design of members subject to combined bending and torsion

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

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

The Steel Construction Institute Silwood Park Ascot Berkshire SL5 70N Telephone: 01 344 23345 Fax: 01 344 22944

P057: Design of Members Subject to Combined Bending and Torsion

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http://sefie.steelbiz.org/DiscussSteelbizContent.aspx?ResourceID=5701


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 and choosing a load path for such 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. After a brief 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 ..


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CONTENTS

Page

SUMMARY

NOTATION

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

2. BASIC THEORY 2.1 Torsion 2.2 Bending 2.3 Combined Bending and Torsion

3. LOAD APPLICATION

4. WORKED EXAMPLES

5. SECTION PROPERTIES

6. CHARTS TO ASSIST IN EVALUATION 6.1 Standard Cases 6.2 Extreme Cases

REFERENCES

APPENDIX A. Evaluation of Torsional Properties

APPENDIX B. Solution of Differential Equations

1 1 2

3 12 13

16

17

89

98 99

109

1 1 1

118

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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 de la torsion est nkcessaire dans les calculs de dimensionnement sont identifiees. La thkorie classique de la torsion des piPces 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 flexion et 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.

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Progretto di Membrature Soggette a Flessione e Torsione Sommario Vengono discusse le cause dell'impegno torsionale degli elementi strutturali e identificate quelle situazioni nelle quali P necessario tenere conto esplicitamente della torsione nei calcoli di pregetto. E presentata la teoria della torsione con riferimento ai profili aperti e 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 gamma significativa di condizioni di carico e di vincolo. E altresi illustrato un metodo semplice che consente di combinare gli effetti della torsione e della flessione, metodo in accord0 con l'approccio delle BS.5950: parte I. Una serie di esempi consente la comprensione del'approccio progettuale per elementi soggetti a flessotorsione nella sua completezza. Si mette in 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 analizan las causas que originan cargas de torsi6n en las piezas de una estructura y se identifican aquellas situaciones en que debe incluirse la 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 diseno y formulas para diferentes tipos de cargas torsoras y condiciones de apoyo. Se incluye, en particular un procedimiento sencillo de combinacion de efectos de flexion y torsion, congruente con el metodo patrocinado por la 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 de resistencia ultima.

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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

Shear modulus of elasticity of steel (taken to be -

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 cross sectional area between the free edges of the cross section and a 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

Uniformly distributed applied torque (torsional moment) or maximum value of varying applied torque Shear acting on cross section

Distance from toe of channel section to point on flange where T~ in the

E 2( 1 +v) ' where

flange is maximum, given by W,, X B'

W n o + Wn2

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W”,

xo, Yo

Y Z

7-

Normalised warping function at a point ‘S’ on cross section

Co-ordinates of the shear centre with respect to the centroid

Perpendicular distance from neutral axis to a point on cross section

Distance from left end of member (origin of co-ordinate system) to 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 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 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

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

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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 applied so 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 loads from floor slabs supported on the top flange of beams, even for channel sections.

Designing to transfer loads by means of torsion should be avoided wherever 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 potential difficulties 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 and joints 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 account in 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 above is an example of a broader principle, which is valid due to the ductility 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 load is 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 obtained in the usual manner by assuming that the loads act through the shear centre and resolving the forces into components parallel to the principal axes. Torsional stresses and 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.

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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.(') Members which are curved on plan or which contain particularly slender plate elements, e.g. cold formed sections, are not considered in 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 affected by torsion tends to be governed by the proportions of 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 column H 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 be used 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).

Open sections Closed sect ions Figure 1.1 Choice of section

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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 Shear centre 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 centre lies on the y-y axis but is eccentric to the x-x axis (Figure 2.ld). When the channel section 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 centre lines of the elements intersect at a single point have the shear centre located at that point (Figure 2.lf).

2.1.2 Torsional resistance The total resistance of a member to torsional loading 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).

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Y I V

\

U’ IF \

V

X -- f- U

- X

I

I Y

(a) Doubly symmetric sections (b) Point symmetric sections

Y I

Y I

I F- X+- I

x -X

Y

(c) Sections symmetric about major axis (d) Sections symmetric about minor axis

Y I I Y Y

X - % yo I

i X-- - .

I S 1 x 0 L-

I 1 x 0 4 i- Y Y

(f) Sections with a single junction

Y Asymmetric sections

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

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Constant torque - ends f r e e t o warp

(a) Uniform torsion

Constant torque - end warping

prevented

(b) Non-uniform torsion

-3- Vary ing torque

n

(c) Non-uniform torque

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 the flanges restrained against warping at the fixed end (Figure 2.2b).

Alternatively, for a simply supported beam with an applied torque at the centre of 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 associated with these additional stresses is 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 condition is closly approximated for very thin walled open sections such as cold formed sections.

Between these two extremes, the members will be in a state of non-uniform torsion and the loading will 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)

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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

torque

Figure 2.3 General orientation

$rt.icai

Positive angle 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).

i.e. Tq = P,e,+ Pxey (2- 1 1 At any cross section the total torsional resistance is given by:

T, = Tp+ T, (2.2)

or T, = G J 4 ’ - E H V (2.3)

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

T GJ - - - 4 ’ - a 2 V (2.4)

where a = [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.

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I-sections

P%

ben Ing

+ + torsion

iPY

p x , f. bend i rrg 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 closed sections 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:

4A h* J = - c (Slt)

where 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:


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1 t Mean perimeter h

I Figure 2.5 Shear flow in a closed section

I -

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 T, in a thin walled section is given by:

Tq Tt = - (2.8)

2Aht

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 - - c - Tq (2.9)

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 surfaces of the element, the two 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: T z GJ

4 = L (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 determining the torsional constant, J , for which its inclusion leads to significantly larger The Tables in Section 5 make due allowance for such fillets.

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

The maximum shear stress T~ in an element of thickness t is given by:

rt = G@' (2.11)

2.1.5 Warping - open sections

General When a uniform torque is applied to a member of open section restrained against warping, the member itself will be in non-uniform torsion and the rate of change of angle of twist will vary along the length of the member. The rotation of 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

warping normal \Stress ow) bending momentin d plane o f flange("Bi moment")

- W - - qry S ressKw) shear

\/hears flange 1

Y Y y

Rotation o f cross sect ion Figure 2.7 Warping stresses in open cross sections

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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. For these reasons the methods given 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 the surface 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 Warping normal stresses

(b) Channel sections

The magnitude of the warping normal stress at any particular point ‘S’ in the cross section is given by:

U, = - EW,,f (2.12)

where W,, = 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 ‘S’ in the cross section is given by:

E SW&’ r, = -

t (2.13)

where S,, = 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).

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T

(a) l sections (b) Channel sections

Figure 2.9 Warping shear stress

2.1.6 End conditions The end conditions of the member influence greatly the torsional stresses along the member. This publication considers the three ideal situations described below. Where the ideal conditions do not apply it may be necessary to interpolate between 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. Warping fixity cannot be provided without also providing torsional fixity. Thus there are 3 possible sets of end conditions relevant for torsional calculations:

(a) Torsion fixed, warping fixed: This is satisfied when twisting about the longitudinal z-axis and warping of the cross section at the end of the member 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 end conditions are ‘warping free’ even when fixity is provided for bending. On the other hand, torsional fixity can be provided relatively simply by standard end connections Figure 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 end only - the torsional equivalent of a cantilever. On the other hand, torsional restraint (though not normally full fixity) can be provided at the unsupported end of a cantilever beam.

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Ideal

_ _ _ _ _ - --- _ _ _ _ _ _ - - Plate

Feasible connection (a) Torsion fixed, warping fixed

Schematic representation

U -U--- L_-_*

Channel 1

I deal (b) Torsion fixed, warping free

Figure 2.10 End conditions

Feasible connection

2.2 Bending Procedures for checking the adequacy of steel members subject to bending are fully 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 case of an I or channel bent about its major principal axis and not provided with full lateral restraint, design is likely to be governed by lateral-torsional

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buckling considerations with the design condition being:

M< Mb (2.14)

where M = equivalent uniform moment

M b = 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 on the nature of the applied loading.

If the beam is stocky, e.g. due to the presence of 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 certain situations where advantage is taken of a favourable moment pattern, 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.

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

V QW For web, Tbw = - I t

For flange, Tbf = - v Qr I T

(2.15)

(2.16)

where V = the applied shear force 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

( 2 1 plastic modulus of section i.e.

Qf = the statical moment for the flange

The derivation of Qw and Q, is given in Appendix A and values are tabulated Section 5.

