Structural Analysis III
Contents
1. 48 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis W 8 A HINGE B C D w E Y HINGE A B C D E W HINGE 10 A b B C D HINGE M 11 EN A B Ge D P HINGE 12 D 5 A B C W 13 A wince B CL D e C i A B Kia HINGE 49 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 3 Statically Determinate Frames 15 B HINGE A A D 16 o E P__ A 17 D 50 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis HINGE HINGE B amp C 18 p A D P HINGE HINGE Ja B Q C Q 19 lt _ W 20 51 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis POINT OF MOMENT APPLICATION M fa nag B 21 A by A B 29 p B 23 A 52 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis B P 24 EE A P C gt D B 25 A HINGE B C x 26 A 53 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis A P B lt2. 27 D B C HINGE 28 AA D E u C B D 29 A 54 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 30 P uy O HINGE 55 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 4 Statically Indeterminate Beams 31 32 33 34 D AB E G 5 N ka HINGE HINGE 35 56 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 57 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 5 Statically Indeterminate Frames P B C 38 A e M B D 39 A P B C A D 40 A E P B 41 el AD 58 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis B C D DE 42 Al F M B D DN LONE 43 A W B C 44 A D To A B ome 45 P D 59 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis P B C E 46 A D FA A SSI YO 47 48 60 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 51 61 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 62 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 56 5
3. Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape B S Solution This frame is quite similar to the previous frame except that D cannot move horizontally This being the case we must have a horizontal reaction acting to the left at D Further since DE x 0 we must have a horizontal reaction at A opposing H Please note this as 1t 1s a common misconception Just because there are no applied horizontal forces does not mean there cannot be any horizontal reactions but if there are they must balance Finally for the reactions then we note that the vertical supports must offer upwards reactions Thus our deflected shape and reactions are 37 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis The BMD SFD and AFD follow directly by applying the techniques covered earlier given the reactions Note especially that joints B and C are effectively closing and that beam BC behaves similar to a fixed fixed beam 38 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 13 Example 10 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape Solution To proceed with this frame we will split it at the hinge tb Vent Veg tv 39 Dr C Caprani Structural Analysis III Chapter 4 Qu
4. The shear force and axial force diagrams follow similarly by considering the forces along or transverse to each member One particularly notable point is that the applied horizontal load splits at B some goes through shear down to A giving A whilst the rest probably smaller puts member BC into compression before travelling down member CE in shear to give H Thus we have 45 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis pia 46 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 Problems 4 3 1 Introduction There is no better way to learn qualitative analysis than by practice So here follows a good variety of determinate and indeterminate structures for analysis For each of the following structures determine the e Reactions e Bending moment diagram e Shear force diagram e Axial force diagram e Deflected shape For the trusses identify the sense of the force if any in each member 47 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 2 Statically Determinate Beams P HINGE 1 E NA B CA D W HINGE 2 3 A B GC DLA HINGE 3 AX B G 5 M P HINGE 4 A B CL D P HINGE HINGE 5 AN B cC DE F A B C HINGE HINGE P HINGE HINGE 7 A E F BC D
5. and C must be the same since member BC does not change length Also we see that we develop an anti clockwise moment reaction at A Next introduce the vertical support at C noting that we now have an upwards vertical reaction at C and proceed as before to get c tJ BAD Notice that the Point of Contraflexure is noted as a dot in the deflected shape drawing and its location is produced across to locate the zero point of bending moment on the column AB The shear force and axial force diagrams are obtained as was done in Example 6 Tini SED AED 33 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 11 Example 8 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape e E A D For this frame we will start by establishing the reactions First since there is no Solution horizontal support at D and since DE 0 we know H 0 Also by considering removal of restraints we will see that the two vertical reactions are upwards to give B lt E gt gt b DS 34 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Bending moments are only caused by forces transverse to a member Thus with no horizontal reactions 1 e no forces transverse to members AB or CD there can be no bending moments in the columns This only leaves the beam BC to act as
