Understanding Structural Analysis - Brohn - 3rd Edition

 

i

Understanding Structural Analysis DAVIDBROHN

MIStructE PhD,CEng,MIStructE Princibal Lecturer n StructuralEngineering, StructuralEngineering, Po\techruc BristolPo\techruc Bristol

OveAruP SirOve Forewordby Sir

GRANADA

London Tbronto SYdneY NewYork

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Contents

Preface edgements Achnouedgements Achnou Part I tructures eterminatetructures The analysis f statically eterminate Staticalndeterminacy ndeterminacy 2 Statical qualitative nalysis f beams 3 The 4 The qualitative nalysis f frames I

Part II 5 6 7 8 9 1O

virtualwork The theorems f virtualwork The flexibilitymethod flexibilitymethod The stiffnessmethod frames The stiffnessmethod grids Momentdistribution Momentdistribution planerames rames Plasticanalysis Plastic analysis f plane

11 yield ines ineanalysis ine analysis freinforced oncrete labs Influenceines Influence 12 The

I

3 22 38 57 73 97 11 4 137 149 17 8 202 22r

Afpendix: Solutions opractice roblems

23 2

Index

282

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Foreword

headat A force is not just a straight ine with an arrow head at one end.That s ust a ifetums convenientabstraction convenient abstractionor or shorthandor what n real ife tums outto out to be a bundle of particles under stress and alwayschanging changing ndmoving ndmovingunder andstrain, strain, always under he provocation slightest rom changing ircumstances. he theory of structures and, n fact, our whole scientificapparatus scientificapparatuss s foundedon foundedon suchabstractions. suchabstractions. They have enabledus enabledus to impose imposesome order on tie chaoswith chaoswith which whichwe someorder we are faced when we look at the unending ndoverwhelrning ndoverwhelrningwonders wondersof of nature which far exceed haveeven ound bat, f we exceedour our powersof comprehension.Wehaveeven imaginaryworld world of science s a true pictue ofreality andact assume hat this imaginary andact nfluenceand accordingly,we accordingly, we can nfluence andchange changeheworld he world we ive n to suchan suchanextent extent fact, doalmostanything tlnt we canabolish canabolishwant want anddrudgery anddrudgeryand, and, n fact,do almostanythingwe ike, hcluding destroying he planetwe depend n, togetherwith togetherwith ts fauna nd lora, in a few weeks weeks-- if we only couldagree where to start. couldagreewhere This whole mechanisticworld-picture, he Cartesian Cartesianor Newtonianworld Newtonianworldof of heaqystrain strain now; now; we 6nd n all disciplineshat disciplineshat t does not science, s under underheaqy doesnot work any more,nature notcollaborate. haveno no ime o elaborate anymore, naturesimplydoes simplydoesnot collaborate. have on tlnt now and here are, anyhow, housands fbooksand fbooksandparnphlets parnphlets ritten theime.But structuresand about that theime. But how does tris affect he theory of structures and he n-holebusiness n-hole structuresare businessof of structural engineer engineering? ing?Our structures are all the time getting better, bigger, lighter lighter andsafer, andsafer, ourmachines our machines remoreefficient, energy.Weare areall he time earning o domore domorewith with ess, smoother, using ess energy.We so what is wrong with tllat? hisbook, mportantstage As David Brohnpointsout n his book, here s acritically mportant stage o be reachedbefore reachedbefore we can even applyour applyour numericalanalysis analysiso a structure, caneven havea structure to apply t to. Whenwe Whenwe have hat our namel), hat we must havea anzlysiswill anzlysis will tell us whetler the structure s capable f doingwhat t is supposed to do. Further, that the skillsrequired o choose his preliminary tructure tructureare are gradually may thosewemay ttainwhen ttain whenwe we have of an entirely different nature rom thosewe masteredour mastered our structural echniques. basisof of these Brohn confineshimself confineshimself o suggesting lnt tie basis theseskills skills s the loadand recognition of the relationshipbetween relationshipbetween lrc load and he resultingbehariour resultingbehariourof of ho'r' a the structure, in other words that we gainan ntuitive understanding f ho'r'a structure will behaveunder behaveunder oad.This will. whenwe haveacsuired acsuiredOre Ore

