-I

i

StructuralAnalysis

DAVID BROHNPhD, CEng, MIStructE

Princibal Lecturer in Structural Engineering,Bristol Po\techruc

Foreword by Sir Ove AruP

Understanding

GRANADALondon Tbronto SYdneY NewYork

Contents

PrefaceAchnou ledgements

Part I

The analysis of statically determinate structuresStatical indeterminacyThe qualitative analysis of beamsThe qualitative analysis of frames

Part II

5 The theorems of virtual work6 The flexibility method7 The stiffness method - frames8 The stiffness method grids9 Moment distribution

1O Plastic analysis of plane Irames11 The yield line analysis ofreinforced concrete slabs12 Influence lines

Afpendix: Solutions to practice broblems

Index

I234

I

3223857

73

9711413714917820222r

232

282

Foreword

A force is not just a straight line with an arrow head at one end. That is just aconvenient abstraction or shorthand for what in real life tums out to be a bundleof particles under stress and strain, always changing and moving under theslightest provocation from changing circumstances. The theory of structuresand, in fact, our whole scientific apparatus is founded on such abstractions.They have enabled us to impose some order on tie chaos with which we arefaced when we look at the unending and overwhelrning wonders of naturewhich far exceed our powers of comprehension. We have even found tbat, if weassume that this imaginary world of science is a true pictue ofreality and actaccordingly, we can influence and change the world we live in to such an extenttlnt we can abolish want and drudgery and, in fact, do almost anything we like,hcluding destroying the planet we depend on, together with its fauna and flora,in a few weeks - if we only could agree where to start.

This whole mechanistic world-picture, the Cartesian or Newtonian world ofscience, is under heaqy strain now; we 6nd in all disciplines that it does notwork any more, nature simply does not collaborate. I have no time to elaborateon tlnt now and there are, anyhow, thousands ofbooks and parnphlets writtenabout that theime. But how does ttris affect the theory of structures and then-hole business of structural engineering? Our structures are all the timegetting better, bigger, lighter and safer, our machines are more efficient,smoother, using less energy. We are all the time learning to do more with less,so what is wrong with tllat?

As David Brohn points out in his book, there is a critically important stage tobe reached before we can even apply our numerical analysis to a structure,namel), that we must have a structure to apply it to. When we have that ouranzlysis will tell us whetler the structure is capable of doing what it is supposedto do. Further, that the skills required to choose this preliminary structure areof an entirely different nature from those we may gradually attain when we havemastered our structural techniques.

Brohn confines himself to suggesting tlnt tie basis of these skills is therecognition of the relationship between tlrc load and the resulting behariour ofthe structure, in other words that we gain an intuitive understanding of ho'r' astructure will behave under load. This will. when we have acsuired Ore

necessary experienc, enabb us to choG the at least approximately rightstructure for a given task just by looking at its shape and proportions on adrawing. This is of course most important, for the increasing use of computershas to a great extent killed tlis understanding which is so essentia.l for rescuingthe art of structura.l design.

The computer has come to stay, we must Live with it, and this book teachesus how to do so whilst remaining masters ofthe proceedings. We must be ableto check that the output from the computer is correct, and where and when touse it, and what its limitations are. If this book did nothing else it would still bethe most important contribution to structual design which has appeared for along time and should be compulsive reading for anyone interested in thesubject. I have been extremely worried by the fact that gifted graduates fromour universities enter the profession with the idea that it is below their dignityto put pen to paper - the computer does it all. Here we have the necessaryremedy for such conceit, and it is high time. Unfortunately I have neither thetime nor the knowledge to do justice to the achievements of Brohn's book, Ihave only a few hours left for the printers' deadline, which is entirely my ownfault. But he has seemingly gone through every known method of structuralanalysis and shows by clear diagrams and explanations how the structure isaffected by the loading, thus at each stage giving the reader this essentialunderstanding of structural behaviour. I hope this book will give rise to a livelydiscussion.

I cannot resist adding a comrnent of my own. Whilst recognising theimportance of what Brohn has done, I do not tlink he has gone far enough.Understanding of structural behaviour is very necessary, but are there notmany more things that are equally or even more necessary?

Every structural designer of repute has declared that structural design is anart as well as just an application of science and technique to a given problem.You could also put it the other way round and say that only if it is a work of art aswell will it be adrnired and add to the reputation of the designer. It isunfortunately impossible to define what art implies, but it has in any casenothing to do with numerical analysis. There are many other matters toconsider as well. The whole purpose of a structural design is to help us to makethe things we need, or fancy we need, orjust fancy. So we must make it veryclear to outselves what we want to achieve with our design, which willobviously affect its shape, the materials we use and all kinds ofother things.If we want to build something is this the right place for it, could not our purposebe better achieved in a different way altogether? Only when we have sorted outall these matters to ou.r and our clients' satisfaction will a structural ana.lysisbecome relevant. Obviously, wbat I would call dzsigz is much more importantthar structural analysis, for that determines what we are going to get for ourefforts. And moreover raial we decide to do is much more important than howto do it, and that opens tJre sluke va.lve for a whole flood of questions, social,political, ethical which tlreaten us all with confusion, or worse, because weare not able to agree on wbat to do-

How to live in peace sitl or:r: neighbours on this pJanet witiout destroying itis the ultimate and now pressiug problem and I wish I knew the answer.

Ove Arun

Preface

This book is aimed at the identi f icat ion oJ the fundamental princiPles of

structural analysis together with the develoPment oI a sound understanding

of structural behaviour. This combination leads to the abi l i ty to arr ive at

a numerical solut ion.

Using a series of structural diagrams as a visual lanSuage ol

structural behaviour that can be understood with the minimum oJ textual

comments, the book aims to develop a qual i tat ive understanding of the

response of the structure to load. I t is ideal ly suited to under8raduates

studying indeterminate framed structures as Part of a core course in civi l

or structural engineerinS' but i t is also suitable, because of i ts

quali tat ive approach, for students of architecture and bui lding technology.

The book is in two parts. Part I ' the f irst lour chapters, deals with

the development ol qual i tat ive ski i ls; that is ' the abi l i ty to Produce a

non-numerical solut ion to the loaded l ine-dia8ram ol a structure. I t is

considered that the abi l i ty to arr ive at the quali tat ive solut ion to framed

structures is a signif icantly imlortant component of the overal l

understanding of structural behaviour.Part I I deals with current methods of structural analysis using the

diagrammatic format to which the student has become accustomed.

The need lor the developrrent of qual i tat ive ski l ls increases with the

increasing use of the computer in design off ices. In the near future, the

computer wi l l replace the majori ty ol analysis and structural desiSncalculat ions. Unfortunately, this wi l l also have the elfect of el iminating

much of the experience and consequent understanding gained by the student

and trainee engineer.This work explains how that understanding is develoPed along with

current analyt ical procedures, PreParing the student for the design olf i 'e

where the computer ls rne source of virtual ly al l numerical design data'

Understandinq structural Analgsis is an inteSrated approach to the

teaching and learning of the PrinciPles ol structural analysis, ol which

this textbook is a major part. The ideas embodied in this book are also

avai lable in an audio/visual series of sel{- learning programmes ol the same

name.

The audio/visual programmes are backed by a suite ol micro-computer

programs which have been used to produce the numerical and SraPhical

solut ions to the Practice problems, included in this text '

The audio/visual programmes and comPuter-aided learning software are

avai lable from:

osE Ltd

197 Botley Road

OXFORDox2 oHE, UKTeL 0865 726625

who should be contacted direct and not through Granada Publishing'

Acknowledgements

The research project upon which this book is based has extended over aperiod ol ten years. In that t ime, many fr iends and col leagues havecontr ibuted to the development of my ideas of the way in which students canbe encouraged to reach a better understanding of structural behaviour.

Bristol Polytechnic has provided both t ime and resources and my Head ofthe Department, Dr Matthew Cusack, has been part icularly support ive.

These ideas would have been stillborn without the continuous slrpport,interest and encouragement of Peter Dunican, senior partner of the Ove ArupPartnership. Many other engineers in that remarkable organisation havehelped me with their advice and constructive cri t icism.

Perhaps the most successful period for the development and testing ofthe quali tat ive approach as a basis for the explanation 01 theories andmethods ol analysis was the year I spent with the Department of Civi l andStructural Engineering at Hong Kong Polytechnic. I owe much to discussionswith Dr Kwan Lai and Dr Norris Hickerson. but most of al l to the resDonse olthe exceptional students.

However, i t has been the extensive and part icularly fruitJulcol laboration with Professor Peter Morice oJ the Department of CiviLEngineering at Southampton University which has led to many of the specif icexplanations and visual sequences in the early part ol the book.

I am indebted to al l of them.

1 The Analvsis of StaticallvDeterminate Structures

The subject of this book is the behaviour and analysis of stat ical lyindeterminate structures. However, this f i rst chapter reviews thebehaviour of deterninate structures, a thorough understanding of whichis essential before the topic of indeterminacy can be tackled. The textassumes a basic knowled8e of mechanics including an understandin8 ofthe principles of overal l equi l ibr ium, bending moments, shear and axialforces.

It is possible to analyse determinate structures by consideration ofequi l ibr ium

- in general terms, the application ol force and moment

eouarions v 1 O. d = 0 and l t = 0.With most real structures, this is not possible as the presence ol

redundant members (secondary load paths) makes i t necessary to considerrelat ive member delormation beJore a solut ion of the structure can be

attained. The number of unknowns which cannot be lound Jrom equil ibr iumconsiderations is known as the degree oJ stat ical indeterminacy.

The design oJ engineering structures usually starts from a need tosostain loads. Init ial ly though, i t requires an understanding ol the way inwhich a proposed system of members can provide the required support, and

how it wi l l deform.It is, however, clear that an understandin8 oi the behaviour of

stat ical ly indeterninate systems is based upon a thorou8h appreciat ion

cf deterr irrate systems.