2.3 Combined bending and torsion

il

The presence of loading which produces simultaneous bending and torsion in member means that some degree of interaction between the two effects will occur. This may be regarded as analagous to the situation in a member subject to 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 the bending moment, inducing additional warping moments and torsional shears.(") Account must also be taken of the additional minor-axis moments produced by the major-axis bending effects acting through the torsional deformations, including the amplification noted above. Any plasticity is liable to have a disproportionate (and so far unquantified) effect on the torsional deformations. The design criterion is therefore taken as a limit on

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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,

MYt

ZY

~

Obyt = -

(2.17)

(2.18)

U, = EW&“ (2.19)

Determination of ubyt involves calculation of M,,, which depends on the major axis moment M , and the amount of twist 4, thus:

M y t = 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 2.1 S . under ‘Warping normal stresses’. Whenever lateral-torsional buckling is a design criterion (i.e. when P b is less than p y ) , the values of U, and oby 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 Deformation and stresses due to combined major axis bending and torsion

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Adopting a similar form of interaction expression to that used in BS 5950: Part for combined axial load and in-plane bending moment, the criteria are:

( i ) Buckling Check:

-+ PY

(2.21)

(ii) ‘Capacity’ Check: ubyt + u w P y (2.22)

M, is the equivalent uniform moment according to BS 5950: Part I given by:

M, = mxMx Equation 2.22 may govern when M, < 1 or when M,, > pyZ,

The use of values of Mb > py Z, 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 the criteria modified to:

(i) Buckling Check:

M, M y + -+- M b

(ii) ‘Capacity’ Check:

includes minor-axis moment this should be added and

(2.23)

(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.

The torsional shear stresses and warping shear stresses should also be amplified in a 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.

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3. LOAD APPLICATION

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 connection is 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 Torsional effects - methods of load application

16


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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 loading could be applied in such a way as to produce negligible 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 are very 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 or from the equations given in Appendix B.

Notes: (i) In Examples 3 , 5 , 6 and 7 it has been assumed that the ends of 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 situations such beams are likely to be part of a frame and the values 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


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

CALCULATION SHEET

Example I

The beam shown b e b w is unrestml'ned abng its length . An eccentrr'c l o a d ;S applted to the bottom flange at t h e centre of the span in s u c h a way 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.

R

I 1 W= 100 kN I - L = 4 0 0 0 m m - 1

' It W

i W

' i negative angk of t w k t due to T ?

18


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+

Loadmq (Factored)

The a&= Steel Construction --- PUB rs* Isheet 2 of E X . I (Rev.

Institute Client Contract No. Silwood Park Ascot Berks SL5 7QN

Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

Job No.

Job Title - -- - - - - - - Worked LxampLe I

Date J u n . '89 Checked by m Date Jun. '89 -

9 Point load W

bistributed Load (sell w t .) W

E c c e n trl 'ci ty Q

19


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I Job No. PUB


CALCULATION SHEET

Try 254x 2S4 .@ 89 kg/m UC Grade 43 SteeL

Add I t i o n a l proper tie3 from this publlcatiol

- -,

W,

rablc 5.;

.l3 I tabcc 6.

20


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Silwood Park Ascot Berks SL5 7QN

Made by Fax: (0990) 22944 Telex: 846843 Telephone: (0990) 23345

Contract No. Client

CALCULATION SHEET

Date Jun. '89 Checked by m Date Jun . l89 -

ti> l3uc k l inq Check (at U . L . 3 )

Check combmed bendinq J a ~ d torston

Myr = MT.@

To coCcuCate 0

21


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

CALCULATION SHEET

To calcc,late M'

Torsional functlbn -0".G.J.a 0.4s

Client Contract No.

Made by

Checked by Dare Jun. '89 Date J

t

t

'un. '89

€9.2.18

€ 9 . 2 . 1 2 '4. 2.19

Gmph 2

Ec+. 2.21

J

22


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The e ? ! N B &S? Steel Construction -- Institute

Job Title

Job No. I Sheet 6 of . I I R e v . - - -- - - - - Worked Excrmpk I


CALCULATION SHEET

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

23


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The Steel Construction --- Institute

Job Title

Job No. m8 #S? [Sheet 7 of E 1 (Rev.

- - - -- - - - Wor.ked Example I

Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

Date Jun. '89 Checked by FM Date Jon. '89

I

24


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The 4cfl Steel Construction --= Institute

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

CALCULATION SHEET

at O< = 0.6

at support -L= 0

A t Support

The Steel Construction --= Institute

Job Title

Job No.

-- - - - - - - Worked Lxampk I Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Made by m Date J u n . '89 Checked by

Date J

L

'on. '89

25


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The =

I



Contract NO. Cl ient

A CALCULATION SHEET 1-

A t ml'dspan

T

Ey.2. It

LC+. 2.11

E?. 2 .l3

26


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Fax: (0990) 22944 Telex: 846843 Telephone: (0990) 23345

Contract No. Client

Made by

CALCULATION SHEET

Date J u n '89 Checked by M Date Jun '89

~y inspection tk maximum combined

\sheor stresses occw- at ttx suppart

0 1 2

I A t Suppor t

In web at 3 , < k W = -21.6N/mrn2

27


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CALCULATION SHEET

These calculations confirm the comparativQLy low lwd-,s of shear str-.

= 1-32'

R e f . 13

(4.2.3)

28


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lJob No. PUB ?S The Steel Construction --B- Institute

ccA Job Title . 7 I Sheet I of h. 2 IRev*

- - -- - - - - Worked Example 2 I Silwood Park Ascot Berks SL5 7QN Client Contract No.

l

m . 3 I,.,.,.,., m * * * =

I I 0 I -.

~elepnone: (UYYUJ ~ 5 5 4 3

CALCULATION SHEET

Made by Fax: (0990) 22944 Telex: 846843 I

Date Jun '49 Checked by m

29


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Contract No. Client

Made by

CALCULATION SHEET

Date J u n ' 8 9 W Date Jun '89 Checked by

Check cornbmed bmdlnq and torslon

= 6 S )

- ( 250 )x 61.1 =.38*2crn; 2s0t 150

E?. 2.21

Ref. 13 (4 2.3)

30


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136 Urn

To caCculate 0

Steel Construction - - Institute -


J(

? I I 100kN

I I L

To Tf 7

B = To t GJ

at centre 01 span t = ~=2000rnrn 2

Ref. 13 (4.3.7.2) table U

E?. 2.5

31


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Silwood Park Ascot Berks SL5 7QN Cl ien t Cont rac t No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by Date

Silwood Park Ascot Berks SL5 7QN Cl ien t Cont rac t No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by lDate Jun . ' 89 ' 89 - - . CALCULATION SHEET Checked by M lDate Jun. '89

I CALCULATION SHEET Checked by M Date '89

E?. 2.20

E q . 2 .l8

€ 4.2.22 E?. 2.17

32


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

CALCULATION SHEET

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

CALCULATION SHEET

Date J u n ' 8

1

1

2

j

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€9 . 2.9

€9.2.2.5

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33


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Steel Construction

Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Date Jun. '89 Checked by M Date Jun. '89

34


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CALCULATION SHEET

Date Jun '89 Checked by B4

Example 3

1 /A 8 1 II 3 1 ' I

' d -I t-e-rsmm

TorsionaI moment Ty = W. e Ty = I~OX?SXIO"=?*S~NITI

l h i s acts in a q o t i v e sense

_ a . T? = - ? 6 k N m

35


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L

36


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The - 1 1 Job No. ce* Job Title

PUB ?5f Isheet 3 of EX .3 IRev. Steel Construction Institute

-- - - - -- - - - Morked Example 3

Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Made by

Checked bl

Chec k

37


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= E . Wno . D" To caCcuCate 0"


CALCULATION SHEET

Date JI, n ' 8 9 Checked by M Date J

I

!

on ' 89

€9. 2.19 '9. 2.1.5

Gmpb 4

€9.2.21

Eq. 2.22

E?. 2 .If

38


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T h e Steel Construction -- Institute Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

Job No.

Job Title PUS ?S;Z Isheet 3 of Ex.3 IRev. -- --

- - -- - - - - Worked Example 3

Date J U n . '8 9 Checked by EM Jun. '89 -

Date

39


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CALCULATION SHEET

Excrmpk 4

W= IO0 kN

3n '89

Client Contract No.

Made by

Checked by -

-

40


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W = I kN/m


CALCULATION SHEET

41


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L


CALCULATION SHEET

L



Cont rac t No. C l ien t

CALCULATION SHEET

Date Jun. ' 8 9 Checked by m Date Jun . '89

3

t

L Cl ien t

Checked by

Made by

Cont rac t No.

Date Jun. ' 8 9 m Date Jun . '89

3

t

:. compact at Least

42


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The Steel Construction -- Institute

Job Title

Job No. PUB ?S? I Sheet 4 of Ex.4 IRev. -- - -- - - - - - War ked Example 4

L



Contract No. Client

CALCULATION SHEET m .

Date Jun . '89 m Date Jun. 'S9

Checked by

Chec k lateral tors Iorml bucklinq (at U.L.S.)

