Civil Engineering Reference
In-Depth Information
stresses on the RC element. The cracking pattern based on the
r
−
d
coordinate after cracking
is shown in Figure 4.11(g).
In the general case of
ρ
f
=
ρ
t
f
t
, when the set of applied stresses
σ
,
σ
t
and
τ
t
is increased
proportionally, the
r
−
d
coordinate will deviate from the 1
−
2 coordinate. The deviation
angle is defined as
β
=
α
r
−
α
1
, which will increase until it reaches the ultimate strength. If
the RC element is reinforced with the same amounts of steel in the
- and
t
-directions, i.e.,
ρ
f
=
ρ
t
f
t
in Figure 4.11(c), the
r
−
d
coordinate will coincide with the 1
−
2 coordinate,
and the deviation angle
will become zero.
After cracking and in the subsequent cracking process under increasing proportional loading,
however, the direction of the subsequent cracks was observed to deviate from the 1
β
−
2
−
coordinate and to move toward the
r
d
coordinate. Tests showed that the observed deviation
will be less than the theoretical deviation. At ultimate load, the observed
β
-angle is about
one-half of the calculated
-angle.
In view of the fact that the direction of subsequent cracks occurs in between the 1-2
coordinate and the
r
β
d
coordinate, two types of theories have been developed, namely,
the rotating angle theories (Hsu, 1991a, 1993) and the fixed angle theories (Pang and Hsu,
1996; Hsu and Zhang, 1997; Hsu, 1996, 1998). The rotating angle theories are based on
the assumption that the direction of cracks is perpendicular to the principal tensile stress in
the concrete element, as shown in Figure 4.11(g). In other words, the direction of cracks is
governed by the
r
−
d
coordinate, and the derivations of all the equilibrium and compatibility
equations are based on the
r
−
d
coordinate.
In contrast, the fixed angle theories are based on the assumption that the direction of
subsequent cracks is perpendicular to the applied principal tensile stress, as shown in Figure
4.11(f). In the fixed angle theories, the direction of cracks is represented by the 1
−
2 coordinate
system, and the derivations of all the equilibrium and compatibility equations are based on the
1
−
−
2 coordinate.
The term fixed angle simply means that the angle
α
1
remains unchanged when the applied
stresses (
σ
,
σ
t
and
τ
t
) are increased proportionally. It does not mean that the applied stresses
(
α
1
angle, when the applied stresses are increased
in a nonproportional manner. Nor does the term imply that the observed crack angles are fixed
in the subsequent cracking process.
σ
,
σ
t
and
τ
t
) can not produce a different
4.3.2 Fixed Angle Theory
As in Section 4.1.1, Equation (4.7), we can derive the matrix
[
T
]
that transforms a set of
concrete stresses in the
c
t
c
−
t
coordinate (
σ
,
σ
and
τ
t
) into a set of concrete stresses in the
c
c
2
c
1-2 coordinate (
12
). The superscript
c
in all six stress symbols emphasizes that these
stresses are intended for the concrete element (Figure 4.11b). The transformation process in
Section 4.1.1 is shown in Figure 12(a), where the angle
σ
1
,
σ
and
τ
α
1
between the two coordinate systems
is positive and rotates counter-clockwise. Now, let us reverse the transformation process to
find the transformation matrix which transforms the stresses in the 1-2 coordinate (
c
c
2
σ
1
,
σ
and
c
c
c
c
τ
12
) into the stresses in the
−
t
coordinate (
σ
,
σ
and
τ
t
). This process is shown in Figure
t
4.12(b), where the angle
α
1
is negative and rotates clockwise.
To reverse the direction of a rotation, it is only necessary to change the angle
−
α
1
in
all the terms in the matrix of Equation (4.7). It can be seen that all the trigonometric functions
remain the same, except that sin(
α
1
into
−
α
1
)
=−
sin
α
1
. Changing the sign of the four terms with
