TX_1~ABS:AT/ADD:TX_2~ABS:AT 25 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 Review ARticle The Comparative Study of Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches Mohammed M. Saleh1*, Dlshad K. Ahmed2, Ali R. Yousef 3 1Department of Civil Engineering, Engineering Collage, Universita della Calabria, Cosenza, Italy, 2Department of Civil Engineering, Cihan University-Erbil, Kurdistan Region, Iraq, 3Department of Civil Engineering, Engineering Collage, Salahaddin University, Erbil, Iraq ABSTRACT This study investigates the capacity of the steel section using both AISC and Eurocode approaches. Three types of steel sections were subjected to biaxial bending by applying loads to both main axes and examined by both approaches. The concept of Fisher was also adopted as an approach. This concept proposed that only the compression flange could withstand lateral loading and the torsional influence could be ignored. The findings suggested that the Eurocode approach is more conservative in the design of steel sections subject to biaxial bending as it takes into account the level at which the load is applied, the type of the section whether rolled or welded and its height-to-width ratio (lateral buckling effect). The AISC approach considers the shear center of the section as the level at which the loads are applied. The conservatism of the results was more pronounced when the section is close to H-section. Fisher`s concept of structural design of biaxial bending of structural steel is more conservative than both AISC and Eurocode approaches of analysis. Keywords: AISC specification, biaxial bending analysis, eurocode, fisher`s concept, steel sections INTRODUCTION Due to many factors, such as good mechanical properties, quick and easy construction and economy, steel structures are commonly used for construction. Overhead crane runway girders are examples of biaxial bending in which bending moments applied to both major and minor axis. When the loads applied through the shear center, twisting would not develop[2] as shown in Figure 1. For this case, AISC 360 Commentary,[1] biaxial bending equation given by the AISC 360 Eq. (H1-1b) by supposing of axial load that equal to one. The interaction equation then reduces to M M M M ry cy rz cz � � 1 (1) where M ry = applied bending moment (y-axis), M cy = nominal moment capacity (y-axis), M rz = applied bending moment (z-axis), M cz = nominal moment capacity (z-axis). For the LRFD method, Mc = Φ b M n , while for ASD method, Mc = M n /Ω b , Φ b = 0.9 = factor for resistance in flexure, Ω b = 1.67 = safety factor for flexure. Regarding to the bending about the strong axis, the Eurocode method focused on several factors that could affect the flexure strength of the beams. The Elastic Critical Moment is valuable for the study of lateral-torsional beam buckles. The maximum bending moment value provided by the beam is described as this quantity, far from imperfections of some kind. 2 2 1E Wcr T z T E I M G I E I L L G I π π  = +    (2) where I T is the torsion constant, Iz is the second moment of area in (weak axis), I W the warping constant, L is the length between laterally braced cross-sections of the beam, and E and G are the longitudinal modulus and the shear modulus of elasticity, respectively. Because of the different loading types and bending moment diagrams, broad equations can be found in practical implementations and are suitable for a range of conditions, the most widely used in the formulation adopted by the commission,[3] acceptable to members subject to bending moment on a strong axis with a mono-symmetric cross-section Corresponding Author: Mohammed M. Saleh, Department of Civil Engineering, Engineering Collage, Universita Della Calabria, Cosenza, Italy. E-mail: mohammed@civil-eng.it Received: Oct 16, 2020 Accepted: Dec 20, 2020 Published: Dec 30, 2020 DOI: 10.24086/cuesj.v4n2y2020.25-32 Copyright © 2020 Mohammed M. Saleh, Dlshad K. Ahmed, Ali R. Yousef. This is an open access article distributed under the Creative Commons Attribution License (CC BY-NC-ND 4.0). Cihan University-Erbil Scientific Journal (CUESJ) Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 26 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 on a weak axis. For the calculation of the elastic critical moment under criteria not covered by Boissonnade et al.