week3 stat indet theory 2012(2)

8/2/2019 Week3 Stat Indet Theory 2012(2)

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2 Statical Indeterminacv

so far in your study of the analysis of structures, the range of structureshas been limited to those which may be solved by the application of thethree equations of statical equiiibrium:

all horizontal forces must balance,all verticai forces must balance,all moments must balance.

Consequently for a solution to be found there can only be threeunknowns.

However, virtually all real structures have more than three unknowns,they are 'statica.lly indeterminate' and cannot be soived by the threeequations of equilibrium alone.

That condition is known as'indeterminacy'and there are two ways ofidentif ying the degree to which a structure is indeterminate: statical andkinematical indeterm.inacy. This chapter deals with the former.Kinematical indeterminacy is specifically related to a particular method ofanalysis, Stiffness, which is discussed in Chapter 7.

Although this chapter will ultimately show how to f ind the number oftimes a structure is statically indeterminate, this number or'degree', isonly of importance in the flexibility method of analysis. The degree is thenumber of equations of compatibility which must be identified to add to thethree eguations of equilibrtum for a solution of all the unknowns.

However, as you will discover, it is the identification that thestructure is or is not indeterminate which is of primary importance, inother words, it.is the qua.litative appreciation of this condition whichshould be uppermost in your mind. The calculation of the degree ofindeterminacy can only be safely carried out once this qualitativeunderstanding has been soundly established.

DEFINEINDETERMINACY

2 dimensionalequilibrium condt'n


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ST AT I C AL I N DE TE RMI N ACYDefinition of the state of staticaf indeterminac,qVirtually all real structures do not satisfy this criterionand are known as statically indeterminate structures.The primary objective of this text is the developmentof an understanding of the relationship betweendeterminate and indeterminate structures and thesolution, in parricular, of the latter.

To begin with we will define the axes. fuiost of ourstructures and most real structural problems areultimately resolved into two dimensions set in thex - z plane. All reactions and forces will bereferred to as:

P horizontal in the x direction7 vertical in the Z direction

moment about the y axis, set norrnal tctheX-zplane.Remember that all these forces must be tnequilibrium at anq point on the structure-These three equations of equilibrium are sufficienlto analyse statically determinate structures.We will look first at a simply supported beamwith a diagonal load w. There are only threeunknowns, the reactions H4: v6 and izg. Thereaction in the horizontal direction at B hasbeen removed by putting restraint B on roilersto allow horizontal movement. The reactionsare shown as:

The action of the joint on the memberNote that ,Y4 would be equal to the horizontalcomponent of the load.

23

SUet q l9 aap.r uu,o,to *luct'qreJara thoso, rn whtoh st,Ye$ rLJukartJwLtht^ Eno uo:ubcrt ol bhc- strucftuosuch aS be-aArt3 t1l6uvtu.Es aUA force-1an bo Atts*uneA bJ w*V dttSAw \uaforr1 t aluilibruu.

@u I Ll B.J Lr A,^ . IAll vqeta( 1" rocs

\7fAll hovEa'tal f.rrn, {


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24

u ea$A s--,*- lve

PtnntA su pport-I1-r- +->H-\i's T v-


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.5T AT I CAL I N DE TE RM I N ACYIt is possible to provide an internal reiease ro rnestructure. Some internal releases are dif ficult tovisualise and are introduced here to preDare you for theiruse later in the text as analytical devices.

This is an internal rnoment release, a hinge. It iscapable of transferring axial and shear forces. Thebending moment is always zero at a hinge. The ioint isnot externaily restrained in position and both membersare free to adopt slope independently of one another.The mornent release, the hinqe, is f airly common instructures; however, the shear release and the axialrele:se q,hnwn hcre :re nnlv likelri fn Annc^r as nert nf :- "' "Jtheoretical, analytical procedure. In ea.ch case twostress resultants are transmitted and one released.

