Civil Engineering Reference
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+
T
τ
1
τ
2
τ
3
τ
2
τ
2
τ
3
τ
1
τ
3
τ
1
1
2
3
τ
1
τ
2
τ
3
t
d
1
h
1
t
d
2
h
3
Figure 6.3
Shear stresses in a multicell section.
6.1.2.2
Equilibrium equations
A prestressed concrete element, as shown in Figure 6.4a, is reinforced
orthogonally with longitudinal and transverse (prestressing or nonprestress-
ing) steel reinforcements. The applied stresses on the element have three
stress components, σ
l
, σ
t
, and τ
lt
. The longitudinal steels are arranged in the
l
l-direction (horizontal axis) with a uniform spacing of
s
l
. The transverse
steels are arranged in the
t
-direction (vertical axis) with a uniform spacing
of
s
as shown in Figure 6.4a. After cracking, the concrete is separated by
diagonal cracks into a series of concrete struts, as shown in Figure 6.4b. The
cracks are oriented at an angle α with respect to the
l
l-axis. The principal
stresses on the concrete strut itself are denoted as σ
d
and σ
r
. According to the
unified theory (Hsu 1993), after transformation, the governing equations
for equilibrium condition are shown as follows:
2
2
σ
=
σ
cos
α σ
+
sin
α ρ
+
f
+
ρ
f
(6.1)
l
d
r
l
l
lp lp
A
l
f
l
n
l
=
s
α
cos α
q
t
A
l
f
l
s
s
q
d
σ
d
σ
d
s
l
s
l
d
sin α
α
s
1
q
q
s
l
1
σ
d
α
l
q
l
A
l
f
l
s
l
n
l
=
σ
d
s
l
α
A
l
f
l
q
1
1
(a)
(b)
Figure 6.4
Equilibrium in element shear. (a) Shear element. (b) Truss element.
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