Bending Analysis of a Cracked Timoshenko Beam Based on the Nonlocal Strain Gradient Theory

Vibration Analysis of Viscoelastic Timoshenko Cracked Beams with Massless Viscoelastic Rotational Spring Models

“…For a standard linear solid beam under the simplesupported boundary conditions, it is supposed that the material and geometric parameters are E 2 = 39:68GPa, E 1 = 14GPa, ρ = 500kg/m 3 , L = 1 m, and b = 0:1 m. Additionally, Poisson's ratio ν = 0:3, the uniform sudden load Q 0 = 10 6 N ⋅ m −1 , and the shear correction factor κ = 10ð1 + νÞ/ð12 + 11νÞ. In order to analyze the effect of viscous coefficient on the vibration properties of the viscoelastic beam, the viscous coefficient is taken as η ∈ 6:9 × ½10 4 , 10 12 GPa ⋅ h according to references [15,16].…”

“…And by using the expression for the rectangular cross section beams in references [11,12], the equivalent stiffness of crack at the location x = x j with the crack depth d j in time domain and Laplace domain are given as, respectively,…”

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

“…Pouretemad et al 46 studied free vibration frequencies of rotating non-uniform multiple cracked nanobeams. Dona 47 and Fu and Yang 48 studied bending and dynamic behaviours of multiple cracked nanobeams using the strain gradient theory.…”

“…Firstly, the beam theories used are the Euler-Bernoulli beam theory 33,[35][36][37][38][39][40][41][42][43]45,47,50 and the Timoshenko beam theory. 20,34,44,46,48,49 Secondly, the crack is modelled as an elastic spring connected by two intact segments to each other at cracked sections. When analysis of cracked nanobeam vibrations, the longitudinal displacement and therefore translational spring, is neglected; the only flexibility constant, and therefore rotational spring, which is related to the bending moment being considered.…”