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Lecture #2

Buckling of aircraft structures

8/13/2019 S6 Lecture 02_ Buckling of Aircraft Structures

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PLATES IN UNI-AXIAL COMPRESSION

Object Buckling factor k

2

20.9

cr  

kE 

b 

 

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PLATES IN SHEAR

Object Buckling factor k

3

2cr  

kE 

b 

 

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CYLINDRICAL RODS IN COMPRESSION

Object Buckling factor k

4

0

0.05

0.1

0.15

0.2

0.25

0.3

0 500 1000 1500

R/h

k 

cr 

kEh 

R 

analytically k=0.6!

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SHPERICAL SHELL UNDER EXTERNAL PRESSURE

Object Buckling factor k

5

0

0.05

0.1

0.15

0.2

0 500 1000 1500

R/h

k  

cr 

kEh 

R 

analytically k=0.6!

8/13/2019 S6 Lecture 02_ Buckling of Aircraft Structures

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• Initial imperfections (the effect is especially large forcylindrical and spherical shells);

• Curved geometry (very actual for aircraft panels);

• Uncertain supporting conditions;

• Inelastic behavior of material (the effect is crucial forlightweight and high-loaded aircraft structures).

FACTORS WHICH AFFECT THE ACCURACY OFBUCKLING ANALYSIS

6

Ways to increase the accuracy of calculations:

• Use empirical methods and formulas (much testing

required);

• Use numerical methods.

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COMPARISON BETWEEN CIRCULAR AND SQUARECROSS SECTIONS

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• Empirical determination of critical stress (Southwell

method).• Usage of plasticity correction factors calculated by

empirical formulas.

• Finite element analysis (FEA).

WAYS TO FIND THE CRITICAL STRESS WHICH ISBEYOUND THE YEILD LIMIT

8

8/13/2019 S6 Lecture 02_ Buckling of Aircraft Structures

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Thus, the graph of against will form a

straight line with a slope of .

The deformed shape of panel is

The view of coefficients is

Near the buckling mode corresponding

to m  and n , the displacement is

SOUTHWELL METHOD FOR PLATES

9

1 1

, sin sinmn 

m n 

m x n y  w x y Aa b 

 

,

mn x 

mn x 

C N A

f m n N  

 

max

mn x 

x cr x  

C N w 

N N 

 

maxw 

 max

x 

w 

N  x cr N 

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PLASTICITY CORRECTION – EMPIRICAL FORMULAS

10

Kan and Sverdlov proposed the following formula to

take into account the plastic behavior of material:

,

where ;

- elastic critical stress (calculated using linearphysical law);

- material ultimate stress.

2

1

1cr u 

 

 

,

u 

cr el 

 ,cr el 

 u 

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PLASTICITY CORRECTION – EMPIRICAL FORMULAS

11

 According to Gerard, the nonlinear material behavior

could be taken into account using the plasticitycorrection factor :

- tangent and secant moduli;

- elastic and inelastic Poisson’s ratios. 

2

0.9

cr 

k E 

b 

 

 

 2

2

1   1 1 1 3

1 2 2 4 4

el s t  

pl s 

E E 

E E 

 

   

,t s 

E E  

,el pl  

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BUCKLING MODES OF AIRCRAFT PANEL

12

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POST-BUCKLING BEHAVIOR

13

2c 

 t 

 ,cr skin 

 str  

The post-buckling

behavior of aircraftpanels is usually

studied using the

concept of “attached

skin” having the

width of  2c :

,2

  cr skin 

str 

c t 

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FLOWCHART OF NONLINEAR BUCKLING FEA

14

Linear

compression

Linear buckling

Transfer of initial

imperfections

Linear physical law is used

Probable modes of buckling are

found, dangerous one is chosen

depending on the area of interest

Nonlinear analysis

Results

The critical force corresponds to

the loss of convergence

The model is slightly modified

according to buckling mode,

nonlinearities are specified

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FLOWCHART OF NONLINEAR BUCKLING FEA

15

Dimensions of the panel:

a  = 60 mm, h  = 22 mm, 1  = 4 mm, 2  = 6 mm.Length is 300 mm.

Supporting conditions: clamped at all sides.

Problem is to find a critical stress for this panel.

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FLOWCHART OF NONLINEAR BUCKLING FEA

16

Linear

compression

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FLOWCHART OF NONLINEAR BUCKLING FEA

17

Linear buckling

1st mode

884 MPa

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FLOWCHART OF NONLINEAR BUCKLING FEA

18

Linear buckling

2nd mode

996 MPa

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FLOWCHART OF NONLINEAR BUCKLING FEA

19

Linear buckling

3rd mode

1210 MPa

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FLOWCHART OF NONLINEAR BUCKLING FEA

1) Rename *.rst file to

"buckling.rst“, move it frombuckling to structural analysis directory.

2) Insert a command object in the environment:

 /prep7upgeom,0.1,1,1,buckling,rst

 /solu

3) Set nonlinearities:large deflection -> on

nonlinear effects -> yes

20

Transfer of initial

imperfections

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FLOWCHART OF NONLINEAR BUCKLING FEA

21

Nonlinear analysis

485 MPa – last substep

converged

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0

5

10

15

20

25

30

35

0 100 200 300 400 500 600

FLOWCHART OF NONLINEAR BUCKLING FEA

Results

Nonlinear FEA

predicted 509 MPa

which is 1.73 times

smaller than elastic

solution.

22

Shortening, mm

Stress, MPa

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Structural layout

dimension

Determinative factor

Distance between ribs and

fuselage framesPrimary buckling of panels

Distance between stringersLocal buckling of skin

between stringers

Distance between spar web

stiffeners

Buckling of spar web under

shear loads

Distance between rivets in

longitudinal joints

Local buckling of skin

between rivets23

EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT

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EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT

The stringer cross section is dramatically affected by

buckling, showing the compromise between buckling

resistance and technological simplicity:

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WHERE TO FIND MORE INFORMATION?

25

Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 9

… Internet is boundless … 

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TOPIC OF THE NEXT LECTURE

26

The concept of thin-walled beam.

Normal stresses

 All materials of our course are available

at department website k102.khai.edu 

1. Go to the page “Библиотека” 

2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”