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In this article, free vibration characteristics of rotating axially functionally graded tapered beams with six different boundary conditions are studied through a finite-element approach. The beam element employed in this study is a two-noded element whose shape functions are obtained in terms of special functions namely basic displacement functions. Application of this beam element provides good results as the shape functions take the effects of variable centrifugal force, cross-sectional area, and mechanical properties of the material. In this study, the effects of material non-homogeneity, taper ratio, rotating speed parameter, hub radius, and tip mass are investigated. The results are given in tabular form so that they could be used as a means of comparison for future studies in this field.

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An extended finite element method (XFEM) is developed to study fracture parameters of cracked metal plates and tubes that are repaired on top of the crack with a composite patch. A MATLAB® stand‐alone code is prepared to model such structures with eight‐noded doubly curved shell elements in the XFEM framework. Crack trajectory studies are performed for a diagonally cracked panel under fatigue loading. Verification studies are investigated on different shell type structures such as a cracked spherical shell and cracked cylindrical pipe with different crack orientations. The effects of using patch repairs with different fibre orientations on the reduction of stress intensity factors (SIFs) is also studied which can be useful for design purposes. XFEM is selected as any crack geometry can be embedded in the finite element mesh configuration with no need to coincide the crack geometry with meshed elements and so re‐meshing with fine mesh generation is not needed in the current method.