Lecture 5 Buckling and Ultimate Strength of Plates 05... · 2018. 4. 18. · Elastic and Inelastic...

OPen INteractive Structural Lab

Topics in Ship Structural Design(Hull Buckling and Ultimate Strength)

Lecture 5 Buckling and Ultimate Strength of Plates

Reference : Ship Structural Design Ch.12

NAOE

Jang, Beom Seon


Facts about MSC Napoli

One of the world’s largest container ships when built (1991)

Built to BV Class and changed to DNV 2002

Last renewal survey carried out in 2004 in Singapore

Built 1991

Length over all 275.66 m

Breadth 37.13 m

Draught 13.50 m

Gross tonnage 53,409 GRT

Capacity 4419 TEU

Slide 2

Failure of MSC Napoli Container ship – DNV Report by Olav Nortun


Accident January 2007 – MSC Napoli

Ship left Antwerp 17 January 2007

heading for Sines in Portugal

18 January - water ingress in engine

room reported

All 26 crew members safely rescued

Ship beached in Lyme Bay near

Branscombe, UK on 19 January

2007

Slide 3



Accident January 2007 – MSC Napoli

The vessels was split into two in

July 2007

Forward part was towed to

Belfast for recycling



B200 Plate panel in uni-axial compression

Unstiffened Plate (Plating between stiffeners)

Elastic and Inelastic Buckling

Post-Buckling and Ultimate strength

DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13

Classification Rule

Johnson-Ostenfeld plasticity correction formula

)N/mm(1000

9.0 2

2

s

tkEel

For plating with longitudinal stiffeners (in direction of compression stress): k=4

Ideal elastic buckling stress


Buckling of a Wide Column

The plate is acting more as a wide column than as a plate. The product EI is

replaced by the plate flexural rigidity D.

The thickness / length ratio plays the same role as the slenderness ratio for

columns.

The width b plays no part, no support along the unloaded edge → It is

inefficient to use

6

12.1 Elastic Plates Subjected to Uniaxial Compression

2

2cr

DbP

a

22 2

2 212(1 )cr

D E t

a t v a

Buckling of wide column

Simple support along the loaded edges

With no support along the unloaded edges

OPen INteractive Structural LabOPen INteractive Structural Lab

Large-Deflection Plate Theory by von Karman

Small-Deflection Plate Theory

Large-Deflection Plate Theory

9.2 Combined Bending and Membrane Stresses-Elastic Range

2

22

2

2

4

4

22

4

4

4

21

2y

wN

yx

wN

x

wNp

Dy

w

yx

w

x

wyxyx

2

22

2

24 2

1

y

wN

yx

wN

x

wNp

Dw yxyx

D

p

y

w

yx

w

x

w

4

4

22

4

4

4

2


Plate is assumed to be free to move inward under the action of the in-plane

compression. → The strain energy of deformation is due to bending only

Buckling of a Simply Supported Plate

From large-deflection plate theory

Since the edges are simply supported, the

deflected shape can be expressed in the form:

which satisfies both the boundary conditions and

the general biharmonic equation.

8


24

2

at ww

D x

sin sinmn mn

m n m n

m x n yw w C

a b

24 2 2

2

2 28mn

m n

ab m nU D C

a b

22 2

2 20 0=

2

a bD w wU

x y

22 2 2

2 22(1 )

w w wv dxdy

x y x y

Buckled shape of long plate

0, xyyax NNptN


Strain Energy Density for plane stress (σz=0)

Reference

Load applied

on dydz area

Elongation in

x-direction

dy

dxdz

dxεx

dyεy

dzεz

dxdydz

dydxdzdxdydzdu

yyxx

yyxx

)(2

1

))((2

1))((

2

11

dxdydz

dydxdzV

du

xyx

xyx

2

1

))((2

1

22

thickness t

dx

dy

dz

dxdydzdu xyxyyxx )(2

1


Strain Energy for plane stress (σz=0)

