2 Virtual work methods

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1 Virtua wrk methds Ntatin A E G H I I m P q Q u V y δ δ δ δ θ θ μ φ crss sectina area f a member Yung's mduus mduus f trsina rigidity hriznta reactin secnd mment f area f a member secnd mment f area f an arch at its crwn ength f a member bending mment in a member due t a unit virtua ad bending mment in a member due t the appied ads aia frce in a member due t the appied ads shear frce in a member due t a unit virtua ad shear frce in a member due t the appied ads aia frce in a member due t a unit virtua ad vertica reactin appied ad hriznta defectin dispacement f an appied ad in its ine f actin; vertica defectin defectin due t the appied ad etensin f a member, ack f fit f a member eement f ength f a member reative rtatin between tw sectins in a member due t the appied ads rtatin due t the appied ads frm factr in shear shear defrmatin due t the appied ads. Intrductin The principe f virtua wrk prvides the mst usefu means f btaining the dispacement f a singe pint in a structure. In cnjunctin with the principes f superpsitin and gemetrica cmpatibiity, the vaues f the redundants in indeterminate structures may then be evauated. The principe may be defined as fws: if a structure in equiibrium under a system f appied frces is subjected t a system f dispacements cmpatibe with the eterna restraints and the gemetry f the structure, the tta wrk

2 6 Structura Anaysis: In Thery and Practice dne by the appied frces during these eterna dispacements equas the wrk dne by the interna frces, crrespnding t the appied frces, during the interna defrmatins, crrespnding t the eterna dispacements. The epressin virtua wrk signifies that the wrk dne is the prduct f a rea ading system and imaginary dispacements r an imaginary ading system and rea dispacements. Thus, in Chapters and and Sectins 6., 6.4, 6.5, 9.8, and. dispacements are btained by cnsidering virtua frces underging rea dispacements, whie in Sectins , , and. equiibrium reatinships are btained by cnsidering rea frces underging virtua dispacements. A rigrus prf f the principe based n equatins f equiibrium has been given by Di aggi. A derivatin f the virtua-wrk epressins fr inear structures is given in the fwing sectin.. Virtua-wrk reatinships y 4 P P/AE θ /R / Q P f f R (i) Q 4 4 (ii) (iii) Figure.

3 Virtua wrk methds 6 T the structure shwn in Figure. (i), the eterna ads are graduay appied. This resuts in the defectin f any pint 4 a distance δ whie each ad mves a distance y in its ine f actin. The ading prduces an interna frce P and an etensin δ in any eement f the structure, a bending mment and a reative rtatin δθ t the ends f any eement, a shear frce Q, and a shear strain φ. The eterna wrk dne during the appicatin f the ads must equa the interna energy stred in the structure frm the principe f cnservatin f energy. Then: y/ P / θ/ Qφ / () T the unaded structure a unit virtua ad is appied at 4 in the directin f δ as shwn in Figure. (ii). This resuts in a frce u, a bending mment m, and a shear frce q in any eement. Nw, whie the virtua ad is sti in psitin, the rea ads are graduay appied t the structure. Again, equating eterna wrk and interna energy: y/ δ P / θ/ Qφ / u/ m θ qφ () Subtracting epressin () frm epressin (): δ u m θ qφ Fr pin-jinted framewrks, with the ading appied at the jints, ny the first term n the right-hand side f the epressin is appicabe. Then: δ u Pu/AE where P is the interna frce in a member due t the appied ads and ; A and E are its ength, area, and mduus f easticity; and u is the interna frce in a member due t the unit virtua ad. Fr rigid frames, ny the ast tw terms n the right-hand side f the epressin are significant. Then: δ m θ qφ m d / Qq d / μag

