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 S...

Author: D. A. Nethercot | P. R. Salter | A. S. Malik | Steel Construction Institute (Great Britain)

352 downloads 2850 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

SCI PUBLICATION 057

Design of Members Subject to Combined Bending anu Iorslon m

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

m -

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

0The Steel Construction Institute 1989 (Reprinted, l99 7)

The Steel Construction Institute Silwood Park Ascot Berkshire SL5 7 0 N Telephone: 0 1 344 23345 Fax: 0 1 344 22944


This publication seeks to provide authoritative guidance for the design of steel structures subjected to combined bending and torsion. In most cases, however, it should be possible to avoid the introduction of significant torsion, by paying attention to detail andchoosing a load path forsuch 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. Afterbrief a 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


CONTENTS Page SUMMARY NOTATION 1.


2.

DESIGNING FOR TORSION 1 .l Introduction 1.2 Scope of the Publication 1.3 Choice of Members

1 1 2

BASICTHEORY 2.1 Torsion 2.2 Bending 2.3 Combined Bending and Torsion

3 12 13

3.

LOADAPPLICATION

16

4.

WORKEDEXAMPLES

17

5.

SECTIONPROPERTIES

89

6.

CHARTS TO ASSIST IN EVALUATION 6.1 Standard Cases 6.2 Extreme Cases

98 99 109

REFERENCES APPENDIX A.

Evaluation of Torsional Properties

APPENDIX B. Solution

of Differential Equations

111 118

iii


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 despiPces 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 flexionet 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


Progretto di Membrature Soggette a Flessione e Torsione Sommario Vengono discusse le cause dell'impegno torsionale degli elementi strutturali e identificate quelle situazioni nelle qualiP necessario tenere conto esplicitamentedella 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 gammasignificativa di condizioni di carico e di vincolo. E altresi illustrato un metodo semplice che consente di combinare glieffetti della torsione e della flessione, metodo in accord0 con l'approcciodelle BS.5950: parte I. Una serie di esempi consentela comprensione del'approccio progettuale per elementi soggetti a flessotorsione nella sua completezza. Si mettein 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 analizanlas causas que originancargas de torsi6n enlas piezas de unaestructura y se identifican aquellas situaciones en que debe incluirsela 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 cargas de 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 deresistencia ultima.

V


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 steel (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 closed section

of flanges; mean perimeter

of


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 crosssectional area between the free edges of the cross sectionanda 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 EH@'' Thickness generally; web thickness Uniformlydistributedappliedtorque(torsionalmoment)or value of varying applied torque

maximum

Shear acting on cross section Distance from toe of channel section to point on flange where W,, X B' flange is maximum, given by W n o + Wn2 vi

T~

in the


W”,

Normalised warping function at a point

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

Distancefrom left end of member(origin of co-ordinatesystem)to transverse section under examination (Figure 2.3)

‘S’

on cross section

Elastic moduli about the major and minor axes Distance from support to point of applied torsional moment (or to end of 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

4 with respect to z Fourth derivative of 4 with respect to z

Third derivative of

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 Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

7-

Combined shear stress Shear stress due to plane bending Pure torsional shear stress Warping shear stress at point on cross section of the cross section

due to restrained warping

vii


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 force passes occur if the section is loaded in such a manner that the resultant 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 loadsfromfloor slabs supported on the top flange of beams, even for channel sections.


Designing to transferloads by means of torsionshould be avoidedwherever 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. Attention to detail, particularly when considering how loads are actually transferred to members,can minimise or even eliminate many potentialdifficulties 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 considered. If the members andjoints 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 accountin 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 aboveis an exampleof a broader principle, which is valid due to theductility 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 thepublication

This publication is concerned with the minority of cases where the loadis 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 obtainedin the usual manner by assuming that the loadsact through the shear centre and resolving the forces into components parallel to theprincipal axes. Torsional stressesand 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


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.(') Memberswhich are curved on plan or which contain particularly slender plate elements, e.g.cold 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 affectedby torsion tends to be governed by the proportionsof axial load to bending moment and the unrestrained length of the section. For members predominantly subject to bending, an 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.When the unrestrained length of the member is high, hollow sections can be advantageous.


Torsional loading also has a significant influence on the initial 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 beused 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).

