Civil Engineering Reference
In-Depth Information
(a)
(b)
m
V
V
y
B
2
C
2
B
1
C
1
m
y
m
cr
V
cr
A
2
A
1
k
f
θ
cr
k
S
D
1
D
2
O
O
θ
y
θ
u
θ
τ
cr
τ
y
τ
u
τ
Figure 4.7
Backbone curves of the flexural member: (a) Bending backbone curve; (b) Shear
backbone curve.
In addition, the critical points A
1
,A
2
,B
1
,B
2
, etc. in the primary curves can be derived accord-
ing to the lateral load
F
, for example
V
cr
=
F
cr
and
m
cr
=
V
cr
L
, which also means that the bend-
ing and shear effects must behave on the same type of branches for the SDOF system. In the
above equation,
x
is the total displacement; the initial elastic stiffness of the SDOF system
k
0
is
satisfied by
k
0
=
L
2
1
k
f
+
1
ð4
:
14Þ
k
s
k
f
=
3
EI
L
,
k
s
=
GA
L
ð4
:
15Þ
When the SDOF system is in the plastic domain and both the bending and shear behaviors are
governed by the hardening branches A
1
B
1
and A
2
B
2
, referring to Eq. (4.5), there are following
relationships:
m
=
Γθ
0
ðÞ=
α
0
f
θ
00
+
m
int
V
=
Ψτ
0
ðÞ=
α
0
s
τ
00
+
V
int
m
cr
<
m
<
m
y
V
cr
<
V
<
V
y
if
,
then
ð4
:
16Þ
where
α
0
f
and
α
0
s
are the stiffnesses of the local plastic mechanisms, RH and SH, on branches
A
1
B
1
and A
2
B
2
;
m
int
and
V
int
are the intercepts of the
m
-
θ
00
and
V
-
τ
00
relationships, respectively.
Once the moment
m
at the RH and the shear force
V
of the SDOF system exceed or equal to
the corresponding yielding capacities
m
y
and
V
y
, the bending and shear behaviors are repre-
sented by the yielding platforms B
1
C
1
and B
2
C
2
. The local plastic mechanisms shown in
Eq. (4.16) are still feasible by satisfying the conditions as:
m
≥
m
y
V
≥
V
y
ð4
:
17Þ
and
α
f
and
α
s
here are specified with smaller values (close to zero) to simulate the yielding
platform behaviors.
When the deformations
θ
and
τ
of the SDOF system exceed
θ
u
and
τ
u
, the bending and shear
behaviors are depicted by the softening branches C
1
D
1
and C
2
D
2
. The tangent stiffnesses
α
f
and
