Mm action: the combinations of F and M corresponding to the full strength of the cross-section. This is the ‘strength’ limit, representing the case where the primary moment acting in conjunction with the axial load accounts for all the cross-section’s capacity. The substance of Fig. 18.10 can be incorporated within the type of interaction formula approach of section 18.2 through the concept of equivalent uniform moment presented in Chapter 16 in the context of the lateral–torsional buckling of beams; its meaning and use for beam-columns are virtually identical. For moment gradient loading member stability is checked using an equivalent moment = mM, as shown in Fig. 18.11. Coincidentally, suitable values of m, based on both test data and rigorous ultimate strength analyses, for the in-plane beam-column case are almost the same as those for laterally unrestrained beams (see section 16.3.6); m may conveniently be represented simply in terms of the moment gradient parameter b. The situation corresponding to the upper boundary or strength failure of Fig. 18. 10 must be checked separately using an appropriate means of determining cross- sectional capacity under F and M. The strength check is superfluous for b =+1 as it can never control, while as b Æ-1 and M Æ M c it becomes increasingly likely that the strength check will govern. The procedure is: (1) check stability using an interaction formula in terms of buckling resistance P c and moment capacity M c with axial load F and equivalent moment , (2) check strength using an interaction procedure in terms of axial capacity P s and moment capacity M c with coincident values of axial load F and maximum applied end moment M 1 (this check is unnecessary if m = 1.0; = M 1 is used in the stability check). Consideration of other cases involving out-of-plane failure or moments about both axes shows that the equivalent uniform moment concept may also be applied. For simplicity the same m values are normally used in design, although minor M M M 520 Members with compression and moments Fig. 18.9 Primary and secondary moments 2 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/ 'F 1.0 cross—section interaction /= -1.0 = 0.0 = 1.0 0,5 L/r = 40 0 0.5 ( ( M 1.0 1.0 cross—section interaction = —1.0 = 0.0 = 1.0 0. 0 0.5 M 1.0 Effect of moment gradient loading 521 Fig. 18.10 Effect of moment gradient on interaction 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/ F M=rnM variations for the different cases can be justified. For biaxial bending, two different values, m x and m y , for bending about the two principal axes may be appropriate.An exception occurs for a column considered pinned at one end about both axes for which b x = b y = 0.0, whatever the sizes of the moments at the top. 18.4 Selection of type of cross-section Several different design cases and types of response for beam-columns are outlined in section 18.2 of this chapter. Selection of a suitable member for use as a beam- column must take account of the differing requirements of these various factors. In addition to the purely structural aspects, practical requirements such as the need to connect the member to adjacent parts of the structure in a simple and efficient fashion must also be borne in mind. A tubular member may appear to be the best solution for a given set of structural conditions of compressive load, end moments, length, etc., but if site connections are required, very careful thought is necessary to ensure that they can be made simply and economically. On the other hand, if the member is one of a set of similar web members for a truss that can be fabricated entirely in the shop and transported to site as a unit, then simple welded connec- tions should be possible and the best structural solution is probably the best overall solution too. Generally speaking when site connections, which will normally be bolted, are required, open sections which facilitate the ready use of, for example, cleats or end- plates are to be preferred. UCs are designed principally to resist axial load but are also capable of carrying significant moments about both axes.Although buckling in 522 Members with compression and moments Fig. 18.11 Concept of equivalent uniform moment applied to primary moments on a beam-column 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 plane of the flanges, rather than the plane of the web, always controls the pure axial load case, the comparatively wide flanges ensure that the strong-axis