Understanding Structural Analysis

I -

i

Understanding Structural Analysis DAVIDBROHN

PhD,CEng,MIStructE Princibal Lecturerin StructuralEngineering, BristolPo\techruc

Forewordby Sir OveAruP

GRANADA London Tbronto SYdneY NewYork

Contents

Preface Achnouledgements Part I The analysisof staticallydeterminatestructures 2 Staticalindeterminacy 3 The qualitativeanalysisof beams 4 The qualitativeanalysisof frames I

Part II 5 6 7 8 9 1O 11 12

The theoremsof virtualwork The flexibilitymethod The stiffnessmethod- frames The stiffnessmethod grids Momentdistribution Plasticanalysisof planeIrames The yieldlineanalysisofreinforcedconcreteslabs Influencelines

I

3 22 38 57 73

97 114 137 149 178 202

22r

Afpendix: Solutionstopracticebroblems

232

Index

282

Foreword

A force is not just a straight line with an arrow headat oneend. That is just a convenientabstractionor shorthandfor whatin reallife tums out to be abundle of particles under stress andstrain, alwayschangingandmovingunderthe slightest provocationfrom changingcircumstances.The theory of structures and, in fact, our whole scientificapparatusis foundedon suchabstractions. They have enabledus to imposesomeorder on tie chaoswith whichwe are faced when we look at the unendingandoverwhelrningwondersof nature which far exceedour powersof comprehension.Wehaveevenfoundtbat, if we assumethat this imaginaryworld of scienceis a true pictue ofreality andact accordingly,we caninfluenceandchangethe world we live in to suchanextent tlnt we canabolishwant anddrudgeryand,in fact, do almostanythingwe like, hcluding destroyingthe planetwe dependon, togetherwith its faunaandflora, in a few weeks - if we only couldagreewhere to start. This whole mechanisticworld-picture, the Cartesianor Newtonianworld of science,is under heaqystrain now; we 6ndin all disciplinesthat it doesnot work any more, nature simplydoesnot collaborate.I havenotime to elaborate on tlnt now andthere are, anyhow,thousandsofbooksandparnphletswritten about that theime.But how doesttris affectthe theory of structuresandthe n-holebusinessof structural engineering?Our structuresare all the time getting better, bigger, lighter andsafer, our machinesare moreefficient, smoother, usingless energy.We are all the time learningto domorewith less, so what is wrong with tllat? As David Brohn points out in hisbook, there is acriticallyimportantstageto be reachedbefore we caneven applyour numericalanalysisto a structure, namel),that we must havea structure to applyit to. Whenwe havethat our anzlysiswill tell us whetler the structure is capableof doingwhatit is supposed to do. Further, that the skills requiredto choosethis preliminarystructureare of an entirely different naturefrom thosewe maygraduallyattainwhenwe have masteredour structuraltechniques. Brohn confineshimselfto suggestingtlnt tie basisof theseskillsis the recognition of the relationshipbetweentlrc loadandthe resultingbehariourof the structure, in other words that we gainanintuitive understandingof ho'r' a structure will behaveunder load.This will. whenwe haveacsuiredOre

necessaryexperienc€,enabb us to choG€ the at leastapproximatelyright structure for a given task just by lookingat its shapeandproportionson a drawing. This is of coursemostimportant,for the increasinguseof computers hasto a great extent killedtlis understandingwhichis soessentia.l for rescuing the art of structura.ldesign. The computer hascometo stay, we must Livewith it, andthis bookteaches us how to do sowhilst remainingmastersofthe proceedings.Wemustbe able to check that the output from the computeris correct, andwhere andwhento use it, andwhat its limitationsare. If this bookdid nothingelseit wouldstill be the most important contributionto structual designwhichhasappearedfor a long time and shouldbe compulsivereadingfor anyoneinterestedin the subject. I havebeenextremely worried by the fact that giftedgraduatesfrom our universities enter the professionwith the ideathat it is belowtheir dignity to put pen to paper- the computerdoesit all. Here we havethe necessary remedy for suchconceit, andit is hightime. UnfortunatelyI haveneitherthe time nor the knowledgeto dojustice to the achievementsof Brohn'sbook, I have only a few hours left for the printers' deadline,whichis entirely my own fault. But he has seeminglygonethroughevery knownmethodof structural analysisand showsby clear diagramsandexplanationshow the structureis affected by the loading,thus at eachstagegivingthe readerthis essential understandingof structural behaviour.I hopethis bookwill give rise to a lively discussion. I cannotresist addinga comrnentof my own. Whilstrecognisingthe importance of what Brohn hasdone,I do not tlink he hasgonefar enough. Understandingof structural behaviouris very necessary,but are there not many more things that are equallyor evenmore necessary? Every structural designerof repute hasdeclaredthat structuraldesignis an art as well asjust an applicationof scienceandtechniqueto a givenproblem. You couldalsoput it the other wayroundandsaythat onlyif it is awork of art as well will it be adrniredandaddto the reputationof the designer.It is unfortunately impossibleto definewhat art implies,but it hasin anycase nothing to do with numericalanalysis.There are manyother mattersto consideraswell. The wholepurposeof a structuraldesignis to helpusto make the things we need, or fancywe need,orjust fancy.So we must makeit very clear to outselves what we want to achievewith our design,whichwill obviously affect its shape,the materialswe use andall kindsofother things. If we want to buildsomethingis this the right placefor it, couldnot our purpose be better achievedin a differentway altogether?Onlywhenwe havesortedout all these matters to ou.randour clients' satisfactionwill a structuralana.lysis become relevant. Obviously,wbat I wouldcalldzsigzis muchmoreimportant thar structural analysis,for that determineswhat we are goingto get for our efforts. And moreover raial we decideto do is muchmoreimportantthanhow to do it, andthat openstJresluke va.lvefor a wholefloodof questions,social, political, ethical which tlreaten us all with confusion,or worse, becausewe are not able to agreeon wbat to doHow to live in peacesitl or:r:neighboursonthis pJanetwitiout destroyingit is the ultimate andnow pressiugproblemandI wish I knew the answer. Ove Arun