This chapter develops the relat ionship between load and delormation for

a range of structures which are amenable to solut ion by the application of

equi l ibr ium alone.Once we have analysed the behaviour of the proposed structures \re are

then able to start an approPriate process of numerical analysis to l lni cJ:

UN'A,RSIA';]:;G SIRUCTURAL ENSIYSJS

how much of each of the various parameters is involved, Jor example, thevalues ol the loads carr ied by each member and, as a consequence, the srzeeach wil l have to be to carry i ts load saJely. \ve can go on to f ind thevalues ol deformations which wil l result Jrom the loadins.

TI]E ANALYSIS AF STF.IICALLY DETERMINATE STRUCTURES

Thus we see that structural analysis must have two.omponents. The f irst is a qual i tat ive understanding and::e second the numerical procedures. I t must bejderstood that qual i tat ive analysis is not in any way a,i : icsri tute lor numerical analysis but should be regarded3: a necessary complement to i t so that the twoallaoaches consti tute a complete whole giving an

-..erstanding of, and an abi l i ty to evaluate, the::: icrural perlormance.

:-e oasic principles of our structural analysis l ie in the:a i

-- f srat ics. Mosr srrucrures are required to be:-:---3 rn a stat ic state. This is not to say that we:_g-::ci analyse dynamic behaviour such as may be caused:.

--: :rhquakes, wind gusts or moving loads but ini t ial ly,a:

-e3s:. we shal l concern ourselves with stat ics. The::aie: c.ane demonstrates the kinds of equi l ibr ium::r ' :- : :cns which we have to satisJy. The vert ical_.. : : : :an ar the ground must balance the total downwards::. :?:: counterbalance and load.

:e:..c-). u'e can see that wind Jorces will tend to make:-Je r _c-e crane structure sl ide sideways and this too,1-r--

- .esistedj by horizontal support lorces at the

a.- :1:s \r 'e cal l ho.izontal equi l ibr ium.

---:_-1. -: :S Clear that the counterbalance weight cannot!c--i=-L= 1- .oldirions of loading on the jib so that any!rE-:i---::ce $ill tend to topple the crane. If we addE :rs :-e : .Cit ional toppl ing ei lect oJ the wind Jorcesr: lei =E:: l :e base oJ the crane wil l have to provide_E\s=!-,= :c :hese out-ol-balance toppling moments.-r1s

-r =* iioment equilibrium.

VELTIC'AL EoUILIARIUM

Fr't^4_-.-.':'

--------)+

I1ARI: ONTAL @UILI 5P,JU M

\kM

QUALiTATIVE

- , t tu

MA- 5reo-3

NLIA,1ER-tc.\L

/\^oMENT Eeut!.tE;ui/At

UN D' ?S' IlE DJ NC STRU CTURAL AN ALYS I S

In our three-dimensional world we can express these

equil ibr ium requirements in the fol lowing way. First ly '

we must ensure that in each of three direct ionst at

an8les to each other' which we wil l label x' Y and z'

the resultant oJ al l forces acting on the structure must

be zero. In other words, reactions must balance loads.

Secondly, as we have seen in the case oI the toppl ingeifect, the tendency for the structure to rotate about

any of these three axes must be resisted. We say that

the resultant moment about each of the three axes x,

Y and z must also be zero. Thus the moment of

reactions must balance the moment oJ loads. This

gives us al l together six condit ions of equi l ibr ium.

In much of the fol lowing explanation of behaviour, and in

many real l i fe engineering situations, we Jind i t ispossible to be sure that the lorces in the direct ion v

are zero and that there are no moments of lorces about

the x and z axes. I f , indeed' this is the case, then our

problem can be reduced to the consideration of three

condit ions of equi l ibr ium only. Such simpli f icat ion is

described as a plane problem because al l forces l ie in

one plane,

we shaltr find it convenient to label the lorces in the .r

direct ion with symbol d' to denote horizontal, and

those in the z direct ion v, to denote vert ical. Also

we shal l use the symbol l , for moments in the plane,

about the Y axis.

6.

7.

8. Vith these symbols we can write down the threeof equil ibrium - all the horizontal lorces must sum tozero, all the vertical forces must sum to zero and the sumoI the moments must also be zero.

tt{

Y =a VarttLt aluilihri.t

H'o Hon2o,{e q.fb/ilrf"1

M.o iqon',t,f ?uilibrt-t

TEE ANALYSIS AF ::;'':'."LY DETERMINATE STRUCTI]RES

:-

\ow it wi l l be remembered that a pure moment,=l led a couple, can be represented by two equal:^d opposire paral le l Jorces ar a dlsrance apart ., . lhis diagram we have the forces ar a olslance: grvlng a moment tv : F x d.

-3: us now consider the moment of this couple::cur two points, A and B in the plane.:: :rsidering I irst the point A we see that as---,= Cownward force tr passes through A i t wi i l-Eie no moment about A and al l that remains::: :e anticlockwise moment ol the upward

::.ae = at a distance d. The moment aboutr

-: YA = a x d anticlockwise. Considering:_e:oinl B the upward forces pass through:*e:olnr and the only moment is caused by:*e :tr \r 'n*ard force r at distance d. This:--:.. i ves an anticlockwise moment

-:: - .

:o,* consider rhe moment about a point Ca-=:_,ae: from the l ine oJ action oJ the upward::r=- E'e can also f ind the total moment of our:r:- = about this point. For the downward force:

=-e -5 an arr iclockwtse momenr due to rhe

]:1:. : : : i . j plus e, and for the upward force F a:-\ :cra.,= :roment due to the lever arm e. The-as*-:-: =omenr 4 C is the force F mult ipl ied byie :F_-:a:;oa j . st i l l , of course, anticlockwise.I_': r : :aie shovn is one oJ the most powerfuli ,= : : , - : - . . : : - : .a l aralysis. namely that . as a couple],as :- =-e :noment about any point in the plane,].e ?:r*--:r :r i : . cf a structure wil l require that the'r:ra

-,:r-3::: cf all Jorces must be zero about any!t!r-: - ---e :iare.

-r* J _' :a :r:rsiCer an actual structure. The beam ABCD

s 3,Trc:r i,5!-!-!ed at A and B and loaded at C. We shallE! '*r=:-E .: :ec-!s of any self-weight and only study those:j.tc :: :-E ::a-ied load t/.

t f,"^-( &".1-

-

.sr , o)

F

e.

II

' l.) rl{ '+:

- :^r, / t,fve ,vs

Metlta^6^ ^a

tiarz.^iaL !,ul,eti'|'

40. I f there are less than three external restraints, nomatter how many internal restraints, the structurefai l . In this example, with horizontal iol ler releaseiboth supports, the structure wil l ro1l, i .e. this is astate of unstable equi l ibr ium. Beware of the soDh:the notion that the structure could be stable underexactly vert ical load. Indeterminacy or stabi l i tv is aproperty oJ the structure rot the loading

STATICAL ]NDE"': i . : : ; : :-- :arvert the whole structure into a ' Iree' we must.

- - : F r^F f . :m- RCnF A , , , r wi l l .e lea5e rhree

-

.

- ' \ rn rnrernal for .es. ,F. "F d1d Yf. Tqu( lhe^f

in. la+arm in:-w iq.

: : : le supports are pinned and the structure cannot be_::-:3i ro a tree because of the absence oJ a Jul ly f ixed

-. : : _ ' : . :he.uppor l ( ma) be reduced to a 5tat i .a l ly

:: : : : . : ,nate condit ion. This two-bay frame is pinned at

_-:: : jcture may be reduced to a stat ical ly determinate::- : , lhe removal of the vert ical and horizontal

-a:rr. is at H and the horizontal restraint at C. The:-:-:r3:-re ABCEDG is then stat ical ly determinate, with-:: : : :a.minate canti lever frarne EFH f ixed to i t . The,:- : , :e is J t imes indeterminate because i t is necessary-: _: : :5e rhree reactions to reduce the structure to a

=: :1- ' . determinate f orm.

- .

- - : :a.e! 'Jrames may be solved in a s imi lar way by-- . - : . , - r ie\ n l \ , . i r i ( a l lv dc crminale sub-frames.

' _--_-:rorev l rame has a hin8e at D and has pinned

do|e( utgko tr.z 3 x

iitu

FHIvH

STA?I CAL I N DETERH!N AC!

\s an alternative means of reducing the structure to aieterminate form, a rol ler release has replaced the pin'oint at D. This part icular release should be noted since:i is frequently employed in analyt ical procedures. Thus.o horizontal reaction can be taken at support D.lecause the horizontal forces must be in equi l ibr ium,::e horizontal reaction at A wil l be equal to the:orrzontal component of the load.

:- is last example of reducing the structure to a:ererminate form is the axial release in the column. Ihe::rding moment capacity at this release is unaffected.: ich releases are rarely employed either in practice or in:-1 r" t i cal proced ur es,

r- : :oment reaction at A has been introduced to the:,:r ial frame turning i t into a ful ly Jixed support, and:e structure is now 2 t imes indeterminate,

l [ : Tra].eCuce this structure to a stat ical ly determinate-r":* : : ' . :eleasing the two reactions at D, yD and dD,T'!e .5 : :a:r icularly important release system because: 5 f ,1:-:est to recognise guali tat ively.

2 x sGtt

28 . - ' : ; : .=: I ' IDING STRUCTURAL 3,VAIYS15

22. To summarise then, indeterminacy may be the resul:

2.

3.

l .

The last two are classif ied as internal indeterminac\.

These three terms are used in other texts. They ha\ '

21, Pin-jointed structures respond to the condit ion o{'ndeterminacy In rhe 5ame wav. The presence ofaddit ional members, here members PQ and PR, wil lthe distr ibution of forces in the remaining members.those members have a large cross-sectional area the_force in them wil l be correspondingly large,

addit ional external reactions,

internal r igidity,

the addit ion of members,

A STAT I C'A LLYlNOLT L P-Ir1IN ATL

ST R-u oT Lt E E-js Lho samo es

A RE0./^/DAV7 STLUoTqzEIs Eho sahto as

A, HlPE'.slATto 37 rul ' fuRE-

same meaning. The term indeterminacy wil l be usec r-

re\L and r .e dcgree oI inder.rminacy referred to as -

t imes indeterminate.