Two-dimensional structures are almost alwaysindeterminate. The exception in its many f orms is thethree-hinqed arch, There are four external reactions, onemore than may be solved by the three equations ofequilibrium. However, the hin3e provides another equationof equilibrium and if moment equilibrium is considered atB the unknown reactions H6 and Vg orovide the fourthequation of equilibrium.

A particular feature of the qualitative understanding ofthe structural analysis and behaviour is the concept ofthe f ree body diagram. Each element or member in astructure must be in equilibrium.

z)

fh-+{ A,fr'tve

Trptcttpt htya

-r


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z6 UNDERSTANDING

A STATtoALLYINDLT E.P.M'IN ATL3T R-u oTL.tEe-is Lho sam.- qsA EEDUNDAUT STzLt)TUEEfs Ehe' sarno asA tlYPep- STATto S7 R.uoTuRE

STRUCTURAL ANALYSISPin-jointed structures respond to the condit-ion ofindeterminacy in the same way. The presence ofaddit.ional members, here members PQ and PR, will affectthe distribution of forces in the remaining members. Ifthose members have a large cross-sectionai area then theforce in them v,ill be correspondingly large.

2t.

-& - E:! .sufports

Lxf,A vAP*1b/4

-7x scaQta$I n4ettttt44ttw(?-Ha-|. ^l-tlo>{ A D{+,A T/os(aErqllq.JActetunatz-o L' fclt4se'tnsflQ4

22. To summarise then, indeterminacy may be the result of:l. additional external reactions,2. internal rigidity,3. the addition of members.

The last two are classif ied as internal indeterminacy.

These three terms are used in other texts. They have thesame meaning. The term indeterminacy will be used in thitext and the degree of indeterminacy referred to as 'n'times indeterminate.

24. It is important to appreciate that structures may beconverted back to a statically determinate conditionfrom the staticaily indeterminate, by a variety ofstrategies. This technique is important because itmodels the decisions made by the structural designerwhen he reduces the highly complex, real structure to asimpler Iorm for the purposes of analysis. The portalframe is I times indeterminate because it has fourexternal reactions. The frame has been released to astatically determinate structure by the introduction of ahinge ln the beam BC.

tt.

.

Insert pin ...


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S TA'!-I CAL I N DE TE RIUII N ACYAs an alternative means of reducing rne structure to adeterminate form, a roller release has replaced the pinjoint at D. This particurar release shourd be noted sinceit is frequently emproyed in analyticar procedures. Thusno horizontal reaction can be taken at support D.Because the horizontal forces must be in equilibrium,the horizontai reaction at A will be equal to thehorizontal component of the load.

Tiris last example of reducing the srructure to adeterminate form is the axial release in the column. Thebending moment capacity at this release is unaffected.Such releases are rarely employed either in practice or rnanalytical procedures.

A moment reaction at A has been introduced to theportal frame turning it into a fully fixed support, andthe structure is now 2 times indetermrnate.

"ls--t(oVp

2 x st-afu


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30

boadtn3f^ovnuyt-6v Erce

UN DE R.STAN DI N G STRUCTU RAL AN ALYS I S29. The concept of a'tree'as a statically determinate

structure is a particularly powerful one in thequalitative analysis of structures because it is naturaland easily recognised in other structures.

30. The effect of a load on a tree is always that of acantilever. The effect of the load goes straight to theground. For this reason it is often preferable for us toconvert our indeterminate structures into an analytical'tree'. The resulting load effects are the simpler toidentif y and analyse.

We will now look at more complex frames, whichrepresent the complexity of structures met by thestructural designer. The two-bay, two-storey frame isfixed at A and J and pinned at H. All other joints are"i-iAl,, nnnnanfarl

Thi< rrrrn-harr f r:me haq croht ellterna I rear-tinns at AH, and J. Therefore the externaJ degree ofindeterminacy is:

8 - 3 (equations of equilibrium) = 5 timesindeter minate.

