Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 3, pp. 553-571, Warsaw 2009 OPTIMIZATION OF ANTI-SYMMETRICAL OPEN CROSS-SECTIONS OF COLD-FORMED THIN-WALLED BEAMS Jerzy Lewiński Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland e-mail: jerzy.lewinski@put.poznan.pl Krzysztof Magnucki Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland Institute of Rail Vehicles, Tabor, Poznań, Poland e-mail: krzysztof.magnucki@put.poznan.pl The paper deals with cold-formed thin-walled beamswith the Z-, S- and Clothoid-section.A short survey of optimal designs of thin-walled beams with open cross-sections is given. Geometric properties of three cross- sections are described. Strength, local and global buckling conditions for thin-walled beams are presented. The optimal design criterion with a dimensionless objective function as a qualitymeasure is defined. Results of numerical calculations for optimal shapes of three cross-sections are presented in tables and figures. Key words: thin-walled beam, open cross-section, global and local buckling, optimal design Notations a,b,c,d – dimensions of cross-sections r – radius of the circular arc t – thickness of the beamwall u – dimensionless parameter of the clothoid A – area of the cross-section H – depth of the beam L – length of the beam JS−V – geometric stiffness for Saint-Venant torsion Jy,Jz – inertia moments 554 J. Lewiński, K. Magnucki Jω – warpingmoment of inertia M0 – loading moment R – principal radius of the clothoid α,β – angles of the S-section λ – relative length of the beam θp – angle to the principal axes ω – warping function Φj – objective function 1. Introduction Shapes of open cross-sections of contemporary cold-formed thin-walled beams are rather complicated. They are usually mono-symmetrical, although some- times anti-symmetrical too. The main constraints in designing thin-walled structures are strength and stability conditions. The beginnings of the opti- mal design of thin-walled structures reach back to 1959. The first paper on optimal design of a thin-walled beamwith an open cross-section (I-section) in pure bending state was presented byKrishnan and Shetty (1959). A complete survey of optimal designproblemsof structures for the secondhalf of the twen- tieth century was given by Gajewski and Życzkowski (1988) and Krużelecki (2004). A bibliography on the problems of topology and shape optimization of structures using FEM and BEM for 1999-2001 was collected by Mackerle (2003). Optimal design criteria for shapes of thin-walled beams cross-sections under strength and local and global stability constraints was presented by Cardoso (2000). Karim and Adeli (1999) presented global optimum design of cold-formed steel hat-shape beams under uniformly distributed load using a neural network model. Variational and parametric design of an open cross- section of a thin-walled beamunder stability constraints was describedbyMa- gnucki and Magnucka-Blandzi (1999), Magnucki and Monczak (2000). Vinot et al. (2001) presented amethodology for optimizing the shape of thin-walled structures.Magnucki (2002) studied optimization of an open cross-section of a thin-walled beamwith flatweb and circular flange analytically and numerical- ly. Knowledge-based global optimization of cold-formed steel columns under pure axial compressionwas presented by Liu et al. (2004). In result of the stu- dy, five anti-symmetrical open cross-sections were proposed. Theoretical and experimental studyon theminimumweight of cold-formed channel thin-walled beamswith andwithout lips were analysed byTian and Lu (2004). Optimum design of cold-formed steel channel beams under uniformly distributed load Optimization of anti-symmetrical open cross-sections... 