Table Of ContentP057: Design of Members Subject to Combined Bending and Torsion
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SCI PUBLICATION 057
Design of Members
Subject to
Combined Bending
m - m
anu I orslon
D. A. NETHERCOT BSc(Eng) PhD CEng FlStructE
University of Nottingham
P. R. SALTER BSc(Eng) CEng MlStructE
(formerly of) The Steel Construction Institute
A. S. Malik BSC MSC
The Steel Construction Institute
ISBN 1 870004 44 2
0T he Steel Construction Institute 1989
(Reprinted, l99 7)
The Steel Construction Institute
Silwood Park
Ascot
Berkshire
SL5 70N
Telephone: 013 44 23345
Fax: 013 44 22944
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P057: Design of Members Subject to Combined Bending and Torsion
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This publication seeks to provide authoritative guidance for the design of steel
structures subjected to combined bending and torsion. In most cases, however, i t
should be possible to avoid the introduction of significant torsion, by paying
attention to detail andch oosing a load path fors uch an alternative. This publication
is, therefore, concerned with the minority of cases where the loads have to be
applied eccentrically with respect to the shear centre. Afterb r iae f discussion of the
background theory, simple methods of evaluating torsional stresses and
deformations are detailed, and worked examples illustrating the use of tables and
charts included in the publication are provided.
The text has been prepared by Professor D A Nethercot of Nottingham University
and Messrs. P R Salter and A S Malik of the Steel Construction Institute and
reviewed by Mr J C Taylor and Dr R Narayanan of The Steel Construction
Institute.
..
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P057: Design of Members Subject to Combined Bending and Torsion
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CONTENTS
Page
SUMMARY
NOTATION
1. DESIGNING FOR TORSION
1 .l Introduction 1
1.2 Scope of the Publication 1
1.3 Choice of Members 2
2. BASICT HEORY
2.1 Torsion 3
2.2 Bending 12
2.3 Combined Bending and Torsion 13
3. LOADA PPLICATION 16
4. WORKEDE XAMPLES 17
5. SECTIONP ROPERTIES 89
6. CHARTS TO ASSIST IN EVALUATION
6.1 Standard Cases 98
6.2 Extreme Cases 99
REFERENCES 109
APPENDIX A. Evaluation of Torsional Properties 1 1 1
APPENDIX B. Solution of Differential Equations 118
iii
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P057: Design of Members Subject to Combined Bending and Torsion
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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 dela torsion est nkcessaire
dans les calculs de dimensionnement sont identifiees. La thkorie classique de la
torsion desp iPces en acier a section ouverte ou fermee est prksentke. Des solutions des
Pquations qui en resultent sont prksentees sous forme de diagrammes de
dimensionnement, d'une part, et sous forme de formules, d'autre part.
Elles permettent de sklectionner les charges de torsion a appliquer et les conditions
d'appuis. Une mkthode simple pour combiner les effets de la torsion et de la flexion
est presentee. Elle est en accord avec la norme BS 5950 : Partie 1.
Le mkthode complkte de dimensionnement en flexione t torsion combinkes est
illustree au moyen d'exemples. Ils montrent que le dimensionnement est souvent
gouverne par la necessitk de restreindre les deformations torsionnelles, sous les
charges de service, a des valeurs acceptables, plutbt que par la resistance ultime.
Berechnung von Bauteilen unter Biegung und Torsion
Zusammenfassung
Die Ursachen f u r Torsionsbeanspruchung von Bauteilen werden besprochen und die
Falle, in denen eine Berucksichtigung der Torsion klarer Bestandteil der statischen
Berechnung sein muJ3. Die elementare Theorie der Torsion von offenen und
geschlossenen Stahlquerschnitten wird vorgestellt. Die Losungen der sich ergebenden
Gleichungen werden fur eine Auswahl von Torsionsbelastungen und
Randbedingungen in Form von Bemessungstafeln und Formeln zur Verfugung
gestellt. Eine einfache Methode fur kombinierte Beanspruchung aus Biegung und
Torsion entsprechend BS 5950, Teil I , wird vorgestellt. Der vollstandige Weg zur
Bemessung bei Biegung und Torsion wird anhand einer Reihe von Beispielen
aufgezeigt. Die Beispiele zeigen, daJ3 die Bemessung oft von der Notwendigkeit
bestimmt wird, die Verdrehung durch die angreifenden Lasten in akzeptablen
Grenzen zu halten, weniger von dem Gesichtspunkt der Bruchfestigkeit.
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P057: Design of Members Subject to Combined Bending and Torsion
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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 qualPi necessario tenere conto esplicitamented ella
torsione nei calcoli di pregetto.
