NCCI: Elastic critical moment for lateral torsional bucklingNCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU NCCI: Elastic critical moment for lateral torsional buckling This NCCI gives the expression of the elastic critical moment for doubly symmetric crosssections. Values of the factors involved in the calculation are given for common cases. For a beam under a uniformly distributed load with end moments or a concentrated load at mid-span with end moments, the values for the factors are given in graphs. Contents 1. 2. 3. 4. Created on Thursday, January 15, 2009 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement General Method for doubly symmetric sections C1 and C2 factors References 2 2 4 12 Page 1 General For doubly symmetric cross-sections. Method for doubly symmetric sections The method given hereafter only applies to uniform straight members for which the crosssection is symmetric about the bending plane.com Created on Thursday. The conditions of restraint at each end are at least : restrained against lateral movement restrained against rotation about the longitudinal axis The elastic critical moment may be calculated from the following formula derived from the buckling theory : M cr = C1 ⎧ π 2 EI z ⎪ ⎛ k ⎞ I w 2 (kL )2 (kL )2 GI t + (C z )2 − C z ⎫ ⎪ ⎜ ⎟ + ⎨ ⎜ ⎟ 2 g 2 g⎬ 2 π EI z ⎪ ⎪ ⎝ kw ⎠ I z ⎭ ⎩ (1) where E G Iz It Iw is the Young modulus (E = 210000 N/mm2) is the shear modulus (G = 80770 N/mm2) is the second moment of area about the weak axis is the torsion constant is the warping constant Page 2 .cticm. It may be downloaded free of charge from the following web site: http://www.all rights reserved. the elastic critical moment Mcr may be calculated by the method given in paragraph 2. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement 2. January 15.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 1. For cases not covered by the method given in paragraph 2. 2009 This material is copyright . the elastic critical moment may be determined by a buckling analysis of the beam provided that the calculation accounts for all the parameters liable to affect the value of Mcr : geometry of the cross-section warping rigidity position of the transverse loading with regard to the shear centre restraint conditions The LTBeam software is specific software for the calculation of the critical moment Mcr. 0 unless less than 1. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement F S S F zg > 0 zg < 0 Figure 2. kw should be taken as 1. k should be taken as not less than 1. January 15.1 Point of application of the transverse load Page 3 .NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU L is the beam length between points which have lateral restraint k and kw are effective length factors zg is the distance between the point of load application and the shear centre. C1 and C2 are coefficients depending on the loading and end restraint conditions (see §3).0.1). It is analogous to the ratio of the buckling length to the system length for a compression member. Note : for doubly symmetric sections.0 can be justified. 2009 This material is copyright . Unless special provision for warping fixity is made.all rights reserved. Created on Thursday. In the general case zg is positive for loads acting towards the shear centre from their point of application (Figure 2. The factor k refers to end rotation on plan. The factor kw refers to end warping. the shear centre coincides with the centroid. The latter expression should be simplified as follows : M cr = C1 π 2 EI z L2 I w L2GI t + I z π 2 EI z (3) For doubly symmetric I-profiles. 2009 This material is copyright . M cr = C1 π 2 EI z ⎧ I w ⎪ L2 ⎨ ⎪ Iz ⎩ + ⎫ L2GI t ⎪ 2 + (C2 zg ) − C2 zg ⎬ 2 π EI z ⎪ ⎭ (2) When the bending moment diagram is linear along a segment of a member delimited by lateral restraints. 