An analytical solution for torsional analysis of constrained open-section members, which can estimate the variation of the axial displacement, the normal and shearing stresses along the beam and across the cross section, is important and is the main goal of this article. A new formulation for constrained torsional analysis of open-section members is presented. In the formulation, cross sections with curved corners can be analyzed. The cross section is decomposed into some straight and curved segments with uniform thickness. This decomposition makes it possible to solve the governing equations in each segment analytically. The method can predict the stress concentration at reentrant corners of the cross section with an acceptable accuracy. Specific formulas for channel beams and tubes with a longitudinal slot are also derived. Accuracy of the presented method is verified by comparing the obtained results with the three-dimensional finite element method solutions with a fine mesh. Numerical investigations show that the obtained formulas give relatively accurate results for thin to moderately thick-walled open cross sections.
MuraokaYHiroyaM. Development of an all-aluminum automotive body. J Mater Process Tech1993; 38(4): 655–674.
2.
CavazzutiMBaldiniABertocchiE. High performance automotive chassis design: a topology optimization based approach. Struct Multidiscip Optim2011; 44(1): 45–56.
3.
SatoKInazumiTYoshitakeA. Effect of material properties of advanced high strength steels on bending crash performance of hat-shaped structure. Int J Impact Eng2013; 54: 1–10.
4.
TimoshenkoSPGoodierJN. Theory of elasticity. New York: McGraw-Hill, 1970.
El FatmiR. Non-uniform warping including the effects of torsion and shear forces. Part I: a general beam theory. Int J Solids Struct2007; 44: 5912–5929.
WangCMReddyJNLeeKH. Shear deformable beams and plates. New York: Elsevier, 2000.
18.
DufortLDrapierSGrédiacM. Closed-form solution for the cross-section warping in short beams under three-point bending. Compos Struct2001; 52: 233–246.
19.
El FatmiR. A non-uniform warping theory for beams. CR Mecanique2007; 335: 467–474.
20.
GjelsvikA. The theory of thin-walled bars. New York: Wiley, 1981.
21.
MurrayNW. Introduction to the theory of thin-walled structures. Oxford: Clarendon Press, 1984.
22.
ProkicA. Thin walled beams with open and closed cross-section. PhD Thesis, University of Belgrade, Belgrade, 1990.
23.
ProkicA. New warping function for thin-walled beams. I: theory. J Struct Eng: ASCE1996; 122: 1437–1442.
24.
ProkicA. New warping function for thin-walled beams. II: finite element method and applications. J Struct Eng: ASCE1996; 122: 1443–1452.
25.
SapountzakisEJMokosVG. Nonuniform torsion of bars of variable cross section. Comput Struct2004; 82: 703–715.
26.
SapountzakisEJMokosVG. Warping shear stresses in nonuniform torsion by BEM. Comput Mech2003; 30: 131–142.
27.
TsipirasVJSapountzakisEJ. Bars under nonuniform torsion—application to steel bars, assessment of EC3 guidelines. Eng Struct2014; 60: 133–147.
28.
CruzJMFMendonçaAV. Torsional analysis of thin-walled beams of open sections by the direct boundary element method. Comput Struct2014; 136: 90–97.
29.
TrefftzE. Über die Wirkung einer Abrundung auf die Torsionsspannungen in der inneren Ecke eines Winkeleisens. ZAMM: Z Angew Math Me1922; 2(4): 263–267.
30.
HuthJH. Torsional stress concentration in angle and square tube fillets. J Appl Mech: T ASME1950; 17(4): 388–390.
31.
ByrneJGCarréBA. Torsional stress concentrations at rounded corners of rectangular hollow sections. J Mech Eng Sci1962; 4(4): 334–340.
32.
ConwayHD. A simple formula for the maximum stress in a twisted angle or channel. Int J Mech Sci1972; 14(9): 615–617.
33.
Ridley-EllisDJOwenJSDaviesG. Theoretical and measured torsional behaviour of rectangular hollow sections. In: Proceedings of the twelfth international offshore and polar engineering conference, Kitakyushu, Japan, 26–31 May 2002. Mountain View, CA: International Society of Offshore and Polar Engineers.
34.
PaoloneARutaGVidoliS. Torsion in multi-cell thin-walled girders. Acta Mech2009; 206(3–4): 163–171.
35.
EcsediIBaksaA. Prandtl’s formulation for the Saint-Venant’s torsion of homogeneous piezoelectric beams. Int J Solids Struct2010; 47: 3076–3083.
36.
HematiyanMREstakhrianE. Saint-Venant torsion of open-section members of uniform thickness. J Strain Anal Eng2011; 46: 56–66.
37.
HematiyanMREstakhrianE. Torsion of functionally graded open-section members. Int J Appl Mech2012; 4: 1250020.
38.
ShiraziAARHHematiyanMR. Closed-form formulation for torsional analysis of beams with open or closed cross sections having a crack. Proc IMechE, Part G: J Aerospace Engineering2013; 227: 1953–1967.
39.
EcsediI. Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars. Mech Res Commun2013; 52: 95–100.
40.
TalebanpourAHematiyanMR. Torsional analysis of piezoelectric hollow bars. Int J Appl Mech2014; 6: 1450019.
41.
BarrettaRFeoLLucianoR. Some closed-form solutions of functionally graded beams undergoing nonuniform torsion. Compos Struct2015; 123: 132–136.
42.
WangCY. Torsion of tubes of arbitrary shape. Int J Solids Struct1998; 35: 719–731.
43.
DoostfatemehAHematiyanMRArghavanS. Closed-form approximate formulas for torsional analysis of hollow tubes with straight and circular edges. J Mech2009; 25: 401–409.
44.
ShahpariS-ZHematiyanMR. Closed-form solutions for torsion analysis of structural beams considering web-flange junctions fillets. J Theor Appl Mech2013; 51: 393–407.
