PSI - Issue 14

Rahul Saini et al. / Procedia Structural Integrity 14 (2019) 362–374 R. Saini, S. Saini, R. Lal, and I. V. Singh / Structural Integrity Procedia 00 (2018) 000–000

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7

�

� � � � � � 0 � � ,

�18�

respectively. The equations �17� and �18� have been solved for unknowns �� , �� , … , �� �� , �� �� using the computer software MATLAB. The thermal stresses raised in the plate due to thermal environment have been obtained as follows: Step 1. The equation ��� have been discretized by using equation (14), ��� � � � � , � � ��� � � � �� � � � � � �� � � � ��� ��� � � � � , ��� � � � � , � � ��� � � � �� � 1 � � � � �� � � � ��� ��� � � � � , ⎬ ⎪ ⎪⎫ �19� Step 2. Substitution of the numerical values for �� , �� , … , �� �� , �� �� (obtained from solving equations �17,18� ) in �19� provide the thermal stresses for C and S-plates, respectively. 3.2. Vibration analysis For vibration analysis, equation (11) has been discretized by using equation (14) for various order derivatives of and at �� grid point � , it leads to ���� � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � ��� ��� � � � � � � � � � 0. �20� Now, the satisfaction of equation (20) at � � 2� internal grid points together with the regularity conditions provides a set of equations in terms of � � 2� unknowns � , � � 1, 2, 3, … , � � 2� . These resulting equations together with the two more equations obtained by satisfying the periphery conditions, give a complete set of � � 2� equations in � � 2� unknowns. For C-plate, the set of these � � 2� homogenous equations can be written as � � � � � � �0�, �21� where , � are the matrices of order � � � 2� , 2 � � � 2� , respectively and � � � , � , … , ��� � � . For a non-trivial solution of equation (21), the frequency determinant must vanish and hence � � � � 0. �22� Similarly, for S-plate, the frequency determinant can be written as � � � � 0, �23� respectively. ��� � ��� � ��� � 0, ⎭⎪⎪