)

The definite integral means here the shaded area between the

two ri-curves and the L/2-line. To got right value of the ben-ding moment it is essential to note the sign of the L/2

-bending moment and the sign of the integral If > , the

longitudinal bonding moment is positive and means a hogging moment. The shaded area is in this case negative, because

but the integration is made in the negative direction, whereby the integral itself will be positive.

In the sane way we get for the moment at section II:

M = M _{(rim - ria)} d

* *

The integral is here negative and means the negative area between the points I and II

Finally we have for section II:

T

Mr +

-This integral is positive since _{> r} and means the area between the points II and 1. The last two integrations are made in the positive direction.

maxima at the points I and II and a minimum at the point lOE. *

The "non-dimensional" moment M may be plotted in the figure *

when a suitable scale for M is chosen.

In cases of loadings similar to those, given in the above

example it is possiblo to state in beforehand that the maximum

shear forces appear at the bulkheads and the maximum bending

moments appear somewhere about the middle of the empty holds.

Therefore it will be sufficient to calculate the shear force

or bending moment at some certain sections only. Such cal-culations can be made directly by means of equations (3), (4)

and (5). In order to facilitate such calculations some notes on integral and double-moment factors for weight are given

below.

and p factors at arbitrari weight diotributions0

Fig. 11:2 shows an arbitrary weight (deadweight, light weight

or total weight) distribution curve. The weight aft of the L/2 - section is and that forward is WF thus WA + W is

the entire weight W. In most cases it is suitable to calculate

at first the factors with respect to L/2. For the L/2 - sec-tion we have:

L/2 = and L/2

_WL

For the integral factor at any section X-X, we obviously have:

+ wx = L/2

-

7

where Wx is the weight between the L/2 - section and the sec-tion X-X. The plus sign is used for secsec-tions forward, and the minus sign for sections aft of L/2.

The double-moment factor at the section x-x can easily be

shown to be:

M

= L/2 + (2 L/2 - i) ± 2 -z-W L

where x is the distance from L/2, positive forward and nega-tivo aft of L/2.

is the statical moment with resDect to X-X of all weights between L/2 and X-X0 This moment is always positive.

i and p functions for a special type of light weight distri-bution.

For tankers, bulkcarriers and other similar ships with

machine-ry aft, and which have no erections amidships, the light weight distribution often may be considered to be constant over a long

part amidships. This is intended to be shown by the weight

distribution curve in figure 11:3. The weight aft of L/2 is

hare WA and that forward of L/2 is W. The constant weight per unit length amidship is

Analogous to the case of a buoyancy in an even keel condition

we can here use the concept "light weight prismatic

coeffi-cient",

LW' For the weight forward of L/2 the prismatic

co-officient is and for the weight aft the coefficient is Thus we have: L o W A L 2

where WF and WA are the light weights forward and aft of L/2 respectively.

The total prismatic coefficient for the light weight is

ex-pressed by:

LW

(p +

The light weight half-section coordinate with respect to L/2

is expressed by:

1f

hLW=

Using the half-section as origin for the light weight for the

and function within the parallel part of the weight distri-bution curve:

Here is

LW the non-dimensional coordinate with respect to the light weight half-section HLW. The coordinate is positive

for-ward and negative aft of HLW.

If the double-moment factor is calculated with respect to L/2 the minimum value at the half-section is to be found by:

i

LWO

LVJL/2

The functions are shown in figure 11:3 and are obviously ana-logous to those of the buoyancy, compare figure 2.

The above formulas can in many casos be used for quick

cal-culations for ships of the typo mentioned. If there is a bridge or similar erection, its weight must be separated from

the light weight in order to make the weight distribution

constant over the middle part. It may be most practical to include such weights in the deadweight calculations.

i = +

-(11:3) LW i LW =

+ -

(11:4)

A simple numerical example.

The last mentioned calculating procedure may be iflustraded by

the following.

A bulk carrier with = 470' has the general arrangement according to figure 11:1. Its light weight, 4.600 tons with C.G.

28,2' aft of L/2,is distributed according to figure 11:3.

A load of 15.300 tons, distributed in accordance with figure

11:1 and the disposion given below, brings the ship into an

even keel condition.

From ydrostatic curves the following should be available at

the displacement in question, 19.900 tons.

cp=0,790

;

h=+0,022

;

L/2=0'2020

The buoyancy double-moment minimum value at the half-section

is:

(0,022)2

= 0,2020 = 0,2014.

