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Lecture #4 Buckling in aircraft structures

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PLATES IN UNI-AXIAL COMPRESSION ObjectBuckling factor k 2

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PLATES IN SHEAR ObjectBuckling factor k 3

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CYLINDRICAL RODS IN COMPRESSION ObjectBuckling factor k 4 analytically k=0.6!

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

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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.

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COMPARISON BETWEEN CIRCULAR AND SQUARE CROSS SECTIONS 7

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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 IS BEYOND THE YEILD LIMIT 8

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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

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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 linear physical law); - material ultimate stress.

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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.

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BUCKLING MODES OF AIRCRAFT PANEL 12

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

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FLOWCHART OF NONLINEAR BUCKLING FEA 14 Linear compression Linear buckling Transfer of initial imperfections Nonlinear analysis Results The critical force corresponds to the loss of convergence The model is slightly modified according to buckling mode, nonlinearities are specified Probable modes of buckling are found, dangerous one is chosen depending on the area of interest Linear physical law is used

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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 1 st mode 884 MPa

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FLOWCHART OF NONLINEAR BUCKLING FEA 18 Linear buckling 2 nd mode 996 MPa

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FLOWCHART OF NONLINEAR BUCKLING FEA 19 Linear buckling 3 rd mode 1210 MPa

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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: /prep7 upgeom,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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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 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

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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:

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WHERE TO FIND MORE INFORMATION? 25 Megson. An Introduction to Aircraft Structural Analysis Chapter 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.eduk102.khai.edu 1. Go to the page “Библиотека”Библиотека 2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”Structural Mechanics (lecturer Vakulenko S.V.)

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