Civil Engineering Reference
In-Depth Information
4.3.4.3
V
c
in Fixed Angle Theory
In the case of a pure shear element with equilibrium at cracks, Equations (4.51)-(4.53) become:
c
12
2sin
c
2
sin
2
ρ
f
y
−
τ
α
1
cos
α
1
=−
σ
α
1
(4.72)
c
c
2
cos
2
ρ
t
f
ty
+
τ
12
2sin
α
1
cos
α
1
=−
σ
α
1
(4.73)
12
(cos
2
c
sin
2
c
2
τ
ty
−
τ
α
1
−
α
1
)
=−
σ
sin
α
1
cos
α
1
(4.74)
The following four steps are used to derive an explicit expression for
τ
ty
. First, the com-
c
2
pressive stress
σ
is eliminated from Equations (4.72) and (4.74) by multiplying Equation
(4.72) by cos
α
1
, and then subtracting the latter from the former.
After simplifying, the resulting equation is:
α
1
and Equation (4.74) by sin
c
12
)
τ
ty
+
τ
=
ρ
f
y
cot
α
1
(
(4.75)
Second, multiply Equation (4.73) by sin
α
1
and Equation (4.74) by cos
α
1
, and then subtract
the latter from the former. After simplifying, the resulting equation is:
c
12
)
(
τ
ty
−
τ
=
ρ
t
f
ty
tan
α
1
(4.76)
Third, multiplying Equations (4.75) and (4.76) and then simplifying gives:
(
c
τ
ty
=
τ
12
)
2
+
ρ
f
y
ρ
t
f
ty
(4.77)
Fourth, expanding Equation (4.77) into a Taylor series and taking the first two terms of the
series gives, to a good approximation:
2
ρ
f
y
ρ
t
f
ty
+
ρ
f
y
ρ
t
f
ty
12
)
2
c
(
τ
τ
ty
=
(4.78)
Equation (4.78) was first derived by Pang and Hsu (1996). The second term on the right-
hand side of Equation (4.78) is the 'contribution of steel' (
V
s
), the same as that derived in
Equation (4.71) for rotating angle theory. The first term is the 'contribution of concrete' (
V
c
).
It can be seen that
V
c
is contributed by
12
, the concrete shear stress along the cracks.
τ
4.3.4.4 Summary for
V
c
Comparison of Equation (4.78) for fixed angle theory and Equation (4.71) for rotating angle
theory shows clearly that the 'contribution of concrete' (
V
c
) is caused by defining the cracks
in the principal 1-2 coordinate of the externally applied stresses (or fixed angle). In the fixed
angle theory, a concrete strut is subjected not only to the axial compressive stress
c
σ
2
, but also
c
c
12
to a concrete shear stress
τ
12
, in the direction of the cracks. This concrete shear stress
τ
is
the source of the
V
c
term observed in tests.
When the cracks are assumed to be governed by the principal
r
d
coordinate of the
concrete element, a concrete strut is subjected only to an axial compressive stress
−
σ
d
.Inthe
rotating angle theory, the
r
d
coordinate rotates with increasing proportional loading in such
a way that the concrete shear stress
−
c
12
vanishes. As a result, the
V
c
term is always zero.
In summary, both the rotating angle theory and the fixed angle theory are useful. The fixed
angle theory is more accurate because it can predict the
V
c
term observed in tests. However, it
τ
