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A moderated thick double curved shallow shell with functionally graded materials subjected to static and dynamic loads are investigated by the meshless methods. Shear deformable plate theory is applied and the governing equation with respect to the middle surface is formulated in the Cartesian coordinate system with five degrees of freedom. The numerical solutions of the partial differential equations are obtained by the Finite Block Method in both strong form formulation (Point Collocation Method) and weak form formulation (Meshless Local Petrov-Galerkin) with Lagrange series interpolation and mapping technique. Two-dimensional plane-stress and plate bending problems are coupled in the governing equations. The second-order partial differentials are employed in the algorithm of strong form of governing equations and the first-order derivatives are required only in the weak form. The stress resultant intensity factors are evaluated by the Crack Opening Displacement (COD) and by the Variation Principle Method (VPM) techniques. The Laplace transform method and Durbin's inverse technique are utilized to deal with the crack problems under dynamic load. Comparisons have been made with analytical and numerical solutions with the finite element method in order to demonstrate the accuracy and convergence of the meshless approaches.

The boundary integral equations (BIE) and displacement discontinuity methods (DDM) are formulated for solution of smooth curved crack problems within 2D elasticity and Reissner's plate bending theory. The equivalence between the direct boundary element method and the indirect boundary element method is shown here. The Chebyshev polynomials of the second kind are employed to solve the integral equations numerically. This enables determination of the stress intensity factors at the crack tips directly5 by the coefficients of Chebyshev polynomials. In the DDM, the coefficients of stress influence for the constant displacement discontinuity element are derived in closed form for the Reissner's plate bending problem. The degree of convergence of BIE is demonstrated with increasing numbers of the collocation points. Comparison is made with the analytical solutions from the stress intensity factor handbook for an embedded circular crack in a flat plate under tensile force. High accuracy of the presented numerical solutions makes possible to use them as benchmarks to evaluate the degree of accuracy for any numerical algorithms.