Non-linear material behaviour - Concrete Structures for Wind Turbines - page 89

The s s e s curves are parallel and therefore E s,eff ¼ E s .

We have the following situation when crack formation is complete:

s s,eff e s and

1 : 3 s sr

g C ¼ E s

b t
g C r s ;

f ctm
E c0m

1 : 3 ð 1 þ a Ed r s Þ b t

e srd ; 1 : 3 þ f ctm e srd ; 1 : 3 ¼

a Ed r s

The characteristic value for the yield point f yk is reached for

f yk
E s

f ctm
E c0m

b t
a Ed r s ¼

f yk
E s

f ctm
E s

b t
g C r s

e smy ¼

d) Yielding of the steel ( e s > e smy ):
Without taking into account strain hardening, the result is

s s,eff ¼ f yk .

Tension stiffening is very much dependent on the stress s sr leading to the appearance of the

first crack. The tension stiffening effect of the concrete between the cracks therefore
increases as the percentage of reinforcement decreases (Figure 3.3).

A section is said to be under-reinforced when the steel reinforcement reaches its yield
point before the tensile strength of the concrete is exceeded. Such a minimal percentage
of steel reinforcement should be avoided because the reinforcement would fail
suddenly upon the appearance of the first crack. In a similar way to DIN 1045-1
[33] 13.1.1 (1), the following condition must be complied with in order to guarantee
ductile behaviour of a structural member:

f ctm
r s

s sr ¼ ð 1 þ a Ed r s Þ < f yk

Fig. 3.3 Tension stiffening effect for different reinforcement percentages