24. l t is important to appreciate that structures may be

converted back to a stat ical ly determinate condit ion

lrom the stat icai ly indeterminate, by a variety oJ

strategies. This technique is important because i tmodels the declsions made by the structural desiSnerwhen he reduces the highly complex, real structure Ia =simpler form for the purposes of analysis. The portal

frame is I t imes indeterminate because i t has four

external reactions. The frame has been released to a

stat ical ly determinate structure by the introduction c:

hinge in the beam BC.

9xtraJqf Prtr

h3t4tv

^Ltnb.g

A DQ

THE AN ALYSI S AF ST ATi CALLY DETERMIN AT E . TRUC?UR'S

[e =: _c-- :econnect the two parts of the frame and welf i0uE f :: . .ecr deJlected shape of our unsymmetrical iytu't'r :_r:=-trtnned portal Jrame- The horizontalfifi{F:c.:-::. =: :re top of the frame, which has been caused!!$ . i i [aa r-: : : \ 'ert ica] component only, is cal ied ther. i{rr ! ! : : : :me. I t always arises in a frame-typeit?irrrr: r -F_-: 3ither the loading is appiiedrla,rr Fi:nE:: : :- \ or the structure i tself is unsymmetrical.b .s D+:-r'. :-- Je noted that the loaded point oJ thisiGr/r-jlr: :e:-:ats both in the direction of the load and"r rF- 1_E-:: : , . !r . that is, horizontal ly.

i ] s $r-:* _::e:al i !g that in this example we have["rio!rLur:p: : - _=)er of new ideas. Firstly, that bendinginlt[Mfimcr]= :|: a_i'::. \'ertical members due to horizontal!0l l ]Er jn : : .---=. \ '3\ ' to that in which vert ical forces@lG :Fl[r ig -. ] : :renls in horizontal members. Secondly,f i l l ]@ r_ T1L5 : -:-e::3f ie analysis we are assuming thatd iFF e-: _r: _:: : . : :ans of the members in the axialdh tr-r i ln r--- i . . ; lC place secondary restraints upon thero.r_x*JE :: : : . :3:. Doints in the structure. Thirdly,[h

-

:!{ fe:-: :r _eei nol occur only in the direction ofrfr @]_g- Le :-- i :rree-part solut ion of the three

-a: : . : - - . i : ion.on\tsts o tnree par l r :1ic aJ:_--:_:: o: bending rnoments, the support

d-. : : - - : : i : - : : :ed .hape. \ ou -nay nor' i r , rc.=c.:

-a ! :a.1 \\ i th any one of the three

r@ rnE4 _e,: : : : : i fve f rom one component of the' is. iJf{ :_.

- . t : : : - a i l three component parts are

,J_,,_"1-:_*

, -R + ++ +,*

_Ll l. t r= 'n*}+ 11{{

3rd ::rs-.:3.: rArlh each olher. A typicali,f,[ r-*:r :::,::: a compuler analysis i5 given,

ffirE r j'r flire ilE!-(:s aI A and DE 3 arc:- --.e.e:c.e there is no

33

8li&A MG :fl! =-'C:Je 1:,: CcllaPse

.frr:r. __: :E 3 :-e-_:::isn because

E fiE I:a:-- -_:s: )e slr,all compared

:rt.TTe-L_. 3EF calid susrain a:rE rre-Ee 3-F .re.e a cable-

E I arE --

:_,- a ,!-aC :a member DE1cle1=.

--

-=

C:aCUmSIanCeS the.Er.r :I: airs.:::oei :,:e

-.: :i basic rules of

{!as:t: =L*-L-Jas. r:ich is ihat al]

Pscrao Act teist 't

!tl|:-sr:r_-s- -::lal

ther do not affect

d Ttc i=--r--r:. T:-ls. :n the context ofeirE:!r: s=-l:-.r-.::i5 frame is a

=_rca F sa,t=a J! linear analvsis. It is

:_=r_-= -s :eccgnised as statically

![ :'E :-ree :ieanDers rn a trlansle.tffE :_r3: fr:_n1r n arial forces which can be

Y< ^f

an' , i l ih. i , m

I . - ,

:enain determinate no matter how::. adCed. The system can be extended

FAs Triar,alo'. Ba

stat r

tal

tB,

LL

STATICAL I N DET'R!4; I; ;'!

The portal frame ABCD with pinned supports at A and D

also has hinge releases at B and C. Therefore there is no.estraint against sway and the structure wil l col laPsesrdeways.

This frame does not appear to be a mechanism because'rnder certain condit ions the member BEF could sustain a

icad. IJ, for example, the member BEF were a cable,

)roperly anchored at B and F, then a load in member DE.ould be supported. However, in these circumstances the

siructure would have to disobey one of the basic rules of

:he behaviour of elast ic structures' which is that al l

Ceflect jons due to the loading must be small compared$ i th the structural dimensions, so that they do not affectaie Seometry oJ the structure. Ihus, in the context of

: ire analysis of elast ic structures, this frame is a

:nechanism and cannot be solved by l inear analysis. I t is

ao&rmr^ato Lvllss

a ctzruisfz, s uP f ores,

I x na.

Lr,./ts 2 x tM,s1aa,t4t^t-e'intuna(1

- -.

: t:S:.ii!'rl N G STRUCTURAL 4,VeIvS15

46,

47,

,?: number of members

3 three equat 'on\ of equi l rbr ium.

We wil l look l i rst at the Warren Sirder truss, which is

stat ical ly determinate internal ly as i t consists of a

series of tr iangular pin-jointed frames. The truss hasstat ical ly determinate external reactions.

If we now add an extra vert ical reaction at the centre

t l "en !he. l ru.Lure rs

ul.

S TATI CAL I N DE T E RM I Ii P.CY

: f \ _' is greater than m + 3 the structure is a

.r.e:r^Eaism.

: : r ' is less than ,n + I then the structure isfrE:eaminate.

[- 1.=-rna the structure shown. The ca]culat ior1 shows:E::r is stat ical ly determinate, but i t should be clear

=: _ris structure wil l col laPse because the l i rst panel

Jemonsrrate\ that rhe4!- grdRU, 'd, uIdL",5. r t ' rJ I:r-En.i iat-Lve solut ion must be judged quali tat ively.

i i - :a.ee-dimensional indeterminacy may be dealt with in:-e same way, although i t should be appreciated that:: :3s: bui ldin8 structures are reduced to a series of l inked:- lo-dimensional Jrames Joa ease of analysis.

T.ara i ra

36 ST RUCTU RAL A,VE.LYS"TS

Consequently, the release of a ful ly f ixed support lreessix reactions. This bent frame, ful ly f ixed at A ard D

is 6 t imes stat ical ly indeterminate. The ' tree' method of

idenr iJyin8 the degree ol indeterminacy ;s even moreefJective with three-dimensional structures.

, , ! , -a i : l ; i iDrNG

:4. i7. 5E. :

-

,4. The effect of the introduction of a pinned support at D,previously ful ly f ixed, is to release three stressresultants, the moments about each axis. This frame is

therefore 3 t imes indeterminate. The structure could be

released to a stat ical ly determinatertreerby the removal

ol the three lorce reactions at D.

If we now return to the frame Jul ly Jixed at A and D

an internal hinge would release three internal stressresultants, the bending moments about each axis. This

frame is now 3 t imes indeterminate.. ,at>'Ezl-Y

h'^9t wtl 3rvr.r^o'n Ar vaaasa.

STATICAL

&Determine the degree oJ indeterminacy oJeach of the structures shown dist inguishingbetween rrte.na-7 and exter.ral condit ions.

r 3 The Qualitative Analysis ofBeams

ln the next two chapters, you wil l study an approach to the analysis of

structures which is i ikely to be unfamil iar to you' The tradit ional

approach to the study of structural analysis has been based, almost

exclusively, on quanti tat ive ( i .e. numerical) methods' on one hand' this is

entirely logical. The structural designer does, eventual ly, need to Put

values to loads and dimensions and determine the numerical value of

reactions, bending moments and so on'

There is, however, a cri t ical ly imPortant stage before that numerical

analysis; the prel iminary analysis required to size the structural members'

The nature of the analysis oJ stat ical ly indeterminate structures is such

that the desiSner must know the size ol al l the structural members befor:e

an analysis is carr ied oL1t. Consequently, i t should be clear that the

detai led, numerical analysis of the strLlcture is a check on that Prel iminary

analysis.This places great imPortance on that stage ol the desiSn procedure' I f

the structure is incorrectly sized' repeated analysis wil l need to be

carried out. This is true re8ardless of whether the analysis is being

carried out bY hand or bY comPuter.

I t is general ly assumed that the ski l ls required for this prel iminar.r

analysis are the inevitable consequence oI studying numerical techniques'

The results of extensive research into the development oJ an understandlng

oJ structural behaviour in undergraduates and trainee engineers suggest

very stronSly that this is not the case.

That prel iminary analysis requires a quite di l lerent set of ski l ls which

are referred to here as 'qual i tat ive'. Essential ly non-numerical, the basrs

of these ski l ls is the recognit ion of the relat ionship between the load and

the result ing behaviour of the structure' In simple strLlctures' that is the

THE QUALITATIVI ;,"'Z:7JI.i OF AEA&S

relat ionship between the load, deflected shaPe and the result ing reactions.

These ski l ls a.e dist inct ly dif ferent from quanti tat ive ski l ls; they

rely on sets of coherent dia8rams rather than mathematical modelsr for

example. However, the most obvious difJerence between the quali tat ive and

the quanti tat ive approach to the analysis of structures is that there is no

obvious sequence for the steps in the quali tat ive approach.

once a part icular numerical ly based analyt ical technique has been

learned, then the sequence of the solut ion wil l almost always be the same.

However, faced with a qual i tat ive analysis i t is not aPparent from which

point the solut ion wil l emerge most elfect ively. I t may be best to start

with the bending moment diagram in one case or the deflected shape in

another.In many structural problems it is necessary to start with al l parts of

the ful l solut ion at the same t ime, which is potential ly very confusing.

Students studying this approach say that they understand each of the steps

in the examples then' as soon as they try one of the practice problems they

are stuck. This suESests that the teaching method is ' in some way,

deficient. This is not necessari ly so. The exPlanation is that qual i tat ive

problems are inherentlq diJf icult . This new aPproach requires a shif t of

att i tude and this takes t ime and practrce.

