Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 101 2 2 1 2 ' '         c c c fin c f (4) where f c is the uniaxial compressive strength of concrete. By following this procedure, the value of  is properly reduced when tensile stresses occur; thus,  < 1 always holds in this case. -1.4 -1 -0.6 -0.2 0.2 -1.4 -1 -0.6 -0.2 0.2  2c /| f c |  1c / | f c |  1c   2c Tension-tension 1 max 1 max 2 max 1         ct f Tension-compression k k f ct 73.0 6.0 max 2 max 1 max 2           c ct f f k k   73.0 lim  Compression-tension   0 73.0 56.66 9 9 8.12 max 2 max 1 2 2 max 2                     k k k k k f c Compression-compression   1 0 1 65.3 1 max 2 max 1 2 max 2              c f c 2  c c 2 1    Figure 1 : Adopted failure envelope [16]. After having determined the nonlinearity index, concrete secant elastic modulus E c can be then calculated as:   2 2 ' ' ' 1 1 2 2 2 2                                    ci ci ci ci c cf cf cf E E E E E E E E D (5) where E ci is the initial value of concrete Young modulus, D is a compressive post-peak nonlinearity parameter that determines the degree of strain softening when concrete crushing occurs (see [14, 16] for details), and E' cf is the secant modulus corresponding to peak stress. When a tensile stress is present, E' cf is simply evaluated as in case of uniaxial compression, i.e. E' cf = E cf = f c /  c0 , while for biaxial compression the following relation is adopted:   ' 1 4 1    cf cf E E A x , (6) A being the ratio between the initial value of concrete Young modulus and the secant one corresponding to peak stress ( E ci / E cf ), while the term x takes into account the dependence on the actual loading and is evaluated through the relation: 2 1 3           c f J x f . (7) The first addend of Eq. 7 represents the failure value of the invariant 2 c J f . Based on the definition of the nonlinearity index  (Eq.3), the following expression can be found:   2 2 2 1 2 1 2 1 3        c fin c fin J . (8)