6. the structure for the BMD e AB For this portion we recognize that we have an increasing force due to the accumulation of load form the UDL as the distance increases Thus we have a doubly increasing moment as the distance changes and so the BMD curves upwards as shown e BC For this section just apply force x distance 24 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis M A eu a ES He lnc dia tence a Def Bud hue The BMD must not step no applied moment and so joins at B to yield os e BA And the shear force diagram follows either form the load types or by looking at the BMD curve to slope line to constant A e 25 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 8 Example 5 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape Solution For this frame we will start by establishing the reactions First since there is no 26 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 27 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 9 Example 6 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape b c la MW Solution To begin we will determine the direction of the vertical re
7. to give AG a gt 8 The shear force diagram is easy to construct by just following the forces moving left to right it is down at A then up at B over the line to a height equal to the applied force The total height at B is the vertical reaction at B and this must sum to the total downward forces at A and C Leacl VA Note also that since V dM dx we see that the negative shear corresponds to a negative slope in the BMD 19 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 5 Example 2 Problem Analyse the following beam for the reactions bending moment and shear force diagrams and draw the deflected shape A c oB Pe PAS Solution Using the same techniques as outlined in Example 1 we can quickly arrive at the deflected shape and reactions eee Va f Ue Based on the reactions we can then examine the two portions of the structure for bending moments e AC For this portion the moment comes mainly from the reaction V However the moment gets progressively smaller than it would have been if just V was acting i e force x distance since the UDL acts in the opposite direction This means the BMD curves as shown below e CB This portion is as studied in Example 1 is found from force x distance 20 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis u 1 my My ar Va paru x ra Ti R Curve a Fxe Again just like i
8. 7 ss HINGE 63 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 6 Trusses B A C E 58 i P D B D ki A H 59 pr E G P E B C D F Ca eae N ZN H G 61 64 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis B C D p o VAN AN A F E C 63 B 2 64 65 65 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 66 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis P AD B C 70 A F E Z 71 72 73 67 Dr C Caprani
9. Structural Analysis III Chapter 4 Qualitative Analysis Chapter 4 Qualitative Analysis 4 1 I i K ntroduction 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 FT BCO sr io o o a de or ein AND gt R de Maternal zao noise huis ba jeca NHWF ND NGS SNP 4 2 Qualitative Analysis Techniques esessssssvvvvesnnnsvnvnesnnnnnnvnsennnnnnvnssnennnnensssnnnnnseneene nN On A L o 42 1 Imroductiont ea Y Y Y ND A eden 4 22 Conditions of Structural Behaviour srrrronnrrvrvrnvnnvennrnvennrrvneernvnsensrnvserrsveeer I 4 2 3 Methods to Aid SO Ut OM tetitas LA 174 Example OS O II 0 4 20 Example ON 42 El dd a ZA 4 2007 O OA 429 Ex A catch od Y Y O TE A YF II L LALL Example a o e S RE OM A21 Example arr 42 3 ERA UM CRE RE ER GM Ga 4214 Example LA FYD CU O 4 3 Problems 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 47 FE Torodd 4 3 2 Statically Determinate Beams erennrnornnrnvennnnnvernnrnnennrrvnresnenesnrrnvsrrrrereerse FO 1 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 3 3 4 3 4 4 3 5 4 3 6 Statically Determinate Pra MES ai iii 50 Statically Indeterminate Beams A los a a O O 56 Statically Indeterminate ETameS 955511 uenaspio spao NODYN YY YE GN YRS FYD YRU OF 58 TAS EY HRN RR GW OY CYR RA m ere 64 Rev 1 2 Dr C Caprani Structural Ana
10. This means that for the joint below M M M A further implication of this is seen in the BMD there is a step in the bending moment for member AB at the joint of value M M M 11 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 19 For frames we normally neglect axial deformation This means that members cannot change length and because deflections are small this means that the member s joints must move perpendicular to the line of the member For example below B can only move along the line BB 20 Trusses do not have bending moment diagrams 21 Remember the axial force sign convention Tension gt Compression lt gt No axial force poe 12 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 22 Positive shear force sign convention makes the letter N up on the left down on the right 13 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 3 Methods to Aid Solution The following are some methods that may help you carry out the analyses 1 To find a support reaction Remove the Restraint offered by the reaction and draw the deflected shape of the resulting structure Apply the support reaction in such to as to bring the structure back to where it should be 2 Use Points of Certainty where you know the deflected position for example at a support the deflection is zero and usual
