Civil Engineering Reference
In-Depth Information
Failure modes diagram for example problem 5.4 (proportional loading,
S
=−
α
r
=
30
◦
)
Figure 5.28
1,
3.
2-D element partially under-reinforced in t
-
direction.
Transverse steel yields before the
crushing of concrete. Longitudinal steel does not yield.
4.
Over-reinforced 2-D element.
Concrete crushes before the yielding of steel in both
directions.
These four modes of failure can be divided graphically by
failure modes diagrams
, Figure
5.28. In this diagram the percentage of longitudinal steel,
ρ
, and the percentage of transverse
steel
ρ
t
are represented by the horizontal axis and the vertical axis, respectively. Figure 5.28
is constructed for Example Problem 5.4, in which a 2-D element is subjected to proportional
loading defined by the normalized Mohr circle in Figure 5.25. This proportional loading has
a pair of principal stress variables of
S
30
◦
=−
1 and
α
1
=
and a set of
m
-coefficients:
m
=
0.5,
m
t
=−
0.866. Such failure modes diagrams for proportional loading were
first constructed by Han and Mau (1988).
To understand Figure 5.28, we must first study the
equal strain condition
for steel in Section
5.4.7.1 and the
balanced condition
of steel and concrete in Section 5.4.7.2. After the equal
strain line OA and the balanced point B are located, the boundaries for the four failure modes
are constructed in Section 5.4.7.3.
0.5 and
m
t
=