(4 .3 .S~ i laterat r e s t r a i n t s

(4.3*? *I ) -IC requirement M 6 Mb

Ref. r3

Ef'rec t ive length L E = 1 x D [stance bet ween

L E = 2000mrn

M m MA-

m = 0 3 7 ( f o r p = O )

Re!. IO UsIng tcrbwhted inbrrnstl'on from Ref.10

(table 18'

M = 0.57 ~'103 = 59 kNm

43


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Silwood Park Ascot Berks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by lDate J u n . ‘89 CALCULATION SHEET Checked by m Date

4 859

44


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CALCULATION SHEET

The a!&!! PUB ?Sf Steel Construction Institute

'Job Title

Job No. ISheet of 4 IRev. -- - - - - - - Worked Example 4

45


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CALCULATION SHEET Checked by

I -1'

II

7 Sectr'on X-X

Sectron Y- Y

r I ln . '89

46


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Silwood Park Ascot Berks SJ5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

I Worked Exmpk ! S 1

Loadlnq

A-ssurninq 45Tx152 (i3 82 U 3 Grade 43

= 1 6 8 m m

m . '89

Silwood Park Ascot Berks SJ5 7QN


Contract No. Client

CALCULATION SHEET

Date J u n . '89 Checked by F.M Date J

1

l

n

!

II

n h

Client

Checked by

Made by

Contract NO.

Date J u n . '89 F.M Date J

1

l

n

!

II

n h

47


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CALCULATION SHEET

Date Jun '89 Checked by W Date 5

n n

6

T

If

f 2

48


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The N e ! Steel Construction ---- Institute - - -

- -- - -


CALCULATION SHEET

Job No.

Job Title PUB ? d? Isheet 4 of Ex,&IAe"'

Worked Example 5 Client

Checked by

Made by

Contract No.

Date 5 UD ' 89 W DateJ d

Q

-L 1

on 'S9

Gmph h

49


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Steel Construction

Silwood Park Ascot Berks SL5 7QN Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

50


Discuss me ...C

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The Steel Construction Institute - wor ~a Silwood Park Ascot Berks SLS 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET I

27-3 t ( I. I t 6 2 . 4 ) 244 266

u n ’89 E-?. 2.12 E?. 2.19

E?. 2.21

E-?. 2.22 9.2. I7

51


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CALCULATION SHEET

consider example S bot using a cbanneC sect ion.

Outer sk in 01 brr'ckwork \

~nnev s k i n of /bloc kwor I(

52


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The <m/ P U . b 2 5 3 Steel Construction =- Institute

Job No. Sheet 2 of Rev. -- Job Title - -- - - - - Worked Example 6 -



Contract No. Client

Made by lDate - J I

CALCULATION SHEET Checked by PM Date

3.0 kWm

8.1 8 kWm

53


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The Steel Construction --- Institute

maH Job Title

Job No. Poa ?5? Sheet 3 of h. Rev.

- - - - -- - - Worked ExampLe 6 Silwood Park Ascot Berks SL5 7QN


Contract No. Client

CALCULATION SHEET Date J U n '89

Checked by m Date h n '89

54


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The 44/ Steel Construction ---- Institute Silwood Park Ascot Berks SLS 7QN

- - - - - -- -

Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

L

Telephone: (0990) 23345

CALCULATION SHEET

Made by Fax: (0990) 22944 Telex: 846843 A I

Date Jon 3 9 Date Checked by m J

l

-

A I Made by

Checked by Date Jon 3 9 Date m J

l

-

Job No.

Job Title PUB ?5? I Sheet 4 of Ex, aRev. Worked Exarnpte 6

Client Icontract No.

a

55


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The e!H Steel Construction ---- Job Title P U B ?W I Institute

Job No. Sheet 3 of h. Rev.

- -- - - - - - Worked E x a m pLe 6 L

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345

Contract No. Client

Made by Fax: (0990) 22944 Telex: 846843 n

CALCULATION SHEET

Date Jun '89 Checked by M Date Jun '89

56


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Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345

Contract No. Client

~ Fax: (0990) 22944 Telex: 846843

' CALCULATION SHEET

Made by Date Ju r~ '89

Checked by Mr Date J - 'on 'S9 E?. 2.12 L?. 2.19

Ecy. 9.22

Ea, .2. v

57


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L

Example f

Outer S kin 01 brickwork

24Sm I Loading



Contract No. Client

CALCULATION SHEET

Date J un '89 Checked by M Date J



Contract No. Client

CALCULATION SHEET

Date J un '89 Checked by M Date J

1

I

I

4.84 kVi

3.0 Cm

1.84 kMT

TU n '89

58


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CALCULATION SHEET

Date Jun '89 Checked by EM Date J

lo=?-

59


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The Q Steel Construction : Institute

- - -- - - - - - I


CALCULATION SHEET



Contract No. Client

CALCULATION SHEET

Date Ju n '89 Checked by M Date J

!

(

Client

Checked by

Made by

Contract No.

Date Ju n '89 M Date J

!

(

60


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The Steel Construction ---- Institute Silwood Park Ascot Berks SL5 7QN Contract No. Ctient

Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

e ? ! Job Title

Job No. PUB ?S7 ISheet 4 of Ex.f IRev' - - -- - - - - Worked Example P

Date Juri ' 89 Checked by BM Date Jun '89

L4.2.21

Ref. l3 :tab Le 3:

F*, 4.3.7.2 E7.2.20

€ 9 . 2. I %

61


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The <CH Steel Construction __ Institute Silwood Park Ascot Berks SLS 7QN


Contract No. Client

CALCULATION SHEET

Job No.

Job Title p ua 757 Isheet \S of €.x . p e v .

- - -- - - - - Worked Example f

Date Jun '19 Checked by M Date Jun '89

23.5 +. 0 . 9 I + 0 . 5 ~ 23.5 = O-jT < I 32. 8 2?3 32.8

T--

m t r i b u e d o n b a small amount. A s a n be seen the second term

4 1 :. 0- k.

LC). 2.22

E?. 2.17

Eq.2.15

Ea. 2.9

62


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Silwood Park Ascot Berks SL5 7QN Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

63


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Example I

J u n '89

Steel Construction == =


64


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The c!! Steel Construction ---=I- Job Title PUB F57 Institute

Job No. (Sheet 2 of E,-. 8 (Rev.

- - -- - - - - Worked Example 1 Silwood Park Ascot Ekrks SL5 7QN Client Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET Checked by Date

A nurn ber 01 ursef need invest I'gstl'ny

un '89 'un ' 8 9

65


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The mw- ' Steel Construction 7 -

Job No.

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

CALCULATION SHEET

66


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I Siiwood Park Ascot Berks SL5 7QN Client Icontract No. I

Telephone: (0990) 23345

CALCULATION SHEET

Made by m . Fax: (0990) 22944 Telex: 846843 1

Date Jar) ' 8 9 Checked by m Date J

E m e n tr l 'city

:. el = 224mm

Case I ,

Case 2 ,

Case 3 ,

shear

61


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The Steel Construction =-=- Institute

Job Title

Job No. f U 3 ?5? Isheet 5 of

-- - - __ - - Worked Exampte 8 -

I Silwood Park Ascot Berks SL5 7QN Client Icontract No. I Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

I Made by

Checked by Date Jun '89

M Date JUP ' 89

-I 2 . 0 krVm -?*S k Nm

68


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L Client Contract No.

Made by lDate Jon ' A9 I - - . . -,

I CALCULATION SHEET Checked by M

Check t w i s t For live Load (at S k s )

To vs iona For L/a = 5000/821 = 6 - 0 9

functl'on 0 . G . J = 0.62 T, . a

= -7.6 kNrn

69


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4e! Job Title lob No. , :Sheet ;T of Ex. 8 IRev' 1 The PUB as? Steel Construction -=-= Institute - - - -- -

- - Worked Exurnpk 8 Silwood Park Ascot Berks SL5 70N Client Contract No.

-

Telephone: (0990) 23345

CALCULATION SHEET

Made by Fax: (0990) 22944 Telex: 846843 I

Date Jun ' 89 Date Jur, 'S9 Checked by

l

Re!. 13 (tab IR 9)

(table 13.

W, E?. 2.2c

Graph U

E q . 2.18

70


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Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET Checked by

Job No.

Job Title The Steel Construction --- Institute

PUB ?a? Isheet 8 of Ex .glRev* - -- - - - - - Worked Example 8

Silwood Park Ascot Berks SL5 7QN Client (Contract No. - -

4

1

l

r

- = 4 7 ~ l C I

un '89 '89

E?. 2.12 E?. 2 .l9

Graph 6,

E a . 2.21

€ ~ . 2 . 2 2 E ? . 2 Sj

71


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CALCULATION SHEET

L Client Contract No.

Made by

Checked by Date J bn ' S 9

m Date J u n '89 c

-

72


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Job No. The Steel Construction --m Institute

e ! ! Job Title P U B j15T Isheet I of Ex.91Rev'

- - - - -- - - Worked Exarnpte 9

Silwood Park Ascot Berks SL5 7QN Client !Contract No. Telephone: (0990) 23345

Made by m Fax: (0990) 22944 Telex: 846843 I

CALCULATION SHEET Date J u n '89

m Date ,?U n '69 Checked by

A S I I I k t W C k Q t S I l l at Im Intervals I l I

I I I

L 4m

73


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CALCULATION SHEET

badinq

Door t f i x i n g

S h e a r at A F V A - W L = 3 S X 4 =?kN 2 2

74


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Institute - -

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

CALCULATION SHEET

sect (on pmpertces tarn Reference IO

Client

Checked by

Made by

Contract No.