moment capacity M cx is not reduced very much by lateral–torsional buckling effects for most practical arrangements. Indeed the condition M b = M cx will often be satisfied. In building frames designed according to the principles of simple construction, the columns are unlikely to be required to carry large moments. This arises from the design process by which compressive loads are accumulated down the building but the moments affecting the design of a particular column lift are only those from the floors at the top and bottom of the storey height under consideration. In such cases preliminary member selection may conveniently be made by adding a small percentage to the actual axial load to allow for the presence of the relatively small moments and then choosing an appropriate trial size from the tables of compres- sive resistance given in Reference 1. For moments about both axes, as in corner columns, a larger percentage to allow for biaxial bending is normally appropriate, while for internal columns in a regular grid with no consideration of pattern loading, the design condition may actually be one of pure axial load. The natural and most economic way to resist moments in columns is to frame the major beams into the column flanges since, even for UCs, M cx will always be com- fortably larger than M cy . For structures designed as a series of two-dimensional frames in which the columns are required to carry quite high moments about one axis but relatively low compressive loads, UBs may well be an appropriate choice of member. The example of this arrangement usually quoted is the single-storey portal building, although here the presence of cranes, producing much higher axial loads, the height, leading to large column slenderness, or a combination of the two, may result in UCs being a more suitable choice. UBs used as columns also suffer from the fact that the d/t values for the webs of many sections are non-compact when the applied loading leads to a set of web stresses that have a mean compres- sive component of more than about 70–100N/mm 2 . 18.5 Basic design procedure When the distribution of moments and forces throughout the structure has been determined, for example, from a frame analysis in the case of continuous construc- tion or by statics for simple construction, the design of a member subject to com- pression and bending consists of checking that a trial member satisfies the design conditions being used by ensuring that it falls within the design boundary defined by the type of diagram shown as Fig. 18.3. BS 5950 and BS 5400 therefore contain sets of interaction formulae which approximate such boundaries, use of which will automatically involve the equivalent procedures for the component load cases of strut design and beam design, to define the end points. Where these procedures permit the use of equivalent uniform moments for the stability check, they also require a separate strength check. Basic design procedure 523 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/ BS 5950: Part 1 requires that stability be checked using (18.7a) (18.7b) The first equation applies when major-axis behaviour is governed by in-plane effects and the second when lateral-torsional buckling controls. Both should normally be checked. In Equation (18.7) the use of p y Z, rather than M c , makes some allowance in the case of plastic and compact sections for the effects of secondary moments as described in section 18.2. For non-compact sections, for which M c = p y Z, no such allowance is made and an unconservative effect is therefore present. Evaluation of Equation (18.7) may be effected quite rapidly if the tabulated values of P cy , P cx , M b and p y Z y given in Reference 1 for all UB, UC, RSJ and SHS are used. In the cases where m values of less than unity are being used it is essential to check that the most highly stressed cross section is capable of sustaining the coincident compres- sion and moment(s). BS 5950: Part 1 covers this with the expression (18.8) Clearly when both M cx and M b values are the same Equation (18.7) is always a more severe check, or in the limit is identical, and only Equation (18.7) need be used. Values of A g p y , M cx and M cy are also tabulated in Reference 1. An an alternative to the use of Equations (18.7) and (18.8), BS 5950: Part 1 permits the use of more exact interaction formulae. For I- or H-sections with equal flanges these are presented in the form: (18.9a) (18.9b) (18.9c) in