Civil Engineering Reference
In-Depth Information
Substituting Equation (5.111) into Equation 2 results in
σ
t
−
ρ
t
f
t
−
ρ
tp
f
tp
ε
t
=
ε
r
+
ε
r
−
ε
d
−
σ
d
(5.112) or
14
ε
t
,
f
t
, and
f
tp
in Equation
14
can be solved simultaneously with the two
stress-strain relationships of Equations
10
and
12
for transverse steel.
The variables
5.4.3.4
σ
d
The following two compatibility equations are more convenient to use in the solution procedure
described in Section 5.4.3.5. From Equations (5.45) and (5.49) we have:
ε
r
and
α
r
as Functions of
ε
,
ε
t
and
ε
r
=
ε
+
ε
t
−
ε
d
(5.113) or
15
ε
t
−
ε
d
ε
−
ε
d
tan
2
α
r
=
(5.114) or
16
5.4.3.5 Solution Procedures
A set of solution procedures is proposed, as shown in the flow chart of Figure 5.16. The
procedures are described as follows:
Step 1: Select a value of strain in the
d
-direction
ε
d
.
Step 2: Assume a value of strain in the
r
-direction
ε
r
.
Step 3: Calculate the softened coefficient
ζ
and the concrete stresses
σ
d
from Equations
8
and
7
, respectively.
Step 4: Solve the strains and stresses in the longitudinal steel
ε
,
f
and
f
p
from Equations
13
,
9
and
11
, and those in the transverse steel
ε
t
,
f
t
, and
f
tp
from Equations
14
,
10
,
and
12
.
Step 5: Calculate the strain
ε
r
=
ε
+
ε
t
−
ε
d
from Equation 15 .If
ε
r
is the same as assumed,
the values obtained for all the strains are correct. If
ε
r
is not the same as assumed, then
ε
r
is assumed, and steps (3) to (5) are repeated.
Step 6: Calculate the angle
another value of
α
r
, the shear stress
τ
t
and the shear strain
γ
t
from Equations 16 ,
3 , and 6 , respectively. This will provide one point on the
τ
t
−
γ
t
curve.
Step 7: Select another value of
ε
d
and repeat steps (2) to (6). Calculation for a series of
ε
d
values will provide the whole
τ
t
−
γ
t
curve.
The above solution procedures have two distinct advantages. First, the variable angle
α
r
does
not appear in the iteration process from Step 2 to Step 5. Second, the calculation of
ε
t
in
Step 4 can easily accommodate the nonlinear stress-strain relationships of reinforcing steel,
including those for prestressing strands. These advantages were derived from an understanding
of the three characteristics (Section 5.4.3.1) of the twelve governing equations. Steps 1 to 3
are proposed because of characteristic 2. Steps 4 and 5 are the results of characteristic 3, and
Step 6 is possible based on characteristic 1.
ε
and
