Crack Paths 2006

The intersection between the nonlinear tension softening and the elastic loading

branch gives the strain Hi, from which follows the reduced Young’s modulus:

i ti t t i 1 i 1 i t i i t i i f E p f 2 f E E f E f (8)

Ÿ

H

a)

b)

(a)

(b)

Figure 1. (a) Saw-tooth diagram for non-linear tension softening (a) and saw-tooth

procedure. (b).

Adopting Eq.(6) and a value p of strength percentage, the corresponding uplifted

stress is:

2

f p C A A2 4 B B ˜ ˜ ˜ ˜

V

.

(9)

t

The intersection point (see Fig. 1b) between the nonlinear tension softening and the

elastic branch corresponding to the linear behavior of the current stage in the saw-tooth

diagram, is the following:

c a 4 b b 2 ˜ ˜ (10)˜

Ÿ ˜ ˜ ˜ c a 2 4 b b 2 iH E f ti

i

a 2

the values of a, b, and c are reported at the end, in the nomenclature list. The value of

ti f given by Eq. (10) can finally be substituted into Eq. (8) to obtain

1iE .

Saw-Tooth Laws for Steel in Tension and compression

W e adopt an elastic perfectly plastic stress-strain diagram for reinforcing steel both

intension and compression, according to EC2 prescriptions [6], (see Fig. 2). The

uplifted post peak curve is the following:

(11)

V

y f p 1