0,790

ht weight distribution over the parallel part is fully determined by the following, which can be easily calculated

once for all.

LW = 1 118 ; hLW = - 0,089

LWL/2

= 0,261 5

The light weight double-moment minimum value is

= 0,2615 ( 0,089)2

0,2544 1,118

5.800

(nDW)L/2 = 15.300 = 0,3791

The double-moment factor with respect to L/2 is:

1,542.900

DwL/2 = 15.300

479 = 0,2146

Due to the above it is now possible to express the buoyancy

and light weight _{and p factors by simple mathematical}

func-tions, valid exactly within the parallel parts and approxi-mately some distance outside the end points of the parallel

parts. The deadweight -r and p factors must be calculated

separately, e.g. according to ecluations (11:1) and (11:2).

= 09790 = 1,266 ;

- 1,118 = 0,894

Thus we have:

The deadweight distribution is according to the details given

below: Weight Lever tons feet Tank aft 100 - 218 -Tank aft 200 - 170 -Hold *5 _{5.200 - 107} -iJloment 21.800 34.000 556.400 ]Dbl .-mom. 612.800 Hold 3 aft L/2 300 - 2 - 600 Hold ₃ fwd L/2 5.000 + 32 + 160.000 Hold 4.200 + 168 ± 705.600 930.100 Fore peak 300 + 215 + 64.500 Total deadweight 15.300 + 20,74 + 317.300 1.542.900

The deadweight aft of L/2 is loo + 200 + tons; thus the integral factor at L/2 is:

= 0,5000 + 1,266 From eq. (14)

= 0,2014 + 1,266

()2

11LW = 0,5000 + 0,894 _LW

0,2544 ± 0,894

As a first example we choose to calculate the shear force at the bulkhead between holds 1 and. 2, which is situated. 134'

forward of L/2; thus its length coordinate with respect to L/2 is + 0,285. With respect to the buoyancy and light weight

half-sections, the bulkhead thus has the following coordinates:

= 0,285 - 0,022 = + 0,263 ; = 0,285 + 0,089 = ± 0,374

The r factors for buoyancy and. light weight are

= 0,5000 + 1 266 X 0,263 = 0,8330

LW = 0,5000 + 0,894 x 0,374 = 0,8344

The deadweight ri factor can be calculated _{by eq. (11:1),}

where W 5.000 tons.

5.00 C)

DW 0,3791 +

1500

= 0,7059

The factor can also be calculated. by:

10.800 T1DW - _{- 15.30d}

where is the total load. aft of the section referred to.

The total-weight integral factor can be calculated according

The factor can also be calculated by:

rw =

x W

10.800 +

₀₉₈₃₄₄

x 4.600

A 19.900

The !non_dimensional _{shear force value is}

= - = 0,7356 - 098330 = - 0,0974

It should be noted that this value is only approximate, because the section falls somewhat outside the parallel part. The

absolute value of the shear force is finally:

T = A T = 19.900 X 0,0974 1.940 tons.

As a second example we choose to calculate the longitudinal

bending moment at the midpoint of hold 2, which is situated

999 _{forw. o± L/2. The coordinate with respect to L/2 is then}

+ 0,210. The corresponding coordinates with respect to the half-sections are

= 0,210 - 0,022 = + 0,188 = 0,210 + 0,089 = + 0,299 ;

From the expressions on page 11/9 we now have

= 0,2014 + 1,266 X _{(0,188)2 = 0,2461} = 0,2544 + 0,894

x

(0,299)2

= 03343

The deadweight p factor can be calculated according to eq. 11:2; where we have:

M

2 -- = 2

x 5.000 x (99 - 32) - 0,0932

-DW = 0,2146 0,210 x 0,2418 + 0,0932 0,2569

The to-tal weight double-moment factor can be calculated according to eq. (3); thus:

4.600 15.300

= 0,3343 x

19900

+ 0,2569 X 19.900 - 0,2748

The "non-dimensional" longitudinal bending moment a-b the sec-tion referred to is -then:

M

= ( - = o,s x (0,2748 - 0,2461) = + 0,01 43

The plus sign indicates a hogging-moment, the value of which

is:

M = L I'i' _{= 19.900} 470 0,0143 = 134.000 tons x ft.

