Calhoun: The NPS Institutional Archive
DSpace Repository
Theses and Dissertations
Thesis and Dissertation Collection
1974
Combined ship hull horizontal and vertical
wave bending moments in irregular seas
Baltazar, Mario Anastacio
Massachusetts Institute of Technology
http://hdl.handle.net/10945/16921
Downloaded from NPS Archive: Calhoun
COMBINED SHIP HULL HORIZONTAL AND VERTICAL
WAVE BENDING MOMENTS IN IRREGULAR SEAS
Mario Anastacio Baltazar
DUDLEY KNOX LIBR'
Ry
NAVAL POSTGRADUATE
(
SCHOOL
MONTEREY. CAUFORN.A
939«
COMBINED SHIP HULL HORIZONTAL AND VERTICAL WAVE
EENDING MOMENTS IN IRREGULAR SEAS
by
Mario Anastacio Baltazar
Lieutenant, Philippine Navy
B.S., United States Naval Academy (1966)
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Ocean Engineer
And the Degree of
Master of Science in Naval Architecture
And Marine Engineering
at the
Massachusetts Institue of Technology
May,
19 7 4
ABSTRACT
by
Mario A. Baltazar
Rational design methods based on a statistical
description of both the ship load and strength have been
proposed by several authors in lieu of the traditional
In general, the effect of
bending moment calculations.
horizontal moments on the primary strength has been
neglected.
It is shown in this paper, as several others
have found, that the horizontal moments can be very
significant.
Two methods are described to obtain the
magnitude of the equivalent combined moment.
An example application is shown for a large tanker
Wave moment trends
by use of the MIT seakeeping program.
with speed, sea state and wave heading are noted. Measured
and predicted values of the vertical bending stress are
compared.
Finally, the parameters of the weibull distribution to describe long-term wave moment amplitudes are
derived.
11
^
ACKNOWLEDGEMENTS
The author would like to express his gratitude to
his thesis supervisor, Professor Alaa Mansour, for the
valuable advice and direction he provided during the conduct of this study.
Likewise, the author would like to thank a friend
who has been generous with his time, Atle Steen, and Professor
Chryssostomidis for his advice.
Finally,
I
wish to thank my wife, Heather, who has
been very patient and understending for all of my thre
years at M.I.T. and especially these last two months.
111
2
4
3
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
iii
NOTATION
vi
LIST OF FIGURES
viii
Chapter
I.
II.
INTRODUCTION
1
PROCEDURE
4
II. 1 Determination of the Equivalent Combined
Wave Bending Moment
III.
4
II. 2 Long-Term Wave Load Analysis
16
II. 3 Details of the Procedure
20
RESULTS
24
111.1 Comparison with Other Published Works
111.
111.
25
Comparison of the Calculated Values of
the Combined Moment Response
....
32
Comparison of Calculated Vertical
Bending Stress and Full-Scale
Measurements
36
Examination of Bending Moment Trends and
44
Wave Load Prediction
111.
IV.
.
DISCUSSION OF RESULTS
58
IV. 1 Evaluation of the Analytical Results
.
58
IV. 2 Comparison of the Values of the Combined
Moment
IV.
3
65
Comparison of Vertical Bending Stress
IV
.
69
Chapter
Page
IV. 4 Bending Moment Trends and Wave Load
Prediction
V.
73
CONCLUSIONS AND RECOMMENDATIONS
75
REFERENCES
80
.....
APPENDIX A
-
SAMPLE COMPUTER OUTPUT DATA
APPENDIX B
-
CALCULATIONS FOR AN AVERAGE VALUE
APPENDIX C
-
CALCULATION OF THE MEAN RMS PEAK-TO-PEAK
VERTICAL BENDING STRESS AND THE
STANDARD DEVIATION WITH RESPECT TO
91
WIND
^VR
84
89
NOTATION
L
= ship's length
H
= wave height
h
= significant wave height
M^
= vertical bending moment
M
= horizontal bending moment
H
M
= equivalent combined bending moment
k..
= vertical section modulus
k„
= horizontal section modulus
K
= section modulus ratio
E
= mean square value of the amplitude of the response
s,k
- parameters of the Weibull distribution
m
= mean value of the response amplitude
V
= amplitude of the vertical moment component
H
= amplitude of the horizontal moment component
V
= ship's speed
L
= effective wave length
RAO
= response amplitude operator
RMS
= root mean square
u
= circular frequency
n
= phase angle of the horizontal moment with respect to a
wave with its crest amidship
e
= phase angle of the vertical moment with respect to
a
<j>
A.
wave with its crest amidship
= phase angle of the equivalent combined moment
= correlation coefficient of the vertical and horizontal
bending moment
vi
aT
= total or combined stress
av
= vertical bending stress
a„
= horizontal bending stress
X
= wave length
S^tw)
= spectral density of the response
H(cj)
= RAO of the ship
S(w)
= sea spectral density
H
vn
LIST OF FIGURES
Figure
Page
11. 1
11. 2
111.1
111.
111.
111.
2
3
4
The Longitudinal Stress Distribution on a
Ship Hull Girder
Section Modulus Ratio
VS. DWT of Tankers
5
.
.
RMS of the Bending Moment Response at
Amidship VS. Heading in Long-Crested
Waves
26
RMS of the Bending Moment Response at
Amidship VS. heading in Short-Crested
Waves for Low Seas
27
RMS of the Bending Moment Response at
Amidship VS. Heading in Short-Crested
Waves for Moderate Seas
28
RMS of the Bending Moment Response at
Amidship VS. Heading in Short-Crested
Waves for High Seas
111. 5
111.
6
29
Correlation Coefficient Between Vertical
Bending Moment and Horizontal Bending
Moment
35
Comparison of predicted and Measured Vertical
Bending Stress for the UNIVERSE IRELAND
38
Relationship Between Significant Wave Height
and Wind Speed From Various Sources
40
Wind-Wave Height Standard Deviations From
Various Sources
40
Mean RMS of the Vertical Bending Response VS.
Significant Wave Height for All Areas in
the Ship's Route
42
.
111.
7
111. 8
111.
9
15
.
....
111. 10 A Comparison of the Vertical Moment Response
in Long-Crested Seas VS. the Combined
Moment in Short-Crested Seas for h = 19.2
Feet
45
vm
Figure
Page
III. 11 A Comparison of Vertical Moment Response in
Long Crested Seas VS. the Combined Moment
in Short-Crested Seas for h = 4 3.3 Feet
.
IV. 1
IV. 2
.
46
Vertical Wave Bending Moment Amplitude in
Regular Waves
59
Horizontal Wave Bending Moment Amplitude
in Regular Waves
59
IV. 3
The Effective Wavelength in Oblique Waves
IV.
Measured and Computed Vertical Wave
Bending Moment Amplitude in Regular
Wave s
64
Measured and Computed Horizontal Wave Bending
Moment Amplitude in Regular Waves
64
Phase Angle Between Horizontal and Vertical
Wave Bending Moment at Midship for Series
60, Block 0.80
67
Phase Angle Between Horizontal and Vertical
Wave Bending Moment at Midship for Series
'......
60, Block 0.80
68
4
IV. 5
IV. 6
IV. 7
IX
.
.
61
.
Chapter
I
INTRODUCTION
The traditional method for determining the load for
primary strength calculations consists of poising the ship
on a trochoidal wave of length equal to that of the ship and
wave height equal to one-twentieth (1/20) of the length.
The static bending moment is then determined by the load-
shear-moment relations from beam theory.
There have been
various proposals to modify the wave height (e.g., H = 0.6L
n
fi
or H = l.l/L) partly to correct for dynamic effects and partly
because it is thought than an L/20 wave is too severe for the
very large ships.
Notwithstanding, this empirically derived
procedure was considered entirely satisfactory until relatively
recently and is still in prevalent use inasmuch as it gives
conservative estimates.
The appearance of large ocean-going tankers and bulk
carriers and novel ship types gave impetus towards the search
for a "rational" procedure for ship structural design.
It
was realized that the application of the standard bending
moment calculations for these ships lead to unrealistic results.
At the same time new knowledge relating to ship
structural design had accumulated and experimental and comp-
utational facilities to advance this knowledge have improved
tremendously
Many marine-oriented societies and organizations
undertook basic research to improve the accuracy of predictions of sea loads and motions, and the development of a
rational design method applicable for all ship types.
scale and model tests data were collected.
Full
The random
nature of both the loads on a ship and its capacity to resist these loads was recognized, so that probability theory
was utilized to develop statistical models for the design of
ships [1-14]*.
The advantages of a probabilistic approach
over the traditional deterministic calculations are discussed
in reference [7].
In the conventional method, the primary strength of
a ship is based solely on the longitudinal stresses caused
by vertical bending, or bending about the horizontal neutral
axis.
The effects of lateral bending, or bending about the
vertical neutral axis, is neglected.
However, for ships
travelling oblique to the waves, the lateral loads reach a
magnitude equal to or greater than the vertical bending
moment.
The combined effect of lateral and longitudinal
bending may be critical for some ship types.
A requisite for an integrated probabilistic design
procedure is a more precise knowledge of the loads acting on
*The numbers in brackets refer co references listed
in the Appendix
.
,
3
a ship.
This paper represents a small portion of the total
effort to completely define the structural loads experienced
by a ship in a seaway.
It is an investigation of the magnitude
of combined horizontal and vertical bending in irregular seas.
Due to their relative phase difference, the amplitudes of the
vertical and horizontal moments generally are not acting at
the same time
In the analysis, the loads are treated as random
variables.
Consideration of the total longitudinal stress at
the deck leads to the definition of an equivalent combined
bending moment.
Two methods are developed to generate the
statistical parameters of the combined load so that the usual
statistical methods to predict long-term extreme values of the
load can be applied.
An example application is then made for
a large tanker which had been previously instrumented so that
full scale data for the vertical bending stresses experienced
in service are available for purposes of comparison.
Such a
comparison serves to check yet another time the validity of
computational techniques to simulate ship motion and loads in
waves, and in particular the MIT 5-D seakeeping program.
The
predicted moment responses are compared and analyzed for trends
Chapter II
PROCEDURE
II. 1
Determination of the Equivalent Combined Wave Bending
Moment
A ship travelling at an angle to the waves will be
subjected to unsymmetrical bending such that the stress at
the deck is not uniform but assumes a maximum value at one
edge, as shown in figure II. 1.
This primary stress may be
thought of as the combined effect of vertical and horizontal
bending and, to a much lesser extent, torsion.
Previous
studies [15, 16] have established that torsional moments are
relatively small, never exceeding 10% of the vertical moment
in any given condition.
In this study the effects due to
torsion are considered negligible, but it is recognized that
torsion loading may be critical for some ships, as those
The contribution to the
with large hatch openings [17].
ship primary stress may be included, if so desired, in a
manner analogous to the method described here [18].
The stress at the deck edge is given by
=
°T
°V
+
°H
+
c
+
Horizontal Bending Stress
Vertical Bending Stress
(sagging)
Combined Stress
Figure II
.
1
The Longitudinal Stress Distribution
on a Ship Hull Girder
where
o
= combined, or total,
o
= stress due to vertical bending
stress
o„ = stress due to horizontal bending
In terms of moments, one can write
T
= *V +
M
H
^H
*V
where
= vertical bending moment
ML
M„ = horizontal bending moment
H
k
k
T
V
h
JJ = vertical section modulus
= ivv
z
=
zz =
horizontal section modulus.
y
multiplying both sides of the equation by k^, one gets
k
°T V
=
"V
+
^ MH
If one defines an equivalent combined bending moment
^
=
rr
K7
,
and the section modulus ratio K = *—-, then one
gets the relation
M
T
= My + KM
(1)
R
It is emphasized here that one cannot simply add
the maximum values of the vertical and horizontal moments to
get the maximum combined effect.
In general the moments at
any point along the length of the ship are not acting in
phase.
Furthermore, it may be of interest to specify the
complete distribution of the primary stress at the deck.
Within the limits of the beam theory, this is done by getting
the stress at one other point across the ship's deck.
A
convenient reference point is at the ship's center line where
the stress due to lateral bending is zero for most ships due
to port-and-starboard symmetry, i.e., a
= a v at the centerline.
In the statistical approach to sea loads, the ex-
citing force is assumed to be a stationary, narrow-band and
gaussian process with zero mean over a short period of time
(e.g.
30 minutes).
The ship is considered to be a linear
system so that the response is also stationary, narrow-band
and gaussian with zero mean over the same period of time.
Then the amplitudes follow the Rayleigh distribution [19, 20].
Over a long period of time, the assumption of a stationary
excitation process cannot be made.
However,
it was found
.
8
that the amplitudes follow closely the Weibull distribution
[8,
9,
It is shown that the Rayleigh is a special case
10].
of the Weibull distribution given by:
f
x
(x)
=
F
wher fy(x) and F
(s/k)(x/k)
x
(x)
(x)
S
1
exp-(x/k)
S
= 1 - exp-(x/k)
S
X10
(2)
X>0
(3)
are the probability density and distrib-
ution functions respectively, and
s
and k are parameters.
To
get the Rayleigh distribution, one sets s=2 and k = /E where
E is the mean square value of the amplitude of the response.
Furthermore it was found that the best-fit Weibull curve is
approximately the exponential distribution obtained from
equations
(2)
and
(3)
by setting
s = 1
and k = m where m is
the mean value of the response amplitude.
The statistics of
the ship bending moment response, therefore, can be specified
if the mean square value, or variance, of the process is
known
The usual method to obtain the variance of the re-
sponse in random seas, attributed to St. Denis and Pierson,
consists of specifying the energy density spectra of the
ocean waves and obtaining the ship response amplitude operator (RAO)
,
or the response per unit amplitude of regular
wave, by model tests or some computational procedure using
the strip theory approach.
From the theory of linear systems,
the spectral density of the response
c£'n
be found from the
relation [21]
S
where
S
R
(u>)
= |H(u>)
2
|
S(o))
= spectral density of the response
(to)
H(co)
= ship response amplitude operator
S(w)
= spectral density of the seaway.
(RAO)
The area under the response spectrum is the variance
of the process and represents the sum of the responses to the
infinitely many sine waves that are thought to compose the
actual sea.
Two methods used to obtain the variance of the wave
bending moments by computer simulation techniques are ex-
plained below.
1)
First Method
Equation
(1)
can be written showing the moments as
a continuous random function of time,
M (t)
T
=
MyU)
+ KM
t:
H
(t)
10
Using the assumption that the sea can be represented by infinitely many harmonic waves of small amplitudes, continuous
frequency and random phase, one can write the vertical and
horizontal moment responses as:
On t
M, 7 (t)
= V cos
n
M
= H cos (u t +
n
n
v
H
(t)
v
+ £
6
n)
;
,
and
n)
where
= amplitude of the vertical moment response to the
V
nth wave
= amplitude of the horizontal moment response to
H
nth wave
a)
= frequency of the nth wave
e
= phase angle of the vertical moment with respect
to a wave with its crest amidship
9
n
= phase angle of the horizontal moment with respect
to a wave with its crest amidship
then
Mm
T
(t)
=
A-
n-1
V cos
n
(to
v
n
t +
£
)
+ K
)
+ KH
n'
Lh
n=l n
+
cos(ft)
t
v
n
6
or
on
Mm
T
(t)
=
A
n=l
cos(ft) t + e
[v
L
n
which can be rewritten as
n
n
n
cos(u
t +
^
n
6
)
n'J
n'
)
11
M„(t) =
(v
cos e
x
I
n
n=l
-
sin
(V
v
n
e
n
+ KH cos
n
+ KH sin
9
n
n
n
6
)
n'
cos u t
n
sin u t
n
)
'
i
Let
X
Y
n
= V cos
n
e
n
= V sin
n
e
n
+ KH cos
n
n
+ KH sin
n
n
6
n
Then,
n=l
Since M
(t)
L
X cos
n
to
n
t + Y sin w t
n
n
(4)
is also a continuous random function of time, it
can also be represented by:
Mm
(t) =
T
i
A comparison of equations
6
n
= tan
Z
Y
n=l
-1 Y
n
(4)
n,
~v~
n
cos
(a)
v
and
and
n
t +
(5)
?
Z
n
e
n
(5)
)
;
shows that
= X
2
n
+ Y
2
n
12
Then
Z
n
= V
Equation
+ K H
n
(6)
+ 2KH V cos(e
v
nn
n
nrr
-
6
(6)
)
gives the amplitude of the combined
moment due to vertical and horizontal bending, with the
phase difference in the occurence of the component peaks
taken into consideration.
The MIT 5-D seakeeping program
calculates the ship motions and the vertical and horizontal
moment responses to regular waves (RAO)
.
A slight modifica-
tion of the program to obtain the equivalent combined moment
response operator was made by use of equation (6).
The RAO's
were then used in the usual power spectral analysis in the
frequency domain to obtain the variance of the responses in
irregular seas.