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. 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 qualitative understanding of the response of the structure to load. It is ideally suited to under8raduates studying indeterminate framed structures as Part of a core course in civil or structural engineerinS' but it is also suitable, because of its 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 ol qualitative skiils; that is' the ability to Produce a non-numerical solution to the loaded line-dia8ram ol a structure. It is 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 II deals with current methods of structural analysis using the diagrammatic format to which the student has become accustomed. The need lor the developrrent of qualitative skills increases with the increasing use of the computer in design offices. In the near future, the computer will replace the majority ol analysis and structural desiSn calculations. Unfortunately, this will also have the elfect of eliminating 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 analytical procedures, PreParing the student for the design olfi'e

design data' where the computer ls rne source of virtually all numerical Analgsis is an inteSrated approach to the Understandinq structural ol which teaching and learning of the PrinciPles ol structural analysis, are also this book in this textbook is a major part. The ideas embodied ol the same available in an audio/visual series of sel{-learning programmes name. The audio/visual programmes are backed by a suite ol micro-computer programs which have been used to produce the numerical and SraPhical solutions to the Practice problems, included in this text' software are The audio/visual programmes and comPuter-aided learning available from:

osE Ltd 197 Botley Road OXFORD

ox2 oHE, UK TeL 0865726625 Publishing' who should be contacted direct and not through Granada

Acknowledgements

The research project upon which this book is based has extended over a period ol ten years. In that time, many friends and colleagues have contributed to the development of my ideas of the way in which students can be encouraged to reach a better understanding of structural behaviour. Bristol Polytechnic has provided both time and resources and my Head of 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 in 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 resDonseol 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 sequencesin the early part ol the book. I am indebted to all of them.

1 The Analvsisof Staticallv DeterminateStructures

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

However, this first chapter reviews the

behaviour of deterninate

structures, a 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 of the principles of overall equilibrium, bending moments, shear and axial forces. It is possible to analyse determinate structures by consideration of equilibrium - in general terms, the application ol force and moment eouarions v 1 O. d = 0 and lt = 0. With most real structures, this is not possible as the presence ol redundant members (secondary load paths) makes it necessary to consider relative member delormation beJore a solution of 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 in 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 are then able to start an approPriate process of numerical analysis to llni

cJ:

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,the srze each will have to be to carry its load saJely. \ve can go on to find the values ol deformations which will result Jrom the loadins.

TI]E ANALYSIS AF STF.IICALLY DETERMINATE STRUCTURES

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

\k

-, M A-

M QUALiTATIVE

ttu 5r eo- 3€

NLIA,1ER-tc.\L

:-e oasic principles of our structural analysis lie in the :a i --f sratics. Mosr srrucrures are required to be :-:---3 rn a static 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 initially, a: -e3s:. we 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 ::.:?:: counterbalance and load.

VELTIC'AL

EoUILIARIUM

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

by horizontal support lorces at the -.esistedj a.€- :1:s \r'e call ho.izontal equilibrium.

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

+

I1ARI: ONTAL @UILI 5P,JUM

---:_-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 add

E :rs :-e :.Citional toppling eilect oJ the wind Jorces r: lei =E::l:e base oJ the crane will have to provide _E\s=!-,= :c :hese out-ol-balance toppling moments. -r1s =* iioment equilibrium. -r /\^oMENT

Eeut!.tE;ui/At

UND' ?S' IlE DJ NC STRU CTURAL ANALYS I S

In our three-dimensional world we can express these equilibrium requirements in the following way.

