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Topics in Ship Structural Design(Hull Buckling and Ultimate Strength)
Lecture 7 Elastic Buckling of Stiffened Panels
Reference : Ship Structural Design Ch.13
NAOE
Jang, Beom Seon
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Comparison between Buckling and Ultimate Strength of plating
Plate capacity interactions between biaxial compression FEA, Buckling, Ultimate strength, a / b = 3, t = 13mm
Reference
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Comparison between Buckling and Ultimate Strength of plating
Plate capacity interactions between biaxial compression FEA, Buckling, Ultimate strength, a / b = 3, t = 21mm
Reference
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Two types of buckling
Stiffened panels can buckle in essentially two different ways: overall buckling and
local buckling.
Overall buckling : stiffeners buckles along with the plating
Local buckling :
1) stiffeners buckle prematurely because of inadequate rigidity
2) plate panels buckle between the stiffeners → shedding extra load into stiffeners
→ eventually stiffeners buckles in the manner of column.
For most ship panels, the buckling is inelastic. “failure” instead of “buckling”
Nevertheless, elastic buckling analysis gives a good indication of the likely modes of
failure and a foundation for the more complex question of the inelastic buckling and
ultimate strengths of stiffened panels.
4
Elastic Buckling of Stiffened Panels
Overall buckling Local buckling
Torsional buckling (tripping) of stiffeners
Plate buckling
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Schematics of various types of stiffed panel buckling
Local buckling
Overall buckling
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General
STEP 1 : Minimum flexural rigidity of stiffeners to avoid overall buckling
→ The minimum value of γx to ensure stiffener buckling does not precede plate buckling
γx :: flexural rigidity of stiffener + plating /the flexural rigidity of the plating
STEP 2 : Calculation of Column Buckling stress
→ Buckling of a column composed of stiffener and plating of effective breadth
STEP 3 : Calculation of stiffener-tripping stress
→ Flexural-torsional buckling of a stiffener with rotational restraint by plating
6
Elastic Buckling of Stiffened Panels
2
3
12(1 )x xx
EI v I
Db bt
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Plate-Stiffener (BEAM) Combination model
Orthotropic plate model
PLATE
STIFFENER
PLATE
STIFFENER
Idealization of continuous stiffened panel
Plate-Stiffener Separation model
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General
Analytical methods of solving buckling problems include two principal types: discrete beam : more versatile and accurate, but more computation
orthotropic plate : only when the stiffeners are very closely spaced.
In this lecture, it is assumed that
the edges of the panel are simply supported
individual elements of the stiffeners are not subject to instability
First principle in regard to stiffener
stiffeners should be sufficiently rigid and stable (∵ stiffener buckling = overall buckling and the plating is left with almost no lateral rigidity
Substantial lateral load ship panels must carry requires sufficiently large stiffness and rigidity.
It is best to first perform an elastic buckling analysis because:
relatively simple, consisting mostly of explicit formulas
for slender panels, it may be one of the governing failure modes
it indicates whether an inelastic analysis is required
8
Elastic Buckling of Stiffened Panels
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Minimum flexural rigidity to avoid overall buckling
Some parameters
the ratio of the flexural rigidity of the combined section to the flexural
rigidity of the plating
the panel aspect ratio
the area ratio
9
13.1 Longitudinally Stiffened Panels – Overall buckling v.s. Plate buckling
2
3
12(1 )x xx
EI v I
Db bt
L a
B B
xx
A
bt
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Minimum flexural rigidity to avoid overall buckling
The minimum value of γx (the less one of the two values) to ensure stiffener
buckling does not precede plate buckling
(1) panel with one central longitudinal stiffener (whichever is less)
(2) panel with two equally spaced longitudinal stiffeners(whichever is less)
10
13.1 Longitudinally Stiffened Panels – Overall buckling v.s. Plate buckling
222.8 (2.5 16 ) 10.8x x
48.8 112 (1 0.5 )x x x
3 243.5 36x x
2228 610 325x x x
0
500
1000
1500
2000
0 2 4 6 8 10
γx
L/B
1.5
0.5
2
3
0
1000
2000
3000
4000
5000
6000
0 5 10
γx
L/B
1.5
0.5
2
3
one central longitudinal stiffener two equally spaced longitudinal stiffener
δx δx
min x
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Minimum flexural rigidity to avoid overall buckling
A more general solution, valid for any number of stiffeners, has been
presented by Klitchiff
where NB=number of panels=1+number of longitudinal stiffeners
11
13.1 Longitudinally Stiffened Panels – Overall buckling v.s. Plate buckling
2 2 2 2 2 2 24(1 ) (1 ) 2x x B B BN N N
Homework 6-1 Plot Klitchiff’s curve versus L/B for different NB and δx and compare with the previous two curves.