[3] ( ) ( ) 0.52 z 222 2 g 2 gz z w T cr 1 2 2 3 j 3 jw z zz k L C z C zE I k I G I M C C z C zk I E IK L       − −     π    = + + −          π             (3) The shape of the bending moment diagram gives the coefficient of C 1 , C 2 , and C 3 , support conditions give the coefficients k z and k w which are called the effective length factor, z g = (z a - z s ), Where z a and z s , comparable to the centroid of the cross-section, are the locations of the point of applied load and the shear center; these amounts are positive if the compressed portion is placed and negative if the tension portion is placed. While AISC requirements lead us to calculate the capacity of the bending moment without taking into account the level of load application,[1] even the steel section is rolled or welded.[4] The lateral load is also added to the compression flange of the member, as shown in Figure 2. In this case, the conceptual solution proposed by Fisher should be used,[5] this concept assumed that only the compression flange can survive lateral loading and the torsional impact can be ignored. In the case of I-shapes, the plastic section modulus of a Z-axis flange is provided by Z Z t z= 2 (4) Where Z z is plastic section modulus about the Z-axis. The moment capacity of one flange about the Y-axis is nt y tM F Z= × (5) Linear additions of the moment terms in[1] as shown in Figure 3 typically lead to results which are too conservative.[6] Based on the definition on biaxial bending, we will extend this to one case of study using both AISC and Eurocodes to investigate and demonstrate the deference between them regarding the level at which the load is applied, the type of the section whether rolled or welded and its height-to-width ratio by evaluating the results using Robot Structural Analysis software. Figure 1: Pure biaxial bending[2] Figure 2: Lateral load applied on the upper flange METHODOLOGY To demonstrate the exact difference between AISC and Eurocode analysis methods regarding biaxial bending, the case of study struggles with the loads added to the upper flange as shown in Figure 1, and Fisher’s concept[5] will be regarded. The case study includes a simply supported beam with a span of 10.0 m, measured center-to-center of support, with the load being applied at midspan, as shown in Figure 4, the beam under consideration is subjected to vertical load and the lateral load with the value 25 kN and 5 kN, respectively. The beam has yielding strength (f y = 355 MPa), modulus of elasticity (E = 21x103 MPa) and detailed properties are shown in Table 1. For the common base of comparison, the load factors in both approaches will be based on that of AISC standards.[7] 1.2 DL + 1.6 LL (6) The applied factored loads (moments) are shown in Table 2. The dead load includes the self-weight of the member as uniformly distributed which is calculated by the software. AISC Methods of Analysis For AISC methods of analysis in the Robot structural analysis software, modification factor should be defined for moment gradient C b in the member definition section which is equal to 1.32 for single point load act on simply supported beams laterally braced at the supports only. The lateral buckling length coefficient that represents the lateral bracing is equal to 1.0 (no lateral bracing for the beam). The results of the analysis are shown in Tables 3-6. Classification of Section for Local Buckling � f b t � � 9 58. Width-to-Thickness ratio for a flange. � fp E Fy � �0 38 9 24. . Limiting slenderness for compact flange. � � f fp � the flange is non-compact. Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 27 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 Table 1: Cross-section properties Cross-section properties Symbul Values Unit Symbol description A x 139.2 cm2 Cross-section area A y 55.2 cm2 Shear area – Y-axis A z 86.88 cm2 Shear area – Z-axis I y 104264.9 cm4 Second moment of area about the Y-axis I z 2443.48 cm4 Second moment of area about the Z-axis Z y 3435.12 cm3 Plastic section modulus about the Y (major) axis S y 2880.246 cm3 Elastic section modulus about the Y-axis Z z 342.6 cm3 Plastic section modulus about the Z (minor) axis S z 212.477 cm3 Elastic section modulus about the Z-axis d 72.4 cm Cross-section height b f 23 cm Cross-section width t f 1.2 cm Thickness of Flange t w 1.2 cm Web thickness Table 2: Internal forces Internal Forces AISC Vertical load Lateral load Applied factored Loads 40 kN 8 kN Applied moment y-axis z-axis 116 kN.m 20 N.m Figure 3: Simple linear interaction for biaxial bending[2] Figure 4: The case of the study � w w h t � � 58 3. Width-to-Thickness ratio for a web. � wp E Fy � �3 76 91 45. . Limiting slenderness for compact web. � � w wp � ∴ the web is compact. Parameters of Lateral Buckling Analysis C b = 1.32 lateral-torsional buckling modification factor, L b =10 m laterally unbraced length of a member (lateral- torsional buckling), c = 1 for doubly symmetric section. r I C S ts z w y = = 5 49. cm, h0 = d – 2tf = 70 cm L r E F p z y = =1 76 1 79. . Limiting laterally unbraced length for the limit state of yielding. 2 2 yts r y y 0 y 0 0.7F1.95r E Jc Jc L 6.76 0.7F S h S h E     = + +        = 5.3 m Limiting laterally unbraced length for the limit state of inelastic lateral- torsional buckling. Q Lb>Lr ∴ F C E L r Jc S h L r cr LTB b b ts y b ts ( ) . .� � � � � � � � � � � � � � � �2 2 0 2 1 0 078 112 13 MPa (7) Fcr: Critical stress (lateral-torsional buckling), J: torsional constant, c: Coefficient, Sy: Elastic section model. M F S ny LTB cr LTB y( ) ( ) .� � � 322 95 KN.m Nominal lateral-torsional buckling strength. Eurocode Method of Analysis In this method of analysis, the level where the load is applied must be defined, set the level of the applied load at Z = 1 in the member definition section beside the shape of bending moment in the load type section, the results of the analysis are shown in Tables 7-9. Class of section Flange: c/t f = 8.68 > 10 235 8 1 f y = . ∴ the flange is classified to the third class. Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 28 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 � LT y y cr W f M � � 2 14. Non-dimension slenderness ratio for lateral- torsional buckling. Lateral buckling curve is d from Table 11. α LT =0.76 Imperfection factor for lateral buckling curves Coefficient for calculation of X LT. � � � � LT LT LT LT � � �� � � 1 0 16 2 2 2 . Reduction factor for lateral- torsional buckling. M W f b Rd LT y y M . .� � � � 1 161 39 kN.m buckling resistance moment. RESULTS The main differences between AISC and Eurocode analysis results are explained in this section. Concern to the nominal strength, based on the results of the analysis, significant differences between the two approaches are demonstrated in Table 10. As shown in Table 10, the difference between the design moment strengths of the Y-axis is 129.26 kN.m, where the section capacity regarding the AISC approach is approximately 80% higher than the Eurocode approach. The main reason for this would be due to the level of applied load, as this condition is not recognized by the AISC standard. AISC specifications take into consideration the level of applied load through the cross-section shear center, while this level would be under or above or just through the shear center considered by the Eurocode approach, this effect is demonstrated in Figure 5. The level of application of the load has a considerable effect on the elastic critical moment M cr . To show this effect more, several sections with deferent depth and width but the same thickness of flange and web as listed below were tested. Section 1: Total depth of beam = 724 mm, width of beam = 230 mm. Section 2: Total depth of beam = 574 mm, width of beam = 250 mm (similar