In Chapter 9 (Lecture 03), Plate bending (Derivation of Plate

Bending Equation), the followings are derived

Reference

2

2

2

2

2)(

1 x

wv

y

wz

v

Ey

yx

wGz

2

2 = +u

x y

x

wzu

y

wzv

yx

wz

2

)2(

dzdxdyduUa b t

txyxyyxx

0 0

2/

2/)(

2

1

2

2

2

2

2)(

1 y

wv

x

wz

v

Ex

2

2

)(y

wzy

2

2

)(x

wzx


Strain Energy Density for plane stress (σz=0)

Reference

dydxyx

wv

y

w

x

wv

y

w

x

wD

dydxyx

wt

vv

vE

y

w

x

wv

y

w

x

w

v

Et

dzdydxyx

wGzz

y

w

x

wv

y

w

v

E

zx

w

y

wv

x

w

v

EduU

a b

a b

a b t

t

22

0 0 2

2

2

22

2

22

2

2

23

0 0 2

2

2

22

2

22

2

2

2

3

222

2

2

2

2

2

2

2

0 0

2/

2/

2

2

2

2

2

2

2

2

)1(222

)1)(1(2

)1(

12

42

)1(12

4)()1(

)()1(2

1

)1(2 v

EG

dydxyx

w

y

w

x

wv

y

w

x

wDU

a b2

2

2

2

2

2

0 0

2

2

2

2

2

)1(22


Work done for plane stress

Reference

dxdyx

wtdxdyNW

a b

a

a

x

b

x

0 0

2

0 0 2

1

dxx

wdx

x

w bb

x

0

2

0

2

2

111

For unit-width strip in Section 9.2 dx

w

dxx

ww

dxx

w

2

1

21)1(

aa for small a



Likewise, the work done by the in-plane compressive stress is

Because of W=U, and hence,

The minimum value of σa is given by taking only one term, say Cmn,

where m and n indicate the number of half-waves in each direction in the

buckled shape.

When n=1, σa gives the smallest value. Hence the plate will buckle into only

one half-wave transversely.

13


42 2

8

amn

m n

b tW C m

a

2 22 2 2

2 2

2 2

mn

m n

a

mn

m n

m na D C

a b

t m C

2

0 02

a bat w

W dxdyx

22 2 2 2

2 2 2( )a cr

a D m n

tm a b

222

2

1( )a cr

D am

a t m b

)/,...,/,/max(...

...)/,...,/,/min( 2211

21

212211 nn

n

nnn dcdcdc

ddd

cccdcdcdc



A buckling coefficient k is generally used. It depends on the type of

boundary support.

For design applications, in which the plate thickness is to be determined, it

is usually written like this:


2mb a

ka mb

2

( )a cr

tKE

b

2

212(1 )

kK

v

2

2( )a cr

Dk

b t

Q: Which critical stress will be higher?, which stiffener

arrangement is better against in-plane compression?

3

212(1- )

EtD



For long simply supported plates it is

usually assumed that k=4.

Assuming v=0.3

15


2

2( ) 4a cr

D

b t

2

( ) 3.62a cr

tE

b

Classification Rule

)N/mm(1000

9.0 2

2

s

tkEel

k=4, s=b (m)

Homework #1 Plot this curve

2mb a

ka mb

2

2( )a cr

Dk

b t

Buckled shape of long plate



16


a/b=1, m=1 a/b=2, m=2

a/b=3, m=3



For a wide plate, in which the aspect ratio(a/b) is less than 1.0, m=1

For a general "wide plate“, in terms of a

because a<b

For design purposes it may be written as:

For v=0.30

17


2

2( )a cr

Dk

a t

2a

k kb

2

( )a cr

tKE

a

222

21

12(1 )

aK

v b

222

2( ) 1a cr

D a

a t b

22

0.905 1a

Kb

a

b

2

b

a

a

bk

)1(12 2

32

v

EtD



Longitudinal stiffeners: a>>b(=s),

Transverse stiffeners: a<<b, a=s, b=B,

Longitudinally stiffened plating have the great advantage over

transversely stiffened plating in ship structures, and the former is used

wherever possible.

