Post-Buckling5.2.4

Post-buckled composite stiffened panel under shear

Buckling vs Post-buckling

• in general buckling does not imply failure especially

for plates

P

Pcr

1

wcenter

beam post-buckling curve is flat and failure strains are reached at low

post-buckling loads

Buckling vs Post-buckling

• post-buckled (stiffened) skins are significantly lighter

but have additional challenges:

– susceptible to skin/stiffener separation failure

where only resin holds the structure (unless

fasteners or other means are used to hold it together)

– susceptible to the creation and growth of

delamination under fatigue loading especially at high

post-buckling factors

skin/stiffener

separation

skin/stiffener

separation

skin/stiffener

separation

Buckling versus Post-Buckling

• If a composite structure is allowed to post-buckle,

the ratio of final failure load to buckling load (post

buckling ratio PB) must be chosen carefully

(especially for shear loads)

• Conservative approach (but not very efficient): Do

not allow buckling below limit load (i.e. PB=1.5)

• PB>5 great design challenge both under static and

fatigue loads

Post-buckling mode

frame

frame

stiffeners

skin

What buckles and when?

What BC’s are the different components supposed to simulate?

Post-Buckling Scenarios

• Skin buckling as a whole (stiffeners only increase the bending stiffness of the skin)

• Skin buckling between the bays (stiffeners act as panel breakers)

• Stiffeners buckle as columns or locally (crippling)

• Frames do not buckle!

Post-buckling scenarios

• Most efficient:– Skin between stiffeners and frames buckles

first

– Stiffeners do not buckle and do not move out

of plane (depending on cross-section they may

rotate => BC implications)

– When required PB value is reached, skin fails

in compression and/or stiffeners fail by crippling

stiffeners impose zero deflection

and slope

stiffeners impose zero deflection

only

Post-Buckling Analysis• governing equations: Large deflection von Karman

equations

• place moment equilibrium eqn into Fz equilibrium eqn:

022

2

22

2

2

2

22

2

2

zyxyxyx

yxyx py

wp

x

wp

y

wN

yx

wN

x

wN

y

M

yx

M

x

M

px, py, pz distributed loads (force/area)

• use the moment-curvature relations to substitute for

the moments

D16=D26=0

Bij=0

yx

wDM

y

wD

x

wDM

y

wD

x

wDM

xy

y

x

2

66

2

2

222

2

12

2

2

122

2

11

2

Derivation of von Karman

equations (large deflections)• to obtain:

1st von Karman equation: Predominantly bending

zyxyxyx p

y

wp

x

wp

y

wN

yx

wN

x

wN

y

wD

yx

wDD

x

wD

2

22

2

2

4

4

2222

4

66124

4

11 2)2(2

Strain Compatibility

y

w

x

w

x

v

y

u

y

w

y

v

x

w

x

u

xyo

yo

xo

2

2

2

1

2

1

2

2

2

22

2

2

3

2

3

2

3

2

32

2

32

2

2

32

2

3

2

2

2

32

2

2

32

2

3

2

2

2

2

1

2

2

1

y

w

x

w

yx

w

yx

w

x

w

yx

w

y

w

yx

v

yx

u

yx

yx

w

y

w

yx

w

yx

v

yx

w

y

w

xyx

v

x

yx

w

x

w

yx

w

yx

u

yx

w

x

w

yyx

u

y

xyo

yo

xo

...

