Civil Engineering Reference
In-Depth Information
α
r
=
60
◦
and
ε
d
=
Figure 5.8
Strain condition when
0
ε
/ε
y
and the cracking strain ratio
ε
r
/ε
y
are also plotted in
The longitudinal steel strain ratio
Figure 5.7 as a function of the angle
α
r
according to Equations (5.65) and (5.66), respectively.
For each of these two equations a range of
ε
d
/ε
y
ratios from 0 to
−
0.25 is given. Again, the
45
◦
, Figure 5.7 gives
effect of the
ε
d
/ε
y
ratio is shown to be small. When
α
r
=
ε
=
ε
y
, and
α
r
is decreased to 30
◦
,
ε
r
=
2
ε
y
to 2.25
ε
y
. When
ε
increases to the range of 3
ε
y
to 3.5
ε
y
, and
ε
r
increases rapidly to the range of 4
ε
y
to 4.75
ε
y
. These strains increase even faster when
α
r
is further decreased.
A strain condition is shown in Figure 5.8 when
60
◦
α
r
=
and
ε
d
=
0. At the first yielding
of the transverse steel (Figure 5.8a) the longitudinal steel strain
ε
is 0.33
ε
y
, and the cracking
strain
ε
y
. These values can be obtained from the two points indicated by (a) in Figure
5.7. When straining increases to the stage of yielding in the longitudinal steel (Figure 5.8b) the
transverse steel strain
ε
r
is 1.33
ε
t
is 3
ε
y
, and the cracking strain
ε
r
is 4
ε
y
. These values can be obtained
from the two points indicated by (b) in Figure 5.7.
The two solid curves for
0 (Figure 5.7) was first
presented by Thurlimann (1979). The trend of these two curves shows that the cracking strain
ratio
ε
r
/ε
y
in the special case of
ε
d
/ε
y
=
ε
r
/ε
y
increases very rapidly after the first yield of steel, when the angle
α
r
moves away
from 45
◦
. It was, therefore, concluded that the limitation of 30
◦
<α
r
<
60
◦
is quite valid from
the viewpoint of crack control.
5.3 Mohr Compatibility Truss Model (MCTM)
5.3.1 Basic Principles of MCTM
The Bernoulli compatibility truss model has been applied to flexural members in Chapter 3.
This bending model satisfies rigorously Navier's three principles of mechanics of materials,
namely, the parallel stress equilibrium condition, the Bernoulli linear compatibility condition
and the uniaxial constitutive laws. The linear constitutive law (i.e. Hooke's law) is used in
Section 3.1, and the nonlinear constitutive law in Section 3.2.
In this chapter we will be dealing with RC 2-D elements. A rigorous analysis of such 2-D
elements should also satisfy Navier's three principles of mechanics of materials. In this case of
2-D elements, Navier's principles should consist of the 2-D equilibrium condition, the Mohr
circular compatibility condition and the 2-D or biaxial constitutive laws.
However, if the 2-D (or biaxial) constitutive law is replaced by a linear 1-D (or uniaxial)
constitutive law, the theory is greatly simplified. This simple model, which is called the
Mohr
