Structural Fracture Analyyypsis of Membrane Type … S3 Structures and... · Structural Fracture Analyyypsis of Membrane Type LNGC Containment Systems Under Sloshing Impacts ... STRUCTURAL

Structural Fracture Analysis of Membrane Type y ypLNGC Containment Systems Under Sloshing Impacts

Jung Min Sohn and Jeom Kee PaikJung Min Sohn and Jeom Kee PaikPusan National University

Elastic Large Deflection Behavior of Plates with Partially Rotation-Restrained Edgesa t a y otat o est a ed dges

Do Kyun Kim and Jeom Kee Paik Pusan National University

The Lloyd’s Register Educational Trust (LRET) y g ( )Marine & Offshore Research Workshop

16-18 February, 2010 at Engineering Auditorium, NUS

STRUCTURAL FRACTURE ANALYSIS OF MEMBRANE TYPE STRUCTURAL FRACTURE ANALYSIS OF MEMBRANE TYPE

LNGC CONTAINMENT SYSTEMS UNDER SLOSHING IMPACTSLNGC CONTAINMENT SYSTEMS UNDER SLOSHING IMPACTS

Jung Min Sohn and Jeom Kee Paik

LRET Research Centre of Excellence,

Pusan National University, KOREA

Structures and GeotechnicsLRET Marine & Offshore Research Workshop16~18 February, 2010

OverviewOverview

�� Background Background

�� ObjectivesObjectives

�� LNGC containment SystemLNGC containment System

�� QuasiQuasi--static Analysisstatic Analysis

�� Dynamic AnalysisDynamic Analysis

�� Conclusions & RemarksConclusions & Remarks

BackgroundBackground

� This trend brings a question if the cargo containment system of LNG

carriers with the existing proportions is strong enough against abnormal

actions which may happen in service.

S

S

S

11223344

11223344

11223344

1122334455

254254K Twin ScrewK Twin Screw

223223K Max Single ScrewK Max Single Screw

205205K Single ScrewK Single Screw

145145K AsK As--builtbuilt

Tank Sloshing Design

Design Loads Structural Failure Analysis

• Fracture of corrugated membrane

• Fracture of insulation system (foam)

Modeling

• Extent of analysis

•Mesh size (corrugation, foam)

• True stress–true strain relation in cryogenic

condition

•Dynamic yield stress

•Dynamic fracture strain

•Mastic

• Sloshing load profile

CFD Simulations

Sloshing–Load Characteristics

• Sloshing load profile with time

• Peak pressure

• Pressure impulse

Sloshing

Frequency

Sloshing Scenarios

• Tank filling level

•Duration of tank motion

•Amplitude of tank motion

• Rolling angle

Design Sloshing Loads

• Probabilistic exceedance curve

First–Fracture Based Structural

Design Curve

Sea

Trials

Objectives(1/2)Objectives(1/2)

� To evaluate strength performance of membrane corrugations in

Mark ⅢⅢⅢⅢ type LNG carrier cargo tanks under either quasi-static

pressure or sloshing impact pressure actions.

� To develop the procedure of strength analysis for membrane

corrugation structures using nonlinear finite element method.

Objectives(2/2)Objectives(2/2)

MARK III Type LNGC Cargo Containment SystemMARK III Type LNGC Cargo Containment System

Nomenclature: Dimensions of the corrugated membrane plate.

t

H

R1

R2

R3

S

S - d

d

2

d

2

d

2

d

2

d

2

ParameterR1

(mm)

R2

(mm)

R3

(mm)

d

(mm)

H

(mm)

t

(mm)

S

(mm)

Large corrugation 9.4 65.4 8.4 77 54.5 1.2 340

Small corrugation 8.4 38.4 8.4 53 36.0 1.2 340

Schematic of membrane (Mark III) type LNG cargo containment system.