I section). Section 3: Total depth of beam = 474 mm, width of beam = 400 mm (similar H section) For all sections assumed that the level of the applied load at the upper part of the beam just on the flange with actual member definition for both approaches, the outcomes for this condition are shown in Figure 6. This figure indicates that Section 3 which is similar to H sections would be the critical situation. To show an explanation for this huge difference between the two approaches, for both approaches the level of applied load places at the shear center of the cross-section (center of the section), and the modification factor C b sets to one. The results of the analysis are shown in Figure 7. The results are getting closer so that the first reason behind this difference in the computation is recognized which the AISC approach does not consider the level of applied Table 3: Cross-section properties required for AISC approach Symbol Values Unit Symbol description A x 139.2 cm2 Cross-section area A y 55.2 cm2 Shear area - Y-axis A z 86.88 cm2 Shear area - Z-axis J 66.82 cm4 Torsional constant Cw 3096768.81 cm6 Warping constant I y 104264.9 cm4 Moment of inertia of a section about the Y-axis I z 2443.48 cm4 Moment of inertia of a section about the Z-axis Z y 3435.12 cm3 Plastic section modulus about the Y (major) axis S y 2880.246 cm3 Elastic section modulus about the Y-axis Z z 342.6 cm3 Plastic section modulus about the Z (minor) axis S z 212.477 cm3 Elastic section modulus about the Z-axis d 72.4 cm Height of cross-section b f 23 cm Width of cross-section t f 1.2 cm Flange thickness t w 1.2 cm Web thickness r y 27.37 cm Radius of gyration - Y-axis r z 4.19 cm Radius of gyration - Z-axis Table 4: Internal forces (AISC approach) Internal forces Symbul Values Unit Symbol description Section M rz −20 kN.m Required flexural strength V ry −4 kN Required shear strength V rz −20 kN Required shear strength Web: c w /t w = 57.53 < 72 235 58 58 f y = . ∴ the web is classified to the first class. Lateral-torsional Buckling Analysis (General Method [6.3.2.2]) L cr,upp = 10 m upper flange, lateral bracing, C1=1.26 / C2=0.55 / C3 =1.73 factors, I w =3096768.8 cm6 warping constant, z g = 36.2 cm From the shear center to the point of applied load. ( ) ( ) ( ) ( ) 0.52 z 2 2 z w Tz 2cr 1 2 g 3 j2 w z z z 2 2 g 3 j k L k I G IE I M C C z C z 223.05k I E IK L C z C z          π  + + = − − =  π      −     kN.m Critical moment for lateral-torsional buckling. Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 29 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 Table 5: Nominal strengths (AISC approach) Nominal strengths Symbul Amounts Unit Symbol description Section Respect to the Y axis M py 1219.47 kN.m Nominal plastic bending moment [F] M ny [YD] 1219.47 kN.m Nominal flexural strength in the limit state of yielding [F3.1] M ny [LTB] 322.95 kN.m Nominal lateral-torsional buckling strength [F3.1] M ny 1[LTB] 244.66 kN.m Nominal lateral-torsional buckling strength (C b =1.0) [F3.1] M ny [FLB] 1203.48 kN.m Nominal strength for local buckling of a compression flange [F3.2] M ny 322.95 kN.m Nominal flexural strength [F3] V nz 1850.54 kN Nominal shear strength [G2.1] Respect to the Z axis M pz 121.62 kN.m Nominal plastic bending moment [F] M nz [YD] 120.69 kN.m Nominal flexural strength in the limit state of yielding [F6] M nz [FLB] 118.53 kN.m Nominal strength for local buckling of a compression flange [F6.2] M nz 118.53 kN.m Nominal flexural strength [F6] V ny 1175.76 kN Nominal shear strength [G2.1] Table 6: Design strengths (AISC approach) Design strengths Symbol Amounts Unit Symbol description Section Respect to the Y axis Fib*M py 1097.52 kN.m Design plastic bending moment [F] Fib*M ny [YD] 1097.52 kN.m Design flexural strength in the limit state of yielding [F3.1] Fib*M ny [LTB] 290.65 kN.m Design lateral-torsional buckling strength [F3.1] Fib*M ny 1[LTD] 220.19 kN.m Design lateral-torsional buckling strength [F3.1] Fib*M ny [FLB] 1083.13 kN.m Design strength for local buckling of a