2

2

2

2

2

yxxy

xyoyoxo

Derivation of von Karman

equations (large deflections)

• use (non-linear) strain compatibility

2

2

2

22

22

2

2

2

2

y

w

x

w

yx

w

yxxy

xyoyoxo

non-linear termsεxo, εyo, γxyo mid-plane

strains

• invert stress-strain eqns to express strains in terms of

Nx, Ny, Nxy

xyoxy

yoxoy

yoxox

AN

AAN

AAN

66

2212

1211

xyxyo

yxyo

yxxo

NA

NAAA

AN

AAA

A

NAAA

AN

AAA

A

66

2

122211

11

2

122211

12

2

122211

12

2

122211

22

1

A16=A26=0

Derivation of von Karman

equations (large deflections)

• substitute in strain compatibility to obtain,

2

2

2

22

22

66

2

2

122

2

112

2

122

2

222

122211

11

y

w

x

w

yx

w

yx

N

Ax

NA

x

NA

y

NA

y

NA

AAA

xyxyyx

• introduce Airy stress function and potential V that identically

satisfy stress equilibrium:

yx

FN

Vx

FN

Vy

FN

xy

y

x

2

2

2

2

2

potential of distributed

loads (=0 in our post-

buckling problem)

Derivation of von Karman

equations (large deflections)

• to obtain

2nd von Karman equation: Predominantly stretching

(and non-linear)

2

2

2

22

2

22

4

66

4

4

1122

4

124

4

222

122211

12

1

y

w

x

w

yx

w

yx

F

Ax

FA

yx

FA

y

FA

AAA

Post-buckling of a square anisotropic plate

under compression

simply-supported with three immovable edges:

w=0 at x=y=0 and x=y=a

u=0 at x=0

v=0 at y=0 and y=a

u=-C at x=a (constant compressive displacement)

a

a

Px

(applied

force)

rigid

x

y

Solve the two von Karman equations

approximately

• assume

a

yK

a

xK

x

a

Py

a

PF

a

y

a

xww

yx

2cos

2cos

22

sinsin

0220

22

11

• substitute in the 2nd von Karman equation

a

x

aw

a

y

aw

a

x

aK

AAA

A

a

y

aK

AAA

A 2cos

2

2cos

2

2cos

162cos

164

42

114

42

114

4

202

122211

11

4

4

022

122211

22

w11, K02, K20 unknown

coefficients

32

32

2

11

11

2

12221120

2

11

22

2

12221102

w

A

AAAK

w

A

AAAK

matching coefficients

of cos(2πy/a) and

cos(2πx/a)

and the 2nd von

Karman eqn is

identically satisfied

Solution (cont’d)

• Py (as a function of Px) and deflection C are determined

by integrating stress-strain eqns and using average BC’s

on u and v: 2

2

1

x

w

x

uxo

yxxo N

AAA

AN

AAA

A2

122211

12

2

122211

22

also

non-linear strain-displacement eq

inverted stress-strain eq

Airy stress function F

dxdy

x

wdxdy

x

F

AAA

Adxdy

y

F

AAA

Adxdy

x

ua aa aa aa a 2

0 00 0

2

2

2

122211

12

0 0

2

2

2

122211

22

0 02

1

yx

FN

Vx

FN

Vy

FN

xy

y

x

2

2

2

2

2

Solution (cont’d)

=-C =0

4

22

11

w

from which:

dxdy

a

y

a

x

aw

a

P

AAA

aA

a

P

AAA

aAyuyaua

yx 2

112

122211

2

12

2

122211

2

22 )sincos(2

1),0(),(

aw

a

P

AAA

aA

a

P

AAA

aAC

yx

8

22

112

122211

12

2

122211

22

similarly, from 2

2

1

y

w

y

vyo

= Poisson’s ratio ν for

isotropic

=0 for in-plane problems

11

2

122211

22

11

11

12

8 A

AAA

aw

A

APP xy

Solution (cont’d)

• substitute now in the 1st von Karman equation

2

2

2

222

2

2

2

2

4

4

2222

4

66124

4

11 2)2(2y

w

x

F

yx

w

yx

F

x

w

y

F

y

wD

yx

wDD

x

wD

• use the following:

a

yw

A

AAA

aa

P

y

F

a

xw

A

AAA

aa

P

x

F

a

y

a

x

aw

y

w

x

w

a

y

a

x

aw

y

w

yx

w

x

w

x

y

2cos

32

2

2cos

32

2

sinsin

sinsin

2

11

22

2

122211

2

2

2

2

11

11

2

122211

2

2

2

2

112

2

2

2

4

114

4

22

4

4

4

Solution (cont’d)

• to substitute, and match coefficients of

sin(πx/a)sin(πy/a) to obtain the governing eq for w11

• which for w11≠0 leads to

=Pcr , buckling load (units of force), exact for square plate

2nd von Karman eq is satisfied

approximately!