Material Properties: Corrugated membrane plateMaterial Properties: Corrugated membrane plate

0 0.2 0.4 0.6

0

400

800

1200

1600

2000

Strain

Stress(MPa)

20°C

-163°C

Stainless steel

Stress versus strain curves

Temperature E(MPa) σY(MPa) α(mm/℃)

20°C189,000

280

1.4 10-5

-163°C 307

Material properties of stainless steel used for the corrugated membrane plate

Corrugated membrane plate

Material Properties: Insulation SystemMaterial Properties: Insulation System

Parameter 20ºC -163ºC

Ex(MPa) 9450 13200

Ey(MPa) 8000 11200

Ez(MPa) 820 1800

Gxy(MPa) 790 2900

Gxz(MPa) 325 700

Gyz(MPa) 260 550

νxy 0.1 0.1

νxz 0.1 0.1

νyz 0.1 0.1

Density (ton/mm3) 6.8 ×10-10 6.8 ×10-10

Parameter 20ºC -163ºC

Ex(MPa) 135 170

Ey(MPa) 180 215

Ez(MPa) 65 95

Gxy(MPa) 7 11

Gxz(MPa) 7 11

Gyz(MPa) 7 11

νxy 0.4 0.4

νxz 0.2 0.2

νyz 0.2 0.2

Density (ton/mm3) 1.25 × 10-10 1.25 × 10-10

Material stiffness properties of the reinforced polyurethane foam

Integrated orthotropic material stiffness properties of the plywood plates.

Analysis for QuasiAnalysis for Quasi--static Pressurestatic Pressure

1. Modeling Extent and Mesh size(1/2)1. Modeling Extent and Mesh size(1/2)

Membrane (Mark III) type LNG cargo containment system with boundary conditions.

????

y

xz


1. Modeling Extent and Mesh size(2/2)1. Modeling Extent and Mesh size(2/2)

X

Y

Z

(a) Corrugated membrane plate (b) Insulation system

Finite element mesh size.

Schematic illustration of the loading condition

A low temperature is maintained in a membrane-type LNG containment

system to keep the gas in a liquid state.

- Service temperature: -163°°°°C ~ 20°°°°C

p

2. Loading Conditions2. Loading Conditions



3. Boundary Conditions(1/2)3. Boundary Conditions(1/2)

1mm

Contact elementLC

1mm

LC

Contact element1mm

LC

Boundary condition (a) Boundary condition (b) Boundary condition (c)

Undeformed line

deformed line

Deformed shape under quasi-static pressure using boundary condition (b).

Various idealised boundary conditions.

Boundary condition (d) Boundary condition (e)

3. Boundary Conditions(2/2)3. Boundary Conditions(2/2)


Various idealised boundary conditions.

LC

1mm

Contact element

Plywood

RPUF

Plywood

RPUF

Triplex

LC

1mm

Plywood

RPUF

Plywood

RPUF

Triplex

Contact element


4. Results(1/5)4. Results(1/5)

Measurement point of the corrugated membrane.

A

Comparison of the deflection (deformation in the z direction) at point A with varied types of boundary conditions.

1

6

2

0.0 -10.0 -20.0 -30.0

0

5

10

15

20

25

30

35Point A

Pressure(bar)

4

5

7

3

1: B.C. (a) at 20 °°°°C2: B.C. (b) at 20 °°°°C3: B.C. (c) at 20 °°°°C4: B.C. (d) at 20 °°°°C5: B.C. (e) at 20 °°°°C6: B.C. (d) at -163 °°°°C7: B.C. (e) at -163 °°°°C

Deflection(mm)

Comparison of the deflection (deformation in the z direction) at point B with varied types of boundary conditions.

1

-0.0 -5.0 -10.0 -15.0 -20.0 -25.0

Point B

Deflection(mm)

0

5

10

15

20

25

30

35

Pressure(bar)

6

2

4

5

7

3




B





C

0.0 -10.0 -20.0 -30.0 -40.0

Point C

Deflection(mm)

5

10

15

20

25

30

35

Pressure(bar)

0

1

6

2 4

5

7

3


Comparison of the deflection (deformation in the z direction) at point C with varied types of boundary conditions.

Schematic of defining the critical load-carrying capacity regarding load versus deformation curve.



