Civil Engineering Reference
In-Depth Information
Figure 5.11
Stress condition in prestressed concrete
Figure 5.11. Detailed derivation of these equations can proceed in the same manner as for
reinforced concrete. If the prestressing steel is absent, i.e.
ρ
p
=
ρ
tp
=
0, these equations, of
course, degenerate into those for reinforced concrete.
Similar to the bending theory in Chapter 3, it is very convenient to neglect the tensile
stress of concrete, i.e.
σ
r
=
0, in the rotating angle shear theory. As a result, the stress-strain
relationship of concrete in tension becomes irrelevant.
5.4.2 Summary of Equations
The three stress equilibrium equations for a PC 2-D element and the three strain compatibility
equations for the corresponding element are first summarized. The three equilibrium equations
are taken from the first type of expression in Section 5.1.2, and the three compatibility equations
are taken from the transformation expression in Section 5.2.1.
Equilibrium equations
σ
−
ρ
f
−
ρ
p
f
p
=
σ
d
sin
2
α
r
(5.94) or
1
σ
t
−
ρ
t
f
t
−
ρ
tp
f
tp
=
σ
d
cos
2
α
r
(5.95) or
2
τ
t
=
(
−
σ
d
)sin
α
r
cos
α
r
(5.96) or
3
where
ρ
p
,
ρ
tp
=
prestressing steel ratios in the
- and
t
-directions, respectively,
f
p
,
f
tp
=
stresses in prestressing steel in the
- and
t
-directions, respectively.
Compatibility equations
ε
=
ε
r
cos
2
α
r
+
ε
d
sin
2
α
r
(5.97) or
4
=
ε
r
sin
2
α
r
+
ε
d
cos
2
ε
t
α
r
(5.98) or
5
γ
t
2
=
(
ε
r
−
ε
d
)sin
α
r
cos
α
r
(5.99) or
6