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Calculation of overall buckling stress
An alternative approach is to calculate the overall buckling stress (σa) cr and
compare it to the plate buckling stress σ0.
In short panels, the equivalent slenderness ratio of each column is the
actual slenderness of the section or in terms of nondimensional parameter.
In long panels, the stiffeners receive some lateral restraint from the sides of
the panel→ bukling in more than one half wave.
the equivalent slenderness ratio < the value given by the former formula
where Cπ is given by whichever is less
12
13.1 Longitudinally Stiffened Panels
0( )a cr 2
0 3.62t
Eb
/ ( )eq x x
L a a
I A bt
/ ( )eq x x
C aL aC
I A bt
1
2(1 1 )
x
x
C
1C
212(1 )(1 )x
xeq
vL a
t
2
3
12(1 )x xx
EI v I
Db bt
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Calculation of overall buckling stress
Then
Using the adjective “slender” to describe a panel in which the overall
buckling stress calculated from elastic theory is less than the yield stress.
Normally, plate buckling precedes overall buckling → When overall buckling
occurs the plate flange of the stiffener will not be fully effective over the
width b.(due to out-of-plane deformation, caused by buckling)
13
13.1 Longitudinally Stiffened Panels
The buckled center portion is
discounted completely and the
original b is replaced by be
2
2( )
( / )a cr
eq
E
L
e a
e
b
b
2
2( / )Y
eq
E
L
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Calculation of overall buckling stress
The effective width is taken to be the width at which the equivalent plate
would buckle at an applied stress of σe
for the original plate
if it is assumed that k is the same in both cases then
For long plate (a/b>1)
The effective width would reach its smallest possible value when σa reached
yield σyield.
14
13.1 Longitudinally Stiffened Panels
2
2e
e
Dk
b t
2
2( )a cr
Dk
b t
( )e a cr
e
b
b
1.9e
e
b t E
b b
min
1.9eb
b
2
2( ) 4a cr
D
b t
Et
b Y
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Calculation of overall buckling stress
The critical value is given by
The axial stress in the stiffener is larger than the external applied stress.
(σa)cr axial stress corresponding to (σe)cr
A suitable procedure
1. Assume some initial value of be
2. Calculate δxe and Ixe, and then evaluate (L/ρe)eq
3. Calculate (σe)cr
4. Using this value, recalculate be from
5. Repeat from step 2 until be has converged
6. Calculate (σa)cr
a single nonlinear equation for be
15
13.1 Longitudinally Stiffened Panels
2
2( )
( / )e cr
e eq
E
L
2
2( )
( / )
et xa cr
x e eq
b A E
bt A L
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w we w f f e w f e
A Adb A A A b t A A b t
at
)()( xeexa AtbAbt
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e
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b b
2
2( )
( / )e cr
e eq
E
L
2
2( )
( / )
et xa cr
x e eq
b A E
bt A L
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Calculation of overall buckling stress
(σe)cr depends on ρe depends on depends on be.
When bt> 2Ax (usual case), decrease of be → increase ρe
→ increase in (σa)cr
The lowest of (σa)cr corresponds to be= b, when plating in fully effective.