However, you wil t discover as you begin to SrasP this approach, that you

find a signif icantly increased confidence in your overal l understanding oJ

structural behaviour which wil l contr ibute to your understanding of problems

of structural desiSn and the methods of analysis studied in later chapters.

Perhaps the Sreatest value of this qual i tat ive approach wil l only be

apparent to you once you start your of l ice training. The use of computers

is increasing rapidly. How can you check that the output is correct? The

checking system must be independent and robust ' You wil l discover that a

sound understanding ol structural behaviour wil l play a siSnif icant part in

the overal l checking procedures which must be constructed to ensure the

correct use oJ the comPuter and the el imination of data errors.

)40 UN DERSTANDING STRUCTURAL EJVEiYSTS

frFr

stvcin

l . V/e wil l start by restat inS the assumptions \rhich underl iethe elast ic analysis of structures. The l irst is that,

for al l loading cases, the structure is within the elast icrange of material behaviour, Srress is direc! lyproport ional to strain and deflect ions direct lyproport ional to load.

The second assumption is that deformations due to theloadin8 do not create secondary bending moments. Thecanti lever bent ABC is subject to two point loads t{rand ,{2. Here the load ,{2 wil l cause bending in thecanti lever BC which wil l result in a horizontaldeflect ion at B, ABh. The secondary moment t /1 x ABhis ignored. In the Engineer's Theory of Bending i t isassumed that deformations due to loading do not.esultin signi l icant changes in the structural geometry.

In the quall tat ive analysis of structures and most

manual numerical analysis, axial loading, which wouldresult in axial strain in the member, is ignored in thedetermination oJ bending moments. The lrame, ABC isloaded in the vert ical direct ion at B.

4. AlthouSh the load i/ would cause the column AB to shorten,i t is assumed in the quali tat ive analysis that there is nosecondary moment in beam BC. General ly, in the elast icanalysis of structures, deJormations are considered to besmall compared with the dimensions of the structure sothat the equations of equilibririm ol the deformedstructure are consistent with those ol the undeformedstructure. I t should be noted however, that thisdeformation Agu :s included in the computer analysis ojslrucrureS.

',

5,

THE QUALrTATil,il -:-','3:lsj.9 AF EEA|"LS

It is important to restate the diagrammatic convention for

the support condit ions and the result ing reactions. The

rol ler support has one reaction, normal to the Cirection

of movement. The pinned support has two lorce reactions

and the lul ly Jixed suppott t . ,vo force and one moment

reaction.

6. The principle of superposit ion is assumed to aPPly as the

structures are within the elast ic range. We wil l

i l lustrate this principle with this example. The load t/

on the two-sPan beam ABC Produces a hogging moment

over support B with a downward reaction yC, to

balance this bending moment at C.

. I f we remove the vert ical reaction at C, the beam becomes

a simply supported beam with canti lever' I f we now aPPly

the reaction at C, vC as a load, the combination oJ

these two condit ions wil t Produce the solut ion to the

original two-sPan beam' This simple example i l lustrates

one ol the most Power{ul desiSn tools in the armoury olthe designer, since it allows the reduction of comPlex

structures into simpler lorms and the examination of the

load ef f ects seParately.

t . The deflect ions may be treated in the same way. The

vert ical deflect ion at C due to the load n must be equal

to the downward deJlection at C due to the lorce

reaction yC since the real deflect ion at C is zero.

4l

Vtr e-'rmrfllnr'--

ffir-i-t

--

\

\

-1f '

\*

tL^. /

"n{--4*1 Ao"

+-

\.u-= N,. Y ' -

FL#-_(q_

-

42 ANALYSIS

This is the f irst example of the quali tat ive analysis ofstructures, the simply supported beam ABC, with acanti lever CD. The ful l solut ion of a qual i tat iveanalysis must always consist oJ:

the deJlected shape,

reactions,

bending moment diagram,

You must become accustomed to the idea that they are ofequal importance and your qual i tat ive analysis wil l not becorrect unti l each of them has been determined and foundto check against the other two,

In the quali tat ive analysis oj beams, the best placeto slart is l ikely to be the deflected shape. We wil lstart the del lected shape by identi fying the pointsthrou8h which we are certajn that the deflected beammust pass, A and C and a downwards deflect ion beneaththe load.

We can now draw the smooth elast ic curve through thesethree polnts. Note that CD is straight. The curve mustbe smooth and with practice a very accufate deflectedshape may be drawn. Try practising the identi f icat ion ofthe deflected shape with a f lexible plast ic ruler,

This simple example of the bending moment diagramii lustrates the need to dist inguish between the structure,which is drawn as the base l ine oJ the bending momentdragram, and the diagram itselJ. The point here is thatthe value of the bending moment over the port ion of thespan CD is zero.

N-B- The bending nonenL diagran js a]wags drawnan ttle bendjng tersjon sjde af the structure.

UNDERSTANDING

lwF--+=+-p

9.

I l .

I0.

THE OUALITAT!Vi })iA'YSIS OF BEAMSThere are certain simple rules which should be borne inmind when carryinB out the quali tat ive analysis,Experience suSgests thar because thi. approach 'sunfamil iar, students tend to become confused, reach asolut ion containing obvious incompatibi l i t ies and are notabie to spot them. Having found a solut ion consist ing ofthe deflected shape, reactions and bending momentdiagram you should take a cool look at each part and seehow it relates to the other, Is there a bending moment atthe support? Unlikely. Is the bending moment at aninternal support ze(o? Very unl ikely. Does the tensionidentiJied by the deflected shape agree with the bendingmoment diagram drawn on the tension side of thestructure? We wil l cal l them 9olde, rules.

Here is the Jirst.

\nd here is another.Bending moments result ing from point loads are

l inear expressions, therefore the bending momentdiagram consists ol straight l ines. ln addit ion thedia8ram can only change direct ion at a load or reaction.

\nd another.That is, where the hinge occurs at the end of a

structure, or at an internal hinge which we wil l studylater in this chapter.

* tf . pavb 6f Ehe,

sEruX,uro rutai3 sEra!'iCafi* loaan3

-

No EE-ND|N9 MoMe^iT

|fe

d

:i . We wil i now carry out a qual i tat ive analysis on thepropped canti lever AB, Ve wil l apply the principle ofsuperposit ion to examine the effect of the ful ly l ixed

support at A, by the notional removal of the momentreact ion at A, MA.

the. loaairp on al\ *ructuyo

44 UNDERSTANDING

:sj+L+"

- F"trrf +^ffiun*"M,c,H

fve rvc

15.

STRUCTURAL ,INAIYSJS

As a simply supported beam there wil l be a clockwiserotat ion at A. The propped canti lever however, wi l l havea zero rotat ion at A and to return to our originalstructure we must apply a antjcTockwise moment at A.

17. I t is now apparent that there wil l be a change ofcurvature in the deflected shape, which is known as thepoint ol contraf lextrre. This provides another point ofreference between the deJlected shape and the bendingmoment diagram because the bending moment is zero atthis point. The member is unstressed at the change ofcurvature.

We can now attempt to draw the bendinB moment

diagram. We know that there wil l be a hogging bendingmoment at the f ixed support at A. This is recognised bythe anticlockwise reaction and the identi f icat ion of

tension on the top of the beam from the del lected shape.

At the other end o{ the beam the upward reaction vBwil l cause a bending moment on the undersjde ol thebeam. The bending moment diagram wil l be a straiShtl ine from B unti l i t meets the l ine of act ion oJ the point

load. The diagram is completed with a strai8ht l ine

between A and the load. This l ine must pass through thepoint of contraf lexure.

Another golden rule.

18.

xU botaina rtowe^,t.'

^ ai^3rois .r*

th c, bato lmo "1; po&.1

f csrfirqf lcxtlro.

t9.

THE QUALITATITa a-',':::.]-: :: ttat4s

24. The next example is the two-span beam ABC. Thedeflected shape must pass through the supports and thedownward deflect ion under the load. Note that astructure wil l alrags deJlect in the dr-rectro, oJ theload provided that i t is the only load on the structure.

21, The smooth curve shows the change of curvature,characterist ic of al l mult i-span beams. The direct ion of

the vert icat reaction at C may be identi f ied by imaginingthat the reaction is removed, The beam would deflect

upwards and the direct ion of the reaction necessary to

bring the beam back to the correct posit ion would beoownwaros.

22, Both vA and vB act upwards and the f irst l ines of

the bending moment diagram may be drawn from a

consideration of the effect of the upward reaction at A

producing tension on the underside oJ span AB and thedownward reaction at C, producing tension on the toP ofspan BC.

The bending moment dia8ram is completed with a

straight l ine between the load and point B. The

coincidence ol the zero moment and the point of

contral lexure may now be reco8nised and checked.

-ffi

46 UNDERS tAl lDtNG Sl RUCTURA, ArtALfSIS

24. The analysis of a three-span beam would proceed in asimilar way.

The Jirst step is to identi fy the points oi certainty onthe deflected shape. The beam must pass over each oJthe supports and deflect below the load, A smooth curvemay then be drawn between these polnts,

The direct ion of the vert ical reactions is identi f ied bytheir notional removal. The reaction at D must actupwards and the reaction at C downwards.

Ihe f irst, most obviolrs points on the bending momentdragram are usually those at each end of the beam. Herethe upward reaction at A and D both produce bendingtension on the underside of spans AB and CDrespectively.

25.

26.

r*

27.

3-

THE QUALITATIVE A:iF.:!s 3 'F

BEAMS

:: complete the bending moment dia8ram you must be

:: le to identi fy the fact that the beam is hogging over__e support at B.

This beam has two points of contraf lexure' in span BC

=.a between point B and the Point load. Note the

:e:-rt i t icat ion of areas ol bending tension on the

::: lected shape and the visual check against the bending

- r:nent dla8ram.

g. -.^:s next example is a three-span beam with a l ixed

;-:Jort at A loaded on the central span with a point load

;. Note that the beam must enter the Jixed support

r-: : zero deformation compared with the unloaded

i:: icrure, Fai lure to draw this clearly is a common

i:r:ce of error in drawin8 the deflected shape.

