Design of members subject to combined bending and torsion PDF
Preview Design of members subject to combined bending and torsion
P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... SCI PUBLICATION 057 Design of Members Subject to Combined Bending m - m anu I orslon D. A. NETHERCOT BSc(Eng) PhD CEng FlStructE University of Nottingham P. R. SALTER BSc(Eng) CEng MlStructE (formerly of) The Steel Construction Institute A. S. Malik BSC MSC The Steel Construction Institute ISBN 1 870004 44 2 0T he Steel Construction Institute 1989 (Reprinted, l99 7) The Steel Construction Institute Silwood Park Ascot Berkshire SL5 70N Telephone: 013 44 23345 Fax: 013 44 22944 Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... This publication seeks to provide authoritative guidance for the design of steel structures subjected to combined bending and torsion. In most cases, however, i t should be possible to avoid the introduction of significant torsion, by paying attention to detail andch oosing a load path fors uch an alternative. This publication is, therefore, concerned with the minority of cases where the loads have to be applied eccentrically with respect to the shear centre. Afterb r iae f discussion of the background theory, simple methods of evaluating torsional stresses and deformations are detailed, and worked examples illustrating the use of tables and charts included in the publication are provided. The text has been prepared by Professor D A Nethercot of Nottingham University and Messrs. P R Salter and A S Malik of the Steel Construction Institute and reviewed by Mr J C Taylor and Dr R Narayanan of The Steel Construction Institute. .. 11 Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... CONTENTS Page SUMMARY NOTATION 1. DESIGNING FOR TORSION 1 .l Introduction 1 1.2 Scope of the Publication 1 1.3 Choice of Members 2 2. BASICT HEORY 2.1 Torsion 3 2.2 Bending 12 2.3 Combined Bending and Torsion 13 3. LOADA PPLICATION 16 4. WORKEDE XAMPLES 17 5. SECTIONP ROPERTIES 89 6. CHARTS TO ASSIST IN EVALUATION 6.1 Standard Cases 98 6.2 Extreme Cases 99 REFERENCES 109 APPENDIX A. Evaluation of Torsional Properties 1 1 1 APPENDIX B. Solution of Differential Equations 118 iii Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... Design of Members Subject to Combined Bending and Torsion The causes of torsional loading on structural members are discussed and those situations in which the explicit consideration of torsion needs to form part of the design calculations are identified. The basic theory of the torsion of both open and closed steel sections is presented. Solutions of the resulting equations in terms of both design charts and formulae for a selection of applied torsional loadings and support conditions are provided. A simple method for combining the effects of torsion and bending, consistent with the approach of BS 5950: Part I is presented. The complete design approach for combined bending and torsion is illustrated by means of a number of worked examples. These show that design will frequently be governed by the need to restrict twisting at working load to acceptable levels, rather than by considerations of ultimate strength. Dimensionnement des Elements Soumis a Flexion et Torsion Combinees Rbume Les raisons conduisant a un chargement par torsion des elements structuraux sont discutees et les situations ou une prise en compte explicite dela torsion est nkcessaire dans les calculs de dimensionnement sont identifiees. La thkorie classique de la torsion desp iPces en acier a section ouverte ou fermee est prksentke. Des solutions des Pquations qui en resultent sont prksentees sous forme de diagrammes de dimensionnement, d'une part, et sous forme de formules, d'autre part. Elles permettent de sklectionner les charges de torsion a appliquer et les conditions d'appuis. Une mkthode simple pour combiner les effets de la torsion et de la flexion est presentee. Elle est en accord avec la norme BS 5950 : Partie 1. Le mkthode complkte de dimensionnement en flexione t torsion combinkes est illustree au moyen d'exemples. Ils montrent que le dimensionnement est souvent gouverne par la necessitk de restreindre les deformations torsionnelles, sous les charges de service, a des valeurs acceptables, plutbt que par la resistance ultime. Berechnung von Bauteilen unter Biegung und Torsion Zusammenfassung Die Ursachen f u r Torsionsbeanspruchung von Bauteilen werden besprochen und die Falle, in denen eine Berucksichtigung der Torsion klarer Bestandteil der statischen Berechnung sein muJ3. Die elementare Theorie der Torsion von offenen und geschlossenen Stahlquerschnitten wird