In the above example an even keel condition is assumed. The

calculating procedure at a trimmed condition is exactly -the

same, except that the buoyancy _{11 and values should be} cal-culated according to equations (21) and (22), where the

co-ordinates now should be taken with respect to the buoyancy half-section in the trimmed condition. 1or determining the

half-section position at trim it should be suitable to have

q1

o

r.

o 1)4

j

r2t TT«4

Para /4/ bdy

C. L /

'z

e

L

LW

t-

Qç_ Qi

Q;

_D"

W0L

',/j_,__w

f?tt.)i..

_{____2c.j}

0.oz

0.01

/-fl

'f

//

E 4 T1

/R

M

-

7/

i

0.7

û.'

o. 0.4 O., 0. Z 0.1

C

//

c.(.

tt,

X

¿

27:3

F.r

+LW

kW

/_- -'S'

<:''

*\\

Ic

4g

Sorne notes on calculating wave-induced shear forces and longitudinal bending moments.

Conventional static view.

,

The conventional static calculating method, where the ship is considered as resting on one wavecrest amidships hogging -or on two crests at ends - sagging - can of course be

trans-ferred to the ideas, dealt with in the foregoing.

Figure 111:1 shows a ship's sectional area curves in hogging

and sagging conditions as well as in the still water

condi-tion. The hogging and sagging curves may be obtained in a conventional manner when the ship is balanced in some

stan-dard wave.

The integral functions and for the three conditions, still water, hogging and sagging, can be calculated and are shown in the - r system. It may be noted that the slope of

a curve amidships equals the inversed value of the "prismatic

coefficientt', that the ship will have in the condition in

question. If the wave height amidships is known, the "prisma-tic coefficient" in the hogging and sagging condition may

easily be deteinined. Thus we have for the prismatic coeffi-cients for hogging and sagging:

A® A®

cp _;

S'sÇ (P

where AH and A are the midship sectional areas in the

hog-ging and saghog-ging conditions.

In the figure the three corresponding double-moment functions are shown. As a curiosity it may be mentioned, that the

-value amidships in the hogging condition can approximately be

In figure 111:1 there are also shown the difference curves and for hogging and sagging in relation to the still water condition.

It might perhaps be possible to get such "standard" difference

curves for different ship forms and conditions, whereby the

integral and double-moment factors in a standard wave may be

obtained by adding the hogging or sagging value.

Modern statistical view.

The statistical methods of analysing full scale measurements

of wave bending moments acting on ships' hulls are now

uni-versally adopted and several papers have been written on the

sub j e ct.

By means of the results published by Bonnet et al., [1],

further treated by Nordenström, [2] [3], the maximum wave-induced longitudinal bending moment for a ship can be said

to be

M8=mL3B2

(111:1)

where L and B are the length and breadth of the ship in an arbitrary length unit and ,' is the density of the sea water

in the corresponding units.

The non-dimensional factor m depends on the ship's length and

according to Nordenström the factor is:

270

-m 2,75 e 10 (111:2)

where L is the ship's length in metres.

M8 now means the wave-induced longitudinal bending momont

as an extreme design value.

Dividing eq. (111:1) by A L we get an expression for the

maximum wave-induced bending moment in non-dimensional form.

Thus with A = ' LB d cp:

* _L

M

wave

m

Concerning wave-induced shear forces there are up to now very

little full scale measurements available. If we, however, assume that there is the same relationship between the

maxi-mum values of wave-induced bending moment and shear force as

in the static standard-wave condition, we can estimate the

maximum wave-induced shear force, based on the above ex-pressions for the statistically estimated maximum bending

moment.

Figure 111:2 shows the difference curve between the sagging

and hogging sectional areas given in figure 111:1.

The "peak-to-peak" shear force, i.e. the difference between the wave-induced shear forces in the hogging and sagging con-ditions, at a distance x from the aft end is then expressed

by:

X

T= J'

ASHdx

o

The maximum "peak-to-peak" bending moment at the section where T = O, is then expressed by

X

M = 1 T dx

max x

o

The relationship between the maximum bending moment, _Mmax and the maximum values T1 and T2 respectively can be given by:

F1 + F2

which also means that small differences between F1 and F2, due to lack of accuracy in balancing the ship in the wave, may be allowed.

In the example shown, which concerns a ship with rp 0,79 in a standard trochoidal wave, we have a1 = 4,1 cm, a2 = 4,3 cm, b = 12,5 cm, F1 15,0 cm2 and. F2 = 15,5 cm2, which gives k = 3,44.

