CONTINUUM MECHANICS for ENGINEERS

(b) Project each of the stress vectors obtained in (a) onto the primed
axes to determine the nine components of [ σ .
ij ′ ]
(c) Verify the result obtained in (b) by a direct application of Eq 3.5-1
of the text.
⎡143
36 114⎤
1 ⎢
⎥
Answer: [ σ ′ ]=
⎢
⎥
MPa
ij 36 166 3
7
⎣
⎢114
3 −15⎦
⎥
3.13 At point P, the stress matrix is given in MPa with respect to axes
Px 1x 2x 3 by
⎡6
4 0⎤
⎡2
1 1⎤
⎢
⎥
⎢ ⎥
Case 1: [ σ Case 2:
ij]=
⎢
4 6 0
⎥ [ σ ij]=
⎢
1 2 1
⎥
⎣
⎢0
0 −2⎦
⎥
⎣
⎢1
1 2⎦
⎥
Determine for each case
(a) the principal stress values
(b) the principal stress directions.
Answer: (a) Case 1: σ (1) = 10 MPa, σ (2) = 2 MPa, σ (3) = –2 MPa
Case 2: σ (1) = 4 MPa, σ (2) = σ (3) = 1 MPa
( 1) eˆ eˆ
(b) Case 1: nˆ
1+ 2 ( 2) eˆ eˆ
=± , nˆ
1−2 ( 3)
=± , nˆ = meˆ3
2
2
( 1) eˆ ˆ ˆ
Case 2: ˆ 1+ e2 + e3
( 2) – eˆ ˆ
n = , ˆ
1+ e2
( 3) – eˆ ˆ ˆ
n = , ˆ
1– e2 + 2e3
n =
3
2
6
3.14 When referred to principal axes at P, the stress matrix in ksi units is
* [ σ ij]=
⎡2
0 0⎤
⎢ ⎥
⎢
0 7 0
⎥
⎣
⎢0
0 12⎦
⎥
If the transformation matrix between the principal axes and axes
Px 1x 2x 3 is
⎡ 3 1
⎢ 5
1
[ aij]= ⎢a
a a
2
⎢ 3 −1
⎣ 5
−
4
−
5
21 22 23
−
4
−
5
where a21, a22, and a23 are to be determined, calculate . σ ij
⎤
⎥
⎥
⎥
⎦
[ ]