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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4. Numerical results The lowest three roots of frequency equations ���� ��� have been computed using MATLAB and retained as the values of the frequency parameter Ω . The critical buckling load � ∗ � has been calculated using Bisection method. The numerical values for TI material properties are taken from reference (Malekzadeh et al., 2011). From the literature, the values of various parameters are taken as follows: material graded index � �� 1� �� �� �� �� in-plane force parameter ∗ � ���� �1�� �� 1�� ��� �� , temperatures � � ���� , � � ����� , ����� � � � ����� , thickness � � ��1 and Poisson’s ratio � � ��� . 4.1. Convergence and comparison study During the computations, the number of grid points has been fixed as 16 since there was no difference up to the fourth place of decimal between two successive values of frequency parameter Ω for varying values of , ∗ and � keeping � � ����� . In this regard, the computer program developed to evaluate Ω was run for these values taking � 6����� � �1� and observe that �Ω � � �� � Ω � � � � ������� for all the three modes � � 1� �� � . In graphical form, the percentage error �1 � Ω � Ω �� ⁄ � � �1�� for a specified plate � � , ∗ � ��� � � � � ����� and � � ����� has been shown in Fig. 2. The deviations obtained for this set of values of the parameters for all the three modes of vibration were maximum. Further, it is worth to mention that for evaluation of the frequencies one has to consider a finite number of terms ( ) in the binomial expansion (5). Being an infinite series, its accuracy will depend upon the value of . In the present analysis, has been taken as 20, as further increase in this value does not improve the result, except at the fifth place of decimal. A comparison of results in the absence of thermal environment �� � �� � �� for different values of in-plane force � ∗ � �1�� �� �� ) and graded index ( � �� � ) with those obtained by DTM (Lal and Ahlawat, 2015) and Rayleigh-Ritz method (Pradhan and Chakraverty, 2015) has been presented in Table 1. Further, by allowing the frequency to approaches zero, the values of � ∗ � in compression, for varying values of has been compared with DTM (Lal and Ahlawat, 2015) and reported in Table 2. A close agreement has been observed which describes the versatility of the present technique.

Fig. 2. Percentage error in Ω: ‘○’ – I mode; ‘ △ ’ – II mode; ‘□’ – III mode (a) C-Plate (b) S-Plate.