OpenClosed sections Figure 1.1 Choice of section

2

sect ions


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 Shearcentre 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 ~ ) Conversely, . for sections which have symmetry about the y-y axis only, the shear centrelies on the y-y axis but is eccentric to thex-x axis (Figure 2.ld). When the channelsection 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 centrelines of the elements intersect at a single point have the shear centre located at that point (Figure 2.lf). 2.1.2 Torsionalresistance The total resistance of a member to torsionalloading 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


Y

V

I

X --

IF \

f-X

’U

U

\V

I

I

Y (b) Point symmetric sections

(a) Doubly symmetric sections

Y I

Y

I I


X+-

I

F-

x

-X

Y

( c ) Sections symmetric about major axis

YI I

(d) Sections symmetric about minor axis

Y

%I

Y

X-

X--

yo

-.

i

Y Asymmetric sections

1 xI 01LY

(f) Sections with a single junction

Figure 2.1 Shear centre ‘S’and centroid ‘c’

4

x 0 4 iY

I S


Constant torque ends f r e e t o warp

(a) Uniform torsion

Constant torque end warping prevented

(b) Non-uniform torsion

Varyingtorque

-3-

n

(c) Non-uniform torque Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Figure 2.2

Uniform and non uniform torsion

In this case the applied torque is resisted entirely by shear stresses distributed throughout the cross section. The ratio of the applied torque to the twist per unit length is equal to the torsional rigidity, GJ, of the member, where G is the shear modulus and J is the torsional constant. J is sometimes called the ‘St Venant’ torsion constant. However, when the member is in a state of non-uniform torsion, the rate of change of the angle of twist varies along the length of the member. An example would be a cantilever with an applied torque at the free end and theflanges restrained against warping at the fixed end (Figure 2.2b). Alternatively, for a simply supported beam with an applied torque at the centreof the span, considerations of symmetry about the centreline of the span mean that the cross section must remain plane during twisting and, therefore, the rate of change of angle of twist must vary throughout the span (Figure 2 . 2 ~ ) . In both these cases the warping deflections due to the bending of the flanges vary along the length of the member. Both direct and shear stresses are generated which are additive to those due to bending and pure torsion respectively. The stiffness of the member associatedwith these additional stressesis proportional to the warping rigidity, E H , where E is the modulus of elasticity and H is the warping constant.

When the torsional rigidity, GJ, of the section is very large compared with the warping rigidity, E H , the member will effectively be in a state of uniform torsion. Closed sections, angles and tee sections behave in this manner as do most flat plates and all circular sections. Conversely, if the torsional rigidity of the section is very small compared with the warping rigidity, the member will effectively be in a state of warping torsion. This conditionis closly approximated for very thin walled open sections such as cold formed sections. Between these two extremes, the memberswill be in a stateof non-uniform torsion and the loadingwill therefore be resisted by a combination of uniform and warping torsion. This is the condition which occurs in hot rolled I, H and channel sections. A more detailed explanation of these effects is given by Trahair.(6)

5


Thus the stresses produced in a member by torsion are:

(1) shear stresses due to pure torsion (2) shear stresses due to warping torsion (3) bending stresses due to warping Thus torsional stresses induced in a member can be identified as pure torsional shear stress, warping shear stress and warping normal stress. Each stress is associated with the angle of twist (4) or its derivatives. Hence, when 4 is determined for different positions along the girder length, the corresponding stresses can be evaluated at each position. In order to determine the direction of these stresses correctly, it is necessary to adopt a standard sign convention as illustrated by Figure 2.3. The longitudinal axis is defined as the z axis. When a member is viewed along the longitudinal axis towards the origin,an anti-clockwise twist is taken to be positive.

Direct ion of


$rt.icai

torque Positive angle Figure 2.3 General orientation

o f twist

The total torque, therefore, at any cross section may be obtained by resolving the applied loads in two directions parallel to the principal axes (x-x and y-y), multiplying by the relevant eccentricity, and then recombining as shown in Figure 2.4(a) and (b).

Tq

i.e.

=

P,e,+ Pxey

(2-1 1

At any cross section the total torsional resistance is given by:

T,

T, = Tp+

(2.2)

or

T,

(2.3)

=

GJ4’-EHV

For equilibrium, the torsional resistance T, must be equal to the applied torque Tq; Hence,

T GJ

- --

where

a =

4’-a2V

(2.4)

[E]’”

a , is the torsional bending constant. (Values of ‘a’are given in Tables in Section 5).

Tp is the pure torsional resistance and T,,, is the warping torsional resistance. The two sets of shear stresses thus produce torsional moments which together balance the applied torque.