555 usingmicroGeneticAlgorithmwas presentedbyLee et al. (2005).Global opti- mization of cold-formed steel thin-walled beams with lipped channel sections were described by Tran and Li (2006). Optimal design of open cross-sections of cold-formed thin-walled beams with respect to the dimensionless objective function as the quality measure was presented byMagnucka-Blandzi andMa- gnucki (2004b), Magnucki and Ostwald (2005a,b), Magnucki et al. (2006a,b), Magnucki and Paczos (2008). Kasperska et al. (2007), Ostwald et al. (2007), Ostwald andMagnucki (2008),Manevich andRaksha (2007) describedbicrite- rial optimal design of open cross-sections of cold-formed beams. Strength, glo- bal and local buckling and optimization problems of cold-formed thin-walled beamswith open cross-sections were collected and described byMagnucki and Ostwald (2005a,b), Ostwald andMagnucki (2008). The present paper provides further development of optimal shaping of anti-symmetrical open cross-sections of cold-formed thin-walled beams in pure bending state. These beams of the length L, depth H, and wall thickness t are simply supported and carry two equal moments M0 applied to the beam ends (Fig.1). The optimization includes three anti-symmetrical cross-sections: Z-section, S-section and clothoid-section. Fig. 1. A scheme of the thin-walled beam 2. Geometric properties of three cross-sections 2.1. Anti-symmetrical Z-section A scheme of the cross-section with principal axes yz is shown in Fig.2. The middle line of the Z-section is a broken line situated symmetrically with respect to the origin O(0,0). Geometric properties of the cross-section are defined by the following di- mensionless parameters x1 = b a x2 = c b x3 = t b x4 = d a (2.1) where: a, b, c, d – sizes of the cross-section, t – thickness of the wall. 556 J. Lewiński, K. Magnucki Fig. 2. A scheme of the Z-section Depth of the beam is H =2a+ t= a(2+x1x3) (2.2) Total area and geometric stiffness for Saint-Venant torsion of the cross-section A=2atf0(xi) JS−V = 2 3 at3f0(xi) (2.3) where f0(xi)=x1(1+x2)+x4+ √ (1−x4)2+ 1 4 x21 The product of inertia with respect to the principal axes yz is zero Jyz =2a 3tx1[−3(x1x2−x4)(2−x1x2−x4)+(1−x1) √ x21+4(1−x4)2] = 0 (2.4) fromwhich x2 = 1 x1 (1+ √ 1−C0) (2.5) where C0 =(2−x4)x4+ 1 3 (1−x1) √ x21+4(1−x4)2 Moments of inertia of the plane area (Fig.2) with respect to the y and z axes are Jy =2a 3tf2(xi) Jz =2a 3tf3(xi) (2.6) where f2(xi)= 1 4 x21 [ x1 (1 3 +x2 ) +x4+ 1 6 √ x21+4(1−x4)2 ] f3(xi)=x1+ 1 3 [2− (1−x1x2)3− (1−x4)3]+ 1 6 (1−x4)2 √ x21+4(1−x4)2 Optimization of anti-symmetrical open cross-sections... 