Eng. 6002 Ship Structures 1Hull Girder Response

Analysis

Lecture 5: The shape of Ocean design waves, wave bending

moments

Overview We can consider the wave forces on a ship to be

quasi-static. This means that they can be treated as a succession of equilibrium states.

When a wave passes by a vessel the worst hogging moment occurs when the midbody is on the crest of a wave, and the bow and stern are in the troughs

Overview The worst sagging moment occurs when the

midbody is on the trough, and the bow and stern are on crests

Furthermore, the highest bending moments occur when the wavelength approaches the vessel length

Overview cont. The design wave for a vessel will therefore

have a wavelength equal to the vessel length.

The wave height (peak to trough) is generally assumed to be 1/20th of the wave length (any larger and the wave will break)

Trochoidal Wave Profile The shape of an ocean wave is often

depicted as a sine wave, but waves at sea can be better describaed as "trochoidal".

A trochoid can be defined as the curve traced out by a point on a circle as the circle is rolled along a line.

Trochoidal Waves cont. The discovery of the trochoidal shape came from

the observation that particles in the water would execute a circular motion as a wave passed without significant net advance in their position.

The motion of the water is forward as the peak of the wave passes, but backward as the trough of the wave passes, arriving again at the same position when the next peak arrives. (Actually, experiments show a slight advance of the water with the waves, but that advance is small compared to the overall circular motion.)

Source: http://www.dddb.com/rotation.html

Trochoidal Waves cont.For a design wave we assume the following

wave is possible LW=LBP, HW=LBP/20

We can see that LW=2R and HW=2r

Trochoidal Waves cont.Which gives

The following formula describes the shape of the waves

20 and,

40,

2

R

rLr

LR BPBP

cos1

sin

rz

rRx

Trochoidal Waves cont.Substituting, we have

To plot the wave, we simply calculate x and z as a function of

cos140

sin402

Lz

LLx

Trochoidal Waves cont.

0 50 100 150 200 250 300 3500

5

x

z

Trochoidal Waves cont. The L/20 rule for wave height has been shown to

be overly conservative for large vessels and a more modern formula is:

Which gives

Note Hughes gives (for L>350 m)

metres)(in 607.0 BPW LH

BPBP LrL

R 303.0,2

metres)(in 227

BP

WL

H

Calculating Wave Bending Moments We can now calculate the wave bending

moments by placing the ship on the design wave and using the Bonjean curves

Calculating Wave Bending MomentsSo, to determine the wave bending moment we:1. Obtain bonjean curves2. At each station determine the still water

buoyant forces (using the design draft)3. At each station determine the total buoyancy

forces using the local draft in that part of the wave

4. The net wave buoyancy forces are the difference between the total and still water buoyancy forces

SWiwiwavei FFFt ,,,

Calculating Wave Bending Moments From here we have a set of buoyancy

forces due to waves, which are in equilibrium (recall Lecture 4)

We calculate the moment at midships from the net effect of forces either fore or aft

Computer application We can also use computer packages (such

as Rhino) to find the bending moments Using a hull model, the buoyant forces on

the fore and aft ends of the hull can be determined by the volume and centroid of the submerged volumes at a specific waterline surface

A similar procedure could be used to determine the wave values, but the waterline surface would be the trochoidal wave profile