Static and free vibration analyses of small - scale functionally graded beams possessing a variable length scale parameter using different beam theories

Aghazadeh, Reza
This study presents static and free vibration analyses of functionally graded (FG) micro - beams on the basis of higher order continuum mechanics used in conjunction with classical and higher order shear deformation beam theories. Unlike conventional ones, higher order elastic theories consider the size effect for the beam. Strain gradient theory (SGT) and modified couple stress theory (MCST) are the two common non-classical continuum approaches capable of capturing the size effect. Shear deformation beam theories consider the effects of shear strain across the thickness. In the base of SGT and generalized beam theories and taking the thermal effects into account, the governing equations and boundary conditions are derived using a variational formulation based on Hamilton’s principle. This new model may be reduced to the non-classical Bernoulli-Euler beam model based on the modified couple stress theory (MCST) when two of the material length scale parameters and extra terms of higher order beam theories are taken to be zero. Numerical analyses using differential quadrature method (DQM) are conducted by considering static bending and free vibration problems of a simply supported FG beam.