11. a simply supported beam giving the BMD as ps Next we note that the columns are in compression by the reactions and transmit the end shears of member BC to ground whilst there is no axial force in the beam since there are no horizontal forces The shear force and axial force diagrams are thus f Lastly we come to draw the deflected shape of this frame However before we do so we recall that there is no bending in the columns and that member BC behaves as if a simply supported beam We examine the bending of member BC in more detail 35 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis In this diagram we have identified the tangents to the end rotations of beam BC and the perpendiculars to these tangents We recall that the right angle rigid joints of the frame remain at right angles and so joints B and C of the frame rotate through 0 However since there is no bending in member AB and since A cannot move pin support B must move to B so that the rotation 6 can occur at B Joint C and member CD behave similarly Finally we note that the distance BB and CC must be the same since member BC does not change length All of this gives 94 4 a 4298 2 y A way to think about it is that the frame sways to the right in order to avoid bending the columns 36 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 12 Example 9 Problem
12. action at C As ever to do this we will remove the restraint and examine what happens ot the structure under gt 28 Dr C Caprani the applied load Structural Analysis III Chapter 4 Qualitative Analysis Given this deflected shape it is obvious that the vertical reaction at C should be upwards to keep C at the correct height Since there are no vertical loads this means that because of SF 0 we must have V acting downwards Since there is no other possible horizontal force by DE 0 we have H acting to the left Thus we have a EA _ Meda BC gt zA Ap fy 3 A Ha fva In the above diagram we have also indicated some points of certainty That of A is easy due to the support However at B we note that the frame should move away from the load but cannot move vertically downwards since member AB does not change length ignoring axial deformation This locates the deflected position of joint B And as indicated in the diagram once the deflected location of joint B is known so is that of joint C because we know that member BC does not change length Finally then to assist us drawing the deflected shape between these points of certainty we recognize that the joint is opening and so is rotating clockwise to give 29 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Fan to Crave of B In the above the tangents to the deflected shape curves are shown at join
13. alitative Analysis We note then that CD is effectively a simply supported beam and this gives the interaction force direction as upwards for CD reflecting the support that the structure ABC offers and downwards for ABC reflecting the push coming from the load on the beam From V we can determine the moment and vertical reactions at A The deflection behaviour of the beam CD is straightforward We examine the deflection behaviour of ABC noting that B moves away from the load downwards and member BC maintains the perpendicular angle to the tangent at B Moreover member BC has not transverse force as so remains straight 1 e does not bend Lastly see that the vertical movement of joints B and C must be the same since member BC does not change length 40 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis The diagram emphasises the point that the horizontal movement at C and D must be equal since the beam CD does not change length With the reactions and deflected shape established the remaining diagrams follow easily using the techniques previously described 41 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 14 Example 11 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape B e D gt 777 E This is a more complex frame than previous frames and so we will
14. begin by cutting Solution the structure back and gradually adding in the extra members This is a bigger scale removal of restraints method where the members are considered as a type of restraint We start with the portion ABC which has been studied previously B C gt j x A a Ha r 42 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis If we now introduce member CE we can see that it must push upwards on joint C to keep C at the horizontal level it should be at since member CE doesn t change length This tells us that we have an upwards vertical reaction at E And since SE 0 we therefore know that V is downwards Also there must be a horizontal reaction at E to keep E from moving right This causes tension on the outside of member CE All this is summarized in the following diagram Notice that we have dotted in where member CD would be if it were connected This tells us that the vertical reaction at D must be upwards as follows 43 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Since we know the sides of the members upon which there is tension we can assess the equilibrium of joint C From this we see that the bending moment in member CE is biggest With this information and the simple force x distance strategy of earlier examples we get 44 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis
15. draw the bending moment diagram on the tension face of the member to be consistent with our convention Remember fixed supports will have a moment reaction pinned supports will not though there may be an external moment applied at a pinned support For unbraced frames only symmetrical such frames symmetrically loaded will not sway Keep in mind deflections are always small and we neglect the self weight of the structures only analyse for the loads shown Deflected shapes are always very smooth curves except at a hinge Rigid joints in frames must keep the same angle as they rotate Det lecteot Ul lpadeot Posirem y Pesttio NA of i Tangent ae Tan Maintains ea dies 9 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 14 Rigid joints can only open or close M LU Open T M h z BMD lt Ut Close Es qa af a BMD 15 At a rigid joint with two members there is only one value of moment M above There is one rare exception to this rule 16 Ata right angle rigid joint the shear becomes the axial and the axial becomes the shear in the alternate members Just use DEF x 0 and DE 0 to see why Vx X Par Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 17 This is not the case for oblique angle joints gt ge Y Ys Fy 18 When more than two members meet at a rigid joint the joint must be in equilibrium
