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Hull Girder Response - Quasi-Static Analysis

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- 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

- 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)

buoyancy curve - b(x)

weight curve - w(x)

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

Sign Convention

Positive

Upwards

+ f

- 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

Net Load -f

Sign Convention

Positive

Upwards

+ f :

Shear Force - Q

+ Q

Positive

Clockwise

+ Q :

- Q

- 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

Shear Force - Q

Sign Convention

+ Q

Positive

Clockwise

+ Q :

- Q

Bending Moment - M

Positive

Sagging

+ M :

- M

- 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.

- Static Analysis - No Waves Present
- Most Warships tend to Sag in this Condition
Putting Deck in Compression

Putting Bottom in Tension

- 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.

- Hogging Wave Condition
- Wave height chosen to represent a “reasonable extreme”
- Typically:

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

- 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.

Excess Weight Amidships - Excess Buoyancy on the Ends

Compression

Tension

Excess Buoyancy Amidships - Excess Weight on the Ends

Tension

Compression

- 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.

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

Can be misleading for long components

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

W

fwd

lcg

aft

FP

- 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

- 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

- 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.

lcg

x

G

1.2h

wa

wf

aft

FP