vorgestellt. Die Losungen der sich ergebenden Gleichungen werden fur eine Auswahl von Torsionsbelastungen und Randbedingungen in Form von Bemessungstafeln und Formeln zur Verfugung gestellt. Eine einfache Methode fur kombinierte Beanspruchung aus Biegung und Torsion entsprechend BS 5950, Teil I , wird vorgestellt. Der vollstandige Weg zur Bemessung bei Biegung und Torsion wird anhand einer Reihe von Beispielen aufgezeigt. Die Beispiele zeigen, daJ3 die Bemessung oft von der Notwendigkeit bestimmt wird, die Verdrehung durch die angreifenden Lasten in akzeptablen Grenzen zu halten, weniger von dem Gesichtspunkt der Bruchfestigkeit. iv Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... Progretto di Membrature Soggette a Flessione e Torsione Sommario Vengono discusse le cause dell'impegno torsionale degli elementi strutturali e identificate quelle situazioni nelle qualPi necessario tenere conto esplicitamented ella torsione nei calcoli di pregetto. E presentata la teoria della torsione con riferimento ai profili apertie a quelli a sezione chiusa. Viene quindi riportata la soluzione delle equazioni che reggono il problema, in forma sia di abachi sia di formule, per una gammas ignificativa di condizioni di carico e di vincolo. E altresi illustrato un metodo semplice che consente di combinare glei ffetti della torsione e della flessione, metodo in accord0 con l'approcciod elle BS.5950: parte I. Una serie di esempi consentlea comprensione del'approccio progettuale per elementi soggetti a flessotorsione nella sua completezza. Si mettei n luce come la necessita di limitare a livelli accettabili la deformazione torsionale sotto i carichi di esercizio governa il progetto in molti casi, mentre la resistenza ultima riveste minore importanza. Diseno de Piezas Sometidas a Flexion y Torsion Combinadas Resumen Se analizanl as causas que originanc argas de torsi6n elna s piezas de unaes tructura y se identifican aquellas situaciones en que debe incluirsel a consideracion explicita de la torsion en 10s calculos de rrn proyecto. Se presenta la teoria basica de torsion en secciones de acero tanto abiertas como cerradas; t a m b i h se suministran soluciones de las ecuaciones resultantes mediante abacos de disenoy formulas para diferentes tipos cdaer gas torsoras y condiciones de apoyo. Se incluye, en particular un procedimiento sencillo de combinacion de efectos de flexiony torsion, congruente con el metodo patrocinado porla BS.5950: Parte I . Se desarrollan una serie de ejemplos que ilwtran e l me'todo de analisis completo para flexion y torsion, y demuestran que a menudo el diseno queda controlado por la necesidad de mantener la torsion en tensiones admisibles a niveles aceptables en lugar de por consideraciones der esistencia ultima. V Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... Area enclosed by mean perimeter of closed section Torsion bending constant equal to [E] Distance between toe of flange and centre line of web of channel section Beam depth Torsional modulus, constant for closed sections Distance from centre of web to shear centre of channel section Modulus of elasticity of steel (205000 N/mm2) Eccentricity of load with respect to the shear centre E Shear modulus of elasticity of stel (taken to be - 2( 1 +v) ' where Poissons ratio v = 0.3, thus E/G = 2.6 and G has an approximate value of 79000 N/mm2) Warping constant for cross section Depth of open section, centre to centre of flanges; mean perimeter of closed section Second moments of area of cross section about the major and minor axes Torsional constant for cross section Length of member Bending moment acting on cross section Applied load Statical moment (about the neutral axis of the entire cross section) of the crosss ectional area between the free edges of the cross sectiona nda plane cutting the cross section across the minimum thickness at the point under examination Value of Q for a point in the flange directly above the vertical face of the web Value of Q for a point at mid-depth of section Plastic modulus about the major and minor axes Warping statical moment at a point 'S' on cross section Flange thickness Pure torsional resistance equal to GJ+' Applied torque (torsional moment) at given location Warping torsional resistance equal to P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... W”, Normalised warping function at a point ‘S’ on cross section xo, Yo Co-ordinates of the shear centre with respect to the centroid Y Perpendicular distance from neutral axis to a point on cross section Z Distancef rom left end of member( origin of co-ordinates ystem)t o transverse section under examination (Figure 2.3) Elastic moduli about the major and minor axes Distance from support to point of applied torsional moment (or to endo f uniformly distributed load over a portion of span), divided by the span length (i.e. aL is the distance, (Y is a fraction of L ) Total angle of twist at a transverse section of member, radians First derivative of 4 with respect to z Second derivative of 4 with respect to z Third derivative of 4 with respect to z Fourth derivative of 4 with respect to z Combined longitudinal stress Longitudinal stress due to plane bending Warping normal stress, i.e. longitudinal stress at a point on cross section due to restrained warping of the cross section 7- Combined shear stress Shear stress due to plane bending Pure torsional shear stress Warping shear stress at point on cross section due to restrained warping of the cross section vii Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... 1. 1. l ,Introduction When a member is subject to torsion it will twist about a longitudinal axis which passes through the shear centre of the cross section. However, torsion will not occur if the section is loaded in such a manner that the resultant force passes through the shear centre. In the majority of design situations, the loads are appliedso that the resultant force passes through the centroid. If the section is doubly symmetric, this automatically eliminates torsion because the centroid and the shear centre coincide. In most cases, the load transfer through the connections of the members applying the loads may be regarded as ensuring that these loads are effectively applied through the shear centre. This is also generally true of loadsf romf loor slabs supported on the top flange of beams, even for channel sections. Designing to transferl oads by means of torsions hould be avoidedw herever possible, as it is not usually an efficient method of resisting loads. When this is not possible, care should be taken to arrange framing so as to minimise any torsion. Attentiotnd o e tailp ,a rticularly when considering how loads are actually transferred to members,c an minimise or even eliminate many potentiald ifficulties associated with torsional effects. Where significant torsional eccentricity is unavoidable, consideration should be given to the use of box girders, comprising either a lattice girder fully triangulated on all faces or hollow rolled or plated sections. The assumptions made when using a computer program to analyse a grillage or three-dimensional framework should also be consideredIf. the members andj oints are assumed to have torsional resistance, then torsional moments will be included in the output. In order to maintain equilibrium with the applied loads, these must then be taken into accounitn designing the joints and the members. If, on the other hand, the members and joints are assumed not to have torsional resistance, no torsional moments will arise and the remaining moments and forces will be in equilibrium with the applied loads. In most cases this approach will be the more practical. However, this assumption should not be used for fatigue analysis. The abovei s an exampleo f a broader principle, which is valid due to thed uctility of steelwork. Unless it is necessary to utilise the torsional resistance of a member, it is not necessary to take account of it. As always, the details of the joints must be made consistent with the assumptions made in the analysis. 1.2 Scope of the publication This publication is concerned with the minority of cases where the loadi s eccentric to the shear centre. Whilst it is important to recognise and deal with such cases when they occur, it is also important not to apply its methods where they are not necessary. When loading is eccentric with respect to the shear centre, the response of the member may conveniently be examined by separating the loading into bending and torsional components. Bending stresses and deflections can be obtainedi n the usual manner by assuming that the loadsa ct through the shear centre and resolving the forces into components parallel to thep rincipal axes. Torsional stressesa nd deformations can be calculated for standard cases using the Tables (Section 5 ) and Graphs (Section 6) in this publication. For non-standard cases and for determination of the torsional effects at other than the critical positions, equations have been provided in Appendix B. The user may then choose between hand or computerised methods of calculation. 