Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

A new displacement-based framework for non-local Timoshenko beams

“…(32b, c), i.e. the local stiffness coincides with the stiffness of the classical TM theory; in this manner, the sought non-local solution will be stiffer than the local solution, consistently with the experimental behavior reported by Lam et al [12]; (ii) the attenuation functions g u (x, n) and g z (x, n) governing pure bending and pure shear non-local effects are given the exponential form (34). Being b 1 = 1 and due to the exponential form (34) of the attenuation functions, the sought non-local solution will revert to the classical local TM beam solution as l ?…”

“…(34). To determine these parameters an error minimization procedure is pursued, based on the experimental data reported in Fig.…”

“…It can be seen that a very good agreement between the experimental bending rigidity ratios and those predicted by the proposed non-local beam model is attained for C ¼ 3:17 Á 10 11 Nm À6 and l = 30 lm in Eq. (34). Figure 12 A certain number of parameters/functions shall be set in the proposed non-local beam model, concerning the local terms, [see b 1 in Eq.…”

“…0 in the attenuation function g(x, n) given by Eq. (34). The numerical solution of the equilibrium Eqs.…”

“…For instance, Eringen's integral theory has been used by Peddieson et al [30], Wang and Shindo [31], Civalek and Demir [32] to derive non-local EB beam models, by Wang and Liew [33], Wang and et al [34] to derive non-local TM beam models, the latter in conjunction with the principle of virtual work. A non-local EB beam model has been presented by Challamel and Wang [35] based on a gradient elastic model and a non-local integral elastic model, where the constitutive relation is expressed by combining local and nonlocal curvatures.…”