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necessaryexperienc€,enabb necessaryexperienc€, enabb us to choG€ the at least leastapproximately approximatelyight ight structure for a given task ust by lookingat lookingat its shapeandproportionson a drawing. This is of coursemost coursemost mportant, or the ncreasing se seof of computers has o a great extent killed lis understanding hich s so soessentia.l essentia.lor rescuing the art of structura.ldesign. structura.ldesign. The computer hascome hascome o stay, we must Livewith it, and his book eaches howto us how to do sowhilst remainingmasters remainingmastersofthe ofthe proceedings.Wemustbeable be able to checkthat check that theits output from from the computers s correct, andwhere and whereand andwhen when o use it, and what limitationsare. limitations are.computer If this book did nothingelse nothing else t would wouldstill still be the most important contribution o structual design designwhich whichhas hasappeared appearedor or a long time and shouldbe shouldbe compulsive eading or anyonenterested n the subject. I havebeen havebeenextremelyworried by the fact that gifted giftedgraduates graduatesrom profession our universities enter the with the idea hat it is below heir dignity to put pen to paper the computerdoes computerdoes t all. Here we have he necessary remedy for suchconceit, and t is high ime. Unfortunately haveneither haveneither he ustice time nor the knowledge o do to the achievements f Brohn'sbook, Brohn'sbook, haveonly have only afew a few hours eft for the printers' deadline,which which s entirely my own fault. But he hasseemingly has seeminglygone hrougheveryknownmethod knownmethodof of structural analysisand analysis and showsby showsby clear diagrams ndexplanations ndexplanations ow he structure s affected by theof loading, husbehaviour. at eachstage eachstagegiving giving reader his rise essential understandingof understanding structural structuralbehaviour. hope hishe book will give to a ively discussion. I cannotresist adding addinga a comrnentof comrnentof my own. Whilst recognising recognisinghe importanceof what Brohn hasdone, Brohnhas done, do not tlink he hasgone hasgone ar enough. Understandingof structural behaviours very necessary, ut are there not many more things that are equallyor equallyor even morenecessary? evenmore necessary? Every structural designer designerof of repute hasdeclared hasdeclared hat structural structuraldesign design s an art as well as ust anapplication f scienceand givenproblem. problem. scienceand echniqueo echnique o a given You couldalso couldalsoput it the other way roundand roundandsay say hat only f it is a work ofart of art as well will it be adrniredand adrniredandadd add o the reputation reputationof of the designer. t is unfortunately impossible o definewhat definewhat art implies,but implies,but it has n anycase nothing nothingto toaswell. do withThe numericalanalysis. numerical analysis. Tahere aremany are manyother other consideras consider well. wholepurpose ofThere structural structuraldesign design s tomatters sto helpus o helpus o make the things we need, or fancywe need,orjust need,orjust fancy.So fancy.Sowe wemust must make t very clear to outselves what we want to achievewith achievewith our design,which whichwill will obviouslyaffect obviously affect ts shape, he materialswe materialswe useandall andall kindsofother things. If we wantto wantto buildsomething sthis s this the right place or it, could couldnot not our purpose be better achievedn a differentway differentwayaltogether?Only whenwe Onlywhen we havesorted havesortedout out all these matters to ou.rand ou.randour our clients'satisfaction clients' satisfactionwill a structural structuralana.lysis becomerelevant. Obviously, Obviously,wbat wbatII would wouldcall call dzsigz dzsigz s muchmore muchmore mportant thar structural analysis, or that determineswhat determineswhat we are going o get for our efforts. And moreoverraial moreover raial we wedecide decide o do s much muchmore more mportant han hanhow how to do it, and hat opens Jresluke Jresluke va.lveor a whole lood loodof of questions, ocial, political, ethical which us allwith all with confusion, r worse,because worse, becausewe are not able o agreeon agree on lreaten wbat to dowbatto

How to live in peace itl or:r: eighbours n his pJanetwitiout destroying destroyingtt is the ultimate and pressiugproblem problemand wish knew he answer. andnow now pressiug Ove Arun

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Preface

This book is aimed at the identification oJ the fundamental princiPles of structural analysis together with the develoPment oI a sound understanding of structural behaviour. This combination leads to the ability to arrive at a numerical solution. ol lanSuageo Using a series of structural diagrams a s a visual lanSuage structural behaviour that can be understood with the minimum oJ textual comments, the book aims to develop a qualitative understanding of the of the structure to load. It is ideally suited to under8raduates responseof response studying indeterminate framed structures a s Part of a core course i n civil becauseo of its or structural engineerinS' but it is also suitable, because qualitative approach, for students of architecture and building technology. The book is in two parts. Part I' the first lour chapters, deals with the development o l qualitative skiils; that is' the ability t o Produce a structure. It is non-numerical solution t o the loaded line-dia8ram ol a structure. considered that the ability to arrive at the qualitative solution to framed structures is a significantly imlortant component of the overall understanding of structural behaviour. Part I I deals with current methods o f structural analysis using the diagrammatic format to which the student has become accustomed. The need lor the developrrent o f qualitative skills increases with the increasing use of the computer in design offices. I n the near future, t h e computer will replace the majority ol analysis and structural desiSn

calculations. Unfortunately, this will also have the elfect of eliminating understandinggained gained by the student much of the experience and consequent understanding and trainee engineer. develoPed along with This work explains how that understanding s develoPedalong current analytical procedures, PreParing he student for the design olfi'e

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design data' where the computer ls rne source o f virtually all numerical Analgsis is an inteSrated approach to th e Understandinq structural analysis, ol which teaching and learning of the PrinciPles ol structural are also this textbook is a major part. The ideas embodied i n this book available in an audio/visual series of sel{-learning programmes o l the same name. The audio/visual programmes are backed b y a suite o l micro-computer programs which have been used t o produce the numerical and SraPhical solutions t o the Practice problems, included in this text' software a r e The audio/visual programmes and comPuter-aided learning available from: osE L t d 197 Botley Road OXFORD

ox2 oHE, U K 0865 26625 TeL0865 TeL Publishing' who should be contacted direct and not through Granada

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Acknowledgements

The research project upon which this book is based has extended over a period

years. ol ten In that time, many friends and colleagues have contributed to the dev elopment of my ideas of the way in which students c a n be encouraged o reach a better understanding of structural behaviour. Bristol Polytechnic has provided both time and resources resourcesand and my Head o f the Department, Dr Matthew Cusack, has been particularly supportive. These ideas would have been stillborn without the continuous slrpport, interest and encouragement of Peter Dunican, senior partner of the Ove Arup Partnership. Many other engineers n that remarkable organisation have helped me with their advice and constructive criticism. Perhaps the most successful period for the development and testing of the qualitative approach as a basis for the explanation 01 theories and methods ol analysis was the year I spent with the Department of Civil and Structural Engineering at Hong Kong Polytechnic. I owe much to discussions with Dr Kwan Lai and Dr Norris Hickerson. but most of all to the resDonse l the exceptional students.

However, it has been the extensive and particularly fruitJul collaboration with Professor Peter Morice oJ the Department of CiviL Engineering at Southampton University which has led to many of the specific explanations and visual sequences n the early part ol the book. I am indebted to all of them.

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1 The Analvsis f Staticallv Determinate tructures

The subject of this book i s the behaviour and analysis of statically indeterminate structures.

However, this first chapter reviews t h e

behaviour of deterninate

structures, a thorough thorough understanding of which

is essential before the topic of indeterminacy can be tackled.

