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Unformatted text preview: AUBURN UNIVERSITY Department of Civil Engineering CIVL 7690 Analysis of Plate and Shell Systems Final, Take Home, Due August 4, 2009 1. A simply supported semi-infinite plate shown below is free at the cut-off edge. A vertical line loading of a sinusoidal distribution is applied at the free edge. Derive a general expression for the deflection and the maximum bending moments. If 0.1 k/in, q = 100 in, a = 1 in, E=29,000 ksi, and 0.3 t μ = = , determine max max max max , , , and x y w M M R 2. A square plate of dimension “a” is simply supported on all four boundaries. The plate is subjected to a linearly varying compressive load, x N , resulting from a pure bending as shown in the sketch. Using the energy method discussed in the class, determine the critical load. Assume the displacement function (a) single half sine waves for both x- and y-directions and (b) a single half sine wave in the x-direction and two half sine waves in the y-direction. y y x ( 29 sin y p y q a π = free edge x a a a N . . ss . . ss . . ss . . ss ∞ 1 1. M. Lévy suggested a solution of the following form for a rectangular plate that has two opposite edges simply supported: 1 sin sin m m m y y w X X a a π π ∞ = = = ∑ (a) where X m is a function of y only. As the line loading, ( 29 p y , is given by a sinusoidal function, m is taken to be equal to one. Since the semi-infinite plate is subjected to a sinusoidal line loading at its left edge ( x =0) and there is no distributed loading on the plate, the governing differential equation is a homogeneous forth order partial differential equation. 4 4 4 4 2 2 4 2 w w w x x y y ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ (b) Taking partial derivatives of (a) and substituting into (b), gives cos w y X y a a π π ∂ = ∂ sin w y X x a π ∂ ′ = ∂ 2 2 2 sin w y X y a a π π ∂ = - ∂ 2 2 sin w y X x a π ∂ ′′ = ∂ 3 3 3 cos w y X y a a π π ∂ = - ∂ 3 3 sin w y X x a π ∂ ′′′ = ∂ 4 4 4 sin w y X y a a π π ∂ = ∂ 4 4...
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- Summer '10
- Civil Engineering, Sin, Department of Civil Engineering, Sine wave, xm