compression flange [F3.2] Fib*M ny 290.65 kN.m Design flexural strength [F3] Fiv*V nz 1665.49 kN Design shear strength [G2.1] Respect to the Z axis Fib*M pz 109.46 kN.m Design plastic bending moment [F] Fib*M nz [YD] 108.62 kN.m Design flexural strength in the limit state of yielding [F6] Fib*M nz [FLB] 106.68 kN.m Design strength for local buckling of a compression flange [F6.2] Fib*M nz 106.68 kN.m Design flexural strength [F6] Fiv*V ny 1058.18 kN Design shear strength [G2.1] Verification formulas UF (H1_1b) 0.59 M ry /(Fib*M ny ) + M rz /(Fib*M nz ) Verified UF (G2_1) 0 V ry /(Fiv*V ny ) Verified UF (G2_1) 0.01 V rz /(Fiv*V nz ) Verified load unless when the load applied just on the top flange and for the case L b > Lr then the square root of Engineers[7] may conservatively takes one which does not take one in the analysis of Robot structural analysis software. In addition, other factors cause. Such as the type of section (rolled or welded) and the depth to width ratio(h/b), which is very effective in the case of lateral torsional buckling Table 11. The imperfection factor α LT which is given in Table 12 covers the effect of lateral-torsional buckling that results in the redaction factor X LT for the design moment capacity as shown in Velikovic et al.[8] Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 30 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 Table 7: Cross-section properties required for Eurocode approach Symbol Values Unit Section A x 139.2 cm2 A y 55.2 cm2 A z 84 cm2 I x 65.509 cm4 I y 104264.9 cm4 I z 2443.48 cm4 W ely 2880.246 cm3 W elz 212.477 cm3 h 72.4 cm b 23 cm t f 1.2 cm t w 1.2 cm r y 27.37 cm r z 4.19 cm A nb 1 (6.2.2.2) E ta 1 (6.2.6.(3)) Table 8: Internal forces Internal forces Symbol Values Unit Symbol description Section M y, Ed 111.84 kN.m Bending moment M y.Ed M z, Ed −18.75 kN.m Bending moment M z.Ed Figure 6: Comparison of design moments about Y-axis with actual conditions Figure 7: Comparison of design moments about y-axis with the load at the shear center for both approaches � � � � LT LT LT LT � � �� � ���� � �� 0 5 1 0 2 2 . . (8) � � � � LT LT LT LT � � �� � 1 2 2 2 (9) M W f b Rd LT y y M . � � � 1 (10) The rolled section performs better than the welded section. Table 2 shows that, because of the lateral load, the difference in the bending moment between two approaches around the Z-axis is about 31.25. Robot Structure Analysis software does not follow Fisher’s concept, however, in most cases, Robot Structure Analysis software does not follow Fisher’s concept, in most cases, the Eurocode approach is closer to the concept of Fisher. Fisher’s concept is based on (2) and (3), where the nominal strength is Figure 5: The influence of the level of the applied load on the capacity of the section Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 31 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 Table 9: Design forces and verifications Design forces Symbol Values Unit Symbol description Section M b, Rd 161.39 kN.m Buckling resistance moment (6.3.2.1) M y, pl, Rd 1219.47 kN.m Plastic resistance moment (6.2.5.(2)) M y, el, Rd 1022.49 kN.m Elastic resistance moment (6.2.5.(2)) M y, c, Rd 1022.49 kN.m Moment resistance (6.2.5.(2)) V y, c, Rd 1131.38 kN Plastic shear resistance (6.2.6.(2)) Respect to the Z axis z M z, pl, Rd 121.62 kN.m Plastic resistance moment (6.2.5.(2)) M z, el, Rd 75.43 kN.m Elastic resistance moment (6.2.5.(2)) M z, c, Rd 75.43 kN.m Moment resistance of a compressed section part (6.2.5.(2)) V z, c, Rd 1721.66 kN Plastic shear resistance (6.2.6.(2)) Verification formulas Global stability check of member UFB[M y M z ] 0.94 M y, Ed /(X LT *M y, Rk /gM1) + M z, Ed /(M z, Rk /gM1) (6.3.3.