Date Jon '89 M Date Jun '89

\

1

I

2

r'

If

1'

1

75


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The ,Job No. P u b ?S?

Steel Construction -- Institute

ISheet 4 of E x ,91Rev’ Job Title

-- - - - - - - Worked Example 9 Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

ci\ Bucklinq check

76


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CALCULATION SHEET

66.6 x I 0'

= E . W n o . 0 "

Job No. P m 75? I Sheet 5 of E)41ReV* Job Title

Worked Example 9 Client

Checked by

Made by

Contract No.

Date J un ' g 9 m Date Job '89

B

c

E?. 2.18

Ea. 2.12 €9.2.19

GrayhCi

€ 9 . 2 . 2 1

77


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t

Ec+.2.22

€9.2. l?


CALCULATION SHEET

Date Jun '49 Checked by EM Date J un ' 8 9

-

78


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7- mern ber 4- B r a c k e t 0

/m intends

79


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Job No.

Job Title The Steel Construction -- PUB ?S? Sheet I of cx.l Rev.

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

CALCULATION SHEET

r 1.1 l : I II

80


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Silwood Park Ascot Berks SLS 7QN 1- Icontract No. Telephone: (0990) 23345 I Fax: (0990) 22944 Telex: 846843 Made by

L

CALCULATION SHEET Checked t -

81


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CALCULATION SHEET

Date Jon ' 8 9 Checked by

bistribut r'on 01 t o r y e

I(e = I

K b = 48 €1,

'on '89

82


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I The d

I Silwood Park Ascot Berks SL5 7QN Client !Contract No.

I Telephone: (0990) 23345 I Fax: (0990) 22944 Telex: 846843 Made by Date

I CALCULATION SHEET lChecked

Kc = I x 1 0 -

: . kc = 1754 kNrn /radian

l J

83


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CALCULATION SHEET

Date J u n '89 Checked by m Date 1

L.

To calculate a

1.

84


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I Silwood Park Ascot Berks SLS 7QN Client [Contract No-- I ~~~~ ~

I Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

I CALCULATION SHEET P e c k e d Il'

Check Column for axial load and bendinq

85


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I I I

The Steel Construction EEE ~ ~~

Institute Silwood Park Ascot Berks SL5 7QN Contract No. Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843 Made by

CALCULATION SHEET

L -- - wwr KU

J

beam

86


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CALCULATION SHEET

TO colluIclte 0

Et+. 2.24

87


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CALCULATION SHEET

c

88


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5. SECTION PROPERTIES

A list of the basic section properties required for design can be found in Steelwork Design Guide to BS 5950, Volume

Tables 5.1 to 5.4 list the additional cross sectional properties that are required in the calculation of angle of twist and torsional stresses for hot rolled I, H, Joist and Channel sections.

The torsional constant, J, the torsional bending constant, a, and the warping constant, H , are properties of the entire cross section. The normalised warping function, W,,, and the warping statical moment, S,,, vary at different points on the cross section to which the term applies.

The formulae needed to evaluate these torsional properties are given in Appendix A .

Values of the statical moments for the flange, Q,, and the web, Qw, are used to calculate flexural shear stresses in the flange and web respectively. The statical moment for the flange, Q,, is the first moment of area of the portion of the flange between the toe of the flange and the edge of the web about the neutral axis of the whole section. The statical moment for the web, Q,, is the first moment of area of half the cross section about the neutral axis of the whole section. (Qw = plastic modulus/2.) Values of Qf and Qw are also given in Tables 5.1 to 5.4.

Tables 5.5 to 5.7 list the additional cross sectional properties for hot rolled structural hollow sections, J and C, described in Section 2.1.3.

89


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Table 5.1 Universal Beams - Torsional Properties

I Designation

Serial Size

mm

914x419

914x305

838x292

762x267

686x254

610x305

610x229

533x210

457x191

457x152

- Mass per

kg/m

metre

-

388 343

289 253 224 201

226 194 176

197 173 147

170 152 140 125

238 179 149

140 125 113 l 0 1

122 109 101 92 82

98 89 82 74 67

82 74 67 60 52 -

-r Torsional Constant

J

cml'

1730 1190

929 627 421 293

51 4 307 222

405 267 161

307 219 169 116

788 341 200

217 155 112 77.2

180 126 102 76.2 51.3

121 90.5 69.2 52.0 37.1

89.3 66.6 47.5 33.6 21.3

Torsional Bending Constant

a

mm

3640 4050

2950 3310 3690 4040

3120 3590 3900

2690 3020 3460

2510 2760 2960 3270

2170 2770 3240

2180 2410 2630 2910

1830 2030 2150 2340 2590

1590 1720 1860 2020 2220

1290 1390 1530 1730 1950

Warping Constant

H

dm6

88.7 75.7

31.2 26.4 22.0 18.4

19.3 15.2 13.0

11.3 9.38 7.41

7.41 6.42 5.72 4.79

14.3 10.1 8.09

3.99 3.45 2.99 2.51

2.32 1.99 1.82 1.60 1.33

1 . l 7 1.04 0.923 0.819 0.706

0.569 0.499 0.429 0.387 0.311

Normal ised Warping Function

W,

cm2

929 920

680 680 674 670

605 599 595

499 494 488

428 424 422 419

469 456 450

342 339 337 334

277 274 273 271 269

216 214 212 21 1 209

171 170 168 169 167

Warping Statical Moment

S,,

cm4

35700 30800

16900 14500 12200 10300

11900 9500 8150

8490 7110 5680

6480 5670 5070 4280

11400 8270 6140

4360 3810 3330 2810

3130 2720 2490 2210 1850

2040 1810 1630 1450 1260

1240 1100 955 858 694

Statical

F 1 ange

01

cm3

3280 2860

2100 1830 1560 1320

1560 1260 1090

1210 1030 835

972 86 1 777 662

1400 l040 867

725 640 564 481

564 496 458 41 1 347

402 362 327 295 259

306 274 241 217 178

Moments

Web

Q,

cm3

8830 7740

6290 5470 4760 41 80

4580 3820 3400

3580 3100 2590

2810 2500 2280 2000

3730 2760 2290

2070 1840 1640 1440

1600 1410 1310 1180 1030

1120 1010 916 828 736

900 811 721 642 547

swo?wo

Continues over next page

90


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. . . Table 5.1 continued

swoI S W 0

S W O S W O

S W 1

Designation r Serial Size

mm

406x178

406x140

356x171

356x127

305x165

305x127

305x102

254x146

254x102

203x133

203x102

178x102

152x89

127x76

- Mass per metre

kg/m -

74 67 60 54

46 39

67 57 51 45

39 33

54 46 40

48 42 37

33 28 25

43 37 31

28 25 22

30 25

23

19

16

13 ~

Torsional Constant

J

cn‘

63.0 46 . O 32.9 22.7

19.2 10.6

55.5 33.1 23.6 15.7

14.9 8.68

34.5 22.3 14.7

31.4 21 .o 14.9

12.1 7.63 4.65

24.1 15.5 8.73

9.64 6.45 4.31

10.2 6.12

6.87

4.37

3.61

2.92


a

mm

1580 1730 1910 2110

1670 1950

1390 1610 1780 1990

1350 1560

1330 1510 1700

913 1020 1120

971 1100 1220

l050 1200 l400

867 958

1050

974 1120

760

770

583

422

vlarp i ng Constant

H

dm6

0.608 0.533 0.464 0.39

0.206 0.155

0.413 0.331 0.286 0.238

0.104 0.081

0.234 0.196 0.164

0.101 0.0842 0.0724

0.0441 0.0353 0.0266

0.103 0 .OR58 0.0662

0.0279 0.0228 0.0183

0.0373 0.0295

0.0153

0.00991

0.0047

0.002

Normalised Warping Function

W,

C d

178 177 175 174

139 138

151 149 147 146

108 107

124 122 121

92.8 91.5 90.5

77.3 76.4 75.7

90.9 89.7 88.7

63.9 63.3 62.8

66.0 65.1

49.3

43.2

32.2

22.7

rarp i ng Statical Moment

SW

CIb

1280 1130 994 839

554 422

1030 834 727 609

362 284

708 599 508

407 344 299

213 172 131

425 358 280

163 135 109

212 170

116

86.6

55 .O

32.9

Statical Moments

F 1 ange

Q C.’

