A second order effect in wave bending moment

A SECOND ORDER EFFECT IN WAVE BENDING MOMENT 0. Grim, Hamburg Ship Model Basin P. Schenzle, Hamburg U n i v e r s i t y

Abstract

A r a t h e r simple method i s introduced to account f o r ship form n o n l i n e a r i t i e s i n heaving and p i t c h i n g motion. Thus i t i s possible to evaluate e s p e c i a l l y the well known d i f f e r e n c e between hogging and sagging wave bending ntoment by c a l c u l a t i n g a small second order s h i f t of the mean value. The consequences are demonstrated f o r a t h i r d generation container s h i p . A p p l i c a t i o n to the i r r e g u l a r seaway i s o u t l i n e d .

Tne t r a d i t i o n a l way of estimating the loading of a ship i n the seaway uses tne f i c t i v e model o f a q u a s i - h y d r o s t a t i c pressure d i s t r i b u t i o n i n a f i x e d t r o c h o i d a l wave f o r the c a l c u l a t i o n of wave bending moment.

Due to the lack i n knowledge about hydroc^ynamical phenomena the naval

a r c h i t e c t s developed r a t h e r sophisticated techniques f o r h y d r o s t a t i c c a l c u l a -t i o n s -taking i n -t o accoun-t -the exac-t geome-try of each ship form. One o f -the consequences was a marked d i f f e r e n c e i n the absolute value o f the so computed

hogging and sagging nwment due to the V-shaped sections at the ships ends.

i^hen b e t t e r unaerstanding o f hydrodynamic phenomena in seakeeping led to gene r a l agreement on l i n e a r i s e d dynamic computation procedures f o r wave bending moment, t h i s asymmetry i n hogging and sagging moment could no longer be evaluated. On the one hand l i n e a r i s a t i o n allows superposition i n the i r r e -gular seaway, on the other hand c e r t a i n nonlinear features of quasi s t a t i c c a l c u l a t i o n s , e s p e c i a l l y h u l l form n o n l i n e a r i t i e s , g o t l o s t . In the l i n e a r i s e d nx)del o f the ship heaving and p i t c h i n g i n a regular wave a harmonicly o s c i l l a t i n g pressure f i e l d i s assumed t o be a c t i n g on the s h i p ' s surface up t o the

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-smooth water l i n e . So the induced forces moments and loads are also narmonicly o s c i l l a t i n g w i t h zero mean.

Here a r a t h e r simple c o r r e c t i o n i s possible to take i n t o account second order steady forces in the area of temporary immersion o f the s h i p ' s surface. This area of local v e r t i c a l r e l a t i v e motion S

s (X, t ) « S„(x) c o s (cJt-t-eCx))

of the ship w i t h respect to the wave surface i s a ' b e l t ' of v a r i a b l e breadth

' ar/ouna the smooth water l i n e o f the ship(Fig.-t).

R g . - f

As the l i n e a r i z e d pressure d i s t r i b u t i o n i s truncated at the smooth water-l i n e a c o r r e c t i o n is necessary t o account f o r the reawater-l immersion and emer-sion of the sections(Fig.2).

(Hydrodynamic pressures are neglected here,because they produc^e"^second order mean f o r c e s . Second order o s c i l l a t i n g forces are not considered h e r e . )

This c o r r e c t i o n y i e l d s an always p o s i t i v e t r i a n g u l a r pressure d i s t r i b u t i o n represented by a second order l i n e force along the w a t e r l i n e always

d i r e c t e d i n s i d e the body of the magnitude:

w i t h a mean value o f :

From t h i s mean l i n e force there f o l l o w r e s u l t a n t f o r c e s i n l o n g i t i d i n a l , transverse and v e r t i c a l d i r e c t i o n according to the l o c a l slope of the ship surface y^(x,z,) .The l o n g i t u d i n a l component i s used i n the Hamburg method f o r c a l c u l a t i n g the resistance increase i n waves IlJ[2].

Here only the v e r t i c a l component i s discussed (P«g.3)

! /

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-which has two consequences:

1. A s l i g h t l y a l t e r e d mean f l o a t i n g p o s i t i o n of the ship by a mean emersion z , and a mean t r i m angle ©

(a small s h i f t of the niean values o f the heaving and p i t c h i n g m o t i o n ) . 2. A mean shear force^and sagging moment Hy

(a small s h i f t of the mean values o f the shear and bending l o a d ) .

The emersion and t r i m are:

Z o ' ^ f ^ d x / f ^ g H ^ ) ( a t center of water plane)

V e r t i c a l shear force and sagging moment are w i t h the r e s u l t i n g load f per u n i t l e n g t h :

Rg.4-Fig.4 shows normalized emersion z^,trim ^ and bendingmoment M^^atiXtfor head waves as a f u n k t i o n o f wave-length c a l c u l a t e d f o r a container ship o f t h e t h i r d generation ( t a b l e 1 ) .

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-Consequences of inean emersion and t r i m are more frequent slamming and less shipping of green water under extreme c o n d i t i o n s .

F i g . 6 shows r e l a t i v e motion a t the bow i n A = L head waves as a f u n c t i o n o f wave amplitude w i t h and w i t h o u t second order mean emersion z ( x ) .

The consequence of the mean sagging moment i s an as^mimetric o s z i 1 1 a t i n g o f the wave Dending moment and an increasing or decreasing o f the maximum r e s u l t i n g bending load depending on whether the smooth water moment N^, i s sagging or hogging.

The a p p l i c a t i o n of these r e s u l t f o r second order steady q u a n t i t i e s to the i r r e g u l a r seaway seams p o s s i b l e . I f the r e l a t i v e motion S(fcj i s a narrow band process then e . g . the second order mean bending moment My i s a slowly vary-ing q u a n t i t y . As the maxima of the narrow band r e l a t i v e motion are Rayleigh d i s t r i b u t e d , the d i s t r i b u t i o n of the second order q u a n t i t y M y i s , l i k e t h a t o f S ^ j o f the exponential type. Now the over a l l mean M ^ o f the

slowly varying mean bending moment My i s o b t a i n e d i n the same way as the mean resistance increase i n the i r r e g u l a r seaway:

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-and the exponential d i s t r i b u t i o n density o f M ^ - i s :

Model t e s t s f o r the experimental evaluation o f the e f f e c t are i n p r e p a r a t i o n .

boese.P.: Eine einfache Methode zur Berechnung der Widerstandserhbhung eines S c h i f f e s im Seegang. I n s t . f . Schiffbau d . Univ. Hamburg,Rept.No 258 Febr. 1970.

Blume,P.,Keil,U..Schenzle,P.:Rechnerische Bestimmung der Widerstandserhbhung eines S c h i f f e s i n regelma3igen Wellen und Vergleich m i t entsprechen-den Model!versuchen. I n s t . f . S c h i f f b a u d.Univ.Hamburg,Rept.No.297, Nov.1973.