Civil Engineering Reference
In-Depth Information
is
rm
l
A
1
A
rm
−1
N
=
ε
E
c
A
1
1
+
−
1
(with
ε
i
≤
ε
≤
ε
i
+ 1
)
(11.17)
(
i
−
1)
s
rm
l
A
1
A
rm
−1
N
r
i
=
ε
i
E
c
A
1
1
+
−
1
(11.18)
where
A
1
and
A
2
=
areas of transformed sections in non-cracked and in fully
cracked states;
A
1
=
A
c
(1
+
αρ
);
A
2
=
A
c
αρ
;
α
=
E
s
/
E
c
;
A
c
=
area of
concrete;
ρ
=
A
s
/
A
c
.
A
rm
=
mean transformed cross-section area.
N
=
axial normal force.
N
r
i
=
value of
N
just before formation of the
i
th crack.
s
rm
=
mean crack spacing.
ε
=
D
/
l
; where
D
is imposed displacement.
ε
i
=
D
i
/
l
; where
D
i
is the imposed displacement at which the
i
th crack
is formed.
The transformed section area
A
1
=
112.6 in
2
;
A
2
=
4.61 in
2
. Using
ζ
=
0.5,
Equation (11.16) gives
A
rm
8.86 in
2
.
The number of cracks at crack stabilization,
n
=
=
l
/
s
rm
=
4 cracks.
Equations (11.12) and (11.14) give:
N
r1
=
39.4 kip;
ε
1
=
84 × 10
−6
. Substitut-
ing the value of
ε
1
in Equation (11.17) gives
N
=
10.0 kip; this is the lower
ordinate plotted for
ε
=
ε
1
.
2, 3 and 4 and solving Equations (11.11), (11.12) and (11.18) for
ε
i
and substitution of this value in Equations (11.17) and (11.18) give all the
values required for plotting the graph in Fig. 11.3. The following is a list of
the values of
Setting
i
=
ε
i
and the corresponding ordinates for
i
=
1, 2, . . . , 4:
Crack number
i
i
N
r
i
(kip)
N
(lesser ordinate, kip)
1
84×10
−
6
39.4
10.0
2
362×10
−
6
43.0
24.6
3
700×10
−
6
47.6
33.4
4
1119 × 10
−
6
53.4
41.1
Discussion of results
If the same example is analysed with a reduced value of the steel area,
A
s
, the
vertical drops in the
N
-value at each crack formation will be larger and the
degradation of the slope of the
N
-
graph will be faster with the successive
crack formations. Furthermore, the value given for
s
rm
should be increased
because of the reduction in
A
s
(see Appendix E). As a result, the number of
cracks will be smaller and the cracks will be wider.
When the steel ratio
ε
ρ
=
A
s
/
A
c
is reduced below a limiting value
ρ
min, y
the
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