which the three expressions cover respectively: (a) Major axis buckling (b) Lateral-torsional buckling (c) Interactive buckling All three should normally be checked. The local capacity of the cross-section should also be checked. Class 1 and class 2 doubly-symmetric sections may be checked using: mM F P MFP mM F P MFP xx x xx yy y yy 105 1 1 1 1 + () - () + + () - () £ . cc ccc cc ccc / / / / F P mM M mM M F P y yy yy c c LT LT bc c c 1+++ È Î Í ˘ ˚ ˙ £ 1 F P mM M F P mM M x xx xx yx y y c cc c cc ++ È Î Í ˘ ˚ ˙ +£105 05 1 F Ap M M M M y x x y ygcc ++Ѐ1 F P mM M mM pZ yy yy c cy LT LT b ++£1 F P mM pZ mM pZ xx yx yy yy c c ++£1 524 Members with compression and moments 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/ (18.10) In Equation (18.10) the denominators in the two terms are a measure of the moment that can be carried in the presence of the axial load F. For fabricated sections, the principles of plastic theory may be applied first to locate the plastic neutral axis for a given combination of F, M x and M y and then to calculate M rx and M ry . This is manageable for uniaxial bending – F and M x or F and M y – but it is tedious for the full three-dimensional case and some use of approxi- mate results 1 may well be preferable. 18.6 Cross-section classification under compression and bending It is assumed in the discussion of the use of the BS 5950: Part 1 procedure that the designer has conducted the necessary section classification checks so as to ensure that the appropriate values of M cx , M cy , etc. are used. When the tabulated data of Reference 1 are being employed, any allowances for non-compactness are included in the listed values of M cx and M cy , but only if P cx and P cy have been taken from the strut tables rather than the beam-column tables will these contain any reduction. The reason is that for pure compression the stress pattern is known, whereas under combined loading the requirement may be to sustain only a very small axial load; to reduce P cx and P cy on the basis of uniform compression in each plate element of the section is much too severe. For simplicity, section classification may initially be conducted under the most severe conditions of pure axial load; if the result is either plastic or compact nothing is to be gained by conducting additional calculations with the actual pattern of stresses. However, if the result is a non-compact section, pos- sibly when checking the web of a UB, then it is normally advisable for economy of both design time and actual material use to repeat the classification calculations more precisely. 18.7 Special design methods for members in portal frames 18.7.1 Design requirements Both the columns and the rafters in the typical pitched roof portal frame represent particular examples of members subject to combined bending and compres- sion. Provided such frames are designed elastically, the methods already described for assessing local cross-sectional capacity and overall buckling resistance may be employed. However, these general approaches fail to take account of some of the special features present in normal portal frame construction, some of which can, when properly allowed for, be shown to enhance buckling resistance significantly. When plastic design is being employed, the requirements for member stability change somewhat. It is no longer sufficient simply to ensure that members can safely M M M M x x z y y z rr Ê Ë ˆ ¯ + Ê Ë Á ˆ ¯ ˜ 1 2 1Ѐ Special design methods for member in portal frames 525 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/ C 25'A symmetrical E about 4- resist the applied moments and thrust; rather for members required to participate in plastic hinge action, the ability to sustain the required moment in the presence of compression during the large rotations necessary for the development of the frame’s collapse mechanism is essential.This requirement is essentially the same as that for a ‘plastic’ cross-section discussed in Chapter 13. The performance require- ment for those members in a plastically designed frame actually required to take part in plastic hinge action is therefore equivalent to the most onerous type of response shown in Fig. 13.4. If they cannot achieve this level of performance, for example because of premature unloading caused by local buckling, then they will prevent the formation of the plastic collapse mechanism assumed as the basis for the design, with the result that the desired load factor will not