The text

assumes a basic knowled8e of mechanics including an understandin8 o f the principles of overall equilibrium, bending moments, shear and axial forces. It is possible to analyse determinate structures by consideration of equilibrium - i n 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 o l redundant members (secondary load paths) makes it necessary to consider relative member delormation beJore a solution o f the structure can be

attained.

The number of unknowns which cannot be lound Jrom equilibrium

considerations is known as the degree oJ statical indeterminacy. The design oJ engineering structures usually starts from a need to sostain loads. Initially though, it requires an understanding ol the way i n which a proposed system of members can provide the required support, and how it will deform. It is, however, clear that an understandin8 oi the behaviour of statically indeterninate cf deterrirrate

systems is based upon a thorou8h appreciation

systems.

This chapter develops the relationship between load and delormation for a range of structures which are amenable to solution by the application of equilibrium alone. Once we have analysed the behaviour of the proposed structures \re a r e then able to start an approPriate process of numerical analysis to llni

cJ:

 

SIRUCTURALENSIYSJS UN'A,RSIA';]:;G SIRUCTURALENSIYSJS how much of each of the various parameters is involved, Jor example, the values ol the loads carried by each member and, as a consequence, he srze each will have to be t o carry its load saJely. \ve can g o on to fi nd the values ol deformations which will result Jrom the loadins.

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TI]E ANALYSIS AF STF.IICALLY DETERMINATE STRUCTURES

Thus we see that structural analysis must have two .omponents. The first is a qualitative understanding understandinga a nd ::e second he numerical procedures. I t must be jderstood that qualitative analysis s not in any way a ,i:icsritute lor numerical analysis but should be regarded 3: a necessa necessary ry complement to it so that the two allaoaches constitute a complete whole giving an -..erstanding of, and an ability to evaluate, t he :::icrural perlormance.

\k

M

QUALiTATIVE

-, M A-

ttu 5reo-3€

NLIA,1ER-tc.\L

:-e oasic principles of our structural analysis i e in the :a i sratics. Mosr srrucrures --f are required t o be :-:---3 r n a static state. This is not to say that w e :_g-::cianalyse :_g-::ci analyse dynamic behaviour such as may be caused : . --::rhquakes, wind gusts or moving loads but initially, a: -e3s:. w e shall concern ourselves with statics. The ::aie: c.ane demonstrates the kinds of equilibrium ::r':-::cns which we have to satisJy. The vertical _..::::an ar the ground must balance the total downwards ::.:?::

VELTIC'AL

EoUILIARIUM

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

by horizontal support lorces at the

-.esistedj :1:s \r'e call ho.izontal equilibrium.

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

+

I1ARI:ONTAL I1ARI: ONTAL @UILI 5P,JUM

---:_-1. - : :S Clear that the counterbalance 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 add

E :rs :-e :.Citional toppling eilect oJ the wind Jorces r:

lei =E::l:e base oJ the crane will have t o provide _E\s= -,= :c :hese out-ol-balance toppling moments. -r1s =* iioment equilibrium. -r

 

/\^oMENT

Eeut .tE;ui/At

ANALYS ALYS I S CTURAL AN STRUCTURAL UND' UND' ?S' IlE DJ NC STRU

In our three-dimensional world we can express these equilibrium requirements i n the following way. Firstly' w e must ensure that i n each of three directionst at an8les to each other' which we will label x'

Y and z '

the resultant oJ all 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 toppling eifect, 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 all together six conditions of equilibrium.

6.

In much of the following explanation of behaviour, and in many real life engineering situations, we Jind it is

possible o be sure that the lorces in the direction v are zero and that there are no moments of lorces about the x and z axes. If, indeed' this is the case, then o u r problem can be reduced to the consideration o f three conditions of equilibrium only. Such simplification i s all forces lie in becauseall described as a plane problem because one plane,

7.

we shaltr ind it convenient to label the lorces in the .r direction with symbol d' to denote horizontal, and those in the z direction v , t o denote vertical. Also we shall use the symbol l , for moments in the plane, about the

t

t {

8.

Y =a VarttLt aluilihri.t

Y axis.

Vith these symbols we can write down the three

of equilibrium all the horizontal orces must sum o forcesmust must sum o zero and he sum zero, all the vertical forces oI the moments momentsmust must alsobe also be zero.

H'o

Hon2o,{e q.fb/ilrf"1

M.o

iqon',t,f ?uilibrt-t

www.engbookspdf.com TEE ANALYSIS AF ::;'':'."LY

 

\ow it will be remembered that a pure moment, =lled a couple, can be represented by two equal :^d opposire parallel Jorces a r 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 Iirst the point A we see that as ---,=Cownward force tr passes hrough A it wiil -Eie no moment about A and all that remains ::::e ::.ae

anticlockwise moment ol the upward = at a distance d. The moment about

DETERMINATE STRUCTI]RES

r

- : Y A = a x d anticlockwise.

Considering

:_e:oinl

B the upward forces pass through

:*e:olnr

and the only moment i s caused by

t

:*e :tr\r'n*ard force r at distance d. This :--:.. i ves an anticlockwise

moment

.s f,"^( &". r , o ) 1-

-. :o,* consider rhe moment about a point C a-=:_,ae: from the line oJ action oJ the upward -::

E'e can also find the total moment of our = about this point. For the downward force

::r=:r:-

I

: =€-e -5 an arriclockwtse momenr due to rhe j plus e, and for the upward force F a ]:1:.:::i. :-\:cra.,= :roment due to the lever arm e. -as*-:-: ie

:aie shovn is one oJ the most powerful

e.

.) rl- F o

-q )

:noment about any point in the plane,

].e ?:r*--:r:ri:.  t r-:

{'+:: <

'l

:: ,-:-..::-:.al aralysis. namely that. as a couple

],as :-€ =-e 'r:ra

i s the force F multiplied by

C :F_-:a:;oa j . still, of course, anticlockwise.