K-e:ause

the bending moment diagram is drawn on the

:=:s:rn side of the structure a check should alwaus be

-:Ce between the deflected shape and the bending-,. : ient diagram. To Jaci l i tate this check, identi fy the

-eas of bending tension (T)' The reactions are

:-:ermined from the deflected shape by their notional

i?:: toval.

:-e most obvious l ines of the bending moment diagram

-:r now be drawn: the sagging moment at A, the-::ging moments at B and C and the sa8ging bending

-.:nent below the load. Note that i t is known that the

, -e of the bending moment diagram from A wil l move to

::ass the basel ine of the structure because the vert ical

--=:cl ion yA is causing anticlockwise moments which wil l

: :aJce the clockwise moment MA that is moving alon8--.e beam lrom A to B. The bending moment diagram-:\ ' rhen be completed with straight l ines as there are-: orher loads between these identi{ ied points on the:

48 UN DERSTATI DI N G ST RU CTU RAL AdAZYSIS

lTilllfi illl;i]llllilXl

.\a 1Mr--\r

tA 'rM\7

33.

32. Practice problens

A-lr,r 'aqrs complete the lul l solut ion by drawing the

three parts, however confident you feel about any one

oJ them, This is part icularly true of the direct ion ol

the .eactiors . There is a temptation to miss this

step and attempt to construct the bending moment

diagram from the deflected shape.Experience shows this to be most unrel iable and

the intermediate step of Producing the diagram of the

reactions is usually cr i t ical in ensurinB coherence

between each of the three diagrams:

deflected shape,

reactlons,

bending moment diaSrams.

External ly appl ied moments are rare ln real

structures. This example is introduced here to

prepare the student for the use of external moments

in the analyt ical procedures to fol low. You may

find them di l f icult to visual ise and this is probably

because they are unfamil iar.

The moment aPplied to the end of a canti lever

wil l cause a constant bending moment and is one of

the occasions when there is a bending r,noment at the

free end of a structure.

The deflected shape is derived from the bending

moment diagram which shows that the deflected

shape must be of single curvature' and since i t is the

top f ibres which are extending in tension, the

deflect ion must be dowrttards.

There is a balancing anticlockwise reaction

moment, MA. Note that there is no vert ical

reaction as there are no loads to balance a reaction

at A.

34.

x.

F.

tt-

'fHE OUALITATIVE ,Ai.IAIYSI.S OF BEAMS

Vhen the load moment is appl ied in the span of thesimply supported beam AB, the Jirst point to recogniseis that the beam nust rotate in the direct ion of themoment. Note that the beam wil l almost always deflectt la9, vert ical ly or horizontal ly, from the undeflectedposit ion at the point oi appl icat ion of the moment. Thepoints oJ certainty on the deflected shape are the zerode{lection at the supports at A and B and the rotationof the beam at the point ol application of the loadmoment.

Because the vert ical equi l ibr ium is only maintained by

the reactions, the vert ical reactions are equal and

opposite and their value is the applied moment divided by

the span. Taking moments at A, the applied moment M

produces a clockwise moment on the structure. The

posit ion of the moment, relat ive to A, has no eJfect on

the value oJ the external moment on the structure. This

clockwise moment must be balanced by the anticlockwise

moment produced by YB.

The bending moment diagram result ing from theapplication of an external moment is unusual in that

there is a discontinuity at the point oJ appl icat ion oJ

the moment. The moment is on both sides oJ the

structure at the sa.a?e point, The sum of the hoggingand sagging moments at the point of appl icat ion ofthe external moment is equal to the value of the

external moment. The slopes of the l ines ol the

bending moment diagram are equal, because theyresult from the same va.lue of the vert ical reaction.

we wil l now apply the moment to the propped canti lever

beam AB. Note the tension on the underside of the beam

at the f ixed support at A.

-5_+.. r +.,YA VE

f f i r '

^ah#-.4 |r lE

U N DERS'I AN DI N G ST RU CTURAL AN ALY S f S

39. The result ing diaSram oi the reactions and bending

moments. Note that the sloPe ol the bending moment

diagram either side oJ the Point ol aPPlication of the

moment is the same, result ing from the constant shear

force across the beam,

40. We wil l now apPly the moment to a three-span beam'

ABCD. Try to draw this del lected shape' You wil l f ind

it di l f icult to recognise the anticlockwise rotat ion at B'

Howeverr i l you ensure tnar the curve is smooth throu8h

the points oi certainty' that rotat ion wil l be appatent '

41. The direct ion of reactions may be found Irom the

dellected shape. Perhaps the most dif f icult part ol this

solut ion is the recognit ion ol tension on the underside ol

the beam at B. Try treating BC as a beam' 'bui l t- in'at

B and C. This wil l reveal more clearly the tension on the

underside at B.

42, The plott ing of the bending moment diagram from a

part ial uni lormly distr ibuted load wil l now be explained'

The reaction at A wil l be greater than that at B'

ffir*

IF

TEE 2UALITATTVE .2,\t:1::_= :: :a-t-.,:sl The characte.istjc paraboJa of the bending momenl

:E8ram result ing from a uniformly distr ibuted loadaDL) is piotted on a new baseline between the

:r iotted values of the bending moment at the start and-: i ish of the uniformly distr ibuted load. Ihe:_raximum height of the parabola is wl2lg plotted-:: ! :a./ to the structure basel ine where I is the_entth ol the uniformly distr ibuted load. The

-rabolic curve may now be drawn to complete the

ae.dlng moment diagram,

A :: lre diagram is drawn to a large scale there are a-: jnber of addit ional geometrical propert ies which wil lia rhe accurate drawing of the parabola.

I and 2. Note that the tangents to the start and:,. ish of the parabola are the slopes of the bending::oment diagram Jrom the reactions to the UDL. Therea i l l be no discontinuity in the diagram unless there is a:oint load at the end of the UDL,

I and 4. Height at + span = f moment at mid-span:: L DL.

--::e tangent to the curve tor lhe naxinun value oJ the

-trding moment in the beam is paral lel to the str:ucture

it wi l l not coincide with the maximum ordinate Jor the': lL unless the load is symmetrical ly placed on the

t

atF

rt!

I L}]le obvioirs limitation ol the qualitative approach is that--Ee may be more than one qua.litative solution fofsrrtures with more than one load, The simply*-Dported beam AB is extended by a canti lever from B toC The beam i9 only part ial ly loaded wirh a uniformly

J:srr ibuted loddi the canti lever is ful ly loaded. There=:e lwo possible solut ions. In this f irst case i t is:.ssumed that the load on the beam is suff icient to createa: upward reaction at A.

A ffi

-1: -:*

V=ru

ffr-U N DERST AN D I N G ST RU CTU RAL AI/AiYSIS

47. Consequently there wil l be a sa88ing bending moment

from A to the start of the uni lormly distr ibuted load'

Note that the plott ing oi the parabolic distr ibution of

bending moments due to the uniformly distr ibuted load

may be apPlied to the canti lever.

48. Another galden rule.

- : - : :

. : - - -J

: --:T!

-

_:{d

*:- f

- -

' - !

I f we now increase the sPan oI the canti lever to tnduce a

downward reaction at A, there wil l be a single curvature

in the deflected shape, because the ef{ect of the load on

span AB is insuff icient to countera't the effect of the

hogging moment from the cantl lever'

50. The bending moment diagram wil l be hoSSing over the

span AB, desPite the sagging effect of the part ial UDL'

49. le".qr"*-" lt"6![-!

: _f , I

, - {

lt"

.IF

l

Jlltrfoofil,

* Tha bu\dt^q./ t\ w

r\, hert r 4 ASrc^ b\4?'r

^ 4istrtbdt t.4 Io41

a lrvrS J cufvc4

is

THE QUALITATIVE ANAIYSIS OF BEAMS

Jl. This next example ol a propped cantilever beam has ahinge in the span at B. Remember that one of the mostimportant features ol a hinge is that the bendlng momenr15 always zero.

Because the support at A is simple, allowing fu.ll

-aidr, as the load causes the beam to deflect

&rrrards, AB will curve and rotate. The shear induced

-C ri- l l turn BC into a canti iever. There wil l be

&-'iauity in the deflected shape at the hinAe,

shows the reactions and bendins moment

lJore ihat the result ing distr ibution of bendingis similar to that Jor a propped canti lever

a hinge. The explanation is that in this example,tEs been Dositioned at the Doint of

i!+ect the equilib.ium of each part ol thefhe equilibrium of part AB is sustained by theEtical shear Jorce sR, This must be balanced

shear force on BC. I t is this shear

-Jr creates the hogging, clockwise moment

ar C-

rample is the three-span beam ABCDE, with a5pan CE at D. The point load tv is to the leJt

in span CE. As CD deflects downwardsIoad, DE wil l rotate without bending.

, there is ngx6ear lorce across the hin8e.

Mc

uo

I i""-'1"^

"j

Til l be undefo.med Jrom the load to the hinqe.

UNDERSTAN DING STRUCTURAL A,^/A'Y.sT.'

55. This diagram shows the reactions and bending moment

dlagram.

55. This next beam has two spans and is ful ly f ixed at the

support A. There is a hin8e at B and a point load in span

CD. Ve can draw the points of certainty on the

deflected shaPe' Note that the beam wil l have zero

rotat ion at A.

57. I t is sometimes helpful ' when there is a hin8e rn a

structure, to imagine i t replaced by a short member'

The force in the member, in this case comPression' helps

to clari fy the resuit ing bending either side oi the hin8e'

In this case, AB wil l act as a canti lever and BC as a

beam suPPorted at B.

J8. This dia8ram shows the reactions and bendin8 moment

diagram. Note that the l inal solut ion for the bendin8

moment dia8ram is similar to that for a beam Fithout

a hinge. ln this example the hinge has been Placed at or

near to the natural point of contraf lexure'

. THE QUALITATIVE A}JALYS'S CF BEAI4Si9. This next example is almost impossible to visual ise. I t

is the application oJ equal and opposite externalmoments either side of the hinge, This is a device usedlater in one of the analyt ical methods.