To maintain internal equilibrium within the elements of the member, warping also produces direct bending stresses, but no direct stresses are produced by pure torsion.

6


P%

+

ben Ing

I-sections

+ torsion

iPY

px,

f.

bend irrg

torsion

(b) Channel sections Figure 2.4 Bending and torsion


The values of T p and T , will generally vary along the length of the member, even where T, remains constant. This will be discussed in more detail in Sections 2.1.4 and 2.1.5. 2.1.3 Torsion of closedsections As explained in Section 2.1.2, the torsional rigidity, GJ, of a closed section is very large compared with its warping rigidity, E H , and hence a closed section may reasonably be regarded as subject to pure torsion only.

The total angle of twist 4 is given by: GJ where

Tq

=

the applied torque

z = the length of member subject to Tq

In a closed section the walls are, in general, relatively thin and pure torsion produces a shear flow around the section which is sensibly constant at any point. For a closed thin-walled section, J is given by: J=-

where

4A h* (Slt)

c

Ah = the area enclosed by the mean perimeter of the section

(Figure 2.5) Z((s/t) = the summation around the mean perimeter of the ratio of length along the perimeter to thickness for each element. For a section of uniform thickness:

Z ( s / t ) = h/t where

h = the mean perimeter.

For a closed thin walled section of uniform thickness:


t

Mean perimeter h

1

I I -

Figure 2.5 Shear flow in a closed section

S

For thick walled hollow sections, more accurate expressions for J are given in Appendix A.8. Value of J for standard hot-rolled hollow sections are given in Tables 5.5 to 5.7. The shear stress


Tt

T,

in a thin walled section is given by:

=

Tq 2Aht

(2.8)

For rolled rectangular and circular hollow sections and other closed sections where the walls are relatively thick, the shear stress given by Equation 2.8 is conservative. A more accurate value may be obtained from the theory of thick walled Tt

-

Tq -

(2.9)

c

where C = is the torsional modulus constant (Appendix A.8). Values of C for standard hot-rolled hollow sections are given in Tables 5.5 to 5.7. For large fabricated box sections, reference should be made to more detailed literature.(')

-

2.1.4 Pure torsion open sections If a torque is applied at the ends of the member in such a way that the ends are free to warp, then the member will only develop pure torsion (Figure 2.2a). The resulting shear stresses vary linearly across the thickness of each element (Figure 2.6)* They are maximum at the surfacesof the element, thetwo values being equal but opposite in direction. The stresses are greatest in the thickest element of the section.

Elsewhere there are also small shear stresses orientated perpendicular to the dominant stresses shown. Although they contribute half the resistance to pure torsion in each plane element, due to their much longer lever arms, they are negligible in value and need not be calculated. (For further details see Reference 14 page 376 Figure 5.37.) Figures 2.6(a) and (b) show the stress patterns for I sections and Channels. The total angle of twist

4 is given by:

4 = T Lz GJ

(2.10)

*Strictly speaking this condition will be violated at the junctions between the web and the flanges, particularly in rolled sections wtih radiused root fillets. It is usual to neglect this effect except when determiningthetorsionalconstant, J , for which its inclusionleadstosignificantlylargerThe Tables in Section 5 make due allowance for such fillets.

8


Figure 2.6 Stress patterns due to pure torsion (Stress diagrams enlarged for clarity)

The maximum shear stress rt =


2.1.5

T~ in

an element of thickness t is given by: (2.11)

G@'

Warping - open sections

General When a uniform torque is applied to a member of open section restrained against warping, the memberitself will be in non-uniform torsion and the rateof change of angle of twist will vary along the length of the member. The rotationof the section with respect to a restrained end will be accompanied by bending of the flanges in their own plane. The direct and shear stresses generated are shown in Figure 2.7.

