COLD FORED STEEL SECTIONS-II 0 COLD FORED STEEL SECTIONS- II 1.0 INTRODUCTION In the last hapter, the speial features and attrations of old formed steel setions for man industrial appliations were presented and disussed. In view of the use of ver thin steel sheet setions, (generall in the 1 mm - 3 mm range), partiular attention has to be paid to bukling of these elements. Stiffened and unstiffened elements were ompared and the onept of effetive width to deal with the rapid design of ompression elements together with suitable design simplifiations, outlined. Finall, the methods adopted for the design of laterall restrained beams and unrestrained beams were disussed. The tehniques of eliminating lateral bukling in pratie, b providing lateral braes or b attahment to floors et were desribed so that the ompression flanges would not bukle laterall. In this hapter the design of olumns for axial ompression, ompression ombined with bending as well as for torsional-flexural bukling will be disussed. The diversit of old formed steel shapes and the multipliit of purposes to whih the are put to, makes it a diffiult task to provide general solution proedures overing all potential uses. Some design aspets are nevertheless inluded to provide a general appreiation of this versatile produt. It is not unusual to design some old formed steel setions on the basis of prototpe tests or b emploing empirial rules. These are also disussed in a summar form herein..0 AXIALLY CORESSED COLUNS As pointed out in the last hapter, loal bukling under ompressive loading is an extremel important feature of thin walled setions. It has been shown that a ompressed plate element with an edge free to deflet does not perform as satisfatoril when ompared with a similar element supported along the two opposite edges. ethods of evaluating the effetive widths for both edge support onditions were presented and disussed. In analsing olumn behaviour, the first step is to determine the effetive area (A eff ) of the ross setion b summing up the total values of effetive areas for all the individual elements. The ultimate load (or squash load) of a short strut is obtained from s A eff. f d Q. A. f d (1) Copright reserved Version II 0-1
COLD FORED STEEL SECTIONS-II where s ultimate load of a short strut A eff sum of the effetive areas of all the individual plate elements Q the ratio of the effetive area to the total area of ross setion at ield stress In a long olumn with doubl - smmetri ross setion, the failure load ( ) is dependent on Euler bukling resistane ( EY ) and the imperfetions present. The method of analsis presented here follows the err-robertson approah presented in the hapter on "Introdution to Column Bukling". Following that approah, the failure load is evaluated from { [ s ( 1 η) EY ] [ s ( 1 η) EY ] 4 s EY } 1 where 0.00 e η 0 r e η 0, for 0 r, e for r > 0 EY the minimum bukling load of olumn and r radius of gration orresponding to EY. π EI min e () (a) (b) 1.0 Q 1 Q 0.9 Q 0.8 Q 0.7 p / f Q 0.6 Q 0.5 Q 0.4 Q 0.3 0 40 80 10 160 00 e / r Fig. 1 Column Strength (non- dimensional) for different Q fators Version II 0-
COLD FORED STEEL SECTIONS-II Fig. 1 shows the mean stress at failure (p / ross setional area) obtained for olumns with variation of e /r for a number of "Q" fators. (The -axis is non dimensionalised using the ield stress, f and "Q" fator is the ratio of effetive ross setional area to full ross setional area). lots suh as Fig. 1 an be emploed diretl for doubl smmetri setions. Load point Neutral axis e s Load point Effetive setion neutral axis A B A B (a) Channel setion loaded through its entroid (b) The move of the neutral axis (due to plate bukling) auses an eentriit e s and a onsequent moment. e s. This would ause an additional ompression on flange AB Fig. Effetive shift in the loading axis in an axiall ompressed olumn.1 Effetive shift of loading axis If a setion is not doubl smmetri (see Fig. ) and has a large redution of effetive widths of elements, then the effetive setion ma be hanged position of entroid. This would indue bending on an initiall onentriall loaded setion, as shown in Fig.. To allow for this behaviour, the movement of effetive neutral axis (e s ) from the geometri neutral axis of the ross setion must be first determined b omparing the gross and effetive setion properties. The ultimate load is evaluated b allowing for the interation of bending and ompression using the following equation: ult. e s (3) where is obtained from equation () and is the bending resistane of the setion for moments ating in the diretion orresponding to the movement of neutral axis; e s is the distane between the effetive entroid and atual entroid of the ross setion.. Torsional - flexural bukling Singl smmetri olumns ma fail either (a) b Euler bukling about an axis perpendiular to the line of smmetr (as detailed in.1 above) or (b) b a ombination of bending about the axis of smmetr and a twist as shown in Fig. 3. This latter tpe of Version II 0-3
