AE 321 Homework 1 Due in class on September 6, 2006
1. How many terms (and why) are represented by:
Aij
Bij ,kl Cij, ji Dijk Apq Bqr
2. How many equations (and why) are represented by:
Aij Bjk Ckl = Dil
Aij , j = 0
Aijk ,l + Bij ,kl = Cijkl
3. Write out t

AE 321 Homework 5 Due in class on October 4
1. The components of a displacement field are (in meters):
ux = (x2 + 20) 104 , uy = 2 yz 103, uz = (z2 xy) 103,
(a) Consider two points (2, 5, 7) and (3, 8, 9) in the undeformed configuration. Find the change i

AE 321 Homework 2 Due in class on September 13
1. Find the rotation matrix [R] of direction cosines to transform: (a) 30 counter clockwise around axis
x3 , (b) 45 clockwise about x3 , followed by 60 counterclockwise about x1
2. Let
A = 2e1 + 4 e2 + 2e3 .

AE 321 Solution to Practice Problems Chapter 3: Strain 1. Given the displacement field
ux = ( x 2 + 20 ) !10"4 m, uy = 2yz !10 "3 m, uz = ( z2 " xy) !10 "3 m
(a)
Before deformation, the distance between P(x, y, z ) = P(2,5,7 ) and Q(x, y, z ) = Q(

AE 321 Practice Problems Chapter 2: Traction and Stress
1. Find the components of the traction on planes defined by n1 = 1
2 , n2 = 1 2 , n3 = 0 and
n1 = 1
2, n2 = !1 2 , n3 = 0 for the following states of stress:
(a)
! 1 1 = ! !1 2 = ! 2 1 = 0

AE 321 Solution of Homework #4
1. (a) The equilibrium equations for the case of no body forces is given by the expression
ij, j =
ij x j
=0
(1)
Since the stress is constant everywhere in the body, the equilibrium equations are satisfied everywhere. (b)

AE 321 Solution of Homework #3
1. The components of the traction at a given plane is given by
Ti( n ) = ij n j = ji n j or T
[ ]= [ ][n],
(n )
(1)
where ij represent the symmetric stress tensor and n j represents the j component of the outward normal unit

AE 321 Solution of Homework #2
1(a) Note that we are rotating the original frame counterclockwise by 30 degrees. Therefore,
= +30 . The rotation matrix R has the form
x1 x1' R = x2 x1' x3 x1' cos = sin 0 3/2 = 1/ 2 0 3/2 = 1/ 2 0
' x1 x2 ' x2 x2 ' x3 x

AE 321 Solution of Homework #1
1.
The number of terms is given by the following expression
3n ,
(1.1)
where n is the number of free indices. Then, Number of free indices n
Aij Bij ,kl C ij , ji Dijk
A pq Bqr
Number of terms
3n
2 4 0 3 2
9 81 1 27 9
2.
The

AE 321 Homework 4 Due in class on September 27
1. Assume that the stress state everywhere in the interior of a body is given in Cartesian coordinates by:
[]
P 0 0 ij = 0 P 0 KPa 0 0 P
Note: This stress state is called isotropic or hydrostatic. (a) Does t

AE 321 Homework 3 Due in class on September 20
1. Find the components of the traction on planes whose normal vector have components
n1 = 1 2 , n2 = 1 2 , n3 = 0 , and n1 = 1 2 , n2 = 1 2 , n3 = 0 for the following states
of stress: (a)
11 = 12 = 21 = 0 13

AE 321 Aerospace Structures I
John Lambros Department of Aerospace Engineering University of Illinois at Urbana-Champaign
Fall 2008
AE 321 0. 1
Evolution of Aircraft Structures
1890s : Octave Chanute (Civil Engineer) 1904 : Wright brothers
1909 : Louis B

AE 321 Practice Problems Chapter 1: Mathematical Preliminaries
1. How many terms (and why) are represented by:
Aij
Bij ,kl Cij, ji Dijk
A pq Bqr
2. How many equations (and why) are represented by:
Aij Bjk Ckl = Dil
Aij , j = 0
Aijk ,l + Bij ,

AE 321 Solution to Practice Problems Chapter 1: Mathematical Preliminaries 1. The number of terms is given by the following expression
3n ,
where n is the number of free indices. Then, Number of free indices n
Aij Bij ,kl C ij , ji Dijk A pq Bqr
(

AE 321 Practice Problems Chapter 3: Strain
1. The components of a displacement field are (in meters):
ux = ( x 2 + 20 ) ! 10"4 , uy = 2yz ! 10 "3, uz = ( z2 " xy ) ! 10 "3,
(a) Consider two points (2, 5, 7) and (3, 8, 9) in the undeformed configur

AE 321 Solution to Practice Problems Chapter 2: Traction and Stress
1.
Determine the components of the traction on a given plane and for a given state of stress. The components of the traction at a given plane is given by
Ti ( n ) = ! ij n j = !