B.C. TemperatureLoad-carrying capacity

(bar)

(a) 20°C 8.29

(b) 20°C 7.93

(c) 20°C 13.2

(d)20°C 6.19

-163°C 17.2

(e)20°C 6.56

-163°C 17.2

Critical load-carrying capacities of the membrane corrugation with varying the boundary condition and temperature

Deflection

Pressure

Critical point



von Mises stress distribution and deformed shape with boundary condition (e) and temperature of -163°°°°C.

CA

B

E

F

D

1

MN

MX

XY Z

.076602202.686

405.295607.905

810.5141013

12161418

16211824

ELEMENT SOLUTION

STEP=1

SUB =30

TIME=.503563

SEQV (NOAVG)

TOP

DMX =32.263

SMN =.076602

SMX =1824

Measurement points.

Analysis for Impact Pressure

1. Modeling of Material1. Modeling of Material

Material C(1/s) q Reference

Mild steel 40 5 Cowper and Symonds

High-tensile steel 3,200 5 Paik and Chung

Stainless steel

(304L)

100 10 Forrestal &Sagartz

5,000

32,000

5.3

4.8 Langdon &Schleyer

39,033 5.136 Hsu & Jones

12,500 4.5 Current study

Strain rate sensitivity function on the dynamic yeild strength and dynamic fracture strain (Cowper-Symonds equation)

Variation of the critical fracture strain used for LS-DYNA FE simulation as a function of mesh size at a quasi-static loading condition

1/

1.0

q

Yd

Y C

σ εσ

= +

&1

1/

1

q

fcd

fc C

ε εε

− = +

&

0 2 4 6

0

1

2

3 LS-DYNA

Mesh size(mm)

fcd

fc

εεεε

εεεε


2. Sloshing Impact Action2. Sloshing Impact Action

10ms0

pmax

Time

Pressure(bar)

5ms

Idealized profile of impact pressure in terms of pressure pulse versus time history.


3. Modeling Extent and Boundary Conditions(1/2)3. Modeling Extent and Boundary Conditions(1/2)

(a) Model I (b) Model II

LS-DYNA modelling for Models I and II.

Part Model I Model II

Mesh size

Membrane 1 mm 2 mm

Insulation 5.67 mm 10.63 mm

Virtual model 6.07 mm 12 mm

Number of

elements

Membrane 38829 10567

Insulation 38700 7424

Virtual model 840 256

Comparison of Models I and II in terms of mesh size and the number of elements


3. Modeling Extent and Boundary Conditions(2/2)3. Modeling Extent and Boundary Conditions(2/2)

0 0.02 0.04 0.06 0.08 0.1

-10

-5

0

5

10

15Model I

Model II

Deflection(mm)

Time(s)

A B

Point C

Peak pressure

End pressure

0 0.02 0.04 0.06 0.08 0.1

0

50

100

150

200

250Model I

Model Il

Kinetic energy(kJ)

Time(s)

Comparison of the deflection behaviour of Models I and II at monitoring points A, B and C at pmax=17.2bar.

Comparison of the kinetic energy in Models I and II at pmax=17.2bar.

4. Sloshing Impact Pressure Design Curves Against Fracture(1/2)4. Sloshing Impact Pressure Design Curves Against Fracture(1/2)


Deflection behaviour and fracture time of the

whole model with pmax=25bar at monitioringpoints A,B and C.

Deflection behaviour and fracture time of the

whole model with pmax=20bar at monitioringpoints A,B and C.

0 0.02 0.04 0.06 0.08 0.1

-3

-2

-1

0

1

2

Time(s)

Deflection(mm)

pmax = 20bar

First-fracture

Point C

A

B

0 0.02 0.04 0.06 0.08 0.1

-4

-2

0

2

Time(s)

Deflection(mm)

pmax = 25bar

First-fracture

Point C

A

B

4.4. Sloshing Impact Pressure Design Curves Against Fracture(2/2)Sloshing Impact Pressure Design Curves Against Fracture(2/2)


CA

B

E

F

D

Measurement points.

0 0.02 0.04 0.06 0.08

0

20

40

60

80

100

24.28bar

75.46bar

Time until first-fracture(s)

Peak pressure(bar)

Peak pressure versus time-until-first-fracture relation.

