Hull Girder Response

Quasi

Static Analysis
Basic Relationships
Model the hull as a Free

Free box beam.
Beam on an elastic foundation
Must maintain overall Static Equilibrium.
Force of Buoyancy = Weight of the Ship
LCB must be in line with the LCG
0 0
L L
g a x dx g m x dx
0 0
L L
g xa x dx g xm x dx
Basic Relationships
From Beam Theory
–
governing equation for bending
moment:
Beam is experiencing bending due to the differences
between the Weight and Buoyancy distributions
2
2
d m
f x
dx
Where
f
(
x
)
is a distributed
vertical load.
( ) ( ) ( )
f x b x w x
Buoyancy
g
a
(
x
)
Net Load
Weight
g
m
(
x
)
Basic Relationships
buoyancy curve

b(x)
weight curve

w(x)
net load curve

f(x) = b(x)

w(x)
Sign Convention
Positive
Upwards
+ f
Basic Relationships
The solution for
M
(
x
) requires two integrations:
The first integration yields the transverse shear force
distribution,
Q
(
x
)
Impose static equilibrium on a differential element
Q
M
f
Q + dQ
M + dM
dx
0
Q f dx Q dQ
dQ
f
dx
0
x
Q x f x dx C
But ships are “Free

Free” Beams

No shear at ends!
Q
(0) = 0 and
Q
(
L
) = 0, so
C
= 0
Finding Shear Distribution
Shear Force

Q
+ Q :
Positive
Clockwise
Sign Convention
Positive
Upwards
+
f
:
Net Load

f
+ Q

Q
Basic Relationships
The second integration yields the longitudinal bending
moment distribution,
M
(
x
):
Sum of the moments about the right hand side = 0
Q
M
f
Q + dQ
M + dM
dx
0
2
dx
M Qdx f dx M dM
0
dM
Q
dx
0
x
M x Q x dx D
Again, ships are “Free

Free” Beams

No moment at ends!
M
(0) = 0 and
M
(
L
) = 0, so
D
= 0
Finding Bending Moment Distribution
Shear Force

Q
Bending Moment

M
+
Q
:
Positive
Clockwise
+
M
:
Positive
Sagging
+
Q

Q

M
Sign Convention
Shear & Moment Curve Characteristics
Zero shear and bending moments at the ends.
Points of zero net load correspond to points of
minimum or maximum shear.
Points of zero shear correspond to points of
minimum or maximum bending moment.
Points of minimum or maximum shear
correspond to inflection points on bending
moment curve.
On ships, there is no shear or bending moments
at the forward or aft ends
.
Still Water Condition
Static Analysis

No Waves Present
Most Warships tend to Sag in this
Condition
Putting Deck in Compression
Putting Bottom in Tension
Quasi

Static Analysis
Simplified way to treat dynamic effect of waves on hull girder
bending
Attempts to choose two “worst case”conditions and analyze them.
Hogging Wave Condition
»
Wave with crest at bow, trough at midships, crest at stern.
Sagging Wave Condition
»
Wave with a trough at bow, crest at midships, trough at stern.
Wave height chosen to represent a “reasonable extreme”
Typically:
Ship is “balanced” on the wave and a static analysis is done.
1.1
BP
H L
Wave Elevation Profiles
The wave usually chosen for this analysis is a
Trochoidal
wave. It has a steeper crest and flatter
trough.
Chosen because it gives a better representation of
an actual sea wave than a sinusoidal wave.
Some use a cnoidal wave for shallow water as it
has even steeper crests.
Trochoidal vs. Sine Wave
20
15
10
5
0
5
10
15
20
0
20
40
60
80
100
120
140
160
180
200
Lenght (ft)
Wave Height (ft)
Trochoidal Wave
Sinusoidal Wave
Sagging Wave
Excess Weight Amidships

Excess Buoyancy on the Ends
Tension
Compression
Hogging Wave
Excess Buoyancy Amidships

Excess Weight on the Ends
Tension
Compression
Weight Curve Generation
The weight curve can be generated by numerous
methods:
Distinct Items (same method as for LCG)
Parabolic approximation
Trapezoidal approximation
Biles Method (similar to trapezoidal)
They all give similar results for shear and bending
moment calculations. Select based on the easiest in
your situation.
Distinct Item Method
ITEM
Material
units
wt/unit
WT
LCG
VCG
LMOM
VMOM
GROUP C  JOINERY WORK
Forward cabin
berth flat
composite
35
0.77
27
10.50
1.25
282.98
33.69
mattress
35
3.00
105
10.50
1.50
1102.50
157.50
shelf p&s
composite w/veneer
12
1.02
12
12.00
2.50
146.88
30.60
verticals p&s
composite w/veneer
34
1.02
35
12.00
1.00
416.16
34.68
desk
composite w/veneer
4
1.28
5
14.50
2.50
74.24
12.80
supports and hardware
5
14.50
2.50
72.50
12.50
hanging locker
composite w/veneer
27
1.28
35
15.00
2.00
518.40
69.12
rod & hardware
10
15.00
3.00
150.00
30.00
cabinet
composite w/veneer
17
1.02
17
16.75
3.00
290.45
52.02
door blkhd
composite w/veneer
25
1.85
46
17.25
2.00
791.43
91.76
drawers
wood
10
5.00
50
15.00
0.50
750.00
25.00
sole
plywood & teak
29
2.50
71
16.40
0.50
1168.50
35.63
overhead
honeycomb/vynal
24
0.50
12
17.00
6.25
204.00
75.00
Each component is located by its l, t and v position
and weight
Can be misleading for long components
Example Weight Curve
120K Bbl TAO Weight Curve
0
20
40
60
80
100
120
100
0
100
200
300
400
500
600
700
Feet from FP (+ Aft)
Distributed Weight (LT/ft)
Weight Curve
Displacement =
LCG =
27450
299.3
LT
ft aft FP
1/19/99
For each weight item, need
W
,
lcg
,
fwd
and
aft
Weight Item Information
fwd
W
aft
lcg
FP
Trapezoid Method
Models weight item as a trapezoid
Best used for
semi

concentrated
weight items
Need the following information:
Item weight
–
W (or mass, M)
Location of weight centroid wrt FP

lcg
Forward boundary wrt FP

fwd
Aft boundary wrt FP

aft
lcg
must be in middle 1/3 of trapezoid
Trapezoid Method
Find
l
and
x
Solve for
w
f
and
w
a
so
trapezoid’s area equals
W
and the centroid is at the
lcg
lcg
x
fwd
aft
l/2
l
w
f
w
a
FP
w
W
l
Wx
l
w
W
l
Wx
l
a
f
6
6
2
2
G
2
l
f
lcg
x
Biles Method
Used for weight items which are nearly
continuous
over
the length of the ship.
Assumes that weight decreases near bow & stern.
Assumes that there is a significant amount of parallel
middle body.
Models the material with two trapezoids and a
rectangle.
Biles Method
l
3
1.2h
l
3
l
3
w
f
w
a
FP
lcg
G
x
aft
l
x
h
w
l
x
h
w
l
w
h
a
f
7
54
6
.
0
7
54
6
.
0
The Three Types of Structure
Characteristics
Primary
Structure
Secondary
Structure
Tertiary Structure
In

plane rigidity
Quasi

infinite
Finite
Small
Loading
In

plane
Normal
Normal
Stresses
Tension,
Compression
and Shear
Bending and
Shear
Bending, Shear
and Membrane
Examples
Hull shell, deck,
blkhd, tank top
Stiffeners on
blkhd, shell
Unstiffened shell
Boundaries
Undetermined
Primary structure
Secondary
Structure
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