3.17 Numerical Examples
77
3.17 Numerical Examples
Example 3.1 A sheet of metal is deformed uniformly in its own plane that the strain
components related to a set of axes xy are.
ε x = −200 × 10
−6
ε y = 1000 × 10
−6
γ xx = 900 × 10
−6
(a) Find the strain components associated with a set of axes x
y
inclined at an
angle of 30° clockwise to the xy set as shown in Fig. 3.8. Also, find the principal
strains and the direction of the axes on which they act.
Solution: (a)
The transformation equations for strains similar to that for stresses can be written as
below:
ε x =
ε x +ε y
2
+
ε x −ε y
2
cos 2θ +
γ xy
2
sin 2θ
ε y =
ε x +ε y
2
−
ε x −ε y
2
cos 2θ −
γ xy
2
sin 2θ
γ x y
2
= −
ε x −ε y
2
sin 2θ +
γ xy
2
cos 2θ
Using Eq. (3.19), we find
2θ = tan
−1
450
600
= 36.8
◦
Radius of Mohr’s = R =
(600) 2 + (450) 2 = 750.
Therefore,
Fig. 3.8 Transformed
co-ordinate system
77
3.17 Numerical Examples
Example 3.1 A sheet of metal is deformed uniformly in its own plane that the strain
components related to a set of axes xy are.
ε x = −200 × 10
−6
ε y = 1000 × 10
−6
γ xx = 900 × 10
−6
(a) Find the strain components associated with a set of axes x
y
inclined at an
angle of 30° clockwise to the xy set as shown in Fig. 3.8. Also, find the principal
strains and the direction of the axes on which they act.
Solution: (a)
The transformation equations for strains similar to that for stresses can be written as
below:
ε x =
ε x +ε y
2
+
ε x −ε y
2
cos 2θ +
γ xy
2
sin 2θ
ε y =
ε x +ε y
2
−
ε x −ε y
2
cos 2θ −
γ xy
2
sin 2θ
γ x y
2
= −
ε x −ε y
2
sin 2θ +
γ xy
2
cos 2θ
Using Eq. (3.19), we find
2θ = tan
−1
450
600
= 36.8
◦
Radius of Mohr’s = R =
(600) 2 + (450) 2 = 750.
Therefore,
Fig. 3.8 Transformed
co-ordinate system