18


2

2

4( )a cr

D

s t

222

2( ) 1a cr

D s

s t B

m=1

k=4

<<1


Reproducing the event in a computer model

Direct wave load calculations

Linear strength analysis

Non-linear strength analysis

Load and strength comparisons

Simulation of crack propagation

Slide 19



Most severe wave for engine room area


Wave crest around midship

Vertical ”g” force

Aft ship out of water

Hull forces: Shear force and moment


Structural arrangement in Engine room zone

21

Failure of MSC Napoli Container ship


Not sufficient buckling capacity

The buckling capacity might not have been checked sufficiently

when the ship was built

Potentially insufficient buckling strength in the engine room

bulkhead



Four stages of progressive collapse Outer shell



Four stages of progressive collapse Inner structure



Four stages of progressive collapse Inner structure



Alternative correcting actions

The likelihood of reoccurrence is very low:

Damage statistics are very good

Little likelihood of such a harsh sea state

The ship’s strength was below the strength of similar ships

Maybe not all ships checked in this area

However – the consequences are major

Increase buckling strength

Minor modifications – small amount of steel to be added

Aft of the engine room bulkhead

Can be done while in service



Solutions for Some Principal Cases

When unloaded edge (A) is replaced

by simply supported, the critical

buckling stress drops more than when

loaded edge (B) is by simply supported.

27

12.2 Other Boundary Conditions

Buckling coefficient k in the design formula

for flat plates in uniaxial compress

2

( )a cr

tKE

b

A, B fixed

B simply supported

A fixed

B fixed

A simply supported

A & B simply

supported

Q: Which edge is more effective to in-

plane buckling? Loaded edge or unloaded

edge?

A : Unloaded edge

B: Loaded edge



28


Buckling stress coefficient k for flat plates

in uniaxial compression

A1, A2 clamped

A1 pinned

A2 clamped

A1, A2 pinned

A1 free

A2 clamped

A1 free

A2 pinned

A1 A2 free

Loaded edges

clamped

Loaded edges

Simply supported

2

2( )a cr

Dk

b t

In general, b≈ 800mm, a≈3300mm, a/b≈3~4

Unloaded edge : clampedLoaded edge : clamped

Unloaded edge : clampedLoaded edge : simply supported

Unloaded edge : pinned and clampedLoaded edge : clamped

Unloaded edge : pinned and clampedLoaded edge : simply supported

Unloaded edge : simply supportedLoaded edge : clamped

Unloaded edge : simply supportedLoaded edge : simply supported

Unloaded edge : free & clampedLoaded edge : clamped

Unloaded edge : free & clampedLoaded edge : simply supported

Unloaded edge : free & pinnedLoaded edge : clamped

Unloaded edge : free & pinnedLoaded edge : simply supported

Unloaded edge : freeLoaded edge : simply supported



29


Buckling stress coefficient k for flat plates

in uniaxial compression

Buckling coefficient k in the design formula

for flat plates in uniaxial compress

A1, A2 clamped

A1 pinned

A2 clamped

A1, A2 pinned

A1 free

A2 clamped

A1 free

A2 pinned

A1 A2 free

Loaded edges

clamped

Loaded edges

Simply supported

2

2( )a cr

Dk

b t

2

( )a cr

tKE

b

A, B fixed

B simply supported

A fixed

B fixed

A simply supported

A & B simply

supported


Clamped Edges

For in-plane loads, as in the case of lateral loads, it is not possible to obtain

finite expressions for the solution of clamped plates.

Numerical solutions by Faxen, Maulbetsch, and Levy.

30


Buckling coefficient k for clamped plates under uniaxial compression


Unloaded Edges Rotationally Restrained

Lundquist and Stowell have

investigated the case in which

the support along the unloaded

edges is intermediate between

simply supported and clamped.