I_€': r : i,=

=omenr 4

I

The

F

-,:r-3:::

cf a structure will require that t h e cf all Jorces must be zero about any

- ---e :iare.

_':a :r:rsiCer an actual structure. The beam ABCD -r* J s 3,Trc:r i,5 - - ed at A and B and loaded at C. We shall E '*r=:-E .::ec- s of any self-weight and only study those :j.tc :: :-E ::a-ied load

t/.

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DI N G STRUCTURAL A.VAIY5I5 UNDERS'IAN UNDERS'IANDI

We can start the qualitative analysisof analysis of the structure by observing hat since there must be equilibrium of moments bout he support supportA A the only orce whichc whichc a n balance he clockwise moment of the load t1l t C will be an upward support reaction at B, giving the necessary anticlockwise moment. Furthermore, since the support is on rollers, only a vertical lorce yB can act at this point.

1 4 . As the lorce 1/B 1/Bhas has to produce a moment about A balancing that of the load , we can observe that it will have to have a magnitude greater than t/ because its lever arm about A is less. Thus if there is to be equilibriumof equilibrium of vertical forces, he vertical support reaction yA at A will necessarily be a downward force

tr."

to balance the excess of yB over ff.

15.

Ve can also alsouse use he requirement hat moments um to zero about any point in space o confirm this result. Taking moments about B we observe hat there is a ,1rat clockwise moment due to the load ,1r at C. A balancing anticlockwisemomJnt s required. Thus y A must act downwards.

Mv46o 6o;;

16.

shouldobserve Lastly, we should observe hat as neither neither the force kr nor the reaction yB has a horizontal component he requirement f horizontal horizontalequilibrium equilibriumensures ensureshat hat the horizontalsuDport eactiona horizontalsuDport eaction a t A is zeto.

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TEE ANALYSJS C- S::::C?LLY

I7.

DETERMINATE STRUCTURES

The concept oJ equilibrium also applies t o any part of the structure. I f w e consider an imaginary cut separating on e part of the structure from the rest, it is clear that rnternal forces at the cut will be required t o maintain equilibrium. Let us suppose hat we separate part o f t h e structure to the right oJ B by introducing a cut at K.

lw

"t' I

cu'f

lE.

I f the portion o{ the beam KCD is to be in overall vertical equilibrium then there will have to be an lnternal vertical force S at the cut point K to balance the load rf.

It must act from below upwards on the part

'l'

tw

--*--D

KCD and have a value equal to the load, This internal force is called rhe shear force-

{v="

I S shear s = w fo"'

The position at which we choose to place the cut K between the support point B and the load point C has no eflect upon the value ol the shear force, which will be constant from B to C. To the right of C there are no external forces and so the shear force reduces to zero i n the portion oi the beam CD, W e can plot the distribution of s as a shear Jorce diagram, drawing i t underneath rhe base line, that being where we have shown the shear force arrow.

,.,

The diagrammatic convention for shear force is that the value of the shear iorce is plotted on one side of t h e base ine, above or below; that is corresponding o t he vrew an observer would have sitting behind the cut K. The load t r would cause the part oJ th-^beam th-^ beam KCD to fall lrelor,, the beam, thus the dia8ram is plotted below the base ine. The arrow on the base ine is required to confirm this convention.

www.engbookspdf.com 1..; J}E?S:}:i )J EG STRUCTIJRAL ANAIYSIs  

21.

ls

a----l--+-r

tW

For a cut K in the beam between A and B w e see that the greater magnitude of v B compared with t { will lead to vertical equilibrium requiring a shearing orce . s from above acting downwards at K, b eing the dilference

/<

4\

between

th"^ W)

Cl4g;

YB and t'r',

' 4 .. zr=o

Again the choice oJ the position o f K between A and B does not aflect the equilibrium equation and thus s will

lw

 _.i_-_o IA a

also have a constant value oJ shear force between these points.

'lva

Yua

Supposew Suppose w e had considered he vertical equilibrium o f leJt-hand portion of the beam from K t o A. You will see that this will lead to an upward shear force 5 to balance

i-l'-

V

the only vertical force on this portion of the beam, the

ls

suPport reaction vA.

Vp, -

24.

_Al_

:- :-

This is a necessary result because, when we close the cut b y putting the two sides of the beam together, the t w o

ls

shear Jorces must cancel each other leaving no external out-of-balance vertical force at the cut section.

Ir

rk

www.engbookspdf.com ww.engbookspdf.com THE ANALYSIS OF S:Ai:CALLY

DETERMINATE STRUCTURES

-: r1lv remains to be clear about the plorting convention

 

::. :re diagram oi shear Jorce distribution. We have have a-:\rn a horizontal arrov/ pointing to the right to indicate --::

$e were looking at the force acting on the ri8ht-

-:-.: side of the cut. We could equally equally well make the =::r* point to the left and look at the force acting on

*+=f,-

:_e -eir-hand side of the cut when clearly the shear force :-:i:am

in this case becomes a vertical reflection of the Firher.iiForam is..)rrcc

but i t i s i'nDortant

:: -...:rde the horizontal arrow to show which way we are

:5-

-:: -:-.now return to the moment equilibrium at the r-:a::rarv cut point K between B and C. We se e -F_ -'e loFd l, Dro.i||ces ] .lockwise moment about . :-:\i as ,^t.,,y$ and lor equilibrium. rhis musr be w \, _:sj:3i bl an internal anticlockwise moment within :-E :e3:n ar the cut K itself. -;:+r ------

_-.ol|oh K ir .,rn m^ke no conrribution ro

ls r:

"

1W

t[tr,

--- - - ---Alzr-o

4czl Ma(x)

{ u6c1=o

cd,,itihrir,m. Thiq internal momenr is called

:E:=:::-::r ll-

As the shear shear force , S

" < t[-

\l

is always

::' . -L-'1rr^

"

No uo,ne*

:esrrained in positi on and both members

fua$fl

slope independently of one another.

