Lecture 7 Elastic Buckling of Stiffened Panelsocw.snu.ac.kr/sites/default/files/NOTE/Lecture 07...

OPen INteractive Structural Lab

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


Comparison between Buckling and Ultimate Strength of plating

Plate capacity interactions between biaxial compression FEA, Buckling, Ultimate strength, a / b = 3, t = 13mm

Reference


Comparison between Buckling and Ultimate Strength of plating

Plate capacity interactions between biaxial compression FEA, Buckling, Ultimate strength, a / b = 3, t = 21mm

Reference


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

OPen INteractive Structural Lab5

Schematics of various types of stiffed panel buckling

Local buckling

Overall buckling


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


2

3

12(1 )x xx

EI v I

Db bt


Plate-Stiffener (BEAM) Combination model

Orthotropic plate model

PLATE

STIFFENER

PLATE

STIFFENER

Idealization of continuous stiffened panel

Plate-Stiffener Separation model


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



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



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


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



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


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.


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



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


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



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


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



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


2

2( )

( / )e cr

e eq

E

L

2

2( )

( / )

et xa cr

x e eq

b A E

bt A L

( 4 )1.9 12 3

w we w f f e w f e

A Adb A A A b t A A b t

at

)()( xeexa AtbAbt

1.9e

e

b t E

b b

2

2( )

( / )e cr

e eq

E

L

2

2( )

( / )

et xa cr

x e eq

b A E

bt A L



(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


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



17


A

tbhtbbhte

teAhtbbhtb

I

thbttbA

2

)(

)(3

)(

332

00

2

11

2

01

2

01

3

11

33

00

11200

Homework 6-2 illustrates the previous graph using the following formula



Possible modes of elastic buckling of stiffened panels

18


be=b

For small stiffener

(σa)cr for be > (σa)cr for b

For large stiffener

(σa)cr for be < (σa)cr for b


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


Stiffener tripping


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.


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


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


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”

3

3

)2

)(2

()2

(

))(2

(

)(

3)(

12)(

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Derivation of Work Done during Buckling

6 X 6 Eigenvalue Problem


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


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


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


Types of stiffener buckling and method of analysis

29


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)


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



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


Effect of web bending



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


0)(2

2

4

42

K

dx

dIGJ

dx

ddEI SPaSZ



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


4 4 2 22

,4 2( ) ( ) 0SZ a sp a

m mEI d GJ I K m

a a

b

DMK R 4



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


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



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


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



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


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



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


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



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


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


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


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


Homework 6-1

Orthogonal array for 8 parameters with 3 levels

41


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

Lecture 7 Elastic Buckling of Stiffened Panelsocw.snu.ac.kr/sites/default/files/NOTE/Lecture 07...

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Transcript of Lecture 7 Elastic Buckling of Stiffened Panelsocw.snu.ac.kr/sites/default/files/NOTE/Lecture 07...