Conclusion & RemarksConclusion & Remarks

� The static strength and dynamic strength characteristics of Mark ⅢⅢⅢⅢ type

LNG carrier cargo tanks are analyzed based on a series of finite element

analysis.

� A custom boundary condition has been developed for the strength

performance of Mark ⅢⅢⅢⅢ type LNG carrier cargo tanks under static pressure.

� Based on a several case study about membrane corrugation under the

sloshing impact pressure action is analyzed and it can offer numerical

information for nonlinear structure analysis.

� To develop the guidance of tank sloshing design based on first-fracture

based structural design curve.

MN

MX

XYZ

MN

MX

XYZ

ELASTIC LARGE DEFLECTION BEHAVIOR OF PLATESELASTIC LARGE DEFLECTION BEHAVIOR OF PLATES

WITH PARTIALLY ROTATIONWITH PARTIALLY ROTATION--RESTRAIND EDGESRESTRAIND EDGES

Do Kyun Kim and Jeom Kee Paik

LRET Research Centre of Excellence,

Pusan National University, KOREA

Structures and GeotechnicsLRET Marine & Offshore Research Workshop16~18 February, 2010

OverviewOverview

•• Aim of this studyAim of this study

•• Governing Differential Equations for PlatesGoverning Differential Equations for Plates

•• Analysis of Elastic Large Deflection Behavior of Simply Analysis of Elastic Large Deflection Behavior of Simply

Supported PlatesSupported Plates

•• Effect of Partially RotationEffect of Partially Rotation--Restrained platesRestrained plates

•• Verification of the MethodsVerification of the Methods

•• ConclusionConclusion

Aim of this studyAim of this study

To investigate the elastic large deflection behavior of plates with partially rotation-

restrained edges in association with the torsional rigidity of the support members and

under compression.

Aim of this studyAim of this study

X Y

Z

Plate surrounded by longitudinal stiffeners and transverse frames in a continuous stiffened-plate ship structure.

The edge condition of the plating in a continuous stiffened-plate structure is neither simply supported nor clamped because the torsional rigidity of the support members at the plate edges is neither zero nor infinite.

Transverse frames

Longitudinals

a

a

a

a

b

b

b

b

b

b

b

b

b

b

B

L

Governing Governing

Differential Equations for PlatesDifferential Equations for Plates

4 4 4

4 2 2 4

2 2 22 2 2

2 2 2 2

2

( ) ( ) ( )2 0o o o

w w wD

x x y y

w w w w w wF F F pt

y x x y x y x y t

∂ ∂ ∂+ + ∂ ∂ ∂ ∂

∂ + ∂ + ∂ +∂ ∂ ∂− − + + =

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

The elastic large deflection behavior of plates can be analyzed by solving the following

compatibility and equilibrium equations.

Governing Differential Equations for PlatesGoverning Differential Equations for Plates

where wo, w = initial and added deflections, F=Airy’s stress function, D= plate bending rigidity,

E= elastic modulus, v= Poisson’s ratio, t= plate thickness, p= lateral pressure.

4 4 4

4 2 2 4

22 2 22 2 2 2 2 2

2 2 2 2 2 2

2

2 0o o o

F F F

x x y y

w w ww w w w w wE

x y x y x y x y x y x y

∂ ∂ ∂+ +

∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ − − + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2 2

2 2 2 21x

F Ez w w

y x y

∂ ∂ ∂= − + ∂ − ∂ ∂

σ νν

The membrane stress components inside the plate shall be obtained after the

elastic large deflection analysis, namely

2 2 2

2 2 2 21y

F Ez w w

x y x

∂ ∂ ∂= − + ∂ − ∂ ∂

σ νν

2 2

2(1 )xy

F Ez w

x y x yτ τ

ν∂ ∂

= = − −∂ ∂ + ∂ ∂

,x yσ σwhere, = normal stresses in the x and y directions, respectively, = shear stress, and z = coordinate in the plate thickness direction with z=0 at plate mid-thickness.