E presentata la teoria della torsione con riferimento ai profili apertie a quelli a
sezione chiusa. Viene quindi riportata la soluzione delle equazioni che reggono il
problema, in forma sia di abachi sia di formule, per una gammas ignificativa di
condizioni di carico e di vincolo.
E altresi illustrato un metodo semplice che consente di combinare glei ffetti della
torsione e della flessione, metodo in accord0 con l'approcciod elle BS.5950: parte I.
Una serie di esempi consentlea comprensione del'approccio progettuale per elementi
soggetti a flessotorsione nella sua completezza. Si mettei n luce come la necessita di
limitare a livelli accettabili la deformazione torsionale sotto i carichi di esercizio
governa il progetto in molti casi, mentre la resistenza ultima riveste minore
importanza.
Diseno de Piezas Sometidas a Flexion y Torsion Combinadas
Resumen
Se analizanl as causas que originanc argas de torsi6n elna s piezas de unaes tructura y
se identifican aquellas situaciones en que debe incluirsel a consideracion explicita de
la torsion en 10s calculos de rrn proyecto.
Se presenta la teoria basica de torsion en secciones de acero tanto abiertas como
cerradas; t a m b i h se suministran soluciones de las ecuaciones resultantes mediante
abacos de disenoy formulas para diferentes tipos cdaer gas torsoras y condiciones de
apoyo. Se incluye, en particular un procedimiento sencillo de combinacion de efectos
de flexiony torsion, congruente con el metodo patrocinado porla BS.5950: Parte I .
Se desarrollan una serie de ejemplos que ilwtran e l me'todo de analisis completo
para flexion y torsion, y demuestran que a menudo el diseno queda controlado por
la necesidad de mantener la torsion en tensiones admisibles a niveles aceptables en
lugar de por consideraciones der esistencia ultima.
V
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P057: Design of Members Subject to Combined Bending and Torsion
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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
E
Shear modulus of elasticity of stel (taken to be -
2( 1 +v) ' where
Poissons ratio v = 0.3, thus E/G = 2.6 and G has an approximate
value of 79000 N/mm2)
Warping constant for cross section
Depth of open section, centre to centre of flanges; mean perimeter of
closed section
Second moments of area of cross section about the major and minor axes
Torsional constant for cross section
Length of member
Bending moment acting on cross section
Applied load
Statical moment (about the neutral axis of the entire cross section) of the
crosss ectional area between the free edges of the cross sectiona nda
plane cutting the cross section across the minimum thickness at the point
under examination
Value of Q for a point in the flange directly above the vertical face of the
web
Value of Q for a point at mid-depth of section
Plastic modulus about the major and minor axes
Warping statical moment at a point 'S' on cross section
Flange thickness
Pure torsional resistance equal to GJ+'
Applied torque (torsional moment) at given location
Warping torsional resistance equal to
P057: Design of Members Subject to Combined Bending and Torsion
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W”,
Normalised warping function at a point ‘S’ on cross section
xo, Yo Co-ordinates of the shear centre with respect to the centroid
Y Perpendicular distance from neutral axis to a point on cross section
Z Distancef rom left end of member( origin of co-ordinates ystem)t o
transverse section under examination (Figure 2.3)
Elastic moduli about the major and minor axes
Distance from support to point of applied torsional moment (or to endo f
uniformly distributed load over a portion of span), divided by the span
length (i.e. aL is the distance, (Y is a fraction of L )
Total angle of twist at a transverse section of member, radians
First derivative of 4 with respect to z
Second derivative of 4 with respect to z
Third derivative of 4 with respect to z
Fourth derivative of 4 with respect to z
Combined longitudinal stress
Longitudinal stress due to plane bending
Warping normal stress, i.e. longitudinal stress at a point on cross section
due to restrained warping of the cross section
7- Combined shear stress
Shear stress due to plane bending
Pure torsional shear stress
Warping shear stress at point on cross section due to restrained warping
of the cross section
vii
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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 appliedso that the resultant force
passes through the centroid. If the section is doubly symmetric, this automatically
eliminates torsion because the centroid and the shear centre coincide.
In most cases, the load transfer through the connections of the members applying
the loads may be regarded as ensuring that these loads are effectively applied
through the shear centre. This is also generally true of loadsf romf loor slabs
supported on the top flange of beams, even for channel sections.
Designing to transferl oads by means of torsions hould be avoidedw herever
possible, as it is not usually an efficient method of resisting loads. When this is not
possible, care should be taken to arrange framing so as to minimise any torsion.
Attentiotnd o e tailp ,a rticularly when considering how loads are actually
transferred to members,c an minimise or even eliminate many potentiald ifficulties
associated with torsional effects.
Where significant torsional eccentricity is unavoidable, consideration should be
given to the use of box girders, comprising either a lattice girder fully triangulated
on all faces or hollow rolled or plated sections.
