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Hull Girder Response - Quasi-Static Analysis. Basic Relationships. Model the hull as a Free-Free box beam. Beam on an elastic foundation Must maintain overall Static Equilibrium. Force of Buoyancy = Weight of the Ship LCB must be in line with the LCG. Basic Relationships.

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Hull girder response quasi static analysis
Hull Girder Response - Quasi-Static Analysis


Basic relationships
Basic Relationships

  • Model the hull as a Free-Free box beam.

    • Beam on an elastic foundation

  • Must maintain overall Static Equilibrium.

    • Force of Buoyancy = Weight of the Ship

    • LCB must be in line with the LCG


Basic relationships1
Basic Relationships

  • From Beam Theory – governing equation for bending moment:

  • Beam is experiencing bending due to the differences between the Weight and Buoyancy distributions

Where f(x) is a distributed vertical load.

Net Load

Buoyancyrga(x)

Weightgm(x)


Basic relationships2
Basic Relationships

buoyancy curve - b(x)

weight curve - w(x)

net load curve - f(x) = b(x) - w(x)

Sign Convention

Positive

Upwards

+ f


Basic relationships3
Basic Relationships

  • The solution for M(x) requires two integrations:

  • The first integration yields the transverse shear force distribution, Q(x)

    • Impose static equilibrium on a differential element

f

M

Q

Q + dQ

M + dM

dx

But ships are “Free-Free” Beams - No shear at ends!Q(0) = 0 and Q(L) = 0, so C = 0


Finding shear distribution
Finding Shear Distribution

Net Load -f

Sign Convention

Positive

Upwards

+ f :

Shear Force - Q

+ Q

Positive

Clockwise

+ Q :

- Q


Basic relationships4
Basic Relationships

  • The second integration yields the longitudinal bending moment distribution, M(x):

    • Sum of the moments about the right hand side = 0

0

f

M

Q

Q + dQ

M + dM

dx

Again, ships are “Free-Free” Beams - No moment at ends!M(0) = 0 and M(L) = 0, so D = 0


Finding bending moment distribution
Finding Bending Moment Distribution

Shear Force - Q

Sign Convention

+ Q

Positive

Clockwise

+ Q :

- Q

Bending Moment - M

Positive

Sagging

+ M :

- M


Shear moment curve characteristics
Shear & Moment Curve Characteristics

  • Zero shear and bending moments at the ends.

  • Points of zero net load correspond to points of minimum or maximum shear.

  • Points of zero shear correspond to points of minimum or maximum bending moment.

  • Points of minimum or maximum shear correspond to inflection points on bending moment curve.

  • On ships, there is no shear or bending moments at the forward or aft ends.


Still water condition
Still Water Condition

  • Static Analysis - No Waves Present

  • Most Warships tend to Sag in this Condition

    Putting Deck in Compression

    Putting Bottom in Tension


Quasi static analysis
Quasi-Static Analysis

  • Simplified way to treat dynamic effect of waves on hull girder bending

  • Attempts to choose two “worst case”conditions and analyze them.

    • Hogging Wave Condition

      • Wave with crest at bow, trough at midships, crest at stern.

    • Sagging Wave Condition

      • Wave with a trough at bow, crest at midships, trough at stern.

  • Wave height chosen to represent a “reasonable extreme”

    • Typically:

  • Ship is “balanced” on the wave and a static analysis is done.


Wave elevation profiles
Wave Elevation Profiles

  • The wave usually chosen for this analysis is a Trochoidal wave. It has a steeper crest and flatter trough.

  • Chosen because it gives a better representation of an actual sea wave than a sinusoidal wave.

  • Some use a cnoidal wave for shallow water as it has even steeper crests.



Sagging wave
Sagging Wave

Excess Weight Amidships - Excess Buoyancy on the Ends

Compression

Tension


Hogging wave
Hogging Wave

Excess Buoyancy Amidships - Excess Weight on the Ends

Tension

Compression


Weight curve generation
Weight Curve Generation

  • The weight curve can be generated by numerous methods:

  • Distinct Items (same method as for LCG)

  • Parabolic approximation

  • Trapezoidal approximation

  • Biles Method (similar to trapezoidal)

  • They all give similar results for shear and bending moment calculations. Select based on the easiest in your situation.


Distinct item method
Distinct Item Method

Each component is located by its l, t and v position and weight

Can be misleading for long components



Weight item information
Weight Item Information

  • For each weight item, need W, lcg, fwd and aft

W

fwd

lcg

aft

FP


Trapezoid method
Trapezoid Method

  • Models weight item as a trapezoid

  • Best used for semi-concentrated weight items

  • Need the following information:

    • Item weight – W (or mass, M)

    • Location of weight centroid wrt FP - lcg

    • Forward boundary wrt FP - fwd

    • Aft boundary wrt FP - aft

  • lcg must be in middle 1/3 of trapezoid


Trapezoid method1
Trapezoid Method

  • Find l and x

  • Solve for wfand wa so trapezoid’s area equals W and the centroid is at the lcg

FP

lcg

x

wa

G

wf

fwd

l/2

l

aft


Biles method
Biles Method

  • Used for weight items which are nearly continuous over the length of the ship.

  • Assumes that weight decreases near bow & stern.

  • Assumes that there is a significant amount of parallel middle body.

  • Models the material with two trapezoids and a rectangle.


Biles method1
Biles Method

lcg

x

G

1.2h

wa

wf

aft

FP