notes_28_plate_buckl

notes_28_plate_buckl - Plate Buckling ref Hughes Chapter 12...

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Plate Buckling ref. Hughes Chapter 12 Buckling of a plate simply supported on loaded edges treated as a wide column results in similar Euler stress, with EI replaced byD*(b): dividing by area (b*t): π 2 D b π 2 D D := Et 3 eqn 9.1.5, 1/b P e := a 2 σ e := a 2 t 12 ( 1 − ν 2 ) implied Buckling of a simply supported plate. i.e. simply supported on all four sides deflected shape is represented by sine waves in x and y: wx , y) := ∑∑ C mn sin m ⋅π⋅ x sin n y m and n are the number of half waves in ( , a b deflection eqn 12.1.3 2 2 2 π 2 a 2 D m + n argument in text for minimum stress a 2 b 2 can specify n = 1, but m not clear σ acr := tm 2 with n = 1, can express in form where k is buckling coefficient in equation eqn 12.1.5 σ acr := k π 2 D N.B. shift to plate width b in denominator b 2 t m := 1 .. 2 n := 1 .. 2 2 a_over_b := 0.5 , 0.6 .. 4 k(a_over_b , m) := m + a_over_b eqn 12.1.6 a_over_b m  k(a_over_b , 1)   k min (a_over_b) := min k(a_over_b , 2)  k(a_over_b , 3)  k(a_over_b , 4)  6 therefore for long plates, simply 5 supported on loaded ends k = 4 k min (a_over_b) 4 σ acr := 4 π 2 D b 2 t 3 0 1 2 3 4 a_over_b very wide i.e. a/b -> 0, approaches Euler as a<<b 2 σ acr (a_over_b) := k min (a_over_b) π 2 D k -> b + a 2 = b 2 + a 2 -> b 2 b 2 t a b ab a 1 notes_29_plate_buckling.mcd
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a square column meets these boundary conditions hence will buckle with each edge forming half sine wave deflection in half sine waves approaching square for large a/b; long plate loaded on end simply supported on all sides Plate loaded on all four sides; σ ax in a direction, σ ay in b direction Again taking two half wave sine series using energy methods, results in combination expression
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.082 taught by Professor Davidburke during the Spring '03 term at MIT.

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notes_28_plate_buckl - Plate Buckling ref Hughes Chapter 12...

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