Reddy Beam Theory Research Articles

Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models

The free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated in this work using various nonlocal beam theories. To this end, the nonlocal elasticity equations of Eringen are incorporated into the various classical beam theories namely as Euler–Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy beam theory (RBT) to consider the size-effects on the vibration analysis of SWCNTs. The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations of each nonlocal beam theory corresponding to four commonly used boundary conditions. Then molecular dynamics (MD) simulation is implemented to obtain fundamental frequencies of nanotubes with different chiralities and values of aspect ratio to compare them with the results obtained by the nonlocal beam models. Through the fitting of the two series of numerical results, appropriate values of nonlocal parameter are derived relevant to each type of chirality, nonlocal beam model, and boundary conditions. It is found that in contrast to the chirality, the type of nonlocal beam model and boundary conditions make difference between the calibrated values of nonlocal parameter corresponding to each one.

Nonlocal theories for bending, buckling and vibration of beams

Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.