This deflected shape is difJicult to arr ive at withoutan attempt at the reactions and bending momentdiagram.

This diagram shows the reactions and bending momentdiagram. In this example, because oJ the curious loading,we appear to have broken one of the golden rules. Infact the moment is st i l l zero at the hinge, but has avalue, the imposed moments, either side. The reactionat A may be determined by considering the equil ibr iumof AB. That downward reaction wil l cause a ho8gingmoment from A to C. Taking moments about C of spanAC, the applied moments cancel out.

tre wil l now study another example oJ alternativesolut ions. The canti lever ABC has two point loads, Hereihe downward load at C is greater than the upward loadat B,

55

!s

t

P.

N^+N--"

67. This results in an anticlockwise moment at A and asingle direct ion oJ curvature. The dotted l ines on thebending moment diagram indicate thernegative' effectof the upward load at B.

Mr^ |9L ' a4A+.

vAt I

iiii,rr*

U N DE RST AN DI NG ST RUCTU RAL ENAIYSIS

63. We wil l now examine the effect of making the load at Bgreater than the load at C. There is now a change in thecurvature in the deflected shape.

This diagram shows the reactions and bending momentdiagram. There is a point of contraf lexure between Aand B and a clockwise moment reaction at A.

Al l real structures have a complex loadingarrangement. However, with a sound understanding olthe quali tat ive method, you wil l learn to understand howto model real loadings and the response oJ realstructures by reducing the complex structures to a seriesoi simpler sub-structures.

Practice ProblensProduce the lul l three-part solut ion for each ofthese structures.

A typical computer analysis Jor each of thesestructures is given.

wl

M|c"+la:-1-,

w^f64.

- f , ^ ^\ \

'j---?'i----.-ff

lF=-'.t+/.;;4.--F-=iR

,/j-"ll!!Y.RRR+

65.

4 The Qualitative Analysis ofFrames

Ihe introduction 01 the second dimension in the plane, the vert icaldirect ion, required for two-dimensional structures, introduces thecomplication of horizontal loads or Jorces acting normal to vert icalmembers. The effect ol loads or Jorces normal to members wil l always

create internal bending moments. The f ixi ty at the joints wil l al low thetransfer of bending moments in vert ical members into horizontal members.You wil l Jind i t useJul to refer back to the last part of Chapter I whichstudied the distr ibution oi bending moments in stat ical ly determinatetwo-dimensional Jrames. Often the solut ion to a stat ical ly indeterminateframe is revealed i f you try to see the effect of the load on the

determinate frames within the indeterminate system, Thus the qLral i tat ive

approach to determinate Jrames may be seen to l ink with the analysis of thedegree of indeterminacy,

When you reach the end of this chapter and attempt to solve the practiceproblems you wil l f ind them di l f icult at f i rst. There are a number ofreasons for this. The I irst and most important aspect oJ this approach to

understanding is that solving problems quali tat ively js dif f icult in i tselJ.Secondly, there is no immediate route into the solut ion. The problems seemopen-ended. In fact they are internal ly coherent and that is the clue tothe solut ion. You must check al l the loads and forces against the ef lects -apply the golden rules and a?waqs produce the Jul l three-part solut ion ofbending moment, dei lected shape and reactions. I f your qual i tat ive solut ionis correct then each oJ these parts to the solut ion wil l check against theolner two,

f J8 UNDERSTANDING STRUCTURAL E]VAI,Y.s15l . We wil l start by examining the simplest two-dimensional

structure' the canti lever bent ABCDE' ful ly l ixed at A'

2. The bent is loaded with two Point loads, , t / r vert ical ly

at B and f l2 horizontal ly at D. Canti lever structures

are always stat ical ly determinate and i t is oJten simp

to start the three-Part qual i tat ive solut ion by drawing

the bending moment dia8ram and using i t to help to

the deflected shape. Note that the bending moment is

constant in thal part of the structure paral lel to the

load hr2, over the member BC.

We know that the beam ABC wil l deflect downward

under load h/r and that load ry2 wil l create a tension

in the top surface of BC. From this we are able to

recognise that joint C wil l deflect downwards. Notewhen drawing the dei lected shaPe in two-dimensional

structures the real shorteninB or extension of members

under axial load must not be represented in the

otherwise a misleading deflected shape could tesult.

Such strains are insiSni l icantly small compared with

bending del lect ions in the quali tat ive analysis.

Consequently the deflected Posit ion ol joint C isrhe l ine CE of the or ig inal structure.

Another 9'o-7 de, ru le.

Virtual ly al l structures with a f ixed support are

deJormed by the loading system just before the

4.

However, you must identi fy the zero rotat ion at the

support as one of the points of certainty on the defl

shape.

X ot afut!{unasqP?ot? / tho 1412'

THE QUALITATIVE ANALYSIS '? 'RANES

The f inal part of the solut ion is the diagram of the:eactrons, drawn, as usual, as the action ol the jojr i: : : the nenber, Thus MA is anticlockwise. I f you:Eve dif f iculty visual ising the support condit ions, try.e-creating the support condit ions and the structureaith a piece of f lexible plast ic.

This is a reminder that the ful l solut ion must always

-icludel

2,

l .

59

the deJlected shape,

bending moment diagram,

l. reactions.

1-ou should start to identiJy the internal:elat ionship between each part of the ful l; .r lut ion; bending tension and deflected shape,":_-oment reactions and bending moment diagram {or"-xample,

::rere is a plott ing routine for the bending moment at: joint or node which is part icularly helpful. We know:om the deJlected shape that there wil l be a:e.sion on the top of the member at A and from the:eaction diagram, that the reaction moment I tA is=lt iclockwise. Remember that we have adopted an

--;=ra-l l sign convention where clockwise momentsare positive.

aol low this routine,very carelul lyl

Go from the joint under consideration along themember and rotate in the direct ion of the siE/?rof the moment. The value oJ the bending momentis plotted on the ordinate narmal to the member.Fortunately this is the same output convention usedby (almostl) al l computer programs lor the analysisof plane frames.

nnnrALl l l l l ' ] \ ,

t ' t/-i.'

A 'L"Ye,-

.:--l-" s +'r- r-I

iDr>- |-- ' - '+-1

I l+-

$ -'' 60 UN DE RS T AN DIN C STRUCTURAL ENEIYSiS9. This next example is the same canti lever with an

anticlockwise load moment at D.

10. The bending moment result in8 from the application of a

l l l l l l l l l l l l l r i l lEl-

I

moment on a canti lever is a constant value thoughout

structure regardless ol the shaPe ol the structure. In

this example, the moment produces bending tension on

the underside oJ the cantilever. Once again, if in doubt'

try re-creating the structure with a piece ol plast ic.

The deflected shapes which result from the application

of moments are dif l icult to Produce intuit ively.

Consequently i t is the relat ionshiP between the

deflected shape and the bending moment diagram and

the convention of drawing the bending moment on the

tensio, side of the structure to which we must turn.

le of the frame fromlmaSine the { ibres on the undersi(A to D extending and it will be apparent that point C

will deflect upwards.

There is only one reaction at A' the clockwise moment

balancing the anticlockwise load moment. Note

part icularly that there are no load reactions.

l l .

1)

.fm

THE QUALTTATIVE ANAI,YSIS O:' IR?.I!ES

We wil l now reverse the two-member frame and extend

it by a horizontal member CD, hinged at the junction Cand supported on a horizontal rol ler supPort at D.

I e have generally ignored the siSnificance ol axial

5ress resultants in the members. However, i t is helpJul

:rre to note that there is an internal compresslon In

:nember BC.

To begin with we identi ly the points of certainty on

:he dellected shape:

the member must enter the lul ly f ixed

support at A without rotat ion from its

original, unstrained posit ion,

point B wil l be pushed downwards by the

compression in BC'

because the r ight-angle wil l be Preservedat B, point C wil l deflect horizontal ly,

the horizontal deflect ion of C wil l rol lsupport D to the r ight.

fl.

l .

2.

x

le can now complete the deflected shape. Member BC

r: l l remain straight because the structural conJisuration

.l iminates bending moments in that member and there is

-.o horizontal reaction at D.

3.

4.

UN DE RST AN DI N G ST RU CTU RAL AN ALYS I S

17. This f igure shows the f inal bending moment diagram.The slope of the bending moment diagram from C to thel ine of the load wil l be the same as the slope from B toA. I t is the compression in member BC which iscreating these bending moments.

Our original canti lever bent is now converted into alrame with two supports, with the addit ion of a rol lersupport at C. We wil l examine the effect of sway bysubjecting the frame to a horizontal load at B.

To identi fy the direct ion of the reaction at C we wil lnotional ly remove i t . Point C would deflect downwardsas a.esult of the horizontal point load at B and theresult inB reaction at C would be upwards.

We can now draw the deJlected shape, beingcareful to maintain the r ight-an8le at B. Thereactions are derived as lol lows:

18.

20.

HA

the horizontal reaction dA wil l balancethe applied load,

the vert ical reaction, vA wil l be equal andopposi te ro the reacr ion vC.

There wil l be an anticlockwise moment atA, 14A, the sign of which is confirmed by thetension on the left-hand lace of the columnAB, at A.

l .

2.

3,

Illilr'*f"

THE QTJALITA'IIVE A ELTSIS OF FPI!,GS

The bending moment diagram may be derived from the

ractions and confirmed with the dellected shaPe. The

rpward reaction at C will cause sagging moments in the

bam BC, IVe know that there is a tension on the

Ht-hand side of the column at A, which completes ihe

HnB moment diagram. Note particularly the

rdirmation of the point ol contraflexure in member AB.

Ire change the sign of the load by reversing the

-lction then all the stress resultants and delormations

-ll have opposite si8ns.

As q,ith the dellected shape above so the reactions and

moment diagram are reversed. Thus we have

ft effect of sway, either to the riSht or the left.

Ye will now remoYe the horizontal Ioad and apply ayertical load on member BC. Vhich way then' will the

srpport at C deflect? To the riSht or the lelt?