4 6 Structura Anaysis: In Thery and Practice where and Q are the bending mment and shear frce at any sectin due t the appied ads and I, G and A are the secnd mment f area, the rigidity mduus, and the area f the sectin; μ is the frm factr; and m and q are the bending mment and shear frce at any sectin due t the unit virtua ad. Usuay the term representing the defectin due t shear can be negected, and the epressin reduces t: δ m d / In a simiar manner, the rtatin θ f any pint 4 f the structure may be btained by appying a unit virtua bending mment at 4 in the directin f θ. Then: θ m d / Qq d / μag where m and q are the bending mment and shear frce at any sectin due t the unit virtua mment.. Sign cnventin Fr a pin-jinted frame, tensie frces are cnsidered psitive and cmpressive frces negative. Increase in the ength f a member is cnsidered psitive and decrease in ength negative. The unit virtua ad is appied t the frame in the anticipated directin f the defectin. If the assumed directin is crrect, the defectin btained wi have a psitive vaue. The defectin btained wi be negative when the unit virtua ad has been appied in the ppsite directin t the actua defectin. Fr a rigid frame, mments prduced by the virtua ad r mment are cnsidered psitive, and mments prduced by the appied ads, which are f ppsite sense, are cnsidered negative. A psitive vaue fr the dispacement indicates that the dispacement is in the same directin as the virtua frce r mment..4 Iustrative eampes Eampe. Determine the hriznta and vertica defectin f pint 4 f the pin-jinted frame shwn in Figure.. A members have a crss-sectina area f 8 in and a mduus f easticity f 9, kips/in.

5 Virtua wrk methds 6 4 k 6 k k (i) (ii) Sutin ember frces u due t a vertica unit ad at 4 are btained frm (i) and are tabuated in Tabe.. The stresses in each member, P/A, due t the rea ad f 6 kips are given by: PA / 6u 8 / u Figure. ember frces u due t a hriznta unit ad at 4 are btained frm (ii) and are tabuated in Tabe.. Tabe. Determinatin f frces and dispacements in Eampe. ember P / A u u Pu / A Pu / A Tta The vertica defectin is given by: v Pu / AE 58 / 9,. 7 in dwnward

6 64 Structura Anaysis: In Thery and Practice The hriznta defectin is given by: h Pu / AE 6 / 9,. 98 in t the right Eampe. Determine the defectin at the free end f the cantiever shwn in Figure.. The crss-sectin is shwn at (i), the mduus f easticity is 9, kips/in, the mduus f rigidity is, kips/in, and the shear stress may be assumed t be unifrmy distributed ver the web area. k k 8 4 k k Q m q k 4 k-ft k-ft Sutin (i) (ii) (iii) Figure. The rigin f crdinates is taken at and the functins and Q derived frm (ii) as: and Q A unit vertica ad is appied at and the functins m and q derived frm (iii) as: m 4 and q. The mment f inertia and the area f the web are given by: I 6( 4) / 5. 6(. 6) / 44in4 A 44. ( 4. ) 54. in

7 Virtua wrk methds 65 Since and Q are zer ver the ength, the vertica defectin at is given by: δ m d / Qq d AG 8 / 8 ( 4) d/( 9, 44) 8 d /(, 5. 4) in 8 Eampe. Determine the vertica defectin f pint 4 f the pin-jinted frame shwn in Figure. if members and are made. in t shrt and members 56 and 5 are made. in t ng. Sutin ember frces u due t a vertica dwnward unit ad at 4 have aready been determined in Eampe. and are tabuated in Tabe.. Tabe. Determinatin f frces and dispacements in Eampe. ember δ u u δ Tta.55 The vertica defectin is given by: v u in upward Eampe.4 Determine the defectin at the free end f the cantiever shwn in Figure.4. The mment f inertia has a cnstant vaue I ver the ength and increases ineary frm I at t I at.

8 66 Structura Anaysis: In Thery and Practice m (i) I I I (iii) (ii) Figure.4 Sutin The rigin f crdinates is taken at and the functin derived frm (i) as:. A unit vertica ad is appied at and the functin m derived frm (ii) as: m. The mment f inertia is given by: I I I /. Since the vaue f is zer ver the ength, the vertica defectin at is given by: δ m d/ ( ) d/ ( / ) d/ /.5 Vume integratin Fr straight prismatic members, m d / m d /. The functin m is aways either cnstant ang the ength f the member r varies ineary. The functin may vary ineary fr rea cncentrated ads r parabicay fr rea distributed ads. Thus, m d may be regarded as the vume f a sid with a crss-sectin defined by the functin and a height defined by the functin m. The vume f this sid is given by the area f crsssectin mutipied by the height f the sid at the centrid f the crss-sectin. The vaue f m d has been tabuated fr varius types f functins and m, and cmmn cases are given in Tabe..