557 The warping function ω(s) for the Z-section (half section) is shown in Fig.3. Fig. 3. Geometric interpretation of the warping function ω(s) The warping function in characteristic points of the Z-section have the following values ω1 =0 ωi = a iω̃i i=2,3,4 (2.7) where ω̃2 = 1 2 x1x4 ω̃3 = ( 1+ 1 2 x4 ) x1 ω̃4 = [ 1+ 1 2 (x1x2+x4) ] x1 The warpingmoment of inertia Jω =2a 5tf5(xi) (2.8) where f5(xi)= 1 3 [x4ω̃ 2 2 +x1(ω̃ 2 2 + ω̃2ω̃3+ ω̃ 2 3)+x1x2(ω̃ 2 3 + ω̃3ω̃4+ ω̃ 2 4)] The centroid and the shear center of the plane area of anti-symmetrical cross- sections are located in the origin O(0,0). 2.2. Anti-symmetrical S-section A scheme of the cross-section with auxiliary axes y1z1 and principal axes yz is shown in Fig.4. The middle line of the S-section is a composite curve (two circles and one line segment) situated symmetrically with respect to the origin O(0,0). 558 J. Lewiński, K. Magnucki Fig. 4. A scheme of the S-section Geometric properties of the cross-section are defined by the following di- mensionless parameters x1 = r a x3 = t r and β (2.9) where: a, r – sizes of the cross-section, β – angle, t – thickness of the wall. Depth of the beam is H =2acosθp+2r+ t=2a [ cosθp+x1 ( 1+ 1 2 x3 )] (2.10) Total area and geometric stiffness for Saint-Venant torsion of the cross-section A=2atf0(xi) JS−V = 2 3 at3f0(xi) (2.11) where f0(xi)= √ 1−x21+x1(π+β−α) cosα=x1 The product of inertia with respect to the auxiliary axes y1z1 Jy1z1 =2a 3tf1(xi) (2.12) where f1(xi)= {1 3 (1−x21)2+x1 [ cosβ+x1+ 1 4 x1(1+cos2β−2x21) ]} x1 Optimization of anti-symmetrical open cross-sections... 559 Moments of inertia of the plane area (Fig.4) with respect to the y1 and z1 auxiliary axes are Jy1 =2a 3tf2(xi) Jz1 =2a 3tf3(xi) (2.13) where f2(xi)= {1 3 √ (1−x21)3+ 1 4 x1[2(π+β−α)+2x1 √ 1−x21− sin2β] } x21 f3(xi)= 1 3 √ (1−x21)5+ + 1 2 x1{(π+β−α)(2+x21)+x1[ √ 1−x21(4−x 2 1)+(4+x1cosβ)sinβ]} The angle θp defining the principal axes is tan2θp =− 2Jy1z1 Jz1 −Jy1 (2.14) and, principal moments of inertia Jy = 1 2 (Jz1+Jy1)− √ 1 4 (Jz1+Jy1)2+J2y1z1 (2.15) Jz = 1 2 (Jz1+Jy1)+ √ 1 4 (Jz1+Jy1)2+J2y1z1 The warping function ω(ϕ) for the S-section (half section) is shown in Fig.5. Fig. 5. Geometric interpretation of the warping function ω(ϕ) The warping function for the S-section is defined as follows ω(ϕ)= [sinα− sin(α+ϕ)+x1ϕ]x1a2 (2.16) 560 J. Lewiński, K. Magnucki The warpingmoment of inertia Jω =2a 5tf5(xi) (2.17) where f5(xi)= (f51−f52+f53+f54)x31 f51 = 1 2 ( x1 √ 1−x21− 1 2 sin2β ) f52 =2[(π+β−α)x1+ √ 1−x21]cosβ f53 = [2sinβ− (π+β−α) √ 1−x21]x1 f54 = 1 6 (π+β−α){9+2[(π+β−α)2−3]x21} 2.3. Anti-symmetrical Clothoid-section A scheme of the cross-section with auxiliary axes y1z1 and principal axes yz is shown in Fig.6. The middle line of the Clothoid-section is a curve situated symmetrically with respect to the origin O(0,0). Fig. 6. A scheme of the Clothoid-section In Cartesian auxiliary coordinates, the curve is parametrized as follows y1 = a √ π u1∫ 0 sin πu2 2 du z1 = a √ π u1∫ 0 cos πu2 2 du (2.18) Optimization of anti-symmetrical open cross-sections... 