Civil Engineering Reference
In-Depth Information
Figure 5.18
Stresses (MPa) and strains for
ε
d
=−
0
.
0002 (close to first yield) in example problem 5.3
It is interesting to note that
γ
t
/
2 can be obtained without knowing the angle
α
r
by using
Equation (5.52), which is directly a simple function of
ε
,
ε
t
and
ε
d
,
(
γ
t
2
=±
ε
−
ε
d
)(
ε
t
−
ε
d
)
(0
=±
.
00155
+
0
.
0002)(0
.
00208
+
0
.
0002)
=
0
.
00200
The stresses in the mild steel and prestressing steel can be calculated from the strains using
the stress-strain relationships:
Equation 9
a
f
=
E
s
ε
=
200 000 (0
.
00155)
=
310 MPa
Equation 10
a
f
t
=
413 MPa (just yielded)
Equation 11
a
f
p
=
E
ps
(
ε
dec
+
ε
)
=
200 000(0
.
005
+
0
.
00155)
=
1310 MPa
f
tp
=
Equation 12
a
0
Mohr's circles for stresses on the concrete element, in the steel grid, and on the PC element,
as well as Mohr's circle for strains, are all plotted in Figure 5.18 for the selected strain of
ε
d
=
−
0.0002. In order to plot these Mohr's circles, the following additional stresses are calculated:
ρ
f
+
ρ
p
f
p
=
0
.
012 (310)
+
0
.
003 (1310)
=
7
.
65 MPa
ρ
t
f
t
=
0
.
012 (413)
=
4
.
96 MPa
σ
−
ρ
f
−
ρ
p
f
p
=
3
.
44
−
7
.
65
=−
4
.
21 MPa
σ
t
−
ρ
t
f
t
=
1
.
72
−
4
.
96
=−
3
.
24 MPa
