CHAP 1 StressStrain Analysis
23
5.
Find the principal stresses and the corresponding principal stress directions for the
following cases of plane stress.
(a)
σ
xx
= 40 MPa,
σ
yy
= 0 MPa,
τ
xy
= 80 MPa
(b)
σ
xx
= 140 MPa,
σ
yy
= 20 MPa,
τ
xy
= −60 MPa
(c)
σ
xx
= −120 MPa,
σ
yy
= 50 MPa,
τ
xy
= 100 MPa
Solution:
(a) The stress matrix becomes
40 80
MPa
80
0
xx
xy
xy
yy
To find the principal stresses, the standard eigen value problem can be written as
0
In
The above problem will have nontrivial solution when the determinant of the coefficient
matrix becomes zero:
40
80
0
80
0
xx
xy
xy
yy
The equation of the determinant becomes:
2
40
80 80
40
6400
0
The above quadratic equation yields two principal stresses, as
1
102.46MPa
and
2
62.46MPa
.
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 Fall '08
 Staff
 Linear Algebra, Finite Element Analysis, Strain, Stress, Eigenvalue, eigenvector and eigenspace, MPa, Eigen value problem

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