On bending consistency of Timoshenko beam using differential and integral nonlocal strain gradient models

In this work, the static bending response of Timoshenko beam under different boundary and loading conditions is analyzed and compared with the application of nonlocal strain gradient models in differential (DNSGM) and integral (INSGM) forms. High-order and standard boundary conditions are introduced for DNSGM, while the relation between strain and nonlocal stress are expressed as integral equations for INSGM. The differential equations for DNSGM and integro-differential equations for INSGM are solved directly with the Laplace transformation. The explicit expression for bending deflection and rotation is derived uniquely with eight unknown constants for both DNSGM and INSGM. The results obtained with current models are validated against to the existing results in literature. On the static bending of Timoshenko beam subjected to different boundary and loading conditions, inconsistent responses occurs for DNSGM, while consistent softening and toughening responses can be obtained for INSGM.