1 Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... The combined bending and torsional effects (Section 2.3) are then checked by means consistent with the methods used in BS 5950: Part 2 . ( l3) This publication is principally concerned with providing guidance for the design of hot rolled open sections. However, guidance is also given on the design of hot rolled tubular sections but reference should be made to more detailed literature for the design of Box Girders.(') Membersw hich are curved on plan or which contain particularly slender plate elements, e.g.c old formed sections, are not consideredin this publication. Detailed guidance on these topics is given in References 2 to 5 . Examples in Section 4 have been provided to illustrate the use of the Tables (Section 5) and Graphs (Section 6) for standard cases. 1.3 Choice of member The initial choice of member in design situations not affectedb y torsion tends to be governed by the proportionso f axial load to bending moment and the unrestrained length of the section. For members predominantly subject to bendinga,n I section such as a universal beam will produce an efficient design. Similarly for members subject to axial loading, a universal columnH section is a reasonable choice.W hen the unrestrained length of the member is high, hollow sections can be advantageous. Torsional loading also has a significant influence on thinei tial choice of section for maximum efficiency. For reasons which are explained later in this publication, I shaped sections are particularly poor at resisting torsion while tubular sections can be very effective. Although H sections are better at resisting torsion than I sections, they are still a poor choice compared with a tubular member. Torsional effects should, therefore, be taken into account early in the design process when the type of member to beu sed is under consideration and not left to the final stages when perhaps an inappropriate type of member has already been selected. Not only may lighter sections result, but the design time will also be reduced. A distinction is made in this publication between open sections such as I and channel section shapes which are poor at resisting torsion and closed sections such as tubular members which are more effective (Figure 1.1). OpenC sloesce tido ns sect ions Figure 1.1 Choice of section 2 Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement P057: Design of Members Subject to Combined Bending and Torsion Discuss me ... 2. When a member is loaded so that the resultant of the applied forces does not pass through the shear centre of the section, the member will be subject to additional stresses due to torsion as well as those due to bending. In the method presented in this publication the effects of torsion and bending are first considered separately and then combined, as explained in Section 2.3. 2.1 Torsion 2.1 .l Shearc entre The shear centre of a cross section lies on the longitudinal axis about which the section would twist if torsion acts on the section. If the resultant force acts through the shear centre, no twist will occur and the torsional stresses will be zero. The shear centre and the centroid are not necessarily coincident. However, in a rolled I or H section, which is symmetrical about both principal axes, the shear centre, S, coincides with the centroid, c (Figure 2.la). This is also true for sections which are point symmetric such as zed sections (Figure 2. lb). For a channel section there is a symmetry about the x-x axis only and, therefore, the shear centre lies on the x-x axis but not the y-y axis (Figure 2 .1~)C. onversely, for sections which have symmetry about the y-y axis only, the shear centrel ies on the y-y axis but is eccentric to thex -x axis (Figure 2.ld). When the channels ection is asymmetric the shear centre is eccentric to both axes (Figure 2.le). Methods of calculating the position of the shear centre of a cross section are given in Appendix A. Special cases such as angles and tees where the centrel ines of the elements intersect at a single point have the shear centre located at that point (Figure 2.lf). 2.1.2 Torsionarl esistance The total resistance of a member to torsionalo ading is composed of the sum of two components known as ‘uniform torsion’ and ‘warping torsion’. In some cases only uniform torsion occurs. When warping is included in the torsional resistance, the member is in a state of ‘non-uniform torsion’. Uniform torsion is also referred to as ‘pure’ or ‘St Venant’ torsion. When uniform torsion occurs, the rate of change of the angle of twist is constant along the member (Figure 2.2a). 3 Created on 3202 MJualyrc 2h0 200911 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
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