PSI - Issue 47

S. Aiello et al. / Procedia Structural Integrity 47 (2023) 668–674 S.Aiello, V. De Biagi, P. Cornetti, B. Chiaia / Structural Integrity Procedia 00 (2019) 000–000

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By defining the conditions at the ends of the crown (r=R2) and the appropriate conditions at the change of geometry (r=R1), two zones were identified for investigation, the center of the plate with the anomalous thickness (r=0) and the zone close to the changing geometry (r → � ). So, the circular plate’s deformation is taken into account. Similarly, to one-dimensional model, also in this case critical temperature is the temperature difference between intrados and extrados for which the normal stresses �� �� are equal to the average tensile stress, ��� (13,14). �� � ��� � �� ���� � �∗� � (8) �� � ��� � �� ���� � �∗� � (9) �� �� � � � � �� � � � � � � � � � �� � �∗ ����� (10) �� �� � � � � � � � � � � � �� � � �� � �∗ ��� (11) � � �� �� � ��� �� � �� � ��� �� �� � � 1 � �� � � �� � � � �� � (12) �� � � �� ���� � �∗ �� � � � �� � ����� ������ ��� � � � � � � � � � � � � �� � � �� � � � �� ������� � � � � (12) �� � ��������������� � � ����� ��������� � � � � � � � � � � � � �� � � �� � � � �� ������� � � � � (13) Where y is the fiber considered, �� and �� are the circumferential and radial moment, is the Poisson ratio, � � � � are function of position r, external radius of the circular plate � , the external radius of the crown � , the anomalous thickness ℎ , the design thickness , the weight and the elastic material properties of the concrete. Table 3. Critical cooling values (°C) for different classes of concrete and different value of H/h rate. Position r close to � , � � 1000 , � � 2000 , = 400 mm. H/h 1.5 2 2.5 3 3.5 4 Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� Tcr �� C20/25 9.62 -3.51 9.2 -13.65 9.18 -27.14 9.25 -44.28 9.32 -65.16 9.37 -89.77 C25/30 10.44 -2.17 10 -11.95 10 -24.88 10.09 -41.32 10.18 -61.34 10.25 -84.96 C30/37 11.44 -0.57 10.98 -9.92 11 -22.22 11.11 -37.85 11.2 -56.89 11.32 -79.35 C35/45 12.43 0.97 11.95 -7.98 11.99 -19.69 12.13 -34.56 12.27 -52.68 12.39 -74.08 C40/50 12.99 1.83 12.5 -6.92 12.55 -18.31 12.71 -32.77 12.86 -50.41 12.99 -71.23

Table 4. Critical cooling values (°C) for different classes of concrete and different value of H/h rate. Position � 0, � � 1000 , � � 2000 , = 400 mm, � � � � � → �� � Tcr �� H/h 1.5 2 2.5 3 3.5 4 C20/25 10.08 9.79 9.94 10.17 10.42 10.65 C25/30 10.88 10.57 10.73 10.98 11.24 11.48 C30/37 11.86 11.52 11.69 11.96 12.24 12.5 C35/45 12.83 12.47 12.65 12.94 13.24 13.51 C40/50 13.38 13.01 13.19 13.49 13.8 13.63