The analysis of the bending stiffness and intensity of cylindrical tubes

Science in China Series E: Technological Sciences◎2007Science in China PressSpringer-VerlagThe analysis of the bending stiffness andintensity of cylindrical tubesSONG YuQuan#, GUAN ZhiPing', NIE YuQin2 & GUAN XiaoFang'1 Superplastic and Plastic Research Institute, Jlin University, Changchun 130025, China;2 Department of Engineering Mechanics, Institute of Mechanical Science and Engineering, Jilin University,Changchun 130025, ChinaBased on the mechanics of material, the bending stiffness and intensity of cylin-drical bar and tube are analyzed. By comparing the cylindrical tube whose ratio ofoutside diameter to internal diameter is 0.7 with the cylindrical bar, it is concludedthat when both of them have the same mass, the section stiffness of the cylindricaltube is three times that of the cylindrical bar; when both of them have the sameexternal diameter, the mass of the cylindrical tube is only 1/2 that of the cylindricalbar, but the section stiffness of the cylindrical tube is 3/4 that of the cylindrical bar.By virtue of the elemental elastic-plastic theory, the yield stress of the liquid-filledcylindrical tube is investigated. Due to the incompressibility of liquid and the strainhardening effect of material, the yield stress of the liquid-filled tube is enlargedcompared with the hollow tube, thus raising its bending intensity. Under the dy-namic load, compared with the hollow tube, the impact resistance of the liquid-filledtube is also raised due to elastic recovery, Because the hydraulic pressures per-pendicular to the inner surface are identical everywhere, the local stress concen-tration resulting from the ovalisation of the tube would be decreased, and the re-sistance to buckling would be improved.bending of cylindrical tube, liquid flled sifness, intensityToday, the hollow structural members are widely used instead of the solid ones in manufacturingindustry, and have a good developing trend in future. In the report of American Iron and SteelInstitute (AISD), the proportion of light hollow members in the typical autos made in NorthAmerica has been raised to 16% from 10% in the past fifeen years'". Now, by the hydroformingtechnology, General Motor Corporation (GM) has made many light hollow members, including theengine cradle, the radiator cradle, the underbeam, the shed cover joist, the inner support, and so on.The hollow members have a high performance/mass ratio, and especially the ones formed by hy-droforming technology have a special advantage regarding the stiffness and bearing capacity due to中国煤化工Received September 3, 2006; accepted October 20, 2006doi: 1.007511431 007-0031-3MHCNMHGCorresponding author (email: syq@jlu.edu.cn)Supported by the Innovation Foundation of Jilin University and“985 Project" of Jili Universitywww.scichina.com www.springerlink.comSci China Ser E Tech Scil June 2007 Ivol. 50 Ino. 3 I 268-278strain hardening effect' 23 . The further improvement of the stiffness and intensity of the hollowmember would be required to economize material and develop the technology. During the 10thFive-Year Plan period in China, the investment on the railroad industry reached to 350 billion yuan,and more than 1000 km railroads are to be built annually and the rails of more than 900000 tonsincluding the replacing rails of about 600000 tons will be used up for repair. The replacement of therails by the hollow members will be significant to sustainable development. In this paper, thebending stiffness and intensity of cylindrical tubes are investigated in the light of the elementalelastic-plastic theory under the condition that material is continuous and homogeneous, and Hook'slaw is obeyed during the elastic deformation and the assumption of constant volume is obeyedduring the plastic deformation. In particular, by analyzing the liquid-filled cylindrical tube, it isconcluded that regarding the improvement of the stiffness and intensity of the members, someliquid-filled members have superiority over the hydroforming members.1 The stiffness of cylindrical bars and tubesRigidity is the capability of resistance to elastic deformation regarding the essence of material, andthe modulus of elasticity E, which is the important mechanical index determining the rigidity ofmaterial, mainly depends on the property of atomic bond, the bond strength and the lattice type.Stiffness is the capability of resistance to elastic deformation regarding the structural member,including section stiffness Tj and member stiffness T. The section stiffness Tj under the pure tensileor compressive load is expressed as the product of E and the cross-sectional area; the bendingsection stiffness is expressed as EI, where I is the moment of inertia of the cross-sectional areaabout the neutral axis; the bending member stiffness T is dependent on the section stiffness EI andthe span of member.After the couples are applied to the cylindrical bar, the material is stretched above the neutrallayer and compressed below it. The neutral axis is the intersecting line between the neutral layerand cross section. The neutral axis on the cross section is taken as z axis of the coordinate system;the symmetric axis perpendicular to the neutral axis is taken as y axis; the axis normal to the crosssection is taken as x axis. According to the geometry of deformation shown in Figure 1, we obtainmn=dx= ρdθ,(1)mn'=(ρ- y)d0,(2)m'n' -mn .