*Buckling of Stiffened Panels (Topic 9)

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* Buckling of Stiffened Panels ( Topic 9)

(Tripping )

Advanced Local Structural Design & Analysis of Marine Structures

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory under Pressure

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short clamped plates (Topic 7)

– Low aspect ratio plates, strength & permanent set (Topic 7A)

[Part III] Buckling of Plates & Stiffened Panels

– Failure modes (Topic 8)

– Tripping (Topic 9) + Post-buckling strength of plate (Topic 9A) – Post-buckling behaviour (Topic 10)

[Theory of Plates and Grillages]

Adv. Local Structural Design & Analysis of Marine Structures (Overview)

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Reminder

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Objectives

The aim of this lecture is:

• To equip you with the knowledge & understanding of tripping of stiffeners.

At the end of this lecture, we should be able to:

• Derivethe formula for the critical stress of tripping a stiffener.

• Calculate the critical stress of tripping a stiffener of rolled or built-up sections.

• Be aware of warping effects on tripping.

#Tripping #Stiffener #StiffenedPanel

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• When a grillage experiences in-plane compression, tripping of stiffeners will occur if

– stiffeners have weak torsional strength

– long slender web (depth (d_w)/thickness (t_w) > 15)

Introduction

• To avoid tripping

– d_w/t_w< 12 is required.

– tripping brackets are fitted

However, it is necessary to estimate the

strength of stiffener against tripping.

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* Tripping of stiffeners

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The word ‘tripping’ is used because by virtual of the plate to which the stiffener is attached. The toe cannot move sideways and therefore any lateral buckling of the stiffener entails some torsion (about the centroid of stiffener) and some lateral bending of the stiffener.

, u z, w

Tripping of stiffeners (1/6)

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There are 3 major components of strain energy in the buckled form:

1. Sideways (lateral) bending strain energy

x dx EI v

U_B _z ^L

2

0 2

2 2

1



^_^ _^ ^_^

where, =

vis lateral deflection due to stiffener buckling.

I_z issecond moment area of the stiffener cross-section about z-axis.

L is the span of the stiffener.

( )² ²₂ ² ¹₂ ₀ ²₂ ²

2 2

L

B z

L L d v v

U M EI EI dx

EI EI dx x

    

= =   =    

2 2

1 1

1 L R

M EI d v dx

M E M d v d w

I R y R EI R dx or dx

L M ML

L L

R R EI EI







=

= = → = → =

   

= =    =   =

Simple beam theory

Tripping of stiffeners (2/6)

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2. Torsional strain energy where

f is angular displacement of the stiffener.

J is torsional constant.

x dx GJ

U_T ^L

2

2 0

1



^_^ ^_ ^_^

=

f

3. Strain energy due to stiffness k (= M(x)/f)

where

k is the stiffness of rotational constraint provided by attached plating

dx k

U

_S

= 

0^L 2 2

1

f

1. Sideways (lateral) bending strain energy

x dx EI v

U_B _z ^L

2

0 2

2 2

1



^_^ _^ ^_^

=

Tripping of stiffeners (3/6)

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• Therearealsosomestrainenergycomponentsduetolongitudinalandtransverse warpingofthestiffenercross-section butformostpurposetheycanbeignored.

• The total strain energy is U = U_B +U_T +U_S

dx k

x dx GJ

x dx EI v

U ⁼ _z



^L^_^ _^ ^_^ ⁺



^L^_^ ^_ ^_^ ⁺



^L

 0

2 2

1 2

2 0 1 2

0 2

2 2

1 f f

[Tip] How to remember?

The effect of

Bending + Torsion + Stiffness

x dx EI v

U_B _z ^L

2

0 2

2 2

1



^_^ _^ ^_^

= dx

GJ x

U_T ^L

2

2 0

1



^_^^_ ^_^

= f

dx k

U_S ⁼



0^L 2 2

1 f

Source: BTS ARMY

Tripping of stiffeners (4/6)

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Work done in compression by critical end stress σ_cr is

where integration is throughout the cross-section A and u is the approach of one end of a filament of area dA to the other end, i.e. the shortening of the stiffener over the span.

Then _dx

u ^L x

2

0 2 2

1



^_^ ^_ ^_^

=  f

x dx u ^L v

2 2 0

1



^_^_^ ^_^

= v =  f

Since and

Hence,

 

_^









 



 







=  dx dA

W _cr ^L x

2 0

2 2

1  f

 dx

dA x

W _cr ^L

2

0 2

2

1

 

^_^ ^_ ^_^

=

   f

x dx I

W _cr _o ^L

2

2 0

1



^_^ ^_ ^_^

=

  f

where I_ois the polar moment of stiffener cross-section.

W =  =F d

 

cr u dA

Tripping of stiffeners (5/6)

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For stability^, W U

dx k

x dx GJ

x dx EI v

x dx

I_o ^L _z ^L ^L ^L

cr



^_^ ^_ ^_^ ⁼



^_^ _^ ^_^ ⁺



^_^ ^_ ^_^ ⁺



 0

2 2

1 2

2 0 1 2

0 2

2 2

1 2

2 0

1 f f f

As υ = ρf, we have

2 2 2 2 2

2 2 2 2

2 2



 







 



 







= 



 







z x x

x

v  f f

This critical buckling stress due to tripping of stiffener can be determined when the buckling shape which satisfies the necessary boundary conditions of the stiffener is found.





