Civil Engineering Reference
In-Depth Information
A method of design based on crack control was pioneered by Gupta (1981, 1984), using
the compatibility equations, (5.56) and (5.57), as well as the equilibrium equations, (5.21) and
(5.22). Assuming
ε
d
=
0,
f
=
E
s
ε
,
f
t
=
E
s
ε
t
, and selecting an
α
r
angle, we can determine
the steel ratios (
ρ
t
) in the longitudinal and transverse directions.
Gupta's design method produces good approximate solutions. However, this method is not
rigorous because the diagonal concrete stress
ρ
,
σ
d
in the equilibrium equations and the diagonal
concrete strain
ε
d
in the compatibility equations do not have a realistic relationship based on
the constitutive law of concrete. A rigorous solution which satisfies Navier's three principles
(equilibrium, compatibility and constitutive laws of materials) will be given by the Mohr
compatibility truss model in Section 5.3.
5.2.4.2 Cracking Condition at Yielding of Longitudinal Steel
The strain compatibility equations can also be used for the analysis of cracking condition at
the yielding of steel. Two cases should be investigated, namely, the yielding of the longitudinal
steel and the yielding of the transverse steel.
As shown in Section 5.2.1, the first two compatibility equations, (5.41) and (5.42), involve
four normal strains
ε
,
ε
t
,
ε
r
and
ε
d
, plus the angle
α
r
. In this analysis, the longitudinal steel
strain is given as
ε
=
ε
y
, and the principal compressive strain
ε
d
is considered a small given
value. Our aim is to express the transverse steel strain
ε
t
and the cracking strain
ε
r
as a function
of
ε
,
α
r
.
The transverse steel strain
ε
d
and the angle
ε
t
is related to the longitudinal steel strain
ε
by Equation (5.49).
Rearranging Equation (5.49) to express
ε
t
gives
ε
−
ε
d
)tan
2
ε
t
=
ε
d
+
(
α
r
(5.59)
Substituting
ε
t
from Equation (5.59) into Equation (5.45) provides the equation for
ε
r
:
ε
−
ε
d
)tan
2
ε
r
=
ε
+
(
α
r
(5.60)
Dividing Equations (5.59) and (5.60) by
ε
y
and setting
ε
=
ε
y
we obtain the nondimensional
equations for
ε
t
and
ε
r
as follows:
1
tan
2
ε
t
ε
y
=
ε
d
ε
y
+
−
ε
d
ε
y
α
r
(5.61)
1
tan
2
ε
r
ε
y
=
−
ε
d
ε
y
1
+
α
r
(5.62)
The transverse steel strain ratio
ε
t
/ε
y
and the cracking strain ratio
ε
r
/ε
y
areplottedinFigure
5.7 as a function of the angle
α
r
according to Equations (5.61) and (5.62), respectively. For
each equation a range of
ε
d
/ε
y
ratios from 0 to
−
0.25 is given. It can be seen that the effect
45
◦
, Figure 5.7 gives
of the
ε
d
/ε
y
ratio is small. When
α
r
=
ε
t
=
ε
y
and
ε
r
=
2
ε
y
to 2.25
ε
y
.
α
r
is increased to 60
◦
,
When
ε
t
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 increased.