Firstly'

we must ensure that in 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 to 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 our problem can be reduced to the consideration of three conditions of equilibrium only. Such simplification is described as a plane problem because all forces lie in one plane,

7.

we shaltr find it convenient to label the lorces in the .r direction with symbol d' to denote horizontal, and those in the z direction v, to denote vertical. Also we shall use the symbol l, for moments in the plane, about the Y axis.

8.

t

t {

Y =a VarttLt aluilihri.t

Vith these symbols we can write down the three

of equilibrium- all the horizontallorces must sum to zero, all the vertical forces must sum to zero and the sum oI the momentsmust also be zero.

H'o

Hon2o,{e q.fb/ilrf"1

M.o

iqon',t,f ?uilibrt-t

TEE ANALYSIS AF ::;'':'."LY

DETERMINATE STRUCTI]RES

\ow it will be remembered that a pure moment, =lled a couple, can be represented by two equal :^ d op po sire pa r allel J or c es ar a dls r anc e ap a r t . ,. 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 through A it wiil -Eie no moment about A and all that remains ::::e ::.ae r

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

-: YA = a x d anticlockwise.

Considering

:_e:oinl

B the upward forces pass through

:*e:olnr

and the only moment is 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

i,=

=omenr 4

is the force F multiplied by C j. :F_-:a:;oa still, of course, anticlockwise.

I_€': r:

:aie shovn is one oJ the most powerful

:: ,-:-..::-:.al ar aly s is . nam ely t hat . as a c o u p l e

],as :-€ =-e

:noment about any point in the plane,

].e ?:r*--:r:ri:. cf a structure will require that the 'r:ra -,:r-3::: cf all Jorces must be zero about any !t!r-: - ---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/.

I

The

{'+:- : < - q)

'l

F

e.

.) rl-Fo

UNDERS'IANDI N G STRUCTURAL A.VAIY5I5

We can start the qualitative analysisof the structure by observingthat since there must be equilibrium of momentsabout the supportA the only lorce which can balance the clockwise moment of the load t1lat 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/Bhas 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 equilibrium of vertical forces, the 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 usethe requirementthat momentssum to zero about any point in spaceto confirm this result. Taking moments about B we observethat there is a clockwise moment due to the load ,1rat C. A balancing anticlockwisemomJnt is required. Thus yA must act downwards.

Mv46o;

16.

Lastly, we should observe that as neither the force kr nor the reaction yB has a horizontal componentthe requirementof horizontalequilibriumensuresthat the horizontalsuDportreaction at A is zeto.

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

I7.

DETERMINATE STRUCTURES

The concept oJ equilibrium also applies to any part of the structure. If we consider an imaginary cut separating one part of the structure from the rest, it is clear that rnternal forces at the cut will be required to maintain

lw

"t'

equilibrium. Let us supposethat we separate part of the structure to the right oJ B by introducing a cut at K.

lE.

I

cu'f

If the portion o{ the beam KCD is to be in overall vertical equilibrium then there will have to be an

tw

lnternal vertical force S at the cut point K to balance the load rf.

It must act from below upwards on the part

'l '

--*--D

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

{v="

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 in the portion oi the beam CD, We can plot the distribution of s as a shear Jorce diagram, drawing it 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 the base line, above or below; that is corresponding to the vrew an observer would have sitting behind the cut K. The load tr would cause the part oJ th-^ beam KCD to fall lrelor,, the beam, thus the dia8ram is plotted below the base line. The arrow on the base line is required to confirm this convention.

I S shear s= w fo"'

1..;!J}E?S:}:i )J EG STRUCTIJRAL ANAIYSIs

21.

For a cut K in the beam between A and B we see that the greater magnitude of vB compared with t{ will lead

ls

a----l--+-r /<

tW

to vertical equilibrium requiring a shearing lorce .s from above acting downwards at K, being the dilference

4\

between

th"^ W)

Cl4g;

YB and t'r',

'4.. zr=o

Again the choice oJ the position of 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

Supposewe had considered the vertical equilibrium of leJt-hand portion of the beam from K to 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.

This is a necessary result because, when we close the cut by putting the two sides of the beam together, the two

_Al_ :-:-

ls

shear Jorces must cancel each other leaving no external out-of-balance

Ir

ork

vertical force at the cut section.

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 a-:\rn a horizontal arrov/ pointing to the right to indicate --:: $e we re l ook ing at t he f or c e ac t ing on t h e r i 8 h t -:-.: side of the cut. We could equally well make the =::r*

point to the left and look at the force acting on

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

in this case becomes a vertical reflection of the Fir her . iiFor am is . . ) r r c c ! but it is i 'n D o r t a n t

*+=f,"