The degree of rotational restraint

is specified in terms of a

coefficient of restraint, defined as

Cy : rotational stiffness of the

supporting structure along the

unloaded edge

31


y

bC

D

Loaded edges

clamped

Loaded edges

Simply

supported

Buckling coefficient k for plates with loaded edges simply supported and longitudinal edges

rotationally restrained


Loaded Edges Rotationally Restrained

The important boundary conditions are those along the longer edges of the

plate. Thus, for short wide plates the edge restraint along the loaded edges

becomes significant.

Similar to end conditions in a column, by using an effective length ae:

for clamped ends ae = 1/2a

for one end simply supported and

the other clamped ae = 0.707a

Using a coefficient of restraint ζ :

Cx : rotational stiffness of the supporting structure along the unloaded edge

The solution to this case is obtained from

in which K1 and K2 are related to the buckling coefficient k.

32


222

2( ) 1 e

a cr

e

aD

a t b

x

aC

D

2 21 21 2 1 2tan tan ( ) 0

2 2

k kK K K K

1,2 ( 4)2

K k k



33


a<b

Buckling coefficient 𝑘 for wide plates in compression elastically restrained on the loaded edges

kbak 2)/(



The corresponding coefficient in the

"design" version of the wide plate

formula

In ship structures the rotational restraint

is usually provided by flange-and-web

type of transverse stiffeners.

In this case ζ is given approximately by

d : depth of the web

I : second moment of area of the stiffener about

the midthickness of the web

J : Saint-Venant’s torsion constant for the

stiffener

34


2

( )a cr

tKE

a

2 2

3 2 2

27

2.6

a Id J

t b b

Buckling coefficient k for plates with loaded edges simply supported and

longitudinal edges rotationally restrained


All Edges Simply Supported

a is parallel to σax and b to σay. Aspect ratio α=a/b.

Applying the energy method yields the following expression for the critical

combination:

U=W

If we denote the square plate critical stress and nondimensional form

35

12.3 Biaxial Compression

22 22

2 2

2=ax ay

cr

m D mn n

b t

2

,1( ) 3.62ax cr

tE

b

m m

ayaxmn

m m

ayaxmn n

mC

b

atn

b

am

a

bC

tW

2

2

22

4222

4

88

2

2

2

22

4

42

2

2

2

22

4

88

m m

mn

m m

mn nm

CDb

ab

b

n

a

mCD

abU

22 2

2 2

,1 ,1

1=

( ) ( ) 4

ayax

ax cr ax cr cr

m mn n



36


Buckling stresses of biaxially loaded simply supported plates

α=3,m=3a

b

σay

σay

ab α=1,m=n=1

ab

σay

α=0.5,m=1,n=2

α=3,m=1a

b

σay

σxα=3,m=3a

b

ab

α=1,m=1

σx

ab σx

α=0.5,m=1

2

,1( ) 3.62ax cr

tE

b

When σax ≈σay, α=1

2

2

62.3

62.3)5.05.0()(

b

tE

b

tEcrayax

m=1 m=2m=3



37


a/b = 3 a/b = 5

Plate under biaxial load


B400 Plate panel in bi-axial compression

For plate panels subject to bi-axial compression the interaction between the

longitudinal and transverse buckling strength ratios is given by

DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13

1

n

cyy

ay

cycxyx

ayax

cxx

ax K

Homework #2 Plot DNV bi-axial interaction curve and compare with theprevious interaction curve (Fig. 12.8)


All Edges Clamped

For plates subjected to approximately equal compressive stresses

(σax ≈σay)the interaction formula is

When α=1,

For square plates(α=1), critical combinations are given for particular

values of σax / σay, including cases in which σay is tensile.

When σax = σay

39


2

2 2

2

3( ) 1.20 3 2ax ay cr

tE

b

2

6.9)(

b

tEcrayax

22

15.10905.0)61.561.5()(

b

tE

b

tEcrayax


Pure Shear

In ship structures the plating is commonly subjected to large shear loads.

