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