Eng. 6002 Ship Structures 1
Hull Girder Response Analysis
LECTURE 4: 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=2pR and HW=2r
Trochoidal Waves cont.
Which gives
LBP
LBP
r p
R
,r 
and, 
2p
40
R 20
 The following formula describes the shape of the
waves
x  R  r sin 
z  r 1  cos  
Trochoidal Waves cont.
Substituting, we have
L
L
x
  sin 
2p
40
L
1  cos  
z
40
To plot the wave, we simply calculate x and z as a
function of 
Trochoidal Waves cont.
z
5
0
0
50
100
150
200
x
250
300
350
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:
HW  0.607 LBP (in metres)
 Which gives
LBP
R
, r  0.303 LBP
2p
 Note Hughes gives (for L>350 m)
227
HW 
(in metres)
LBP
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 Moments
So, to determine the wave bending moment we:
1. Obtain bonjean curves
2. 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
Fi ,wave  Fi ,wt  Fi ,SW
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