Functionally Graded Materials (FGM) have been been of great importance to many researchers for a long time because of their wide range of applications in structural mechanics. The newly-created materials have promising applications in many fields such as space projects, the energy sector, communications projects, the defense industries, biomedical sectors and miscellaneous others. This study, which includes static, free vibration, dynamic, thermal and post-buckling analyses of beam structures, are conducted with the aid of newly developed consistent shape functions as well as co-rotational approach, arc-length control method, and implicit average constant acceleration Newmark method to solve engineering beam structures.