2.081J/16.230J Plates and Shells Homework #2 Due date: class on Monday February 27 P ROBLEM 1 (a) Consider a simply supported square (a × a) plate loaded by a uniform pres sure q0 . Find an approximate solution, i.e. the relation between the load intensity q0 and the central deflection of the plate w0 . Use the Raleigh-Ritz method (δΠ = 0) and try sinusoidal shape function. Moreover, the Gaussian curvature vanishes if the edge of the plate are straight. a 0 x simply supported a y q0 D Π= 2 Z S 2 (w,xx +w,yy ) dS − Z S q w dS ³π y´ ³π x´ sin a a (b) [Extra Credit] Derive an approximate solution for the clamped plate. Try this shape: ∙ µ ¶¸ ∙ µ ¶¸ 2π x 2π y w0 1 − cos w (x, y) = 1 − cos 4 a a w (x, y) = w0 sin 1 a 0 x fixed support a y q0 P ROBLEM 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. q0 ⎛π x⎞ ⎛π y⎞ q (x, y) = q0 sin ⎜⎜⎜ ⎟⎟⎟sin ⎜⎜⎜ ⎟⎟⎟ ⎝ a⎠ ⎝ a ⎠ [Hint] The stress formula for plate is: σ αβ = z Mαβ h3 /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: (σ αz )av = Qαz 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