Civil Engineering Reference
In-Depth Information
Figure 4.9
Mohr circle for strains
respectively. The projections of these two displacements, AD and DE, on the 1-axis are
(
γ
t
/2)cos
α
1
(sin
α
1
) and (
γ
t
/2)sin
α
1
(cos
α
1
), respectively. The sum of these two displacement
projections, AE
=
(
γ
t
/2)2cos
α
1
sin
α
1
,isthestrain
ε
1
due to the shear strain
γ
t
, because it has
an original length OA of unity.
Summing the strain
ε
1
due to
ε
ε
t
(Figure 4.8a) and that due to
γ
t
(Figure 4.8b), the
and
γ
12
/2 in Equations (4.35)
and (4.36) can similarly be demonstrated by direct geometric relationships.
ε
1
is expressed by Equation (4.34). The expressions for
ε
2
and
total
4.2.3 Mohr Strain Circle
Referring to element A at the right-hand support of a typical reinforced concrete beam,
as shown in Figure 4.3, the deformation of this 2-D element is indicated in Figure 4.9(a),
and the transformation of strains in this element is illustrated by a Mohr strain circle in
Figure 4.9(b).
Recalling the double angle trigonometric relationships of Equations (4.9)-(4.12), and sub-
stituting these equations into Equations (4.34)-(4.36) results in:
ε
+
ε
t
2
+
ε
−
ε
t
2
α
1
+
γ
t
2
ε
1
=
cos 2
sin 2
α
1
(4.39)
ε
+
ε
t
2
−
ε
−
ε
t
2
α
1
−
γ
t
2
ε
2
=
cos 2
sin 2
α
1
(4.40)
γ
12
2
=−
ε
−
ε
t
α
1
+
γ
t
2
sin 2
cos 2
α
1
(4.41)
2