τ

Governing Differential Equations for PlatesGoverning Differential Equations for Plates

Analysis of Elastic Large DeflectionAnalysis of Elastic Large Deflection

Behavior of Simply Supported PlatesBehavior of Simply Supported Plates

Configuration of PlatesConfiguration of Plates

Geometry and loading conditionsGeometry and loading conditions

1

MN

MX

X

Y

Z

-.999934

-.777719-.555505

-.33329-.111075

.11114.333355

.55557.777785

1

MAY 23 2009

20:30:49

NODAL SOLUTION

SUB =1

FREQ=371.042

UZ (AVG)

RSYS=0

DMX =1

SMN =-.999934

SMX =1

where, a= plate length, b= plate breadth, = applied longitudinal stress , = applied transverse stress (+: tension, -: compression), p= lateral pressure.

σ xav

σ yav

rcyσ

rtyσ

rcxσ

rtxσ

Comp.

Tens.

Tens.

x

y

a t a ta −2a t

b− 2

bt

bt

bt

0

1

wo/ w

opl

a/2 ab

wopl

Initial deflectionInitial deflection

Welding residual stressWelding residual stress

Initial imperfectionInitial imperfection

rtxσσσσ

*

maxxσσσσ

maxxσσσσ

rcxσσσσ

: Residual stress distribution

: Membrane stress distribution

due to applied loads

accounting for the effect of

residual stress

: Total membrane stress

distribution

tb

tb

tbb 2−−−−

sin sino omn

m x n yw A

a b=

π π

sin sinmn

m x n yw A

a b

π π=

where

m, n = bucking mode half-wave number in the x- and y-direction

Aomn = buckling mode initial deflection amplitude

Amn = unknown amplitude of the added deflection function

22 2 24 4 4 2 2 2 2 2 2

4 2 2 4 2 2 2 2 2 22 2 0o o ow w wF F F w w w w w w

Ex x y y x y x y x y x y x y x y

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Analysis of Elastic Large Deflection Behavior of Simply SupporteAnalysis of Elastic Large Deflection Behavior of Simply Supported Platesd Plates

2 2 44 4 4

4 2 2 4 2 2

( 2 ) 2 22 cos cos

2

mn mn omnm n EA A AF F F m x n y

x x y y a b a b

+∂ ∂ ∂ + + = + ∂ ∂ ∂ ∂

π π π

2 2 2 2

2 2 2 2

( 2 ) 2 2cos cos

32

mn mn omnP

EA A A n a m x m b n yF

m b a n a b

+= +

π π

P HF F F= +

2 2

( ) ( )2 2

H xav rx yav ry

y xF σ σ σ σ= + + +

2 2 2 2 2 2

2 2 2 2

( 2 ) 2 2( ) ( ) cos cos

2 2 32

mn mn omnxav rx yav ry

EA A Ay x n a m x m b n y

m b a n a b

+= + + + + +

π πσ σ σ σ

The particular solution, Fp of the stress function, F is obtained by solving Equation as follows.

The homogeneous solution, FH of the stress function, F, which satisfies the loading condition is

given by treating the welding-induced residual stress as an initial stress parameter, namely


0rtx t

rx rcx t t

rtx t

for y b

for b y b b

for b b y b

σσ σ

σ

≤ <

= ≤ < − − ≤ ≤

0rty t

ry rcy t t

rty t

for x a

for a x a a

for a a x a

σσ σ

σ

≤ <

= ≤ < − − ≤ ≤

2 , 2rcyrcx

t t

rcx rtx rcy rty

b b a aσσ

σ σ σ σ= =

− −

Welding residual stress Welding residual stress

rcyσ

rtyσ

rcxσ

rtxσ

Comp.

Tens.

