Civil Engineering Reference
In-Depth Information
According to these definitions, the coefficients
m
,
m
t
and
m
t
have a physical meaning.
They are simply the applied stresses
σ
1
. These
normalized stresses can be represented by a normalized Mohr circle in Figure 5.22(d). The
values of the
m
-coefficients are given by the abscissas and ordinates of points A and B. The
normalized principal stresses of
σ
,
σ
t
and
τ
t
normalized by the principal stress
σ
1
is, of course, equal to unity, and the normalized principal
stress of
0.5. These two normalized principal stresses are indicated by
the points C and D, respectively. The 2
σ
2
is the ratio
S
=−
α
1
angle remains 60
◦
. In short, the normalized Mohr
circle characterizes a given set of proportional loading, either in the
−
t
coordinate (
m
,
m
t
and
m
t
) or in the principal 1
α
1
).
The relationship between the set of three coefficients (
m
,
m
t
and
m
t
)inthe
−
2 coordinate (
S
and
−
t
coor-
α
1
)inthe1
−
dinate and the set of two principal stress variables (
S
and
2 coordinate can be
obtained through the coordinate transformation relationship. Taking the inverse transformation
of Equation (4.7) and setting
τ
12
=
0, the set of stresses (
σ
,
σ
t
and
τ
t
)inthe
−
t
coordi-
nate can be expressed by the set of principal stresses (
σ
1
and
σ
2
)inthe1
−
2 coordinate as
follows:
σ
=
σ
1
cos
2
α
1
+
σ
2
sin
2
α
1
(5.118)
=
σ
1
sin
2
α
1
+
σ
2
cos
2
σ
t
α
1
(5.119)
τ
t
=
(
σ
1
−
σ
2
)sin
α
1
cos
α
1
(5.120)
σ
1
and the definitions of
m
-coefficients (Equations
5.115-5.117) into Equations (5.118)-(5.120), and then cancelling the principal stress
Substituting the definition of
σ
2
=
S
σ
1
result
in:
cos
2
S
sin
2
m
=
α
1
+
α
1
(5.121)
sin
2
S
cos
2
m
t
=
α
1
+
α
1
(5.122)
m
t
=
(1
−
S
)sin
α
1
cos
α
1
(5.123)
α
1
are given, then the three
m
-coefficients can be
calculated from Equations (5.121)-(5.123). Take, for example, the case of
S
If the two principal stress variables
S
and
α
1
=
=
0.5 and
30
◦
as shown in Figure 5.23(a):
sin 30
◦
=
cos 30
◦
=
0
.
50
0
.
866
sin
2
30
◦
=
cos
2
30
◦
=
sin 30
◦
cos 30
◦
=
0
.
25
0
.
75
0
.
433
cos
2
S
sin
2
m
=
α
1
+
α
1
=
0
.
75
+
0
.
50(0
.
25)
=
0
.
875
sin
2
S
cos
2
m
t
=
α
1
+
α
1
=
0
.
25
+
0
.
50(0
.
75)
=
0
.
625
m
t
=
(1
−
S
)sin
α
1
cos
α
1
=
(1
−
0
.
5)(0
.
433)
=
0
.
2165
These calculated
m
-coefficients are also shown in Figure 5.23(a), together with the normal-
ized Mohr circles.
In addition, Figure 5.23(b) and (c) give two more cases of
S
and
α
1
, their corresponding
m
-coefficients and their normalized Mohr circles. Comparison of the three cases illustrates
how the normalized Mohr circles are affected by
S
and
α
1
.