Comparison of predicted and experimental re-
sults have shown generally good agreement [15, 16, 21, 22, 23]
and have led to widespread acceptance of the superposition
method of St. Denis and Pierson.
2)
Second Method
Considering the wave loads as random variables, it is
generally accepted that the bending moments M^ and M
are sta-
tionary, narrow-band, gaussian processes with zero means for
short periods of time.
Then M
and gaussian with zero mean,
EM
= E
is also stationary, narrow-band
i.e., in statistical notations:
^
+ KE M
=0
13
Furthermore,
= E
E M„
K
+
2
2
m nY
"
14
= E
H]
+ K E M
^
+ 2KE
H
M
H
but the covariance is defined as
E MyM
where m^ = m
H
A/hHK" ^)
2
e
]
m "
[( h
h
™uf}}
= mean values =
fL„ - correlation coefficient of the vertical and
horizontal bending moments
therefore,
A
MI
2k
3
/ vh(
e
K
7
+ K E
M
R]
+
H
For a zero-mean process, the mean square is equal to the
variance.
(7)
14
As in the first method, the value of the section
modulus ratio,
K,
must be known for a given ship or esti-
mated during preliminary design.
An attempt was made to
examine the section modulus ratio of several sizes of tankers
to determine if there was a trend with the growth in the size
of these ships.
As shown in figure II. 2, the points are
widely scattered with no apparent trend.
Since the value of
K depends on the construction of the ship
(framing system,
scantlings, etc), perhaps the scatter of points is indicative of the present uncertainty about both sea loads and the
strength of ships, and the rather arbitrary way a "safety
factor" is arrived at.
Likewise, the value of the correlation coefficient,
/>
,
of the vertical and horizontal moments must be deter-
Implicit in the use of equation
mined.
that an average value of
/O
(7)
is the assumption
already takes into account the
phase difference of the peaks of M^ and M
in all conditions
at sea.
With this method, the bending moment response amplitude operators may be determined by experiment or computer
techniques.
Spectral analysis could then be performed to
obtain the variances of M^ and M
moment M
,
and finally the combined
CO
C
o
H
15
ro
O
X
H
2=
O
^^
CO
M
3
cfl
H
H
s
o
CN
C/2
•
>
M
M
o
•H
cu
4-1
u
3
P^
M
•H
Pn
CO
CO
3
H
3
t3
O
S
3
O
•i-l
4-1
o
CD
W
3
O O
S
-W
>
^
4J
C
O
o
CU
CO
Pi
^
.
16
Long-Term Wave Load Analysis
II. 2
After the statistics of the short-term response are
determined for the range of sea states that make up the
actual seaway, the expected long-term loads may be predicted.
Several methods to obtain the long-term distribution of wave
loads and the prediction of extreme values have been proposed
[6,
9,
11,
18,
23].
Mansour [18] has described procedures
for the analysis of failure under extreme loads under two
conditions:
1.
Short-term analysis
-
when the route of the ship
is not known and a "severest condition" criterion
is imposed.
2.
Long-term analysis
-
If the ship's route is known
and if that route is more or less permanent.
In this study, a procedure is discussed to determine
the value of the parameter k = m of equations
(2)
and
(3)
for a ship travelling oblique to waves in all sea conditions
described by the significant wave height and period (h,T) as
The following assumptions are made:
in [24].
1.
wave headings are uniformly distributed over all
directions
17
2.
of the sea.
The speed is essentially determined by the state
The speed-significant wave height relation is
adopted from [14] as follows:
Speed, V
significant wave height, h
knots
ft,
less than 12.5
12.5 to 21.5
greater than 21.5
16
12
6
The procedure followed is described below:
1.
Obtain the RMS response to all sea states (i.e.,
range of sea spectra)
in all conditions of speed
and wave headings by some analytical method as
the "MIT 5-D seakeeping program."
2.
For each speed, get the average of the RMS
response over all wave directions.
3.
Use Bretschneider
'
s
two -parameter family of sea
spectra to get the response to any sea described
by a particular (h,T) pair.
This is most easily
done by scaling the responses to the sea spectra
of step
(1)
.
We call this matrix of the response
R with elements
4.
r.
..
ID
For each wave height in [24], determine the
probability (i.e., relative frequency of occurence)
.
.
18
of each observed period.
We call the resulting
matrix of probabilities P with elements
5.
p.
.
Determine the average RMS response for each
significant height by the relation
m. =
i
7
r
j
.
.
ID
where the subscripts
x
p.
i
and
columns of a matrix.
.
ID
j
refer to rows and
The resulting column
matrix is called M.
6.
Determine the probability of occurrence of each
significant wave height.
matrix Q with elements
7.
We call this column
q.
The mean or expected value of the RMS of the
response for a particular area is given by
mean RMS =
J
m.
x q.
l
Then the desired parameter m = 1.25 x (mean RMS
of the response)
analysis [18].
in the case of a short-term
For the long-term analysis, a
time factor is additionally introduced to get the
19
weighted mean of the RMS of the response for all
areas in the ship's route.
Steps 4-6 is equivalent to getting the probability
of occurrence of each
scribed.
(h,T)
pair in any one area being de-
If we call this matrix P'
mean RMS =
7
h
i
Y
h
r
.
with elements
.
iiJ
p!
.
ID
,
then
x p'.
r
i]J
.
D
The data in [24] have been computerized and could
be used in the above calculations.
20
II. 3
Details of the Procedure
The methods for wave load analysis and prediction, as
explained in the foregoing sections of this chapter, were
applied in the analysis of a large tanker.
The UNIVERSE
IRELAND was one of five vessels of different sizes instrumented
for full-scale measurement of stresses experienced in actual
service.
However, the strain gage arrangement on deck was
such that the stress due to lateral bending was cancelled out,
There-
so that only vertical bending stresses were recorded.
fore, the results derived for the response of primary interest,
the combined moment, could not be verified.
The description
and analysis of the results of the 3^-year project sponsored
by the American Bureau of Shipping (ABS)
is reported in
[25]
and provide the basis for comparison with the analytically
derived values of vertical bending stress at amidship.
The general characteristics of the UNIVERSE IRELAND
are listed below:
Full load displacement, tons
Length overall, ft
Length between perpendiculars, ft
Breadth, ft
Depth, ft
ft
Design draft (keel)
(kwl)
Block coefficient
Section modulus, top, in 2 -ft
Section modulus, side, in 2 -ft
,
...
375,811.
1135.17
1083.
174.87
10 5.
81-0-5
0.86
566,794.
617,350.
The ship's lines and other data pertaining to the
ship necessary to generate the input for the computer analysis
21
were obtained from the ABS
.
It was necessary to use an
approximate lightship weight distribution because of an
apparent discrepancy in the distribution curve provided.
Since the still water bending moment, at least, is very
sensitive to the distribution of the weight, a probable
error was anticipated in the calculated values.
For ex-
ample, the maximum still water bending moment calculated
by the computer exceeded the value determined by the ABS
by about 11.5%.
The MIT 5-D seakeeping program is fully described
Only minor modifications were necessary to do the
in [26].
quasi-static combined bending moment analysis.
The re-
sponse to long-crested irregular seas were used in conjunction with an auxiliary program to obtain the responses to
short-crested seas.
This program uses the cosine-squared
spreading function
s(/<)
=\
cos
2
(/*)
where/tt= angle between the wind direction and the wave
components.
The principal wind directions were given ex-
actly the same values as the wave-to-course headings.
The computer was run for ship's speed of
16 knots,
nine wave headings from
to 180
6,
12 and
in increments of
22
22.5
,
and nine sea states
height and period.
described by the significant
The computer output consisted of the
root mean square of the response, the significant height
(H
/?
)
and the average of the l/10th highest (H no
)
of the
response measured peak-to-peak for all combinations of the
three speeds, nine wave directions and nine sea states.
Since it was desired to compare the values of the vertical,
horizontal and combined wave moments, output was obtained
for all three responses.
Furthermore, to be as accurate
as possible, the three responses were calculated for twenty
stations specified along the length of the ship, even if
eventually only the responses amidships were utilized in
the analysis.
The sea data were carefully chosen to match the
actual seaway experienced by the ship during the instru-
mentation period.
This was done by obtaining sea data from
[24] for the areas lying along the service route of the
UNIVERSE IRELAND, which made regular runs between the Persian
Gulf and Western Europe by way of the cape of Good Hope [25].
Although the seakeeping program can accept and use an actual
sea spectrum or simulate developing or decaying seas by the
Bretschneider two-parameter spectrum, the Pierson-Moskowitz
one-parameter family for fully developed seas was chosen.
Nine sea states with significant heights ranging from 3.6 feet
to 43.3 feet
(corresponding to observed periods of 1.5 seconds
23
to 18.5 seconds) were used.
Spectral ordinates for forty
frequencies were calculated while twenty five points were
used to describe the response amplitude operators covering
the wave frequency range 0.1
<
- —
< 9.0
with most of the
Li
points spaced closely around the expected spectral peak at
- = 1.0.
A
is the wavelength and L is the length of the
Li
ship.
The response to regular waves (RAO's) were not
printed in order to save on the cost of running the computer.
Chapter III
RESULTS
The following general areas of study were explored:
1.
Comparison of the derived results with those
in other published works on the subject.
2.
Comparison of the variance of the combined
moment obtained from the two methods described
in the previous chapter.
3.
Comparison of theoretical calculations for the
vertical bending stress with full-scale
measurements.
4.
Examination of bending moments trends and wave
load prediction.
The results of the above studies are presented in
the following sections.
i
24
25
III.l
Comparison of Results with Other Published Works
The calculated value of the root-mean- square (RMS)
of the bending moment responses are plotted in figures III.l
to III.
4
as a function of the wave headings.
For contrast
this was done for the responses to long-crested and short-
crested fully developed seas with significant wave heights
of 7.4 ft., 19.2 ft. and
4
3.3 ft.
It was intended to show
the response to a low, moderate and high sea states.
Since
only the value of the horizontal moment in "realistic" seas
are of interest, the RMS of the response were plotted only
for short-crested seas.
The graphs were prepared for a
representative ship's speed.
The general behavior of the
response did not change with speed.
Faltinsen [22] describes the comparison between
theory and experiments of tne wave-induced horizontal and
vertical forces and moments of a series 60 hull form with a
block coefficient of 0.80 and a Froude member of 0.15 moving
in regular waves with different headings at an angle to the
direction of the waves.
The predictions made by the new
strip theory developed by Salvesen, Tuck, and Faltinsen,
which is used in the MIT 5-D seakeeping program proved very
satisfactory.
[27].
The theory is described by Salvesen et al, in
For vertical bending moments, the theory predicted
higher values than the experiments for
-
<
0.7 in quartering
and following waves and lower values for X/L
waves.
<
0.6 in bow
This trend may also be observed by comparing the RMS
0]
26
(D
u
to
13
60
C
T3
CD
>
a
•H
.n
en
T3
•H
E
<
to
a)
in
ro
tH
•u
>
cfl
cfl
03
T3
c
O
4J
ex
a)
co
I-l
co
cu
•
cu
u
H
M
H
Pi
0)
QJ
e
60
•H
PH
u
I
4-1
!-J
a
C_)
0"\
3
Q)
o
a
60
O
j
c
•H
Bending
Heading
the
of
in
<r
to
/"N
O
c
CO
CN
o
60 O-
C W
•H 0)
T3 OS
c
CU
oa
c
o
u
1
4-1
UJ
4-1
c
0)
w S
S O
M S
m
o
H
X
N^
CO
VS
— —— —
!
—j —7
—
1.!
,
'
-
'
'
'
——
'
!
—
V
,
—B
L
-1?|
.
.
1
'
—
.
—
-
-
'
1
o>
01
to
27
60
01
w
60
c
#:if?: =tRT
.....
:::i
...J.
::.::
'
'
— ——
J_J_
;
+^
'
iiji
|
!
1
#=4t
i
i
-lil
__—
H~
'
'
'
4m
\Wl
t
-^ —
-H4
|LL
—
jftt
i
1
1
--4---ffl+
|
\
1
j
.
.
j
i
-f-
"P"
H
'
1
—4—-U
;
rH
'
II
i
I
I
—
I
-i —H-jfff
j
--I "TTTTTT
t
.
i
'
•
'
*•
,
-=rrr
!
.
—
:
-iii
"
'
_i
'
;
—
1
i
Li.
'
1
—
!
|
—
'
;
_J
:
,
i
}.
j
ill'
-ti-Uljl;
nj
j
0)
:
s
1
-^
•
— M—
*
-
-rH
r
'
!
'
1
'
-
-J
——
(
i
*
i
i
-j
ea
iH
!
-t— %4^
—
:
)
III
'iii 44+"
-H-h-
1
——
—
-1
-4+J
\
-f+f _
|t)|
c/o
>
tttt
llll
'
i
1
1
.
-
.
'
..
•^T—
-*——'!
'
'
,
LLj_ 4-
^rr~
M
iLil
:te
4
—
— T^
-ft4 ^*T^7
J
/. .-
4
Bp
.
/
m—
J
4-1
-l
—
tfl
Vj
01
U-l
o
i—
1
to
\t—
C
O
—
i4—
-
-::
H
tft
/
j
|
s s
o
<
^7-
-i
J
01
W
•H
+|
trt /iU-
*H-L
r
ca
j
r
J4
U5
T3
t?=
:i^ ^t
'
/iL
-j4lf
r-^
1
#
p.
•H
^r
M
L ^-
/r
1
i
—
—
—
4—
i
K
i
--
y
.<.... J_)_L
A- -y4 r—
[n
^
;
-
i
-^ =nz
V-te
rVr-
1
|
/]
=5=f
—
'
/
1
t-
|j|
'
1
j
:
!
'
_^
llll
'
1
.
i
ttttH-
(1)
4-i
'
:
,
1
Tl
-M
T+T
'
-
-t-
....
'
ii^ibr
'
-t+Jf
!
..i
y*
|
1
'
•
:
ILUj
1
.
~it-^_
•
h-H-
tHt
--»-(-
1
-t-
'
4-lii
i
ftffl
r—
T-j
—
i
Mi
1
|~i
i
:
_L_|_L
-r+—U—
i
|
1
'
i
^—
tffjjtt
j
-
'
-J-4-M-
---•
T ;, T
-' f
1
-|
"4±SE
-.-gipEf
-
:
44-
—
'
'
— 5- 4^ —
+
4-i
-JTtT
L
1
+Hf
|)4f _i^ IIU
1
-
-j
4h
J
__niBl
4
-H-H-
1
i
•*—
—
~rtr-
t: +
m
-HH-
M
I
III
II
—
-
i
'
'
ttfH
:t|S|
11
1
1
1
=
j:
:
1
'
i
'
1
CN
•
M
M
H
—
r it
CO
0)
(X
0)
CO
c/>
01
Pi -a
01
4J
4J
0)
C
CO
M
3
0)
0)
,
!
:
-4+
:
.
;
—
|
i
|
~—
'
'
p
i
i
_.
,
1
,
I'li
4
_,
*
i
i
'
1
-
r
-7— ^
'
i
i
i
i
,
rt
i
:.
/
I
'
1
i
1
'
f
-++|
*t=
J
i
—
—A*
'
i
-i~
—....
1"f->'-
tHt
—
ffllffl
4- ::|ffi
-1: 4-- —
1
'
1
....
tifr —-- l-t-j
r TT
——
|
.
'
:
'
1
4i
M-
*
—J
•
.
—
(-U-U-
1'ij
—
—H —
j-{-4-
—
'
;
'
1
1
j
-
r i
i
ttr^
[t=|
m
—
h=
#t
i
|].i_
—
;
-H--
•
Ji+J- 4-H-
itti
-t— *•*-'
',
-
-
—
•H ^3
T3 CO
^ut
13
0)
\
Ol
4f —
1
1
.
i
ii±n
H
pmJ
—l
'iii
'
'
'
it
—
±t=
IT)
1
•
—
txa
::it
j
1
^4
4-j-rf
itir
Ljl
....
::::
~±
i,p4
-
-::
T-f-TT
:: :-
T
1
CO
O
^s
X
C
o
3
60 a.
CO
<-
•H 01
T3 OS
4J
u
CQ
C
01
to
2
orf
1
U-l
0)
e
o
S
7~rr
O
;
:
0)
13
ffff
"•
~-
C
r:::
!.!
<3-
O
rH
X
*w^
C^l
~:
-rl
f
FF
LT1
iw
CO
O
01
oi
;.::
.
C
*J
w
S
j
"Hrt"
60
^3
T3
-•
1
C
+4
i
1
>-~—
-—H-
I
4->
CQ -H
\---
j
2
^
C O
i-H
-
1
60
•H
>4
E H
o u
60
T*—
*
4444-
f4
H"
c^
-
\;
-rj-h-
i
1
-
'
'
"T"
-M-^
::
.
h^-4-
\
I-
1
!
-it: "_:
..
