Civil Engineering Reference
In-Depth Information
Adding Equations (4.20) and (4.21) results in
σ
1
,
2
−
σ
+
σ
t
2
2
σ
−
σ
t
2
2
12
2
r
2
+
τ
=
+
τ
=
(4.22)
t
Note that a term
r
2
is added to Equation (4.22). By equating both the left- and the right-hand
sides of Equation (4.22) to
r
2
, we can see that Equation (4.22) is the equation of a circle with a
radius
r
, centered on the
σ
+
σ
t
)/2 from the origin, as shown in Figure
4.2(a). The left-hand side of Equation (4.22) represents points C (
σ
-axis at a distance of (
σ
1
,
τ
12
) and D (
σ
2
,
−
τ
12
)
on the circle and the right-hand side represents points A (
σ
,
τ
t
) and B (
σ
t
,
−
τ
t
). The angle
between radius for C and radius for A is the angle
φ
. Angle
φ
does not appear in Equation
(4.22), and, therefore, can have any magnitude.
In summary, Figure 4.2(a) shows that the abscissas and ordinates for points C (
σ
1
,
τ
12
) and
D(
σ
2
,
−
τ
12
) on the circle can be expressed in terms of the abscissas and ordinates of points
A(
apart based on geometric (i.e. circular) relationship.
Notice that points C and D represent a set of stresses (
σ
,
τ
t
) and B (
σ
t
,
−
τ
t
) at an angle
φ
σ
1
,σ
2
,
τ
12
) in the 1-2 coordinate, and
points A and B represent a set of stresses (
σ
,
σ
t
,
τ
t
)inthe
−
t
coordinate. Therefore, the
first set of stresses (
σ
1
,
σ
2
,
τ
12
) can be expressed in terms of the second set of stresses (
σ
,
σ
t
,
τ
t
) by the three equations (4.16)-(4.18).
Now we can compare the stress transformation relationship, (Equations 4.13-4.15), to the
geometric (i.e. circular) relationship (Equations 4.16-4.18). Obviously, the two sets equations
are similar, but not identical. Two differences can be noted: (1) all the angles in the transfor-
mation equations are twice those in the geometric relationship; (2) all the signs of the angles
are opposite in the two set of equations.
In order to bring these two sets of equations into direct correspondence, we can eliminate
these two differences by taking
φ
=−
2
α
1
(4.23)
The resulting Mohr stress circle is shown in Figure 4.2(b).
In Mohr's graphical representation of stress transformation (Figure 4.2b), the angles in
Mohr's stress circle are always twice as great and measured in the opposite direction, compared
with the actual stress field in an element. Since the angle
α
1
in the
−
t
coordinate rotates
counter-clockwise, the angle 2
α
1
in the Mohr circle must rotate clockwise. In addition, one
must always remember that, although the algebraic analogy is precisely true, the geometric
analogy is not, due to the double angle relationship.
In the Mohr stress circle (Figure 4.2b), the point B presents some confusion with regard to
the negative sign of the shear stress
τ
t
. This can be explained by rotating the 1-2 axes by 90
◦
as shown in Figure 4.1(d). It can be seen that the shear stress
τ
12
on the top 1-face is positive
and pointing leftward. In contrast, the shear stress
τ
t
on the top
t
-face as shown in Figure
4.1(a) is positive and pointing rightward. Obviously, in the case of finding a shear stress at 90
◦
,
a positive shear stress
τ
12
in the rotating 1-2 coordinate must be treated as a negative shear
−
τ
12
stress
τ
t
in the stationary
t
coordinate. To be consistent with the sign convention of
τ
t
(calculated from stress transformation), we can define
as
τ
t
=−
τ
t
(4.24)