273 244 218 185

152 118

228 188 166 140

113 90 .O

164 141 121

123 . 106

93.5

79.6 65.5 50 .5

111 95.1 74 .B

61.5 51.2 41.9

61 .5 49.9

44.6

33.5

24.3

17.0

Web

0,

cm’

752 673 597 524

444 360

606 505 447 387

327 270

422 361 312

353 305 270

240 204 169

284 243 198

177 153 131

157 130

116

85.6

62 .O

42.5

91


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Table 5.2 Universal Columns - Torsional Properties

T ~

Mass per

kg/m

metre

~

634 551 467 393 340 287 235

477

202 177 153 129

283 240 198 158 137 118 97

167 132 107 89 73

86 71 60 52 46

37 30 23

~

Torsional Constant

J

cm"

13700 9240 5820 3550 2340 1440 812

5700

560 383 25 1 153

2030 1270 734 379 250 160 91 .l

625 322 173 104 51.3

138 81.5 46.6 32.0 22.2

19.5 10.5 4.87


a

mm

856 935 1040 1180 1310 1490 1750

l040

1820 2030 2290 2650

898 1010 1170 1400 1570 1780 2100

821 977 1160 1340 1590

772 892 1040 1160 1290

730 870 1070 --

Warping Constant

H

dm6

38.8 31 .l 24.3 19.0 15.5 12.3 9.54

23.8

7.14 6.07 5.09 4.16

6.33 5.01 3.86 2.86 2.38 1.97 1.55

1.62 1 .l8 0 .894 0.716 0.557

0.317 0.25 0.195 0.166 0.142

0.04 0.0306 0.0214

Normal ised Warp i ng Function

W"0

cmz

422 406 390 376 366 356 346

397

325 320 316 31 1

258 250 242 235 231 227 223

170 164 159 155 152

105 102 100.0 98.7 97.6

58.0 56.6 55.5


S.",

cm4

34400 28700 23300 18900 15800 13000 10300

22400

8220 7100 6040 5010

9170 7500 5970 4560 3870 3250 2610

3570 2710 2110 1730 1370

1130 915 728 630 546

258 203 144

Statical

F 1 ange

Of

cm'

2890 2470 2070 1720 1470 1240 1000

1880

847 742 638 536

1050 883 721 565 486 414 339

504 394 31 4 262 21 2

205 171 138 121 105

64.2 51.6 37.2

Moments

Web

CL

cm3

7120 6040 5000 4110 3500 2910 2340

4850

1990 1730 1480 1240

2550 2120 1720 1340 1150 976 195

1210 937 743 614 494

489 401 326 284 249

155 124 92.1

W n o W"0

swo?swo

92


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s re

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Table 5.3 Joists - Torsional Properties

wno Twno

swoTswo

Designation

Serial Size

mm

254x203

254x114

203x152

152x127

127x114

127x76

114x114

102x102

102x44

89x89

76x76

__ Mass per

kg/r

metre

__

81.85

37.20

52.09

37.20

29.76 26.79

16.37

26.19

23.07

7.44

19.35

14.67 12.65 __

Torsional Constant

J

cm'

153

25.5

64.9

34.2

20.9 16.9

6.69

19.0

14.4

1.25

11.6

6.83 4.67

Torsional

Constant Constant Bending warping

a

dm6 mm

H

728 0.312

632 0.0393

532 0.0709

373 0.0183

317 0.00787 348 0.00807

284 0.00209

286 0.00599

241 0.00321

191 0.000177

l 8 8 0.00158

163 0.000597 182 0.000699

Normal ised Warping Function

W,

cmz

119

68.9

71 .l

44.2

33 .O 33 .O

22.4

29.6

23.2

10.6

17.6

13.6 12.9


S W 7

CR4

1200

253

446

185

109 107

40.7

90.3

60.7

7.1

38.7

22.7 20.7

Statical Moments 1 F 1 ange

0,

cm3

228

84.3

112

54.5

35.0 35.5

20.3

30.0

22.2

6.04

16.0

10.3 10.4

Web

OW

cm3

538

230

270

139

90.5 85.9

51.8

75.6

56.7

17.6

41.4

27.1 24.4

93


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Table 5.4 Channels - Torsional Properties

Designation

Serial Mass Size per

metre

mm kg/m

132x102 65.54

381x102 55.10

305x102 46.18 305x89 41.69

!54x89 35.74 !54x76 28.29

!29x89 32.76 !29x76 26.06

!03~89 29.78 !03x76 23.82

178x89 26.81

152x89 23.84

127x64 14.90

102x51 10.42

16x38 6.70

Constant

61 .O 960

46.0 9 30

35.4 785 27.6 720

22.9 626 12.3 641

20.4 578 11.4 586

17.8 529 10.4 529

15.1 480 8.13 494

12.4 429 5.94 461

4.92 315

2.55 228

l .23 146

I W C

_-

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0 -

arp i ng onstanl

H

dm6

.217

.l53

.OW2

.0551

.0347

.0194

.0263

.0151

.0192

.0112

.0134

.00764

.008a1

.00486

.001a8

.00051?

.000101

Normalised Warping Functions

W,

cm1 -

133

113

87 .8 79.4

63.0 56.7

55.4 49.4

48.1 42.5

41 .l 36.6

34.4 31 .l

21 .o

13.6

7.3, -

W",

cm1 -

65 .O

62.7

52.1 42.7

38.4 30.9

35.7 29.3

32.7 27.2

29.3 24.5

25.7 21.3

14.5

E .89

4.96

Warping Statical Moments

-

S W l

cm4 -

720

573

395 296

224 145

189 125

156 105

125 82.1

97.1 60.7

34.6

14.9

5.32 -

-

sw2

cm" -

148

897

'56 '10

41 02

1 1 81 .O

84.2 62.0

61.6 45.2

42.9 32.3

18.1

8.49

2.92 -

SW3

cm4

-274

-198

-128 -105

- 70.4 - 50.9

- 55.3 - 40.5

- 42.1 - 31.0

- 30.8 - 22.6

- 21.5 - 16.1

- 9.04

- 4.24

- 1.46

listance rom entre Nf web o shear centre

eo

mm

31.3

34.4

35.9 29.3

31.9 25.5

33.2 27.0

34.3 28.4

35.4 29.3

36.5 29.7

24.6

18.9

14.3

S W 1

,tatical Moment!

F 1 ange 01

cm'

319

277

201 161

133 93.1

118 85.4

101 75.9

84.7 61.6

68.2 46.6

31.9

16.5

8.02

- Web 0,

C.' ~

603

466

319 279

207 159

174 135

143 113

115 87.7

88.8 65.0

44.7

24.4

11.7

94


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s of

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ater

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all

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s re

serv

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the

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Table 5.5 Circular Hollow Sections - Torsional Properties

r Designation

Outside Diam

mm

21.3

26.9

33.7

42.4

48.3

60.3

76.1

88.9

114.3

139.7

168.3

193.7

219.1

Thick ness

mm

3.2 ]

3.2 ]

2.6 ] 3.2 ) 4.0 ]

2.6 ) 3.2 ) 4.0 ]

3.2 4.0 5.0

3.2 4.0 5.0

3.2 4.0 5.0

3.2 4.0 5.0

3.6 5.0 6.3

5.0 6.3 8.0 10.0

5.0 6.3 8.0 10.0

5.0+ 6.3 8.0 10.0 12.5 16.09

5.0+ 6.3 8.0 10.0 12.5 16.09 20.09

r; Mass per metre

kg/m

1.43

1.87

1.99 2.41 2.93

2.55 3.09 3.79

3.56 4.37 5.34

4.51 5.55 6.82

5.75 7.11 8.77

6.76 8.38 10.3

9.83 13.5 16.8

16.6 20.7 26.0 32.0

20.1 25.2 31.6 39.0

23.3 29.1 36.6 45.3 55.9 70.1

26.4 33.1 41.6 51.6 63.7 80.1 98.2

orsional onstant

J

C d

1 .54

3.41

6.19 7.21 8.38

. 12.9 15.2 18.0

23.2 27.5 32.3

46.9 56.3 67.0

97.6 118 142

158 193 233

384 514 625

961 1180 1440 1720

1710 2110 2590 3130

2640 3260 4030 4880 5870 7110

3860 4770 5920 7200 8690 l0600 12500

orsional lodul us :onstant

C

ca'

1.44

2.53

3.67 4.28 4.97

6.10 7.19 8.48

9.59 11.4 13.4

15.6 18.7 22.2

25.6 31 . O 37.3

35.6 43.3 52.4

67.2 89.9 109

138 169 206 247

203 250 308 372

273 337 416 504 606 735

352 436 540 657 793 967 140

Continues.. ..

r Designation T Outsidc

Dian

m m

244.5

273.0

323,9

355.6

406.4

457 .O

508.0

- Thick ness

ma -

6.3 8.0 10.0 12.5 16.0 20. 09

6.3 8.0 10.0 12.5 16.0

25. 09

6.3+ 8.0 10.0 12.5 16.0

25.01.

20.09

20.01.