. : :::?: -

 ll@r"

*

r, -:-:-: '-- r _:-.:5:

_:,e:se, the hinge, is fairly common i n :::'ar.

(?**\( f,

the shear release and th-'axial

L-

.ut,s e,. _: - -::e. are only likely to aPPearas aPPear as part of a '1" ,.r,:lr _ ::- =-=-\:ical procedure. I n each case t wo i,_:"?ri :5-_1_:: ::e lransmitted and one released'

snea fitaxi*

el\r,

--€

Na a x1.\l firco

r u b-t::-er: :r: a' &,':fr.'---:-:&,':fr.'---:-:-

Uawfet

::i.rciures are almost always l-:

e\ceDtion in its many forms is the

r- -. lere are f our ext ernal reactions, on e .llirn ::-_ -.. _; ::.r'ec b the three equations oJ iae hinge provides another equatlon €r, r,:r- r-- ::;:1::. ti'tf.*.r"-::::

:-: : ::ioinent equilibrium i s considered at _:::::t.s HCand VC orovide the fourth

l l ylrL--,:r,| _ii' t-{-_: -

l a -:_.

mt"ii 'r_ : ::-,

, u 0 tt-i -' " € -

l :::--:

li'llu dr',{:1-j-r

:_i.

lllG vi*|:

:

r:

frl _lr:-irE _: "s:

:

::l e q u a l i ta ti ve

:,: :.,i

behaviour

element -:j:a-. itbrium. :E - =ar -3cn

'+\.

/" UA

u n d e rsta n d i n S of is the concePt or member

ol

re+j1w

in a

vB\

*"* \'

4.r-

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, :: 1'::

A NDING STRUCTU AL AN AL YS S

13.

I t is convenient to separate the concept of indeterrt into external and internal ind eterrninacy and we wil, first at external indeterminacy. A third support ha-. added to the simply supported beam. This means

beam cannot be solved by the three equations of equilibrium, The structure is then said to be: st at ical lg indexer njnat e. The subsequentstudy subsequentstudy of the analysis of structures \ provide us with the means to determine the value o: fourth reaction, but for the time being in this

s ^ J-i /A

#n,,v{

/w.

ar F . oncer neo w it l' t he. o- r . eot

---

/

a_d lat er t he degr e:

indeterminacy. Similarly, the cantilever may be made indeter

' rP

by the introduction of a vertical reaction at B i ^+^

r

nr

^n ^a.l

-'n+i l a\/ar

The three-hinped arch has been fixed at the crown. Although this diagram shows a load on the arch it si be emphaslsed hat the degree of indeterminacy is 3 property of the structure

and not of the loadin8.

1 6 . Because any ioading will cause the ribs of th e arch i: to spread, inducing a horizontal reaction, there will always be four unknown aeactions n a riSid arch, the vertical and horizontal reactions at each support.

www.engbookspdf.com STATICAL STATI CAL I NDETERMIN NDETERMINCY CY  

Tlllh rj.:c--ar

importance of the condition oi

2:

der"u-eq

is that the structural properties of an

ririb..,t'i.:-e ri@6.,

srructure will affect the distribution oJ -r1s is not so with a statically determinate

@. -.-:;.

iirtcl [r-

:

rcf

rc

"k thnu-

s,rpporred beam, supportin8 a unilormly

ffirc

,cad will produce the same bendin8 moment :e€-Jdless oJ the varyinS dimensions of the

af -L

T:€:rsiribution

b.. flwllll

aE:e:aninate structure are unafJected by the

Ea- i 6rr io see the importance of indeterminacy.

c

lrlfr

cE'EE s{.T[,ort. Imagine the beam at the support

q

a_'a F?. re

oteM

so deep that virtually virtually

no .load was

enc supports. The hogging moment at B

/,\

E g=:z:e. ihan the sagging moment in the spans, 3

hE

BC would be actins almost as cantilevers.

-<

-7

fiansferred t o the ends of the beam

U*g*.es:s

1E ].-aa s'ould be carried by the centre support.

[0 ffi

crcr--*ances

b S

W

aeam shown has a greater structural depth

*, m..c::

mlllfl

ffi

:: i;re slructure.

&Eugn:s

lb

of forces and reactions i n a

h

rmel

the hog8ing bending moment at B

compared with the sagging moment in

lFI6icEBC. 1fre

Er:g:e e\amples illustrate the dilemma for the th e

re@i

::-:rer.

SrrE

:+:_:::

He must know the size oJ the rhe analysis is carried ou1. yet t he

sd

a[G -r-e.-f,ers is the object object ol the analysis. The

lnr

rs alE: :re designer must always make an

Lfiifr-

.+

&a-= ive

tdlnrman=--re.

analysis of the structure before

numerical analysis can be carried out.

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"

I

)0

iJRDERSTANDINGSTRUCTURAL A?VEIYSIS iJRDERSTANDINGSTRUCTURAL

The concept oJ a 'tree' as a statically determinate structure is a particularly powerJul one in the qualitative analysis of structures because t is na and easily recoSnised n other structures.

 

Y^

34. 34 .

The effect of a l oad on a tree is always that of a cantilever, The efJect oJ the load goes straight to Sround. For this reason t is ojten preferable for u s convert our indeterminate structures into an anal .E Itree'. The resulting load effects are the simpler to identiJy and analyse.

We will now look at more complex frames, which represent the complexity oJ structures met by the structural desi gner. The two-bay, two-storey fixed at A and f and pinned a t H. All other joints rigidly connected.

