About the Torsional Constant for thin-walled rod with open. cross-section. Duan Jin1,a, Li Yun-gui1

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1 Internationa Forum on Energy, Environment Science and Materias (IFEESM 17) bout the Torsiona Constant for thin-waed rod with open cross-section Duan Jin1,a, Li Yun-gui1 1 China State Construction Technica Center, Beijing, 11, PR China a duanjin78@16.com Key words: open thin-waed rod, torsiona constant, finite eement method bstract: This paper presents theoreticay the torsiona inertia moment for open thin-waed rod, based on the theory of thin-waed she. In the deduction, the rigid contour hypothesis is adopted and the cross-sectiona warp is taken into account. With the above torsiona inertia moment obtained and using the simiar method for cosed beam eements in Reference [1], the stiffness matrix of thin-waed beam eement with open cross-section is derived with the warping of cross-section considered. Other than the traditiona stiffness matrix of beam eement having tweve DOFs, i.e. six for each node, this stiffness matrix is corresponding to fourteen DOFs, i.e. seven for each node. This matrix coud be used for the finite eement anaysis of open thin-waed rod, with section warping considered. Introduction In buiding structures, the thin-waed rod with open cross-section is widey used. It is usuay anayzed by theoretica soution or finite eement method (FEM) [1-] for simpe structure or compicated structure respectivey. Either way, there is an important or even critica property of the cross-section needing to be determined first. It is the torsiona inertia moment for open thin-waed cross-section. In this paper, the torsiona inertia moment for open thin-waed rod is derived theoreticay based on the theory of thin-waed she. In the deduction, the rigid contour hypothesis is adopted and the cross-sectiona warp is taken into account. With the above torsiona inertia moment obtained and using the simiar method for cosed beam eements in Reference [1], the stiffness matrix of thin-waed beam eement with open cross-section is derived with the warping of cross-section considered. Other than the traditiona stiffness matrix of beam eement having tweve DOFs, i.e. six for each node, this stiffness matrix is corresponding to fourteen DOFs, i.e. seven for each node. This matrix coud be used for the finite eement anaysis of open thin-waed rod, with section warping considered. The torsiona constant for thin-waed rod with open cross-section Figure 1 gives an iustration of the cross section of an open thin-waed rod. There are three sets of coordinate system, i.e. Cartesian coordinate system ocated at the centroid of cross-section, contour coordinate system whose origina ocation is moving aong the midcourt ine of the thin-waed cross section, and coordinate system ocated at the shear center, whose axes are parae to that of contour coordinate system. Detaiedy speaking, seect arbitrary point P at the midcourt ine C as the origina point, and denote the vectors Copyright 18, the uthors. Pubished by tantis Press. This is an open access artice under the CC BY-NC icense ( 9

2 aong the wa thickness orientation and the tangentia midcourt ine as n and s respectivey, then the contour coordinate system is obtained. Offset the origina point of contour coordinate system to the shear point, and denote the axes parae to n and s as n% and s% respectivey, then another coordinate system is obtained. Wa thickness t Centroid z s s% P n y x Midcourt ine denoted as C q O r Shear center n% Cy Cz The origin of contour coordinate s Figure 1 Cross section of an open thin-waed rod ccording to Reference [5], the warping function of the cross-section is as foowing: ω (n, s ) = ω ω (1) Where ω denotes the contour warping function, equaing the sectoria area of the midcourt ine, ω denotes the thickness warping function, see the foowing Equation (): s ω = rc ( s )ds ω, ω = nqc ( s ) () Where ω denotes the contour warping function at s=; rc, qc denotes r and q respectivey corresponding to the midcourt ine The foowing Equation () coud be obtained from Equation (1) and (): ω ω = rc ( s ) n, = qc ( s ) s n () The foowing equations coud be obtained for pure torsiona probem at cyindrica coordinate system xns: u x = ω (n, s ) χ, un = q ( s )θ x, us = r ( s )θ x (4) The shearing strain is as foow: u x un γ xn = n x = ( qc q ) θ x, x γ = u x us = ( r ( s ) (r ( s ) n) ) θ c x, x xs s x (5) Observing Figure 1, the foowing equation coud be obtained:

