Civil Engineering Reference
In-Depth Information
6.2.2.2 Solution Algorithm
To develop an efficient solution algorithm, the first two equilibrium equations 1 and 2
are summed and subtracted to obtain the following two equations, which are used as the
convergence criteria for the solution procedure:
−
σ
2
1
ρ
f
+
ρ
t
f
t
=
(
σ
+
σ
t
)
+
σ
20
−
σ
2
cos 2
1
12
sin 2
ρ
f
−
ρ
t
f
t
=
(
σ
−
σ
t
)
−
σ
α
1
+
2
τ
α
1
21
The iterative procedure of FA-STM is illustrated by the flow chart in Figure 6.26. First
select a ¯
γ
12
. After completing the DO-loops in the
flow chart using Equations
20
-
21
, the shear stress
ε
2
value and assume two values for ¯
ε
1
and
τ
t
and the shear strain
γ
t
for a selected
value of
ε
2
are calculated from equations
3
and
6
a
, respectively. By selecting a series of ¯
ε
2
values, we can plot the
τ
t
−
γ
t
curves up to the peak point. Curves relating any two variables
can similarly be plotted.
6.2.3 Example Problem 6.2
6.2.3.1 Problem Statement
45
◦
, has been tested by Pang and
Hsu (1995). This 2-D element will be used to illustrate the step-by-step calculation procedure
of FA-STM. The properties of concrete and steel and the loadings are as follows:
A RC 2-D element B2, as shown in Figure 6.2(b) with
α
1
=
Longitudinal steel:
ρ
=
1.789%,
f
y
=
446.5 MPa, and
E
s
=
200 000 MPa.
Transverse steel:
ρ
t
=
1.193%,
f
ty
=
462.6 MPa, and
E
s
=
192 400 MPa.
Properties of concrete:
f
c
=
44.1 MPa and
ε
o
=
0.00235.
Applied stresses:
σ
=
0MPa,
σ
t
=
0 MPa, and
τ
t
increased until failure.
Notice that the
ρ
/ρ
t
ratio (
=
1.5) in this panel is not far from unity, Therefore, the contribution
c
of concrete shear stress,
τ
12
, will not be large.
Analyze the behavior of this 2-D element B2 by the fixed angle softened truss model (FA-
STM).Plotthe
c
2
c
τ
t
−
γ
t
curve, the
σ
−
ε
2
curve, the
¯
τ
12
−
γ
12
curve, the
f
−
ε
curve,
the
f
−
ε
¯
curve, the
f
t
−
ε
t
curve, and the
f
t
−
ε
t
curve. Compare these curves with those
¯
predicted by SMM.
6.2.3.2 Solution
The calculations according to the flow chart in Figure 6.26 are best done by computer, because
the procedure involves a nested DO-loop. The calculations were terminated when the error is
less than 0.1% and when the concrete compressive strain
ε
2
reaches a limiting value
ε
lim
of
−
0.005.
The computer-calculated results are recorded in Table 6.2 for three critical points: point
1 at first yield; point 2 at second yield; point 3 at the peak strength of
τ
t
. For comparison,
the SMM-predicted results are also given in Table 6.2. The three critical points are labeled as
point 1
∗
at first yield, point 2
∗
at second yield, point 3
∗
at the peak strength of
τ
t
.