COLD FORED STEEL SECTIONS-II behaviour is known as Torsional-flexural behaviour. urel torsional and purel flexural failure does not our in a general ase. osition after Flexural - Torsional bukling Axis of smmetr Original position θ Shear entre v Fig. 3 Column displaements during Flexural - Torsional bukling Theoretial methods for the analsis of this problem was desribed in the hapters on Beam Columns. Analsis of torsional-flexural behaviour of old formed setions is tedious and time onsuming for pratial design. Codes deal with this problem b simplified design methods or b empirial methods based on experimental data. As an illustration, the following design proedure, suggested in BS5950, art 5 is detailed below as being suitable for setions with at least one axis of smmetr (sa x - axis) and subjeted to flexural torsional bukling. Effetive length multipliation fators (known as α fators) are tabulated for a number of setion geometries. These α fators are emploed to obtain inreased effetive lengths, whih together with the design analsis presribed in.1 above an be used to obtain torsional bukling resistane of a olumn. For For EY EY > TF TF,, α α 1 EY TF (4) α values an be omputed as follows: where EY is the elasti flexural bukling load (in Newtons) for a olumn about the - Version II 0-4
( 7 ) COLD FORED STEEL SECTIONS-II π EI axis, i.e. e e effetive length ( in mm) orresponding to the minimum radius of gration TF torsional flexural bukling load (in Newtons) of a olumn given b 1 { 4 β } (5) 1 TF ( EX T ) ( EX T ) EX T β where EX Elasti flexural bukling load of the olumn (in Newtons ) about the x- axis given b π EI x e T Torsional bukling load of a olumn ( In Newtons) given b T 1 GJ r 0 π E Γ e x0 β is a onstant given b β 1 (7) r0 In these equations, r o polar radius of gration about the shear entre (in mm) given b (6) where r ( r r ) 1 (8) 0 x x0 r x, r are the radii of gration (in mm) about the x and - axis G is the shear modulus (N/mm ) x 0 is the distane from shear entre to the entroid measured along the x axis (mm) J St Venants' Torsion onstant (mm 4 ) whih ma be taken as b t, summed up for 3 all elements, where b flat width of the element and t thikness (both of them measure in mm) I x the moment of inertia about the x axis (mm 4 ) Γ Warping onstant for all setion. 3 Version II 0-5
COLD FORED STEEL SECTIONS-II.3 Torsion Behaviour Cold formed setions are mainl formed with "open" setions and do not have high resistane to torsion. Hene the appliation of load whih would ause torsion should be avoided where possible. Generall speaking, b adjusting the method of load appliation, it is possible to restrain twisting so that torsion does not our to an signifiant extent. In general, when examining torsional behaviour of thin walled setions, the total torsion ma be regarded as being made up of two effets: St. Venant's Torsion or ure Torsion Warping torsion. St.Venant's torsion produes shear stresses, whih var linearl through the material thikness. Warping torsion produes in-plane bending of the elements of a ross setion, thus induing diret (i.e. normal) stresses and the angle of twist inreases linearl. Sine old formed setions are thin walled, the have ver little resistane to St. Venant's Torsion and will twist substantiall. The extent of warping torsion in a thin walled beam is ver muh dependent on the warping restraint afforded b the supports as well as the loading onditions and the tpe of setion. If the beam ends are restrained from warping, then short beams exhibit high resistane to warping torsion and the total torque ating on suh a beam will be almost ompletel devoted to overoming warping resistane, the St Venant's Torsion being negligible. Conversel, the resistane to warping torsion beomes low for long beams and warping stresses and degrees of twist beome ver large. A detailed theoretial treatment of beams subjet to bending and torsion is given in another hapter. As stated previousl, partiular are and attention should be paid to the detailing of the onnetions and the method of load appliation so that the design for torsion does not pose a serious problem. 3.0 COBINED BENDING AND CORESSION Compression members whih are also subjet to bending will have to be designed to take into aount the effets of interation. The following heks are suggested for members whih have at least one axis of smmetr: (i) the loal apait at points of greatest bending moment and axial load and (ii) an overall bukling hek. 3.1 Loal Capait Chek The loal apait hek is asertained b satisfing the following at the points of greatest bending moment and axial load: F s x x 1 (9) Version II 0-6