The shearing load can cause buckling since it gives rise to in-plane

compressive stress.

40

12.4 Other Types of In-plane Loads

For the case of pure shear, in-plane

compressive stress is equal to the shear stress

and acts at 45° to the shear axis.

In shear buckling, the coefficients are denoted

as ks and Ks.

For simply supported plates

For clamped plates

2

2cr s

Dk

b t

2

cr s

tK E

b

2=5.35+4( / )sk b a

2=8.98+5.6( / )sk b a

24 2 t ww

D x y

Buckling of an infinitely long, simply supported plate

tNNpN xyyx ,0


For simply supported plates

Pure Shear


B300 Plate panel in shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)

2

2

2

434.5),N/mm(1000

9.0

l

sk

s

tEk ttel

2=5.35+4( / )sk b a2

2cr s

Dk

b t

22

2

2

90.0)1(12

b

tEk

b

tEk

vsscr


Pure Shear

ks and Ks are given for various types of boundary conditions. Because of the

symmetry of the pure shear loading , the choice of a and b is independent of

the load.


Buckling coefficient of flat plates in shear

Buckling coefficient of flat plates in shear (Design formula)


Biaxial Compression and Shear

For long plates ks is given approximately by:

– All edges simply supported:

– All edges clamped:

where

43


1/2 1/21/2 1/2

= 2 1 2 2 1 6ay ayax ax

s

e e e e

k

1/2 1/21/2 1/2

4 8 4= 4 4 8

33 3

ay ayax axs

e e e e

k

2

2e

D

b t


In-plane Bending

σb denotes the largest or edge value of the applied stress.

44


Some approximate formulas to calculate the values of kb

– simply supported edges:

for

for

– clamped edges:

for

– one unloaded edge clamped; the others simply supported

for

– unloaded edges clamped; loaded edges simply supported

for

2

2( )b cr b

Dk

b t

/ 2 / 3a b

/ 2 / 3a b

/ 1a b

/ 1/ 2a b

/ 0.4a b

2 215.87 1.87( / ) 8.6( / )bk b a a b

23.9bk

41.8bk

25bk

40bk


In-plane Bending

The figure illustrates the case in which the bending is unsymmetric.

For simply supported edges the value of kb is given approximately by

( simply supported edges only)

45


25 +4bk 3

2

b

a


Combined In-plane Loads: Interaction Formulas

Uniaxial compression and in-plane bending

(σa)cr : critical values of axial loading

(σb)cr : critical values of and in-plane bend

Uniaxial load(compressive or tensile) and shear

For convenience we adopt the symbol R to denote a critical load ratio.

In the present case the strength ratios are

The interaction formula is

In-plane bending and shear

46


1.75

1( ) ( )

a b

a cr b cr

( )

ac

a cr

R

s

cr

R

2 1 1c sR R

( )

bb

b cr

R

21 0.61 1

1.6c sR R

2 2 1 ( >1/2)b sR R



Biaxial compression, in-plane bending, and shear

The two compression strength ratios are

By performing a series of four-variable curve-fitting solutions,

47


( )

axx

ax cr

R

( )

ay

y

ay cr

R

2

4

2

0.625(1 0.6 / )1 1

1(1 0.625 ) 1

(1 )

y s

xbx

x

R R

RRR

R

B500 Plate panel in bi-axial compression and shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)

1

n

cyy

ay

cycxyx

ayax

cxx

ax

qqK

q

2

1

a

aq



48


Interaction curves for biaxial compression, in-plane bending, and shear drawn for α=2



49


2

4

2

0.625(1 0.6 / )1 1

1(1 0.625 ) 1

(1 )

y s

xbx

x

R R

RRR

R

B500 Plate panel in bi-axial compression and shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)

1

n

cyy

ay

cycxyx

ayax

cxx

ax

qqK

q

2

1

a

aq

Homework #3 Plot DNV bi-axial interaction curve likethe right figure and compare with the following curvefor Rb=0


Plates Without Residual Stress

Uniaxially loaded, simply supported square plate, with sides free to

pull in. some typical initial distortion in the form of a half wave in

each direction.

