A finite element method is developed to study the size-dependent static bending,
elastic buckling and free vibration of Timoshenko beam-based plane frame using
two-phase local/nonlocal integral model. The explicit expression for the stiffness
matrix, geometric stiffness matrix and mass matrix is derived. Coordinate
transformation is employed to obtain the stiffness matrix, the geometric stiffness
matrix and mass matrix for a plane frame element oriented around an arbitrary
angle. The effects of the nonlocal parameters on static bending, elastic buckling and
free vibration of Timoshenko beams as well as a clamped square frame are
studied numerically under different boundary and loading conditions. The
bending deflections obtained are validated against existing analytical and
asymptotic results in literature. The numerical results show clearly that, with
the application of the two-phase local/nonlocal integral model, a consistent
softening effect can be obtained for both Timoshenko beams and plane frames.
Keywords
nonlocal integral model, finite element formulation,
Timoshenko beam, plane frame, mechanical response
