Buckling in aircraft structures Lecture #4 Buckling in aircraft structures

PLATES IN UNI-AXIAL COMPRESSION Object Buckling factor k 2

PLATES IN SHEAR Object Buckling factor k 3

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

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

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

COMPARISON BETWEEN CIRCULAR AND SQUARE CROSS SECTIONS 7

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

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

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

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

BUCKLING MODES OF AIRCRAFT PANEL 12

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

FLOWCHART OF NONLINEAR BUCKLING FEA 14

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

FLOWCHART OF NONLINEAR BUCKLING FEA Linear compression 16

FLOWCHART OF NONLINEAR BUCKLING FEA 1st mode 884 MPa 17

FLOWCHART OF NONLINEAR BUCKLING FEA 2nd mode 996 MPa 18

FLOWCHART OF NONLINEAR BUCKLING FEA 3rd mode 1210 MPa 19

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

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

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

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

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

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

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