3 2 . This two-bay frame has eight external reactions at H, and J,

Therelore the pxLprn-.z degree of

indeterminacy i s : 8 - J (equations of equilibrium) = 5 times indeterminate. ,l 'l '

A

tvn

M7

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r=

32

: r;:: i::ilrrlvc

ANALYSIS

The top Jrame BCDEF is reduced to a statically determinate Jor m by the removal of the moments

 

and F . The lower frame by the removal of t he restraint at H. The structure is therelore 3 times indeterminate.

This next exampie of a two-storey frame, has a o indeterminate sub-frame ABEFC, supporting a n indeterminate sub-frame BCDE. We will now system of hi nges to release the sub-structures t o statically determinate f orms.

W e will introduce hinges at B, C and E. Note thar introduction of a hinge between the tlree is, in fact, two separate hinge releases. Each o f sub-frames i s now a determinate three-hineed frame and the structure is 4 t imes indeterminate.

f==lf= =l4 0 . I f there are less than three external restraints, n o matter how many internal restraints, the structure fail. In this example, with horizontal ioller releasei both supports, the structure will ro1l, .e. this is a

>^r

,/t ,fve Metlta^6^

^a

state of unstable

equilibrium. Beware o f the soDh: the notion that the structure could be stable under

,vs

tiarz.^iaL

exactly vertical load. Indeterminacy o r stabilitv is a ,ul,eti'|'

property o J the structure rot

the loading

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STATICAL NDE"':i.:: ; : : --

:arvert the whole whole structure into a 'Iree' we must . - -: F r^F f.:m- RCnF A , , , r will .elea5e r h r e e

- . - ' \ r n rnrernal for.es. , F . " F d1d Yf. ^f

in.la+ar m

in:- w

Tqu( l h e

iq.

do|e( utgko

tr.z

3 x

: : :le supports are pinned and the structure cannot be _::-:3i ro a tree because becauseof of the absence absenceoJ oJ a Jully fixed -.::

_ ' : . :he.upporl( ma) be reduced to a 5tati.ally

:::::.:,nate

_-:::jcture ::-

two-bay frame is pinned at condition. This two-bay

may be reduced to a statically determinate

:, lhe removal of the vertical and horizontal

-a:rr.is

at H and the horizontal restraint at C. The

:-:-:r3:-re ABCEDG is then statically determinate, with -:: :::a.minate cantilever frarne EFH fixed to it. The ,:-

:,:e is J times indeterminate because t is necessary

-: _: ::5e rhree reactions to reduce the structure t o a =:

iitu

-.

:1-'. determinate f orm.

--::a.e 'Jrames may be solved in a similar way by . , - r i e \ nl \,.iri( allv dc crminale sub-frames. --. - : ' _--_-:rorev lrame has a hin8e at D and has pinned

FHI vH

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STA?I CAL I N NDETERH DETERH N C

29

\ s a n alternative means of reducing the structure t o a ieterminate form, a roller release has replaced the pin 'oint at D. This particular release should be noted since : i is frequentl frequently y employed in analytical procedures. Thus . o horizontal reaction can be taken at support D. lecause the horizontal forces must be in equilibrium, ::e horizontal reaction at A will 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 i s unaffected. :ich releases releasesare are rarely employed either in practice or i n :-€1 r" i cal proced ur es,

r- ::oment reaction at A has been introduced to the :,:rial frame turning it into a fully Jixed support, an d : e structure is now 2 times indeterminate,

2 x sGtti

-:roment dia8ram with the most obvious points: l.

bending tension on the left of the Jixed support at A, reduced in value by the h^r

i?^n+21

r a2^+i^n

' ' *"""' ' '

t .,A,

hogging bending in member CD,

3 . bending tension on the right-hand face of the column EC . moment diagram is comPleted and comPared The bendin8 bendin8moment moment a t C bendingmoment rirh the reactions. Note that the bending :n member CB, is equal to the sum of the bending :noments n cD and cE at c.

Y9

H4

I e can now add the effect ol the Point load in column the -1,8. This ' simply supported' effect is ' added' t o moment at A and B .

HE

parti

i: moment bendinBmoment \ote particularly that the slope in the bendinB diagram i n column A B is related to the horizontal reaction at A. Similarly' the change n slope in BC is

V fot^

.elated to the downward reaction at A'

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f

70

UNDERST UNDE RSTAN ANDI DI NG STRUCTU RAL AIVE'YST.S

This gozder ruie

 

{f

f,'," sloye i bf,ldlnl

\J J

t^dtrrc*C

will remind you to carry out this

important check on the qualitative solution.

Lhc444/4u

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

Iil

u

t

lr

ri rill

lll , u

t,rl

l:ii

l;

ti t

49 .

'fya t v o

M

HinSes n frames may produce unusual unusualsolutions. solutions. This orthogonalframe orthogonal frame ABCD has hasa a hinge at C. The structure sways o the right to release h e moment momentat at B . Th e

dirirt

solution s unusual unusualn n that there s a zero bendins

lErG5B{

at the internal, rigidly jointed connection at B, The horizontal

fb

etpL

rays

abt

reaction at D is

zero because moments aDout C, considering member CD, must be zero. Thus th e horizontal reaction at A is zero and the bending moment at B, due only to the horizontal reaction at A is zero.

Ve will now study the frame ABCDE, Ioaded with a

.-lcdly

load at D exactly over the reaction at E. There is a

dEe

hinge at B.

iqect

d is

d

tjil

)t -

Thisstructure This structurehas hasa a n unusual quilibrium n that the vertical reaction at A is zero. Taking moments momentsabout about E,

The expli

in CE. n

there is no out-of-balance moment due to the load since the line of action passes hrough E. AIso the horizontal

l

member.

reaction at A is zero because becauseof of the hingeat hinge at B. Since there s neitherhorizontal horizontalnor nor vertical reactiona t A there are no bending bendingmoments moments n either AB or BC. The bending moment due to the load bendingmoment load r.,in r.,in member memberCD, CD, is resistedonly resisted only by member memberCE CE .

