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Figure 3. (a) Tension softening model, (b) Estimated fracture energy, (c) and (d) Estimated model parameters, ft and sv

concrete, however, cannot be measured experimentally because of the smeared micro crack around the crack tip. In this report two methods were applied to define the crack advance: (1) equivalent crack length : If the material is assumed to be linear elastic, a compliancecrack length relation can be calculated analytically and/or numerically. From this so-called compliance-method the equivalent crack length can be obtained so that the measured compliance corresponds to the calculated compliance-crack length relation. In the present study, the relation was determined by using a finite element analysis. (2) fictitious crack length : A numerical load-displacement curve can be simulated by means of the fictitious crack model when an appropriate tension softening property for the material being tested is given. Since a certain position in the curve corresponds to a softening phase in the ligament (see. Fig.7), a fictitious crack advance can be defined so that the position in the experimental curve, where unloading begins, coincides with the simulated curve. The tension softening model used in the simulation was bilinear model (Fig.3(a)), and the model parameters were determined by optimizing the experimental load-displacement envelope curve based on the least square method [3].