L
r
1
j
1
"Tr
i
1
W
——
1
—
'
—-
7
_
p—
*
-
'
i
—
—
th
'
— —
111
,
1
0)
28
0)
u
60
60
m
-T — per 1
1
•
'
1
'
4=r
urn -H— tOT
'
^+i
t-
:
i-Ji.
H
T3
—
,
4-j-f-J-
Efl
r-—-
ttrr
0)
T'j
.
i
—
-44-trrf-
1
•
:
—
t
t
::
1
--+
-h—
33
t
-X
.
.
3
.
,
+-H
Hrrr
!
+
-
>
a
.
.
4
fl
-r
j-4-H
-+-
—
—I4j-
h-
44
\±—
4=
j
1
,
60
o
n
C
:o
•H
X)
t-f,
fl
4 J-M-
i
0)
•
i
-f—
t-*?r
PC
u
-1
+fc|
——
.4
CO
>
to
CO
0)
i
—
—
i
t-i
i4+
a- co
•H
,e cu
—
"
-
;
1
CO
-u
-a
•H
n
6
cu
;
-i;
-|+H-
1
p-t
—--/
:;•:
<; -a
•
t
frrr4
o
1
*->
-J--J-T
tnt
>
L-
1
£
—
—f-
1
;
—
....
ill
7=r=
^— -H-
"tt^ —
—H-
!
-44
-I-4-1-
'
1
-i
Mm H
-^
-U—
4f(t
1
.
44-
-
M
'
,
i
1
!
1
-f-
}
^fct/
—
-H+r
l
111
,
1
i~
h—
1
-++f+
[
i
i
4—
±(4
1
"~—
—
-
—
Tn+
•-N
CO
o
c
o
(0
4J
M a
C
•H
0)
T3
B£5
C
a>
IM
4J
«
c
w
e
o
^^
ty!
1
4J
0)
m
o
^H
X
\^
1
!
[
{
..
-
-t-M4
ii
1
W
'ii
o
'•'pill
1
—
-+
—
:;--
—
—r;
-
-j
TfFj
•u *o
Cn
C
CU
cu
jj
S
co
O
0)
U-
m
-d-
C
,G
CO
r|
trfj
-
;.-.-
-
-
—
X!
iw
CO
:.:::
—
—
p— __
.:::
....
2
.....
::
erf
t::.
-
- -
•
1
~rrr
:
..:.
-
=rn
in
;•-;
•ro
u
;;r
CM
- -
•
t
1
—
'||j
o
:•::
O
a
CU
u
....
1
w
Vj
i
1!
u
o
•H
T3
::'
*
FFh
CO
Di 12
60
•H
CU
,
"
FT
>
CU
«
4+tt
1
::::
-'
-
1
i
CO
C
1
i
OJ
M
60
1
--*-++-
*
CO
cu
S
J-dL
'
*
:
o
CO
CH
~rrr -rtt
!
,
^
CO
C
:
U4
:::
CO
l
;
=E
<D
111
..L_
—
-
.-
j
^jJl
'
\
j
1
-44--H+
r^+—
-
...u -M-l"
ii
*-i
0J
a.
^-i
|
— r;
-j-L-i-
--
1
1
S
ca
c
O
r
i
1
-
-
—
!
-|
1
i
—
ii
'
^^
i=\
'
-r_-
'
'
/
j
-
'
'
Mi
-U-I4
!
.
—I
-1
4444-
-
.
—
=
1
jl;|
|
/
,
\-j-\
^trr
—
-
—h
/
l
1
m — -^
#
jf-
4-L
M
M
M
'
i
j
1
-:
f:
#:
—
-i+i|
,
4f
+)-
•
__
-i— -J
1
1-4+
rt4
r
'
n
t
i
1
i-L
,
'
rm
cd
-H
•\
co
ID
11)
29
l-i
60
<U
v
%«•
60
c
•H
CO
(!)
SS
a;
>
cfl
S
60
c
o
CO
H
•H
T3
Hea
VS
Seas
idship
h
m
on
H
e
<
60
-H
4-1
cfl
<r
•
H
H
l-l
0)
co
C
o
a
CO
O
00
<D
a;
l-l
Pi
3
60
•H
U
o
U-l
co
cu
>
CO
S
T3
4-1
ID
C
4J
Pn
Mome
Short-Cres
Bending
in
en
the
to
S
PS
CO
•—
C
o
CO
e
60 a,
o
a
(J
CO
•H a>
-a BS
1
4J
14-1
0)
4J
S3
a
CU
Cfl
=
A
o
pel
^
in
o
rH
X
v—
30
values of
ML^
in short-crested seas for wave headings of 0°
(following seas) and 180
and
4.
(head seas)
in figures III. 2,
3
For horizontal bending moments, appreciable diff-
erence existed between theory and experiments only for beam
seas.
One of the early investigations of the effect of
horizontal moments in oblique seas is described by Numata
[16].
It was established in this study that lateral moments
reach values that are substantial and that peak bending
moment exists in the region of the speed for synchronous
pitching and heaving motions.
Wahab [15] shows that the
importance of the horizontal bending moment increases with
increasing ship length.
For the sea condition used in his
investigation, the significant midship horizontal moment for
a ship of 200 meter-length was 46 per cent of the significant
vertical moment while for a 400 meter-length ship the percentage increased to 69.
L
,
The concept of effective wavelength,
described by Numata was helpful to explain the results
obtained from the computer analysis.
However, only one sea
spectrum was used in his study and some conclusions were
reached that are not necessarily true for all conditions at
sea for any given ship.
Inasmuch as the RAO of the UNIVERSE
IRELAND was not recorded by the computer, the figures in
[15] proved very useful.
The ship studied in that work is
the same as the model described by Faltinsen and has general
characteristics similar to the UNIVERSE IRELAND.
The respcnse
31
of the model to regular waves could be assumed to be sub-
stantially identical in shape and trends.
The calculation
of amidship forces and bending moments in [15] also made
use of a single spectrum representing a severe sea.
32
III. 2
Comparison of the Calculated Values of the Combined
Moment Response
Table III-l is a summary of the results obtained
for the RMS value of the combined moment response for the
vessels' service speeds of 16 and
6
The first set
knots.
of values are those calculated from the RAO developed by the
computer by use of equation
(6)
of the first method.
second set of values were calculated from equation
the second method using the value/3
(7)
The
of
= 0.32 obtained from
[18] as reprinted from the published result of a Det Norske
Veritas Research Report.
figure III.
5.
The reprinted source is shown as
The values of P
represent the correlation
coefficient of the most probable largest values of the
vertical and horizontal moment for one ship life of about
twenty years in the North Atlantic.
They were meant to be
reference values and not for use in design.
in M
values is shown for each sea state.
The difference
It is immediately
apparent that the results obtained by both methods are not
very different.
The third set of values was calculated also
by use of equation
(7)
but using a value of
/^
obtained by
averaging the responses in short-crested seas by the first
method for all wind-wave directions and for three representative spectra for low, moderate and high seas.
The mean
value of p. 7ZJ obtained in this manner was 0.532, significantly
VH
The difference in the results for
higher than the ISSC value.
'
M
is also indicated for comparison.
33
0)
o
a)
CO
b^s
a)
oo
M-l
4-1
+
•H
T3
13
a)
2
ST
oo
sr
oo
o O
CO
sr
CO
CM
1
1
1
1
1
CM
1
+
o
o
rH
o O o rH m o O
CM m m o rH
o rH
m
CO
sr
oo
<r
rH
vO
CM
CM
rH
rH
+
+
I
I
1
oo
CM
CM
Sl-
r».
CJV
rH
VO
O
rH
St
CM
CM
av
OV
r^
a\
CO
00
CO
rH
CM
+ + +
1
<D
01
O C
J4J
O
O
,-v
CO
rH
rH
vO
ex co
to
<u
Pd
i3
m
o
•
CJV
CM
a\
CJv
r-.
r-»
rr
ii
oo
<f
o oo
00
sT
o
m
CO
o
a\
O
O
rH
CM
o
O
O
CO
in
rH
o\
CO
rH
CM
m
rH
CM
rs
00
co
rH
NC cocl_>
S'-'
CM
CM
vO
rH
vO
00
CM
rH
O
CJV
o m vO rH
o
CM
m
sr Ov
00 CM
rH
CO
CO sr co
rH
m
f» o
CM
m
m
O CM
rH rH
r-»
CJV
r-~
is.
CJV
cj
u
8-3
CM
CJv
CM
00
m O
CJv
O
cjv
CO
vO
00
CO
r~
co
CM
rH
OV
rH
rH
o
rH
cjv
oo
VO
vO
CM
sT
rH
vO
oo
r--
vO
I
I
1
1
H
rH
rH
1
I
1
1
1
1
1
1
CO
rH
CJv
r>»
vO
CM
CM
00
sT
rH
rH
<D
+
4-1
•H
1
+
1
m
m
sr
i
i
oo
X)
a
c3
iw
0) /-^.
oi
n
o fitn
JZ o
•
J-l
o.
0)
01
2
t3
fj
CM
T3
o
II
j
<U<s^J
O
w w
2 CO
H
S Nw'
rH
m 00
r^
m
o
00
r^
in
m
O
O
rH
r^
CO
m
sr
o
CO
rH
vO
CM
O
rH
CM
CO
m vO
rH
sT
CJv
CM
CM
rs
oo
00
sr
oo
CO
CM
rH
00
00
OV
vO
st
VO
«l
rH
o
CM
in m
m CM
m
CM
rH
rH
C\ 00
CM
en o
CM
CT\
sj-
r^
CO
CJV
IS.
o vO
CM
oo
m o m
cjv
r-«
VO
in
rH
vo
co
CO
rH
m
00
CJV
CJV
rH
rH
rH
0)
to
o a
o
•u
a,
tu
to
<D
Pd
m
O vO
rH
vO
m m
vO
m
O m
rH
o 00
s*
rH
in
CM
CO
vO
00
vo
r-^
>d-
vO
CM
4J
CO
CT\
CJv
JC
2
vO
CO
H 2
CM
VO
<3"
CO
a\
ovO
in
r-.
m
vO
00
rH
rH
ov
VO
O
o
m
o
rH
av
00
CM
vO
r^
vo
00
CO
CM
CM
CM
rH
rH
CM
in
s]-
rH
ST
VO
St
rH
rH
CO
CM
CTV
CO
m oo
o
o
vo
o
CO
m
p^
m
r-
vO
CM
CO
O
sr
cjv
rH
rH
st
CM
m
O
CM vO
CO
CM
rH
rH
CM
CO
CO
rH
CJv
vo
VO
sr
vo
in
sr
CM
rs
sr
CO
vO
co
CO
sr
m
O H
rH rH
rs.
•U •-s
C w
n)
O "—^
i*-i
•H
M-l
vO
ST
CO
t-~
4-1
C
oo
60 -H
•H a)
CO JE
r^
O
rH
vO
CM
CO
sr
rH
CTi
<r
rH
CM
o
o
CO
-3"
CO
VO
si-
vO
CO
CO
St
CO
rs.
rH
o
o
CO
>v
4J y-v
•H
CO
O 4J
O O
rH C
0)
>
^
^i
vO
rH
II
>
vO
M
>
<
~—
V
n
f
1
1
'
—
1
1
:
1
11
——
1
!
:
—
1
1
!
—& —"—
'
——
—
-
1
i
1,
— —
-
-
.
1
,'
j
34
,
-u.
4
^T+*.
- _. U.
'
'
'
;
'
l
:
l
!
i
i
III
!
I
!
1
1
1
1
1
1
;
1
II
1
—
I
-
'
III
1
!
J
;
,
'
||
i
!
.
i
i
|
;4igg 44— T~~~
I,,
#fe4— rm
H-
^*_
<.:.;.:.
1
4*^7
j
—
TTTT
1
!
!
1
i
;
|
]
|
r-
H
1
1
frw
"B4-I4-H
J
1
1
'II'
fc~r
—-- ted
t
'
III!
+-
:
1
v^fr
-^—
JiLwil
W
1—
i
r
1
|
i
1
TTTT
II
1
i
KH+
5sE
j*v
1
H
-LL4.
1""
w +-$
H—
J
j
'
1
I
H"
'
.
'
1
1
.
|
'
i
i
•2
./^--±
--it
I
I
.
'
;;-!-_
9
,
10
/->
o
"wjUl -H-H
•:::;
III
1
*
L__
4U4- 44U.
TMT
——~+4
— —m
£
III
,
,.
Tli
1
1
1
HI
1
:z
£
1
;:tr
j
1
1
1
"5
2
,
CJ
,
—
'
1
M
*.
5
:::
±
::::=
j
.
Wm
"S
ffi
4_
T
Ml
h~i~j-
—
-.
1
—
•
'
'
1
1.
-*-
1
++++1 4+H-
4
-r-t-
4-f-H-
i
1
-M-U-
Mil
={jt
—
i
c ^/
^
fa—
ii
+v
,-
—
.
/
//
i
7M^
_j
TT7?
1
'
^^ 4—
y-l-? 1
/
'
/
i
'•-:~
-
h
-\
-
7*^T^-
-
,
,
||
-L
|
•
Ul
'/»
Ml
'TPT
1
Wr
\
Mil
1
n
1
-.
nil
1
I
j
1
-LLU
1
I
1
—
I
fe
-;—
'
wfr jjji-
——r-^
•
i
-----
i
-
I
—
rrrr
H
i
'
:4^'
1
fftr
4-J-J4-
t
M4
.
t
<
_
-444-I
1
—
|
{'
-iii
44-ii
.
1
+—
rtrr
1
.
xr.
;
\-\-
i
4rr ^._^
:,
jn*
-
H-j-j
.....
.
-
....
.
.
.
y.
::::
j
|
1
l
Ll-j":^
::;:
rjHr
mi
44-
151 S Ml n\Ci .:kjt "kg/
W.4 u
:
-
[
[
[
j
••^h
tffi
10
—
J-0.
....
U-i
1
1
|
....
:r::
—
f
::::
...
--
—
-
....
....
::
—
'
._..-./
:
|
1
-j
•jjjt 44
—
——
:
]
N4
....
/
^rp
:
..
,
-TTTT
-ftH-
i-
i
1
t1t trr
-
4- -j\f
^^
i
44 4=
11
.::::
J±te
'i
||gg
f:*
t§M
o —
!--
'
.?/..:
-
XLj-j.
1
>
1
11 tK- jfjf
j
,
tHt
/ •'A','
-j
—r--
1
,
ij
'
##ff
I
1
'
nrr ::
\
!
fl
|||| ; |l|-k
"i
::fl4|j|
.^
,
4+4r
:
1
1
—
/-*
i.i
,
'
ji)|
:::
'
'
m^ M m P ^
5 jilt
TTT~ Ml]
1
,
r^-'
-•
....
-ftp
.
|
-4
4i
-
--H
!
j
— f
H
h"
ii
.
ilTi'
...
H+f ~n~
;
.
1
::::f:tr::
44
*•"
"T
i
—-+
,
4-j
:
1
^-+4
-
±55t >
j|U 44-U- -H
~^r
r.
-
'
1
.
-jiff
2
i
i
cD
2
o
t
'
1
.
-A
'
—
....
trrr
/
I
~,
-i-i-M-
:;-
:
i
-t-W— 4+4J-
j
M
1
1
o
:
4444-
1
1
LUi
::::
jL
i
1
'
i
1
I
1
•^
1
1
-H+f
—
1
7 ::::: -w-
i
!
:
M-4-
(—(
"Tt"
...
1
j
I
::::: :: 4::
^
—i
1
1
— -,.
1
——
I
""TT
iHt
-44-+U+
,
!
-ft~
/A
L^L^ _i_
J-+—
i
—
.^^-
1
—X4
-rjH-
*
j
4l~
""TT
I-
::
-j-H-!-
::#4^
^ 8p
—
t—
—
Tl
_
i
\
^
-»•
^
-U^L
i
i
1
::ij:4^U-
H.
,
1
,
!
,
7t
."
>
>°+:^
V
/
!
</•
£o
w
'
^1
"/
y
tf->
/• /
1
:
j
—
if-
1
;li
!'<
p-
j
i
-----
;
—
i:..
1
i
1
.
i
-
10
:
icsn i~X~
1
H ||i+
'j
,
30
i
m
....
J
{_;
tHt^ ittt
—
_j
[_f.
.
.
j
.
j
—
-~'
{
rr-
*
....
4+4
4^
4U44.
1
'
.
