Homework 2

Theory of Plates

Homework 2

1. A 2 feet square panel of glass (E = 3 Msi, ν = 0.3) is simply supported on all edges and designed to resist a wind loading, which can be approximated by

q(x,y )=16q o π2sin πx a sin πy

a , (q o = 5 psf, a = 2 ft)

(a) Find the expressions for M x , M y , and M xy in the plate. (b) Prove that the principal moment direction is 45o .

(c) Prove that the maximum principal moment M max and the minimum principal moment M min along the plate diagonal x = y are

M max =16q o π4ν-cos 2πx a ⎛ ⎝ ⎫ ⎭ ⎪ ,

M min =16q o π41-νcos 2πx a ⎛ ⎝ ⎫

⎭ ⎪ (d) What is the minimum thickness of the plate if the maximum stress in the panel can not exceed 500 psi?

(e) For the small-deflection theory to be valid, the maximum plate deflection should be less than one fifth of the plate thickness. What is the minimum required thickness of the plate under this condition? (10 Points)

2. A simply supported rectangular plate is subjected to a uniform load of intensity q o =300 psf. Determine the required plate thickness so that the maximum bending stress in the plate does not

exceed 800 psi. Assume

E =30⨯106 ksi and ν=0.4. (Use the first term approximation in Fourier series expansion). (10 Points)

3. A square plate is simply supported on

four edges and is subjected to

intersecting line loads of uniform

intensity q o . Calculate the deflection at the center of the plate by using the first term approximation. (10 Points)

4. A simply supported rectangular plate is subjected to a concentrated load P. Under the concentrated load, there is an

elastic spring with modulus k. (a) Find the deflection of the

plate at the center. (b) If the stiffness of the spring approach 0 and ∞, what are the condition of the support and the deflection of the plate at center using one term

approximation. (10 Points)

5. A simply supported square plate is subjected to a distributed load as shown. There is an elastic spring with modulus k at the center of the plate. Find the deflection of the plate at the center using one term approximation. (10 Points)

6. A simply supported square plate is supported by an elastic foundation of modulus k and is subjected to a partially uniform line load of intensity q o . Find the deflection at the

center of the plate. (10 Points)

7. A simply supported square plate is supported by an elastic foundation of modulus k and a rigid bar at the center. The plate is subjected to two partially

uniform line load of intensity q o . Find the reaction at the rigid bar. (10 Points)

8. A square plate is simply supported on four

edges and subjected to a diagonal line load

of intensity q o . Find: (i) the expression of w for the plate, (ii) the deflection at the center

of the plate. (10 Points)

9. A rectangular plate resting on elastic foundation with foundation modulus k is simply supported at all the edge. The plate is subjected to a line load as shown. Find the deflection of the plate. (10 Points)