"Wp-it-A1

&-

64 UN DE RST AN DI N G STRUCTURAL AN ALYSI S

25. There are a number of ways in which we can examine thehorizontal deflect ion at C, i .e. sway. This part icularapproach, that ol introducing an art i f ic ial reaction at Cto !\e\e\i\o.(\zo\ta\ mo\efi\e\\, \s \\e same s\a\eal \Newil l employ later in the study of one of the rrajoranalyt ical methods, Moment Distr ibution.

I,,1![

u+-

vo

26. In drawing the deflected shape, joint B wil l not movehorizontal ly but wi l l rotate, because of the art i f ic ialhorizontal restraint at C. I f \re imagine a notionalremoval of the harizantal restraint at A (not shown)then i t is apparent that the restraining horizontalreaction ryA wil l be posit ive, to the r ight. This mustbe balanced by the art i f ic ial reaction a1 C, dC ,i .e. to the left .

To remove the effect of the art i l ic ial horizontalrestraint at C, we apply an equal and opposite force tothe original structure. This is an example of theprinciple of superposit ion. Thus the combination of thereactjon diagrams lor the art i f ic ial ly restrainedstructure and the application of the value of theart iJicial restraint at C as a load, wi l l produce thereal, original structural condit ions of a rol ler supportat C.

This is the l inal bending moment diaqram, thecombination oJ restrained and ,sway'moment diaqrams.Note that the application oJ the principle ofsuperposit ion, that is the combination of these diaqrams,cannot predict the constant bending moment in memberAB. I t is only because we recognise the absence of thehorizantal reaction at A. The slope in the bendingdiagram in the columns oJ frames is alr lags due tothe presence of a hortzo-ral react ion.

$**f:.$-

-

E \irlHEI

za-

THE QUALITATIVE !] ; ; : : ' : : : ' 'RA'1ASThe frame wil l deJlect horizontal ly to the r ight, theei tecr ot the ' load' H^. balan.rns rhe art i f ic 'a l.eaction. Notice that there is single curvature in the:olumn AB. For there to be a change in curvature in the:r lumn oJ a lrame there must be a horizontal reaction to::eate the zero moment and the consequent point of: :r :raf lexure. The recognit ion of the effect of the

- - Tontal react ion is parr i .u lar ly important as i r is a

:_:auent source of errors in the quali tat ive analysis of

I = n i l l now solve the pin-supported portal frame ABCD.

:: jetermine the direct ion of the horizontal reaction aDt.: \ ! i l l notional ly remove i t . Support D wil l rol l to the

- :-r because the beam BC wil i bend and as the_ : r -angle at B must be mainrained. B musT def le(L to:-: r ight. This alone would induce a horizontal: : : lect ion at D. However, the r iSht-an8le at C must also

:e.naintained which wil l rotate CD, inc.easing the-:.-zontal deflect ion at D.

:: .eturn support D to i ts original posit ion the- ' : /ontal react ion al D musT be negat ive, i .e. to the

.::r . To satisfy horizontal equi l ibr ium, the horizontal- : :ct ion at A must be equal ro

"O. Because ' rA and

-

dre equal . dnd the columns AB and CD are the

.=me height, the bending moments at B and C wil l be:3ual.

-: :cause of the later appl icat ion to the method of

' l rment Distr ibution, and because i t is a good example of

:_e principle of superposit ion, we wil l examine the

:.rblem of sway with the introduction oJ an art i f ic ial-:straint in the horizontal direct ion at C. This: jal i tat ive solut ion depends upon the relat ive vazues oJ:-e bending moments induced at B and C for the:r: i f ic ial ly restrained Jrame. You must acceDt at thls

::age that the f ixed-end moment at B is ereater than the

-oment at C, for the case of the point load closer to B.-r is ls analogous to the distr ibution of moments in a

: . \ed-end beam.

65

rJ sw

66 U N DE RST AN DI NG STRUCTU RAL AN ALYS I S

33, This wil l result in a greater bending moment at B than atC for the restrained Jrame. Consequently the horizontalreaction at A is greater than the horizontal reaction atD, To balance the system in the horizontal direct ion theart i f ic ial reaction, aC, must be negative, to the left .

34. To remove the art i f ic ial reaction f lC we must apply anequal force in the opposite direct ion, We now have thetwo basic sets of diagrams for the application of theprincipie of superposit ion to Jind the f inal distr ibutionof bending moments,

The f inal bending moment diagram Jor the or igrnal ,unrestrained frame is the combination of the restrainedand sway solut ions. As with the previous example, thecombination of the quali tat ive bending moment diagramswil l not identi ly the equali ty of the bending moments at Band C. This can only be recognised from the horizontalequi l ibr ium of aA and ,?D.

We wil l now examine the quali tat ive solut lonof a structure with three mernbers at a joint.First ly, we wil l identi fy the Doints of certaintvon the deflected shapel

3,

l . vert ical restraint at A,

rotat ion of joint B with the Dreservationof the r ight-angles between the membersat joint B,

must pass through the supports at C and D.

mfr#tr5ffr r

twl' . - , . - , . -L

--Y-

t llD )k

'\\

35.

36.

2,

: a=

-_-: i_ '

;

!f

rE-:a:

-=^ :::

_: ::

-1i:a

1-,: --a_e

-

: , : : S

67: :Jmpleted deflected lorm reveals that there wil j be

..-s-cn on the top surface of AB and bending tension on:E

-eft-hand side of the column BD. The direct ion ofG reactions shown in this f igure is determined by theE{bnal ' .emoval and replacementi technique described

--e

bending moment diagram can now be plotted,

--:rting with the obvious pointsl

hogging bendinS in AB result ing from thedownward reaction at A,

the upward reaction at C wil l producesagginS in span BC,

The horizontal reaction at D, wi l lproduce tension on the left-hand sideof the column BD.

\ote the discontinuity of the bending moments inmember ABC.

Let us examine the moments at joint B. These havebeen drawn here as Lhe

-ct ion of the joinL on Lhe

--e,nbe.r. Thus the joint will exert a clockwisereaction on member BA. Note part icularly the

relat ionship between the bending moment diagramand the direct ion ol the moments shown here. They

are related via the plott ing routine described inFigure 8. Thus the sum of bending moments UBAand MBD musr equal MBC since the joint must bein overal l moment equi l ibr ium.

1.

c

M&Mtso

::E &ecdI

*1*tq

ILhd

lgtg

h;

z hoq

t hrrdE

Ib b.rm{5r rl|e rt

-EnbJEcrrt i'i

!c can rD!f8: This IEnent al

bte parti

-agram in

atactioar al

rel,ated to

68 UN DERS?ANDI N G STRUCTURAL AA'A'TSiS

40. Remember that bending moments are Plotted

l. from the reference Point,

2, along the member'

3. rotating in the direiiion ol the sitn of thebending moment,

4. plottinB on the ordinate drawn normal tothe members.

41. We will now study the qualitative analysis ol the

frame ABCDE. The point load tr is locatedhorizontally in column AB. Support A is fullyfixed' E pinned and D is on a horizontal roller'

42. We may apply the Principle of suPerPosition to thepoint load by assuming that it is first applied to asimply supported'beam', which in turn transfersthe suDDort 'reactionsr to A and B.

43. We will start the solution by identifying the points

of certainty on the dellected shaPe. rvith Practice'and the study of the deflected shapes oftwo-dimensional structure!' the detlected shaPe

shown may be drawn'

,]

I

!

THE QUALI'rATM elVA:!S--j- :,: ::-:-t::

:-:re direction of the reactions is identified from rhe

:ef lected shape. We can srart ro draw the be;ding-:roment dia8ram with the most obvious points:

l .

3.

69

bending tension on the left of the Jixedsupport at A, reduced in value by theh^r i?^n+21 ra2^+i^n t

' ' * """ ' ' ' . ,A,

hogging bending in member CD,

bending tension on the r ight-hand face ofthe column EC.

The bendin8 moment diagram is comPleted and comPared

rirh the reactions. Note that the bending moment at C

:n member CB, is equal to the sum of the bending

:noments in cD and cE at c.

I e can now add the effect ol the Point load in column

-1,8. This 'simply supported' effect is 'added' to the

moment at A and B.

\ote part icularly that the slope in the bendinB moment

diagram in column AB is related to the horizontal

reaction at A. Similarly ' the change in slope in BC is

.elated to the downward reaction at A'

!c>i

i:Y9

HEH4

Vfot^

f 70

{f f,'," sloye i Lhc-bf,ldlnl t^dtrrc*C 444/4u

\JJ

is alwatt,r cssoqrtU wr'6a forco NoR 4AL boUo sErwtwe..

UNDE RST AN DI N G STRUCTU RAL AIVE'YST.S

This gozder ruie wil l remind you to carry out thisimportant check on the quali tat ive solut ion.

HinSes in frames may produce unusual solutions. Thisorthogonal frame ABCD has a hinge at C. The structuresways to the right to release the moment at B. Thesolution is unusual in that there is a zero bendinsat the internal, rigidly jointed connection at B, Thehorizontal reaction at D is zero because moments aDoutC, considering member CD, must be zero. Thus thehorizontal reaction at A is zero and the bending momentat B, due only to the horizontal reaction at A is zero.

Ve wil l now study the frame ABCDE, Ioaded with aload at D exactly over the reaction at E. There is ahinge at B.

This structure has an unusual equil ibrium in that thevertical reaction at A is zero. Taking moments about E,there is no out-of-balance moment due to the load sincethe line of action passes through E. AIso the horizontalreaction at A is zero because of the hinge at B. Sincethere is neither horizontal nor vertical reaction at Athere are no bending moments in either AB or BC. Thebending moment due to the load r.,i in member CD, isresisted only by member CE.

Iilutlrril

lll ,ut , r ll : i il;tit !tjill

' fya tvo

M

49.

) t -

fb etpLdirirtrays abtlErG5B{

.-lcdly ddEe is ciqect d

The expliin CE. nmember.

aS i 3'a!,

lis

trture

txhent

:

bout

ment

IL

poinr

rt E,inceftal

F

he

:":r what happens to the deformation in BC? We can see--:rat joint C must rotate, which appears to require a:ownwards reaction at A, through a shear force at B, foreguil ibr ium.