Warping stresses are also generated in members of open section when the applied torque varies along the length; even if the ends are free to warp. For an I section member, the action of warping resistance can be visualised as follows. The torque Tq is resisted by a couple comprising forces equal to the shear forces in each flange, and acting at a lever arm equal to the depth between the centroids of the flanges. If each flange is now treated as a beam, the bending

warpingnormal \Stress ow)

d

bending momentin plane o f flange("Bi moment")

- W -- q y r

ressKw) shear flange \/hears

S

1

Y Y

y

Rotation o f cross sect ion Figure 2.7

Warping stresses in open crosssections

9


moments produced by the above forces lead to direct stresses,U,, in the flanges as shown in Figure 2.7. For a section with a low value of GJIEH the above provides a reasonable approximation, but in general it over-estimates the direct stresses whilst underestimating the shear stresses, because it neglects the shear stresses due to pure torsion. Also the above treatment is only applicable to I or H sections and cannot readily be applied to a channel section. Forthese reasons the methodsgiven in this publication have been developed. Warping Stresses


(i) Warping normal stresses These are direct stresses (tension or compression) resulting from the bending of the element due to torsion. In the case of an I beam, the stresses occur in the flanges. They act perpendicular to thesurface of the cross section and are constant across the thickness but vary along the length of an element. Figures 2.8(a) and (b) show the stress patterns in I and channel sections.

(a) I sections

Figure 2.8

(b) Channel sections

Warpingnormalstresses

The magnitude of the warping normal stressat any particular point cross section is given by: U,

where W,,

=

= - EW,,f

‘S’

in the

(2.12)

the normalised warping function at the particular point ‘S’ in the cross section (see Tables 5.1 to 5.7 or formulae given in Appendix A).

(ii) Warping shear stresses These are in-plane shear stresses that are constant across the thickness of the element but vary in magnitude along the length of the element and act in a direction parallel to the edge of the element. Figure 2.9 shows the stress patterns for I sections and channels. The magnitude of the warping shear stress at any point section is given by: r, =

where S,,

10

=

-

E SW&’ t

‘S’

in the cross

(2.13)

the warping statical moment at the particular points ‘S’ in the cross section (see tables 5.1 to 5.7 or formulae given in Appendix A).


T (a) l sections Figure 2.9

(b) Channel sections

Warping shear stress


2.1.6 Endconditions The endconditions of the member influence greatly the torsional stresses along the member. This publication considers the three ideal situations described below. Where the idealconditions do not apply it may be necessary to interpolatebetween conditions or conservatively to assume the worst condition. Torsional fixity must be provided at least at one point in the length of a member, otherwise it will simply twist bodily when a torque is applied. Warpingfixity cannot be provided without also providing torsional fixity. Thus there are3 possible sets of end conditions relevant for torsional calculations: (a) Torsion fixed, warping fixed: This is satisfied when twisting about the of the member longitudinal z-axis and warping of the cross section at the end are prevented. In this situation 4 = 4‘ = 0 at the end. Such a condition may be achieved as shown in Figure 2.10(a). (Note: This torsional end condition is also called ‘Fixed’.) (b) Torsion fixed, warping free: This is satisfied when the cross section at the end of the member is prevented from twisting but is allowed to warp freely. In this situation c) = c)” = 0 at the end. Such a condition may be achieved as shown in Figure 2.10(b). (Note: This torsional end condition is also called ‘Pinned’.) (c) Torsion free, warpingfree:This is achieved when the end is free to warp and twist. The unsupported end of a cantilever illustrates this condition. (Note: This torsional end condition is also called ‘Free’.) Effective warping fixity is not easily provided. A connection providing fixity for bending about both axes is not sufficient. It is also necessary to restrain the flange by means of details such as those shown in Figure 2.10(a), where plates or channel sections are added to provide warping fixity. It is worth considering the fact that provision of warping fixity does not produce such a large reduction in torsional stresses as is obtained from fixity for bending. Thus it may be more practical to assume that the endconditions are ‘warping free’ even when fixity is provided for bending. On the other hand, torsional fixity can be provided relatively simply by standard end connectionsFigure 2.10(b). It should be noted that end conditions for torsion calculations may be quite different from those for bending. A beam may be supported at both ends, but torsionally restrained at one endonly - the torsionalequivalent of a cantilever. On the other hand, torsional restraint (though not normally full fixity) can beprovided at the unsupported end of a cantilever beam.

