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# THEORY OF STRUCTURES CVEN 345

Texas A&M

GPA 3.53

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This 70 page Class Notes was uploaded by Carolyn Kuhn on Wednesday October 21, 2015. The Class Notes belongs to CVEN 345 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/226125/cven-345-texas-a-m-university in Civil Engineering at Texas A&M University.

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Date Created: 10/21/15

cam08 3THTICALLY magsreamsng 157 D E 535 w 3 km 151 Z 100 m so kN 20 kNm r V w mwmnmmw wwmww 3 z 9 I 2 800 lblft CVEN 345 CLASS NUMBER DATE PAGE Ev CVEN 345 CLASS NUMBER DATE PAGE 21 500 lb ft V V I I I M 3 I V I H I 19 maximt 1 quot39 3 HuffKT 3 39 s r V 39 a 3 339 539quot I kLiw J 39 i 10 It quotI I 0 ft 1 500 ib M a u a Aw quot172111 45 WW7 W WWW 239 Al quot L WWW O N w u wquot mm 4 5 I muggyhvtww fitweft gh n39m lhct v klg Inn4 39 Mgr 2 3 Jim W 939 iquot E J m quotm 4 401m v r 4 a Hm I l im z x7645 3m Qt 39 o AA CVEN 345 CLASS NUMBER DATE 0 2 PAGEL OW Iv CVEN 345 CLASS NUMBER DATE 0 L 08 PAGE 3 5 I T 2 A do 39 i f A A PE j F 22144 200 SHEETS r 22141 so SHEETS 22142 100 SHEETS AM 5 322141 50 SHEETS 22142 109 SHEEYS 5 22144 208 SETS 2214 50 SHEETS 22342 100 SHEETS Mquot 22444 200 SHEETS 22314 50 SHEETS 22d 42 168 SREETS Amp 22 44 200 SHEETS 22141 50 SHEETS 22142 300 SREETS MM 2239 44 200 SHEETS BA t f H 1 a6 8 Km 076 h 02 5CBMX HZHC39OB PH O 9 CVEN 345 CLASS NUMBER DATE Q o Cgt8 PAGE gt AVXHAL ogceyg quot EMER S 1 3mm FMW W CVEN 345 CLASS NUMBER DATE 12 gt 06 PAGE 2 A A MonGDT ENERGY STORED V CVEN 345 CLASS NUMBER DATE PAGE i am i M Wmm am L MWhmmm 25 f S Eg w g j w w 3 E gMZ A CVEN 345 CLASS NUMBER DATE PAGE UL Section 104 De ections by the WorkEnergy Method Real Wor 369 O n Work Energy Applied to a Truss To establish an equation that can be used to compute the de ection of a point on a truss due to a load P that increases linearly from zero to a nal a value P we substitute Equations 105 and 109 into Equation 1015 to give 16 P F2L 6 1016 t 2 2AE where P and 5 are collinear and the summation sign 2 indicates that the 3 energy in all bars must be summed The use of Equation 1016 to com pute the horizontal displacement of joint B of the truss in Figure 105 is illustrated in Example 101 As shown in Figure 105 joint B displaces both horizontally and ver tically Since the applied load of 30 kips is horizontal we are able to com pute the horizontal component of displacement However we are not able id to compute the vertical component of the displacement of joint B by the method of real work because the applied force does not act in the vertical a direction The method of virtual work which we discuss next permits us to m compute a single displacement component in any direction of any joint for any type of loading and thereby overcomes the major limitations of the 4 method of real work P a 39 Using the method of real work determine the horizontal de ection 5 of E X A M P L E 1 O 39 1 joint B of the truss shown in Figure 105 For all bars A 24 in2 and E 30000 kipsinz The de ected shape is shown by the dashed lines Solution a Since the applied force of P 30 kips acts in the direction of the required 39n displacement the method of real work is valid and Equation 1016 applies P in 5 1016 I 2 6 M lt gt ire re Values of bar force F are shown on the truss in Figure 105 IS 11 AB BC AC 30 50225 12 40220 12 30215 12 Ii11 2 quot 224 30000 224 30000 224 30000 is 2V 5 06 in Ans Figure 105 v gt Virtual Work Trusses Virtual Work Method Virtual work is a procedure for computing a single component of de ec tion at any point on a structure The method is applicable to many types of structures from simple beams to complex plates and shells Moreover the method permits the designer to include in de ection computations the in uence of support settlements temperature change and fabrication errors i To compute a component of de ection by the method of Virtual work the designer applies a force to the structure at the point and in the direction of the desired displacement This force is often called a dummy load because like a ventriloquist s dummy or puppet the displacement it will undergo is produced by other effects These other effects include the real loads temperature change support settlements and so forth The dummy load and the reactions and internal forces it creates are termed a Q stem Forces work displacements or energy associated with the Q sys f r39ri39Will be subscripted with a Q Although the analyst is free to assign any arbitrary value to a dummy load typically we use a lkip or a lkN force to com pute a linear displacement and a l kip ft or a 1 kN m moment to determine a rotation or slope With the dummy load in place the actual