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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3. Method of solution Let � , � , … , � be the grid points in range �0, 1� of the plate. According to GDQR (Wu et al. 2002), the �� order derivative of transverse displacement � � with respect to at the �� point � is given by � � � � � � ��� � � �ℎ � �� � � � � � ��� � ℎ ��� � � � � � � ℎ ��� � � � � � � � ��� � �� ��� � , � � 1,2, … , �, �14� where, � �� are the weighting coefficients of �� order derivative at point � and ℎ �� � � are the Hermite interpolation functions. Here, for convenience the unknowns ’s have been replaced by ’s in equation (14) as follows: � � , � � , � , � , … , ��� , ��� , � , � � � � � � , � , … , ��� , ��� � . The co-ordinates of grid points have been generated as per the Chebyshev Gauss-Lobatto grid distribution given by � � 1 2 �1 � ��� � � � 1 1 ��� , � 1, 2, … , � 3.1. Thermal analysis Discretizing equation (7) at the internal grid points � , � 2, �, … , � � 1� , using equation (14) by putting � for , one gets ���� � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � ��� ��� � � � � � � � � � � 0, �15� where the superscript over � i.e. � � represents the unknowns � , � , … , ��� , ��� for thermal displacement as �� , �� , … , �� �� , �� �� . The satisfaction of equation (15) at � � 2� internal grid points together with the regularity conditions � � / � ��� � � � � / � � ��� � 0 , provides a set of equations in � 2 unknowns � � , � � 1, 2, �, … , � � 2� . The resulting system of equations can be written in matrix form as, � � � � � 0, �16� where is a matrix of order � � � 2� and is a column vector given by � �� , �� , … , �� �� � � . By satisfying the periphery conditions for a C-plate (8), a set of two homogenous equations is obtained. These equations together with the field equation (16) give a complete set of � � 2� equations in � � 2� unknowns, which can be written as � � � � � � �0� , �17� where � is the matrix of order 2 � � � 2� . Similarly, for S-plate (9), a set of two non-homogenous equations have been obtained which with the field equation (16) give a complete set of � � 2� equations in � � 2� unknowns and can be written as