A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation

A new higher-order beam theory for buckling and free vibration responses of laminated composite and functionally graded porous beams

“…Tables 10 and 11 present the NFF and NCBL of beams, respectively. The results obtained from the proposed theory are compared to those obtained from FBT by Chen et al, 37,39 Jamshidi and 65 Quasi-3D 8.509 15.468 -Arghavani, 13 HBT by Nguyen et al, 47 Karamanli et al, 67 and Quasi-3D by Fang et al 64 and Karamanli et al, 67 The outcomes demonstrate that the proposed results agree with those of previous studies, evidencing the efficiency of the proposed theory. Figures 6 and 7 display the NFF and NCBL of SPD and ASP beams with respect to L/h ratio.…”

“…The Legendre-Ritz functions are utilised to solve the problem. Numerical examples are conducted to verify the proposed theory and investigate the impact of porosity distribution, boundary 47 HBT 0.1007 0.0508 -Chen et al 37 FBT 0.1008 0.0508 0.0204 Karamanli et al 67 Quasi-3D 0.0953 --Fang et al 64 Quasi-3D 0.0875 --APD Present HBT 0.0917 0.0462 0.0185 Nguyen et al 47 HBT 0.0917 0.0462 -Chen et al 37 FBT 0.0917 0.0462 0.0185 Karamanli et al 67 Quasi-3D 0.0870 --Fang et al 64 Quasi-3D 0.0870 --C-C SPD Present HBT 0.5902 0.3159 0.1291 Nguyen et al 47 HBT 0.5900 0.3159 -Chen et al 37 FBT 0.5944 0.3166 0.1291 Karamanli et al 67 Quasi-3D 0.5654 --Fang et al 64 Quasi-3D 0.5653 --APD Present HBT 0.5476 0.2888 0.1174 Nguyen et al 47 HBT 0.5476 0.2888 Chen et al 37 FBT 0.5475 0.2888 0.1274 Karamanli et al 67 Quasi-3D 0.5248 -Fang et al 64 Quasi-3D 0.5248 - 67 HBT 0.04643 0.01295 0.0033349 Chen et al 39 FBT --0.003338 Jamshidi and Arghavani 13 FBT --0.003339 Karamanli et al 67 Quasi-3D 0.04686 0.01299 0.0033378 APD Present HBT 0.039743 0.010800 0.002760 Nguyen et al 47 HBT -0.010800 0.002760 Karamanli et al 67 HBT 0.03973 0.01080 0.0027601 Chen et al 39 FBT --0.002869 Karamanli et al 67 Quasi-3D 0.04058 0.01096 0.0027951 C-F SPD Present HBT 0.012958 0.003336 0.000840 Nguyen et al 47 HBT -0.003335 0.000840 Karamanli et al 67 HBT 0.01295 0.00333 0.0008400 Chen et al 39 FBT -0.00334 0.000840 Jamshidi and Arghavani 13 FBT --0.000840 Karamanli et al 67 Quasi-3D 0.01317 0.00336 0.0008440 APD Present HBT 0.010801 0.002760 0.000694 Nguyen et al 47 HBT -0.002760 0.000694 Karamanli et al 67 HBT 0.01080 0.00276 0.0006934 Chen et al 39 FBT -0.00276 0.000694 Karamanli et al 67…”

“…where i 2 = − 1 ; ω is the eigen-frequency, u 0 j , w 0 j , and φ 0 j are unknown parameters need to be determined; ς j ( x ) are the Legendre-Ritz functions which presented in Nguyen et al 47 for three typical boundary conditions (BCs): clamped-free (C-F), simply-supported (S-S), and clamped-clamped (C-C) as displayed in Table 1.…”

“…43,44 Levinson 45 and Murty 46 introduced polynomial shear functions to analyse the behaviour of beams, and this approach has been adopted in numerous studies. For example, Nguyen et al 47–49 examined static, free vibration, and buckling behaviours of FGP beams, while Özütok Madenci 50 presented the static analysis of LC beams. Shi and Lam 51 analysed the free vibration of composite beams using FEM.…”

“…Figure 1. Two porous beams: (a) SPD: Porosity is symmetric distribution and (b) APD: Porosity is asymmetric distribution 47. …”