40
Comparison of the RMS Midship Combined Moment Response
as Computed by Two Methods
—
35
-8
Series 60, All Headings Included,
Q=l
C B =0.6
F^-OZO
C B - 0.7
<V°
8 Fn r
20
o.\5
\.Q
OX5L
AFT OP
FP
P
05
£5?^^
-
—
-
-
"
o.o
100
100
10
300
L
(ml
400
AMIDSHIPS
•
OS
^r--^^^^
—
oo
ZOO
(OO
10
015L
AFT
OF
300
l{
m
WO
)
FP
P
05
—
^un
oo
too
zoo
300
Figure III.
u(m}
AOO
5
Correlation Coefficient Between Vertical Bending Moment
and Horizontal Bending Moment
36
Comparison of the Calculated Vertical Bending Stress
III. 3
and Full-scale measurements
;
Strain gages were installed on the port and starboard
sides at amidship and several other stations along the length
of the UNIVERSE IRELAND to measure stresses on the deck.
The
arrangement was such that only vertical bending stresses were
measured.
Therefore, though the combined moment could not be
verified, a comparison between full scale data and the com-
puter-derived values was possible.
This comparison by
weather group as specified by wind measurements (Beaufort
scale)
is shown in figure III. 6.
sented in table III.
2.
The conversion from bending moment
to stress is done by the relation:
Section Modulus.
A summary is also pre-
Bending Moment = Stress x
The values compared are the average RMS
values of the peak-to-peak stress for each wind scale.
full scale data were read off figure 11 of [25].
The
The con-
version from wind to wave was done by the graphical relation
shown in figure III.
in table III.
7
[23].
This conversion is summarized
3.
The use of nine sea spectra with each significant
wave height enables the calculation of the standard deviation of the predicted values of the vertical stress.
For
comparison with the measured data, these standard deviations
had to be modified to obtain values based on wind measurements.
This was accomplished by the relation [23]:
8
37
S
where
S.
2
l
V
2
"
+
S
2
- AH
J
tan
2
6
2
= standard deviation of the response with respect
to the wind
S„ = standard deviation of the response with respect
to waves
S_ = standard deviation of the wave height with re-
spect to the wind, as shown in figure III.
from C23 J.
AH = wave height increment
8
= average slope of the response vs.
height plotted in figure III.
For comparison, table III.
4
significant
9.
of the standard deviations fo:
all the cases studied is drawn up.
)
38
RMS
to-Peak
ihip Vertical
ing Stress
•age
10
8
(91)
(319M628) (791)
(387' (208) (112) (46)
(2)
(2)
Beaufort No
(No.
of
Records
39
o
•H
•H
•H
.C
O
3
o
CO
O
CO
CM
CM
csi
r-i
ex
^—
-d-
<r
<| -H
vO
co
sr
vD
cn
vo
TJ
E
<D
CO
4-1
fn
SD
m *
o
oo
vo
ff
en
CO
<r
t-H
r-~
o>
as
en
o
co
•o-
CM
<r
o
r^
i-^
<r
N
O
CU
J-l
H
O
u
X.
on
3
O
u
H
1
en
CI)
3
.-1
c
CO
•u
>
M
T3
CO
H
iH
CO
<D
0)
W
Ph
3
o
CO
pi
CO
§
CO
3
^
^
a.
r^
r^
r^
oo
o
o
CN
as
as
co
o
CN
oo
csi
cfl
co
CO
<r
in
cu
a) .C
M) •u
M •H
P
•
•H
X)
O
S
CD
>
CO
X)
t-i
<
0)
u
CO
ai
u
to
o
•H
T3
W)
o
c
^!
•r-l
CO
-H
O
CO
SO
CO
C>
r~~
11
u
-o
Ph
c
S
<u
4-1
o
m
--<
c
w
•H
•H
•H
4-1
P-l
CO
e
o
u
vO
o
CSl
vO
CS|
oo
as
OS
r^-
CN
O
<f
CM
csl
in
T-i
CO
OS
CO
^
CS|
m
O
n)
M U
CO
00
CO
CO
^1
<D
c
o
^
Qi
>
T3
H
3
cn
CO
(U
vH
CO
o m
m o
m
m O
CN
m CN
o
<
OS
m ^o
o
r-~
r^-
CSl
vO
-tf
m
O
O CO
CSl
v£>
r~-
<!
oo
os
t~-
D.
^
^
S3
o
M
O
14-1
3
CO
0)
i-H
CN
CO
m
vo
87
40
feo
to
AO
'JO
>A1«WD
SfEEV
50
CO
10
\CtJ<STS
,
Figure III.
Relationship Between Significant Wave Height
and Wind Speed from Various Sources
il
10
*^
Z
n-Jf^'W:
#
6
e cAuf°«-T
i
a
-S
V
4
5
•m
>>
•O^CVA
0.
"1
"J
30
40
lO
"jo
V>
to
It
10
Figure III.
Wind-Wave Height Standard Deviations
from Various Sources
41
>
o o
-*
oo
CM
CM
CO
-3-
^r
CM
CM
CM
CM
ci
^ m
\o
n
00
CM
00
VO
CO
oo
<*
CM
CM
00
CM
vO
vO
r^
00
<y>
i-i
CO
u-1
13
u
CO
4J
J3
C60
ca
.C
o
3
o
e
•H
(U
ca
o
CM
(3
ct)
0)
!>
oo
.-I
C3
00
13
4-1
CM
•H
CM
CM
o
CO
O
m
CO
CO
vo
13
CO
u
CO
T3
o
u
0)
.13
60
•H
•H
>
CO
O
CD
rs
i
o
i
•a
vO
r^
r-~
O o o
o>
o
00
St
O
CM
00
CM
13
CO
•u
s
CM
J3
O O
CM
CM
vO
CM
o
pi
C
&
O
>.
u
CO
13
>
(3
CO
13
CD
c^
CO
in
oo
CO
-sT
<f
<r
OS
3
13
CO
J-l
o
m
in
m co
m m m
O
vO
CO
cri
CO
O
^i
CO
4J
o
^3
60
•H
s
i
c
o
CO
u
0)
o
oo
CM
CM
od
CO
o
CO
J3
c
0)
vO
H
CJ\
CO
CM
CM
oo
OV
CO
0)
•H
O
z
u
u
o
l*j
CO
CD
«
m
vo
00
CO
~
42
4J
ti
to
o
•H
1-1
4J
S 3
SO O
H
erf
to
CO
CO ex
>
CO
c
O
CO
0)
CX.C
CO
J-l
CI)
Crf
C
•H
&0
-j
u
3
C
CO
•h
-a
n)
cu
c u
QJ
<
pa
60
n)
to
o
•H
Vj
•U
O
u
y-i
0)
>
-u
60
,C -H
OJ
4-1
0)
K
O
01
>
co
rt
2 3
erf
(3
nl
0J
S
\
ID
c
Hi
v©
00
<T>
Ul
U>
o
N>
1-1
rt
a
o
2
/—
Ul
o
o
Ul
»-J
Ul
O
O
O
on
N>
Ul
Ul
Ul
O
O O
O
4»
.p-
^J
Ul
N>
Ul
.p-
P-
o
o o
o
fD
CD
en
fr
ij
e
en
h•—'
w
Co
fD
Cu
rt
Cu
i-i
M
CD
CD
Cu
l-t
Ifl
CO
d
H(D
Hrt
3*
05
O
3
?r
T)
prO
M
P00
VO
H
to
-P~
cn
VD
^J
OJ
<D
U)
00
^J
LO
00
on
U>
VO
on
o
*-
Ul
*VO
Ui
Ui
O
3
O
en
N-'
ro
rt
rt
N
rt
O
CD
H-
O
3
O
It
CL
P-
fD
i-i
n
CD
IG
M
p-
fD
P.
/—
.
-P-
O
U>
ho
N>
VX3
-^1
VO
Ui
-^1
Ul
00
on
ON
00
h-
3
00
1
.p-
.p-
4S
U>
UJ
u>
Ui
h-'
•vj
H- Cu
-P~
4>
o
cn
Hi
P(D
Cu
CD
ro
Ui
H>
H
VO
-P-
u>
^J
00
Ul
Ul
ON
on
K>
H
*~
V©
oo
-o
Ul
~j
-p-
o
Ul
U>
N3
ro
oj
o
>£>
T3
tn
^
H-
3
o
C
o
ro
<
W
§*s
o
&
M
H
M
3*
3
3 Cu
o
H
CD
fD
rt
rt
ptJ
CD
Hrt
Hi
o
H-
<
T3
fD
O
o
fD
si
.
44
Examination of Bending Moment Trends and Wave Load
Prediction
III.
4
Since the traditional primary strength calculations use
the vertical bending moment in a long-crested sea of arbirtary
height as a basis for ship load, it is of interest to compare
the values of the vertical bending response in long-crested head
seas with the combined moment in short-crested seas for the
higher range of the wave significant heights.
Figures III. 10
and III. 11 are drawn for sea spectra representing fully developed
moderate and severe seas, respectively moderate seas, the combined moment for head seas exceeded the long-crested vertical
moment by about thirty sever per cent (37%) and in severe seas,
the difference was sixteen per cent (16%)
Similarly, it was deemed desirable to compare the relative values of the vertical and horizontal moments for all
conditions at sea.
This is done in figures III
the responses in short-crested seas.
2,
4
and
for
6
One can see that the
horizontal moment is not small compared to vertical moment;
and in low to moderate seas, the former is at least as great as
the latter.
The above mentioned comparison of the response in
long-crested and short-crested head seas reflect the relative
contributions of the vertical and horizontal moments to the
total load.
Table III.
5
illustrates the change with speed in the
bending moment response for low, moderate and high seas.
The
tabulated values are the mean RMS peak-to-mean responses in
short-crested seas averaged over all wind-wave directions.
CO
<u
01
45
u
60
CU
60
CO
>
4-1
en
y-i
to
<u
c-g
CO
•
CT.
T3
.-1
<U
II
4-1
4=
CO
<D
U
U O
C_>
60
C
C
o
H
H
—I
H
4-1
CO
CO
O)
J
CO
C
T3
•H
CD
4-1
•
0)
to
CO
CJ
C ^
o u
0)
P.
^4
CO
1
4-1
3
01
60
&,
u
o
4-1
,c
CO
H
H
c
cu
C
S
o
-H
£ w
fi
rH
CU
e
o o
co
•H
S
4-)
M
QJ
T3
0)
>
a
1+4
•H
JD
O B
o
c o
o
CO
H
(-1
cfl
a
e
o
o
c
a>
co
c
o
a E
•H O
* S
CO
C 60 m
•H C
g •H m
T3
o
H
w C
*n
X
^
4-1
1
4-1
CU
(Jj
—
———
'
1
1
'
1
7r
—
1
-
_ —
~f
^
.
:1
t
:
1
!
—
•
—
'
'
1
.1
:
—
T ^—
L
11
<
11
cn
01
01
46
U
60
0)
60
c
:::±:: p::::::::}
-
"ft
.:
:::$:: :::::±:3±
M
^4
- -U||
T+rf
--
-U4I
wM
—
tttt...
_:.
PTTTj
55 ff- TOT
iLU.
4__4
-*tr
1
'
1
itliiiJiLL
r-
gig
4-M--p :1
1
j-j-U
—
4=#
'
1
|
-
-
'
1
i
\f\
1
:::
—
U-
4-4.
fth
1
1
4ttr
1
+H+
i
^44
1
1
in
44-
'
1
-u-
4~ H
._
11
•—
^
tn
—
-f-H-j-
J
444~ :
w
>
irh^
-•>'•
p+
|ii( b..
'
'
!
>
:
mi
1
1
I
-
0)
jjli
trfr
TTT-
ffff
1
1
(l)
1
—
-
'
1
'
1
jfff
-J-J-f-j-
t4
-a
a)
00
01
1—1
CO
TJ
i
|
iirr
-
|i
mF
1
j
4-4-
\
1
J-H+
'
!
il
44::::::i±:
4-
'
::::::
1
|
4¥-
1
1
r
n
,
Jl
xr
j
:
Trr -H4
Mi
ii
1:4+
:::
Bi
:
l
44^
-
:±::: :::±:::£:
—L-L4-U-
1
Tir
t
|
1
——
.-:
LU-i
-
—
j 11
4^
444
j-pj-
j
—
rV-
1"
1
1
tin
|ii
iii
-
X
4+jj-
H
——
!
j
*-
II'
Ttir
~f
j\
1
1
I'M
.4
1
.
j
1
1
1
5
—
TtTtT-Xt
!
r
111
i^ Trit
~TTT
.
—
4
1/
;
,
4+U L~^4
^
+4+ -4
i±±ttt±J
-M-
§
irJ .....J
-1?
:::±±
l-u.—
-~^
4T
;
H
ifjl
-LLu
1
+444-
ft
4*t
TTTT
....
1
M4t^-:--j Mr-
5 44
i
.
-;-;:
]
,
;
Hti
1
4-j
"ftiX
7
^i
S5 —— —
—
J
i
m
Wr .-
1
,
'—
ffip
---
|
4.4
-4i44
.
1../..
1
T^ir
—
4rr-
i
i
ail
-j-l
.
i
-U
ft
—
±ttt
n
1
1
•
-+ +
-
|||g|
5555
--
^o
p.
•H
x:
C
cn
01
c
E
O
u
T3
•H
60
iw
2
•H
•O
CO
C
g
Oi
S
1
4J
cn
pei
C
M
J
m
^4
X
~—
-
y^f
fff4 4
- |
cn
QJ
Pn
H
4"T
u
0)
J
Pi
r
-U
4J
C
t-l
1
0)
6
H
O
&
w
c
•H
CO
O U
c
iJ
a
E
01
O
1
-
>-i
> S
:
<44
TD
O
a)
a
•rl
1
c
;
'
iitl
fHf —
—*-
|i
ij
1
4^X
:::
L-J-M-f-
III
+H
^^
cn
ffi Trrir
*ftt
8
....
rt-j
,
,
...
••
'
M
f
:;:;j
iffi
H:m
s
•H
Htf 4+1 —^—
Hj± Sff
:
1
i
1
II
4-4U.
mf
44 TO
P===
I
TTtH
_:
:
-
M
crj
CX
rH-l
...
QJ
iJ
ft
1
!
....
-(Trr.....I
-a
•H
;
/:-:t:
1
iii
1
i
c
^2
....
::::
#
;
j
frtTi
LU-LILl!
....
"nTr 4+ 4
;
1
CO
(U
S
-
;
1
H
ri
r^
;
4-
tttl H1
1
4-
;
4.
444-
—
—
rO
01
|||
4t
.
44-
±ttt
|
1
in
.
60
XT
11
:
r _«
1,1
1
4+4 'tT/T ^3 4±4
-
.;
4-
.
c
•H
3
1-
'
1
4
:
1
•
±t±t
V-i
||(|
II II
4-f-J-j-
1
-
01
44t± tet) M-H-
1
;:
;
m
tn
-1
-1
-1
—
---
--
'
1
u
C
c
tn
J+i
ii
H
.
...
111
T4-j-H
Trr
:r.
7
4II4
'
:...
-
1
-HH-ff
.:/.
144 "III
— Nil 7
44-ff
J ...
-r
—
-
'
44-1
j± H4i *tTT TTTT
7
Ill
1
1
ff
1
m
—
— Hr*-i
VAT
4..
..:.:
4
—
i
j
(
-
MM
1
.....
1
60
1
'
<
<
ill!
nil
!
i
1
-t-i-j-i-
ittt
rjr\ ^r+r rt— "Ttj"
'
'
1
— ^
44
|
—
———
-:.C
k
1
m
TT++-
1
;
4fe
11
CO
.
'
1
i
1
<r
41
,
1
-trr
-ti
1
|
01
"trr
ittt
ij
j
——
—
1
...
:
-
....
-
Mil
1
.
1
i
T|--H-|
.J4x—
—
n
S-i
.....
;
1
-
1
—
#
4--f—
nil
.1
1
-U
1
-i
.11 4W-H
•
cn
-.
[
\-
1
;
<n
4J
|
-W4-
.
1
|
'
-:::::
--
_L
4~rtT~T-
T.
|
j
|
:±ffc
-
_,
^^
—ill
\
C
—
-4 T--i:
.-
1
'
[
1
i
r
P
....
h
t
-
TTtt
44
OJ
,
MX.
4!4
1
-44- -j-i-4-i-44—
-
t
F
TTTT
TTTT
....
:-
;
1
1
1
.
hi]
4J
144
;
Trr 4
e
u
47
£
3
3 S
•H
rH
S
oo
60 /~s
3
P<
15d
0)
0)
Cfl
a
o
3
m
j-i
i
u
rH co
VH
60
3 X) ^^
•H 3
f-
w
00
CM
CO
VO
CM
CO
CM
CM
CM
u-1
00
00
CM
u~>
m
<y\
CM
00
rH
<r
m
00
CM
00
m
r~
CM
o-
CM
rH
r-«
CM
00
m
m
VO
vO
<r
CM
r^
CM
vO
VO
T~{
rH
<-i
o
O
r-{
r~-
r~
s
o
4J
c
<u
e
o
<u
•a
3
4J
S
•H
rH
P-,
6
CO
<
•u
3
3
§
o
E
m
3
3
60
CO
3
•H CO
•
H
M
M
01
rH
^3
3
H
vH
x)
0\
CM
3
VO
eu
CM
CO
o>
3
0J
3
J!
