xtxo cii e= o2b2t (!+ C 2c3 Jx \b) H= (4c3_6oc2+3a2c+a2b)_e2Ix 021y +c2b2t(+) H= 2 / a(b+2ba+4bc+6oc) b t l+4c2(3bo÷3o2+4bc+2oc+c2) 1 2(2b+a+2c) immediately adjacent to the flange, is a possibility. Methods for assessing the likeli- hood of both types of failure are given in BS 5950: Part 1, and tabulated data to assist in the evaluation of the formulae required are provided for rolled sections in Reference 2. The parallel approach for cold-formed sections is discussed in section 16.7. In cases where the web is found to be incapable of resisting the required level of load, additional strength may be provided through the use of stiffeners. The design of load-carrying stiffeners (to resist web buckling) and bearing stiffeners is covered in both BS 5950 and BS 5400. However, web stiffeners may be required to resist shear buckling, to provide torsional support at bearings or for other reasons; a full treatment of their design is provided in Chapter 17. 16.3.6 Lateral–torsional buckling Beams for which none of the conditions listed in Table 16.6 are met (explanation of these requirements is delayed until section 16.3.7 so that the basic ideas and parameters governing lateral– torsional buckling may be presented first) are liable to have their load-carrying capacity governed by the type of failure illustrated in Fig. 16.4. Lateral–torsional instability is normally associated with beams subject to 442 Beams Fig. 16.3 (continued) Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ x C N vertical loading buckling out of the plane of the applied loads by deflecting sideways and twisting; behaviour analogous to the flexural buckling of struts. The presence of both lateral and torsional deformations does cause both the governing mathematics and the resulting design treatment to be rather more complex. The design of a beam taking into account lateral– torsional buckling consists essentially of assessing the maximum moment that can safely be carried from a knowledge of the section’s material and geometrical properties, the support condi- tions provided and the arrangement of the applied loading. Codes of practice, such as BS 5400: Part 3, BS 5950: Parts 1 and 5, include detailed guidance on the subject. Essentially the basic steps required to check a trial section (using BS 5950: Part I for a UB as an example) are: (1) assess the beam’s effective length L E from a knowledge of the support condi- tions provided (clause 4.3.5) (2) determine beam slenderness l LT using the geometrical parameters u (tabulated in Reference 2), L E /r y , v (Table 19 of BS 5950: Part 1) using values of x (tabu- lated in Reference 2). (3) obtain corresponding bending strength p b (Table 16) (4) calculate buckling resistance moment M b = p b ¥ the appropriate section modulus, S x (class 1 or 2), Z x (class 3), Z x,eff (class 4). Basic design 443 Table 16.6 Types of beam not susceptible to lateral – torsional buckling loading produces bending about the minor axis beam provided with closely spaced or continuous lateral restraint closed section Fig. 16.4 Lateral– torsional buckling Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ The central feature in the above process is the determination of a measure of the beam’s lateral–torsional buckling strength (p b ) in terms of a parameter (l LT ) which represents those factors which control this strength. Modifications to the basic process permit the method to be used for unequal flanged sections including tees, fabricated Is for which the section properties must be calculated, sections contain- ing slender plate elements, members with properties that vary along their length, closed sections and flats.Various techniques for allowing for the form of the applied loading are also possible; some care is required in their use. The relationship between p b and l LT of BS 5950: Part 1 (and between s li /s yc and l LT ÷(s yc /355) in BS 5400: Part 3) assumes the beam between lateral restraints to be subject to uniform moment. Other patterns,such as a linear moment gradient reduc- ing from a maximum at one end or the parabolic distribution produced by a uniform load, are generally less severe in terms of their effect on lateral stability; a given beam is likely to be able to withstand a larger peak moment before becoming lat- erally unstable. One means of allowing for this in design is to adjust the beam’s slen- derness by a factor n, the value of which has been selected so as to ensure that the resulting value of p b correctly reflects the enhanced strength due to the non-uniform moment loading. An alternative approach consists of basing l LT on the geometrical and support conditions alone but making allowance for the beneficial effects of non- uniform moment by comparing the resulting value of M b with a suitably adjusted value of design moment . is taken as a factor m times the maximum moment within the beam M max ; m = 1.0 for uniform moment and m < 1.0 for non-uniform moment. Provided that suitably chosen values of m and n are used, both methods can be made to yield identical results; the difference arises simply in the way in which the correction is made, whether on the slenderness axis