11


Schematic

representation

Ideal


__ ________-_---Plate Feasibleconnection

U -U-

L_-_*

Channel

1

(a) Torsion fixed, warping fixed

I deal

Feasible connection

(b) Torsion fixed, warping free

Figure 2.10 Endconditions

2.2

Bending

Procedures for checking the adequacyof steel members subject to bending fully are documented in the appropriate Sections of BS 5950:Purt I.(13)Thus Clause 4.3 deals with laterally unrestrained members, including allowances for:

(1) The pattern of moments, Clause 4.3.7 (2) End restraint, Clause 4.3.5 (3) Cantilevers, Clause 4.3.6 (4) Destabilising load conditions, Clause .4.3.4 ( 5 ) Angle sections, Clause 4.3.8 Interaction of shear and bending is covered in Clause 4.2, whilst Clause 4.9 deals with moments applied about both principal axes. In the caseof an I or channel bent about its majorprincipal axis and not provided with full lateral restraint, design is likely to be governed by lateral-torsional 12


buckling considerations with the design condition being:

M< where

(2.14)

Mb

M

=

equivalent uniform moment

Mb

=

lateral-torsional buckling resistance moment

When determining either Mor Mb advantage may be taken of the beneficial effects of non-uniform moments within the beam segment under consideration according to Clause 4.3.7.6, with the exact procedure to be followed for a particular case being principally dependent o n the nature of the applied loading. If the beam is stocky, e.g. dueto the presenceof closely spaced lateral restraints, or if it is bent about its minor axis or for almost all situations involving the use of closed sections, design will be governed by the moment capacity M , at the most highly stressed cross section. Determination of M , is covered by Clauses 4.2.5 and 4.2.6.

Laterally unrestrained beams must also satisfy the provisions of Clause 4.2. In certainsituationswhereadvantage is taken of afavourablemomentpattern, resulting in an Mvalue in Equation 2.14 becoming less than the maximum moment in the beam, local cross sectional capacity may be the governing condition.

I n addition to direct bending stresses, shear stresses Tb due to plane bending are also present. These shear stresses can be determined from the following:


Forweb,

Tbw

For flange,Tbf where

=

V QW It

(2.15)

vI Qr T

(2.16)

= -

V = theappliedshearforce I = the second moment of area of the whole section t = the web thickness T = the flange thickness Qw = the statical moment for the web i.e.

plastic modulus of section

(

2

1

Qf = the statical moment for the flange The derivation of Section 5.

2.3

Qw

and Q, is given in Appendix A and values are tabulated il

Combinedbendingandtorsion

The presence of loading which produces simultaneous bending and torsion in member means that some degreeof interaction between the two effectswill occur. This may be regarded as analagousto thesituation in a member subjectto bending and compression, for which the axial load acting through the lateral deflections caused by the bending loads induces additional moments, which in turn amplify the deflections. In the case of bending plus torsion, the angle of twist 4 caused by the torsion is amplified by thebendingmoment, inducingadditionalwarpingmomentsand torsional shears.(") Account must also be taken of the additionalminor-axis moments produced by the major-axis bending effects acting throughthe torsional deformations, including the amplification noted above. Any plasticity is liable to have a disproportionate (and so far unquantified) effect on the torsional deformations. Thedesign criterion is therefore taken asa limit on

13


the maximum stress at the most highly stressed cross-section to the design strength p y . This check then effectively becomes the “capacity” check in terms of BS 5950: Part I. Neglecting for the time being the effects of amplification, and assuming that the loads produce bending about the major axis together with torsion, longitudinal direct stresses will arise from three causes as illustrated in Figure 2.11. Assuming elastic behaviour, these may be determined from:

M,

(2.17)

~

2, Obyt

U,

MYt

(2.18)

= -

ZY = EW&“

(2.19)

Determination of u b y t involves calculation of M,,, which depends on the major axis moment M , and the amount of twist 4, thus: = 4Mx

(2.20) Determination of a b , depends directly on M , , torsional deformations having a negligible effect on this quantity. The calculation of U, has already been covered in Section 2.1S . under ‘Warping normal stresses’. Myt


Whenever lateral-torsional buckling is a design criterion (i.e. when P b is less than p y ) , the values of U, and o b y t will be amplified by the interaction of torsion and lateral-torsional buckling.

Point on cross section at which peak stress occurs

Deformation

OCbx Figure 2.11

14

Deformation and stresses due to combined major axis bending and torsion


Adopting asimilar form of interaction expressionto thatused in BS 5950: Part for combined axial load and in-plane bending moment, the criteria are:

(i) Buckling Check:

-+

(2.21)

PY

(ii) ‘Capacity’ Check: (2.22) M, is the equivalent uniform moment according to BS 5950: Part I given by: ubyt + u w

Py

M, = mxMx

Equation 2.22 may govern when

M,

< 1 or when M,, > pyZ,

The use of values of M b > pyZ, is justified by the very local nature of the peak stresses and is in general accord with the method of BS 5950: Part I Clause 4.9 for biaxial bending without torsion. If the applied loading also includes minor-axis moment this should be added and the criteria modified to: (i) Buckling Check: M, -+-