16
13.1 Longitudinally Stiffened Panels
2
2( )
( / )e cr
e eq
E
L
be=0.75b
be=0.5bSmall stiffener
Large stiffener
2
2( )
( / )
et xa cr
x e eq
b A E
bt A L
2
2( )
( / )e cr
e eq
E
L
Small stiffener
Large stiffener
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Calculation of overall buckling stress
17
13.1 Longitudinally Stiffened Panels
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Homework 6-2 illustrates the previous graph using the following formula
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Calculation of overall buckling stress
Possible modes of elastic buckling of stiffened panels
18
13.1 Longitudinally Stiffened Panels
be=b
For small stiffener
(σa)cr for be > (σa)cr for b
For large stiffener
(σa)cr for be < (σa)cr for b
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Local buckling of stiffener (tripping)
A stiffener may buckle by twisting about its line of attachment to the plating,
“tripping”. The direction of the tripping alternates as shown.
Tripping and plat buckling do interact but they can occur in either order.
Tripping failure is regarded as collapse, the tripping leads no stiffening and
overall buckling follows immediately. Elastic tripping is a quite sudden
phenomenon → a most undesirable mode of buckling
Open sections used in ship panels have relatively little torsional rigidity.
Closed cross section : difficulties in fabrication, inspection and control of
corrosion.
19
13.1 Longitudinally Stiffened Panels
Stiffener tripping
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Permanent Means of Access (PMA)
IMO introduced PMA regulations for securing access to cargo holds and ballast
tanks in oil tankers and bulk carriers.
Overall and close-up inspections and thickness measurements of the critical hull
structural parts.
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Ultimate Strength Assessment of PMA
Rule for Local Support Members in CSR
Objective : Ultimate strength assessment of PMA using nonlinear FE analysis
Establish evaluation procedure for CSR Tanker design
Research :
No specified rule for PMA structure in CSR
Local support member scantling rule in CSR → over scantling
Establish a ultimate strength assessment procedure.
Propose a evaluation criteria based on CSR cargo hold analysis.
SECTION 10- BUCKLING AND ULTIMATE STRENGTH
2.2 Plates and Local Support Members
2.2.1 Proportions of plate panels and local support members
CSR req. : hxtw+bfxtf
1150x20.5+288x16.5
Proposed :
1150x12.0+150x15.0h/2
bf
tf
tw
b
tfb
bf
b
h
tp
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Ultimate Strength Assessment of PMA
FE Modeling &
Boundary Condition
Linear Buckling
AnalysisInitial
Imperfection
Nonlinear
FE Analysis
Check Failure
Mode
Ultimate Strength
from P-δ curveCRS Cargo Hold
Analysis
Compare Capacity
Curve with Stress
Result : Proposed scantling satisfies required strength.