3 r ( s ) = rc ( s ) n, q ( s ) = qc ( s ) (6) Substitute Equation (6) into (5), the foowing Equation (7) coud be derived: γ xn =, γ xs = nθ x, x (7) The potentia variationa formua for thin-waed rod is as foow [1-]: δπ = Gγ xsδγ xs dd = GJθ x, xδθ x, x d (8) Where J denotes the torsiona inertia moment, see the foowing equation for detais: s t / s 1 (n) dnds = t ds s1 t / s1 J = (n) d = (9) It is obviousy that J denotes St.Venant torsiona constant. Deducing the above equations in rectanguar coordinate system, then J = [( z Cz ω, y ) ( y C y ω, z ) ]d (1) Considering the fact that torsiona inertia moment is the inherent nature of the cross-section and has nothing to do with the coordinate system, the foowing equation coud be derived: J =J = [( z Cz ω, y ) ( y C y ω, z ) ]d (11) The stiffness matrix of thin-waed beam eement with open cross-section Use the simiar method for cosed beam eements, see Reference [1], together with the above Equation (11), the inear stiffness matrix of open thin-waed beam eement coud be derived as foow: K11 K 1 K = 1 K K E 1 EI z 11 K = (1) 1 EI y 6GJ 1 EIω 5 4 EI y sym 4 EI z GJ 6 EIω 1 GJ 4 EIω 15 (1) 1

4 E K 1 = 1 EI z 1 EI y 6GJ 1 EIω 5 EI y EI z GJ 6 EIω 1 E 1 EI z 1 K = E 1 EI z K = 1 EI y 6GJ 1 EIω 5 EI z GJ 6 EIω 1 6GJ 1 EIω 5 4 EI y sym 1 EI y EI y 4 EI z GJ 6 EIω 1 GJ EIω GJ 6 EIω 1 GJ EIω (14) GJ 6 EIω 1 GJ 4 EIω 15 (15) (16) Where I y and I z denotes the principa moment of inertia according to axis v and z respectivey;

5 J denotes the torsiona moment of inertia for open thin-waed rod, see Equation (11); Iω denotes the cross section s warping moment of inertia, see the foowing equation: Iω = ω d (17) Where ω denotes the warping function of cross-section [4,5,6] Summary In this paper, the torsiona inertia moment for open thin-waed rod is derived theoreticay based on the theory of thin-waed she. In the deduction, the rigid contour hypothesis is adopted and the cross-sectiona warp is taken into account. With the above torsiona inertia moment obtained and using the simiar method for cosed beam eements in Reference [1], the stiffness matrix of thin-waed beam eement with open cross-section is derived with the warping of cross-section considered. Other than the traditiona stiffness matrix of beam eement having tweve DOFs, i.e. six for each node, this stiffness matrix is corresponding to fourteen DOFs, i.e. seven for each node. This matrix coud be used for the finite eement anaysis of open thin-waed rod, with section warping considered. Reference [1] Bathe K J, Boourchi S. Large Dispacement naysis of Three-Dimensiona Beam Structures. Int. J. Num. Mech. Engng., 1979, 14: [] Bathe K J. Finite Eement Procedures. NJ: Prentice-Ha, 1996 [] Wang X C. Finite Eement Method. Beijing, China: Tsinghua University Press, (in Chinese) [4] Vasov V Z. Thin waed eastic beams. Israe Program for Scientific Transation, Jerusaem, 1961 [5] Hu H C & Xie B M. Statics for Eastic Thin-waed Rod. 1955, Beijing, China: Science Press (in Chinese, Transated from Russian) [6] Gjesvik. The theory of thin waed bars. New York: Wiey, 1981

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