Steel Design

STEEL BEAM jut come out side(prenominal)ly excessive circularise Dr. A Aziz Saim 2010 EC3 unhindered prick 1 Non-dimensional low density fall behaviour same to amenable/buckling of columns. M Wyfy Material yielding (in- matted deflection) MEd MEd expansible member buckling Mcr Lcr 1. 0 Dr. A Aziz Saim 2010 EC3 Non-dimensional fineness Un tranquil peter ? LT 2 squint-eyed torsional buckling squinty torsional buckling askance torsional buckling is the member buckling manner associated with sl block offer broadcasts loaded most their major(ip) axis, without continuous squint simmpleness.If continuous asquint simmpleness is provided to the beam, then asquint torsional buckling provide be prevented and failure willing number in an other mode, familiarly in- woodworking devisee change form (and/or crop). Dr. A Aziz Saim 2010 EC3 unbuttoned prick 3 Eurocode 3 Eurocode 3 states, as with BS 5950, that twain incubatesectional and member b terminaling resis tance moldiness be substantiate MEd ? Mc ,Rd cross-section(prenominal) double back (In- blueprinte diversion) MEd ? Mb,Rd Dr. A Aziz Saim 2010 EC3 crazy circularize Member buckling check 4 Dr. A Aziz Saim 2010 EC3 unchecked publicise 5 squintly ungoverned jibeThe concept of beam in this Lecture 3 is considering beams in which either no squinty restraint or only intermittent lateral restraint is provided to the compression lip Dr. A Aziz Saim 2010 EC3 worked up s abrogate 6 Lateral Torsional Buckling Dr. A Aziz Saim 2010 EC3 ungoverned post 7 Lateral Torsional Buckling Figure 3-1 shows an crazy beam subjected to load increment. The compression brim worked up and beam is not uncompromising enough. There is a tendency for the beam to deform sideways and twist about the longitudinal axis. The failure mode which whitethorn occur to the beam is called lateral torsional buckling.Dr. A Aziz Saim 2010 EC3 excessive diffuse 8 ?Involves both deflection and twisting rot ary motion ?Out-of plane buckling. crook Resistance M c, Rd ? M pl ? W pl f y ?M0 overdue to the effect of LTB, the bending resistance of cross section become less. Failure whitethorn occurs earlier then expected Dr. A Aziz Saim 2010 EC3 mad charge 9 faces of Laterally sick transmit Dr. A Aziz Saim 2010 EC3 huffy impart 10 Restrained slam Comparsion Dr. A Aziz Saim 2010 EC3 Unrestrained glint 11 Intermittent Lateral Restrained Dr. A Aziz Saim 2010 EC3 Unrestrained broadcast 12Torsional restraint unremarkably both rims ar held in their congress positions by external members during bending. may be provided by load bearing stiffeners or provision of adequate end community details. See Figure 3-4. Dr. A Aziz Saim 2010 EC3 Unrestrained slam 13 pecker without torsional restraint Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 14 Can be discounted when Minor axis bending CHS, SHS, circular or full- metier bar Fully laterally restrained beams ? LT 0. 2 (or 0. 4 in around carapaces) Unrestrained length cross-section(prenominal)al figure End restrained condition The signification along the beam Loading emphasis or compression Unrestrained Beam 16Dr. A Aziz Saim 2010 EC3 Lateral torsional buckling resistance Checks should be carried out on all crazy departments of beams (between the points where lateral restraint equals). Lateral restraint Lateral restraint Lcr = 1. 0 L Lateral restraint Beam on plan Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 17 Three manners to check LTB in EC3 The primary rule adopts the lateral torsional buckling foreshortens inclined by equations 6. 56 and 6. 57, and is set out in article 6. 3. 2. 2 (general case) and cla use 6. 3. 2. 3 (for rolled sections and equivalent welded sections). The imprimatur is a simplified judgment order for beams with restraints in buildings, and is set out in clause 6. 3. 2. 4. The third is a general method for lateral and lateral torsional buckling of morphological components, effrontery in clause 6. 3. 4. Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 18 Eurocode 3 states, as with BS 5950, that both cross-section(a) and member bending resistance must be verified MEd ? Mc ,Rd Cross-section check (In-plane bending) MEd ? Mb,Rd Dr. A Aziz Saim 2010 EC3 Unrestrained Beam Member buckling check 19 Lateral-torsional buckling Eurocode 3 purpose approach for lateral torsional buckling is analogous to the olumn buckling treatment. The design buckling resistance Mb,Rd of a laterally unrestrained beam (or segment of beam) should be taken as Mb,Rd ? ?LT Wy fy ? M1 reducing means for LTB Lateral torsional buckling resistance Mb,Rd = ?LT Wy fy ? M1 Equation (6. 55) Wy will be Wpl,y or Wel,y ?LT Dr. A Aziz Saim 2010 EC3 is the simplification fixings for lateral torsional buckling Unrestrained Beam 21 Buckling curves general case (Cl 6. 3. 2. 2) Lateral torsional buckling curves for the general case are turn inn below (as in Eq (6. 56)) ?LT ? 1 2 ? LT ? ?LT ? ?2 LT but ? LT ? 1. 0 ?LT ? 0. 5 1 ? ?LT (? LT ? 0. ) ? ?2 LT tableland length blot element from knock back 6. 3 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 22 smirch factor ? LT Imperfection factors ? LT for 4 buckling curves (refer slacken 6. 3) Buckling curve Imperfection factor ? LT a 0. 21 b 0. 34 c 0. 49 d 0. 76 Buckling curve selection For the general case, refer to Table 6. 4 Cross-section Rolled I-sections Welded Isections Limits h/b ? 2 h/b 2 h/b ? 2 h/b 2 Buckling curve a b c d d another(prenominal) crosssections Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 24 LTB curves 4 buckling curves for LTB (a, b, c and d) 1. 2 Reduction factor ? LT . 0 0. 8 0. 6 0. 4 0. 2 0. 0 0 0. 5 1 1. 5 thin a crimp b Curve c Curve d 2 2. 5 0. 2 Dr. A Aziz Saim 2010 EC3 Non-dimensional sparsity Unrestrained Beam ?LT 25 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 26 lateral torsional buckling sparseness ? LT Mcr ? Wy f y Mcr chewy scathing buckling minute Dr. A Aziz Saim 2010 EC3 Unrestrai ned Beam 27 Non-dimensional slenderness omen lateral torsional buckling slenderness ? LT ? Wy f y Mcr Buckling curves as for compression (except curve a0) Wy depends on section classification Mcr is the expandable critical LTB aftermath Dr. A Aziz Saim 2010 EC3Unrestrained Beam 28 BS EN 1993-1-1 does not give a method for determining the expandable critical moment for lateraltorsional buckling Mcr May use LTBeam software (can be downloaded from CTICM website) Or whitethorn use method presented by L. Gardner . Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 29 Mcr on a lower floor homogeneous moment For typical end conditions, and beneath uniform moment the live critical lateral torsional buckling moment Mcr is Mcr ,0 G IT Iw Iz Lcr ? EIz ? 2 Lcr 2 ? Iw Lcr GIT ? ? ? 2 ? ? EIz ? ? Iz 2 0. 5 is the shear modulus is the torsion constant is the falsify constant is the inor axis second moment of area is the buckling length of the beam Unrestrained Beam 30 Dr. A Aziz Saim 2010 EC3 Mcr under non-uniform moment Numerical solutions have been reckon for a number of other fill conditions. For uniform twice-symmetric cross-sections, loaded through the shear centre at the level of the centroidal axis, and with the standard conditions of restraint described, Mcr whitethorn be calculated by ? EIz Mcr ? C1 2 Lcr 2 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam ? Iw Lcr GIT ? ? ? 2 ? ? EIz ? ? Iz 2 0. 5 31 C1 factor end momentsFor end moment effect C1 may be approximated by the equation below, though other approximations also exist. C1= 1. 88 1. 40y + 0. 52y2 but C1 ? 2. 70 where y is the ratio of the end moments (defined in the following table). Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 32 C1 factor transverse loading Loading and support conditions Bending moment plat Value of C1 1. 132 1. 285 1. 365 1. 565 1. 046 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 33 approach pattern office for LTB flesh procedure for LTB 1. Determine BMD and SFD from design load s 2. Select section and hold back geometry 3. forkify cross-section ( form 1, 2, 3 or 4) 4.Determine effective (buckling) length Lcr depends on bourn conditions and load level 5. Calculate Mcr and Wyfy Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 34 Design procedure for LTB 6. Non-dimensional slenderness ? LT ? Wy fy Mcr 7. Determine disfigurement factor ? LT 8. Calculate buckling step-down factor ? LT 9. Design buckling resistance 10. Check Mb,Rd ? ?LT Wy fy ? M1 MEd ? 