Civil Engineering Reference
In-Depth Information
Based on the equilibrium (plasticity) truss model, the 2-D element is reinforced with 1.03%
of steel in both the longitudinal and the transverse directions (Example Problem 5.1). The
steel is designed to have the same grade in both directions
f
y
=
413 MPa. In the Mohr
compatibility truss model (Example Problem 5.2) the strength of concrete is assumed to be
f
c
=
27.6 MPa (4000 psi). In order to use the RA-STM, the constitutive laws of concrete and steel
are specified as follows: the compression stress-strain curve of softened concrete is expressed
by Equations 7
a
, 7
b
and 8 . The tensile strength of concrete is neglected (assume
f
ty
=
0);
and the stress-train curve of bare mild steel bars, Equations 9
a
and 9
b
(or 10
a
and 10
b
),
is adopted.
σ
r
=
5.4.6.2 Solution
For the given set of applied stresses the principal stress variables
σ
1
, S and
α
1
are:
σ
−
σ
t
2
2
2
2
σ
+
σ
t
2
2
.
13
−
2
.
13
.
13
+
2
.
13
2
σ
1
=
+
+
τ
t
=
+
+
3
.
69
2
2
2
=
0
+
4
.
26
=
4
.
26 MPa
σ
2
=
0
−
4
.
26
=−
4
.
26 MPa
σ
2
σ
1
=
−
.
4
26
=
26
=−
S
1
4
.
(
σ
−
σ
t
)
2
2
.
13
−
(
−
2
.
13)
cot 2
α
1
=
=
=
0
.
577
τ
t
2(3
.
69)
60
◦
,
30
◦
.
2
α
1
=
and
α
1
=
The three
m
-coefficients are:
σ
σ
1
=
2
.
13
m
=
26
=
0
.
5
4
.
σ
t
σ
1
=
−
2
.
13
m
t
=
=−
0
.
5
.
4
26
τ
t
σ
1
=
3
.
69
m
t
=
26
=
0
.
866
4
.
The normalized Mohr circle for this case of proportional loading is shown in Figure 5.25.
This normalized Mohr circle is characterized by
S
30
◦
in the 1
=−
1 and
α
1
=
−
2 coordinate,
and
m
=
0.5,
m
t
=−
0.5 and
m
t
=
−
0.886 in the
t
coordinate.
1. Select
0.000275 (for the first yield condition)
The stress and strain conditions at first yield have been given in Figure 5.10 (see Example
Problem 5.2, Section 5.3.4) based on the Mohr compatibility truss model. At this load stage
ε
d
ε
d
=−
=−
0.000265,
ε
r
=
0.00345 and
σ
1
=
3.65 MPa. In this softened truss model, however,
ε
d
should be slightly greater than
0.000265 (in an absolute sense), because of the slightly
nonlinear stress-strain relationship of concrete at low level of loading. After several cycles of
trial-and-error process we select the values of
−
ε
d
=−
0.000275, and assume
ε
r
=
0
.
00345
and
σ
1
=
3
.
65 MPa