H
rH
CM
00
VO
<r
<r
o
o
rH
o m
m
o\
o>
CO
CO
rH
VO
CO
CO
t-~
-*
vO
«*
VO
CM
VO
O
CT\
est
o
CM
•-{
m o
00
00
00
<Ti
rH
rH
o>
vO
<r
vO
vO
o\
CO
CO
PC
3 6
X>
w o
s
ai
o
3
l*H
3
O -H
n
u a)
o a.
0)
X
W
i-~
N
U
O
XI
a;
M-l
O O
vO
CO
r^
CM
wsf
3
XI -H
vw
vO
CO
pa
5
a,
en
rH
CO
VO
i
3 M
0)
3
CO rH
3
0)
60
J! 3
4J M
M
3 M
O
c
3
CO
CTv
3
O
0)
XJ
U1
•h
co
60
w
ri
CX 60
15
XI
0)
3
rH 3
60 PQ
3
CO
c
o
4J
1
u
<4-<
^S
VO
00
vO
00
VO
VO
vO
oo
CO
r^
CM
r^
vO
CO
VO
r~-
o
r»»
t
•H rH
CO 3 >3
r^
CM
-a-
CO
00
CO
r~~
<r
co
<-{
<-i
vO
CM
vO
m
O
O
CO -H
3
u
3
>
TJ
(1)
en
u
vO
O)
U
ex
a
ED
A!
4J
3
3
O
Xi
00
•H
•H
O)
14H
,3
•H
3
60
•H
CO
CM
CO
o>
CO
^N
•
U
M-|
<D ^-^
>
a
j
II
II
J3
48
In order to illustrate the procedure to obtain the
parameter k for the long-term distribution of the wave response, a "short-term" analysis was done for the worst
condition in the ship's route (area
7
in [24]).
This is the
North Atlantic in the months of December to February and the
The results obtained are
tabulated data of [24] were used.
given as tables III.
6
to III.
9
for the combined moment.
For
this analysis, the result is
mean RMS = 367820 ft-tons
Since this is a "short-term" analysis, one can then say
m = 1.25 (mean RMS) = 459775 ft. tons.
The equivalent values for the vertical and horizontal moments were also computed.
These are:
mean RMS response, M^ = 19 730 7 ft-tons
mean RMS response, M„ =
2
310 4 3 ft-tons
Note that the expected value for the horizontal moment is
larger because the ship encounters more low-to-moderate
seas in any length of time than severe seas.
49
The mean value of the RMS of the response was also
calculated for all the areas in the route of the ship, i.e.,
a "long-term"
analysis was conducted.
Both
IVL^
and M
were
calculated and the results are summarized in table III. 10.
6
6
50
Table III.
Mean RMS of the Combined Moment Response in Short-Crested Seas
Averaged Over Uniformly Distributed Wind-Wave Directions
Velocity
(knots)
Significant
Wave Height
RMS Mv
(ft-tons)
RMS MH
(ft-tons)
RMS M T
(ft-tons)
(ft)
V=16
V=12
V=6
3.6
13386.3
21411.2
26762.9
7.4
54212.8
79641.4
115467.5
10.7
115913.9
159110.8
240620.1
14.6
212240.4
253534.2
405516.0
19.2
334744.1
346676.0
586564.9
24.3
463645.8
426305.6
756493.1
30.0
591299.8
488621.4
911865.7
36.4
712415.3
537815.8
1050905.6
43.3
924782.1
573471.8
1,167,628.3
3.6
11689.7
19464.6
24197.6
7.4
51434.1
78228.2
112,590.6
10.7
112157.7
160,233.2
238895.4
14.6
206043.0
258953.5
403795.1
19.2
327646.2
358380.9
585867.4
24.3
456173.6
442364.9
757601.8
30.0
583590.6
509395.5
914721.5
36.4
705103.0
561731.7
1055359.7
43.3
810988.6
597580.8
1172221.1
3.6
9831.6
17268.6
22238.0
7.4
49947.6
77878.1
112551.8
10.7
110104.1
157927.6
236043.6
14.6
201175.2
258401.3
398461.4
19.2
320621.7
360908.4
580125.1
24.3
449758.8
449322.7
756008.3
30.0
696331.0
518255.7
913820.7
36.4
696331.0
573820.7
1056466.6
43.3
801856.7
6]
5268.
1175724.0
"
ON
CO
•-\
<!
CO
CO
r^
00
CO
St
rH
CM
CM
rH
CO
CO
CM
CO
m m CM rH 00 r^ St
CM
st
rH
st
m
m
vO
ON
00
vO
O
O
st
st
CO
ON
m
st CO
CN
m
m
00
CO
r—
VO
on
m
m
ON
vO
ON
ON
CM
CO
00
m
CO
CO
CO
vO
O
o
CN
in
CO
rH
on
vO
CO
St
CO
o
CO
CO
CO
st
st
m m m VO vO vO
O
in
00
CO
ON
m
m vO
CM
St
<•
!H
in
H
CO
CO
vO
m
m
St
CM
vO
O
o
CO
CN
CN
CM
CO
CM
rH
r^
CM
oo
CO
O
r-»
ON
O
00
CM
on
iH
on
r-^
00
O
O
CN
l-s
00
in
oo
00
CO
vO
on
r^
CM
l-~
r-»
CN
CO
CM
r~-
r-~
r~-
r^.
CO
rH
vO
i-l
o
O m
m vo
00
o a\
st
on
vO
00
o
o
CM
vo
ON
CO
CN
vO
st
r-~
r-~
CM
ON
00
vO
a\
vo
vO
st
CO
r^.
rH
st
st
st
m m
rH
o
vo
VO
O O
00
CO
ON
m
m
vO
ON
VO
VO
CO
ON
vO
CO
CM
O
CO
st
CO
r^
o p» m st st st st CO CN vO m m
CM
vO
m
m CO rH
O CN vO
r^
rH
^ o m
m
m
O st
o st
o st
rH
CO
rH
st
st
o vo
rH
CM
r^
CM
st
st
vo
oo
00
ON
VO
O
rH
r^
VO
00
CN
vo
oo
CN
o st
o st
CO
p^
CO
st
st
st
VO
VO
r^
m m VO
CO
r-~
r-»
<-\
r^~
r^.
r-\
m
a>
ON
r^
r-~
CO
ON
r-.
ON
r~
00
ON
CO
r^
r-~
on
<r
oo
a\
VO
VO
st
st
CM
00
co
vO
ON
<t
vO
00
CM
CM
st
CN
CM
CO
CM
<r
CM
vo
CO
on
CO
CO
vO
st
m
CN
CN
CM
CO
m
st
on
CO
H
vO
o
o 00
St
<-l
st
co
st
00
<-t
CM
o
r-«
<-\
CN
m
O prH
CM
O
m VO
r^
rH
CO
ON
m
CM
st
st
st
CO
m
o CO VO o ON O in VD
CM
00
CO
rH
CM
CM
ON
r^
VO
st
00
CO
ON
m
CM
in
st
00
o
00
rH
CO
st
CN
CM
m
O
CO
vO
st
CO
oo
CM
O st st CN
CM
CO
st
CM
CO
m
CO
vO
CM
CO
r-~
r-.
<-i
r^-
l»»
m
ON
m
CO
o CO st
o
vD
CM
o
r^
m VO VO
ON
oo
ON
T-i
r-t
r-~
r-«
t-i
•-t
rs
co
t-^
rH
CM
CO
vO
ON
i^
ON
r^
r^
m
ON
in
m
00
st
CO
m
CO
CM
00
VO
CN
CN
<r
st
vo
r~.
r-.
ON
m
m
m
O
m vO
00
vo
st
vo
i^~
CM
r^
vo
<-t
tfl
0)
m vO
m
O
CO
on
<r
vO
m
m
CO
vO
m
ON
rH
m
rH
CM
CM
w
C
o
CN
CO
00
CO
m
o
X
•H
)-i
r-«
*J
•
tfl
M
H
M
CD
rH
.O
C3
H
<t
s
r~
CM
rH
oo
ex
CO
<u
01
O
vO
st
in
X.
^1
rH
en
Pi
H
r-~
t^-
CO
oo
oo
00
vO
o
CM
O CO
st
m
O
r^
in
m
rH
CO
iH
vO
vO
vO
eg
X
ON
CO
VO
rH
ON
CM
O st m 00
CO
m ON
o>
CM
st
o
r>
o O
rH
on
m
m
o
CO
CO
VO
o st
CO
CO
st
st
CO
ON
CO
vo
st
CO
CM
in
CO
CM
st
st
rH
in
00
r^
CO
CN
CO
CO
VO
ON
r~-
CN
CN
CM
rH
r^
VO
CO
vO
st
ON
CO
CM
CM
CM
CO
rs-
r>.
T-{
1-^
r~-
<-{
rH
CO
CO
CO
r^.
rH
rH
in
ON
CM
00
on
CO
CO
CO
vO
st
r~
CO
CO
vD
r^
r~-
t-~
VO
T~\
r-{
ON
vo
ON
CO
O
CO
CO
<-{
rH
on
rH
st
CN
VO
rH
^
VO
m
o
O
CO
O
^ CM CN
CM
CM
00
ON
ON
CN
00
CN
00
ON
rH
00
vO
st
rH
CM
st
CM
vO
st
vO
r^
o-
VO
St
on
CO
oo
vO
CO
vO
vO
st
00
CO
st
CM
CM
vO
st
rH
r^
r~
CM
m
m
CM
CM
rH
ON
o
rH
CN
m
00
VO
O
vo
CO
m
m
CO
CN
O rH
rH
rH
rH
ON
st
CN
st
rH
CM
CO
st
00
m VO
o rH
ON
o O o O O o o o o O O
rH
rH
m
o
oo
r~~
st
00
st
CO
CN
iH
f~>.
r~~
rH
rH
rH
m 00
H CN
CO
m m
r^
vo
vo
o
O
rH
00
m
ON
VO
o
m m VO m
00
rH
CO
m st VO 00
st
r~
oo
rH
VO
m ON
o
rH
in
CO
ON
CN
CM
o
CO
CO
rH
on
CN
m
CN
o
st
CO
st
CO
CO
m st
00
r^
st
m
CN
m
ON
CO
CO
st
st
st
st
m m
r-~
m o
CM
m
CN
o
rH
vO
VO
CO
CN
r~-
m
r->
00
CO
r-~
on
CN
r~.
r-~
|-~
r^.
CO
st
CO
st
m
^ 00
o
m
CO
st
vO
CO
m
m m
CO
o
vO
vO
00
CO
•-i
VO
ON
r-~
CO
CO
CM
VO
CO
st
rH
rH
CO
rH
st
rH
m
0)
o
u
CD
>
rt
3
XI
o
u
-h
1)
PL,
r~.
<u
13
<-i
O
CD
C_>
•
>
CO
S
-u
jC
oo
•H
cu
33
CM
r-\
51
co
CO
o\
00
co
00
m
on
r~
00
oo
00
vO
00
vO
vO
in
00
00
rH
CO
CO
CO
vO
vO
vO
P»-
oo
CM
rH
CM
r^
.-1
CM
rH
vO
r00
r^
r^
H
o 00
00
00
00
on
on
st
vO
vO
On
as
CM
st
ON
vO
CM
CM
CM
r^
CO
00
CM
CM
oo
r^
CO
o
CM
rH
m 00
o
o m
CO
VO
on
ON
CO
m
00
CM
CM
O o
m 00
o
m
O r^
on
a\
o
rH
-*
ON
rH
st
r»
r-«
m
00
•*
st
o
CM
CM
CO
CM
rH
r
P*)
T
-
'
rH
r^
on
.H
rH
vO
ON
<f
CM
rH
m
00
m
rH
o
CM
vo
-*
m
cm
O
iH
oo
on
on
oo
CM
in
oo
oo
on
CM
on
oo
o
CM
CO
as
r^-
sr
r-~
as
rH
<r
co
sr
>h
oo
m
o o
vO
on
vO
<r
st
oo
CM
CM
00
CO
t^.
O
i-l
on
st
ON
on
r--
CM
m
CO
CO
vO
CM
CO
CO
CM
m
CO
vO
CM
m
o
O
vO
cm
H VO
rH
m
m
o o
o
o
o
o
CO
rH
st
CM
rH
CM
in
vO
O
rCO
00
riH
CM
as
O
m
o
iH
CM
co
CM
oo
oo
as
On
ON
o
CO
00
cm
in
m vo
st
st
m
m
m
m
r^
rH
r^
CO
on
rH
as
m
CO
oo
rH
st
co
rCM
m
o
CM
vO
VO
as
CO
o CO
vO
m <r
00
VO
oo
CM
VO
o
m
o sr
CO
CM
oo
oo
as
oo
ON
o o
CO
00
m
CO
in
rH
CM
CM
o
oo
CO
ON
vO
o
m
co
oo
rH
oo
m
rH
vo
rH
m VO
co
co
co
CM
as
as
m
m
<r
00
CO
CM
VO
m
CM
as
o
o
as
00
00
CM
ON
o
rH
00
vO
m
0)
D
C
ras
00
00
r-.
CTv
U
C
o
o
r*.
1^.
cu
CO
O
O
00
vO
CO
st-
00
vO
r-»
m
rH
vo
on
m
r-~
rH
r-»
!"•
in
sr
oo
in
CM
st
o
st
CO
rH
CO
-J-
CM
vO
ON
CM
vO
r~-
rH
rH
m
vo
CM
vO
CO
m CM ON
O
O St
CM
st
O vO
rH
rCM
as
\o
O
CO
co
st
m m st
m
o co ON oo
vO
in
00
CM
vO
CM
8
r-«
-d-
•H
CM
VO
CO
vO
st
vO
St
St
vO
00
r-»
r-~
00
CO
ON
VO
O
rH
rH
sr
m
m vO
st
st
co
st
r^
vO
CO
sr
CM
00
sr
vO
o
m
o o
vo
o
r^
m
o ON
00
00
ON
o
rH
rH
rH
CM
CM
ON
ON
CM
CM
rH
r~~
l">
00
ON
o
ON
r-»
ON
St
m
rH
st
VO
CO
CO
CM
r~.
ON
rH
r^
00
vO
ON
st
m
CM
r-*
vO
vO
oo
co
oo
ON
r>
CO
CM
CM
CM
oo
m
r^
in
r^
o
VO
m
rH
m CO st
o
CM
m
ON
CO
m CM
CM
r-~
r-»
CM
CM
ON
CO
ON
St
ON
52
r-»
r-»
in
m
CO
o o O CM
O o
o
o o
rH
rH
ON
r^
r~-
ON
00
u
o
00
<J-
o rH
o
m
r^
CM
in
m
CO
rH
<}•
CO
00
00
ON
rH
m o
m ooo
00
rH
00
00
CO
CO
r^
r~
o-
rH
CM
rH
CM
CO
-sT
r~-
<r
CM
m
o
ON
rH
00
CO
o
co
CO
CM
CO
O
m
CM
oo
rH
r^
<f
CO
On
00
m
o
o
o
o
CM
<r
<r
m
<r
vO
CM
ON
CM
00
o
O
o O
o
o o o
o
r-»
CM
r~-
ON
^*
vO
ON
r^
vO
CM
00
CO
On
CO
00
00
CO
vO
m CO
00
o CM
rH
rH
rH
rH
CM
o o O o o
O o o o O
<J-
00
-d-
CM
CM
vO
co
CO
rH
<r
CM
vO
m
o
rH
rH
rH
o o
o o o o
O o
o o o o o o o o
00
ON
CO
vO
rH
-d-
rH
r^
CO
m
ON
ON
o
r^
m On
m
o
rH
r^
rH
vO
vO
m
CO
rH
r^
00
CO
in
CO
rH
rH
CM
o
o
o o O o o o
r~.
r^
00
m
o
vO
CO
CO
CM
rH
o
>*
<f
o
CO
m r^ CM o
O
o
o
O
o
o O
o
VO
r-.
m
ON
CO
rH
rH
CO
vO
vO
m
<r
<r
o
rH
CO
CO
in
00
CM
H
<r
vo
ON
O o
O
CM
CO
rH
rH
<r
<r
o
o
o
o
o
o
o
o
O
o O o o o o O o o O o o o o o o
l-~
<r
-<r
m
CO
vO
00
m
<f
rH
rH
o O o
O
o
o o
o o o O o
o
o O
o o o o O o
CO
m CO
oo
CM
O
r^
r^
<r
o
r^
r^
00
rH
vo
vo
CM
O
O
rH
CM
O o
o
O
O
O O
o O o
en
CN
vO
vO
oo
vO
00
rH
rH
rH
~tf
m rH
o
<
CM
oCM
00
l~-
IT)
m
00
00
o
r^
o
ON
rH
CO
rH
CM
m
o
ON
CM
ON
o O
o
rH
o O
O
O
o
o
o
o
o o O O o o o O
on
00
in
CO
OJ
r-^
r~-
f-t
<-\
vO
r-~
CM
rH
ON
<r
<r
om vO
o <r
«*
00
CO
00
CM
m
<f
rH
r^
CO
rH
rH
m
m
ON
ON
CM
CO
<r
<r
CM
o CM
rH
o rH H
H CO CO <r m om
o o
o o O
O
o
o
o
o
o
o
o
o o o o O o o o o o o
I~~
[-.
r~»
r^.
r»
rH
o
rH
CM
r^
rH
m o tH
rH
<r
<r
o
o
O
o
o o
o
o
o o
o
o
o
ON
oo
00
rH
CO
rH
m
ON
in
<r
CO
00
CO
m CM
<r
m
m ON
CO
rH
rH
CO
CO
vO
in
in
ON
-*
rH
00
CM
CM
CM
o O o
o
o
O
o
O
o
o
O
o
o
o o o o o
o
o
o
o O o o o o
o o o
o
CO
vO
CO
CM
CO
ON
on
CM
vO
o rH
o
co
m
m 00
rH
on
r^
rH
CM
VO
CO
CM
r^.