M y

+

(2.23)

Mb

(ii) ‘Capacity’Check: Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

(2.24)

Uby

=

MyfZy

Note that the applied minor axis moment M , is not amplified by the bending-torsion interaction. In Equations 2.23 and 2.24 the various stresses refer to the same point in the cross section at the same point in the length of the member.

in a The torsional shear stresses and warping shear stresses should also be amplified similar manner as follows: 7,1

=

(T,+T,)(~+O.~M,/M~)

(2.25)

and added to the shear stresses due to plane bending. At points of high coincident bending stress and shear stress, a check using BS 5950:Part I Clause 4.2.6 should also be made. The design approach is illustrated by means of a series of worked examples in Section 4. In several cases these show the limit state of acceptable twist at working load as being the governing condition.

15


3.

LOADAPPLICATION

In the previous discussion on torsional theory it has been assumed that loads are applied in such a way that they can move freely with the beam as it twists. This is often a conservative assumption because the system applying the load may be attached in such a way to the member under consideration as to reduce the torsional effect. In such cases the stiffness of the loading system should be taken into account in the analysis. For example, consider a load applied to the member through a column in such a way as to cause a torsional loading. Torsional effects can be greatly reduced by ensuring that a moment connectionis provided and taking into account the bending stiffness of the column, see Figure 3.l(a) and Example 10, Section 4. In a similar fashion, consider torsional loading applied to the main beam shown in Figure 3.l(b). In this case, the torsional effects can be reduced almost to zero by taking account of the bending stiffness of secondary beams which frame into the main beam with moment connections. These effects are explained in greater detail by Johnston.('2)


P

Figure 3.1

16

Torsional effects

-

methods of load application


4.

WORKED EXAMPLES

A number of worked examples have been provided to demonstrate the methods proposed and the use of the Tables (Section 5 ) and Graphs (Section 6 ) given in this publication. It should not be assumed that the solutions adopted are the only or even the best method of dealing with the problem. For example, it may be that a tubular section would be a better member to use in some situations than the universal beam or column section used. Alternatively, the loadingcould be applied in such a way as to producenegligible torsional effects. These decisions can only be made by the designer by considering the structure as a whole along with any architectural constraints which have been imposed.


Examples 1 to 3 have been kept very simple in order to demonstrate the method clearly. As such they are somewhat unrealistic design situations. More complex and realistic problems are dealt with in Examples 4 to 10. It will be seen from Examples 1, 2 and 7 that the shear stresses are relatively insignificant and for normal situations, can be ignored. Only in cases where the span of the member is very short and the rotations arevery high will shear become the governing criterion. In such cases the first and third derivative of 4 (ie 4‘ and @”)will be required to calculate the pure and warping shear stresses. The equations for 4’ and @“ are given in Appendix B. In normal design situations the size of the member will be governed by twist or by the interaction of bending and warping normal stresses. In this case the value of 4 and its second derivative 4‘’ may be obtained from the Graphs provided in Section 6 o r from the equations given in Appendix B. Notes: (i) In Examples 3 , 5 , 6 and 7 it has been assumed that the endsof the beams are ‘fully fixed’ for bending. This has been done to produce simple examples illustrating the treatment of torsion. It should not be taken to imply that bending moments can be determined like this in practice.

In real situationssuch beams are likely to be partof a frame and thevalues of the end moments would depend on the pattern of loading, the relative stiffness of the members joined and the type of connection used. (ii) In Examples 3, 5 and 6 it has been assumed that the ends of the beams are ‘fixed’ against warping. Means of achieving this are illustrated in Section 2.1.6. However, in most practical cases, it is difficult to achieve the condition of ‘warping fixed’. Hence it is usually better to design for the ‘warping free’ condition.

17


Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Example The

I

beam shown

bebw

i s unrestml'ned eccentrr'c l o a d ;S applted to the bottom flange a t the centre of the span in s u c h a w a y thht it does not mvrde any L a t e r a c restmint to the me ber. The end condl'trbns are a s s ' u m e d to be simp& supported for bendinq a n d F i x e d against torslbn b o t free for For the factored loads shown , cherk wQrpin% t c l d e q u a c y of the trial sectrbn. abng its

length . An


R

I

1

I-

L= 4000mm

'It W

18

i

W

W= 100kN

- 1

' i

negative angk of t w k t due to T?