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QUALI'rATIVE aS

ANALYSI:

"

7I

"':1:S

:":r what happens o the deformation i n B C? W e can see

i 3'a ,

--:rat oint C must rotate, which appears to require a :ownwards reaction at A , through a shear force at B , for eguilibrium.

lis

T'neexplanation T'ne explanation is that the structure sways to the left

trture

and joint C sways and moves downwards as member CE sways about E . This allows C D and CE to bend, but

txhent

leaves B C and AB straight.

: bout ment IL

poinr

L

orce in memberCE. memberCE. B u t compressiveorce Clearly here s a compressive there s only one vertjcal reaction at E. We need o

w

inspect he force equillbriumof equillbriumof this suppor support. t.

Nce -,,f\ lY g

)).

The explanation is that there is an internal shear force

rt E,

in CE . It is show shown n here as the action of the joint on the

ince

member.

ftal F

he

www.engbookspdf.com 72

UNDE RS?A.NDIN RS?A.NDIN G S TRUCTURAL AMAI,YSI"5

56.

The resultant of the exterra.i

reaction to the shear and

axial Jorce is equal to the vertical reaction at A.  

\

0*.:b-

) 5E

I

T 3.

-

+"

57, 58.

problens

Produce the full three-part these problems.

{*

lI +

n "f'J'n I

-&

f]

Practice

"{a*

solution to each of

Computer solutions for

similar structural dimensions and loadings are Biven.

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t h"

Part I

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5 The Theorems f Virtual Work

Tl9 k9l l. the soiution f .1.,"19t9" M. " gti911l_, d.19 I]let.

abiliry to delermine structural dg[gllnations.

In one meLhod meLhodof of analysis,

the deformations provide the basis of equations of compatibility which, in addition to the three equ ations of statical equilibriu m, a.llow he solutron of the unknown effects and the lull solution to the distribution oJ internal forces and moments upon which the subseq uent uentstructural structural design s based. In addition however, the deflection of a structure may well be a design criterion. l.

All structures must satisJy two basic states of loadingl

Serviceability when the structure is subject to its working load. It i s at this state that the deJl ection of the structure i s checked.

2.

Ultimate load, where the failure strength of a structure is compared with the serviceability load multiplied by a load factor.

There are two basi c approaches o th e anal ysis of str uctural deformations, .fteiq ClStgy gl9 u-tltual wotk- The latter has the adv advantage antage of being able to deal with congi1lo : qlhgr ltat b9 e lvltlrn 9 glagtic range - a limitation o s rai 'i elergy. Only virtual work will be studied here as it is generally agreed to be the more powerful of the two. The student should bear in mind that although virtual work calculations may be the key to the numer ical solution of a problem in structural

analysis, the math ematics co ntent is relatively trivial.

The difficulty

encountered is the application of the concepts oI virtual work to the problem in hand, Once that can be understood qualitatively, the subsequent calculations although time-consuming, are straightforward . Without thrs qualitative grasp, methods like virtual work may quickly become an exercise in mathematical manipulation and the overall sens e of the method and th e behaviour of the structure, lost,

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76

DERSTAN RSTAN I NG ST PUCTURAL edetyS./ S UN UNDE

l.

_,

lw

\A

I

s.--l--

In a real structure two conditions must always be

Similarly, we

satisfied: l. --3-ft-

\ and B. Th

The sysrem of forces acting on the

slppor:t con

structure and the internal forces must be

capable of sq

r n equilibr iun.

R.oal stru r heor em to

d PPL

statically indeterminate structures using the problems in Chapter 6.

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6 The FlexibilitvMethod

The Flexibilit y Met hod is one of t he t wo main approaches approaches o the analysis o J redundant st ruct ures, t he ot her being t he St ilf ness Met hod. Bot h met noos are based on ident ical t heories of mat erial behaviour and dif f er only in t he mat hemat ical t reat ment of t he basic st ruct ural dat a. Neit her met hod t s part icularly suit ed t o t he hand analysis of real st ruct ures since t he conclusion t o bot h met hods oI analysis is a set of simult aneous equat ions, dependent upon t he degree of indet erminacy oJ t he st ruct ure. In a real st ruct ural Jrame, t he degree of indet erminacy is likely t o be large and alt hou8h theoret theoret ically possible, his hand analysis s t oo lengt hy and error-prone t o be a f easible design of f ice procedure, If such an analysis is. equired, a comput er program based on t he st if f ness met hod is t he mosr likely course of act ion wit h t he ready availabilit y of such proven sof t ware. Alt ernat ively, in real st ruct ures t he symrnet rical shape of t he f rame

ay

allow an approximat e analysis of f orces and moment s or t he use of an int erat ive met hod such as moment dist ribut ion. This is based on t he st if Jness approach and will will be discussed discussed n Chapt er 9. The analyst must always bear in mind, however, t hat he will, will, at best , be approximat ing t o t he behaviour of t he real st ruct u. e. He will be creat inR an analyt analyt ical model which is a t heoret ical model of t he complex, real st ruct ure, The advant age of t he llexibllit y met hod is that it cont ribut es signiJicant ly t o an underst andingof anding of t he real st ruct ure ure because he procedure of t he met hod, t hat oJ reducing reducing a complex st ruct ure to a st at ically det erminat e f orm, is closely relat ed t o t he f undament al design decisions oJ rat ionalising t he real, complex and highly redundant st ruct ure int o a simpler f orm suit able f or analysis. To some ext ent t hat simplif ying procedure is likely t o be adopt ed even when a comput er is used. Three-dimensionalst Three-dimensional st ruct ures are usually reduced reduced t o a series of linked t wo-dimensional f rames, rames, f or example. These frames may be reduced to a

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IJNDERSTANDINGTRUCTURAL 'VA'YSTS

I i1€

beams' T he series of sub-frames or lurther simplified to continuous of these simplification of the real structure which allows the creation

re5_-5

The: i

the behaviour of analytical forms mus, not yield a solution less safe than this understanding hat yo u the real structure and ultimately i t is towards

ihe c

horr

must see your study of the flexibility method directed' lor which th e The flexibility method is based on structural deformations

loacsg et i

Theorem of Virtual Forces is used'

equa

i he stari the fl detern{ structrl deterni applied bendin vert lca to a ca w]In a

Now in which reactt(

positio

the val beam

www.engbookspdf.com THE FLEXIBILITY METHAD

 

Tlre two principles o n which the method of flexibility rests wi.ll be illustrated with the propped cantilever A B.