of the p b versus l LT relationship for the n-factor method or on the strength axis for the m-factor method. Figure 16.5 illustrates both concepts, although for the purpose of the figure the m- factor method has been shown as an enhancement of p b by 1/m rather than a reduc- tion in the requirement of checking M b against = mM max . BS 5950: Part 1 uses the m-factor method for all cases, while BS 5400: Part 3 includes only the n-factor method. When the m-factor method is used the buckling check is conducted in terms of a moment less than the maximum moment in the beam segment M max ; then a separate check that the capacity of the beam cross-section M c is at least equal to M max must also be made. In cases where is taken as M max , then the buckling check will be more severe than (or in the ease of a stocky beam for which M b = M c , identical to) the cross-section capacity check. Allowance for non-uniform moment loading on cantilevers is normally treated somewhat differently. For example, the set of effective length factors given in Table 14 of Reference 1 includes allowances for the variation from the arrangement used as the basis for the strength– slenderness relationship due to both the lateral support conditions and the form of the applied loading. When a cantilever is subdivided by one or more intermediate lateral restraints positioned between its root and tip, then segments other than the tip segment should be treated as ordinary beam segments when assessing lateral– torsional buckling strength. Similarly a cantilever subject to M M M MM 444 Beams Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ effective design point using n—factor method (Pb' nXLT) LI an end moment such as horizontal wind load acting on a façade, should be regarded as an ordinary beam since it does not have the benefit of non-uniform moment loading. For more complex arrangements that cannot reasonably be approximated by one of the standard cases covered by correction factors,codes normally permit the direct use of the elastic critical moment M E . Values of M E may conveniently be obtained from summaries of research data. 6 For example, BS 5950: Part 1 permits l LT to be calculated from (16.3) As an example of the use of this approach Fig. 16.6 shows how significantly higher load-carrying capacities may be obtained for a cantilever with a tip load applied to its bottom flange, a case not specifically covered by BS 5950: Part 1. 16.3.7 Fully restrained beams The design of beams is considerably simplified if lateral– torsional buckling effects do not have to be considered explicitly – a situation which will occur if one or more of the conditions of Table 16.6 are met. In these cases the beam’s buckling resistance moment M b may be taken as its moment capacity M c and, in the absence of any reductions in M c due to local buck- lp LT y p E =÷ () ÷ () 2 Ep M M// Basic design 445 Fig. 16.5 Design modifications using m-factor or n-factor methods Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 1.0 0.8 0.6 0,4 0.2 UB 610 x 229 x 101 at bottom flange load at centroid 0.20.4 0.60.8 1.01.2 1.4 1.61.8 2.0 2.22.4 2.6 XLI ling,high shear or torsion,it should be designed for its full in-plane bending strength. Certain of the conditions corresponding to the case where a beam may be regarded as ‘fully restrained’ are virtually self-evident but others require either judgement or calculation. Lateral– torsional buckling cannot occur in beams loaded in their weaker princi- pal plane; under the action of increasing load they will collapse simply by plastic action and excessive in-plane deformation. Much the same is true for rectangular box sections even when bent about their strong axis. Figure 16.7, which is based on elastic critical load theory analogous to the Euler buckling of struts, shows that typical RHS beams will be of the order of ten times more stable than UB or UC sections of the same area. The limits on l below which buckling will not affect M b of Table 38 of BS 5950: Part 1, are sufficiently high (l = 340, 225 and 170 for D/B ratios of 2, 3 and 4, and p y = 275N/mm 2 ) that only in very rare cases will lateral– torsional buckling be a design consideration. Situations in which the form of construction employed automatically provides some degree of lateral restraint or for which a bracing system is to be used to enhance a beam’s strength require careful consideration.The fundamental require- ment of any form of restraint if it is to be capable of increasing the strength of the main member is that it limits the buckling type deformations. An appreciation of exactly how the main member would buckle if unbraced is a prerequisite for the provision of an effective system. Since lateral–torsional buckling involves both lateral deflection and twist, as shown in Fig. 16.4, either or both deformations may be addressed. Clauses 4.3.2 and 4.3.3 of BS 5950: Part 1 set out the principles gov- erning the action of bracing designed to provide either lateral restraint or torsional restraint. In common with most approaches to bracing design these clauses assume 446 Beams Fig. 16.6 Lateral– torsional buckling of a tip-loaded