loads called the P system are applied to the structure Forces deformations work and energy asso ciated with the P system will be subscripted with a P As the structure deforms under the actual loads external virtual work WQ is done by the dummy load or loads as it moves through the real displacement of the structure In accordance with the principle of conservation of energy an equivalent quantity of virtual strain energy UQ is stored in the structure that is WQ UQ 1017 The Virtual strain energy stored in the structure equals the product of the internal forces produced by the dummy load and the distortions changes in length of axially loaded bars for example of the elements of the structure produced by the real loads ie the P system Analysis of Trusses by Virtual Work To clarify the variables that appear in the expressions for work and energy in Equation 1017 we will apply the method of virtual work to the one 370 WQMQ bar truss in Figure 1060 to determine the horizontal displacement SF of the roller at B The bar which carries axial load only has a crosssectional area A and modulus of elasticity E Figure 1066 shows the bar force F P the elongation of the bar ALP and the horizontal displacement 8 of joint B produced by the P system the actual load Since the bar is in tension it elongates an amount ALP where F PL AL 108 f AE Assuming that the horizontal load at joint B is applied slowly so that all work is converted to strain energy and increases from zero to a nal value P we can use Equation 105 to express the real work WP done by force P as WP 506 1 1018 Although a vertical reaction PU develops at B it does no work as the roller displaces because it acts normal to the displacement of joint B A plot of the de ection of joint B versus the applied load P is shown in Fig ure 10627 As we established in Section 102 the triangular area WP under the loadde ection curve represents the real work done on the structure by load P As a result of the real work done by P strain energy U P of equal mag nitude is stored in bar AB Using Equation 107 we can express this strain energy as UP FP ALP 1019 A plot of the strain energy stored in the bar as a function of the bar force F P and the elongation ALP of the bar is shown in Figure 106c In accordance with the conservation of energy WP equals U P so the shaded areas WP and U P under the sloping lines in Figure 106 afid c must be equal We next consider the work done on the strain energy stored in the bar by applying in sequence the dummy load Q followed by the real load P Fig ure 106d shows the bar force F Q the bar deformation ALQ and the hori zontal displacement 69 of joint B produced by the dummy load Q Assum ing that the dummy load is applied slowly and increases from zero to its nal value Q we can express the real work WD done by the dummy load as WD iQBQ 1020a DECIIOD 1U3 VII39lZLlal WOI KZ l39russes 37f Figure 106 Graphical representation of work and energy in the method of virtual work a Psystem forces and deformations produced by real load P b graphical repre sentation of real work WP done by force P as roller in a moves from B to B c graphi cal representation of real strain energy UP stored in bar AB as it elongates an amount ALP U WP d forces and displacements produced by dummy load Q e graphical representation of real work WD done by dummy load Q f graphical representation of real strain energy UD stored in bar AB by dummy load g forces and deformations produced by forces Q and P acting together h graphical representation of total work W done by Q and P i graphical representation of total strain energy U stored in bar by Q and P 372 bar force de ected position ALP de ection elongation b c BI gtP c 5p a Pv 8 is E FQ U Q Aasiliii WD Aiiiii D gt 0 59 r ALQ de ection elongation e f elongation i g The loadde ection curve associated with the dummy load is shown in Figure 1066 The triangular area under the sloping line represents the real work WD done by the dummy load Q The corresponding strain energy U D stored in the bar as it elongates is equal to UD FQ ALQ 102022 Figure 106f shows the strain energy stored in the structure due to the elongation of bar AB by the dummy load In accordance with the princi ple of conservation of energy WD must equal UD Therefore the cross hatched triangular areas in Figure 1069 and f are equal With the dummy load in place we now imagine that the real load P is applied see Figure 106g Because we assume that behavior is elastic the principle of superposition requires that the nal deformations bar forces reactions and so forth but not the work or the strain energy as we will shortly establish equal the sum of those produced by Q and P acting separately see Figure 106a and d Figure 106h shows the total work W done by forces Q and