16. ly the structure moves away from the applied load though there are rare exceptions 3 For more complex structures remove excess members supports joints and reintroduce one at a time and observe the effect each additional feature has 14 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 4 Example 1 Problem Analyse the following beam for the reactions bending moment and shear force diagrams and draw the deflected shape e Solution Firstly we identify the Points of Certainty e It cannot move horizontally or vertically at A e It cannot move vertically at B e It will probably move downwards at C away from the load This gives the following points through which the deflected shape must pass gt B re fel za X Noting that the deflected shape is always a smooth curve except at hinges of which there are none here we join the three points with a smooth curve 15 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Also we know there is tension 7 above on the outside of the curve and so we include this in our drawing This helps inform us of the bending moment diagram always draw it on the tension face To find the direction of the reactions we will remove each restraint in turn and follow the above steps to see how the beam deflects when the restraint is removed e For Ay AS ee ke oo He cr Since there is no moveme
17. lysis III Chapter 4 Qualitative Analysis 4 1 Introduction 4 1 1 Background The ability to see and interpret structural behaviour is a core ability of a structural engineer At the initial stage of a structural scheme design we are not interested in numbers or amounts only the sense of a load effect Some examples of what we mean by sense are e Is there tension on the top or bottom of a beam e Does the tip of a cantilever deflect up or down e Is the moment reaction clockwise or anti clockwise Getting this level of analysis right is not only the first step but the most important step If we don t get this level right then the answers to a more complicated analysis will be meaningless The ability to get the right answers to this level is called Structural Intuition The better your structural intuition the better you will be a designer This ability reduces errors both in design practice but also whilst in college since you will already see the answer it is easier to catch errors in calculations 3 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 1 2 Reading Material Some good books on structural behaviour are e Brohn D Understanding Structural Analysis 4th Edn New Paradigm Solutions 2005 e Jennings A Structures from theory to practice Spon Press 2004 e Ji T and Bell A Seeing and Touching Structural Concepts Taylor amp Francis 2008 e Hil
18. n Example 1 we recognize that we have no steps in the BMD and so join the moment diagrams for the two portions at C to get yne deb wu Curve The shear force diagram is explained in the diagram pers g Nu m er af dy ea Aa Up ter T Beep P ne M TE VA poche Hon 21 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 6 Example 3 Problem Analyse the following beam for the reactions bending moment and shear force diagrams and draw the deflected shape Solution Again applying the techniques of Example 1 give the following deflected shape and reactions Note that for portion BC we recognize that there is no bending of the member However this does no mean that the member does not move it does and keeps a straight line extending the tangent to the deflected curve just to the left of B The bending moment and shear force diagrams are then found as per Example 2 22 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis A D 8 E 23 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 7 Example 4 Problem Analyse the following beam for the reactions bending moment and shear force diagrams and draw the deflected shape F a 2 Pet Solution Again using points of certainty and removal of restraints we arrive at T 2 Pet fp y Ve This allows us to look at the two portions of
19. nt of the beam when H is released H 0 e For Vi 16 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Since when Va is removed it moves upwards it must be that Vi acts downwards in the actual structure to keep the beam at A where it must remain e For Vz Since when Vz is removed the beam moves downwards Vg acts upwards in the actual structure to ensure that B remains where it should Thus the reactions are With this information is now becomes easier to establish the bending moment and shear force diagrams Starting with the bending moment diagram for the portion AB of the beam we take a cut somewhere to the right of A 17 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Mii ju yA aout LX Puli x distence As may be seen the effect of V is to cause an anti clockwise rotation of the segment which must therefore be resisted by a clockwise internal bending moment Mx as shown This means since the arrow comes from the tension face that tension is on the top of the beam and is increasing as the distance increases the force remaining constant Similarly we examine the portion BC 18 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis There are no applied moments so the bending moment diagram does not have any steps in 1t This means that the two portions that we have identified above must meet over B
