Civil Engineering Reference
In-Depth Information
Similarly
1
2
(
ε
I
+
1
2
(
ε
I
−
ε
b
=
+
+
ε
II
)
ε
II
) cos 2(
θ
α
)
(14.28)
and
1
2
(
ε
I
+
1
2
(
ε
I
−
ε
c
=
ε
II
)
+
ε
II
) cos 2(
θ
+
α
+
β
)
(14.29)
Therefore if
ε
a
,
ε
b
and
ε
c
are measured in given directions, i.e. given angles
α
and
β
,
then
ε
I
,
ε
II
and
θ
are the only unknowns in Eqs (14.27), (14.28) and (14.29).
Having determined the principal strains we obtain the principal stresses using
relationships derived in Section 7.8. Thus
1
E
(
σ
I
−
ε
I
=
νσ
II
)
(14.30)
and
1
E
(
σ
II
−
ε
II
=
νσ
I
)
(14.31)
Solving Eqs (14.30) and (14.31) for
σ
I
and
σ
II
we have
E
σ
I
=
ν
2
(
ε
I
+
νε
II
)
(14.32)
−
1
and
E
σ
II
=
ν
2
(
ε
II
+
νε
I
)
(14.33)
1
−
Fora45
◦
45
◦
and the principal strains may be obtained using the
geometry of Mohr's circle of strain. Suppose that the arm 'a' of the rosette is inclined
at some unknown angle
θ
to the maximum principal strain as in Fig. 14.16(b). Then
=
=
rosette
α
β
Q(
ε
a
)
2
u
C
NM
O
ε
II
ε
I
ε
90
°
P(
ε
b
)
F
IGURE
14.17
Mohr's circle of strain for a
45
◦
strain gauge rosette
R(
ε
c
)
