PSI - Issue 79

Pranaw Parihar et al. / Procedia Structural Integrity 79 (2026) 404–412

408

k  

R k ( x )   4  α = 1

n s  j = 1

n m  i = 1

n t  k = 1

u h ( x )

R i ( x ) u i +

R j ( x ) [ H ( x ) − H ( x j )] a j +

[ β α ( x ) − β α ( x k )] b α

(7)

=

where all symbols used in Eq. (7) have same meaning as presented in Singh et al. (2019). Upon substituting Eq. (7) into Eq. (4) the strain are obtained as,

R k ( x )   4  α = 1

k  

n s  j = 1

n m  i = 1

n t  k = 1

u h ( x )

R i ( x ) u i +

R j ( x ) [ H ( x ) − H ( x j )] a j +

[ β α ( x ) − β α ( x k )] b α

(8)

=

Further, upon substituting Eq. (7) into Eq. (6) the discrete finite element equation for free vibration analysis ob tained as,  K − ω 2 M  r = 0 (9) where, K and M denote the global sti ff ness and mass matrices, respectively. In this section, the free vibration analysis of BDFGM plates is carried out using XIGA framework in conjunction with Reddy’s HSDT theory. A square BDFGM plate with dimensions (2 across 2 a ), thickness (2 h ), with centre crack of length, as illustrated in Fig. (), is considered. Unless otherwise specified, the material properties of the plate are taken from Table 1. Cubic NURBS basis functions are employed throughout the analysis. Standard Gauss quadrature with full integration is used for standard elements, while the integration of enriched elements are performed following the approach of Singh et al. (2019). The material properties are evaluated at the Gauss integration points. Various boundary conditions (BCs) are considered, where S, F, and C denote simply supported, free, and clamped edges, respectively. The enforcement of boundary conditions also follows the methodology described by Singh et al. (2019). The natural frequencies are normalized as: 5. Results and Discussions Initially, the XIGA-HSDT formulation is validated for a bi-directional FGM plate. Since no reference results are available for the vibration analysis of bi-directional FGM plates, the validation is performed using an FGM plate with material properties varying only through the thickness direction, which can be achieved by setting ( q = 0 ) (see Eq. 1). The normalized natural frequencies (NNFs) obtained from the present HSDT-XIGA formulation are then compared with the results reported in the literature Huang et al. (2012). For the validation study, a square bi-directional FGM plate ( a / b = 50, b / h = 0 . 01) with a central crack ( d / b = 0 . 3) under SSSS boundary conditions is considered. The volume fraction coe ffi cients are taken as ( n = 0 . 2) and ( q = 0). The first four lowest NNFs are computed for di ff erent sets of control points and presented in Table ?? . From Table 2, it can be observed that the NNFs found to be in agreement with the reference results. Furthermore, the NNFs remain nearly unchanged when the number of control points exceeds (28 × 28). Based on these observations, the HSDT XIGA framework is considered both validated and numerically converged. Accordingly, all subsequent examples are 5.1. Validation and convergence study