Control of cracking - Concrete Structures: Stresses and Deformations - page 387

The value of M varies in the stage of crack formation as shown in Fig.
11.2(b). Just before formation of the second crack M 10.1 kN-m;

substituting this value in Equations (11.9) and (11.10) gives ( y CT

= M r2 =

0.9 d =

0.225 m):

1496 × 10 −6

ε s2 = w m = 0.45 mm (0.018 in).

ed limit by increasing the steel
area, A s . On the other hand, the mean crack width can become much larger if
A s is reduced below a minimum at which

The value of w m can be reduced to any speci fi

f y ; where f y is the yield
strength of the steel. The minimum value of the steel ratio required to avoid
this situation is discussed in Section 11.6.
This procedure can be employed to determine

σ s2 = E s ε s2 =

∆ T i for i = 2, 3, . . . , n .

Substitution of the value of ∆ T i in Equation (11.8) gives the larger of two M -

ordinates corresponding to T i , required to construct the graph in Fig.

11.2(b); Equation (11.7) gives the lesser ordinate. The values of

∆

∆ T 1 and

the two M -ordinates corresponding to the fi rst crack can be determined by

Equations (11.2), (11.4) and (11.7).
The results of the above analysis are plotted in Fig. 11.2(b); for com-
parison, the dashed line OB is included to represent the case when concrete in
tension is ignored. The values of

∆ T i and the corresponding ordinates are

listed below:

Crack number
M r i (larger ordinate kN-m)
M (lesser ordinate kN-m)

1
9.7
5.8

2
10.1
7.2

3
10.5
8.2

4
11.0
9.0

5
11.6
9.8

6
12.2
10.6

7
12.9
11.4

8
13.6
12.2

11.3.2 Example of a member subjected to axial force
(worked out in British units)
It is required to study the variation of N versus

D / l ) for a member of
length l subjected to an imposed end displacement D (Fig. 11.3) in the range 0
≤

ε ( =

D s / l with D s being the displacement at which stabilized
cracking occurs. Assume that yielding of the reinforcement does not occur in
this range. Consider average crack spacing s rm

ε ≤ ε s ; where ε s =

12 in (300 mm); the value of
the tensile strength of concrete at which successive cracks form is: 3

=

f ct i = f ct1 [1 + 350( ε i − ε 1 )] (11.11)

where f ct i is the tensile strength of concrete at the location of the i th crack; =

D i / l with D i being the imposed displacement at which the i th crack is formed.
The cross-section geometrical data are given in Fig. 11.3. Other data are:
f ct1

ε i

29 000 ksi (200 GPa).
The equations derived in Section 11.3.1 apply for a member subjected to an

= 0.35 ksi (2.4 MPa); E c = 4150 ksi (28.6 GPa); E s =