99

The Jir st step i n the analytical proceoure t s to determine the degree 01 indeterminacy. There is a potential hor izontal reaction at A; however, for this par ticular loading arrangement i t i s zero. There being Jour notional externat reactions the propped cantilever i s _ 4 3

equations of equilibrium

-

I x statica statically lly indeterminate.

The second step is to release the srructure t o a statically determinate form. The reason Jor this

is that

the flexibility method depends upon the ability to determine the deformations of the released determinate structure. The Theorem of Virtuat Forces i s used to

determine the deformalions and this method can only b e applied t o a statically determinate structure i f the real bending moments and other load effects are known. Th e

s--U-,' <

\+*-'xa,"

ver tical reaction at B is removed r educing the structure

to a cantilever beam. The structure deflects downwards with a deflection of AR at B . Now imagine that we know the value of the reaction which has been removed, yB, and we apply it at B. Th e

r eaction y B will return the structure to the or iginal position at B, exactlg, because that is what defines

the value of that reaction, the Jact that it holds t he beam in that position with a zero ver tical deflection at

+-

f Va

"-E{

V6

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STRUCTURAL ANALYS S UN UNDEP,STANDING DEP,STANDINGSTRUCTURAL

$-+"To""

4.

However, we do not know the value of yB.

It i s th e

object of the analytical procedure. If, however, we apply a unit load or action in the direction of the

removed reaction the.ate

S

of deflection for the

application oJ a unit load applied at this point,

fr.".'

i.e. d B B per unir load, may be lound. We also t-v determine the deJormation due to the applied load in the direction of the removed restraint.

These

are the second and third stage in the analytical procedurel

2 . Determine deformation due to the application oi the design load Aa i n the direction of the removed restraint.

3 . Determine deformation(s) resulting Irom the application of unit load(s) and/or reactions, 6BuBu.

Not" particularly that the unit loads

are applied in the direction of the removed restraint. To complete the analytical procedure, we apply th e principle of superposition. This principle allows us to add separately identified load effects together, th e final solution being the sum of the parts.

Abr* Vr.fe,er=O

Th e

separate effects are the removal and replacement of vB.

One efJect is the release producing th e

deflection at B, ABv. The other effect is the deflection at B due to the unit load at B.

If we

multiply this rate of deJlection by the unknown reaction yB it is clear that these two deJlections ABv and yB. 6gugu must have the same ya_tue,

6.

The last stage in the analytical procedure is the setting up and solving of the equation(s)oJ equation(s) oJ compatibility (there may be more than one degree of indeterminacy). The equation of compatibility is equated to zero because he true deflection at B of the original structure is zero.

 

t0l

THE ELEXIBTLITY METAC} --: ieforriarions which result from the anal) tical :' :€edure are given a sign appropriate to the overail ::r' \' ention. In this case AR_. would be negative and ::-, Bv positive. The choice of direction for the Ltnit :.:a and thus the direction of the Lrnkno wn eaction y B

S--= j

tve L+'

i a

: ::bitrary because n a more complex structure structure the true :.:ection will not be as apparent as it is in this simple : =mple.

36

+--

For this propped cantilever however, it is

::i ious that

Y B will act upwards.

: - ? s o l u r i o no n o f t h e e q u a T i o no f c o r n p a u b i l r t ) i c r h a t v B : )ositive. This means that the choice of direction f o r ---e unit load (Figure 4), is correct confirming that t h e : :e.tron of the reaction /B is upwards. l' the stgn been negative then the direction of the reaction -:d : f,uid have been opposite to that chosen or the unit load.

Agu + Vg. f,"ner- o (-v.)* "'

VE +vo)=o Yg rs +vo

1 e must remember that the evaluation of the unknown :eaction is not, normally, the only object of the :nalysis, Usually, the distribution of bending moments is may be apPlied superpositionmay :equired. The principle of superposition ' ,rere oo. The bending moment diagrams which result :rom the application oI the point load and reaction at B bendinSmoment 3re combined to produce the final bendinSmoment iiagram.

I

ITiTfirfi-

Ilirr-*

rc

\' iost real structures are highly redundant a n d w e will now examine the application of the flexibility method to the beam ABC which is 2 x indeterminate. S t e p l. l.

Reduce the s stt r u . L u r e t o a staticallv

determinate formi remove yB and vC.

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ta2

UNDERST UNDE RSTANDI ANDI NG STRUCTURAL E?r'AI,Y.S15

ll.

Step 2. Determlne delormations due to to the application ol the real load(s) n the direction of the removed restraints: ABv and ACv. Try to see the flexibility method as collecting delormations due to the real and unit loads at a particular point, in a particular

To

direction.

12.

Step 3. Apply unit loads and determine deformations in the direction

of the .e,,noyed eactiors.

Now we

can see the reason for the rather cumbersome system ol subscripts. It allows a clear delinition oJ a particular deJormation. Note the complication oJ

rne

more than one degree of indeterminacy. Th e

b ya

application of the unit ioad at one point contributes

Deattl

to the deformation at another.

13.

Interpretation ol subscriprs. The

17. Fo r

capital Creek symbol A is used or deflections due to the real loads and the lower case 6 lor deflections .esulting lrom th e application of the unit loads.

Deo L' tj1.. apph