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Elastic Lateral Torsional Buckling of PMA I
Objective : Establish analytical method to predict lateral torsional buckling of PMA
Research :
Rayleigh – Riz method
Assumption of deflection and strain of PMA section
Total strain energy and work done during buckling
Solve an eigen-value problem
Assumption of sectional displacementPMA Structure
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Elastic Lateral Torsional Buckling of PMA II
Compatibility constraints
Flexural Out-of-Plane Displacement of web plate
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Elastic Lateral Torsional Buckling of PMA III
Derivation of Strain Energy
“At the limit of stability, the second variation of total potential
energy is zero”
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Derivation of Work Done during Buckling
6 X 6 Eigenvalue Problem
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0
50
100
150
200
250
300
3000 5000 7000 9000 11000 13000 15000
Ela
sti
c T
rip
pin
g S
tre
ss
(MP
a)
Span Length (mm)
FEM without Plate (Case A)
FEM with Plate Pinned joint (Case B)
Uniform with a0 and a1 (Case I)
Rothwell with a0 and a1 (Case II)
Uniform with only a0 (Case III)
RothWell with only a0 Case (IV)
Elastic Lateral Torsional Buckling of PMA IV
Comparison with FE analysis
ResultsExtended Plate Rotational Restraint
Y Displacement Constrained
Z Displacement Constrained
Y & Z Displacement Constrained
X Displacement Constrained
YseCafor4
XseCafor
2
1
2
2
2
Bp
p
Bp
p
Bspringpo
Db
l
Db
l
k
Proposed
Method
FEM
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Elastic Lateral Torsional Buckling of PMA V
Tripping v.s. web plate local buckling for varying mid-flat-bar location
hw
αhw
m=1 m=5
m=1
m=5
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Elastic Lateral Torsional Buckling of PMA VI
Thick bottom plate (16t)
Thin bottom plate (13t)
Web Local
Buckling
Plate BucklingLateral Torsional
Buckling
Lateral Torsional
Buckling
Web Local Buckling v.s. Bottom Plate Buckling
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Types of stiffener buckling and method of analysis
29
13.1 Longitudinally Stiffened Panels
Flexural-torsional buckling : torsional buckling of a stiffener may also be caused by a bending moments → compression in the flange.
Flexural buckling of columns : Bending about the axis of least resistance
Torsional buckling of columns : Twisting without bending.
Flexural-torsional buckling of columns : subjected to compression → simultaneous twisting and bending.
Lateral-torsional buckling of beams : subjected to bending, → simultaneous twisting and bending.
Local buckling : Buckling of a thin-walled part of the cross-section (plate-buckling, shell-buckling)
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Types of stiffener buckling and method of analysis
Flexural-torsional buckling : torsional buckling of a stiffener may also be c
aused by a bending moments
→ compression in the flange.
→ nonlinear coupling between the flexural and torsional response
Two different methods for dealing nonlinear elastic buckling
"folded plate" analysis based on finite difference methods
• simpler
• is restricted to certain boundary conditions and simple forms of structural ge
ometry
• basically limited to elastic buckling of bifurcation type
nonlinear frame (finite element) analysis
• necessarily computer-based, but quite economical
• is much more general and can deal with other forms of nonlinearity
30
13.1 Longitudinally Stiffened Panels
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Stiffener buckling due to axial compression
A stiffener acts essentially as a column, but tripping or torsional
buckling differs from that of a column in three ways
the rotation occurs about an enforced axis-the line of attachment to the
plating.
the plate offers some restraint against this rotation.
it is not necessarily rigid body rotation.
31
13.1 Longitudinally Stiffened Panels
Effect of web bending
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Stiffener buckling due to axial compression
The governing differential equation for the rotation is:
where
ISZ = moment of inertia of the stiffener
d = stiffener web height
Isp = polar moment of inertia of the stiffener about the center of
rotation
K ϕ = distributed rotational restraint which plating exerts on the
stiffener.
Φ(x) = a buckled shaped in which the rotation Φ varies sinusoidally
in m half waves over the length a.
σa,T = the elastic tripping stress, the minimum value of applied in-
plane stress σa that would cause tripping
32
13.1 Longitudinally Stiffened Panels
0)(2
2
4
42
K
dx
dIGJ
dx
ddEI SPaSZ
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Stiffener buckling due to axial compression
σa,T satisfies the following, where m is a positive integer.
Kϕ is a function of σa and m.
Kϕ is dependent on σa because plate buckling can diminish or eliminate Kϕ.
Kϕ is dependent on m.
The rotational restraint coefficient is offered by the plating
→ to be defined by the plate’s flexural rigidity which causes a total distributed
restraining moment MR=2M along the line of the stiffener attachment.
If a>>b, unit strip of plating across the span b, ϕ = ½ Mb/D.
This assumes that the buckled displacement of the stiffener is entirely due
to rigid body rotation.