11 22 11 12 66 22

11 2 211 12 66 2211 22 12 11 22

12

11

16 2 21

2 23

1

xA A D D D D P

wD D D DA A A A A

a A

A

22 211 22 12 11 22 3 12

11 11 12 66 22 11

11 22 11

32( 2 ) 1 0

16x

A A A A A Aw D D D D P w

a A A a A

Solution (end)

• For out-of-plane deflections to be possible, must have

1

cr

x

P

P

• the applied load Px must exceed the plate buckling load

Results-Implications

• use as example,

(±45)/(0/90)/(±45) square plate of side 25.4 cm

Material is plain weave fabric with properties:

Ex=Ey=68.94 GPa

νxy=0.05

Gxy=5.17 GPa

ply thickness = 0.19 mm

Load versus center deflection

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5

center defl/plate thickness (w11/h)

P/Pcr

a

a

In-plane compression load

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8

Nx/(Pcr/a)

y/a

=1P

Pcr1.2 2 5

• the center of the plate sees very little load!

Implication

• after buckling approximate in-plane load

beff

a

a

peak Nx value

• there is an effective width beff at the edges of the

panel over which the load is concentrated

Determination of effective width

• total applied force must equal the force created by

the load applied over the effective width:

effxx bNdyN max2

a

y

a

w

A

AAA

a

P

y

FN x

x

2cos

2

32

22

11

22

2

122211

2

2

xx PdyN

22

11

22

2

122211max

2

32

a

w

A

AAA

a

PN x

x

Solving for beff

2211

11

11

12

311212

1

AA

A

P

P

A

Aab

x

cr

eff

• Note that for quasi-isotropic layup,

x

cr

eff

P

Pab

13.12

1

beff=0.303a for Px>>Pcr and QI layup

3.0

4

1

3

12

2211

11

12

11

12

AA

A

A

A

Effective width beff

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pcr/Px

beff/a

beff

a

a

A12

A11=0.05

0.30.7

• weak dependence on A12/A11

Significance for design

• beff provides an idea of where failure is expected under

compression and where reinforcement may be needed

• For (conservative) failure predictions obtain the

maximum compressive stress

h

N x maxmax (h is plate thickness)

• and compare to allowable ultimate compression stress

ult max for no failure

• plus one more use in crippling of stiffener flanges

How good is the analysis compared to

reality?

• For a meaningful comparison, two conditions must be

met:

– Boundary conditions in the “test case” must be the same as in

the model

– Sufficient number of terms must be included in the solution (as

opposed to only one term used here for simplicity)

• As it is hard to find test results with exactly these

boundary conditions, FE results are used to validate the

analysis

• Two different laminates: (a) 45 degree dominated

(b) Quasi-isotropic

Analysis model versus FE (45-dominated

laminate)

• Results from R. Kroese MS Thesis TUDelft 2013

• Excellent agreement between FE and Analysis

end displa-

cement

Analysis model versus FE (Quasi-isotropic

laminate)

• Results from R. Kroese MS Thesis TUDelft 2013

• Excellent agreement between FE and Analysis

end displa-

cement

Boundary conditions discussion

• If the stiffeners in a stiffened panel have very high EI and

very low GJ, and one end is attached to a frame with

also very high EI and low GJ, this analytical model would

be quite accurate

• If not, the model may be conservative or unconservative

• Must also use enough terms in the series and account

for rectangular as opposed to square panels

stiffener

frame

skin between

stiffeners