Civil Engineering Reference
In-Depth Information
(a)
(b)
B
1
C
1
B
2
C
2
m
V
D
1
D
2
V
cr
A
2
A
1
O
τ
O
θ
-V
cr
(
m
t
, θ
t
)
(
V
t
, τ
t
)
Figure 4.15
Loading branches: (a) Bending behavior; (b) Shear behavior.
(a)
(b)
V
m
E
1
E
2
G
2
G
1
V
cr
J
2
H
2
τ
O
θ
O
F
2
F
1
H
1
-V
cr
(
m
t
, θ
t
)
(
V
t
, τ
t
)
Figure 4.16
Unloading branches: (a) Bending behavior; (b) Shear behavior.
place. While for the shear behavior, the tangent stiffness for pre-yield cracked unloading
branches E
2
F
2
can be calculated by
=
k
s
−
τ
−
τ
cr
τ
y
−
τ
cr
E
2
F
2
s
α
ð
k
s
−
k
s
2
Þ
ð4
:
104Þ
where
k
s
is the slope connecting origin to the crack point as shown in Figure 4.16(b);
k
s
2
is
the slope connecting the yield point to the crack point in the opposite side;
τ
is the maximum
deformation attained in the loading direction and is also where unloading takes place. However,
the tangent stiffness
α
s
for the post-yield unloading branch G
2
H
2
should be computed accord-
ing to components above and below the cracking shear capacity
V
cr
. Then, the tangent stiffness
α
s
above the cracking shear level is given by Eq. (4.105), and below the cracking shear level by
Eq. (4.106). The tangent stiffnesses
α
00
f
and
α
00
s
of the RH and SH behaviors can be calculated
according to Eq. (4.5).