9 a a a m a b c c Tabe. Vume integras c d α c β c d e / / ac ac/ a ( c d)/ ac/ a ( c 4 d e )/6 ac/ ac/ a ( c d)/6 ac ( β)/6 a ( c d )/6 ac/ ac/6 a ( c d)/6 ac ( α)/6 a ( d e )/6 c ( a b)/ c(a b)/6 a ( c d )/6 b ( c d )/6 ac ( β )/6 bc ( α )/6 a ( c d )/6 b ( d e )/6 Virtua wrk methds 67

10 68 Structura Anaysis: In Thery and Practice Eampe.5 Determine the defectin at the free end f the cantiever shwn in Figure.5. (i) m / (ii) (iii) Figure.5 Sutin The rigin f crdinates is taken at the free end and the functins and m derived frm (i) and (ii) as: and m The defectin at the free end is given by: δ m d / d/ / Aternativey, the sid defined by the functins and m is shwn at (iii); its vume is: / / / and the defectin at the free end is given by: δ /.

11 Virtua wrk methds 69 Aternativey, frm Tabe., the vaue f m d / is given by: δ ac/ / Eampe.6 Determine the rtatin at the free end f the cantiever shwn in Figure.6. w b/in w b/in w / w / (i) m b-in (iii) (ii) Sutin Figure.6 The functins and m are derived frm (i) and (ii) as: w / and m The rtatin at the free end is given by: θ m d/ w d/ w/ 6. Aternativey, the sid defined by these functins is shwn at (iii); its vume is: / 6 w / 6

12 7 Structura Anaysis: In Thery and Practice and the rtatin at the free end is given by: θ w /. 6 Frm Tabe., the vaue f m d / is given by: θ c ( 4de)/ 6 w ( / w/ )/ 6 w/ 6.6 Sutin f indeterminate structures The principe f superpsitin may be defined as fws: the tta dispacements and interna stresses in a inear structure crrespnding t a system f appied frces are the sum f the dispacements and stresses crrespnding t each ad appied separatey. The principe f gemetrica cmpatibiity may be defined as fws: the dispacement f any pint in a structure due t a system f appied frces must be cmpatibe with the defrmatins f the individua members. The tw abve principes may be used t evauate the redundants in indeterminate structures. The first stage in the anaysis is t cut back the structure t a determinate cnditin and appy the eterna ads. The dispacements crrespnding t and at the pint f appicatin f the remved redundants may be determined by the virtua-wrk reatins. T the unaded cut-back structure, each redundant frce is appied in turn and the dispacements again determined. The tta dispacement at each pint is the sum f the dispacements due t the appied ads and the redundants and must be cmpatibe with the defrmatins f the individua members. Thus, a series f cmpatibiity equatins is btained equa in number t the number f redundants. These equatins are sved simutaneusy t btain the redundants and the remaining frces btained frm equatins f equiibrium. Eampe.7 Determine the reactin in the prp f the prpped cantiever shwn in Figure.7 (a) when the prp is firm and rigid, (b) when the prp is rigid and settes an amunt y, and (c) when the prp is eastic. w b/in V w b/in /4 V m w / / (i) (ii) Figure.7

13 Virtua wrk methds 7 Sutin (a) The structure is ne degree redundant, and the reactin in the prp is seected as the redundant and remved as shwn at (i). The defectin f the free end f the cantiever in the ine f actin f V is: w / 6/ 4 w4/ 8 δ T the cut-back structure, the redundant V is appied as shwn at (ii). The defectin f the free end f the cantiever in the ine f actin f V is: V / / V/ δ The tta defectin f in the rigina structure is: Thus: and δ δ δ w 4 / 8 V / V w/ 8 (b) The tta defectin f in the rigina structure is: Thus: and δ δ δ y w 4 / 8 V / y V w/ 8y/ (c) The tta defectin f in the rigina structure is: δ δ δ VL/ AE where L, A, and E are the ength, crss-sectin, and mduus f easticity f the prp.