33
13.1 Longitudinally Stiffened Panels
4 4 2 22
,4 2( ) ( ) 0SZ a sp a
m mEI d GJ I K m
a a
b
DMK R 4
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Stiffener buckling due to axial compression
In practice some of the sideways displacement of
the stiffener flange occurs due to bending of the
web. Even when the stiffener web is slender.
Cr is the factor by which the plate rotational restraint
is reduced due to web bending.
where (for σa=0)
Cα is a correction factor because of the effect of
plate aspect ratio.
34
13.1 Longitudinally Stiffened Panels
rC 3
1
1 (2 / 3)( / ) ( / )r
w
Ct t d b
4= r
DK C C
b
3
1
1 0.4( / ) ( / )r
w
Ct t d b
2
21
mC
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Stiffener buckling due to axial compression
Cα : correction factor for aspect ratio and plate buckling effects because
plate stiffness decreases as the applied stress σa approaches plate buckling
stress σ0.
Cα=1-(m/α)2 when σa = σ0. if m=α (aspect ratio), Cα= 0. and Kϕ =0.
Tripping occurs in a single half-wave, m=1 → the same as number of half-
wave for square or short panel. α≈ 1 → complete loss of rotational restraint.
For stiffened panels of usual proportions, tripping occurs in a single half-
wave, and hence it is mainly square or short panels in which the loss of
stiffness can occur.
35
13.1 Longitudinally Stiffened Panels
2
2
0
21- 1a m
C
3
1
1 0.4( / ) ( / )r
w
Ct t d b
2
2
0
21- 1a m
C
2
3 2
0
24 11 1
1 0.4
a
w
D mK
b t d
t b
2m
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Stiffener buckling due to axial compression
by substituting Kϕ and solving for σa.
Here, regard m continuous variable. Find m which gives the lowest σa,T
m can be predicted and trial-and-error approach can be avoided.
36
13.1 Longitudinally Stiffened Panels
3
1
1 0.4( / ) ( / )r
w
Ct t d b
2
3 2
0
24 11 1
1 0.4
a
w
D mK
b t d
t b
2
2
2
2
2
2
22
4
3,
4
2
1
,...2,1
,b
m
a
b
DCdEI
a
mGJ
tbCI
m
Minimumr
sz
rsp
Ta
4 4 2 22
,4 2( ) ( ) 0SZ a sp a
m mEI d GJ I K m
a a
0,
dm
d Ta4
2
4 r
SZ
DCam
EI d b
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Stiffener buckling due to axial compression
After estimating m,
(a) Solution when m≥2,
(b) Solution when m=1,
ex) estimated m≈1.3 → m=1
estimated m≈1.7 → m=1 or m=2
(c) Lower bound solution for quick check
- ignore plate restraint effect Cα =0, i.e. σa = σ0 → Kϕ =0
- ignore GJ term since it is small.
37
13.1 Longitudinally Stiffened Panels
2
, 3 2
4
414
2
r SZ ra T
rsp
DC EI d DC bGJ
C b t bI
2 22 2
, 3 2 2
4
41( )
2
SZ ra T
rsp
EI d DCGJ a b
C b t a bI
2
2
,
1a T SZ
sp
dE I
I a
42
4 r
SZ
DCam
EI d b
2
2
2
2
2
2
22
4
3,
4
2
1
,...2,1
,b
m
a
b
DCdEI
a
mGJ
tbCI
m
Minimumr
sz
rsp
Ta
2
2
2
2
2
2
22
4
3,
4
2
1
,...2,1
,b
m
a
b
DCdEI
a
mGJ
tbCI
m
Minimumr
sz
rsp
Ta
1m
2
2
0
21- 1a m
C
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Stiffener buckling due to axial compression
ISZ and Isp can be expressed in terms of stiffener areas
Also, by defining a fractional flange area f=Af/Ax, it becomes
a : adverse effect of panel length
bf : strengthening effect, but cannot be increased indefinitely due to possibili
ty of flange buckling.
bf/tf <14 for mild steel to avoid the flange buckling.