00
m
00
m
00
r^
CM
o
CM
vO
CM
r->
r~~
vO
-<r
rH
<r
<r
<r
ON
CO
CO
CM
CM
rH
<f
r>»
«tf
x
•H
U
s
u-l
Cfl
H
00
CO
r^
00
CM
rH
CM
vO
O
<f
rH
CO
00
r»
00
ON
<r
ON
o
O
00
rH
vO
ON
CM
<r
m
o <f CM
o 00 r^
CO
00
00
CO
CM
VO
o
o o
O
CM
00
CO
rH
rH
on
ON
ON
ON
VO
o
o
o
CM
r^
m CO rH rH o o o o o O
O
o
o O O O o o O O o o o o o o
CM
CO
cfl
,0
>N
en
4-1
•H
8
o
M
00
CO
rH
<t
CM
ON
CO
CM
in
r-~-
r^.
CM
CM
vO
o on
CO
CO
00
00
00
ON
m
<r
CO
rH
in
ON
m
r^
r^
CM
O
CO
rH
m
O
o O o O o
on
-d-
Cm
CM
i-T-H
m
o
r-«
CM
r-~
<f
ON
r^
on
r-~
r--
-3-
vO
vO
1-^
<r
<r
ON
m
m
CO
r-~
00
o
o
o
H
o
o
r-^
r-^
rH
rH
rH
CM
CM
o o O o o
o
o o
o O
o
o
o o o o o
in
r^
o\
o <f
O o
r^
ON
rH
00
<r
CM
vO
ON
CO
00
CM
rH
ON
m
m
ON
r^
rH
00
rH
CO
CM
rH
m
CM
o
CO
m H CM <r
<r
CM
o<r
o
o O o o o o O O o
o o o o o o o o o o
o\
vO
X
rH
m o ON
CO
CM
vO
rH
00
m
vO
00
<f
vO
CM
rH
o O o O
o
o O o o
CM
m
o
o
o
o
vO
00
ON
CM
m
o
o
o
ON
CM
<r
<r
ON
CM
rH
CM
rH
rH
CM
rH
CO
rH
CO
OJ
CO
vO
vO
<r
m
rH
r^
m
rH
CO
<r
m
O o o
o
o
o o
o
O
o
o o o o o
rH
0)
/
O
/
01
>
*o
m o
Sc -h
01
P-.
d)
o
u
>
ccj
2
-u
.c
60
•H
0)
X
rH
CO
CM
m vO r^ CO ON rH
o
o O o o <c
o o O O o o o
rH
53
54
cm
i*»
H
o o o
o o o <r
o
o
o
o
o
rH
iH
CT\
00
.0005289
.0005298
.0003532
.0010008
.0007653
.0012952
.0005298
.0013541
.0018839
.0011186
.0011774
P^
vO
.0001177
.0001766
.0001766
.002767
m
.0020016
.0015307
.0017073
.0027081
.0013541
.0005298
.0004121
.0007653
.0002944
.0002355
.0000588
.0001766
.0000588
flj
a
V
a
o
<r
00
M
H
CO
Z>
iH
ja
ca
H
p~
r»
H
H O O O O O O
O
o
o
CM
X
.0002944
.0001177
<u
>£>
r*»
*o
iH
iH
/
/
o
"J
/
/
0)
n)
/
/
T3
o
*73
0)
Ph
/
/
/
/
/
/
/
/
/
/
/
/
/
.0001177
/
SI
TJ
>
.0000589
o
o
a>
> U
3
oo
<-•
a>
X
CO
iH
CTi
iH
H
CJ>
cm
o-
On
CTv
55
ON
o o
CM
r~
vO
CO
CM
ON
CM
St
r-t
on
CM
vO
vo
m
i-~
vO
st
VO
r-4
r-»
m
o o
00
vO
CO
rH
CM
00
r~-
CM
rH
o
r~
o
CM
CM
CM
m
m
CM
vo
vO
vO
CO
CM
rH
<f
rH
St
m
CM
rH
on
ON
On
CO
CO
CM
00
i-l
m
oo
rH
CO
m
CN
CO
vO
•*
CO
CO
vO
CM
rH
<-h
m CO m
o
o
St
m CM
o CO
CO
rH
st
<r
m
rH
rH
CM
t-H
CM
00
CO
r>»
<-{
rH
r^
CM
CO
in
CO
00
St
00
ON
CM
CM
r^
r^
vO
r~
CM
00
rM
r-t
ON
CO
r^»
r-~
r~.
CM
oo
r^
<-i
CM
O
co
m m
CM
o CM
<*
m St
00
^ o
<-*
r-i
CM
CO
vO
CM
vO
CO
vO
vO
r^.
r^
oo
vO
rH
oo
rH
ON
O
CO
m
CM
CN
r-i
rH
O
o
rH
cd
3
u
Ol
03
vO
rH
CM
on
00
<r
vO
CO
rH
P^
LO
CM
CM
on
vO
ON
CO
ON
r^
-tf
r-~
rH
CO
o rH r^ ON ON m CM St
in
CO
CO
VO
oo
ON
m
00
sr
m
o m
O o o
CM
CO
oo
00
CM
o
o
CM
CM
st
m m VO CM
in
oo
ON
CO
vO
rH
St
CM
CM
CM
rH
T-i
00
o
,0
CO
0>
S
Pi
CJ
m
CM
<-i
r-{
r-\
St
m
CM
m
ON
T-H
rH
CM
CM
ON
CM
CO
a)
o
0.1
O
>+h
£
(0
13
•U
C
O
s
H
C
01
6
o
o
XI
CO
4-1
CM
on
00
CO
rH
CM
rH
rH
on
rH
r^
rH
CM
ON
ON
m
t~-
<T
O
O
CM
o
vO
ON
NO
ON
CO
rH
St
CM
CO
CM
r-\
r^.
ON
ON
CO
CO
CO
ON
CO
m O
m ON
o
rH
T-\
VO
CO
CM
rH
00
CO
CO
00
00
St
CO
rH
CO
rH
CO
P-.
<-{
{"v
m
m
St
CO
rH
<T
CO
m
00
rH
00
<-\
oo
r^.
ON
ON
vO
CM
r-
St
St
CO
in
rH
rH
si-
CM
0)
C
-H
.O
r-»
e
o
M
O
U
UH
Ol
cs
0)
m
st
st
n
o
u
hh
UH
CO
01
CO
o
m
co
00
r-
00
<r
00
r^»
on
CM
00
<r
vO
rH
CO
rH
on
CO
rH
00
rH
ON
CO
vo
CM
o
00
st
r-
vD
m
CO
CM
<r
st
o
CM
3
r-»
00
r-
m
st
m
o
ON
ON
v^"
00
vO
oo
vD
ON
St
r-H
in
CM
r-i
ON
CO
o
ON
00
St
St
ON
in
CO
p~
co
vO
m r^
O
rH
m o
VO
St
vO
VO
CM
rH
00
CO
m CO
CO
o o
m
o
o
St
m
m
o
o
rH
vO
vO
m
rH
rH
rH
m
ON
vO
r-^
VO
CM
O
H m m
CM
CM
r-H
CO
CO
r^
CO
VO
rH
CM
00
00
St
ON
CO
St
rH
St
m
m
fH
to
•H
ca
>
(0
•
>, CO
ca
c
T-i
3
CM
CM
00
<; -a
•
m
CO
rH
CM
on
CO
•
•
oo
rH
CO
vO
oo
r~-
vO
o
cu
t
St
rH
ON
CO
CO
CM
VD
CO
CO
rH
<t
m o o
CO
in
m ON oo
00
<-t
CM
St
O O
T-{
CM
o o
vO
m
St
St
CO
vO
in
rH
oo
CM
o
rCO
oo
m
<-{
in
o
m
VO
ON
CM
CO
St
•-i
r-t
m
rH
-u
B
C
01
x:
co
-H
H s
•M
r^
<r
CO
CO
CM
vO
vo
rH
vO
r-.
vO
vO
ON
rH
CM
CO
in
NO
CO
<f
01
X
I
Vl
o
o
co
O
<r
oCM
o-
CO
CM
m o
r^.
<t
r-\
CM
rH
r^»
Xi
0)
T3
o
CJ
01
>
cd
2
-o
o
M
-h
0)
CM/
0)
-a
o
Ol
>
CO
3
u
JJ
XI
60
•H
0)
K
CM
CO <r
m vO r^ 00 ON o
o rH
o o O O o o O O O o rH
r-{
56
m
rH
•
•
o o o o o
o O 00
m
m
<y>
oo
m
o
o
CO
in
vO
•
•
00
-a-
O
m
m
rH
vo
r»
t-^
o\
•
•
•
en
CTi
r-.
•
•
.-i
ov
ovo
rH
rH
•*
rH
en
O
rH
m
m
CO
rH
CO
rH
r^
•
•
•
•
r-~
rH
CO
rH
CN
rH
CN
rH
CN
m
«*
•
•
oo
rH
m
rH
co
rH
vO
CM
<r
CN
•
•
•
co
CO
CM
cn
CO
vO
vO
CM
co
r--
•
•
•
•
rH
co
CN
cn
CN
rH
co
.-i
m
rH
CO
<c
o
CN
m o
rH
vO
O
rH
o o o o o
ov
•
00
-d-
co
oo
o o o o o
CN
VO
vo
C
o
u
•
CXv
•
•
m
•
en
rH
en
CM
rH
vO
m
r-.
CO
vO
•
o
CN
<r
CM
vO
•
o
Civ
CO
VO
O o o
o o O o o
o o o o o
rH
CN
O o O o o o o o
o
in
o>
co
rH
CO
•
•
•
r-l
CN
rH
rH
CO
CN
VO
r--.
oo
t-(
rH
rH
ON
rH
•
o
m
01
•o
X
o o o o o
a)
X)
o
c_>
0)
>
CO
TJ
O
DE -H
n
0)
PM,
o
0)
>
/
/
/
/
/
/
nj
o
4-i
3 ^
t>0
•rH
o
rr
o
OV
rH
crv
CN
a\
CO
o-
o>
o->
57
Table III. 10
Mean RMS Value of the Response for all Areas in the
Route of the UNIVERSE IRELAND
E[RMS]of M^ Response
E[RMS]of Mp Response
Time Factor
7
154254
299294
2
11
126213
246724
2
18
109398
218546
1
20
66961
138544
1
23
110971
220161
1
28
80454
170798
1
29
92992
189706
1
34
108754
203516
1
41
146576
282585
2
42
123456
246480
2
45
161230
260578
2
Area
Time Weighted Average:
= 124,562 ft-tons
(RMS)
V
(RMS)
m^ = 155,702 ft-tons
iiL
= 646,813 ft-tons
—
= 4.15
= 517,450 ft-tons
*V
Chapter IV
DISCUSSION OF RESULTS
IV. 1
Evaluation of the analytical Results
The results obtained in this study agree
with the
previous findings in other studies that the lateral moment
can be very significant for ships in waves.
Some definite
trends of the moment responses can be observed from figures
III.l to III.
4.
From the results in short-crested seas
shown in figures III.
2,
3
and
4,
one sees that horizontal
and vertical moments change very little with wave direction
in moderate seas.
On the other hand, there is a broad peak
away from the head and following seas for all moment responses
in low sea states.
M^ and M
In severe seas, the maximum response for
are in the head and following seas while M
almost constant regardless of wave heading.
stays
For the long
crested seas of figure III.l, one can see that the vertical
moment peaks at about 60
for low to moderate sea states and
assumes extreme values at head and following seas in high
seas.
cases.
The equivalent combined moment has two peaks in all
The peaks occur in wave headings near 60
and 120
with a shift away from the beam seas case as the sea grows
more severe.
58
—
59
—
*»vf (mcctiom
*»v* O^CCT'O*
JO*
O
— mvc ou«aio*
*>*
*
W»v£ 0>«CCTiON
TO*
O
m**t ou*cco"
oo"
e
.
raOM >c«
tfli
^
OiBCCTiOH so* toon at*
1*1
*
0>efc"o-
Mvt
to"
fl
'
WV^
wave bending moment
amplitude in regular bow waves
wave
Vertical
Vertical
amplitude
in
moment
beam and
bending
regular
quartering waves
Figure IV. la
Figure IV lb
.
Fn. 015
Fn
=
015
Mt*V.J*(.-'
—
—
IWI o»tc'«» uo"
WAVf. MKCUCh '.SO'
MV( >»tzio* '10*
»o"
»«f£?<0«
»8*
0
J
"0*
«o»* •*» 1*1
'»OM »C 1*1
•£;t H>*
ttlVf
«*>( t>*fCTlQN
«**vf
•
14
1
J
»
II
P
/
,A-7^
•
:.
. •
ct
,
Figure IV. 2a
o*cct»o«
JO'
0t»CCfON
SO'
—
w«A»C OiBECtiOM
»o'
w»vt OWCCt-OM
fO'
s»
Horizontal wave bendint* moment
in regular bow waves
amplitude
wwc
-— *»vt
Horizontal wave bending moment
amplitude in regular beam and
quartering waves
Figure IV. 2b
60
One observes from figure III.
2
and III.
3
that the
horizontal moments exceed the vertical moments in almost all
headings for low and moderate seas.
However, in severe seas,
for which structural design is based, one sees the customary
relation of the vertical moment exceeding the horizontal
moment, especially in head and following seas.
The reason
for this phenomenon is that lateral bending is more respon-
sive to higher frequency excitation than vertical bending.
One can see from figures IV.
1
and IV.
2
that the spectral peak
of the lateral moment response for all wave headings are gen-
erally situated at higher frequencies than the corresponding
response curves for the vertical moments.
One can note fur-
ther that the spectral ordinates of the RAO for wave headings
70
,
110
are higher for the lateral moment than the
and 130
Therefore, when the energy of the exciting
vertical moment.
waves are concentrated in the higher frequency range (i.e.,
in low sea states)
moment.
,
the lateral moment exceed the vertical
In high sea states, the reverse trend is true.
For a similar reason the peaks of the response curves
for the vertical moment occur away from head and following
seas in low sea states.
The shift in the location of the
spectral peak of the RAO's in the direction of higher circular frequency,
w,
can be explained by considering the
effective wavelength in oblique seas.
It is generally known
that the wavelength nearly equal to the ship length causes
3
61
k
'VAjfWE
Figure IV
.
The Effective Wavelength in Oblique Waves
62
the maximum midship bending moments.
In oblique seas the
ship senses the effective wavelength, L
figure IV.
3.
,
as shown in
The relation with the actual wavelength is
where
L
= L/cos
L
= effective wavelength
£
L = actual wavelength
f>
As p approaches 90
= heading angle
from either
wavelength decreases while the value of L
or 180
,
the actual
stays fixed.
Therefore the peak frequency of the response goes up, thereby
shifting the RAO spectrum nearer the sea spectral energy
density peak in low sea states.
The theoretical calculations predict greater vertical
moment response in following seas than in head seas, contrary
to experimental results.
figure IV.
4,
Wahab's results [15], shown as
confirm this apparent discrepancy between the
theory and experiments.
Finally, one notes that the combined moment in
"realistic" seas is significantly higher than either of the
63
components in all cases but, because of the relative phase
angles, is not equal to the sum of their individual peaks.
64
E^
,/
i
\
•
S
f
J
1
V
AY
g
/
I
!•
J
Ay
1
O
O.JD
i
u
1
.1
v.
\ j?<^
E u
Sv\
>v^v
»
\
\\
i
1-3 S
2
n
t>
u
:-
**
i
1
i
\
J3
0)
3
n
3
60
•H
H
60
Pn
a
£
i
3
1
w
/
-<
1*
" E
i
^^
o
c
i0
s
s *
2
C^ *5^
Si
;
/
?.
i
*s
C3
.....
a)
u
3
60
65
Comparison of the Values of the Combined Moment
IV. 2
Table III.l shows the comparison of the response
RMS value of the combined moment, M
methods described in chapter two.