COLD FORED STEEL SECTIONS-II F applied axial load s short strut apait defined b A eff. d (eqn. 1) x, applied bending moments about x and axis x oment resistane of the beam about x axis in the absene of F and oment resistane of the beam about axis in the absene of F and x. 3. Overall bukling hek For members not subjet to lateral bukling, the following relationship should be satisfied: F C bx x F x 1 EX C b F 1 EY 1 (10) For beams subjet to lateral bukling, the following relationship should be satisfied: F where x b C b F 1 EY 1 (11) axial bukling resistane in the absene of moments (see eq. ) EX, EY flexural bukling load in ompression for bending about the x- axis and for bending about the -axis respetivel. C bx, C b C b fators (defined in the previous hapter) with regard to moment variation about x and axis respetivel. b lateral bukling resistane moment about the x axis defined in the previous hapter. 4.0 TENSION EBERS If a member is onneted in suh a wa as to eliminate an moments due to onnetion eentriit, the member ma be designed as a simple tension member. Where a member is onneted eentriall to its axis, then the resulting moment has to be allowed for. The tensile apait of a member ( t ) ma be evaluated from t A e. (1) where A e is the effetive area of the setion making due allowane for the tpe of member Version II 0-7
COLD FORED STEEL SECTIONS-II (angle, plain hannel, Tee setion et) and the tpe of onnetion (eg. onneted through one leg onl or through the flange or web of a T- setion). p is design strength (N/mm ) Guidane on alulation of A e is provided in Codes of ratie (eg. BS 5950, art 5). The area of the tension member should invariabl be alulated as its gross area less dedutions for holes or openings. (The area to be deduted from the gross setional area of a member should be the maximum sum of the setional areas of the holes in an ross setion at right angles to the diretion of applied stress). Referene is also made to the hapter on "Tension embers" where provision for enhanement of strength due to strain hardening has been inorporated for hot rolled steel setions. The Indian ode IS: 801-1975 is in the proess of revision and it is probable that a similar enhanement will be allowed for old rolled steel setions also. When a member is subjeted to both ombined bending and axial tension, the apait of the member should be asertained from the following: Ft t and and x x x x 1 1 1 (13) (14) (15) where F t applied load t tensile apait (see eqn. 1) x,, x and are as defined previousl. 5.0 DESIGN ON THE BASIS OF TESTING While it is possible to design man old formed steel members on the basis of analsis, the ver large variet of shapes that an be formed and the omplex interations that our make it frequentl uneonomial to design members and sstems ompletel on theoretial basis. The behaviour of a omponent or sstem an often be asertained eonomiall b a test and suitable modifiations inorporated, where neessar. artiular are should be taken while testing omponents, that the tests model the atual loading onditions as losel as possible. For example, while these tests ma be used suessfull to assess the material work hardening muh aution will be needed when examining the effets of loal bukling. There is a possibilit of these tests giving Version II 0-8
COLD FORED STEEL SECTIONS-II misleading information or even no information regarding neutral axis movement. The speimen lengths ma be too short to pik up ertain tpes of bukling behaviour. Testing is probabl the onl realisti method of assessing the strength and harateristis of onnetions. Evaluating onnetion behaviour is important as onnetions pla a ruial role in the strength and stiffness of a struture. In testing omplete strutures or assemblies, it is vital to ensure that the test set up reflets the in-servie onditions as auratel as possible. The method of load appliation, the tpe of supports, the restraints from adjaent strutures and the flexibilit of onnetions are all fators to be onsidered arefull and modeled auratel. Testing b an independent agen (suh as Universities) is widel used b manufaturers of mass produed omponents to ensure onsisten of qualit. The manufaturers also provide load/span tables for their produts, whih an be emploed b strutural designers and arhitets who do not have detailed knowledge of design proedures. An advantage to the manufaturers in designing on the basis of proof testing is that the load/span tables obtained are generall more advantageous than those obtained b analtial methods; the also reassure the ustomers about the validit of their load/span tables. 6.0 EIRICAL ETHODS Some ommonl used members suh as Z purlins are sometimes designed b time-tested empirial rules; suh rules are emploed when theoretial analsis ma be impratial or not justified and when prototpe test data are not available. (embers designed b proven theoretial methods or b prototpe testing need not ompl with the empirial rules). As an illustration the empirial rules permitted b BS 5950, art 5 is explained below. B/5 B t L /45 L /60 Fig. 4 Z urlins Version II 0-9