1. 0 Mb,Rd for each unrestrained portion Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 35 LTB pillow slip General arrangement Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 36 LTB exercising Design loading is as follows 425. 1 kN A B C 319. 6 kN D 2. 5 m 3. 2 m 5. 1 mLoading Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 37 LTB physical exertion 267. 1 kN A B D 52. 5 kN SF C 477. 6 kN Shear force diagram B A C D BM 1194 kNm 1362 kNm Bending moment diagram Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 38 LTB grammatical cas e For the purposes of this example, lateral torsional buckling curves for the general case will be utilised. Lateral torsional buckling checks to be carried out on segments BC and CD. By inspection, segment AB is not critical. rise 762? 267? 173 UB in grade S 275 steel. Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 39 LTB practice b z tw h d y y r z tf h = 762. 2 mm b = 266. 7 mm tw = 14. 3 mm tf = 21. 6 mm r = 16. mm A = 22000 mm2 Wy,pl = 6198? 103 mm3 Iz = 68. 50? 106 mm4 It = 2670? 103 mm4 Iw = 9390? 109 mm6 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 40 LTB slip For a nominal material onerousness (tf = 21. 6 mm and tw = 14. 3 mm) of between 16 mm and 40 mm the nominal determine of yield strength fy for grade S 275 steel (to EN 10025-2) is 265 N/mm2. From clause 3. 2. 6 N/mm2. E = 210000 N/mm2 and G ? 81000 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 41 LTB Example Cross-section classification (clause 5. 5. 2) e ? 235 / fy ? 235 / 265 ? 0. 94 Outstand flanges (Table 5. 2, mainsh eet 2) cf = (b tw 2r) / 2 = 109. 7 mm cf / tf = 109. 7 / 21. 6 = 5. 8 Limit for frame 1 flange = 9e = 8. 48 5. 08 ? Flange is kind 1 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 42 LTB Example Web intimate part in bending (Table 5. 2, sheet 1) cw = h 2tf 2r = 686. 0 mm cw / tw= 686. 0 / 14. 3 = 48. 0 Limit for Class 1 web = 72 e = 67. 8 48. 0 ? Web is Class 1 Overall cross-section classification is therefore Class 1. Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 43 LTB Example Bending resistance of cross-section (clause 6. 2. 5) Mc ,y,Rd ? Wpl,y fy ? M0 for Class 1 and 2 sec tions 6198 ? 103 ? 265 ? ? 1642 ? 106 Nmm 1. 0 ? 1642 kNm ? 1362 kNm ? Cross-section resistance in bending is OK.Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 44 LTB Example Lateral torsional buckling check (clause 6. 3. 2. 2) ingredient BC MEd ? 1362 kNm Mb ,Rd ? ? LT Wy fy ? M1 where Wy = Wpl,y for Class 1 and 2 sections Determine Mcr for segment BC (Lcr = 3200 mm) Dr. A Aziz Saim 2010 EC3 ? EIz Mcr ? C1 2 Lcr 2 ? Iw Lcr GIT ? ? ? 2 ? ? EIz ? ? Iz Unrestrained Beam 2 0. 5 45 LTB Example For end moment loading C1 may be approximated from C1 = 1. 88 1. 40y + 0. 52y2 but C1 ? 2. 70 1194 y is the ratio of the end moments ? ? 0. 88 1362 ? C1 ? 1. 05 ? 2 ? 210000 ? 68. 5 ? 106 Mcr ? 1. 05 ? 32002 ? 9390 ? 109 32002 ? 81000 ? 2670 ? 103 ? ? ? 68. 5 ? 106 ? 2 ? 210000 ? 68. 5 ? 106 ? ? 0. 5 = 5699106 Nmm = 5699 kNm Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 46 LTB Example Non-dimensional lateral torsional slenderness for segment BC ? LT ? Wy fy Mcr 6198 ? 103 ? 265 ? ? 0. 54 6 5699 ? 10 Select buckling curve and imperfection factor ? LT From Table 6. 4 h/b = 762. 2/266. 7 = 2. 85 For a rolled I-section with h/b 2, use buckling curve b Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 47 LTB Example From Table 6. 3 of EN 1993-1-1 For buckling curve b, ? LT = 0. 34 Calculate reduction factor for lateral torsional buckling, ? LT element BC ?LT ? 1 ? LT ? ? 2 LT LT but ? LT ? 1. 0 where ? LT ? 0. 5 1 ? ?LT (? LT ? 0. 2) ? ?2 LT Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 48 LTB Example ?LT = 0. 51+0. 34(0. 54-0. 2) + 0. 542 = 0. 70 ? ? LT ? 1 0. 70 ? 0. 70 ? 0. 54 2 2 ? 0. 87 Lateral torsional buckling resistance Mb,Rd Segment BC Mb,Rd ? ? LT Wy fy ? M1 265 ? 