Timoshenko beam theory

About: Timoshenko beam theory is a research topic. Over the lifetime, 9426 publications have been published within this topic receiving 200570 citations.

Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs)

Abstract: This paper studies the nonlinear bending behavior of a novel class of multi-layer polymer nanocomposite beams reinforced with graphene platelets (GPLs) that are non-uniformly distributed along the thickness direction. Nonlinear governing equation is established based on Timoshenko beam theory and von Karman nonlinear strain-displacement relationship. The effective Young's modulus of the nanocomposites is determined by modified Halpin-Tsai micromechanics model. Ritz method is employed to reduce the governing differential equation into an algebraic system from which the static bending solutions can be obtained. A comprehensive parametric study is then conducted, with a particular focus on the influences of distribution pattern, weight fraction, geometry and size of GPLs together with the total number of layers on the linear and nonlinear bending performances of the beams. Numerical results demonstrate the significantly improved bending performance through the addition of a very small amount of GPLs into polymer matrix as reinforcements. It is found that dispersing more GPLs that are in square shape with fewer single graphene layers near the top and bottom surfaces of the beam is the most effective way to reduce bending deflections. Beams with a higher weight fraction of GPLs that are symmetrically distributed in such a way are also less sensitive to the nonlinear deformation.

A non-classical Mindlin plate model based on a modified couple stress theory

Abstract: A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.