This article presents a torsional analysis of solid elliptical, hollow circular, and actual bone sections of orthotropic and functionally graded material. The formulation of the governing equation is done using the Saint-Venant torsion theory. A classical power law is considered for the modelling of functionally graded material. Five different radial basis functions-based meshless methods are used for the discretization of the governing differential equations. MATLAB code is developed to solve the discretized partial differential equations. A convergence and validation study has been carried out to demonstrate the effectiveness and accuracy of the present method. Numerical examples for torsional rigidity and shear stresses are presented for circular, elliptical, and bone-shaped irregular sections made up of orthotropic and functionally graded materials. Finally, the proposed radial basis function-based meshless method is applied to the modelling and torsional analysis of an actual bone cross-section.
ElyJFZienkiewiczOC. Torsion of compound bars—A relaxation solution. Int J Mech Sci1960; 1: 356–365.
6.
ChenY. Torsion of non-homogeneous bars. J Frankl Inst1964; 277: 50–54.
7.
DanaoLAMCabreraRM. Torsion of a rectangular prismatic bar: Solution using a power fit model. Philipp Eng J2007; 28: 77–98.
8.
RooneyFJFerrariM. Torsion and flexure of inhomogeneous elements. Compos Eng1995; 5: 901–911.
9.
KrahulaJLLauterbachGF. Finite element solution for Saint-Venant torsion. AIAA J1969; 7: 2200–2203.
10.
TarnJQChangHH. Torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and end effects. Int J Solids Struct2008; 45: 303–319.
11.
LiZKoJMNiYQ. Torsional rigidity of reinforced concrete bars with arbitrary sectional shape. Finite Elem Anal Des2000; 35: 349–361.
12.
SaygunAOmurtagMHOrakdogenE, et al.A simplified solution of the torsional rigidity of the composite beams by using FEM. Adv Struct Eng2007; 10: 467–473.
13.
SwansonSR. Torsion of laminated rectangular rods. Compos Struct1998; 42: 23–31.
14.
SavoiaMTulliniN. Torsional response of inhomogeneous and multi-layered composite beams. Compos Struct1993; 25: 587–594.
EcsediI. Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars. Eur J Mech ASolids2009; 28: 985–990.
20.
GorzelańczykP. Method of fundamental solutions and random numbers for the torsion of bars with multiply connected cross sections. Comput Assist Mech Eng Sci2010; 17: 99–112.
21.
HassenpflugWC. Torsion of uniform bars with polygon cross-section. Comput Math Appl2003; 46: 313–392.
22.
KołodziejJAGorzelańczykP. Application of method of fundamental solutions for elasto-plastic torsion of prismatic rods. Eng Anal Bound Elem2012; 36: 81–86.
23.
RongqiaoXJianshengHWeiqiuC. Saint-Venant torsion of orthotropic bars with inhomogeneous rectangular cross section. Compos Struct2010; 92: 1449–1457.
24.
FrancůJNováčkováPJaníčekP. Torsion of a non-circular bar. Eng Mech2012; 19: 45–60.
25.
JogCSMokashiIS. A finite element method for the Saint-Venant torsion and bending problems for prismatic beams. Comput Struct2014; 135: 62–72.
26.
AbdelkaderKToufikZMohamedBJ. Torsional stress in non-circular cross sections by the finite element method. Adv Mech Eng2015; 7: 1–20.
27.
BeheshtiA. A numerical analysis of Saint-Venant torsion in strain-gradient bars. Eur J Mech ASolids2018; 70: 181–190.
28.
TsiatasGCBabouskosNG. Elastic-plastic analysis of functionally graded bars under torsional loading. Compos Struct2017; 176: 254–267.
29.
KatsikadelisJTTsiatasGC. Saint - Venant torsion of non-homogeneous anisotropic bars. J Appl Comput Mech2016; 2: 42–53.
30.
ChenKHKaoJHChenJT, et al.A new error estimation technique for solving torsion bar problem with inclusion by using BEM. Eng Anal Bound Elem2020; 115: 168–211.
31.
NikmehrOLashkarbolokM. A numerical investigation on the torsional rigidity of bars with functionally graded material (FGM) cross sections weakened by cracks. Iran J Sci Technol - Trans Civ Eng2019; 43: 117–123.
32.
DarılmazKOrakdöğenEGirginK. Saint-Venant torsion of arbitrarily shaped orthotropic composite or FGM sections by a hybrid finite element approach. Acta Mech2018; 229: 1387–1398.
33.
EcsediIBaksaA. Saint-Venant torsion of cylindrical orthotropic elliptical cross section. Mech Res Commun2019; 99: 42–46.
34.
ChenHGomezJPinderaMJ. Saint Venant’s torsion of homogeneous and composite bars by the finite volume method. Compos Struct2020; 242: 112–128.
35.
UscilowskaAFraskaA. A mesh-free numerical method for the estimation of the torsional stiffness of long bones. J Mech Mater Struct2011; 6: 265–276.
36.
LiuGR. Mesh Free Methods: Moving beyond the Finite Element Method. 2nd ed. CRC Press, 2009.
37.
HardyRL. Multiquadric equations of topography and other irregular surfaces. J Geophys Res1971; 76: 1905–1915.
38.
FerreiraAJM. Polyharmonic (thin-plate) splines in the analysis of composite plates. Int J Mech Sci2004; 46: 1549–1569.
39.
FerreiraAJMRoqueCMCJorgeRMN. Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Comput Struct2005; 83: 2225–2237.
40.
SinghJShuklaKK. Nonlinear flexural analysis of laminated composite plates using RBF based meshless method. Compos Struct2012; 94: 1714–1720.
41.
SolankiMKMishraSKSinghJ. Meshfree approach for linear and nonlinear analysis of sandwich plates: A critical review of twenty plate theories. Eng Anal Bound Elem2016; 69: 93–103.
42.
ShuklaVSinghJ. Modeling and analysis of cross-ply and angle-ply laminated plates under patch loads using RBF based meshfree method and new HSDT. Comput Math Appl2020; 79: 2240–2257.
43.
SinghJSinghSShuklaKK. Meshless analysis of laminated composite and sandwich plates subjected to various types of loads. Int J Comput Methods Eng Sci Mech2014; 15: 158–171.
44.
DarilmazKOrakdogenEGirginK. Torsional rigidity of arbitrarily shaped composite sections by hybrid finite element approach. Steel and Compos Struct2007; 7: 241–251.