0. 87 ? 6198 ? 10 ? 1 . 0 3 ? 1425 ? 106 Nmm ? 1425 kNm Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 49 LTB Example MEd 1362 ? ? 0. 96 ? 1. 0 ? Segment BC is OK Mb,Rd 1425 Lateral torsional buckling check (clause 6. 3. 2. 2) Segment CD MEd ? 1362 kNm Mb ,Rd ? ? LT Wy fy ? M1 where Wy = Wpl,y for Class 1 and 2 sectionsDetermine Mcr for segment CD (Lcr = 5100 mm) Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 50 LTB Example ? EIz Mcr ? C1 2 Lcr 2 ? Iw Lcr GIT ? ? ? 2 ? Iz ? EIz ? ? 2 0. 5 Determine y from Table 0 y is the ratio of the end moments ? ?0 1362 ? C1 ? 1. 88 ? 2 ? 210000 ? 68. 5 ? 106 Mcr ? 1. 88 51002 ? 9390 ? 109 51002 ? 81000 ? 2670 ? 103 ? ? ? ? 68. 5 ? 106 ? 2 ? 210000 ? 68. 5 ? 106 ? ? 0. 5 = 4311? 106 Nmm = 4311 kNm Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 51 LTB Example Non-dimensional lateral torsional slenderness for segment CD ? LT ? Wy fy Mcr 6198 ? 103 ? 265 ? ? 0. 62 6 4311? 10 The buckling curve and imperfection factor ?LT are as for segment BC. Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 52 LTB Example Calculate reduction factor for lateral torsional buckling, ? LT Segment CD ?LT ? 1 ? LT ? ? 2 LT 2 LT but ? LT ? 1. 0 where ? LT ? 0. 5 1 ? ?LT (? LT ? 0. 2) ? ?2 LT = 0. 51+0. 34(0. 62-0. 2) + 0. 622 = 0. 76 ? ? LT Dr. A Aziz Saim 2010 EC3 ? 1 0. 76 ? 0. 76 ? 0. 62 2 Unrestrained Beam 2 ? 0. 83 53 LTB Example Lateral torsional buckling resistance Mb,Rd Segment CD Mb,Rd ? ?LT Wy fy ? M1 265 ? 0. 83 ? 6198 ? 10 ? 1. 0 3 ? 1360 ? 106 Nmm ? 1360 kNm MEd 1362 ? ? 1. 00 Mb,Rd 1360 Segment CD is critical and marginally fails LTB check.Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 54 white summon Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 55 Simplified assessment of ? LT For ho t-rolled doubly symmetric I and H sections without destabilising loads,? may be conservatively simplified to LT ? LT ? 1 0. 9 ? z ? C1 ?z 1 0. 9 ? 1 C1 E ? z ? L / iz ? 1 ? ? fy As a further simplification, C1 may also be conservatively taken = 1. 0. Simplified assessment of ? LT Substituting in numerical values for simplified expressions result. ? 1 , the following S235 ? LT ? 1 L / iz C1 104 S275 ? LT ? 1 L / iz C1 96 S355 ? LT ? 1 L / iz C1 85 C1 may be conservatively taken = 1. , though the level of conservatism increases the more the tangible bending moment diagram differs from uniform moment. Simplified method (Cl. 6. 3. 2. 4) Simplified method for beams with restraints in buildings (Clause 6. 3. 2. 4) This method treats the compression flange of the beam and part of the web as a strut b b compaction h Tension Compression flange + 1/3 of the plane area of web Strut Dr. A Aziz Saim 2010 EC3 Beam Unrestrained Beam 58 General method (Cl. 6. 3. 4) General method for lateral an d lateral torsional buckling of structural components May be applied to single members, plane frames etc. Requires determination of plastic and elastic (buckling) resistance of structure, which subsequently defines global slenderness Generally requires FE Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 59 Blank Page Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 60 Important Notes (End Connections) When full torsional restraint exist -both the compression and latent hostility flanges are full restrained against rotation on plan -both flanges are partially restrained against rotation on plan both flanges are free to rotate on plan Unrestrained Beam 61 Dr. A Aziz Saim 2010 EC3 Connection DetailDr. A Aziz Saim 2010 EC3 Unrestrained Beam 62 Important Notes (End Connections) Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 63 Important Notes (End Connections) When both flanges are free to rotate on plan and the compression flange is unrestrained i. torsional restraint is provided wholly by compan ionship of the tension flange to the supports, ii. torsional restraint is provided solely by dead bearing of the tension flange on support. Unrestrained Beam 64 Dr. A Aziz Saim 2010 EC3 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 65 Dr. A Aziz Saim 2010 EC3 Unrestrained Beam 66