CONTINUUM MECHANICS for ENGINEERS

⎡7
3 0⎤
⎢ ⎥
Answer: [ σ ksi
ij]=
⎢
3 7 4
⎥
⎣
⎢0
4 7⎦
⎥
3.15 The stress matrix in MPa when referred to axes Px 1x 2x 3 is
[ σ ij]=
⎡ 3 −10
0⎤
⎢
⎥
⎢
−10
0 30
⎥
⎣
⎢ 0 30 −27⎦
⎥
Determine
(a) the principal stresses, σI, σII, σIII (b) the principal stress directions.
Answers: (a) σI = 23 MPa, σII = 0 MPa, σIII = –47 MPa
( )
(b) ˆ = ê + ê +
n 1
( )
nˆ 2
–0.394 1
0.788 2
0.473ê3 = + +
0.913ê 1 0.274ê2 0.304ê3 ( )
nˆ = + –
3.16 At point P, the stress matrix relative to axes Px1x2x3 is given in MPa by
3
0.110ê2 0.551ê2 0.827ê3 ⎡ 5 a −a⎤
⎢
⎥
[ σ ij]=
⎢
a 0 b
⎥
⎣
⎢−a
b 0⎦
⎥
where a and b are unspecified. At the same point relative to axes
* * *
Px x x the matrix is
1 2 3
⎡σ
I 0 0⎤
* ⎢
⎥
[ σ ij]=
⎢
0 2 0
⎥
⎣
⎢ 0 0 σ ⎦
⎥
III
If the magnitude of the maximum shear stress at P is 5.5 MPa, determine
σI and σIII. Answer: σI = 7 MPa, σIII = –4 MPa
3.17 The state of stress at point P is given in ksi with respect to axes Px1x2x3 by the matrix
[ σ ij]=
⎡1
0 2⎤
⎢ ⎥
⎢
0 1 0
⎥
⎣
⎢2
0 −2⎦
⎥