32.

There is both "EXTERNAL" indeterminacy as well as "INTERNAL"

indeterminacy possible in 2D frames ......... (and trusses) ...........

Creating "TREES" in 2Dframes .................

It depends on the geometry of the assembly of the frame, so thatwe could have 'frames on frames' and this creates significantinternal indeterminacy as well. See how we can treat this ............


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STATT CAL T N DE TE RI4I N ACY33. To convert the whole structure inro a ,tree, we must'cut' the top frame BCDF. A cut will release threeunknown internai forces, VF, HF and uO. Thus thetotal degree of indeterminacy is:J+3=8.

3de,tt,ruihaka, trcz- I x

31

If all the supports are pinnedreduced to a tree because ofsupport, the supports may bedeterminate condition. ThisA, G and H.

and the structure cannot bethe absence of a fully fixedreduced to a staticallytwo-bay frame is pinned at

The structure may be reduced to a statically determinateform by the removal of the verticai and horizontalrestraints at H and the horizontal restraint at G. Thesub-frame ABCEDG is then statically determinate, witnthe determinate cantilever frame EFH fixed to it. Thestructure is 3 times indeterminate because it is necessarvto release three reactions to reduce the structure to astatical.ly determinate form.

Two-storey frames may be solved rn a simiiar way bycreat ing a series of statically determ inate sub_f rames.This two-storey frame has a hinge at D and has pinnedsupports at A and H.

TOTAL degree of INDETERMINACY isfound by ADDING ...... "External"

degree to "Internal" degree ...........

NOTE:Internal hinges ONLY HELP US WITH ANOTHER momentequation WHEN THEY ENABLE US TO 'SPLIT' the structureinto two SEPARATE SIDES !

INTERNAL hinge cannot/does not'SPLIT' structure into two sides !!


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IJ N DE RS T AN DI N G S T RUCTU RAL AN ALY S I S37. The top f rame BCDEF is reduced to a staticaliy

determinate form by the removal of the moments at Band F. The lower frame by the removal of the horizontalrestraint at H. The structure is therefore 3 timesindeterminate.

38. This next example of a two-storey frame, has anindeterminate sub-frame ABEFG, supporting anindeterminate sub-frame BCDE. We will now employsystem of hinges to release the sub-structures tostatically determinate forms.

We will introduce hinges at B, C and E. Note that theintroduction of a hinge between the three members atis, in fact, two separate hinge releases. Each of thesub-f rames is now a determinate three-.hinged portalframe and the structure is 4 times indeterminate.

If there are less than three externa.l restraints, nomatter how many interna.l restraints, the structure willfail. In this example, with horizontal roller releases atboth supports, the structure will roll, i.e. this is astate of unstable equilibrium. Beware of the soph.istry ofthe notion that the structure could be stable under anexactly vertical load. Indeterminacy or stability is aproperty of the structure not the loading arrangement.

t-t=l-

h/+|va 'vb

39.R

40.

Mcchcnsw:' ^o horzottla,L y%frett",("MECHANISMS" denote uncontrolled 'movement'possibilities ....................


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STATI CAL I NDE TE RMI N ACYThe portal frame ABCD with pinned supports at A and Dalso has hinge releases at B and C. Therefore there is norestralnt against sway and the structure will collapsesideways.

This frame does not appear to be a mechanism becauseunder certain conditions the member BEF could sustain aIoad. If, for example, the member BEF were a cable,properly anchored at B and F, then a load in member DEcould be supported. However, in these clrcumstances thestructure wou.ld have to disobey one of the basic rules ofthe behaviour of elastic structures, which is that alldeflections due to the loading must be small comparedwith the structural dimensions, so that they do not affectthe geometry of the structure. Thus, in the context ofthe analysis of elastic structures, this frame rs amechanism and cannot be solved by linear analysis. It isknown as a 'pseudo-mechanism'.structures

The pin-jointed frame is recognised as staticallydeterminate by the three members in a triangle,resuiting .in the three unknown axial forces which can besolved by the three equations of equilibr.ium.