Tens.

x

y

a t a ta −2a t

b− 2

bt

bt

bt

Where,Where,


Idealized welding residual stress distribution in the plating

4 4 4

4 2 2 40 0

2 2 22 2 2

2 2 2 2

2

( ) ( ) ( )2

sin sin 0

b a

o o o

w w wD

x x y y

w w w w w wF F F pt

y x x y x y x y t

m x n xdxdy

a b

∂ ∂ ∂+ + ∂ ∂ ∂ ∂

∂ + ∂ + ∂ +∂ ∂ ∂ − − + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

× =

∫ ∫

π π

2 2 24 4 4 2 2 2

4 2 2 4 2 2 2 2

( ) ( ) ( )2 2 0o o ow w w w w ww w w F F F p

D tx x y y y x x y x y x y t

∂ + ∂ + ∂ +∂ ∂ ∂ ∂ ∂ ∂+ + − − + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

The application of the Galerkin methodThe application of the Galerkin method


3 2

1 2 3 4 0mn mn mnC A C A C A C+ + + =

2 4 4

1 3 316

E m b n aC

a b

π = +

2 4 4

2 3 3

3

16

omnEA m b n aC

a b

= +

π

22 2 4 4 2 2 2 2 2

3 3 3( ) ( )

8

omnxav rex yav rey

EA m b n a m b n a D m n mb naC

a b a b t ab na mb

= + + + + + + +

π πσ σ σ σ

22( ) sin

2

trex rcx rtx rcx t

n bbb

b n b

πσ σ σ σ

π = + − −

2 2

4 4

16( ) ( )omn xav rex yav rey

m b n a abC A p

a b t

= + + + −

σ σ σ σ

π

22( ) sin

2

trey rcy rty rcy t

m aaa

b m a

πσ σ σ σ

π = + − −

sin sino omn

m x n yw A

a b=

π π

sin sinmn

m x n yw A

a b

π π=


2 2 2 2 2 2 2 2

2 2 2 2 2 2

( / 1/ ) [( 1) / 1/ ]

/ / ( 1) / /

m a b m a b

m a c b m a c b

+ + +≤

+ + +

1=m

When When σσσσσσσσxavxav and and σσσσσσσσyavyav are both nonare both non--zero compressive, c = zero compressive, c = σσσσσσσσyavyav//σσσσσσσσxavxav

When When σσσσσσσσxavxav is tensile or zero, for any value of is tensile or zero, for any value of σσσσσσσσyavyav


( 1)a

m mb≤ +

When When σσσσσσσσxavxav is compressive and is compressive and σσσσσσσσyavyav is tensile or zerois tensile or zero

Effect of PartiallyEffect of Partially

RotiationRotiation--Restrained platesRestrained plates

LL L

GJCbD

ζ = SS S

GJC

aDζ =

1.0LL

PL

JC

J= ≤ 1.0S

S

PS

JC

J= ≤

3

3PL

btJ =

3

3PS

atJ =

Effect of Partially RotationEffect of Partially Rotation--Restrained PlatesRestrained Plates

hwx

twx

t

b

hwxtwx

t

b

tfx

bfx

hwx twx

t

b

tfx

bfx

hwy

twy

t

a

hwytwy

t

a

tfy

bfy

hwy twy

t

a

tfy

bfy

where , = rotational restraint parameters for the longitudinal or transverse support

members, = torsion constant of the longitudinal stiffener,

= torsion constant of the transverse frame, .

,L S

ζ ζζ ζζ ζζ ζ3 3( ) / 3L wx wx fx fxJ h t b t= +

3 3( ) / 3S wy wy fy fyJ h t b t= + / [2(1 )]G E ν= +

Nomenclature: Geometrical dimensions for the support members

2 2

0 [1 ( / ) ]yk b a= +

22 2 4 4 2 2 2 2 2

3 3 3( ) ( )

8

omnxav rex yav rey

EA m b n a m b n a D m n mb naC

a b a b t ab na mb

= + + + + + + +

π πσ σ σ σ

2

0 0 0[ / ( ) / ]xk a m b m b a= +

For simply supported at all (four) edgesFor simply supported at all (four) edges

For rotationally restrained at all (four) edgesFor rotationally restrained at all (four) edges

22 2 4 4 2 2 2 2 2

3 3 3( ) ( )

8

omnxav rex yav rey

EA m b n a m b n a m n mb naC

a b a b t ab na mb

D = + + + + + + +

π πσ σ σ σ

0 0

x y

x y

k k

k k×

kkxx = k= kx1x1 + k+ kx2x2 –– kkx0x0

kkyy = k= ky1y1 + k+ ky2y2 –– kky0y0

= buckling coefficient of = buckling coefficient of σσσσσσσσxx at all edges S.S.at all edges S.S.