(4)) Ratio RAT 0.94 Efficiency ratio Section OK Table 10: Comparison between AISC and Eurocode results Approaches AISC Eurocode Design strength Notation y- axis (kN.m) z- axis (kN.m) y- axis (kN.m) z-axis (kN.m) Design buckling resisting moment M b , R d x x 161.39 x Design lateral-torsional buckling strength F ib *M ny 290.65 x x x Moment resistance of a compressed section part M z, c , R d x x x 75.43 Design strength for local buckling of a compression flange F ib *M nz x 106.68 x x Table 11: Imperfection factor for lateral torsional buckling[8] Buckling curve a 0 a b c d Imperfection factor 0.13 0.21 0.34 0.49 0.76 Z Z cm t y= = = 2 342 6 2 171 3 3 . . (11) M Z f Nmm kNm nt t y � � � � � �171300 355 60811500 60 81. . . (12) The design moment capacity for section 1 is ∅M nz =0.9(M nt )=0.9(60.81)=54.72 kN.m, and for sections Table 12: Buckling curves for lateral-torsional buckling (General method) Section Limits Buckling curve I or H sections rolled h/b≤2 a h/b≤2 b I or H sections welded h/b≤2 c h/b≤2 d Other sections ---- d Figure 8: Comparison between the three approaches for design moment about Z-axis Saleh, et al.: Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches 32 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2020, 4 (2): 25-32 2 and 3 are 63.1 kN.m and 155.9 kN.m, respectively. The comparison between the three approaches is shown in Figure 8 Fisher’s concept is more conservative than AISC and Eurocode methods of analysis. As shown, it is safer to use Fisher’s concept or Eurocode approach in Robot structure Analysis software, Figuure 9 where the safety factor is defined as the ratio of the moment capacity of the section to the applied moment. CONCLUSIONS Based on the results of steel sections which analyzed by AISC and Eurocode standards, and Fisher’s concept, the following points can be drawn: 1. Eurocode method of analysis results in more conservative design strength about Y-axis of the section as it takes into consideration the level where the loads are applied. On contrary, the AISC method of analysis does not take this effect into consideration unless when the load applied just on the top flange and for the case L b > Lr then the square root of equation (7) may conservatively takes one which does not take one in the analysis of Robot structural analysis software 2. Eurocode considers the lateral-torsional effect in terms of height/width ratio and the type of the section, rolled or welded, which results in the safety factor, which is not covered by AISC standards. The effect of lateral-torsional buckling and member type is much clear when the section is close to H-section 3. For the three types of sections (1, 2, and 3) considered in this study, Eurocode predicts design strength with safety factors of 1.98, 1.94, and 1.41, respectively, while such safety factors were 1.59, 1.54, and 1.26 for AISC approach. This concludes the adoption of Eurocode method of analysis in structural design of biaxial bending case when using Robot Structure Analysis software 4. Fisher’s concept of structural design of biaxial bending of structural steel is more conservative than both AISC and Eurocode approaches of analysis. REFERENCES 1. A. Ansi. AISC 360-16, Specification for Structural Steel Buildings, 2016. 2. A. Aghayere and J. Vigil. Structural Steel Design: Pearson India, 2008. 3. N. Boissonnade, R. Greiner, J. P. Jaspart and J. Lindner. Rules for Member Stability in EN 1993-1-1: Background Documentation and Design Guidelines, 2006. 4. L. Laim, J. P. C. Rodrigues and L. S. da Silva. Experimental and numerical analysis on the structural behaviour of cold-formed steel beams. Thin Walled Structures, vol. 72, pp. 1-13, 2013. 5. M. Fisher. Steel Design Guide 7: Industrial Buildings, Roofs to Anchor Rods. Chicago, IL: American Institute of Steel Construction, 2005. 6. Y. L. Pi and N. S. Trahair. Inelastic torsion of steel I-beams. Journal of Structural Engineering, vol. 121, pp. 609-620, 1995. 7. A. Engineers. Minimum Design Loads for Buildings and Other Structures, Standard Asce/Sei 7-10. Reston, VA: American Society of Civil Engineers, 2013. 8. M. Velikovic, L. S. da Silva, R. Simões, F. Wald, J. P. Jaspart, K. Weynand. Eurocodes: Background and Applications Design of Steel Buildings Worked Examples, Joint Research Center, 2015. Figure 9: The difference in the factor of safety for biaxial bending manner