,
as obtained by the two
It can be readily seen
that the results are not significantly different, the average for all sea spectra being on the order of eight per cent
(8%)
.
This confirms the statement that the average value of
the correlation coefficient,
/^
,
already considers the phase
difference of the vertical and horizontal bending moment
peaks.
As would be expected, the third set of values for M
were closer to the computer values.
two values of
/?
It is realized that the
that were used were derived in a different
manner and perhaps should not be directly compared.
In de-
riving the second value, it was found that the c^^.__ J-cient
varies strongly with the state of the sea and the wave di-
rection and therefore that no single value will be "true" for
all conditions at sea.
If this method is to be adopted,
there is need to study the correlation coefficient further.
There is a dearth of published material on the subject.
The calculated value of the combined moment also de-
pends on the relative phase angle of the vertical and horizontal moments.
Therefore, it is necessary that the pre-
dicted values be at least as accurate as the predictions for
the vertical and horizontal moment responses.
Comparison of
theoretical and experimental phase angles shows that the
66
agreement is in general quite good [22],
The maximum dis-
crepancies occurred for wave headings of 50
the wave lengths A/L = 1.2 and - = 1.5.
and 110
(Figures IV.
for
6
and IV.
Jj
In view of the above mentioned considerations it
would seem to be indicated that both method
1
and method
to obtain the combined moment are equally "good".
The first
method is probably more convenient for use in analytical
studies.
2
7)
6
S.
T.
F.
Theory
67
Experiments, Wahab
A
Wave Direction 50
Head Sea = 180 )
e-<j>
(
50
A/l
10
1.5 1.2
1.0
.8
.7
.6
.5
1.0
-50
/
-IOC
/
A.
Figure IV
.
Phase Angle Between Horizontal and Vertical
Wave Bending Moment at Midship
for Series 60, Block 0.80
Experiments, Wahab
o
S.T.F.
68
D
Theory
Wave Directions 110
(Head Sea
=
180°
)
X/L
Z.O
1.5
1.1 1.0
.6
•7
-6
-5
.4-
.2)
1
1.0
3.0
2.0
40
\
\
VK
Figure IV
.
Phase Angle Between Horizontal and Vertical
Wave Bending Moment at Midship
for Series 60, Block 0.80
wVL/9
69
IV.
3
Comparison Study of the Vertical Bending Stress
Data on the full-scale measurement of vertical
bending stresses on the deck of the UNIVERSE IRELAND can be
found in [25].
A comparison between calculated and measured
values was done by plotting the average RMS peak-to-peak
amidship bending stresses.
The analytical values were ob-
tained by calculating the response to each of nine sea
spectra corresponding to the observed periods in the tabulated data of [24
3.
The significant wave heights are the
wind-to-wave conversion shown in table III.
3.
The average
RMS peak-to-peak response is the weighted mean of the cal-
culated response to each sea specified by the significant
height.
This is best described by the relation
?
mean RMS =
\
i=l
E h2
y
•
p
(T)
where
Ev
p(T)
= RMS peak-to-peak bending stress
= relative frequency of occurrence of each
observed period for a given significant
wave height.
70
The wind-wave relation as described by the Pierson-
Moskowitz random and ITTC 1966 line, the corrected Roll curve
for the North Atlantic and the corrected Yamanouchi curve for
the North Pacific were examined.
ized in table III.
The conversion is summar-
3.
The use of the Pierson-Moskowitz Random line pro-
duced the worst case in the comparison study.
While the
wind-to-wave relation is intended to convert wind measurements to significant wave heights for seas in random stages
of partial development, there is a bias in favor of seas that
are near full development [25].
Consequently, the values
based on this wind-to-wave conversion, coupled with the known
tendency of the theory to over-predict [21], are iu^cn higher
than the measured stresses.
The Roll and Yamanouchi curves
were based on observed wave heights and corrected for significant heights.
They are believed to be better representation
of an actual sea.
Since the ship route is neither entirely
in the North Atlantic nor in the North Pacific, one sees fairly
good agreement with results based on both the Roll and
Yamanouchi relations for the lower half of the sea states as
described by the beaufort scale.
It can be seen that theoret-
ical results over-predicted the average values of the RMS
stress, specially in the higher sea states.
probable causes of this:
There are several
71
1.
Since the largest divergence of compared values are
consistently in the more severe seas, one is led
to believe that the assumption of linear motions
for the ship, in particular resonance in roll,
exaggerates the analytical values.
Obviously,
heave and pitch are large in heavy seas.
The
divergence caused by roll resonance gets bigger
as the energy of the waves are concentrated more
and more in the lower frequencies.
Faltinsen
[2 2]
used a first approximation of roll at resonance
as input values.
It is also to be noted that
the ship was not provided with bilge keels in
the computer analysis.
It is not known if the
acutal ship has these dampeners.
2.
Loukakis [21] suggested that deflections of the
hull can relieve some bending stresses, so that
even if bending moment predictions are accurate
the resulting conversion into stresses may result
in over-prediction.
3.
The inexact weight load distribution used in the
analysis could have caused some of the over prediction.
A qualitative measure of this effect can be
72
gained by the excess in value in the calmer seas
when the contribution of inertia effects to the
dynamic loads are relatively greater.
4.
Exaggeration of the reported occurrence of the
higher period waves.
In high seas, when the ship
is pitching and heaving,
it is understandable
that such errors could be made by untrained observers.
In this study, this effect would be
very minor.
5.
Inaccuracy in the measured data due to insufficient number of records in extreme weather.
For direct comparison of the scatter of the derived
values of the stress with the measured values, it was necessary to determine the standard deviations with respect to
wind measurements.
One can see from table III.
4
that the
calculated response to the nine sea spectra with the same
significant wave height were more widely scattered than the
measured RMS peak-to-peak, values.
73
IV. 4
Bending Moment Trends and Wave Load Prediction
One can observe from figures III. 10 and III. 11 that
the combined moment in short-crested seas are considerably
higher than the vertical moment in head seas.
The difference
in severe seas is in the order of 15-20 per cent.
Therefore,
at least for this ship, calculations based on vertical bend-
ing only could lead to disastrous results.
Comparison of the horizontal and vertical bending
moments in short-crested seas have been partially discussed
in section IV. 1.
It was explained that in low-to-moderate
seas the lateral bending response to wave excitation is
greater than vertical bending.
One consequence of this is
its possible significance in fatigue failures.
Another is
that over any length of time the expected wave bending load
will be greater for lateral moments than for vertical moments.
This is so because the ship will encounter more low-to-moderate
seas than high seas in its lifetime.
The result for the
short-term analysis for the North Atlantic confirms this
statement.
The statistics reported in [24] were collected
for a period of seven years.
Table III.
5
shows that in short-crested seas the
bending moment varies only slightly with speed.
This state-
ment holds true for all the moment responses, with
M^.
having the greatest variation for all three sea states shown.
74
This indicates that reducing speed in a seaway reduces
dynamic stresses due to slamming, etc. but will not
significantly reduce the primary stresses due to wave
bending loads.
Chapter V
CONCLUSIONS AND RECOMMENDATIONS
The subject of the combined effect of horizontal and
vertical wave moments has been investigated.
A number of
conclusions can be drawn from the results of the study.
1.
The results obtained by the MIT Seakeeping
Program agreed well with published results of
theoretical and experimental investigations
of ship motions and loads in a seaway.
The
following bending moment trends were -^served:
i.
The horizontal moments experienced by the UNIVERSE
IRELAND are not negligible in comparison with the
vertical moments in almost
all conditions at sea.
In general, the relative significance of the lat-
eral moments increase with the size of the ships
[15].
In low sea states, the lateral moment
exceed vertical moments
ii.
The variation with wave headings of the moment
response seem to depend on the state of the sea.
For long-crested seas, the maximum vertical
moment occurs at a wave heading near 60
to moderate seas and at
75
and 180
for low
in severe
76
seas.
The horizontal and combined moments, on
the other hand, exhibit two peaks near 60
120
in all sea conditions.
and
For short-crested
waves, all moment responses show a broad peak
away from
and 180
in low seas.
The vertical
and horizontal moments flatten out in moderate
seas, and the vertical and combined moment
assume their biggest values at
and 180
in
severe seas.
The above behavior of the moments with respect to
wave headings can be explained by the character of
the RAO.
2.
The values of the combined moment obtained by the
two methods described in this paper were not very
different.
Therefore, both methods can be con-
sidered valid.
The first method involves develop-
ing the RAO for the equivalent combined moment by
a strip-theory based computational method and per-
forming the usual power spectral analysis.
Invest-
igations on the correlation between analytical and
experimental values have led to the general acceptance of the method first introduced by St. Denis
and Pierson.
However, it is felt that further
investigations and refinements are necessary before the analytically derived values can be used
77
in actual design.
The second method makes use
of the correlation coefficient,
/^
,
to take into
account the phase difference of the vertical and
horizontal moments.
There is also further need
to investigate this important statistical
parameter.
3.
The comparison study of the predicted and measured
vertical bending stresses showed good agreement
in the lower half of the weather spectra en-
countered in service.
In higher seas,
the excess-
ive predicted values are believed to be caused by
non-linear motions, contrary to the assumption of
the theory.
No doubt the addition oi bilge keels
in the analysis of the ship would have reduced the
errors.
For the ship's route, both the Roll and Yamanouchi
corrected wind-to-wave relations seemed to be valid
representations of the seaway.
In view of the ex-
pected higher-than-actual values by theory, Roll's
curve is deemed the better approximation of the
service route if used in conjuction with the tabulated data of [24], as was done in this study.
4.
It was found that there was a significant diff-
erence between the values of the vertical moment
78
response in long-crested head seas and the
combined moment response in short-crested seas.
Again, it is stated that this may not be true
for other ships, particularly the smaller ones.
For large tankers with the general characteristics of the UNIVERSE IRELAND, the contribution
of the horizontal moment to the longitudinal
stresses must be considered in the calculation
of the primary strength.
Speed is not a significant factor in the magnitude
of the wave loads acting on ships.
It remains to be said that the absolute values of the
combined moment, or the total longitudinal stress, as pre-
dicted by the methods in this study could not be compared
with experimental results.
However, the moment trends were
studied and found to be reasonable and, on the whole, in
conformity with other researchers
1
results.
has confidence on these derived values.
Therefore, one
The theoretical vert-
ical bending stress predictions could have been closer to
measurements but, again, one has firm ideas about the probable reasons for the divergence and therefore one is not
completely dissatisfied.
The procedure followed in this study is fairly in-
volved and the cost is not cheap.
For further analysis
79
of ships in related studies, it is recommended that the RAO,
or an approximation, be obtained in the cheapest manner.
Hopefully, with some anticipation of the expected results,
more realistic values can be obtained.
REFERENCES
1.
Egil Abrahamsen, "Structural Design Analysis of Large
Ships", Transactions, SNAME, Volume 77, 1969.
2.
A.
3.
Dan Hoffman, "Analysis of Ship Structural Loading in a
Seaway", Marine Technology, SNAME, Volume 9, No. 2,
April, 1972.
4.
Egil Abrahamsen, "Recent Developments in the Practical
Philosophy of Ship Structural Design" SNAME Spring
Meeting, Montreal, Canada, July, 19 67.
Mansour, "Statistical Approach to Wave Bending
Moment", Department of Ocean Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts,
Report No. 7 3-12, November, 19 71.
,
5.
A. Mansour,
"On Including the Secondary Stresses in the
Statistical Analysis of Ships" Department of Ocean
Engineering, MIT, Cambridge, Massachuset 4
Report No. 73-17, August, 1973.
,
"
6.
"Methods of Computing the Probability of
Failure under Extreme Values of Bending Moment"
Journal of Ship Research, SNAME; Volume 16, No. 2;
A. Mansour,
June, 19 72.
7.
A. Mansour and D.
Faulkner, "On Applying the Statistical
Approach to Extreme Sea Loads and Ship Hull Strength"
The Royal Institution of Naval Architects, November,
1973.
8.
9.
10.
A.
Mansour, "Probabilistic Design Concepts in Ship
Structural Safety and Reliability", Transactions,
SNAME, Volume 80, 1972.
Nils Nordenstrom, "A Method to Predict Long-term Distributions
of Waves and Wave- Induced Motions and Loads on Ships
and Other Floating Structures", Det Norske Veritas,
Publication No. 81, April, 1973.
E.
Abrahamsen, N. Nordenstrom and EMQ Roren, "Design and
Reliability of Ship Structures", SNAME Spring Meeting,
Washington D. C. April, 1970.
,
80
81
11.
Jun-ichi Fukuda, "Theoretical Determination of Design
Wave Bending Moments" Japan Shipbuilding and Marine
Engineering, Volume 2, No. 3, May 196 7.
,
12.
EV Lewis, D Hoffman, WM Maclean, R Van Hooff and
R. B. Zubaly, Load Criteria for Ship Structural
Design", Ship Structure Committee Report SSC-240,
1973.
13.
Edward GU Band, "Analysis of Ship Data to Predict
Long-Term Trends of Hull Bending Moments" American
Bureau of Shipping, New York, November 1966.
,
14.
A.
Mansour, "Approximate Probablistic Method to Ship
Longitudinal Strength" Department of Ocean Engineering,
MIT, Cambridge, Massachusetts, Report No. 73-8,
May, 197 3.
,
Wahab, "Amidship Forces and Moments on a Cg = 0.80
Series 60 Model in Waves From Various Directions",
Netherlands Ship Research Center TNO, November, 1967.
15.
IR.
16.
Edward Numata, "Longitudinal Bending and Torsional
r.jadings
Moments Acting on a Ship Model at Obi
to Waves", Journal of Ship Research, SNAME, June, 1960.
17.
A.
18.
"5th International Ship Structures Congress Committee
Reports", 197 3, Reports of Committee 3 and 10.
19.
S.
R.
Mansour, "On the Torsion Loading of a Ship",
Department of Ocean Engineering, MIT, Cambridge,
Massachusetts, Report No. 70-12, October, 1969.
Crandall and Marks, "Random Vibrations", Academic
Press,
1963.
Davenport and W. Root, "Random Signals and Noise",
McGraw-Hill Book Co., Inc., New York, 1958.
20.
W.
21.
Theodore A. Loukakis, "Computer Aided Prediction of
Seakeeping Performance in Ship Design," Department
of Naval Architecture and Marine Engineering, MIT,
Cambridge, Massachusetts, August, 1970.
22.
0.
Faltinsen, "Comparison Between Theory and Experiments
of Wave-Induced Loads for Series 60 Hull with
C B = 0.80", Det Norske Veritas, Research Department
Report, No. 70-27-S, July, 1970.
82
23.
D.
Hoffman, J. Williamson and EV Lewis, "Correlation
of Model and Full-Scale Results in Predicting
Wave Bending Moment Trends," Ship Structure Committee
Report SSC-233, 1972.
24.
N.
Hogben and F. Lumb, "Ocean Wave Statistics", National
Physical Laboratory, Ministry of Technology, London,
1967.
25.
Roberts S. Little, EV Lewis and Fred C. Bailey, "A
Statistical Study of Wave-Induced Bending Moments
on Large Ocean-Going Tankers and Bulk Carriers"
Transactions, SNAME 1971, New York, N.Y.
,
26.
Atle Steen, "5-D Seakeeping Program - User's Manual",
Department of Engineering, MIT, Cambridge,
Massachusetts
27.
Nils Salvesen, E. 0. Tuch, and Odd Faltinsen, "Ship
Motions and Sea Loads", Transactions, SNAME, Volume
78,
28.
M.
1970.
Abkowitz, "Stability and Motion Control of Ocean
Vehicles", MIT Press, M.I.T., Cambridge, Massachusetts
A.
1969.
29.
John P. Comstock, ed.
"Principles of Naval
SNAME, New York, 1967.
30.
D.
31.
Walter H. Michel, "Sea Spectra Simplified", Marine
Technology, SNAME, Volume 5, No. 1, January, 19 68.
32.
MA Abkowitz, LA Vassilopoulos, FH Sellars, "Recent
Developments in Seakeeping Research and its
Application to Design", Transactions, SNAME,
Volume 74, 19 66.
33.
Atle Steen, "Consideration of Combined Vertical and
Lateral Wave Bending Moments in Determining Maximum
Plane Stress Experienced by a Ship in a Seaway"
unpublished term paper, Department of Ocean
Engineering, MIT, May 11, 1973.
34.
B.
V.
35.
B.
V.
,
.
._
cnitecture"
Hoffman, "Analysis of Ship Structural Loading in a
Seaway", Marine Technology, SNAME, Volume 9, No. 2,
April, 1972.
Korvin-Kroukousky "Theory of Sea-keeping",
SNAME, New York, N.Y., 1961.