P as point B displaces horizontally an amount 8 5Q 8P Figure 106i shows the total strain energy U stored in the structure by the action of forces Q and P To clarify the physical signi cance of virtual work and virtual strain energy we subdivide the areas in Figure 106h and i that represent the total work and total strain energy into the following three areas 1 Triangular areas WD and U D shown in vertical crosshatching 2 Triangular areas WP and U P shown in horizontal crosshatching 3 Two rectangular areas labeled WQ and UQ Since WD UD WP UP and W U by the principle of conservation of energy it follows that the two rectangular areas WQ and U Q which rep resent the external Virtual work and the virtual strain energy respectively must be equal and we can write WQ UQ 1017 As shown in Figure 106h we can express WQ as WQ Qsp 1021a where Q equals the magnitude of the dummy load and 5 the displace ment or component of displacement in the direction of Q produced by the Psystem As indicated in Figure 106iwe can express UQ as UQ FQ ALP 102w SCCUOII 103 Virtual Work 39l russes 33 where F Q is the bar force produced by the dummy load Q and ALP is the change in length of the bar produced by the Psystem Substituting Equations 1021a and 10211 into Equation 1017 we can write the virtual work equation for the one bar truss as Q 5 FQ ALP 1022 By adding summation signs to each side of Equation 1022 we produce Equation 1023 the general virtual work equation for the analysis of any type of truss A I 2961 ZFQ ALP 1023 The summation sign on the left side of Equation 1023 indicates that in certain cases see Example 107 for example more than one external Q force contributes to the virtual work The summation sign on the right side of Equation 1023 is added because most trusses contain more than one bar Equation 1023 shows that both the internal and external forces are supplied by the Q system and that the displacements and deformations of the structure are supplied by the P system The term virtual signi es that the displacements of the dummy load are produced by an outside effect ie the P system When the bar deformations are produced by load we can use Equa tion 108 to express the bar deformations ALP in terms of the bar force F P p and the properties of the members For this case we can write Equation 1023 as FL 2 5 EFQX E 1024 We will illustrate the use of Equation 1024 by computing the de ection of joint B in the simple twobar truss shown in Example 102 Since the direction of the resultant displacement at B is unknown we do not know how to orient the dummy load to compute it Therefore we will carry out the analysis in two separate computations First we compute the component of displacement in the x direction using a horizontal dummy load see Figure 10717 Then we compute the y component of displacement using a vertical dummy load see Figure 107c If we wish to establish the magnitude and direction of the actual displacement the components can be combined by vector addition 37 Under the action of the 30kip load joint B of the truss in Figure 107a displaces to B the de ected shape is shown by the dashed lines Using virtual work compute the components of displacement of joint B For all barsA 21112 and E 30000 kipsinZ Solution To compute the horizontal displacement 5x of joint B we apply a dummy load of l kip horizontally at B Figure 107b shows the reactions and bar forces FQ produced by the dummy load With the dummy load in place we apply the real load of 30 kips to joint B indicated by the dashed arrow The 30kip load produces bar fOICes F P which deform the truss Although both the dummy and the real loading now act dependently on the structure for clarity we show the forces and deformations produced by the real load P 30 kips separately on the sketch in Figure 107a Wwwa 1024 lt 3 40gtlt16 x12 3 230J000 55Q20X12 1 6xgt 39 v 3 260000 5x 05 in gt Ans To compute the vertical displacement 8 of joint B we apply a dummy load of 1 kip vertically at joint B see Figure 107c and then apply the real load Since the value of FQ in bar AB is zero see Figure 107c no energy is stored in that bar and we only have to evaluate the strain energy stored in bar BC Using Equation 1024 we compute 1024 0 1 4016lt12 012 in l IMNEQ 230000 9 AIM amp Figure 107 a Real loads Psystem producing bar 1 Dummy load Q sys tern producing F Q forces used to compute the horizontal displacement of B The dashed arrow indicates the actual load that creates the forces FP shown in a c Dummy load Q system used to compute the vertical displacement of B Section 105 Virtual Work Trusses 3 7 39539 EXAMPLE 102 As you can see if a bar is unstressed in either the P system or the Q system its contribution to the virtual strain energy stored in a truss is zero NOTE The use