Lecture #4 Buckling in aircraft structures. PLATES IN UNI-AXIAL COMPRESSION ObjectBuckling factor k 2.

Lecture #4Buckling in aircraft structures

PLATES IN UNI-AXIAL COMPRESSIONObject Buckling factor k

2

2

0.9cr

kE

b

PLATES IN SHEARObject Buckling factor k

3

2cr

kE

b

CYLINDRICAL RODS IN COMPRESSIONObject 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

kEhR

analytically k=0.6!

SHPERICAL SHELL UNDER EXTERNAL PRESSUREObject Buckling factor k

5

0

0.05

0.1

0.15

0.2

0 500 1000 1500

R/h

k

cr

kEhR

analytically k=0.6!

• Initial imperfections (the effect is especially large for cylindrical and spherical shells);• Curved geometry (very actual for aircraft panels);• Uncertain supporting conditions;• Inelastic behavior of material (the effect is crucial for lightweight and high-loaded aircraft structures).

FACTORS WHICH AFFECT THE ACCURACY OF BUCKLING ANALYSIS

6

Ways to increase the accuracy of calculations:

• Use empirical methods and formulas (much testing required);• Use numerical methods.

COMPARISON BETWEEN CIRCULAR AND SQUARE CROSS SECTIONS

7

• 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 IS BEYOND THE YEILD LIMIT

8

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 correspondingto m and n, the displacement is

SOUTHWELL METHOD FOR PLATES

9

1 1

, sin sinmnm n

m x n yw x y A

a b

,mn x

mnx

C NA

f m n N

maxmn x

xcr x

C Nw

N N

maxw

max

x

wN

x crN

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 linear physical law); - material ultimate stress.

2

11cr u

,

u

cr el

,cr el

u

PLASTICITY CORRECTION – EMPIRICAL FORMULAS

11

According to Gerard, the nonlinear material behavior could be taken into account using the plasticity correction factor :

- tangent and secant moduli;

- elastic and inelastic Poisson’s ratios.

2

0.9cr

k E

b

2

2

1 1 1 1 31 2 2 4 4

el s t

pl s

E EE E

,t sE E

,el pl

BUCKLING MODES OF AIRCRAFT PANEL

12

POST-BUCKLING BEHAVIOR

13

2c

t

,cr skin

str

The post-buckling behavior of aircraft panels is usually studied using the concept of “attached skin” having the width of 2c:

,2 cr skin

str

c t

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

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.

FLOWCHART OF NONLINEAR BUCKLING FEA

16

Linear compression

FLOWCHART OF NONLINEAR BUCKLING FEA

17

Linear buckling

1st mode884 MPa

FLOWCHART OF NONLINEAR BUCKLING FEA

18

Linear buckling

2nd mode996 MPa

FLOWCHART OF NONLINEAR BUCKLING FEA

19

Linear buckling

3rd mode1210 MPa

FLOWCHART OF NONLINEAR BUCKLING FEA

1) Rename *.rst file to "buckling.rst“,

move it from buckling 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 -> onnonlinear effects -> yes

20

Transfer of initial imperfections

FLOWCHART OF NONLINEAR BUCKLING FEA

21

Nonlinear analysis

485 MPa – last substep converged

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

Structural layout dimension

Determinative factor

Distance between ribs and fuselage frames

Primary 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 rivets

23

EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT

24

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:

WHERE TO FIND MORE INFORMATION?

25

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

… Internet is boundless …

TOPIC OF THE NEXT LECTURE

26

The concept of thin-walled beam.Normal stresses

All materials of our course are availableat department website k102.khai.edu

1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”