Plate slenderness

The relationship between the applied load (σa) and the axial

shortening

12.6 Ultimate Strength of Plates

50

Plate strength without welding (σr=0)

Et

b Y



Slender plate (β>2.4)

Buckling stress is well below yield stress and below the curve of collapse

stress.

After buckling (σa) a greater proportion of the load is taken by the region of

plating near the sides → Non-uniform compressive stress distribution

Deflected shape of the buckled portion → overall stiffness of the plate

(dσa/dεa) is reduced.

The center region becomes more pronounced and the maximum stress at

the sides increases. When the maximum stress = yield stress → collapse.


51Plate strength without welding (σr=0) Post-buckling stress distribution

Ultimate strength

Large margin between buckling and collapse



Plates of intermediate slenderness (1<β<2.4)

Buckling stress ≈ yield stress

For a rigorous analysis, elasto-plastic large deflection theory to be used.

As applied stress increases → magnification of the initial distortion → loss

of stiffness → some local yield → stress redistribution → yielding of the

sides → sudden collapse.

Pitched roof : allows large axial shortening with minimum strain

energy.


Typical post buckling behavior



Sturdy plates (1>β)

The initial distortion is smaller and the magnification is less because the

elastic buckling stress is very large.

Plates can carry a load equal to the full “squash load” σa,u= σY.

After the peak load, the load carrying capacity remains

approximately constant up to very large strains.


Plate strength without welding (σr=0)


Plates With Residual Stress

Departure from linearity occur at the stress which is less σa

less than for a stress-free plate.

Sturdy plate (1>β) : no load shedding, but large

reduction in stiffness → regarded collapse.

Intermediately slender and slender plate (1<β) : the

loss of ultimate strength ≈ σr


2=

2

r

Yb

t

Plate strength with welding (σr=0.1σY)

Middle part

σa+ σr = σY

Edge part

σa- σY = σY

Idealized residual stress

distribution


Effects of Other Parameters

Restraint at Sides

Clamping the sides of a plate increase the elastic buckling stress by

75%, however, the increase in buckling stress even in slender plate ≈

10% at most.

Stiffeners surrounding the panel is not clamped edge → this restraint

can be ignored.

Initial Deformation

The effect of initial deformation removes sharp knuckle in curve of σa

and εa. The increasing lateral deflection causes a progressive reduction

in the in-plane stiffness of the plate.

However, the ultimate strength is slightly decreased.

Shear stress

In –plane shear stress tends to lower the resistance to longitudinal

compression.

Reduced yield stress rτσY


55

2

31

Y

r

Yxyyyxxeq 222 3


Ultimate Strength of Uniaxial Loaded Plates

Plating of uniaxially loaded, longitudinally stiffened,

initial deformation (δp < 0.2βt), residual stress (σr ≈ 0.1σY)

side constrained to remain straight but free to pull in

For sturdy plate, first loss of stiffness is taken as collapse.

For plates of greater slenderness : loss of stiffness is gradual.

Secant modulus ratio


56

Design curves of ultimate strength and secant modulus

22

2 75.21,

4.10225.0

E

ET s


Ultimate Strength of Uniaxial Loaded Plates

Faulkner’s formula for the ultimate strength of unwedded plates : good

agreement with extensive experimental data.

The effect of residual stress → strength reduction factor Rr


57Curves for ultimate strength of plates

2

, 12

Y

ua

Restrained : the sides remain straight and do not pull in.

Unrestrained : both types of transverse deformations can occur.

Stress relived (σr=0), average welding (σr≤0.1σr), heavily welded (σr≤0.33σr)

E

ER

Y

tsrr

1

00.1for and5.2for

)5.21(12

tsts

ts

EEE

E

E

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