561 where a is the scale parameter determining the outer size of the curve, u – dimensionless parameter (0¬u¬u1). The principal curvature radius R= a2 s (2.19) where arc length s= a √ πu1. Geometric properties of the cross-section are defined by the following di- mensionless parameters x1 =u1 x3 = t a (2.20) Depth of the beam is H =2d+ t (2.21) where: u1 is the upper integration limit, as in (2.18), deciding on the ”depth” of convolutions of the curve, t – wall thickness. The total area of the clothoid cross-section is A=2 ∫ A dA=2t ∫ OP ds=2 √ πatu1 (2.22) The moments of inertia of the plane area with respect to the z1 and y1 axes are Iz1 = ∫ A y21 dA=2at √ π u1∫ 0 a2π ( u∫ 0 sin πv2 2 dv ) du=2a3t √ π3 u1∫ 0 [s(u)]2 du (2.23) where s(u)= u∫ 0 sin πu2 2 du (2.24) and Iy1 = ∫ A z21 dA=2a 3t √ π3 u1∫ 0 [c(u)]2 du (2.25) where c(u)= u∫ 0 cos πu2 2 du (2.26) 562 J. Lewiński, K. Magnucki The principal axes and principal moments of inertia are defined by the same expressions as for the S-section, i.e. (2.14) and (2.15). The warping function of the clothoid section (Fig.7) takes the following form ω=2 (1 2 zpyp− zp∫ 0 y1 dz1 ) (2.27) where zp = a √ π up∫ 0 cos πu2 2 du yp = a √ π up∫ 0 sin πu2 2 du Fig. 7. Geometric interpretation of the clothoid warping function ω(u) According to earlier definitions (2.23) and (2.25) zpyp =πa 2c(up)s(up) dz1 = a √ πcos πu2 2 du (2.28) Hence, the warping function, being a function of the parameter up, may be formulated as follows ω(up)=πa 2 [ c(up)s(up)−2 up∫ 0 cos πu2 2 s(u) du ] (2.29) Optimization of anti-symmetrical open cross-sections... 563 Finally, the warpingmoment of inertia of the clothoid section is calculated as follows Iω = ∫ A ω2 dA=2t up∫ 0 ω2(u) ds (2.30) 3. Formulation of the optimization problem 3.1. Optimization criterion The minimal mass and maximal safe load are usually a basic objective in structure designing. The optimization criterion according to the papers of Magnucka-Blandzi and Magnucki (2004a,b), Magnucki et al. (2006a,b), has been formulated in the following form max xi {Φ1(xi),Φ2(xi),Φ3(xi),Φ4(xi)}=Φmax (3.1) and the objective function Φj(xi)= Mj E √ A3 (3.2) where Mj are the allowable moments defined from the strength condition (j = 1), lateral buckling condition (j = 2), local buckling condition of the flange (j=3), and local buckling condition of the web. 3.2. Constraints Strength and buckling are main problems in thin-walled structures desi- gning. Lateral buckling strengths of a cold-formed Z-section beam was pre- sented by Pi et al. (1999). Li (2004) described lateral-torsion buckling of the cold-formed Z-beam. The effects of warping stress on the lateral torsional buckling, and local and distortional buckling of cold-formed Z-beamswere de- scribed by Chu et al. (2004, 2006). Stasiewicz et al. (2004) described local buckling of a bent flange of a thin-walled beam. Analytical and numerical analysis of the stress state and global elastic buckling of a thin-walled beam with amono-symmetrical open cross-sectionwas presented byMagnucki et al. (2004). Critical stresses for open cylindrical shells with free edges were calcu- lated byMagnucka-Blandzi andMagnucki (2004b),Magnucki andMackiewicz (2006) and Joniak et al. (2008). Ventsel and Krauthammer (2001) collected and described strength and buckling problems of thin plates and shells. 