1. Eng. 6002 Ship Structures 1 Hull Girder Response Analysis
Lecture 5: The shape of Ocean design waves, wave bending moments

2. 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

3. 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

4. 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)

5. 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.

6. 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:

7. Trochoidal Waves cont.
For a design wave we assume the following wave is possible
LW=LBP, HW=LBP/20
We can see that LW=2pR and HW=2r

8. Trochoidal Waves cont. Which gives
The following formula describes the shape of the waves

9. Trochoidal Waves cont. Substituting, we have
To plot the wave, we simply calculate x and z as a function of q

10. Trochoidal Waves cont.

11. 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)

12. 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

13. Calculating Wave Bending Moments
So, to determine the wave bending moment we:
Obtain bonjean curves
At each station determine the still water buoyant forces (using the design draft)
At each station determine the total buoyancy forces using the local draft in that part of the wave
The net wave buoyancy forces are the difference between the total and still water buoyancy forces

14. 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

15. 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