= buckling coefficient of = buckling coefficient of σσσσσσσσyy at all edges S.S.at all edges S.S.

= buckling coefficient of = buckling coefficient of σσσσσσσσxx at all edges restrainedat all edges restrained

= buckling coefficient of = buckling coefficient of σσσσσσσσyy at all edges restrainedat all edges restrained

Effect of Partially RotationEffect of Partially Rotation--Restrained PlatesRestrained Plates

Verification of the MethodsVerification of the Methods

XY Z

- Uniform displacementin x-direction

- Rotation y=0

- Rotation z=0x

zy

Uz=0Uz=0

Ux=0Ux=0

Uy=0Uy=0

Monitoringpoint

- Uniform displacement

in y-direction- Rotation x=0- Rotation z=0


Boundary condition for FEABoundary condition for FEA

XY Z

XY Z

XYZ

a/b=1, Mesh size = 71mm



Mesh modelingMesh modeling


a/b Case Longitudinal stiffener size ζζζζL Transverse stiffener size ζζζζS

1

1 250××××12+150××××15 0.1642

650××××12+150××××15 0.28522 400××××12+150××××15 0.2096

3 500××××12+150××××15 0.2398

3

4 250××××12+150××××15 0.1642

1200××××12+150××××15 0.15055 400××××12+150××××15 0.2096

6 500××××12+150××××15 0.2398

5

7 250××××12+150××××15 0.1642

1200××××12+150××××15 0.09038 400××××12+150××××15 0.2096

9 500××××12+150××××15 0.2398


Various dimensions of the longitudinal and transverse support members of a plate under uniaxial compression

LL L

GJCbD

ζ = SS S

GJC

aDζ =

where , = rotational restraint parameters for the longitudinal or transverse support

members

,L S

ζ ζζ ζζ ζζ ζ


Longitudinal compression Longitudinal compression

0.0 0.4 0.8 1.2 1.6

0.0

0.4

0.8

1.2

1.6

2.0

FEM

Theory

Clamped

Partiallyrotation-restrained

Simply supported

a/b = 3

ζζζζL=0.2398 with 500××××12+150××××15 (mm)ζζζζS=0.1505 with 1200××××12+150××××15 (mm)

σσ σσxav/ σσ σσxE

(w+wo)/t

Comparison of the results of the developed method and the ANSYS FEA for a plate under longitudinalcompression, together with the behavior of simply supported, clamped and restrained plates.

FEA (ANSYS)

Developed method (Theory)


Transversal compression Transversal compression

( )+o

w w

t

a/b = 3

ζζζζL=0.0524 with 250××××12+150××××15 (mm)ζζζζS=0.1018 with 1200××××12+150××××15 (mm)

Clamped Simply supported

0.0 0.4 0.8 1.2 1.6

0.0

0.4

0.8

1.2

1.6

2.0

yav

yE

σσσσ

σσσσ

Partiallyrotation-restrained

FEM

Theory

w/t

σσ σσyav/ σσ σσ

yE

(wo+w)/t(w+wo)/t

Comparison of the results of the developed method and the ANSYS FEA for a plate under transversalcompression, together with the behavior of simply supported, clamped and restrained plates.

FEA (ANSYS)

Developed method (Theory)

0.1642

0.1505

ConclusionConclusion

� Developed an analytical method for predicting the elastic large

deflection behavior of plates with partially rotation-restrained edges

� Modifying the analysis procedure for simply supported plates that

uses the governing differential compatibility and equilibrium equations

� The validity of the developed method was confirmed by comparison

with nonlinear FEA with various configuration of plates and loading

ratio

� The insights and developments obtained in this study can be very

efficiently used for the robust design of ship structures

ConclusionConclusion

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