20. other program for the analysis of trusses is TrussMaster developed by the lecturer for the purposes of teaching structural behaviour of trusses This is available on the college computers and a User Manual is also available at www colincaprani com 5 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 Qualitative Analysis Techniques 4 2 1 Introduction Qualitative Analysis is not a linear process For some problems we might start with reactions and proceed through bending moments to a displaced shape whilst for others we may begin with a displaced shape work out reactions and then find the bending moment diagram The approach to use will depend on the problem and there are set rules or procedures that you can follow to be guaranteed to arrive at the correct solution On a more positive note since the structure will only behave in one distinct manner there can only be one correct solution By definition therefore incorrect solutions will contain inherent incompatibilities For example all aspects of a frame s solution may agree e g reactions bending moment diagram etc but it may require a rigid joint to have different rotations Since this is impossible we know that this cannot be the right answer Therefore if we have a proposed solution we must ensure that it does not violate any of the conditions of structural behaviour If 1t does then it is not the correct solution In other wo
21. rds your answer will tell you if it is wrong or not 6 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 2 Conditions of Structural Behaviour There are certainties about structural behaviour that we can rely on when attempting to analyse a structure Most of these are plainly obvious but a few may not be 1 Remember the very basics moment force x distance 2 Know your support types and the type of restraint they offer Symbol Name Movements Roller s Horizontal Ox X 2 eze bo Vertical Roller X dy O Pin X Xe Fixed X X A beam continuous over the Ox X 0 support and can rotate _ Vertical Support oS BET 7 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 3 Recall the shapes of BMD and SFD under the different types of loading rectangular triangular parabolic SF Ras LOAD nn mm ci FU ll au 1 ge pa AM ame sa u AU TT UI ted gt TUTO ARAN 4 Remember shear is rate of change of moment 5 No transverse load or end shear force on a frame member means there is constant BM along the member constant may egual zero 6 There is zero bending moment at a hinge 8 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 7 10 11 12 13 UAleectec fr Members with no bending moments remain straight i e no bending but may still move Always
22. son B Basic Structural Behaviour Understanding Structures from Models Thomas Telford 1993 e Pippard A J S The Experimental Study of Structures Edward Arnold amp Co London 1947 e LStruct E Qualitative Analysis of Structures London 1989 Due to its importance the Ove Arup Foundation sponsored the report The Teaching of Structural Analysis by Prof lan May and Dr David Johnson It is accessible here http www jbm org uk uploads StructuralAnalysiswithCover pdf A summarized version of the report appeared in The Structural Engineer Vol 81 No 7 2003 p 33 37 available at this link http www istructe org thestructuralengineer Abstract asp PID 7904 4 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 1 3 Software In developing your structural intuition it is very helpful to model structures using a appropriate computer program especially when the structure behaves counter intuitively Most structural analysis programs today are extremely complex with many options and capabilities and this can often obscure the modelling process An appropriate program for a few reasons is LinPro freely available from www line co ba You should install LinPro on your own computer Also it is installed on the computers in Rm 392 The program is intuitive to use and comes with a reasonable help file If you have any difficulties using the program please ask the lecturer An
23. t B to demonstrate that the joint is rotating but keeping its angle the same at B The BMD is easily established considering free body diagrams of each member along with the simple force x distance Recalling that shear is transverse forces to the member line in considering the shear of member AB we need only consider the applied load and H to get 30 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis a Ha For the shear in member BC we must first consider that the transverse force besides Vo gets there through member AB as an axial force caused by V to get Vec Fat ty Tension NY Va Mere We combine the two solutions above to get the final shear and axial force diagrams A e lU EL SED RED D 31 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis 4 2 10 Example 7 Problem Analyse the following frame for the reactions bending moment shear and axial force diagrams and draw the deflected shape B c ym A E Solution This frame is the same as that of Example 6 except for the support type at A Thus we will see the influence of fixing a support on a structure Firstly proceed as we did before and remove the support at C to get amp e 32 Dr C Caprani Structural Analysis III Chapter 4 Qualitative Analysis Notice that the diagram emphasises that the horizontal displacements at B
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