3. • moment diagram It can be demonstrated that the C1 and C2 factors depend on the ratio : κ= EI w GI t L2 (5) The values given in this document have been calculated with the assumption that κ = 0. January 15. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement I (h − t f ) Iw = z 4 where h tf 2 (4) is the total depth of the cross-section is the flange thickness 3. Page 4 . • support conditions.1 C1 and C2 factors General The C1 and C2 factors depend on various parameters : • section properties. C2 zg = 0.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU In the common case of normal support conditions at the ends (fork supports). This assumption leads to conservative values of C1. the warping constant Iw may be calculated as follows : Created on Thursday. k and kw are taken equal to 1.all rights reserved. or when the transverse load is applied in the shear centre. January 15.14 1.00 -0. 2009 This material is copyright .NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 3.25 0.1 Member with end moments Created on Thursday.00 Values of C1 for end moment loading (for k = 1) C1 1.all rights reserved.2 Member with end moments only The factor C1 may be determined from Table 3.57 2.1 for a member with end moment loading.52 1.75 -1.25 -0. M ψM -1 ≤ ψ ≤ +1 Figure 3.33 2.50 +0.50 -0.00 +0.75 +0.05 2.00 1. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement Table 3.77 2.1 ψ +1.31 1.55 Page 5 . 683 1. Two cases are considered: Case a) end moments with a uniformly distributed load Case b) end moments with a concentrated load at mid-span The moment distribution may be defined using two parameters : ψ is the ratio of end moments.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 3.3 Member with transverse loading Table 3. and so : -1 ≤ ψ ≤ 1 (ψ = 1 for a uniform moment) μ is the ratio of the moment due to transverse load to the maximum end moment M Case a) μ = qL2 8M Page 6 . January 15.348 Created on Thursday.127 0. 2009 This material is copyright . By definition.2.454 2. M is the maximum end moment. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement 0.4 Member with end moments and transverse loading For combined loading of end moments and transverse loads as shown in Figure 3.630 1. values of C1 and C2 may be obtained from the curves given hereafter. Table 3.all rights reserved.578 1.554 1.2 Values of factors C1 and C2 for cases with transverse loading (for k = 1) Bending moment diagram C1 C2 Loading and support conditions 1.645 Note : the critical moment Mcr is calculated for the section with the maximal moment along the member 3.2 gives values of C1 and C2 for some cases of a member with transverse loading. NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU Case b) μ = FL 4M Sign convention for μ : μ>0 μ<0 if M and the transverse load (q or F). 2009 This material is copyright . each supposed acting alone.all rights reserved. q M ψM M F L (b) ψM Created on Thursday.2 End moments with a transverse load Page 7 .g. January 15. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement L (a) Figure 3. bend the beam in the same direction (e. as shown in the figure below) otherwise The values of C1 and C2 have been determined for k = 1 and kw = 1. 9 -1 4.4 -0.3 1.6 -0.0 -0.0 -1 -0.6 -0.8 -0.1 -1.8 3.5 -1.2 1.2 0 0.5 -0.8 1 1.4 0.6 1. January 15.8 ψM 1 μ<0 Figure 3.0 -0.0 -1 -0.0 -1.5 -0.6 0.1 1.4 0.8 -2 2.7 -0.6 0.5 -0.3 0.4 2.5 0.5 -0.7 -1.2 0.all rights reserved.2 0.2 -1.1 0 μ 2 Created on Thursday.3 End moments and uniformly distributed load – Factor C1 Page 8 .0 1.0 1.8 M ψM -0.0 C1 2.8 1 ψM M ψ M ψM μ>0 5.6 3.4 -0.2 2 1.2 M 0.1 C1 4.6 -1.4 -0.5 2.7 0.5 0.3 -0.2 -1.4 0. 