Civil Engineering Reference
In-Depth Information
Figure 5.21
Relationship between applied stresses and principal stress variables
Figure 5.21(b). These three principal stress variables are defined as:
σ
1
=
larger principal stress, algebraically, in the 1-direction - always positive;
=
σ
2
/σ
1
, ratio of smaller to larger principal stresses - positive when
σ
2
is tension, negative
S
when
σ
2
is compression;
α
1
=
orientation angle, or angle between the direction of larger principal stress (1-axis) and
the longitudinal axis (
-axis).
When an element is subjected to a
proportional 2-D loading
, the larger principal stress
σ
1
in the 1
−
2 coordinate increases while the other two variables
S
and
α
1
remain constant.
The set of proportional stresses
σ
,
σ
t
and
τ
t
in the
−
t
coordinate can also be defined in
terms of the principal stress
σ
1
as follows:
σ
=
m
σ
1
(5.115)
σ
t
=
m
t
σ
1
(5.116)
τ
t
=
m
t
σ
1
(5.117)
The set of three coefficients
m
,
m
t
and
m
t
in the
−
t
coordinate should remain constant
under proportional loading.
Take, for example, a given set of applied proportional stresses (
σ
=
1.25 MPa,
σ
t
=−
0.25
MPa and
τ
t
=
1.30 MPa) shown in Figure 5.22(a). Then the set of principal stress variables,
σ
1
,
S
and
α
1
, can be calculated as follows:
σ
−
σ
t
2
1
2
2
σ
+
σ
t
2
1
.
25
−
0
.
25
.
25
+
0
.
25
2
σ
1
=
+
+
τ
t
=
+
+
1
.
30
2
2
2
=
0
.
50
+
1
.
50
=
2
.
00 MPa
σ
2
=
0
.
50
−
1
.
50
=−
1
.
00 MPa
σ
2
σ
1
=
−
1
.
00
S
=
00
=−
0
.
5
2
.
