Free vibration of radially graded hollow cylinders subject to axial force via a higher-order shear deformation beam theory

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

“…Since 𝑃𝑤 gives a bending moment for a cantilever deviating from its original position subjected to a compressive force 𝑃, the contribution of the term 𝜎 0 𝑢 in (1) in beams may be explained as a HOM according to (34). More precisely, when the HOM is considered, it causes a slight decrease or increase of the bending stiffness for compressive (𝑃 > 0) or tensile (𝑃 < 0) axial force, respectively.…”

“…In particular, Ma et al. developed a higher‐order beam theory to treat the free vibration of axially‐loaded FG tube‐beams [34].…”

“…The effect of the direction of axial force on the free vibration and buckling has been analyzed using the Timoshenko and higher-order beam theories [28][29][30][31][32][33]. In particular, Ma et al developed a higher-order beam theory to treat the free vibration of axially-loaded FG tube-beams [34].…”

A modified lower-order theory for FG beam with circular cross-section

“…In the higher-order shear deformation (HSD) model, the warping effect on the cross-section is taken into consideration and the shear stress free condition is satisfied. Since then, a variety of theories based on HSD models have been proposed by many scholars (Reddy et al, 2001;Aydogdu, 2009;Şimşek, 2010;El Meiche et al, 2011;Shi and Voyiadjis, 2011;Huang et al, 2013;Belabed et al, 2014;Pradhan and Chakraverty, 2014;Duan and Li, 2016a;Duan and Li, 2016b;Geng et al, 2017;Pei et al, 2018;Pei et al, 2019;Huang, 2020;Ma et al, 2021), and can be divided into two types according to the differential order of the governing equations, i.e., the higher-order beam theory (HBT) and the lower-order beam theory (LBT).…”

“…The LBT for FG beams with circular-section based on the third-order HSD model was proposed by Huang (Huang et al, 2013;Huang, 2020) through the differential equilibrium relationship of forces, in which the axial displacement can be considered to be obtained by trial and error. Ma (Ma et al, 2021) also proposed the LBT to study the buckling and vibration behavior of axially-loaded hollow cylindrical FG pipes. Li (Li et al, 2019) investigated the vibration control and analysis of a rotating FG beam with a lumped mass and bonded piezoelectric films in temperature field.…”