8.0 10.0 12.5 16.0

25.09 20. 09

10.0 12.5 16.0

25.09 32.09

10.0 12.5 16.0 20.09 25. O f 32. 09

20. 09

40.09

10.0+ 12.5+ 16.0+ 20. o+l 25.0+4 32.04 40.0+4 50.0d -

Mass per metre

kg/a

37 .O 46.7 57.8 71 .5 90.2

I l l

41 .4 52.3 64.9 80.3 101 125 153

49.3 62.3 77.4 96.0 121 150 184

68.6 85.2 106 134 166 204

97.8 121 154 191 235 295

110 137 174 216 266 335 1111

123 153 194 241 298 376 U62 565

o r s i ona' .onstant

J cm4

6690 8320 10100 12300 15100 17900

9390 11700 14300 17400 21 400 25600 30300

15900 19800 24300 29700 36800 44300 52800

26400 32400 39700 49300 59600 71400

49000 60100 74900 90900 09000 133000

70200 86300 08000 31000 59000 94000 130000

97000 20000 50000 83000 122000 :72000 i24000 182000

orsional odulus onstant

C

cm3

547 681 830 1010 1230 1470

688 857 1050 1270 1570 1880 2220

979 1220 1500 1830 2270 2730 3260

1480 1820 2230 2770 3350 401 0

2410 2960 3690 4470 5380 6540

3070 3780 4720 5750 6950 8490 10100

3820 4710 5900 7200 8730 10700 12800 15000

+ Sections marked thus are not included in

) Sections aarked thus are rolled in grade

9 Sections aarked thus are r o l l e d i n grade

BS 4848: Par t 2

43C only

50C only

95


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ater

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s re

serv

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s do

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the

term

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Table 5.6 Rectangular Hollow Sections - Torsional Properties

Designation

Size

mm

50x25

50x30

60x40

80x40

90x50

00x50

00x60

20x60

20x80

Thick ness

mm

2.5+: 3.0+: 3.2+:

2.5+ 3.0+ 3.2 4.0+ 5.0+

2.5+ 3.0+ 3.2 4.0 5.0+ 6.3+

3 .O+ 3.2 4.0 5 .0+ 6.3+ 8 .O+

3.0+ 3.6 5 .0 6.3+ 8.0+

3.0+ 3.2 4.0 5.0 6.3+ 8.0+

3 .O+ 3.6 5 .0 6.3 8 .O+

3.6 5 .0 6.3 8 .0+

5 .0 6 .3 8.0

10.0

T Mass per metre

kg

2.72 3.22 3.41

2.92 3.45 3.66 4.46 5.40

3.71 4.39 4.66 5.72 6.97 8.49

5.34 5.67 6.97 8.54

10.5 12.8

6.28 7.46

10.1 12.5 15.3

6.75 7.18 8.86

10.9 13.4 16.6

7.22 8 .59

11.7 14.4 17.8

9.72 13.3 16.4 20.4

14.8 18.4 22.9 27.9

rors i ona 1 :onstant

J

cm"

8.41 9.64

10.1

11.7 13.5 14.2 16.6 19.0

25.0 29.2 30.8 36.6 43.0 49.7

43.7 46.1 55.1 65.0 75.8 86.3

76.4 89.3

116 138 161

88.3 93.3

113 135 160 187

121 142 187 224 266

183 242 290 344

401 486 586 688

'ors i ona lodu 1 us :onstant

C

cm3

4.62 5.21 5.42

5.74 6.52 6.81 7.79 8.71

9.74 11.2 11 .8 13.7 15.8 17.7

15.3 16.1 18.9 21.9 24.9 27.6

22.4 25.9 32.9 38.2 43.4

25.0 26.4 31.4 37.0 43.0 49.1

30.7 35.6 45.9 53 .9 62.4

43.3 56 .0 66 . O 76.8

77.9 93 . O 10 26

Designation r Size

mm

150x100

160x80

200x100

250x150

300x200

400x200

450x250

500x300

Thick ness

mm

5.0 6.3 8.0

10.0 12.5+)

5.0 6.3 8 . 0

10.0 12.5+)

5.0 6.3 8.0

10.0 12.5 16.0

6.3 8.0

10.0 12.5 16.0

6.3 8.0

10.0 12.5 16.0

8.0+ 10.0 12.5 16.0

10.0 12.5 16.0

10.0+ 12.5+ 16.0+ 20.0+

Mass per metre

kg

18.7 23.3 29.1 35.7 43.6

18.0 22.3 27.9 34.2 41.6

22.7 28.3 35.4 43.6 53.4 66.4

38.2 48.0 59.3 73.0 91.5

48.1 60.5 75.0 92.6 17

73.1 90.7 12 42

06 32 67

22 52 92 37

'orsiona :onstant

J

cm"

806 985

1200 1430 1680

599 729 882

1040 1210

1200 1470 1800 2150 2540 2990

4050 501 0 6080 7320 8860

8470 10500 12900 15700 19200

15700 19200 23400 28800

33200 40700 50500

52400 64300 80200 97300

rors i ona lodul us :onstant

C

cm3

127 153 184 215 246

106 127 151 175 199

172 208 251 296 342 393

413 506 606 717 85 1

681 840

1020 1220 1470

1140 1380 1660 2010

1990 2410 2950

2700 3280 4050 4840

+ Sections marked thus are not included in

) Sections marked thus are ro l led in grade BS 4848: Part 2

43C only

Continues . . . .

96


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ition

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gree

men

t


Discuss me ...C

reat

ed o

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Mar

ch 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

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f thi

s do

cum

ent i

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t to

the

term

s an

d co

nditi

ons

of th

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Agr

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Table 5.7 Square Hollow Sections - Torsional Properties

r Designation Size

mm

20x20

25x25

30x30

40x40

50x50

60x60

70x70

80x80

90x90

00x100

Thick ness

mm

2.0 ) 2.5+)

2 . O + ) 2.5+) 3 . O + ) 3.2+)

2.5+) 3.0+) 3.2 )

2.5+ 3.0+ 3.2 4.0 5.0+

2.5+ 3.0+ 3.2 4.0 5.0 6.3+

3.0+ 3.2 4.0 5.0 6.3+ 8.0+

3.0+ 3.6 5.0 6.3+ 8.0+

3 . O + 3 . 6 5 . 0 6 .3 8.0+

3.6 5.0 6.3 8 . O +

4.0 5 .0 6.3 8 . 0

10.0

Mass per

Constant metre Torsional

1.12 1.22 1.35 I 1.41 l .43 1.74 2.04 2.15

2.14 2.51 2.65

2.92 3.45 3.66 4.46 5.40

3.71 4.39 4.6 5.72 6.97 8.49

5.34 5.67 6.97 8.54

10.5 12.8

6.28 7.46

10.1 12.5 15.3

7.22 8.59

11.7 14.4 17.8

9.72 13.3 16.4 20.4

12.0 14.8 18.4 22.9 27.9

2.52 2.97 3.36 3.49

5.40 6.17 6.45

13.6 15.7 16.5 19.5 22.6

27.4 32.0 33.8 40.4 47.6 55.3

56.9 60.1 72.4 86.3

102.0 119.0

92.1 108.0 142.0 169.0 200.0

139.0 164.0 217.0 261 . O 312.0

237 .O 315.0 381 .O 459.0

361 439 533 646 761

lorsional lodu 1 us :onstant

C Cm’

1.07 1.21

1.81 2.09 2.31 2.38

3.22 3.61 3.75

6.23 7.11 7.43 8.56 9.65

10.2 11.8 12.4 14.5 16.7 18.9

17.7 18.6 22.1 25.8 29.7 33 .5

24.8 28.7 36.8 43.0 49.4

33.1 38.5 49.8 58.8 68.5

49.7 64.9 77.1 90.7

68.2 81.9 97.9

116 134

Continues ......

Designation

Size

I R

120x120

140x140

150x150

180x180

200x200

250x250

300x300

350x350

400x400

- Thick ness

RB -

5.0 6.3 8.0

10.0 12.5+]

5 .O+ 5.6+ 6.3+ 7.1+ 8 .O+

10.0* 12.5+

5.0 6.3 8.0

10.0 12.5 16.0

6.3 8.0

10.0 12.5 16.0

6.3 8.0

10.0 12.5 16.0

6.3 8.0

10.0 12.5 16.0

8.0+ 10.0 12.5 16.0

8.0+ 10.0 12.5 16.0

10.0 12.5 16.0+ 20 .o+ -

Mass per metre

kg/m

18.0 22.3 27.9 34.2 41.6

21 .l 23.5 26.3 29.4 32.9 40.4 49.5

22.7 28.3 35.4 43.6 53.4 66.4

34.2 43 .O 53 .O 65.2 81.4

38.2 48 .O 59.3 73 .O 91.5

48.1 60.5 75.0 92.6

117

73 .l 90.7

112 142

85.7 106 132 167

122 152 192 237

orsional onstant

J Cm4

775 949

1160 1380 1620

1251 1385 1538 1706 1889 2269 2695

1550 1910 2350 2830 3370 4030

3360 4160 5040 6060 7340

4650 5770 7020 8480

10300

9230 11500 14100 17100 21 100

20200 24800 30300 37600

32400 39800 48900 60900

60000 73800 92300 ,12000

‘orsional lodulus :onstant

C cn’

122 147 176 206 237

170 107 206 227 249 294 342

197 240 291 345 403 468

355 434 519 613 725

444 545 655 779 929

712 880

1070 1280 1550

1290 1580 1900 2330

1790 2190 2660 3260

2900 3530 4360 5240

+ Sections narked thus are not included in

) Sections marked thus are rolled in grade BS 4848: Part 2

43C only

97


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6. CHARTS TO ASSIST IN EVALUATION

6.1 Standard cases The standard cases listed in Table 6.1 have been presented graphically in order to facilitate rapid evaluation of the functions:

4GJ V G J a T,a T,

and ~

at a distance z along the member

The graphs 1 to 9 have been obtained by evaluating the functions given for the relevant case number in Appendix B for the appropriate values of a L and z , where aL is the distance along the member at which the torque is applied.