Obviously the coefficient k only depends o± the shape of the

where k1 and k2 are non-dimensional coefficients.

In figure 111:2 there is also shown the shear force curve

T The two areas F and F between the T -curve and the

x-x 1 2 x

axis obviously both represent the maximum bending moment at

the section _Xml

If the scale for the Tx_curve is chosen so that the first

maximum shear force T1 is represented by a1 length units, and the second. one 12 is represented. by a2 length units, and.

furthermore, the length scale is chosen so that the ship's

length is represented. by b length units we obviously have for the factors k1 and k2:

a1b

a2b

k2=

where F is the area of F1 or and must be measured. in a

square unit corresponding to the length unit, chosen for

and L.

The mean value o± k1 and k2 may be written:

For design purpose it seems reasonable to choose the coef±'i-cient k somewhere between 3 arid 4, where the lower value may

be taken for very full ships and the higher one for finer

ships. For normal ships a value of 3,5 is suggested to be sufficiently accurate for use in practice.

Thus we have for the maximum wave-induced shear force near

the ship's quarter lengths in non-dimensional form:

* *

T

wave 3,5 M wave

*

where M should be taken from eq. (111:3). wave

In order to facilitate the practical use of equations (111:3)

and (111:4) the non-dimensional factor m is given below for

different ship lengths in feet.

Note 1 The wave-induced bending moment according to eq. (111:3) and consequently also the shear force according to

eq. (111:4) means the numerical mean value of the hogging and

sagging conditions. The full scale measurements show that for full ships with c.p = 0,80 there is very little difference

between the hogging and sagging values. For finer ships with = 0,68 the hogging value is abt 10 % lower and the sagging

value abt 10 % higher than the mean value, obtained from eq. (111:3).

Note 2. The statistical maximum bending moment value

accor-ding to eq. (111:3) may be assumed ± occur about amidships, similar to the maximum value for the ship in an extreme con-dition in the standard wave. It may be mentioned, however, that the longitudinal distribution of the "statistical maxi-mum values" may be assumed to follow a rather flat curve,

L 300' 400' 500' 600' 700' 800' 900'

plished concerning the longitudinal distribution of the shear force "statistical maximum values". Based on results from

model tests, [4] [5], figure 111:4 may give a very rough guid-ance for estimation of the longitudinal distribution of the

"statistical maximum values" of wave-induced bending moments and shear forces.

The above mentioned may be illustrated by continuation of the

numerical example given on page 11/7. See also figure 11:1.

The ship has the following data:

L = 470' ô = 0,782

B = 66'9"

cp = 0,790

d = 28'-4 = 0,990

From the table on page 111/5 we get ni = 0,00162 for L = 47Qt

From equation (111:3) the non-dimensional maximum wave-induced

bending moment is obtained:

*

x

470

x

0,790 - 0,0214

M 0,001 62

28,375 0,990 wave

From equation (111:4) we get the maximum wave-induced shear

force in non-dimensional form:

*

T

wave )( 0,0214 = 0,0749

Figure 111:5 shows the still-water shear force, T' , and

bending moment, M*svi, _{according to figure 11:1. There are also} shown the wave-induced shear force and bending moment with the

above maximum values and longitudinally distributed according to figure 111:4, T'wave and M'vjave The curves T4tOt and

are the sum of the absolute values of the still water curves and the wave-induced curves. Finally the real dimensional

values are found by:

* *

Tt0t = T

Bennet, R., Ivarsson, A., Nordenstr5m, N.: Results from Full Scale Measurements and. Predictions of Wave Bending

Moments Acting on Ships. Report No. 32 from Swedish

Shipbuilding Research Foundation. Gothenburg, 1962.

Nordenström, N.: On Estimation of Long Term Distribu-tions of Wave Induced Midship Bending Moments in Ships.

Report from the Division of Ship Design, Chalmers

Uni-versity of Technology. Gothenburg, 1963.

3.. Nordenström, N.: Statistics and Wave Loads. Report from

the Division of Ship Design, Chalmers University of

Technology. Gothenburg 1964.

4. Lötveit, M. MUrer, C., Vedeler, B.9 Christensen, H,:

Wave Loads on a T2 Tanker Model. European Shipbuilding, No. 1, 1961.

5.. Wac}mik, G., Schwarz, F.: Experimental Determination

of Bending Moments and Shear Forces in a Multi-segmented Ship Model Moving in Waves. mt. Shipbuilding IProgress No. 101 , _1963..