Civil Engineering Reference
In-Depth Information
Figure 6.12
Stresses and strains between two cracks
Rearranging Equation (6.57) gives:
1
ρ
σ
c
(
x
)
f
so
=
f
s
(
x
)
+
(6.58)
where
A
c
, the percentage of steel based on the net concrete section. Equation (6.58)
states that, at any section between the two cracks, the sum of the steel stress
f
s
(
x
) and the
concrete stress
ρ
=
A
s
/
σ
c
(
x
) divided by
ρ
must be equal to the steel stress at the cracked section
f
so
.
The steel strain ¯
ε
s
(
x
) and the concrete strain
ε
c
(
x
) are also sketched in Figure 6.12(c).
The steel strain ¯
ε
s
(
x
) decreases from a maximum at the crack to a minimum at the midpoint
between the two cracks. In contrast, the concrete strain
ε
c
(
x
) should be zero at the crack and
increases to a maximum at the midpoint. The difference between ¯
ε
c
(
x
) is caused
by the slip between the steel bar and the surrounding concrete. The slip results in the bond
stresses sketched in Figure 6.12(d) and the gaps that constitute the cracks.
Let us recall that the solution of the equilibrium and compatibility equations, (6.1)
-
(6.6),
requires the constitutive relationship between the
smeared (or average) tensile stress of con-
crete
ε
s
(
x
) and
c
1
σ
and the
smeared (or average) tensile strain of concrete
¯
ε
1
, in the principal 1-direction.
The smeared strain of concrete ¯
ε
1
should be measured along a length that traverses several
cracks. This smeared strain ¯
ε
1
is not the average value of the concrete strain
ε
c
(
x
)shown
in Figure 6.12(c), because the smeared strain ¯
ε
1
includes not only the strain of the con-
crete itself, but also the strain contributed by the crack widths. Hence, the smeared strain
¯
ε
1
must be obtained from averaging the steel strain ¯
ε
s
(
x
) along the steel bar, not
ε
c
(
x
). In
other words, the smeared strain ¯
ε
s
and the
smeared strain of concrete including the crack widths. With this understanding in mind the
smeared tensile strain of concrete
,¯
ε
1
represents both the smeared strain of steel ¯
ε
1
=
ε
s
, is obtained by averaging the strain ¯
¯
ε
s
(
x
)from