 







 +



 





 + 



 







=



L o

L L

L z

cr

x dx I

dx k

x dx GJ

x dx z

EI

0

2

0 2 0

2 0

2 2 2 2

f

f f f



Tripping of stiffeners (6/6)

( )

U W Total Potential Energy

 = +

W = U

^or

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* Simply supported stiffener ends

w/o rotation about x-axis but free to warp

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[Reminder of the previous learning] Column collapse (1/3)

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Simply supported stiffener ends without rotation about x-axis but free to warp

Assume buckling shape is

L x a_m m f = sin

which satisfies boundary conditions: _f ₌ ₀ and ²f /x² = 0 at x = 0 and x = L.

Then,

L x m L

a m x ^m



f = cos



 and

L x m L

a m

x ^m



f ₂ sin

2 2 2

2 = −



since ₀^L^sin² ^{m x}_L^ ^dx ⁼ ₀^L^cos² ^{m x}_L^ ^dx ⁼ ^L₂

when _cr /m = 0 minimum σ_cr occurs. This leads to

we have ^cr ^z ^I^o

m k L L GJ

z m

EI ₂ ₂ /

2 2

2 



 



 + +

= 

 

1

4 / 1 2  

 



= 

z EI

k m L

 z

Integer

2 2 2

2

2 2

0 2 0 0 0

L L L L

cr EI zz dx GJ dx k dx Io dx

x x x

  

 ^ ^^ ^ ^^ ^  ^{ } ^^ ^ ^

 =     +     +          

Simply supported stiffener ends - w/o rotation about x-axis but free to warp (1/4 )

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Example

Find the expression for the tripping of flat bar stiffener due to buckling.

Simply supported stiffener ends - w/o rotation about x-axis but free to warp (2/4 )

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Section propertiesof a flat bar stiffener:

• The second moment of area about the base is

3

3 w w y

t I = d

• The second moment of area about z-axis is

12

3 w w z

t I = d

• The torsional constant is

3

3 w wt J = d

• The polar moment of area about the base is

3 12

3

3 3

3

w w w

w w

w z

y o

t d t

d t

I d I

I = + = + 

• The centroid of cross-sectional area above the base is z = ₂¹ dw

Example (Solution)

Simply supported stiffener ends - w/o rotation about x-axis but free to warp (3/4)

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The stiffness (plate)

(

²

)

3

1 3 −

= b k Et

( )

2 1 G E

= v

+

Substituting the above information into the equation for critical tripping of stiffener gives

(

²

)

³ ² ²

2 2 3

2 2 2 2

1

16  

 

m t d b

a Et d

G t a

m Et

w w w

w w

cr  + −

 

 + 

= for flat bar

Example (Solution)

3

3 w w y

t I = d

12

3 w w z

t I = d

3

3 w wt

J = d z = ₂¹ d_w

o z

cr I

m k L L GJ

z m

EI ₂ ₂ /

2 2

2 



 



 + +

= 

 

3

w w o

I = d t

The span L = a

b = plate breadth

Simply supported stiffener ends - w/o rotation about x-axis but free to warp (4/4 )

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* Built-in stiffener ends w/o warping

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• Assume buckling shape is 



 

 −

= L

x

a_m m

f 1 cos²

= 0

f and

which satisfies boundary conditions:

0 / =

f x at x = 0 and x = L.

Built-in Stiffener Ends w/o Warping

L x m L

a m

x ^m



f 2

sin

= 2



L x m L

a m

x ^m



f 2

cos

4 ₂

2 2 2

2 =



Then, and 

cos 2

sin 0

2 0

2 L

L dx x dx m

L x

m ^L

L =



=



^ ^ ^⁰^L^_^{ −}¹ ^cos²^m_L^^x^_^²^dx ⁼ ³₂^L

Since, and

Then cr ¹⁶ z ² ²₂ ² ⁴ ³ ₂² ₂ ^{/ 4}

( )

o

m L

EI z GJ k I

L m

 



 

=  + + 

 

When ^_cr ^/ ^^m ⁼ ⁰ minimum σ_cr occurs. This leads to

3 1 2

4 / 1 2  

 



= 

z EI

k m L

 z

Integer Comparing this with that of simply supported stiffener, the term for k is 3 times greater than that for simply supported case. This means that the built-in ends give more stable ability and the buckling form is in higher order(i.e. large number of waves).

C.L.

1

4 / 1 2  

 



= 

z EI

k m L

 z

Integer

S.S.

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* Design Considerations

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Design considerations of plate and stiffened panel

Following calculations should be carried out to ensure whether the strength of the plate and stiffened panel under local buckling load are acceptable or not.

Local buckling load

• Check column buckling strength of stiffened panel against plate-induced failure (PIF) or stiffener-induced failure (SIF)

• Check buckling strength of stiffeners against tripping

• Check torsional buckling strength of primary member against overall buckling

• Check bending strength of plate-stiffener combination (PSC) against local bending

• Check plate strength against local load

• Check bending and shear strength of primary members

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• We have investigated the Tripping of Stiffeners .

Learning Outcomes (Review)

• Now we are able to:

- Derive the formula for the critical stress of tripping a stiffener.

- Calculate the critical stress of tripping a stiffener of rolled or built-up section.

- Be aware of post-buckling behaviour of plate (This will be continued in Topic 9A)

• Details can be referred to topics 9 in the lecture notes.

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short clamped plates (Topic 7)

– Additional (Low aspect ratio plates, strength & permanent set)

[Part III] Buckling of Stiffened Panels

– Failure modes (Topic 8)

– Tripping (Topic 9) + Post-buckling strength of plate (Topic 9A) – Post-buckling behaviour (Topic 10)

Adv. Marine Structures / Adv. Structural Design & Analysis (Next Lecture)

[Theory of Plates and Grillages]

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Kam Sa Hab Ni Da

Questions?