homework2

# homework2 - = q sin ⎜ ⎜ ⎝ ⎛ π a x ⎠ ⎟ ⎟ ⎟...

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2.081J/16.230J Plates and Shells Homework #2 Due date: c las s on Monday February 27 PROBLEM 1 (a) Consider a simply supported square ( a × a ) plate loaded by a uniform pres- sure q 0 . Find an approximate solution, i.e. the relation between the load intensity q 0 and the central de ection of the plate w 0 . Use the Raleigh-Ritz method ( δ Π =0 ) and try sinusoidal shape function. Moreover, the Gaussian curvature vanishes if the edge of the plate are straight. x a 0 a simply supported y 0 q Z Z Π = D ( w, xx + w, yy ) 2 dS qw dS 2 S S ³ ´ ³ ´ πx πy w ( x, y )= w 0 sin sin a a (b) [ Extra Credit ] Derive an approximate solution for the clamped plate. Try this shape: µ ¶¸ µ ¶¸ w 0 2 πx 2 πy w ( x, y )= 1 cos 1 cos 4 a a 1

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x a 0 fixed support a y 0 q PROBLEM 2 (a) Find the location and magnitude of the maximum in-plane stress compo- nents σ αβ in the problem of simply supported square plate loaded by a sinusoidal pressures, solved in the class. 0 q (
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Unformatted text preview: , ) = q sin ⎜ ⎜ ⎝ ⎛ π a x ⎠ ⎟ ⎟ ⎟ ⎞ sin ⎜ ⎜ ⎝ ⎜ ⎛ π a y ⎠ ⎟ ⎟ ⎟ ⎞ q x y ⎜ [ Hint ] The stress formula for plate is: z M αβ σ αβ = h 3 / 12 (b) Find the magnitude of the maximum out-of-plane average shear stress ( σ zx ) av and ( σ zy ) av . [ Hint ] The average stress formula for plate is: Q αz ( σ αz ) av = h (c) Assuming that out-of-plane shear is distributed in a parabolic way over the plate thickness (similarly to beams), what is the maximum shear stress at the plate middle surface? 2...
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homework2 - = q sin ⎜ ⎜ ⎝ ⎛ π a x ⎠ ⎟ ⎟ ⎟...

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