Meslw1rc6 ho su1! rLrtrqtE

Pse"tao ,ucLl4*N'sttl

'. BcTriatgtosutelll4otovaYnaha

at-abtalluldr.gcy ^16to

33!l

Ine structure willmany triangies areindefiniteiy.remain determinateadded. The system

no matter howcan be extended

Kinematic mechanism willcollapse ..............


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d oEzYtal4a,g'o Lrq ssa otoruinate, supborts1t

2J= m+3J - t\u^bEN3owt12y = nwnb* l_nemborl

UN DERSTANDI NG STRUCTURAL AN ALYSISWe will look first at the Warren girder truss, which isstaticaily determinate internally as it consists of aseries of triangular pin-jointed f rames. The truss hasstatically determinate external reactions.

46. If we now add an extra vertical reaction at the centrethen the structure is staticaily indetermrnate, n = l.To emphasise the fact that this is an externaiindeterminacy the actual frame may be ignored.

47. If we now introduce additional members BE and EH inthe truss the structure is 3 times indeterminate overari.It wili always be simpler to separate internal andexternai degrees of indeterminacy, for pin_jointedf rames.

48. Although this text is written to emphasise the gualitativeunderstanding of structural ana.lysis, the actualquantitive vaiue is usefully exam.ined in order to showhow unreliable it is un.less accompanied by an intuitivegrasp of behaviour.

For pin-jointed frames the relationship definingdeterminacy is stated thus:

where j =2xj=r*3,

number of jointsnumber of members

3 = three equations of equilibrium.The three-member frame is thus calculated to bedeterminate.

45.

I x rnd.

Lrut" 2 x tMwe.r*rno1e,intenatg

External truss indeterminacycan be 'seen' if we only look atthe truss outline + supports


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ST AT I CAL I N DE TE RM I N ACYIf 2 x j is greater than n + 3 the structure is amechanism.If 2 x j is less than ,m + 3 then the structure is.rndeterminate.

Exarnine the structure shown. The calcul ation showsthat it is statically determinate, but it should be clearthat this structure wili collapse because the first panellacks diagonal bracing. This demonstrates that thequantitative solution must be judged qualitatively.

Three-dimensional indeterminacy may be dealt with inthe same way, although it should be appreciated thatmost building structures are reduced to a series of linkedtwo-dimensional frames for ease of anaiysis.

There are six equations of equilibrium in threedimensions: forces and moments relative to the threeAXCS.

This cantilever ABCD is set in three dimensions. CD ispara.lle.l to the y axis. Six reactions are required forthe equilibrium of the frame at the 'built-in' suDDort at

H -a^F


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35

htngu wiil 3tvc,ti\jtnAf YelC4Se,qboqt za.t^ a,xts.

STRUCTURAL ANALYSISConsequently, the release of a fully fixed support freessix reactions. This bent frame, fully fixed at A and Dis 5 times statically indeterminate. The ,tree, method ofidentifying the degree of indeterminacy is even moreeffective with three_dimensional structures.

The effect of the introduction of a pinned support at D,previously fuily fixed, .is to release three srressresultants, the moments about each axis. This frame istherefore 3 times indeterminate. The structure could bereleased to a statically determinate rtree, by the removalof the three force reactions at D.

If we now return to the frame fully fixed at A and Dan tnternal hinge wou.ld release three interna.l stressresultants, the bending moments about each axis. Thisframe is now i times indeterminate.

UNDERSTANDING53.

54.

55.

^tr\rrHt"


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STATI CAL I NDETE RI"II N ACY56, 57, 59. Practice probferns

Determine the degreeeach of the structuresbetween rnternaf and

of indeterminacy ofshown distinguishingexternaf conditions.

week3 stat indet theory 2012(2)

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