A modified uncoupled lower-order theory for FG beams

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

“…It should be pointed out that, the displacement field assumption Eq. 4 in the present work is different from the displacement field in the theory for beams with rectangular cross-section (Geng et al, 2017;Pei et al, 2019) due to the bidirectional warping function g (y,z).…”

“…where N, M and Q are, respectively, the membrane force, the moment and the shear force well known in Engineering, P is called the higher-order moment, R z and R y are two higherorder shear forces in the present work which are different from the R z in the theory for beams with rectangular crosssection (Geng et al, 2017;Pei et al, 2019). It should be noted that the three factors, i.e., 1, z and g(y, z), are extracted from the three terms in Eq.…”

“…where E 1 and E 2 denote the Young's modulus of the metal (material-1) and ceramic (material-2) constituents of the FG beam respectively, k is the non-negative variable parameter (power-law exponent). Poisson's ratio is assumed to be a constant, and this can be found in the relevant references (Pradhan and Chakraverty, 2014;Pei et al, 2019). Thus, the Young's modulus of metal with respect to r/R is depicted for various k, as shown in Figure 2.…”

“…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).…”

“…For example, there are coupled expressions of bending and stretch in (Aydogdu and Taskin, 2007;Thai and Vo, 2012;Huang, 2020). For the uncoupled beam theory (Geng et al, 2017;Pei et al, 2019), the bending content is decoupled with the stretching content in the constitutive relations, and hence, the governing equations are greatly simplified. Motivated by the great importance to overcome disadvantages, this paper begins with the fundamentals of beam problems to develop the modified uncoupled LBT for FG beams with circular cross-section.…”