,
Gnedenko, "The Theory of Probability", Chelsea
Publishing Company, New York, N.Y., 1962.
83
36.
H.
Cramer, "The Elements of Probability Theory",
John Wiley and Sons, New York, N.Y., 1955.
Appendix A
SAMPLE COMPUTER OUTPUT DATA
Irregular Wave Results
I.
-
Long Crested Unidirectional Seas
Heading Angle = 22.5 degrees
Ship Speed = 12 kts.
Significant Wave Height = 19.2 feet
Peak Spectral Frequency = 0.5197
Motions at Origin:
RMS
Heave
Pitch
Sway
Roll
Yaw
0.5974
.2456
.2137
.4386
.1301
H(l/3)
H(l/10)
H(l/1000)
2.3897
0.9822
0.8549
1.7546
0.5204
3.0469
1.2524
1.0901
2.2371
0.6635
4.6003
1.8908
1.6458
3.3775
1.0017
H(l/3)
H(l/10)
H(l/1000)
Dynamic Loadings
Vertical B.M,
Sta
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
RMS
0.0
51.1
1454.4
8894.8
41643.6
89928.7
144749.5
199801.3
251274.3
296087.0
329147-6
347970.3
353323.6
344392.0
321096.7
284297.1
180498.0
65363.6
0.0
204.4
5817.9
35579.2
166574.5
359715.0
578998.0
799205.5
1005097.2
1184348.0
1316590.0
1391881.0
1413294.0
1377568.0
1284387.0
1137188.0
721992.2
261454.4
260
7
7417 9
45363 5
212382 6
458636.6
738222.5
1018987.0
1281499.0
1510043.0
1678653.0
1774648.0
1801950.0
1756399
1637593
1449915
920540
333354
84
0.0
393.6
11199.6
68490.0
320656
692451
1114571
1538470
1934812
2279869
2534437
2679371
2720591
2651818
2472444
2189088
1389835
503299
85
Vertical B.M.
Sta.
19
20
21
22
RMS
(cont.)
H(l/3)
97052.3
13065.0
15963.1
14735.9
24263.0
3266.2
3990.7
3683.9
H(l/10)
123741,
16657
20353
18788,
H(l/1000)
186825.7
25150.2
30729.0
28366.7
Horizontal B.M.
Sta.
RMS
H(l/3)
H(l/10)
1
0.0
0.0
0.0
2
3
4
5
6
7
8
9
4568.3
15469.4
24915.5
52746.4
89010.3
129957.7
170987.7
208736.5
238483.0
258256.3
268522.5
269648.6
260647.3
242403.2
214430.1
132947.2
48485.1
22369.9
6595.0
456.7
118.8
18273.5
61877.6
99662.0
210985.7
356041.5
519831.0
683951.0
834946.2
953932.0
1033025.5
1074090.0
1078594.0
1042589.5
969613.0
857720.5
531789.0
193940.3
89479.8
26380.2
1826.8
475.5
23298.8
78894.0
127069.1
269006.8
453952.9
662784.5
872037.5
1064556.0
1216263.0
1317107.0
1369464.0
1375208.0
1329301.0
1236256.0
1093593.0
678031.0
247274.0
114086.7
33634.7
2329.2
606.2
10
11
12
13
14
15
16
17
18
19
20
21
22
H(l/1000)
0.0
35176.6
119114.4
191849.4
406147.5
685379.8
1000674.6
1316605.0
1.0
,9.0
1988574.0
2067623.0
2076294.0
2006984.0
1866504.0
1651111.0
1023693.7
373335.2
172248.6
50781.8
3516.7
915.3
Combined B.M.
Sta.
1
2
3
4
5
RMS
0.0
4177.5
15031.4
30139.6
86181.8
H(l/3)
H(l/10)
H(l/1000)
0.0
0.0
0.0
16710.1
60125.7
120558.4
344727.5
21305.4
76660.
153712.0
439527.5
32167.0
115742.1
232075.0
663600.3
86
Combined B.M.
Sta.
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
RMS
164735.4
253106.8
342644.4
424373.1
492267.6
540433.3
566890.0
571413.7
554399.9
513669.5
452582.7
281527.8
99230.1
37856.3
6591.3
4085.7
3671.7
(cont.)
H(l/3)
658941.7
1012427.2
1370577.0
1697492.0
1969070.0
2161733.0
2267560.0
2285655.0
2213199.0
2054678.0
1810331.0
1126111.0
396920.7
151425.4
27805.4
16342.9
14687.0
H(l/10)
H(l/1000)
840150.7
1290844.0
1747486.0
2164303.0
2510565.0
2756210.0
2891139.0
2914210.0
2821829.0
2619714.0
2308172.0
1435792.0
506073.9
193067.4
35451.8
20837.2
18725.9
1268462.0
1948922.0
2638362.0
3267672.0
3790460.0
4161336.0
4365052.0
4399885.0
4260409.0
3955255.0
3484887.0
2167764.0
764072.3
291494.0
53525.4
31460.1
28272.5
87
II.
Irregular Wave Results
-
Short-crested Multi-Directional
Seas
Ship Speed = 12 kts
Significant Wave Height =
4
3.3 feet
Peak Spectral Frequency = 0.3461
Principal Wind Direction = 22.5 degrees
Motions at Origin:
RMS
Heave
Pitch
Sway
Roll
Yaw
6.2544
1.1184
3.5923
4.1180
0.6277
H(l/3)
H(l/10)
H(l/1000)
25.0175
4.4734
14.3690
16.4720
2.5109
31.8974
5.7036
18.3205
21.0018
3.2014
48.1588
8.6113
27.6603
31.7086
4.8334
H(l/3)
H(l/10)
Dynamic Loadings
Vertical
Sta
9
10
11
12
13
14
15
16
17
RMS
660410.1
770714.8
852958.4
902672.4
916354.1
891059.9
827132.9
728960.0
456989.5
B.,M.
2641640.0
3082859.0
3411833.0
3610689.0
3665416.0
3564239.0
3308531.0
2915840.0
1827958.0
•
H(l/1000)
3368092..0
3930646..0
4350088, .0
4603629..0
4673406..0
4544406. .0
4219378..0
3717696. ,0
2330646..0
5085158.0
5934504.0
6567779.0
6950577.0
7055927.0
6861161.0
6368923.0
5612991.0
3518819.0
Horizontal B.M.
Sta
RMS
H(l/3)
H(l/10)
H(l/1000)
9
440864.
518902,
575644.
606.50,
609593,
584602,
535193,
1763456.0
2075610.0
2302577.0
2424600
2438373
2338410
2140775
1855452
1093937
2248406,
2646403,
2935786,
3091365,
3108925,
2981473,
2729488,
2365701,
1394770,
3394653,
3995549,
4432461,
4667354,
4693867,
4501440,
4120991,
3571745,
2105830,
10
11
12
13
14
15
16
17
463863
273484
88
Combined B.M.
Sta
RMS
H(l/3)
9
958354.,0
1121474..0
1242414..0
1310208..0
1320890..0
1273295..0
1170845..0
1020343..5
616199,,8
3833416.0
4485896.0
4969656.0
5240832.0
5283560.0
5093180.0
4683380.0
4081374.0
2464799.0
10
11
12
13
14
15
16
17
Hd/10)
4887606..0
5719517.,0
6336311..0
6682061. ,0
6736539..0
6493804, ,0
5971309..0
5203752..0
3142619,,0
H(l/1000)
7379326.0
8635349.0
9566587.0
10088601.0
10170852.0
9804371.0
9015506.0
7856645.0
4744738.0
Appendix B
CALCULATIONS FOR AN AVERAGE VALUE OF
pVH
r
For comparison with the value of the correlation
coefficient of
an average of
/0
/0 VH
= 0.32 as obtained from figure III. 5,
was calculated from the moment responses
computed by equation
(6)
M
T
of the first method.
-
E
VH
2K E
V
- K2 E M
H
k
V
M
H
As the long-term extreme wave moments were not ex-
trapolated, the mean
Attt
value was obtained by averaging
over wind/wave directions and for representative sea states
The results of the calculation are summarized below:
89
90
V = 16 knots
Short-crested seas responses
h=7.4 ft.
h=24.3 ft.
h=43.3 ft.
wind/wave direction
/VH
o°
n
22.5°
45.°
67.5°
90° o
112.5
1350
157.5°
180°
average
^VR
0.771
VH
1
= 0.536
VH
.799
.633
.743
.676
.587
.460
.293
.214
.692
.700
.697
.650
.555
.441
.384
.414
.468
.590
.598
.603
0.575
0.556
0.477
mean for low, moderate and high seas
TVH
1
.367
.334
.368
.395
Appendix C
CALCULATION OF THE MEAN RMS PEAK-TO-PEAK
VERTICAL BENDING STRESS AND THE
STANDARD DEVIATION WITH RESPECT TO WIND
The calculated bending moment was converted into stress
by the relation:
Stress = Moment
*
Section Modulus
where the section modulus of the ship at the deck is given as
566,794 in
2
-
ft [253.
The computer output gave the RMS of the response.
To
get the peak-to-peak RMS value, the data was multiplied by
the factor 2/2.
The mean value of the response to a sea described by
its significant height is given by:
9
my =
I
V.
p(T.)
i=l
where
tcl
= mean RMS of the peak-to-peak response to a sea
specified by the significant height.
V.
= RMS of the peak-to-peak response to the ith
sea spectra
91
92
p^T.^ = probability of occurrence of the ith sea spectra.
p(_T.)
is approximated by the relative frequency of
occurrence of the different observed wave periods for a
given wave height in the tabulated data of [24].
Average values for all the areas in the ship's route
The calculation is illustrated in table A.l
were obtained.
for the one case of h = 13.2 feet (wave height code 05 in
[24])
.
The standard deviation of the vertical bending stress
with respect to wave measurements is calculated from the
relation
S
2
2
J^i
"
%f
*
P( T i)
where S2 = standard deviation of the bending stress with
respect to waves.
For direct comparison with experiments
based on
wind measurements, the following relation is used:
S
2
l
= S
2
2
+
S
fs
[
2
3
-
M
12
s
2
tan e
[23]
93
3
E
O 3
o\
l-l
VO
00
cn
CM
W
(^
ID
00
ON
cn
m
o
on
<r
st
cn
rH
en
vO
o r~
m
oo
st
CM
ON
rH
VO
VO
vO
ON
cn
CM
ON
rH
r~-
o
vO
en
CM
st
O
vO
o
VO
rH
O
ON
rH
o
CM
rH
r--.
o rH
O m o
sa-
CM
o
m
vO
ON
cn
cn
CM
O
o
st
CM
•H
CO
cn
CM
<4-l
rH
00
CM
O
rH
rH
rH
CM
o
CN
rH
ON
cn
cn
<!
o
St
CM
in
rH
VO
vO
rH
•
m
o
o
ON
VD
in
m
CM
Cn
•H
3
o
4J
cn
rH
4J
•
IM
rH
1
m
Jl
H
in
o
r-^
St
CM
m m
cn
st
vO
oo
00
cn
rH
CM
<t
ON
cn
ON
o
st
rH
cn
CM
m
o
CO
m
vO
rH
o
rH
00
r^
cn
CM
r-»
A
st
vD
in
CM
cn
cn
C
o
in
o
4-1
1
U
St
vD
•vf
On
On
st
CM
rH
vo
4-1
cn
CM
<r
on
rH
vO
m
<r
m
r^-
cn
r~-
cn
rH
00
CM
o
r^
rH
Xi
60
•H
VD
m
vO
rH
rH
r^
st
st
rH
ON
00
st
oo
in
r~.
r~.
o
st
CM
•
CM
m
rH
ss
CM
CO
m
I
00
O r^
m o
vO
VO
ON
cn
CM
•vf
rH
ON
VO
-tf
m
rH
m
cn
CM
rH
CM
r^
St
rH
cn
rH
m
CM
m
CM
r-~
r^
in
vO
CM
00
rH
CM
cn
rH
rH
r-~
m
CM
CM
St
CD
rH
s
cn
<-\
m
rH
r^
CM
rH
vO
on
vO
CM
st
rH
o
o
CM
o <t
CM
m
rH
cn
CM
rH
r»
St
VO
CO
vO
o
vO
CM
oo
I*-
H
r~-
cn
O
00
M-l
cn
r-~
CO
3
o
O
VO
CM
vO
cn
!-i
i-H
m
o O
m
o CM
st
m rH
rH
rH
CM
ON
rH
on
cn
r^
rH
St
CM
00
00
m
r--
O
u-i
o
o
cn
ON
CJ
•H
•H
II
J-J
4-1
4J
Vj
01
•H
rH
VD
U
CO
CJ
U
3
CN
o H
CM
rH
CJ
o
rH
rH
CM
rH
rH
cn
cn
ON
CM
r~
CM
<r
o rH
o
vO
cn
<t
cn
m
rH
st
vO
cn
CM
ON
St
ON
CM
rH
rH
r^
CM
00
rH
m
o
rH
On
CM
vD
00
CM
st
vO
X
ex
VD
rH
rH
cn
CM
B
oo
VD
cn
rH
rH
o
/
-a
o a)
•h -a
u o
o
CM
/
/
/
/
/
3
cn
/
c_:/
o)
04
OJ
/
/
^
^
<*>
"*
cn
CM
Ov
CM
-<r
cn
00
CM
00
rH
r-.
rH
rH
rH
St
CM
St
m
st
r.
R
3
rH
O
o
3
OJ
u
3
o
o
ex
Cfl
U
r--
IH
P5
CM
•3
•
O
o
00
BSJ
§
II
cu
01
co
u 3
.C o
60 IX
•H ca
01
JS
cfl
01
1
cfl
0)
3
ON
St
cn
Pi VM rH
CII
1
M M
cfl
4-1
3
o
1
o
4-1
cfl
CO
0)
cn
1
o
4-1
3
CO
CO
0)
PM
3
co
u
cfl
0)
cfl
o
cn
m
t^-
cfl
Pn
p4
cn
st
r-»
CU
K
rH
St
00
CO
sr
u
>
rH
4J
VD
CO
CM
Cfl
vO
ON
•H
cm
CM
>
rH
> Ph
to
oo
s s
OJ
tfl
01
CJ
00
vO
rH
rH
rH
3
pq
cfl
01
cn
VO
cn
00
cn
00
ON
00
<r
CM
3
•H
Tl
0)
rH
l-i
3
01
E
o
rH
E
O
vO
00
CM
vO
CO
cn
rH
00
4J
rH
rH
in
cn
r^
rH
rH
V-i
4J
r~-
g
in
r^
cn
cn
0)
CO
3
S>
II
II
vD
QJ
ON
ON
C/3
4J
vw
-3
O
CX
o
cn
01
Pi
Pn
00
s
pd
0)
S
94
The numerical calculation is illustrated below for
the Roll wind- to-wave relation.
=
**
Jl^'i
Beaufort No
h(ft)
m
1
7.2
123079.
2
3
4
5
V
158320.
7.6
186316.
8.0
229732.
9.0
305014.
10.0
"
Sv) 2
2/
r ^
(ton-ft)
s
3.465xl0
5.616xl0
6.36xl0
5.188xl0
7.572xl0
12.4
443627.
1.322xl0
7
15.0
541361.
2.506xl0
9
10
18.2
803314.
964491.
22.0
26.0
1156994.
S
Beaufort No
h(ft)
3.
5.
7.
s
9
)
(kpsi)
279x10
530x10
.239
.878
.296
.0992
.315
.0810
.285
.1184
.344
.2064
.454
.391:
5
.5120
.716
.8637
.929
499x10
1.1712
1.082
9
9
10
10
10
10
l0
(kpsi)
s
0.573
9
9
6
8
K Ti
= S 2 J+ r S 2 " ah 1 tan 2 G
71" J
2
[ 3
2
l
2
s
(kpsi)
H(ft)
2
Un6 l
ft
s
(kpsi)
j
s
1
(kpsi)
1
7.2
.0573
.8
.115692
2.5
.374
2
7.6
.0878
.4
.115692
2.5
.414
3
8.0
.0992
.6
.115692
2.5
.435
4
9.0
.810
1.0
.115692
2.8
.430
5
10.0
.1184
2.5
.115692
3.4
.516
6
12.4
.2064
2.5
.115692
4.3
.668
7
15.0
.3912
3.5
.115692
5.1
.758
8
18.2
.5120
3.8
.115692
6.1
.997
9
22.0
.8637
4.0
.115692
7.0
1.225
10
26.0
1.1712
4.0
.115692
7.8
1.403
153756
Baltazar
Combined ship hull
horizontal and vertical
wave bending moments
in
rregular seas.
'
<
Thesis
B205
74
D
CPLAY
...
Baltazar
Combined ship hull
horizontal and vertical
wave bending moments in
irregular seas.
.
:
wO
!hesB205
Combined ship
hull horizontal
and
vertic
3 2768 001 91251 2
DUDLEY KNOX LIBRARY
^