1. Buckling in aircraft structures
Lecture #4
Buckling in aircraft structures

2. PLATES IN UNI-AXIAL COMPRESSION
Object
Buckling factor k 2

3. PLATES IN SHEAR
Object
Buckling factor k 3

4. CYLINDRICAL RODS IN COMPRESSION
Object
Buckling factor k
analytically k=0.6! 4

5. SHPERICAL SHELL UNDER EXTERNAL PRESSURE
Object
Buckling factor k
analytically k=0.6! 5

6. FACTORS WHICH AFFECT THE ACCURACY OF BUCKLING ANALYSIS
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).
Ways to increase the accuracy of calculations:
Use empirical methods and formulas (much testing required);
Use numerical methods. 6

7. COMPARISON BETWEEN CIRCULAR AND SQUARE CROSS SECTIONS7

8. WAYS TO FIND THE CRITICAL STRESS WHICH IS BEYOND THE YEILD LIMIT
Empirical determination of critical stress (Southwell method).
Usage of plasticity correction factors calculated by empirical formulas.
Finite element analysis (FEA). 8

9. SOUTHWELL METHOD FOR PLATES
The deformed shape of panel is
The view of coefficients is
Near the buckling mode corresponding
to m and n, the displacement is
Thus, the graph of against will form a
straight line with a slope of 9

10. PLASTICITY CORRECTION – EMPIRICAL FORMULAS
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. 10

11. PLASTICITY CORRECTION – EMPIRICAL FORMULAS
According to Gerard, the nonlinear material behavior could be taken into account using the plasticity correction factor h:
- tangent and secant moduli;
- elastic and inelastic Poisson’s ratios. 11

12. BUCKLING MODES OF AIRCRAFT PANEL12

13. POST-BUCKLING BEHAVIOR
The post-buckling behavior of aircraft panels is usually studied using the concept of “attached skin” having the width of 2c: 13

14. FLOWCHART OF NONLINEAR BUCKLING FEA14

15. FLOWCHART OF NONLINEAR BUCKLING FEA
Dimensions of the panel:
a = 60 mm, h = 22 mm, d1 = 4 mm, d2 = 6 mm.
Length is 300 mm.
Supporting conditions: clamped at all sides.
Problem is to find a critical stress for this panel. 15

16. FLOWCHART OF NONLINEAR BUCKLING FEA
Linear compression 16

17. FLOWCHART OF NONLINEAR BUCKLING FEA
1st mode
884 MPa 17

18. FLOWCHART OF NONLINEAR BUCKLING FEA
2nd mode
996 MPa 18

19. FLOWCHART OF NONLINEAR BUCKLING FEA
3rd mode
1210 MPa 19

20. Transfer of initial imperfections
FLOWCHART OF NONLINEAR BUCKLING FEA
Transfer of initial imperfections
1) Rename *.rst file to "buckling.rst“, move it from buckling to structural analysis directory.
2) Insert a command object in the environment:
/prep7
upgeom,0.1,1,1,buckling,rst
/solu
3) Set nonlinearities:
large deflection -> on
nonlinear effects -> yes 20

21. FLOWCHART OF NONLINEAR BUCKLING FEA
Nonlinear analysis
485 MPa – last substep converged 21

22. FLOWCHART OF NONLINEAR BUCKLING FEA
Results
Shortening, mm
Nonlinear FEA predicted 509 MPa which is 1.73 times smaller than elastic solution.
Stress, MPa 22

23. Structural layout dimension
EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT
Structural layout dimension
Determinative factor
Distance between ribs and fuselage frames
Primary buckling of panels
Distance between stringers
Local 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

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: 24

25. … Internet is boundless …
WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 9
… Internet is boundless … 25

26. The concept of thin-walled beam.
TOPIC OF THE NEXT LECTURE
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.)” 26