Civil Engineering Reference
In-Depth Information
Figure 4.5
Geometric relationship for trignometric function
Substituting sin2
α
1
and cos2
α
1
from Equations (4.26) and (4.27) into Equations (4.13) and
(4.14) results in:
σ
−
σ
t
2
2
σ
+
σ
t
2
2
t
σ
1
=
+
+
τ
(4.28)
σ
−
σ
t
2
2
σ
+
σ
t
2
2
t
σ
2
=
−
+
τ
(4.29)
The expressions for these two principal stresses in Equations (4.28) and (4.29) can be
combined into one equation:
σ
−
σ
t
2
2
σ
1
,
2
−
σ
+
σ
t
2
2
=±
+
τ
t
=±
r
(4.30)
τ
12
equal
to zero and taking the square root. The square root term in Equation (4.30) is the radius
r
of
the Mohr circle.
Equations (4.28) and (4.29) can be expressed graphically by the Mohr circle in Figure
4.6(c). For this example we choose an element that is subjected to biaxial tension and shear
(Figure 4.6a). The principal tensile stress
Equation (4.30) can actually be obtained directly from Equation (4.22) by setting
σ
1
is much larger, in magnitude, than the principal
compressive stress
σ
2
(Figure 4.6b).
σ
1
and
σ
2
are represented by the two points C and D lying
on the
σ
-axis of the circle. It can be seen that
σ
1
is the sum of the average stress (
σ
+
σ
t
)/2
and the radius
r
, while
σ
2
is the difference of these two terms.
4.2 Strains in 2-D Elements
4.2.1 Strain Transformation
In Section 4.1.1 we have studied the principle of transformation for stresses. This same
principle will now be applied to strains. A strain is defined as a displacement per unit length.
The definitions of strains
t
coordinate are illustrated in Figure
4.7(a) and (b). They are indicated as positive, using the basic sign convention described in
Section 4.1.1. The strain
ε
,
ε
t
and
γ
t
(or
γ
t
)inthe
−
ε
indicated in Figure 4.7(a) is positive, because both the displacement
