Civil Engineering Reference
In-Depth Information
The solution of the quadratic equation is
B
4
SC
B
2
1
2
S
σ
1
=
±
−
(5.133) or
17
It should be noted that the symbol
S
in Equation (5.130) is the same as the definition of
S
=
σ
2
/σ
1
. This can be easily proven by inserting Equations (5.121)-(5.123) into Equation
(5.130). Equation 17 will be used in the solution procedure in Section 5.4.5.3, and in the
Example Problem 5.4 (Section 5.4.6).
In the case of a nonprestressed element (
ρ
p
=
ρ
tp
=
0), and equal yield strengths of mild
steel in both directions (
f
=
=
f
t
f
y
), Equation (5.133) can be simplified to the following
form:
(
m
ρ
t
+
ρ
ρ
t
(
m
ρ
t
+
f
y
2
S
m
2
σ
1
=
m
t
ρ
)
±
m
t
ρ
)
2
−
4(m
m
t
−
t
)
(5.134)
Equation (5.134) was first derived by Han and Mau (1988). It is valid when both the longi-
tudinal and the transverse steel are assumed to yield according to the equilibrium (plasticity)
truss model.
5.4.5.3 Method of Solution
The twelve equations,
3
,
6
, and
7
-
16
, which have been successfully used to solve the case
of 2-D elements under sequential loading (with constant normal stresses) in Section 5.4.4, can
also be applied to the case of elements subjected to proportional loadings. Minor modifications,
however, need to be made in Equations
14
and
15
. Equation
14
for the longitudinal steel
strains
ε
is still valid for proportional loading, except that the applied longitudinal stress
σ
should be replaced by
m
σ
1
:
m
σ
1
−
ρ
f
−
ρ
p
f
p
ε
=
ε
r
+
ε
r
−
ε
d
−
σ
d
(5.135) or
13
P
The letter P in the equation number
13
P
indicates that it is derived specifically for propor-
tional loading. Similarly, Equation
14
P
for the transverse steel strain
ε
t
is still valid, except
that the applied transverse stresses
σ
t
should be replaced by
m
σ
1
:
m
t
σ
1
−
ρ
t
f
t
−
ρ
tp
f
tp
ε
t
=
ε
r
+
ε
r
−
ε
d
−
σ
d
(5.136) or
14
P
σ
1
has been introduced.
Therefore, an additional equation is required. This new equation is furnished by Equation
(5.133) or
17
. Based on the thirteen equations,
3
,
6
, and
7
to
17
, a solution procedure
is proposed as shown by the flow chart in Figure 5.24. This procedure is somewhat more
complex than the flow chart shown in Figure 5.16 for sequential loading (with constant
normal stresses), because of the additional iteration cycles required by the variable
In Equations
13
P
and
14
P
we notice that a new unknown variable
σ
1
for the
proportional loading. For this trial-and-error procedure with a nested DO-loop for
σ
1
,itwould
be convenient to write a computer program and to take advantage of the power and speed of
a computer.
