Civil Engineering Reference
In-Depth Information
w
L
O
C
x
α
d
L
cot
α
r
h
α
r
b
a
c
α
r
d
T
T
h
r
A
B
q
y
(a) BOX SECTION SUBJECTED TO TORSION
w
L
C
O
x
α
d
α
r
L
cot
α
r
θ
L
D
A
B
s
y
(b) DEFORMATION OF TOP WALL
Figure 7.3
Bending of a concrete strut in the wall of a box section subjected to torsion
Imposing an axis
s
through OB along the direction of the diagonal concrete struts, the slope
of the curve OD can be obtained by differentiating
w
with respect to
s
. Utilizing the chain
rule, the slope d
w/
d
s
is:
d
d
s
=
∂w
d
x
d
s
+
∂w
d
y
d
s
=
(
θ
y
)sin
α
r
+
(
θ
x
) cos
α
r
(7.20)
∂
x
∂
y
The curvature of the concrete struts
ψ
is the second derivative of
w
with respect to
s
,
resulting in
d
2
s
=
∂
d
d
s
d
s
+
∂
d
d
s
d
2
w
d
x
d
y
d
s
=
(
θ
cos
α
r
)sin
α
r
+
(
θ
sin
α
r
) cos
α
r
(7.21)
∂
∂
x
y
Hence,
ψ
=
θ
sin 2
α
r
(7.22) or [9]