564 J. Lewiński, K. Magnucki The space of feasible solutions for optimal shapes of cross-sections of thin- walled beams is restrained. The strength condition has the following form M0 ¬M1 M1 =2 Jz H σall (3.3) where σall is the allowable stress. The global stability condition (lateral buckling condition) for a simply supported beam in pure bending state has the following form M0 ¬M2 M2 = M (Globl) CR cs1 (3.4) where cs1 is the safety coefficient, and the lateral bucklingmoment for a simply supported thin-walled beam in pure bending state is (Magnucki andOstwald, 2005a,b) M (Globl) CR = πE L √ JyJS−V 2(1+ν) [ 1+2(1+ν) π2 L2 Jω JS−V ] (3.5) The local stability conditions for theZ-beam are as follows: • for the bent flange, according to Magnucki and Ostwald (2005a,b) and Stasiewicz et al. (2004) σ(Z−flange)max ¬ σ (Z−flange) CR cs2 σ (Z−flange) CR = 1+x2 1+3x2 x23G (3.6) where σ (Z−flange) CR is the critical stress, G=E/[2(1+ν)] – shearmodulus of elasticity, E – Young’s modulus, ν – Poisson’s ratio, cs2 – safety coefficient. Taking into account the classical theory of plates, the local stability condition for the bent flange may be written down as M0 ¬M3 M3 = σ (Z−flange) CR cs2 Jz a−ef = 2a2t cs2 G 1+x2 1+3x2 x23 f3(xi) 1− ẽf (3.7) where ẽf is the dimensionless parameter of the centroid location of the flange ẽf = x2 2(1+x2) f1(xi) Optimization of anti-symmetrical open cross-sections... 565 • for the flat web according to Ventsel and Krauthammer (2001) σ(Z−web)max ¬ σ (Z−web) CR cs2 σ (Z−web) CR = 2π2 1−ν2 E (x1x3)2 x21+4(1−x4)2 (3.8) where σ (Z−web) CR is the critical stress. Taking into account the classical theory of beams, the local stability condition for the flat web may be put down as M0 ¬M4 (3.9) M4 = σ (Z−web) CR cs2 Jz a−d = 4π2a2t cs2(1−ν2) E (x1x3)2 x21+4(1−x4)2 f3(xi) 1−x4 The local stability conditions for the S-beam take the following forms: • for the open circular cylindrical shell, regarding the results ofMagnucka Blandzi andMagnucki (2004a),Magnucki andMackiewicz (2006), Joniak et al. (2008) σ(S−shell)max ¬ σ (S−shell) CR Cs2 σ (S−shell) CR =αC E 12.7 √ 3(1−ν2) x3 (3.10) where σ (S−shell) CR is the critical stress and αC – coefficient αC =1+0.8 ( β− π 2 )4 Taking into account the classical theory of beams, the local stability condition for the circular cylindrical flangemay be written down as M0 ¬M3 M3 = σ (S−shell) CR cs2 Jz a = 2at2 12.7cs2 √ 3(1−ν2) EαC f3(xi) x1 (3.11) • for the flat web Ventsel and Krauthammer (2001) σ(S−web)max ¬ σ (S−web) CR cs2 σ (S−web) CR = π2 2(1−ν2) E (x1x3)2 1−x1 (3.12) where σ (S−web) CR is the critical stress. 566 J. Lewiński, K. Magnucki Taking into account the classical theory of beams, the local stability condition for the flat web is M0 ¬M4 (3.13) M4 = σ (S−web) CR cs2 Jz a−d = 4π2a2t cs2(1−ν2) E (x1x3)2 x21+4(1−x4)2 f3(xi) 1−x4 The local stability conditions for theClothoid-beam are as follows: • for the open cylindrical shell, according to Magnucka Blandzi and Ma- gnucki (2004b), Magnucki andMackiewicz (2006), Joniak et al. (2008) σ (Cl−shell) edge ¬ σ (Cl−shell) CR,edge cs2 σ (Cl−shell) CR,edge = E 12.7 √ 3(1−ν2) t Redge (3.14) where σ (Cl−shell) CR,edge is the critical stress. Taking into account the classical theory of beams, the local stability condition for the circular cylindrical flange assumes the form M0 ¬M3 (3.15) M3 = σ (Cl−shell) CR,edge cs2 Jz d = 2at2 12.7cs2 √ 3(1−ν2) EαC f3(xi) x1 • for the cylindrical shell σ (Cl−shell) local ¬ σ (Cl−shell) CR,local cs2 σ (Cl−shell) CR,local = E √ 3(1−ν2) t R(y) (3.16) where σ (Cl−shell) CR,local is the critical stress. Taking into account the classical theory of beams, the local stability condition for the flat web is M0 ¬M4 M4 = σ (Cl−web) CR,local cs2 Jz y(u) (3.17) Optimization of anti-symmetrical open cross-sections... 