PSI - Issue 47

802 Luciano Feo et al. / Procedia Structural Integrity 47 (2023) 800–811 Author name / Structural Integrity Procedia 00 (2019) 000–000 3 where � , � , � and � are the total fibers volume, the single fiber volume and the cementitious matrix volume, respectively, and � is the fibers volume ration. The kinematics of the model is inspired by the well-known “cracked-hinge” approach formulated by Olesen (2001). In particular, as shown in Figure 1, the sample behaves rigidly, except in a central portion of length where flexibility is shaped by � independent elastic springs or “strips” aligned with the longitudinal axis. Thus, the two rigid blocks at the beam ends can rotate around the neutral axis of the cracked-hinge section, whose position is described by � . By choosing the rotation as the main generalized displacement parameter, the beam deflection at mid-span section, , can be expressed as: � 2 (2) With this assumption, the average value of the matrix axial stress, � , � , for each k -layer, will depend on the average axial strain � before crack formation and on the crack-opening displacement � after cracking. These two quantities can be expressed as: � � 2 � ∙ � � � � � (3) � � 2 � ∙ � � � � � (4) in which, � , � and � are the rotation of the two rigid blocks at the j-th step of the incremental analysis, the position of the neutral axis and the position of the k-th strip in the cross-section, respectively (Figure 1). As for the steel fibers, the action � was introduced to take account of the bridging effect. For each j-th step of the (5) being � , � , � and � , � the position, the area and the axial stress of the k-th fiber in the cross-section, respectively. Considering that only a low number of fibers, � � , will cross the crack, the position of the neutral axis, � , � , obtained by the equilibrium equation along the longitudinal axis, and the corresponding bending moment are written for each value of the imposed rotation angle, , as reported in Eqs.6-7. ��∙�∙�� � , � � � ��� � � ; � , � ; � ���� � , � � � , � ; � , � ; � � � � � ��� � 0 (6) � ���∙�∙�� � , � � � ��� � � ; � , � ; � � ∙ � ℎ 2 � � ���� � , � � � , � ; � , � ; � � � � � ��� ∙ � ℎ 2 � � , � � (7) incremental analysis, it can be evaluated as: � , � � � , � ; � , � ; � �� � ∙ � , � � � , � ; � , � ; � �