Table 6.1 Standard Cases

Graph Function Distance Torsional loading App. B . Description (2) along Case No. member

l ~ 4GJ CUL Tq f

Concentrated torque on member with torsion fixed, warping frcc at Tq a 4 * \r;

c 3 4"GJa U L ' L ( 1 - U ) ends.

2 (YL Torque applied at

3 42J f f L

Tq varying values of a L

Concentrated torque on member with torsion

Tq a T,+ t S fixed, warping fixed at * =/I *

Y G J a U L L C l d l ) ends. 4 0 Torque applied at

Tq varying values of a L

SA 4GJ 0.SL Uniform torque on member with ends torsion fixed, warping frcc.

7-4

4 6~ V G J a 0.SL

Tq

SB +GJ 0.SL Quarter point torques on member with ends torsion fixed, warping free.

a=0.25, 0.5 and 0.75.

Tq Tq 3 I T q T q 3 / 3 / + 0.SL -L- - 4 4 4 a L * Sum of case 3 for

. . . -

6B V G J a

7-4

sc 4GJ 0.5L Third point torques on member with torsion fixed, warping free at ends.

and 5

Tq T q T q 2 / 2 / + x Y -

SD 4GJ L13 - r + c Sum of case 3 for a=f

7-q

6C Y G J a L13 Tq

7A +GJ 0.SL Tq Uniform torque on

torsion fixed, warping Tq a 7 6 member with ends

W L

fixed.

98


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Table 6.1 continued

Graph Function Distance Torsional loading App. B. Description (2) along Case No. member

7B 4GJ ~ 0.5L $ 4 4 Quarter point torques on member with ends

fixed. 7-q a il c cc a; E - torsion fixed, warping

~ 0 4 4 4 4 Sum of case 5 for a= Tq 0.25, 0.5 and 0.75

8B &GJa L L L L

7c 4GJ 0.5L Third point torques on member with ends torsion fixed, warping fixed. Sum of case 5 for a j and f

T,a

Tsa

Tq

TQ Tq 2 / 2 / - -

I' \i sj - 7D 4!!J L13

j, + - L 3

8C &GJa - 0

9A '?GJ L Tq

7

End torque on member with one end torsion fixed, warping fixed and the other end torsion free, warping free. Use case 7 with a=1.0

Uniform torque on member with one end torsion fixed, warping fixed and the other end torsion free, warping free. Use case 8 with a=1.0

8

6.2 Extreme cases Cases not included in Table 6.1 can conservatively be evaluated by using the Extreme Cases listed in Table 6.2.

No graphs are provided, but the relevant formulae are given in Appendix B for Cases l , 2 and 9.

Table 6.2 Extreme Cases.

Torsional loading Appendix B Description Case No.

M 1

Concentrated torques at ends of member, both ends warping free, left end torsion fixed, right end torsion free.

2 Concentrated torques at ends of member, with ends torsion fixed, warping fixed.

Uniform torque on member with one end torsion fixed and warping fixed, other end torsion fixed and warping free.

9

99


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CL 0 0 GRAPH I


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GRAPH 2


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L Q

GRAPH 3


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L a

GRAPH 4


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3

P 0

GRAPH 5


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GRAPH 6 6

5

4

L - a

3

2

I

n 0-05 0. I 0 -2 0 . 2 5 0.3 0.35 0.4 0 . 4 5 0.5

-(2/"GJa 'c


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GRAPH 7

5-0 L - a

4 -0

3.0

2.0

I -0

0.0 0. I


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GRAPH 8

0.35 0.4 0.45


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0 00

c

GRAPH 9 3.0

2 . 5

2 -0

L - a

1-5

I v 0

0.0 0.2 0.4 0.6 0.8


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REFERENCES

3

4

5

10

11

12

Structural action in steel box girders CIRIA Guide 3, 1977.

VLASOV, V.Z. Thin walled elastic beams. Israel Program for Scientific Translation Ltd. Jerusalem 1961.

DABROWSKI, R. Curved thin walled girders: theory and analysis Cement and Concrete Association 1972.

KHAN, A.H. and TOTTENHAM, H. The method of bimoment distribution for the analysis of continuous thin walled structures subject to torsion. Proceedings of the Institution of Civil Engineers, Part 2 Volume 63, pp 843-863, December 1977.

TIMOSHENKO, S.P. Theory of torsion and buckling of thin walled members of open cross section. Journal of the Franklin Institute. March, April, May, Volume 239. Philadelphia 1945.

TRAHAIR , N. S. The behaviour and design of steel structures. Chapman and Hall. London 1977.

TIMOSHENKO, S.P. and GOODIER, J.N. Theory of elasticity. Third edition. McGraw Hill 1970.

SHANLEY, F.R. Strength of materials. McGraw Hill 1957.

JOHNSTON, B.G. and EL DARWISH, I.A. Torsion of structural shapes. Proceedings of the American Society of Civil Engineers. Journal of the Structural Division, Vol. 91, No. ST1, pp 203-228, February 1965.

Steelwork design guide to BS 5950: Part 1 1985, Volume l Section properties and member capacities. The Steel Construction Institute, 1985.

PASTOR, T.P. and DE WOLF, J.T. Beams with torsional and flexural loads. Proceedings of the American Society of Civil Engineers. Journal of the Structural Division. Volume 105. No ST3. pp 527-538, March 1979.

JOHNSTON, B. G . Design of W shapes for combined bending and torsion. Engineering Journal. American Institute of Steel Construction, pp 65-85, 1982.

109


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13 BRITISH STANDARDS INSTITUTION BS 5950: Part 1: 1985. Structural use of steelwork in building part 1: Code of Practice for design in simple and continuous construction: hot rolled sections. BSI, 1985.

14 McGUIRE, W. Steel Structures Prentice Hall 1968.

110


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APPENDIX A Evaluation of torsional properties

A.l Definitions In the analysis of cross sections subject to torsion, a number of special section properties are used as follows:

H warping constant J torsional constant

W,, normalised warping function SW, warping statical moment

H and J are properties of the entire cross section, while S,, and W,,, apply to specific points on a cross section. The value of S,, at point 1, for example, is denoted SW, .

Tangent line

Y Figure A1

For a generalised shape (see Figure A l ) these torsional section properties may be defined as follows:

1 3

J = -1: t 3 ds

H = Iob W:, t ds

where p . is perpendicular distance to tangent line from shear centre (see Figure Al)

111


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A.2 Symmetrical I and H sections

I

Y Figure A2

For symmetrical I and H sections (see Figure A2) the following expressions may be used:

J = $ [ 2 B T 3 + ( D - 2 T ) t 3 ] (but see A4 for rolled sections)

hB 4

W", = -

hB2T S,, = -

16

where Zy is the second moment of area of the section about the minor axis.

For sections with tapered flanges, the flange thickness may be taken as the average value when using the above expressions.

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A.3 Statical moments for I and H sections In addition the terms Q, and Qw may be calculated thus:

Qr = Aryr A 2 Q, = -yW

where Af = the area of half the flange as shown in Figure A3(a) yf = the distance from the neutral axis to the centroid of the area Af as

shown in Figure A3(a) A = the total cross sectional area

y , = the distance from the neutral axis to the centroid of the area above the neutral axis as shown in Figure A3(b)

Figure A3

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A.4 Torsional constant for rolled I and H sections The value of J may be more accurately calculated as follows for rolled 1 and H sections with parallel flanges (see Figure A 2 ) :

J = t B T 3 + f ( D - 2 T ) t 3 + 2 a 1 D I 4 - 0 . 4 2 T 4

where a1 = - 0.042 + 0.2204-+ 0.1355-- 0.0865-- O.0725--, t r tr t2 T T T 2 T

(T+r)*-tt[r+(t/4)] 2r+ T

D1 =

For joist sections with sloping flanges (see Figure A4).

J = (?)(TI + T2)(TI2+ T22)+ <tT2'+ (D-:T2) ~ ' + ~ c Q D ; - ~ V , T , ~

( F + m)2+ t[r + (t/4)] F + r + m

0 2

2(m- TI ) S = B V, = 0.10504 + 0.1s + 0.08480S2 + 0.06746s' + 0.05153S4

For 8" taper* t r tr t L

a2 = -0 .0772+0.2485-+0.1281--0 .0815~-0 .0837~ T2 T2 T2 T22

Figure A4

B Y

0 1 0

-- - -

0 , i I 0

*Note: Flanges of BS4 joists have a taper of 8". The expressions given above for a2 have been obtained by linear interpolation between the values of the constants for a taper of 16$% and those for parallel flanges. This is a simplification of the more exact procedure given by Johnston and El Darwi~h(~) but is accurate enough for design purposes. The values of J given in the Tables in this publication (Section 5) and in the Steelwork Design Guide, Volume I("'), are based on these constants.

