Hull girder response quasi static analysis
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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

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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

  • 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 Relationships

buoyancy curve - b(x)

weight curve - w(x)

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

Sign Convention

Positive

Upwards

+ f


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

Net Load -f

Sign Convention

Positive

Upwards

+ f :

Shear Force - Q

+ Q

Positive

Clockwise

+ Q :

- Q


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

Shear Force - Q

Sign Convention

+ Q

Positive

Clockwise

+ Q :

- Q

Bending Moment - M

Positive

Sagging

+ M :

- M


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

  • Static Analysis - No Waves Present

  • Most Warships tend to Sag in this Condition

    Putting Deck in Compression

    Putting Bottom in Tension


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

  • 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.


Trochoidal vs. Sine Wave


Sagging Wave

Excess Weight Amidships - Excess Buoyancy on the Ends

Compression

Tension


Hogging Wave

Excess Buoyancy Amidships - Excess Weight on the Ends

Tension

Compression


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

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

Can be misleading for long components


Example Weight Curve


Weight Item Information

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

W

fwd

lcg

aft

FP


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 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

  • 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 Method

lcg

x

G

1.2h

wa

wf

aft

FP


The Three Types of Structure


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