The SteelConstruction Institute

---

a &= --

Job No.

PUB rs*

Job Title

Silwood Park Ascot Contract BerksSL5 7QN Telephone: (0990) 23345 Fax:(0990) 22944 Telex: 846843

Client

CALCULATION SHEET

Checked by

Worked

Isheet

2

of

E X . I (Rev.

LxampLe I No.

Made by

Jun. '89 Date Jun. '89 -

Date

m

+


Loadmq (Factored) Point

load

bistributed

E c c e n trl'ci ty

W Load (sell w t .)

W

9

Q

19


IJob No.

PUB


CALCULATION SHEET

Try

254x 2S4

-

-,

.@ 89 kg/m UC Grade 43 SteeL


W,

Add I ti o n a l

proper tie3 from this publlcatiol rablc 5.;

.l3

Itabcc 6.

20



CALCULATION SHEET

Client

Contract No.

Made by

m

Checked by

Date Date

Jun. '89 Jun .l89 -

bendinq a ~ dtorston ti> l3uc k linq Check (at U . L . 3 )


Check combmed

Myr To

=

J

MT.@

coCcuCate

0

21


I

Institute o n t r a SL5 ct Silwood Park Ascot CBerks 7QN (0990) 23345 Fax: (0990) 22944 T e l e x : 846843

No.

Client

Telephone:

Made by

Dare

Jun. '89

Date

J'un. '89

Checked by

CALCULATION SHEET

€9.2.18

€9.2.12

t'4. 2.19

To

calcc,late


Torsional

M'

functlbn

-0".G.J.a

0.4s

Gmph 2

t

Ec+. 2.21

J

22

--



e?! --

-

Job No.

Job Title

ISheet 6 of N B &S? Worked Excrmpk I

.I I R e v .



CALCULATION SHEET

. . 93 t IO t 98 = 201 N / i m f ( 265 m

23



----


CALCULATION SHEET

Job No. Job Title

m8 #S?

[Sheet

7

of

E 1 (Rev.

Wor.ked Example I No.

Contract Client Made by Checked by

FM

Date

Jun. '89

Date

Jon. '89


I

24



4cfl --= --


CALCULATION SHEET


at

O

Design of members subject to combined bending and torsion

Bending to Break

Design of Curved Members for Machines

Bending of curved tubes

Circuit-bending

Frege and Russell Combined

Members Only

Design of Metallic Cold-Formed Thin-Walled Members

The Bending and Stretching of Plates

Bending the Rules

Badiou: A Subject To Truth

Combined Production of Heat and Power (Cogeneration)

L2-torsion of Hyperbolic Manifolds of Finite

Issues in Design and Technology Teaching (Issues in Subject Teaching)

Subject, Voice and Ergativity

Heidegger and the Subject

Theory of the Subject

Theory of the Subject

Bending the Rules

Combined Arms Breaching Operations

Subject Seven

The Reidemeister Torsion of 3-Manifolds

Tired Church Members

The Reidemeister torsion of 3-manifolds

The Torsion Problem of Curved Beams

Torsion and Shear Stresses in Ships

The Reidemeister Torsion of 3-Manifolds

Anatomies of the subject: Spinoza and Deleuze

Combined and Hybrid Adsorbents: Fundamentals and Applications

Subject Seven

Design of members subject to combined bending and torsion

Bending to Break

Design of Curved Members for Machines

Bending of curved tubes

Circuit-bending

Frege and Russell Combined

Members Only

Design of Metallic Cold-Formed Thin-Walled Members

The Bending and Stretching of Plates

Bending the Rules

Badiou: A Subject To Truth

Combined Production of Heat and Power (Cogeneration)

L2-torsion of Hyperbolic Manifolds of Finite

Issues in Design and Technology Teaching (Issues in Subject Teaching)

Subject, Voice and Ergativity

Heidegger and the Subject

Theory of the Subject

Theory of the Subject

Bending the Rules

Combined Arms Breaching Operations

Subject Seven

The Reidemeister Torsion of 3-Manifolds

Tired Church Members

The Reidemeister torsion of 3-manifolds

The Torsion Problem of Curved Beams

Torsion and Shear Stresses in Ships

The Reidemeister Torsion of 3-Manifolds

Anatomies of the subject: Spinoza and Deleuze

Combined and Hybrid Adsorbents: Fundamentals and Applications

Subject Seven

Recommend Documents