The assumptions made when using a computer program to analyse a grillage or
three-dimensional framework should also be consideredIf. the members andj oints
are assumed to have torsional resistance, then torsional moments will be included
in the output. In order to maintain equilibrium with the applied loads, these must
then be taken into accounitn designing the joints and the members. If, on the other
hand, the members and joints are assumed not to have torsional resistance, no
torsional moments will arise and the remaining moments and forces will be in
equilibrium with the applied loads. In most cases this approach will be the more
practical. However, this assumption should not be used for fatigue analysis.
The abovei s an exampleo f a broader principle, which is valid due to thed uctility of
steelwork. Unless it is necessary to utilise the torsional resistance of a member, it is
not necessary to take account of it.
As always, the details of the joints must be made consistent with the assumptions
made in the analysis.
1.2 Scope of the publication
This publication is concerned with the minority of cases where the loadi s eccentric
to the shear centre. Whilst it is important to recognise and deal with such cases
when they occur, it is also important not to apply its methods where they are not
necessary.
When loading is eccentric with respect to the shear centre, the response of the
member may conveniently be examined by separating the loading into bending and
torsional components.
Bending stresses and deflections can be obtainedi n the usual manner by assuming
that the loadsa ct through the shear centre and resolving the forces into components
parallel to thep rincipal axes. Torsional stressesa nd deformations can be calculated
for standard cases using the Tables (Section 5 ) and Graphs (Section 6) in this
publication. For non-standard cases and for determination of the torsional effects
at other than the critical positions, equations have been provided in Appendix B.
The user may then choose between hand or computerised methods of calculation.
1
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P057: Design of Members Subject to Combined Bending and Torsion
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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.(') Membersw hich are curved on plan or which contain
particularly slender plate elements, e.g.c old formed sections, are not consideredin
this publication. Detailed guidance on these topics is given in References 2 to 5 .
Examples in Section 4 have been provided to illustrate the use of the Tables
(Section 5) and Graphs (Section 6) for standard cases.
1.3 Choice of member
The initial choice of member in design situations not affectedb y torsion tends to be
governed by the proportionso f axial load to bending moment and the unrestrained
length of the section. For members predominantly subject to bendinga,n I section
such as a universal beam will produce an efficient design. Similarly for members
subject to axial loading, a universal columnH section is a reasonable choice.W hen
the unrestrained length of the member is high, hollow sections can be
advantageous.
Torsional loading also has a significant influence on thinei tial choice of section for
maximum efficiency. For reasons which are explained later in this publication,
I shaped sections are particularly poor at resisting torsion while tubular sections
can be very effective. Although H sections are better at resisting torsion than
I sections, they are still a poor choice compared with a tubular member.
Torsional effects should, therefore, be taken into account early in the design
process when the type of member to beu sed is under consideration and not left to
the final stages when perhaps an inappropriate type of member has already been
selected. Not only may lighter sections result, but the design time will also be
reduced.
A distinction is made in this publication between open sections such as I and
channel section shapes which are poor at resisting torsion and closed sections such
as tubular members which are more effective (Figure 1.1).
OpenC sloesce tido ns sect ions
Figure 1.1 Choice of section
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P057: Design of Members Subject to Combined Bending and Torsion
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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 Shearc entre
The shear centre of a cross section lies on the longitudinal axis about which the
section would twist if torsion acts on the section. If the resultant force acts through
the shear centre, no twist will occur and the torsional stresses will be zero.
The shear centre and the centroid are not necessarily coincident. However, in a
rolled I or H section, which is symmetrical about both principal axes, the shear
centre, S, coincides with the centroid, c (Figure 2.la). This is also true for sections
which are point symmetric such as zed sections (Figure 2. lb).
For a channel section there is a symmetry about the x-x axis only and, therefore,
the shear centre lies on the x-x axis but not the y-y axis (Figure 2 .1~)C. onversely,
for sections which have symmetry about the y-y axis only, the shear centrel ies on
the y-y axis but is eccentric to thex -x axis (Figure 2.ld). When the channels ection
is asymmetric the shear centre is eccentric to both axes (Figure 2.le).
Methods of calculating the position of the shear centre of a cross section are given
in Appendix A. Special cases such as angles and tees where the centrel ines of the
elements intersect at a single point have the shear centre located at that point
(Figure 2.lf).
2.1.2 Torsionarl esistance
The total resistance of a member to torsionalo ading is composed of the sum of two
components known as ‘uniform torsion’ and ‘warping torsion’.
In some cases only uniform torsion occurs. When warping is included in the
torsional resistance, the member is in a state of ‘non-uniform torsion’. Uniform
torsion is also referred to as ‘pure’ or ‘St Venant’ torsion.
When uniform torsion occurs, the rate of change of the angle of twist is constant
along the member (Figure 2.2a).
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