of a lkip dummy load in Figure 1072 and c was arbi trary and the same results could have been achieved by applying a dummy force of any valuemFor example if the dummy load in Figure 1071 were doubled to 2 kips the bar forces FQ would be twice as large as those shown on the gure When the forces produced by the 2kip dummy are substituted into Equation 1024 the external work a direct function of Q and the internal strain energy a direct function of F Q will both double As a result the computation produces the same value of de ection as that produced by the l kip dummy Positive values of 8 and 5y indicate that both displacements are in the same direction as the dummy loads If the solution of the virtual work equation produces a negative value of displacement the direction of the displacement is opposite in sense to the direction of the dummy load Therefore it is not necessary to guess the actual direction of the dis placement being computed The direction of the dummy force may be selected arbitrarily and the Sign of the answer will automatically indi cate the correct direction of the displacement A positive sign signi es the displacement is in the direction of the dummy force a negative sign indicates the displacement is opposite in sense to the direction of the dummy load To evaluate the expression for virtual strain energy F QF PLAE on the right side of Equation 1024 particularly when a truss is composed of many bars many engineers use a table to organize the computations see Table 101 in Example 103 Terms in columns 6 of the Table 101 equal the product of F Q FF and L divided by A If this product is divided by E the strain energy stored in the bar is established The total virtual strain energy stored in the truss equals the sum of the terms in column 6 divided by E The value of the sum is written at the bottom of column 6 If E is a constant for all bars it can be omitted from the summation and then introduced in the nal step of the de ec tion computation If the value of either F Q or F P for any bar is zero the strain energy in that bar is zero and the bar can be omitted from the summation If several displacement components are required more columns for FQ produced by other dummy loads are added to the table Extra col umns for FF are also required when de ections are computed for several loadings 376 Compute the horizontal displacement 8x of joint B of the truss shown in Figure 108a Given E 30000 kipsinz area of bars AD and BC 5 inz area of all other bars 4 in2 Solution The F P bar forces produced by the P system are shown in Figure 108a and the F Q bar forces and reactions produced by a dummy load of 1 kip directed horizontally at joint B are shown in Figure 1081 Table 101 lists the terms required to evaluate the strain energy U Q given by the right side of Equation 1024 Since E is constant it is factored out of the summa tion and not included in the table Substituting EFQFP IA 1025 into Equation 1024 and multiplying the right side by 12 to convert feet to inches give 1024 1 30000 1025 13 1 1ltip5x 8x 041 in gt Ans FQ FF 6 A FngLA Member kips kips in2 kiposzftin2 1 2 3 5 6 AR 1 80 20o 7 4 400 m BC 0 100 25 I1 5 0 CD 0 80 20 39 HZ 4 0 we AD 100 2542 5 625 BD 0 60 1547 4 EFQFPLA 1025 DELHUM LU L 2039 L 20 4 Figure 108 a P system actually loads 27 Q system 377 V 11 Luau VV UlA Ll ubbe EXAMPLE 103 CVEN 345 CLASS NUMBER DATE 1 9 gt O gtPAGE PW zra 0 Hoe AG CSzr 1 AG 1 EM O OOKZ A665 gt CVEN 345 CLASS NUMBER DATE PAGE a gx 3 m CVEN 345 CLASS NUMBER DATE PAGE I POP Q0 FE 39 L MB ukN mkN L I A C E gt 39 J 18m 3HO D 1 5 J H 4Panels24 9em 5K TEMMJOmE 12 km 439 8M KY2 Z39f m 4ch 436 a 224 QN C 1134 EG quot83 Q Z P 201Ni IZ39JW y F Ci39gaquot 36 39 79 F H 1 02M 52 2 43048 F H 4 753 Q P 7mg E ME 4 fw41 quotr WDFj ZMg O 47 r 6 kips 4 kip I l or F MA CB 9 m WWE 2MP 239 O Em E ZMP O EH F m EFH 3 6 LL 0 53 If quot 4 x F v 053 gig505224M S 00 LU 3 va Eng we 1 S24 OQWV9 pwrw 4ampqu mq JQamp mwwu 34ww gs Nx2 omou x u i 2amp9W mummw 3 mid mmmzbz mm o 3m Zmgtu CVEN 345 CLASS NUMBER DATE PAGE Z 7 i 39 206 200 A CVEN 345 CLASS NUMBER DATE PAGE i3 quot7 39 erWAm W W gje A th 8 O 4 42m v EL ZfFHO FL mom EH 0sokm A 39 i V44 SOM 2 MM 1 5 ISOANC a I i W M4 60 MW j 2 4L45 gal I i 150 2H 0 2160 Zoo 4 f v VAL 39 Bangkok 5 6 3 ofdmi 45oawzm MAL C39 400 i V Z 04L4Q E 39 CVEN 345 CLASS NUMBER 6 DATE JL O8 PAGE 32239 ZK i jA TB g 5 T4s39l 39 13quot 31 VALUE w I SLOPE 53394quot mmp39egHe quot pom R TE CVEN 345 CLASS NUMBER DATE PAGE quot 4 f M 132 CVEN 345 CLASS NUMBER DATE PAGE 24451 v y v a P 2 4quot I 39 24 1 T w Aliza r2 CVEN 345 CLASS NUMBER DATE PAGE A x 39 A 0 CVEN 345 C ASS NUMBER DATE PAGE 1 M 30 i C M813 5 2 39044 1 Mm quotIOMLWEcMELM CVEN 345 CLASS NUMBER DATE PAGE

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