cantilever Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ ratio of Mr to for box section P 0 0 P 0 P z,, b /7 /1 0 0 N.) 0 C a- 0 - - - 0 C C- (-7, 0 0) 0 0 II N / \ I II 'l I that the restraints will effectively prevent movement at the braced cross-sections, thereby acting as if they were rigid supports. In practice, bracing will possess a finite stiffness. A more fundamental discussion of the topic, which explains the exact nature of bracing stiffness and bracing strength, may be found in References 7 and 8. Noticeably absent from the code clauses is a quantitative definition of ‘adequate stiffness’, although it has subsequently been suggested that a bracing system that is 25 times stiffer than the braced beam would meet this requirement. Examination of Reference 7 shows that while such a check does cover the majority of cases, it is still possible to provide arrangements in which even much stiffer bracing cannot supply full restraint. Basic design 447 Fig. 16.7 Effect of type of cross-section on theoretical elastic critical moment Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 16.4 Lateral bracing For design to BS 5950:Part 1,unless the engineer is prepared to supplement the code rules with some degree of working from first principles, only restraints capable of acting as rigid supports are acceptable. Despite the absence of a specific stiffness requirement,adherence to the strength requirement together with an awareness that adequate stiffness is also necessary, avoiding obviously very flexible yet strong arrangements, should lead to satisfactory designs. Doubtful cases will merit exami- nation in a more fundamental way. 7,8 Where properly designed restraint systems are used the limits on l LT for M b = M c (or more correctly p b = p y ) are given in Table 16.7. For beams in plastically-designed structures it is vital that premature failure due to plastic lateral– torsional buckling does not impair the formation of the full plastic collapse mechanism and the attainment of the plastic collapse load. Clause 5.3.3 provides a basic limit on L/r y to ensure satisfactory behaviour; it is not necessarily compatible with the elastic design rules of section 4 of the code since acceptable behaviour can include the provision of adequate rotation capacity at moments slightly below M p . The expression of clause 5.3.3 of BS 5950: Part 1, (16.4) makes no allowance for either of two potentially beneficial effects: (1) moment gradient (2) restraint against lateral deflection provided by secondary structural members attached to one flange as by the purlins on the top flange of a portal frame rafter. The first effect may be included in Equation (16.4) by adding the correction term L r fpx m y cy £ + ()() [] 38 130 275 36 22 1 2 /// 448 Beams Table 16.7 Maximum values of l LT for which p b = p y for rolled sections p y (N/mm 2 ) Value of l LT up to which p b = p y 245 37 265 35 275 34 325 32 340 31 365 30 415 28 430 27 450 26 Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ compression boom of Brown, 9 the basis of which is the original work on plastic instability of Horne. 10 This is covered explicitly in clause 5.3.3.A method of allowing for both effects when the beam segment being checked is either elastic or partially plastic is given in Appendix G of BS 5950: Part 1;alternatively the effect of intermittent tension flange restraint alone may be allowed for by replacing L m with an enhanced value L s obtained from clause 5.3.4 of BS 5950: Part 1. In both cases the presence of a change in cross-section, for example, as produced by the type of haunch usually used in portal frame construction, may be allowed for. When the restraint is such that lateral deflection of the beam’s compression flange is prevented at intervals, then Equation (16.4) applies between the points of effective lateral restraint. A discussion of the application of this and other approaches for checking the stability of both rafters and columns in portal frames designed according to the principles of either elastic or plastic theory is given in section 18.7. 16.5 Bracing action in bridges – U-frame design The main longitudinal beams in several forms of bridge construction will, by virtue of the structural arrangement employed, receive a significant measure of restraint against lateral– torsional buckling by a device commonly referred to as U-frame action. Figure 16.8 illustrates the original concept based on the half-through girder form of construction. (See Chapter 4 for a discussion of different bridge types.) In a simply-supported span, the top (compression) flanges of the main girders, although laterally unbraced in the sense that no bracing may be attached directly to them, cannot buckle freely in the manner of Fig. 16.4 since their lower flanges are restrained by the deck. Buckling must therefore involve some distortion of the girder web into the mode given in Fig. 16.8 (assuming that the end frames prevent lateral movement of the top flange). An approximate way of dealing with this is to regard each longitudinal girder as a truss in which the tension chord is fully Bracing action in bridges – U-frame design 449 Fig. 