hull girder response - quasi-static analysis. basic relationships model the hull as a free-free box...
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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 Lg a x dx g m x dx
0 0
L Lg x a 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 mf x
dx Where f(x) is a distributed
vertical load.
( ) ( ) ( )f x b x w x
Buoyancyg a(x)
Net Load Weightg m(x)
Basic Relationships
buoyancy curve - b(x)
weight curve - w(x)
net load curve - f(x) = b(x) - w(x)
Sign Convention
PositiveUpwards+ 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
QM
f
Q + dQ M + dM
dx
0Q f dx Q dQ dQ
fdx
0
xQ 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 :PositiveClockwise
Sign Convention
PositiveUpwards+ 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
QM
f
Q + dQ M + dM
dx
02
dxM Qdx f dx M dM
0
dMQ
dx
0
xM 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 :PositiveClockwise
+ M : PositiveSagging
+ 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 BPH 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)
Wa
ve
He
igh
t (f
t)
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 VMOMGROUP 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
-1000100200300400500600700
Feet from FP (+ Aft)
Dis
trib
ute
d W
eig
ht
(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 wf and wa so trapezoid’s area equals W and the centroid is at the lcg lcg
x
fwd
aft
l/2l
wf
wa
FP
wW
l
Wx
l
wW
l
Wx
l
a
f
6
6
2
2
G
2lflcgx
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
wfwa
FP
lcg
G
x
aft
l
xhw
l
xhw
l
wh af 7
546.0
7
546.0
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