be attained. Put simply, the requirement for member stability in plastically-designed structures is to impose limits on slenderness and axial load level, for example, that ensure stable behaviour while the member is carrying a moment equal to its plastic moment capacity suitably reduced so as to allow for the presence of axial load. For portal frames, advantage may be taken of the special forms of restraint inherent in that form of construction by, for example, purlins and sheeting rails attached to the outside flanges of the rafters and columns respectively. Figure 18.12 illustrates a typical collapse moment diagram for a single-bay pin- base portal subject to gravity load only (dead load + imposed load), this being the usual governing load case in the UK.The frame is assumed to be typical of UK prac- tice with columns of somewhat heavier section than the rafters and haunches of approximately 10% of the clear span and twice the rafter depth at the eaves. It is further assumed that the purlins and siderails which support the cladding and are attached to the outer flanges of the columns and rafters provide positional restraint to the frame, i.e. prevent lateral movement of the flange,at these points.Four regions in which member stability must be ensured may be identified: (1) full column height AB (2) haunch, which should remain elastic throughout its length 526 Members with compression and moments Fig. 18.12 Moment distribution for dead plus imposed load condition 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/ 903.5 788.5 0.150 24.6 A (3) eaves region of rafter for which the lower unbraced flange is in compression due to the moments, from end of haunch (4) apex region of the rafter between top compression flange restraints. 18.7.2 Column stability Figure 18.13 provides a more detailed view of the column AB, including both the bracing provided by the siderails and the distribution of moment over the column Special design methods for members in portal frames 527 Fig. 18.13 Member stability – column 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/ height. Assuming the presence of a plastic hinge immediately below the haunch, the design requirement is to ensure stability up to the formation of the collapse mechanism. According to clause 5.3.2 of BS 5950: Part 1, torsional restraint must be provided no more than D/2, where D is the overall column depth, measured along the column axis, from the underside of the haunch.This may conveniently be achieved by means of the knee brace arrangement of Fig. 18.14. The simplest means of ensuring adequate stability for the region adjacent to this braced point is to provide another torsional restraint within a distance of not more than L m , where L m is taken as equal to L u obtained from clause 5.3.3 as (18.11) Noting that the mean axial stress in the column f c is normally small, that p y is around 275N/mm 2 for S275 steel and that x has values between about 20 and 45 for UBs, gives a range of values for L u /r y of between 30 and 68. Placing a second torsional restraint at this distance from the first therefore ensures the stability of the upper part of the column. L r fpx y y u c £ + ()() [] 38 130 275 36 22 1 2 /// 528 Members with compression and moments Fig. 18.14 Effective torsional restraints Below this region the distribution of moment in the column normally ensures that the remainder of the length is elastic. Its stability may therefore be checked using the procedures of section 18.5. Frequently no additional intermediate restraints are necessary, the elastic stability condition being much less onerous than the plastic one. Equation (18.11) is effectively a fit to the limiting slenderness boundary of the column design charts 3 that were in regular use until the advent of BS 5950: Part 1, based on the work of Horne, 4 which recognized that for lengths of members between torsional restraints subject to moment gradient, longer unbraced lengths could be permitted than for the basic case of uniform moment. Equation (18.11) may therefore be modified to recognize this by means of the coefficients proposed by Brown. 5 Figure 18.15 illustrates the concept and gives the relevant additional for- mulae. For a 533 ¥ 210 UB82 of S275 steel for which x = 41.6 and assuming f c = 15N/mm 2 , the key values become: 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/ [...]... full height N ry = 4. 