AE 321 Homework 7 Due in class on November 1, 2013 Problem 1. Consider a rectangular plate of length, l, height, h, and width, t. The plate is composed of a linearly elastic, isotropic, homogeneous material.
(A) (B)
(C)
The body is subjected to a uniform pressure over the faces at x=0 and x=l. The other faces remain traction-free. The body is subjected to the same uniform pressure as in (A), but now the body is constrained by frictionless contact along the rigid walls at y=0 and h=h. The faces at z=0 and z=t are traction-free. The body is subjected to the same uniform end pressure as in (A) and (B), but now all lateral faces are constrained by frictionless contact with rigid walls so that v=w=0.
For the cases (A), (B), (C) find the effective modulus, defined as: E eff xx
xx
in terms of the
elastic constants E and .
Problem 2. The state of stress in the doubly supported beam made of a homogeneous and isotropic material, shown in the figure below, is given by:
3 l
2
2 x p 2 h h
x
3 y 1 4 p 2 h
2
xy
2
y 3 2 y 2 y h 5 h h
y 2 p 2 2 h h
y
1
3 y
3
x h
Find the displacements u 1 and u2 assuming known elastic properties.
p
y
h
x
l Hint: Place the coordinate frame origin at the center of the beam to facilitate your calculations. Problem 3.
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A common engineering practice is to use a strain gauge rosette (Figure b) at angles 0, 60 and 120 which is called a 60 rosette. It contains 3 folded wires (Figure a) at angles 0, 60, 120.
Then, using the transformation of coordinates equations
we obtain the plane strain tensor at the point where the 60 rosette is attached to. Using the rosette in the figure, the following strain readings were obtained at a point of a steel part: a = 190 10 , b = 200 10 , c = -300 10 . If E = 200 GPa and = 0.3, determine: .
-6
.
-6
.
-6
(a)
The 2D (in-plane) principal stresses and their directions.
(b)
The true maximum shear stress.
Problem 4. Consider a composite rectangular plate with evenly distributed continuous fibers along three directions, which are 60 apart (quasi-isotropic material) as shown below.
In the x1-x2 plane the 2D stress-strain relationship is given by: 0 11 11 C11 C 12
C 22 12 12 0
C 22 0
0 22
C 44 12
Show that for this composite C11 , C12 , C22 , C44 are not independent and that: C11 C 22 and C44 C11 C 12 . This material is called “quasi-isotropic” because it behaves as isotropic in a limited number of directions.
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