38
13.1 Longitudinally Stiffened Panels
2
3
wsp f
AI d A
2
2 3 4
3
fxf
f w f
sz f
x x
AAA
A A bI b
A A
22
,
(1 )
1 2
f
a T
bEf f
f a
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Transversely Stiffened Panels
The strength of the stiffened panel is determined by the buckling strength of the plate
between stiffeners.
The required minimum value of γy given is
where
NL= number of plate panels= L/a, NL-1 = number of stiffeners
EIy= flexural rigidity of one transverse stiffener,
including a plate flange of full width a.
39
13.2 Transversely Stiffened Panels
22 2 2 2
2 4
(4 1) 1 2 1
2 5 1
L L L
y
L
N N N
N
Da
EI y
y 22
b
L
B
L
OPen INteractive Structural Lab
Homework 6-1 due date 21th November
Homework 6-1 Analyze the effect of a, b, d, bf, tf, tw, h, tplate using ANOVA
table and an orthogonal array. Recommend to use Minitab.a=2400±10%, b=800±10%, d=450mm, bf,=250mm
tf,= tw=tplate=15 mm±10%,
40
2
2
2
2
2
2
22
4
3,
4
2
1
,...2,1
,b
m
a
b
DCdEI
a
mGJ
tbCI
m
Minimumr
sz
rsp
Ta
33
33dtbt
J wff
2
3
wsp f
AI d A
2
2 3 4
3
fxf
f w f
sz f
x x
AAA
A A bI b
A A
OPen INteractive Structural Lab
Homework 6-1
Orthogonal array for 8 parameters with 3 levels
41
OPen INteractive Structural Lab
A sample of the use of ANOVA table and Orthogonal Array
42
Span Space tp hw tw bf tf
3870.0 733.5 12.8 339.3 7.7 135.0 10.8
3870.0 733.5 12.8 339.3 8.5 150.0 12.0
3870.0 733.5 12.8 339.3 9.4 165.0 13.2
3870.0 815.0 14.3 377.0 7.7 135.0 10.8
3870.0 815.0 14.3 377.0 8.5 150.0 12.0
3870.0 815.0 14.3 377.0 9.4 165.0 13.2
3870.0 896.5 15.7 414.7 7.7 135.0 10.8
3870.0 896.5 15.7 414.7 8.5 150.0 12.0
3870.0 896.5 15.7 414.7 9.4 165.0 13.2
4300.0 733.5 14.3 414.7 7.7 150.0 13.2
4300.0 733.5 14.3 414.7 8.5 165.0 10.8
4300.0 733.5 14.3 414.7 9.4 135.0 12.0
4300.0 815.0 15.7 339.3 7.7 150.0 13.2
4300.0 815.0 15.7 339.3 8.5 165.0 10.8
4300.0 815.0 15.7 339.3 9.4 135.0 12.0
4300.0 896.5 12.8 377.0 7.7 150.0 13.2
4300.0 896.5 12.8 377.0 8.5 165.0 10.8
4300.0 896.5 12.8 377.0 9.4 135.0 12.0
4730.0 733.5 15.7 377.0 7.7 165.0 12.0
4730.0 733.5 15.7 377.0 8.5 135.0 13.2
4730.0 733.5 15.7 377.0 9.4 150.0 10.8
4730.0 815.0 12.8 414.7 7.7 165.0 12.0
4730.0 815.0 12.8 414.7 8.5 135.0 13.2
4730.0 815.0 12.8 414.7 9.4 150.0 10.8
4730.0 896.5 14.3 339.3 7.7 165.0 12.0
4730.0 896.5 14.3 339.3 8.5 135.0 13.2
4730.0 896.5 14.3 339.3 9.4 150.0 10.8
Selection of Stiffened Plate Parameters
Orthogonal Array L27, 37
DNV RP-C201 U.F v.s. Scantlings
DNV PULS v.s. Scantlings