567 4. Numerical solution of the optimization problem Optimization of three anti-symmetrical open cross-sections has been per- formed for a family of cold-formed thin-walled beams: σall/E = 0.0015, ν = 0.3, cs1 = 1.5, cs2 = 2.1, with relative lengths λ = L/H = 7.5, 10.0, 12.5, 15.0, 17.5, 20.0. The results of numerical calculations for the Z-beamare specified in Table 1, for the S-beam in Table 2, and for the Clothoid-beam in Table 3. Table 1.Optimal parameters for the Z-beam λ 7.5 10.0 12.5 15.0 17.5 20.0 x1,opt 0.3897 0.5262 0.6607 0.7908 0.9161 1.0359 x2,opt 0.7259 0.4581 0.3051 0.2077 0.1400 0.0903 x3,opt 0.1175 0.1131 0.1098 0.1072 0.1052 0.1036 x4,opt 0.0458 0.0595 0.0725 0.0850 0.0964 0.1072 Φmax 0.0020725 0.001825 0.001645 0.001507 0.0013940 0.001301 Table 2.Optimal parameters for the S-beam λ 7.5 10.0 11.15 12.5 15.0 17.5 20.0 x1,opt 0.3696 0.3696 0.3696 0.4145 0.4971 0.5789 0.6598 βopt π π π π π π π x3,opt 0.047951 0.047951 0.047951 0.04641 0.04383 0.04155 0.0395 Φmax 0.003049 0.003049 0.003049 0.002857 0.002573 0.002356 0.002187 Table 3.Optimal parameters for the Clothoid-beam λ 7.5 10.0 12.5 15.0 17.5 20.0 x1,opt 2.6 2.6 2.6 2.6 2.6 2.6 x3,opt 0.014353 0.014353 0.014353 0.014353 0.014353 0.014353 Φmax 0.003553 0.003553 0.003553 0.003553 0.003553 0.003553 5. Conclusions The criterion of effective shaping (optimal design) with dimensionless objecti- ve functions (26) enables sorting and comparing beams with arbitrary cross- sections. This criterion is a quality measure of the cross-sections of beams. 568 J. Lewiński, K. Magnucki Fig. 8. Dimensionless objective function Φ max for three considered types of beams According to the plots in Fig.8, the following conclusions may be drawn: • In the case of the Z-Section, the lateral buckling is decisive for the beam of relative length 7.5¬λ. • For the S-Section of the relative length λ¬ 11.15, the lateral buckling imposes no constraint – the condition remains inactive. It is active only for λ> 11.15. • In the case of theClothoid-Section, the lateral buckling remains inactive within the whole considered range of the relative length λ. The beams with the Clothoid-section are definitely better than those with Z- or S-sections. References 1. 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Vinot P., Cogan S., Piranda J., 2001, Shape optimization of thin-walled beam-like structures,Thin-Walled Structures, 39, 611-630 Optymalizacja otwartych antysymetrycznych przekrojów belek cienkościennych walcowanych na zimno Streszczenie Wartykule rozważane są belki o przekrojachpoprzecznychwkształcie Z-, S- oraz w kształcie klotoidy. Zamieszczono krótki przegląd zagadnień optymalnego projek- towania belek cienkościennych o przekrojach otwartych. Opisano właściwości geome- tryczne trzech rozważanych przekrojów. Zapisano warunki wytrzymałości oraz lokal- nej i ogólnej stateczności belek cienkościennych. Sformułowanokryteriumoptymaliza- cyjne z wykorzystaniembezwymiarowej funkcji celu będącej miarą jakości przekroju. Wyniki numerycznychobliczeńoptymalnychzarysówprzekrojówpoprzecznychprzed- stawiono w tablicach i na rysunkach. Manuscript received April 3, 2009; accepted for print April 22, 2009