(3)mnlxSubstituting eqs. (1) and (2) into eq. (3), we obtain(4)PSubstuting eq. (4) into Hook's law, we obtainEσ=-=y.(5)p'1.1 The bending moment中国煤化工When the bar is subjected to the couple M at the ends, th,MYHC N M H Gasses dependentSONG YuQuan et al. Sci China Ser E-Tech Scil June 2007 Ivol. 50 Ino.3 I 268-278269▲”n'ndFigure 1 Geometry of deformation of bending tube.on bending moment at the cross section. According to the equilibrium condition,[ odA, the totalforce of the internal forces σdA at the cross section, is equal to the axial force N applied to theCross section, which is zero, namelyN= I σdA=0.(6)ESubstituting eq, (5) into (6), we obtain」dA=-=s:=0. S2=」,ydA is dfined as thestatic moment of the cross sectional area about the z axis. Because of Sz = 0 resulting from eq. (6),the neutral axis z must be coincident with the centroid of cross section of the bar.According to the equilibrium condition, the total moment of the intermal forces σdA about yaxis is equal to the bending moment My applied to the cross section, which is zero, namelyM,= I zσdA=-=」yzdA=0. I yzdA, characterized by the symbol Ix, is defined as themoment of inertia of the cross-sectional area about y and z axis, and IV = 0. The y and z axes arerequired to be the principal inertial axes according to the properties of areas. Because we select thesymmetric axis perpendicular to the neutral axis as y axis, the condition of Iy =0 can be met.According to the equilibrium condition, the total moment of the internal forces σdA about zaxis is equal to the bending moment Mz applied to the cross section, which is -M, where the upwardbending moment is defined as positive direction. We obtainM= I yodA.(7)Subtituting eq. (5) into (7), we have M: =-[,ydA=-5y, where 1:=」,y?dA isde-fined as the moment of inertia of the cross-sectional中国煤化工s z. Thus, we canobtain the relationship between the bending moment a.MYHCNMHGature:270SONG YuQuan et al. Sci China Ser E- Tech Scil June 2007 Ivol. 50 | no.31 268-2781M(8)ρ EI,As shown in Figure 2(a), regarding the cross section of cylindrical bar, dA=2Vr2 - y2dy.Therefore, the moment Iz of inertia of the cross-sectional area about z axis is expressed as(9)4As shown in Figure 2(b), regarding the cross section of cylindrical tube with extermal diameterof 2rpb and intermal diameter of 2rga , the moment Ig of inertia of the cross-sectional area aboutz axis is similarly expressed as(第-品)Igz=-(10)4z少(a)(b)Figure 2 The cross section of the cylindrical bar and tube. (a) The cylindrical bar; (b) the cylindrical tube.Regarding the structural member with the length of I subjected to the moment M of the couple,its bending deformation is characterized by the turn angle θ between the two ends of the member,and the larger the turm angle θ , the larger the extent of its deformation. According to the rela-dtionship dθ =二, the geometry of deformation shown in Figure 1 and eq. (8), we express the turmρangle θ as θ=dx=pM-dx =Mwhere I is the moment of inertia of thebop(x)°J0EIEIIL’cross sectional area about z axis. The bending section stiffness EI is characterized by the symbol T,andEis the bending member sifness characterized by the symbol T, namely T;= EI andlT=-EI. Therefore, regarding the cylindrical bar, its bending section stiffness Tj and its bendingmember stiffness T are respectively expressed asπr°E中国煤化工(11)MHCNMHGSONG YuQuan et al. Sci China Ser E-Tech Scil June 2007 Ivol. 50 Ino.3| 268-278271_πr;*ET,=(12)Similarly, regarding the cylindrical tube, its bending section stiffness T。and its bendingmember siffness Tg are respectively expressed asEr(r&b-ra)Ty=4(13)En(5-)Tg=一4l(14)1.2 The stiffness/mass ratioRegarding the cylindrical bar and tube, the mass of unit length are respectively r"d andπ(r2 -r2 )d , where d is the density of material. Then, dividing eqs. (1) and (13) by the twoexpressions of mass, we obtain the ratios Tm and Tgm of the section stiffness to the mass of unitlength regarding the cylindrical bar and tube respectively: ._rζE(15)Id(rpb +rq)ETgm =-(16)4dLet Tgm /Tm =k. k is defined as the tube/bar ratio about the siffnessmass ratio. By eqs. (15) and(16), we obtaink=.(17)Let the mass of unit length of bar be equal to that of tube, namely,ζ=届-品(18)Let n characterize rga/rgb . Then0