Bending analysis of a cracked Timoshenko beam based on the nonlocal strain gradient theory

A size-dependent cracked Timoshenko beam model is established based on the nonlocal strain gradient theory and flexibility crack model. Expressions of the higher-order bending moment and shear force are derived. Analytical expressions of the deflection and rotation angle of the cross section of a simply supported microbeam with an arbitrary number of cracks subjected to uniform loading are obtained. The effects of the nonlocal parameter, the material length scale parameter, the presence of the crack, and the slenderness ratio on the bending behaviors of the cracked microbeam are examined. It is found that the material length scale parameter plays an important role in the cracked microbeam bending behavior, while the nonlocal parameter is not decisive. Furthermore, the cracked microbeam also exhibits a stiffening or softening effect depending on the values of the two scale parameters; if the two parameters are equal, the bending deformation of the nonlocal cracked microbeam may not be reduced to that of the classical elastic cracked Timoshenko beam. Additionally, the influence of the size effect on beam stiffening and softening becomes more significant as the slenderness ratio decreases.