38 cm u = 0 .86 5 Sx = 2060 cm3 x = 41.6 Steelwork Design Guide vol 1 A = 104 cm2 py = 355 N/mm2 ly = 5600/43 .8 = 1 28 Table 9 4.7.2 l/x = 1 28/ 41.6 = 3. 08 v = 0.91 lLT = 0 .86 5 ¥ 0.91 ¥ 1 28 = 101 pb = 139 N/mm2 Mb = 139 ¥ 2060000 = 286 ¥ 106 Nmm = 286 kNm Table 19 4.3.6.7 Table 16 4.3.6.4 535 Steel Designers' Manual - 6th Edition (2003) 536 Worked examples The Steel Construction Institute... = 75 .8 cm2 Steelwork Design Guide Vol 1 Sy = 303 cm3 ly = 3600/51.9 = 69.4 Use Table 24 curve c for pc 4.7.2 Table 23 For py = 275 N/mm2 and l = 69.4 value of pc = 183 N/mm2 Pcy = 183 ¥ 7 580 = 1 387 ¥ 103 N = 1 387 kN 940 16 + = 0. 68 + 0.29 1 387 275 ¥ 199000 ¥ 10 -6 = 0.97 \ Adopt 203 ¥ 203 ¥ 60 UC 4.7.4 4 .8. 3.3.1 533 Steel Designers' Manual - 6th Edition (2003) 534 Worked examples Subject The Steel. .. document call 01344 87 2775 or go to http://shop.steelbiz.org/ Chapter ref DAN DAN Checked by GWO Made by 18 Sheet no 3 Table 18 1.6j 5.6 m b= 5.6 - 1.6 = 0.72 5.6 mLT = 0 .86 0 .86 ¥ 530 M = = 0.63 721 Mb 160 P = = 0.05 Pc 33 28 0.05 + 0.63 = 0. 68 OK 4 .8. 3.3.1 Check lower part of column for moment of 0.72 ¥ 530 = 382 kNm ly = 4000/43 .8 = 91 4.7.2 l/x = 91/41.6 = 2.2 v = 0.96 lLT = 0 .86 5 ¥ 0.96 ¥ 91 = 76... Discussion, 32, Sept 1965, 125–34 5 Brown B.A (1 988 ) The requirements for restraint in plastic design to BS 5950 Steel Construction Today, 2, 184 –96 6 Morris L.J (1 981 & 1 983 ) A commentary on portal frame design The Structural Engineer, 59A, No 12, 394–404 and 61A, No 6 181 –9 7 Morris L.J & Plum D.R (1 988 ) Structural Steelwork Design to BS 5950 Longman, Harlow, Essex 8 Horne M.R & Ajmani J.L (1972) Failure... The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 87 2775 or go to http://shop.steelbiz.org/ 5/ 3 Ê My ˆ +Á ˜ Ë Mry ¯ Sheet no 2 Checked by GWO Check local capacity using “more exact” method for plastic section Ê Mx ˆ Ë Mrx ¯ 18 4 .8. 2.3 5/ 3 £1 F/Pz = 664/1970 = 0.337 Mrx = Mry = 87 kNm Ê 24.4 ˆ Ë 87 ¯ 5/ 3 19.6 ˆ +Ê Ë 87 ¯ 5/ 3 = 0.120 + 0. 083 Steelwork... restraint provided by the 1.0 pm r' 0.0 —0.75 —1.0 jI 20 < x 30 K = 2.3 + 0.03x—xf/3O00 30 < x < 50 K = 0 .8 + 0. 08 a,— K0 S275 1O)f/2000 ( 180 +x)/300 steel flm= 0.44 + x/270 — f/200 5355 steel Fig 18. 15 (x — m= 0.47 + x/270 — f/250 Modification to Equation ( 18. 11) to allow for moment gradient \ Steel Designers' Manual - 6th Edition (2003) 530 Members with compression and moments sheeting rails This topic has... 0 .88 these are: 8. 4 and 34.3 Actual b/T = 5. 98 Actual d/t = 41.2 \ d/t greater than limit for pure compression However, actual loading is principally bending for which limit is 100e = 88 \ without performing a rigorous check (by locating plastic neutral axis position etc.) it is clear that section will meet the limit for principally bending Section compact \ Adopt 533 ¥ 210 ¥ 82 UC Steel Designers' Manual. .. = 33 28 ¥ 103 N = 33 28 kN Table 19 4.3.6.7 Table 18 4.3.6.4 Table 24 4.7.4 Steel Designers' Manual - 6th Edition (2003) Worked examples The Steel Construction Institute Silwood Park, Ascot, Berks SL5 7QN Subject BEAM-COLUMN EXAMPLE 2 ROLLED UNIVERSAL BEAM Design code BS 5950: Part 1 BS5950: Part 1 _ 530 kNrn This material is copyright - all rights reserved Reproduced under licence from The Steel. .. beam-columns Steel Designers' Manual - 6th Edition (2003) 532 Members with compression and moments 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 87 2775 or go to http://shop.steelbiz.org/ References to Chapter 18 1 The Steel Construction Institute (SCI) (2001) Steelwork Design... 4.3.6.4 537 Steel Designers' Manual - 6th Edition (2003) 5 38 Worked examples The Steel Construction Institute Silwood Park, Ascot, Berks SL5 7QN Subject Chapter ref BEAM-COLUMN EXAMPLE 2 ROLLED UNIVERSAL BEAM Design code BS 5950: Part 1 Made by DAN Sheet no 4 Checked by GWO pc = 1 78 N/mm2 Table 24 Pcy = 185 1 kN This material is copyright - all rights reserved Reproduced under licence from The Steel Construction . of p c = 183 N/mm 2 P cy = 183 ¥ 7 580 = 1 387 ¥ 10 3 N 4.7.4 = 1 387 kN 4 .8. 3.3.1 Adopt 203 ¥ 203 ¥ 60UC 940 1 387 16 275 199000 10 0 68 029 097 6 + ¥¥ =+ = - . Steel Designers' Manual - 6th. height. r y = 4.38cm u = 0 .86 5 Steelwork Design Guide S x = 2060cm 3 x = 41.6 vol 1 A = 104cm 2 p y = 355N/mm 2 Table 9 l y = 5600/43 .8 = 1 28 4.7.2 l/x = 1 28/ 41.6 = 3. 08 v = 0.91 Table 19 l LT = 0 .86 5 ¥. document call 01344 87 2775 or go to http://shop.steelbiz.org/ References to Chapter 18 1. The Steel Construction Institute (SCI) (2001) Steelwork Design Guide to BS 5950: Part 1: 2000, Vol. 1,