CONTINUUM MECHANICS for ENGINEERS

(b) Verify the result determined in (a) by a Mohr’s circle construction
similar to that shown in Figure E3.8-1.
Answer: σ N = 14.04 MPa, σ S = 5.28 MPa
3.23 Sketch the Mohr’s circles for the simple states of stress given by
⎡σo
0 σo⎤⎡σo0
0⎤
⎢
⎥
⎢
⎥
(a) [ σ σ (b)
ij]=
⎢
0 o 0
⎥ [ σ ij]=
⎢
0 2σ
o 0
⎥
⎣
⎢σ
σ ⎦
⎥
o 0 o
⎣
⎢ 0 0 −σ
⎦
⎥
o
and determine the maximum shear stress in each case.
Answer: (a) (σ S) max = σ o, (b) (σ S) max =
3
2 σ o
3.24 Relative to axes Ox 1x 2x 3, the state of stress at O is represented by the
matrix
[ σ ij]=
⎡ 6 – 3 0⎤
⎢
⎥
⎢
– 3 6 0
⎥
⎣
⎢ 0 0 0⎦
⎥
Show that, relative to principal axes , the stress matrix is
Oxxx
* * *
[ σ ij]=
⎡3
0 0⎤
⎢ ⎥
⎢
0 9 0
⎥
⎣
⎢0
0 0⎦
⎥
and that these axes result from a rotation of 45° about the x3 axis.
Verify these results by Eq 3.9-3.
3.25 The stress matrix representation at P is given by
⎡29
0 0⎤
⎢
⎥
[ σ ij]=
⎢
0 −26
6
⎥
⎣
⎢ 0 6 9⎦
⎥
Decompose this matrix into its spherical and deviator parts, and determine
the principal deviator stress values.
Answer: SI = 25 ksi, SII = 6 ksi, SIII = –31 ksi
3.26 Let the second invariant of the stress deviator be expressed in terms
of its principal values, that is, by
ksi
1 2 3
ksi
ksi
II S = S I S II + S II S III + S III S I