T'ne explanation is that the structure sways to the leftand joint C sways and moves downwards as member CEsways about E. This al lows CD and CE to bend, butleaves BC and AB straight.

Clearly there is a compressive force in member CE. Butthere is only one vertjcal reaction at E. We need toinspect the force equil lbrium of this support.

The explanation is that there is an internal shear force

in CE. I t is shown here as the action of the joint on themember.

QUALI'rATIVE ANALYSI: " "':1:S

7I

L

wNce

- , , f \lYg

)).

72 U N DE RS?A.NDI N G S TRUCTURAL AMAI,YSI"5

56. The resultant of the exterra.i reaction to the shear andaxial Jorce is equal to the vert ical reaction at A.

57, 58. Practice problensProduce the full three-part solution to each ofthese problems. Computer solutions forsimilar structural dimensions and loadings are

Biven.

\

)5E 0*.:b-

"f'J'nf] "{a*

{*n+-IT+"

l+II

-&

3.

th"

Part II

5 The Theorems of VirtualWork

Tl9 k9l l. the soiution of "!gti911l_,!d.19!I]let. .1.,"19t9" M.abil i ry to delermine structural dg[gl lnations. In one meLhod of analysis,the deformations provide the basis of equations of compatibi l i ty which, inaddit ion to the three equations of stat ical equi l ibr ium, a. l low the solutronof the unknown effects and the lul l solut ion to the distr ibution oJ internalforces and moments upon which the subsequent structural design is based.

In addit ion however, the deflect ion of a structure may well be a designcri terion. Al l structures must satisJy two basic states of loadingl

l . Serviceabil i ty when the structure is subject to i ts working load.It is at this state that the deJlection of the structure is checked.

2. Ult imate load, where the fai lure strength of a structure is comparedwith the serviceabil i ty load mult ipl ied by a load factor.

There are two basic approaches to the analysis of structuraldeformations, .fteiq ClStgy gl9 u-tltual wotk- The latter has the advantageof being able to deal with congi1lo!: qlhgr ltat !b9!e lvltlrn !!9 glagticrange

- a l imitat ion o s!rai! ' i elergy. Only virtual work wil l be studied

here as i t is general ly agreed to be the more powerful of the two.The student should bear in mind that although virtual work calculat ions

may be the key to the numerical solut ion of a problem in structuralanalysis, the mathematics content is relat ively tr ivial. The dif f iculty

encountered is the application of the concepts oI virtual work to theproblem in hand, Once that can be understood quali tat ively, the subsequentcalculat ions although t ime-consuming, are straightforward. Without thrsquali tat ive grasp, methods l ike virtual work may quickly become an exercisein mathematical manipulat ion and the overal l sense of the method and thebehaviour of the structure, lost,

76 UN DE RSTAN DI NG ST PUCTURAL edetyS./ S

_, lws.--l--\A I

--3--ft-

R.oal stru

,f,-

TEE THEOREMS AF VIR:' : | ' -- ; t '7

Simi\arly, ve can create a state ol vir.-,ja-aisplacenenL by the introduction of a hinge at

.\ and B. This arrangement is compatible with the

-:upport condit ions, but the structure rs no ionger

capable of support ing the load. The introduction

o two hinSes has creared a mechanism.

Consequently this system satisf ies compatrbj l i tqbLlt not equj l jbr:rum'

The virtual state is an analyt ical device.

Because only one state needs to be g4ti l f ied i l is

more easi ly evaluated. When we combine real and

virtual states we can develop theorems of Sreat Powerand application in the solut ion of structural problems'

This chapter is concerned with the proof oJ the

.aeorens oI virtual watk. The two forms of the

theorem discussed above are ol Part icular interest:

Virtual Displacements, and

Virtual Forces.

\\e wil l use as a basis lor the development of the

iheorems the deformable, Pinned structure ABCD,r,hich is subject to two loads, h/r and t{r: act ingat node D, which is a hinge'

The effect of these twq loads i5 to c-qq99 e-.deflection, at D. T\e sllqq-t!!9jliq-equilibrium anq Ye ha.I9a de{ormed shape cqlBal-i l lg } ' i th.the supPortconditions. This is a rea.l state.

77

ilRTUAL DISPLACEM\rgNT

6."'.ri

THE)RE! oF V ls-TuN-DISPL\CEJT,ENT6

lHEo PE&\ oF VJRTUALFoe.6

f,

rad a(le&tun

74.

TH&REM OFn//n]D5F rna i lU/ UIN U rt LUE=-

JEIIS/Pfl A N IE M1 IEAI?E-uououJ

N rc4 iWtnt .pt "s

N rt&l a#iadr to.,.E

A, raat a4tz*tn

d vfintuo( dlefiL

UNDE RSTAN DI N G S T RUCTU RA L AN ALY S T S

9, Ve will consider first the Theorem of yirtua_zDispfacenents.

10. The real structural system has imposed upon it avirtual displacement. The direction of this virtualdisplacement is unrelated to the direction of thereal deformation. This is a virtual or notionaldisplacement in the sense that, despite theintroduction of the yirtual displacement, the internalforces, and the external loads ttt and t{2 areunchanged by this virtual deformation.

ll. In order to make the diagrammatic presentationas clear as possible we will adhere to the conventionshown here.

12. As we have assumed that the imposition of thevirtual deflection has no effect upon the externalloads, equalJy i t has no effect upon the balancinginternal lorces, i /1, N2 and .V3 in members AD,BD and CD respectively. Note that because allmembers are pinned at each end and the loadingrestricted to node loading the resistance of thestructure is limited to axial forces.

-

q3_R3!th cxtern

-ctidr3r ICG

-

-ti.n

r!'""'st{caEl to {

-ctidlrlttual q

TI{E THEOREMS OF VIRTUAL WARK

i lhe Theorem oJ Vllt-uAl,Dipbrements the:rpression for the vit!]lal-\or.$ymhol

-1 5-qtl&l to--

-.h: \-1\u_e, g!_\\e-Lea\-!9r_c9 -ir,-!hldine.q!i-o!,q!\h\e

vi{+q-aL_dlgplacerogI _!:9_:-lhe corlrpone_q!-ol-tb_llel

force)_multiplig{ b+ -ths -value-ol the virtqg]

displacement.

Ti'e,9.x-t_el11r- lirt-uq!-y!41'q -d*e-Lelrn{9d*bv-E5or+i+gihe externq!!_oj-cej I{ AJltg-..dllgc.i!9!,gll;hc-virJualdisplacetnent The component of force lr'r in the

dlrect ion of the virtual displacement is equal totL x cos 0r, where 0r is the angle between the l ine

ol act ion oJ lorce td I and the direct ion of the virtual

displacement.

For more than one external load actin8 at D we can

determine the resultant Jorce R,,/ which is in the

direct ion of the virtual displacement.-

The externa! yiltllal wolk don-e_ bJ the- rea l ,1o-ads is

equal to the resultant component ol th real loads !!l the

direction oi the virtuaL displacernent multjplied by t!evirtual displacement.

l:*Wu'

.W= Wr , ca?l .d)

rq

w,rff1w,^ lvL t

80 UNDERSTANDING I\NALYSIS

17. In a similar way, we may f ind the resultant of theinternal axial forces RM in the direct ion of thevirtual displacement, The angle is that between thedirect ion of the internal Jorce and the direct ion of thevirtual displacement.

Rx

"v;s t3*,,Eotxs

e".{ Nj . c.s. .J.

^v R"

.r^IL KN,Rw+ Rrv'o

18. Since in the real state the structure is in equi l ibr iumthe sum of R,ir and RM, the resultants of appl ied loadsand internal forces in the direct ion of the virtualdisplacement, must be equal to zero.

ENTEPNAL VOR?VAU WOFI(

e! = eH.6

-j\ egui[ibriumRv =-Rry

Exb VW= I4 MWW=(4

19. The internal vi Ual work, rhe symbol for which is uis equal !o l!9 rq:-u,!fert _-o.{1!_g 1ryelltq! rolces in thedirection of the virtual displacement mult ipl ied by thevrr Ludr !

'JPrdLcr ' re '

'

r ,

20. However since Rrl/ and RN are in equi l ibr ium, theactual values ol the resultant of the external Jorce, andinternal forces wil l be equal. Therefore, the internalvirtual work must equal the external virtual work, sinceboth Rrr ' and RN are mult ipl ied by the same virtual

deformation,

Ie can nor

I : a,I I-.E-

&f*

!n ']hr]6

THE TIIEOREMS AF VIRTi]}.: i:'::

Ve can now slate the Theorem oi Vir tual Displacements.

\ Ie wil l examine the application of the Theorem ofI ir tual Displacements to the ult imate load analysis: i the f ixed-end beam ABC loaded at the centreri th the load H, such as to cause col lapse. Note::rat the ult imate resistance moment at B is twice:rat at A and C.

Ae wil l consider a virtual displacement system which.s compatible with the supports, by the introduction:i hinges at A, B and C. Point B is assumed to,:rdergo a virtual displacement oJ 6 and joints A,3 and C undergo a virtual rotat ion which may bee\pressed in terms of the virtual displacement and:ie spaD dimension r, . Because the angular.ctat ions are small we can assume that the arc of:1e . r rc le radius

" /2 is the same as rhe tangent.lngle 01 and 02 are expressed in radians.

Tris is the bending moment diagram at ult imate-lad, due to the action oJ the ult jmate load ,r ' .;r is system is in for:ce equil ib. ium. That is the

-rr imate moments at A, B and C are in Jorce::ui l ibr ium with the applied, ult imate load ,/ .

/,/

IiIRTUAL sJs?" +aatf@..tfi$vrJ

A*u4 quihibrtua stlstart+ tt'r'44^4 f4'>J

8lIt

I ThcbtaA fi M|RT@ALDis plaee uet*1L1 .1lJ svt4

82f

THbPeu epVfl RT UALt roR@85

B

P.ul stata

UNDER.sTANDING STRUCTURAL ANA LYS IS

25. To f ind the relat ionship between the ult imate loadtr and the moments of resistance M__, we determinethe internal and external virtual work. The internalvrrtual work due to a moment is the mom6it-

In thi