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Overview

For the purpose of analysis, the primary level ofresponse of a ship is modelled as a hollow, thin-wall box beam known as the hull girder

Can use simple beam theory, where: Longitudinal position, loads and deflections have a single

value at any cross section The hull girder remains elastic with small deflections,

and the strain due to bending varies linearly over thecross section (about a neutral axis)

Static equilibrium applies

Horizontal and vertical bending of hull girder may besuperimposed

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Load, Shear and Bending

Overall staticequilibrium requiresthat the totalbuoyancy force equals

the weight of the ship,and l.c.b coincideswith l.c.g

ntdisplaceme

onacceleratinalgravitatio

seawaterofdensity

ondistributimass)(

areasectional-crossimmersed)(

:where

)()(0 0

g

xm

xa

gdxxmgdxxagL L

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Distribution of Weights The weight will not equal the

buoyancy at each location alongthe ship.

The weights are a combinationof lightship and cargo weights(more of less fixed).

The buoyancy forces are

determined by the shape of thehull and the position of thevessel in the water (draft andtrim).

The net buoyancy will adjustitself until it exactly counteractsthe net weight force.

Local segments of the vessel

may have more or less weightthan the local buoyancy. Thedifference will be made up by atransfer of shear forces alongthe vessel.

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Beam Theory

The governingequation for thebending moment,M(x), is:

beamon theloadingtheis)(where

)(2

2

xf

xfdx

Md

For a ship f(x) is a net distributed force, given

by the resultant of the weight and buoyancyforces : f(x) = b(x) w(x)

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Figure 3.1 (a-d), Hughes

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Beam Theory Cont.

To solve for M(x)wefirst need thetransverse shearforce, Q(x).

Summing themoments about adifferential elementgives:

x

dxxfxQ0

)()(

x

dxxQxM

0

)()(

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Sign Conventions

Positive shear causes clockwise rotation of anelement

Positive bending moment corresponds toconcave upwards, or sagging

Negative bending moment corresponds toconcave downwards, or hogging

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Shear Force and Bending Moment

Curves

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Shear Force and Bending Moment

Curves

Features;

Zero load corresponds to max (or min) shearforce

In general the shear force is zero nearamidships and has peaks near quarter points

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Shear Force and Bending Moment

Curves

Features;

Zero shear corresponds to max (or min)bending moment

In general the bending moment will bemaximum near amidships

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Still Water vs Wave Loading cont.

The still water buoyancy distribution isdetermined from the static and momentequilibrium equations (described

previously in this lecture) So we need to know the mass distributionm(x)(or at least the displacement andlocation of l.c.g)

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Bonjean Curves

The local buoyancy per metre can bedetermined from the cross-sectional areaof the hull at discrete locations

The cross-sectional area depends on thelocal draft and is found using bonjeancurves

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Bonjean Curves cont.

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Bonjean Curves cont.

There is one bonjean curve for eachstation. There are 21 stations from FP toAP, so we can divide the LBP into 20

segments

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Bonjean Curves cont.

At each station a curveof the cross-sectionalarea is drawn

Bonjean curves areshown on the profile ofthe vessel and we usethem to determine the

buoyancy distributionat an waterline

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Bonjean Curves cont.

The total displacementat a given draft/trim isfound by summing thecontribution of each

segment

The buoyant line load(used for calculatingthe buoyant force at

each station) is thengiven by i

]m[20

320

0

i

ii

LBPTa

gii

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Assignment #1

For the three station profiles shown below,draw the bonjean curves

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Next Class

Estimation of weight distribution

Calculation of still water bending moment