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Chapter 6 less HW solution

Chapter 6 less HW solution - Chapter 6 Buckling of Plate...

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Chapter 6 Buckling of Plate Elements 1. Differential Equation of Plate Buckling, a Linear Theory Consider an isolated freebody of a plate element in the deformed configuration (necessary for stability problems examining equilibrium in the deformed configuration, neighboring equilibrium). The plate material is assumed to be isotropic, homogeneous and obey Hooke’s law. The plate is assumed to be prismatic (constant thickness) and forces expressed per unit width of the plate are assumed constant. Fig. 6-1 In-plane forces on plate element z F for x N : 2 2 0 x x w w w N dx dy N dy x x x + ↓ - ↑= x y dy dx y N w y w x w y w x x N y N x N xy N yx N yx N xy N t w w dx x x x + w w dx y x y + w w dy x y x + w w dy y y y + 1
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for y N : 2 2 0 y y w w w N dy dx N dx y y y + ↓ - ↑= for xy N : 2 0 xy xy w w w N dx dy N dy y x y y + ↓ - ↑= ∂ ∂ for yx N : 2 0 yx yx w w w N dy dx N dx x x y x + ↓ - ↑= ∂ ∂ Since xy yx N N = (this can be readily proved by taking the in-plane moment at a corner), it follows Fig. 6-2 Bending shear z F : 2 2 2 2 2 2 x y xy w w w N N N dxdy x y x y + + ∂ ∂ (a) y x Q Q dxdy x y + (b) (b) From (a) and (b), one obtains 2 2 2 2 2 2 0 y x x y xy Q Q w w w N N N x y x y x y + + + + = ∂ ∂ (6-1) y Q x x Q x x Q Q dx x + z y y y Q Q dy y + 2
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Fig. 6-3 Moment components It should be noted that the positive direction of xy M in Fig. 6-3 is reversed from that given by Timoshenko and Woinowsky-Krieger (2nd edition, McGraw-Hill, 1959). xy yx M M = 0 x M = (The condition that the sum of moments about the x-axis must vanish yields.) 0 2 y xy y x y M M Q Q dxdydy dydx dxdy Q dxdy dxdydy y x x y + - - - = Neglecting the higher order terms, the above equation reduces to 0 y xy y M M Q y x + - = (6-2) Similarly, moments about the y-axis leads to 0 yx x x M M Q x y + - = (6-3) x y z y M yx M xy M x M x x M M dx x + yx yx M M dy y + xy xy M M dx x + y y M M dy y + 3
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Differentiating Eqs. (6-2) and (6-3) and substituting the results into Eq. (6-1), yields 2 2 2 2 2 2 2 2 2 2 2 2 0 xy y x x y xy M M M w w w N N N x x y y x y x y + + + + + = ∂ ∂ ∂ ∂ (6-4) If one considers (at least temporarily) x y xy N , N , and N are known, then Eq. (6-4) contains four unknowns, x y xy M , M , M , and w . In order to determine these quantities uniquely, one needs three additional relationships. These three additional equations may be obtained from the geometric compatibility conditions.
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