114


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A.5 Channel sections For channel sections symmetrical about their major-axis (see Figure A5) the following expressions may be used

J = [ 2 B T 3 + ( D - 2 T ) t 3 ] (but see A.7 for rolled channel sections)

(B-t /2-eo)h 2 W",, =

(B-t/2-eo)2 h T 4 SW, =

( B - t /2 - 2eo) h ( B - t/2) T 4

(B- t /2-2eo) h (B - t /2) T

SW2 =

SW3 = --

4 8

(B-t/2-3eo) h' (B-t /2)2 T H = + e; I ,

6

(B-t /2)2 T 2( B - t/2) T + ht/3

where e, =

Figure A5

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A.6 Statical moments for channel sections In addition the terms Q, and Q, may be calculated thus:

Q, = A r ~ f

where Af = the area of one flange as shown in Figure A6(a) y , = the distance from the neutral axis to the centroid of the area

A , as shown in Figure A6(a) A = the total cross sectional area

y , = the distance from the neutral axis to the centroid of the area above the neutral axis as shown in Figure A6(b).

Centroids of shaded area

Figure A6

A.7 Torsional constant for rolled channels The value of J may be more accurately calculated as follows for rolled channel sections (see Figure A5):

J = (T) ( T I + T2) (TI2+ T22) +$tT23+ B-t

+2a4D44-2VsT14- 0.21 T;

where D4 = 2{ [3r + t + F , ] - [2(2r + t)(2r + F l ) ] ” 2 }

F1 = T2 - r [S + 1 - (1 + S)”2]

V , = 0.10504+0.1S+0.0848S2+0.06746S3+0.05153S4

t r tr t 2 T2 T2 T2 T ,

CY^ = -0 .1128+0.2829-+0.1320--0 .092~-0 .0951~

Note: BS4 Channel sections are rolled with a 5“ taper to the flanges. The expression given above for a4 has been obtained by linear interpolation between the values of the constants for a taper of 163% and those for parallel flanges. This is a simplification of the more exact procedure given by Johnston and El Darwish(’) but is accurate enough for design purposes. The values of J given in the Tables in this publication (Section 5 ) and in the Steelwork Design Guide Volume l(1o), are based on these constants.

116


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A.8 Structural hollow sections For structural hollow sections the torsional constant J and the torsional modulus constant C may be calculated from the following expressions:

For circular hollow sections:

J = 21

c = 2 2

For square and rectangular hollow sections:

J c = - t + Klt

in which Ah is the area enclosed by the mean perimeter, given by:

A h = ( B - t ) (D-?)-R,2(4-v)

2Ahf K = - h

and the mean perimeter h is given by:

h = 2[(B-t)+(D-t)]-2RC(4-~)

where B = the breadth of the section D = the depth of the section

R, = the average of the internal and external corner radii I = the second moment of area

t = the thickness of the section Z = the elastic modulus

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APPENDIX B Solutions of differential equations Note : The l o a d i n g i s i n d i c a t e d i n T a b l e s 6 . 1 and 6.2

q i e . A p p l i e d t o r a u e T = T

CAS NO

I

-

1

~

2

-

3

L

LIMITS

-

o l z l a l

'ARAMETEI

0

0'

9' '

0"'

9

0'

0'

0'"

9

0'

9' '

6'"

9

9'

9' '

9"'

EXPRESSION

- Tz GJ

T GJ -

0

0

- Ta L [ t a n h - cosh ' - t a n h - + - - s i n h '1 L Z

C J 2a 2a a

T L - ( t a n h - s i n h + 1 - cosh '1 GJ 2a

- [ t a n h - cosh g - s i n h ') CJa 2a

- [ t a n h - s i n h ' - cosh '1 CJa' 2a

T L

T L

aL

5 { ( I - U ) + [ -+- - cosh p] s i n h i } s i n h -

tanh - a

G J

GJa L t a n h

UL s i n h - - - cosh F] cosh f }

[ t m h

- cosh F] s i n h f

s i n h - aL

L a tanh - a

aL s i n h -

[- U + tanh - cosh f - s i n h a s i n h

aL GJ L

s i n h - UL

-~ CJa [ tanh i

[ tanh i a L

s i n h - -~ GJa2

118


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Appendix B continued

CAS1 No

4

5

LIMITS

o l z l a t

ARMETEI:

0

0'

0' '

0"'

0

0'

0' '

0"'

0

0'

0' '

0"'

Where

EXPRESSION

B [z [ ; - ;; ] + cosh ' - t a n h - s i n h ' - l]

[z [ ; - ;; ] + s i n h ' - t a n h - cosh '3

[- 1 + cosh - - t a n h - s i n h '1

L' z 2' L GJ L 2a

T a L' 1 22 GJ L 2a

T a CJa L 2a

T

- L

_ - z L

f [ s i n h - t a n h - cosh '1 L 2a

- Ta GJ

T 1 GJ (K1 + 1 ) -___

T 1 GJa (Kl + 1) -~

[K1. K, + K,] [cosh ' - l] - s i n h - + -] z z a a

[K1. K, + K,] s i n h - cosh ' + l]

[K1. K, + K,] cosh L - s i n h ']

T 1 GJa' (K1 + 1) -~ {[K1. K, + K,] s i n h ' - cosh ']

~~

-~ Ta 1 2 1 aL UL z z (K,+ KS cosh - + [- (cosh - - 1) + cosh -1 s i n h - - -]

GJ (1+l/K1) a ~1 a a

- - T GJ (l+l/Ki)

{K5 s i n h + [z1(cosh - 1) + cosh -3 cosh -l] z 1 UL aL

- - T CJa (l+l/Kl)

{K5 cosh - + [ i , (cosh - - 1) + cosh -1 s i n h ] 2 1 UL aL

T 1 -~ {Ks s i n h - + [ i l ( cosh - - 1 ) + cosh -1 cosh ' ] z 1 aL aL CJa2 (l+l/Kl) L

1 a L (1 - cosh -1 + - 1

L s i n h -

[cosh - - l] + s i n h aL

[- [cosh - + cosh - cosh - - cosh - - l] + - (U-l) - s i n h L aL L aL L

aL L L . L [cosh - - cosh - + - s i n h - ]

K,=- 1 cosh - - l] +

at. a a

s i n h - L L s i n h -

cosh -

t a n h g1 - L K, = - [ si;h + sinh a - - UL

aL cosh -

- tanh - L ] +

L tanh -

119


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CASE NO.

6

7

-

LIMITS

0 5 z 5 a l

ARAMETEF

0

0'

0' '

Q'''

0

Q'

Q'

Q"'

Q

9'

6' '

0"'

EXPRESSION

L 1 + cosh -

L a cosh - - 2 - - s i n h s i n h -

a

2 a a L 2CJa

L

2CJa'

[ K s [cosh i - 1) - s i n h - + - CJ a a

[ K s s i n h

- cosh + 1 CJ

CJa' - cosh ; [ K s s i n h

K 7 - [cosh - - l ] [ t a n h - cosh ') + [cosh - - l ] s i n h ' + aL L aL

CJ a a S r l & 1- [cosh a - 1) [ t a n h - s i n h '1 + [cosh - - l ] cosh

aL L a L

r l

L 1 - [cosh a - l ] [ t a n h a cosh i) + [cosh ; - l ] s i n h - aL L aL CJa : I

L J

r l 5 1- [cosh a L ; - l ] [ t a n h L

Where K c s i n h - - tanh - cosh + tanh - aL L L

K, = tanh - cosh !Z - tanh - - s i n h - L L UL

J

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CAS1

NO

a

9

LIMITS

O l Z l U l

'ARAMETEI

9

9'

9' '

9"'

9

9'

9' '

9"'

Where

9

9'

9' '

9"'

EXPRESSION

- - [ K 8 [ cosh - l ] - - s i n h - + - [- - -)l l a a UL z Z U L z GJ aL a a a 2a

l a - - [K8 s i n h g - cosh - + [- - 'l] GJ U4 a a a a a

- - [ K 8 cosh - s i n h - l] T a GJa u L a a

Z a L

l a z UL -- [ K 8 s i n h - - - cosh '1 GJa2 u L a a

- - [ K s - [ s i n h - - -) t a n h - cosh + ( s i n h - - -1 s i n h '1 l a a UL UL L UL UL

GJ UL a a a a

- - [- [ s i n h - - -1 tanh - s l n h - + [ s i n h - - -1 cosh '1 l a UL UL L . z UL UL

- - [- [ s i n h - - -1 tanh - cosh g + [ s i n h - - -1 s i n h '1 GJa uL a a a a

[ - [ s i n h - - -1 tanh - s i n h - + [ s i n h - - -1 cosh '1

G J UL a a a a

l a UL UL L UL UL

l a UL UL L . z UL UL -- GJa' UL a a a a

L UL K 8 = t a n h - (- - s i n h -1 + cosh - UL UL

a a

L . UL UL UL L U 2 L2 K 9 = tanh - s i n h - - cosh - - - tanh - + 1 + 7

a a 2a

L a GJa L

- - l a GJa' L

Z

KI, [- tanh - cosh g + s i n h -1 + - z cosh a L

a cosh - a

L -

Klo [- tar lh - s l n h - + cosh -1 + - L . z s i n h a

a cosh - L a

Where Klo = - [ ::2 1

- l 1

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design of members subject to combined bending and torsion

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