2009 This material is copyright .NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 3. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement 1.3 -1.5 0.4 0.2 0 ψ 0.5 μ 0 1. 2 0.4 -1.9 -1.0 -1 -0.4 0.0 -1 -0.2 0.1 -1.4 -0.2 0 0.2 1 0.8 1.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 0.5 M ψM ψ 0.5 -0.6 -0.6 0.2 -0.6 -0.8 1 M ψM μ<0 Figure 3.8 1 M ψM Created on Thursday.4 0.4 -0.3 0. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement μ>0 2.6 -1.7 0.1 0.4 μ 2 1.4 End moments and uniformly distributed load – Factor C2 Page 9 .0 -1. January 15.9 -2 -0.5 C2 0.2 μ 1.0 -1 -1.8 -1.2 0.5 0.5 -0. 2009 This material is copyright .7 0.3 -1.8 -0.5 0.6 -0.6 0.1 0.4 -0.6 0.3 0.3 -0.all rights reserved.2 1.5 C2 2.1 0.9 0.8 -0.7 -1.4 -0.5 -1.2 -0.2 0 ψM M ψ 0.8 0. 5 -1 -0.0 C1 3.8 -0.5 2.all rights reserved.9 -0.7 -1.4 0.7 0.1 0 μ 0.2 0.1 0.3 -0.2 0.2 -1.2 0.2 0 0.7 2.3 2 1.9 1.5 -1.8 M ψM -0.4 -0.8 1 ψ M ψM μ>0 4.4 -1.6 -0.6 0.8 -1.8 1 M ψM μ<0 Figure 3.6 1.5 End moments and point load at mid-span – Factor C1 Page 10 .0 C1 2.3 -1. January 15.0 1.1 -1.4 0.5 -0.0 -1 -0.6 -0.2 0 ψ 0.0 -1 -0.6 0.5 0.3 0.2 0.5 0.4 0.1 1.5 0.0 -0.4 ψM M -0.1 1.6 -0.8 1.2 μ 3.6 -1.5 1.2 1 1 1.4 0.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 3.8 -2 2. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement C 1.2 2 Created on Thursday.4 -0. 2009 This material is copyright .5 -0.0 0 -0. all rights reserved.8 1 M ψM μ<0 Figure 3.5 1.2 0.2 0. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement μ>0 2.9 0.7 0.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 0.4 ψM M -0.3 μ 0.5 -0.2 -1.5 C2 2 0.4 -1.6 -0.8 -0.5 -1.1 -1.4 0.9 -0.3 1.2 0 ψ 0. January 15.5 0.7 -2 -1 -1.8 0.3 -0.8 1.8 M ψM -0.4 -0.5 C2 2.6 -0.4 0.0 μ -1.1 0.8 -0.2 1 0.5 -1.1 -0.2 0 ψ 0.0 -1 -0.4 0.7 0.2 0.6 -0.6 0.2 0.8 1 M ψM Created on Thursday.4 1.6 -1.0 -1 -0. 2009 This material is copyright .2 0.4 -0.6 0.3 0.5 -0.6 0.0 -0.1 0.6 End moments and point load at mid-span – Factor C2 Page 11 . H. J.P. Revue Construction Métallique n°3-1974. European Committee for Standardisation. Calcul de la résistance ultime au déversement dans le cas de la flexion déviée. CTICM. Revue Construction Métallique n°2-2002. Theory of elastic stability. 1 References ENV 1993-1-1 Eurocode 3 : Design of steel structures – Part 1. 2 Timoshenko. M. 3 Djalaly.1 : General rules and rules for buildings. Y. and Gere. Mc Graw-Hill. 2nd Edition. CTICM. Déversement élastique d’une poutre à section bi-symétrique soumise à des moments d’extrémité et une charge repartee ou concentrée.all rights reserved. Created on Thursday. January 15.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU 4. 1961. S. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement Page 12 . 2009 This material is copyright . 4 Galéa. Germany 5. Spain Resource approved by Technical Coordinator TRANSLATED DOCUMENT This Translation made and checked by: Translated resource approved by: Page 13 . Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement Company CTICM CTICM SCI Date Alain Bureau Yvan Galéa D C Iles 2/3/05 G W Owens A Bureau A Olsson C Mueller J Chica G W Owens SCI CTICM SBI RWTH Labein SCI 1/3/05 1/3/05 1/3/05 1/3/05 1/3/05 21/4/06 2. Sweden 4. UK Created on Thursday.all rights reserved.NCCI: Elastic critical moment for lateral torsional buckling NCCI: Elastic critical moment for lateral torsional buckling SN003a-EN-EU Quality Record RESOURCE TITLE Reference(s) ORIGINAL DOCUMENT NCCI: Elastic critical moment for lateral torsional buckling Name Created by Technical content checked by Editorial content checked by Technical content endorsed by the following STEEL Partners: 1. 2009 This material is copyright . France 3. January 15.
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