COLD FORED STEEL SECTIONS-II 6.1 Z urlins A Z purlin used for supporting the roofing sheet is skethed in Fig. 4. In designing Z purlins with lips using the simplified empirial rules the following reommendations are to be omplied with: Unfatored loads should be used for designing purlins Imposed loads should be taken to be at least 0.6 kn /mm Claddings and fixings should be heked for adequa to provide lateral restraint to the purlin and should be apable of arring the omponent of load in the plane of the roof slope. The purlin should be onsidered to arr the load normal to roof slope (and a nominal axial load due to wind or restraint fores) These rules appl to purlins up to 8 m span in roof slopes up to 1 Antisag bars should be provided to ensure that laterall unsupported length of the purlin does not exeed 3.8 m. These should be anhored to rigid apex support or their fores should be transferred diagonall to main frames. urlin leats should provide adequate torsional restraint. 6. Design rules The following design rules appl with referene to Fig. 4 The overall depth should be greater than 100 t and not less than L /45. Overall width of ompression flange / thikness ratio should be less than 35. Lip width should be greater than B /5 Setion odulus WL 3 m for simpl supported purlins 1400 ο and WL 1800 3 m for ontinuous or semi rigidl jointed purlins. In the above, L span of the purlin (in mm) W Normal omponent of unfatored (distributed dead loadimposed load) in kn B Width of the ompression flange in mm T thikness of the purlin in mm. The net allowable wind uplift in a diretion normal to roof when purlins are restrained is taken as 50% of the (dead imposed) load. 7.0 CONCLUDING REARKS In the two preeeding hapters on old rolled steel, a detailed disussion of design of elements made from it has been provided, the major differenes between the hot rolled steel produts and old rolled steel produts outlined and the prinipal advantages of Version II 0-10
COLD FORED STEEL SECTIONS-II using the latter in onstrution summarized. Design methods, inluding methods based on prototpe testing and empirial design proedures have been disussed in detail. Thin steel produts are extensivel used in building industr in the western world and this range from purlins and lintels to roof sheeting and deking. Light steel frame onstrution is often emploed in house building and is based on industrialized manufature of standardized omponents, whih ensure a high qualit of materials of onstrution. The most striking benefit of all forms of light steel framing is their speed of onstrution, ease of handling and savings in site supervision and elimination of wastage in site, all of whih ontribute to overall eonom. In the Indian ontext, industrialized methods of prodution and deliver of old rolled steel produts to site have the potential to build substantiall more houses than is otherwise possible, with the same ash flow, thus freeing apital and finanial resoures for other projets. Other advantages inlude elimination of shrinkage and movement raks, greater environmental aeptabilit and less weather dependen. roperl onstruted light steel frames are adaptable to future requirements and will provide high aousti performane and a high degree of thermal insulation. rovided the sheets are pre-galvanised, the members provide adequate orrosion protetion when used lose to the boundaries of the building envelope. The design life of these produts exeeds 60 ears. 8.0 REFERENCES 1. Rhodes, J. and Lawson, R.., Design of Strutures using Cold Formed Steel Setions, The Steel Constrution Institute, Asot, 199.. Chung, K. F Worked examples to BS 5950, art 5, 1987, The Steel Constrution Institute, Asot, 1993. 3. Wei-Wen Yu, Cold Formed Steel Strutures b, Graw Hill Book Compan, 1973. Version II 0-11
COLD FORED STEEL SECTIONS-II Strutural Steel Design rojet Job No. Sheet 1 of Rev. Job title: Column Design Worked Example. 1 ade b RS Date April 000 CALCULATION SHEET Cheked b RN Date April 000 COLUN DESIGN Design a olumn of length.7 m for an axial load of 550 kn. Axial load 550 kn Length of the olumn, L.7 m Effetive length, e 0.85L 0.85.7.3 m Tr 00 80 5 4.0 mm Lipped Channel setion aterial roperties: E 05 kn/mm f 40 N/ mm p 40 / 1.15 08.7 N/mm Setion roperties: A 1576 315 mm I xx 903 10 4 mm 4 I [14 10 4 1576 4.8 ] 44 10 4 mm 4 44 10 r min 37.4 mm 1576 Load fator Q 0.95 ( from worked example 1 ) 4 Cl. 6.. BS 5950: art 5 The short strut resistane, s 0.95 1576 40/ 1.15 65 kn 550 kn < 65 kn O. K Axial bukling resistane Chek for maximum allowable slenderness e.3 10 r 37.4 3 61.5 < 180 O. K Version II 0-1
COLD FORED STEEL SECTIONS-II Strutural Steel Design rojet Job No. Sheet of Rev. Job title: Column design Worked Example. 1 ade b RS Date April 000 CALCULATION SHEET Cheked b RN Date April 000 In a double setion, torsional flexural bukling is not ritial and thus α 1 odified slenderness ratio, α r e Ref. Table 6 π E p 1 61.5 98.5 π.05 10 08.7 0.6 5 98.5 s 0.91 0.91 65 569 kn > O. K Version II 0-13