Civil Engineering Reference
In-Depth Information
σ
=
σ
sin
2
α σ
+
cos
2
α ρ
+
f
+
ρ
f
(6.2)
t
d
r
t
t
tp tp
(6.3)
τ
= −
(
σ
+
σ
)sin cos
α
α
lti
d
r
(6.4)
T
= τ
(
2
0
A t
)
lti
d
where:
σ
l
, σ
t
, and τ
lti
are the three homogenized stress components of the com-
posite element (Figure 6.4a)
σ
d
and σ
r
are the concrete stresses in
d
- and
r
-directions, respectively
(Figure 6.4b, where
r
-direction is perpendicular to
d
-direction and
not shown)
α is the angle between
l
and
d
axes
ƒ
l
and ƒ
t
are the stresses in steel in the
l
- and
t
-directions, respectively
ƒ
lp
and ƒ
tp
are the stresses in the prestressing steel in the
l
- and
t
-directions,
respectively
ρ
l
and ρ
t
are the steel ratio in the
l
- and
t
-directions, respectively
ρ
lp
and ρ
tp
are the prestressing steel ratio in the
l
- and
t
-directions,
respectively
T
is the external torque
A
0
is the cross-sectional area bounded by the centerline of the shear
flow zone
t
d
is the shear flow zone thickness
It should be noted that, for a multicell box under pure torsion, σ
l
= σ
t
= σ
r
= 0
and, assuming a structural section has
N
cells (Figure 6.3), a set of simul-
taneous equations for cell
i
can be obtained.
τ
= −
σ
sin
α
cos
α
(6.5)
lti
di
i
i
T
= τ
(
2
0
A t
)
(6.6)
i
lti
i di
6.1.2.3
Compatibility equations
Similarly, the governing equations for compatibility condition were based
on the unified theory (Hsu 1993) and later extended by Fu and Yang (1996)
and Fu and Tang (2001). It should be noted that for a multicell box under
pure torsion, a set of simultaneous equations for cell
i
was simplified as
γ
lti
= −
(
ε
+
ε
)sin
α
cos
α
(6.7)
di
ri
i
i
2
p
A
0
i
θ
=
θ
=
γ
(6.8)
i
lti
2
0
i
ψ
i
=
sin 2
θ
α
i
(6.9)
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