16.8 Buckling of main beams of half-through girder Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ (a) U—frame unit load unit load (b) laterally restrained and the web members, by virtue of their lateral bending stiff- ness, inhibit lateral movement of the top chord. It is then only a small step to regard this top chord as a strut provided with a series of intermediate elastic spring restraints against buckling in the horizontal plane.The stiffness of each support cor- responds to the stiffness of the U-frame comprising the two vertical web stiffeners and the cross-girder and deck shown in Fig. 16.9. The elastic critical load for the top chord is (16.5) in which L E is the effective length of the strut. If the strut receives continuous support of stiffness (1/d L R ) per unit length, in which L R is the distance between U-frame restraints, and buckles in a single half- wave, this load will be given by (16.6) which gives a minimum value when (16.7) giving (16.8) or (16.9) If lateral movement at the ends of the girder is not prevented by sufficiently stiff end U-frames, the mode will be as shown in Fig. 16.10. The effective length is then: (16.10) In clause 9.6.4.1.1.2 of BS 5400: Part 3 the effective length is given by L k EI L EcR = () d 025. LEIL ER = () pd 025. LEIL ER =÷ ()( ) pd/ . 2 025 PEIL cr R = () 2 05 / . d LEIL= () pd R 025. PEILLL cr R = () + () ppd 2222 // PEIL cr E 2 = p 2 / 450 Beams Fig. 16.9 U-frame restraint action. (a) Components of U-frame. (b) U-frame elastic support stiffness Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ [...]... Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ A 4 57 ¥ 152 ¥ 74 UB provides Mcx of 429 kNm Now check lateral-torsional buckling strength for segments AB, BC & CD Steelwork Design Guide Vol 1 AB b = 0/406 = 0.0 mLT = 0. 57 — M = 0. 57 ¥ 406 = 231.4 kNm Table 18 For LE = 3.0 m, Mb = 288 kNm \ 288 > 231.4 kNm OK Steelwork... from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ Chapter ref BEAM EXAMPLE 1 LATERALLY RESTRAINED UNIVERSAL BEAM From Table 5, limit is span/360 \ d OK \ Use 4 57 ¥ 152 ¥ 67UB Grade 43 Made by DAN 16 Sheet no 3 Checked by GWO 2.5.1 Steel Designers' Manual - 6th Edition (2003) Worked examples The Steel Construction... 125 UB u = 0. 873 ry = 4.98 cm x = 34.0 Sx = 3680 cm3 l = LE / ry = 72 00 / 49.8 = 145 Steelwork Design Guide Vol 1 4.3 .7. 5 l/x = 145/34 = 4.26 v = 0.85 Table 19 l LT = u v l \ l LT = 0. 873 ¥ 0.85 ¥ 145 = 108 Pb = 116 N/mm2 Mb = Sxpb = 3680 ¥ 103 ¥ 116 = 4 27 ¥ 106 N mm = 427kN m > 369kN m OK Table 20 4.3 .7. 3 465 Steel Designers' Manual - 6th Edition (2003) 466 Worked examples Subject The Steel Construction... Ascot, Berks SL5 7QN Design code BS 5950: Part 1 Made by Sheet no 2 Checked by GWO Assuming use of S 275 steel and no material greater than 16 mm thick, Table 9 = 275 N/mm2 take py This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ DAN... girders 470 Steel Designers' Manual - 6th Edition (2003) Initial choice of cross-section for plate girders in buildings 471 transverse This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ stiffener fitlet weld —fl. d A—A N Fig 17. 1 Elevation... licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ 17. 4 Design of plate girders used in buildings to BS 5950: Part 1: 2000 17. 4.1 General Any cross-section of a plate girder will normally be subjected to a combination of shear force and bending moment, present in varying proportions BS 5950: Part 1: 20001... from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ (a) (b) Fig 16.16 Vierendeel-type action in beam with web openings: (a) overall view, (b) detail of deformed region bottom tee (0) b 2o b 2a (b) Fig 16. 17 Castellated beam: (a) basic concept, (b) details of normal UK module geometry Steel Designers' Manual. .. Institution (2000) Part 1: Code of practice for design in simple and continuous construction: hot rolled sections BS 5950, BSI, London Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ References... Reproduced under licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ Fig 16.15 4 57 be/2 I -1 Effective cross-section (strength and stiffness) are then calculated for this effective cross-section as illustrated in Fig 16.15 Tabulated information in BS 5950: Part 5 for steel of yield strength 280 N/mm2... Vol 1 BC b = 377 /406 mLT — M = 0.93 = 0. 97 Table 18 = 0. 97 ¥ 406 = 393.8 kNm But for LE = 3.0 m, Mb = 288 kNm Not OK Try 4 57 ¥ 191 ¥ 82 UB this provides Mb of 396 kNm > 393.8 kNm OK CD Satisfactory by inspection OK A 4 57 ¥ 191 ¥ 82 UB provides sufficient resistance to lateraltorsional buckling for each segment and thus for the beam as a whole Steelwork Design Guide Vol 1 Steel Designers' Manual - 6th . The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ 16.4 Lateral bracing For design to BS 5950 :Part. document call 01344 872 775 or go to http://shop.steelbiz.org/ ratio of Mr to for box section P 0 0 P 0 P z,, b /7 /1 0 0 N.) 0 C a- 0 - - - 0 C C- ( -7, 0 0) 0 0 II. Reproduced under licence from The Steel Construction Institute on 12/2/20 07 To buy a hardcopy version of this document call 01344 872 775 or go to http://shop.steelbiz.org/ 4,' for base