14 7 Structura Anaysis: In Thery and Practice Thus: w 4 / 8 V / VL/ AE and V w4/ 8( / L/ AE) Eampe.8 The parabic arch shwn in Figure.8 has a secnd mment f area that varies directy as the secant f the spe f the arch rib. The vaue f the thrust required t restre the arch t its rigina span is H, and the vaue f the thrust required t reduce the defectin f t zer is H. Determine the rati f H t H, negecting the effects f aia and shearing frces. /4 y H (i) H H / / H y y H / / H (ii) Figure.8 Sutin The equatin f the arch ais, taking the rigin f crdinates at, is: y ( )/ The secnd mment f area at any sectin is given by: I I sec I d/d s where I is the secnd mment f area at the crwn.

15 Virtua wrk methds 7 The hriznta defectin at due t is btained by cnsidering a virtua unit ad appied hrizntay inwards at. Then frm (i) the defectin due t is btained by integrating ver the ength f the arch: m ds/ m d/ / y d/ / ( ) / 5 9 d / The hriznta defectin at due t H is: / Hy d/ / H 4 ( ) d / H / The tta hriznta defectin f in the rigina structure is: Thus, H 5/ The vertica defectin f due t is btained by cnsidering a virtua unit ad appied verticay upwards at. Then, frm (ii): / y d/ 4 / d/ / 48 The vertica defectin at due t H is: / y H y d/ / H ( ) d / 5H/ 9

16 74 Structura Anaysis: In Thery and Practice The tta vertica defectin f in the rigina structure is: y y y Thus, and H 4/ 5 H / H 8/ 5. 4 Eampe.9 The parabic arch shwn in Figure.9 has a secnd mment f area that varies directy as the secant f the spe f the arch rib. Determine the bending mment at, negecting the effects f aia and shearing frces. H a /4 V H V y y / (i) (ii) (iii) y Sutin Figure.9 The redundant frces cnsist f the reactins H and V at, and the cut-back structure is a curved cantiever. The equatin f the arch ais is: y a( ) The hriznta defectin f due t cnditin (i) is btained by cnsidering a virtua-unit ad appied hrizntay inwards at. Then, ver the span frm t / : m y

17 Virtua wrk methds 75 and / Thus the hriznta defectin at is: / (/ ) y d / / a ( / / ) d/ a 4 / 64 The hriznta defectin f due t cnditin (ii) is: H y d/ Ha ( 4) d/ Ha 5 / The hriznta defectin f due t cnditin (iii) is: V y d/ Va ( ) d/ Va 4 / The tta hriznta defectin f in the rigina structure is: s Thus: / V/ 6 Ha/ 5 () The vertica defectin f due t cnditin (i) is btained by cnsidering a virtua-unit ad appied verticay upwards at. Then, ver the span frm t /: and m /

18 76 Structura Anaysis: In Thery and Practice Thus, the vertica defectin f is: / y ( / )( ) d/ / ( / / ) d/ 5 / 48 The vertica defectin f due t cnditin (ii) is: y H y d / Ha 4 / The vertica defectin f due t cnditin (iii) is: y V d / V / The tta vertica defectin f in the rigina structure is: y y y y s Thus, 5/ 6V Ha/ 4 () Sving () and () simutaneusy: H 5/ 6a and V 5/ 48 Thus, the bending mment at is: V/ Ha/ 4 5/ 96

19 Virtua wrk methds 77 Suppementary prbems S. Determine the defectin at the free end f the cantiever shwn in Figure S.. w b/in a b Figure S. S. Determine the vertica defectin at the center f the pin-jinted truss shwn in Figure S. if the tp chrd members are a. percent in ecess f the required Figure S. S. The pin-jinted truss shwn in Figure S. has a vertica ad f kips appied at pane pint. Determine the resuting vertica defectin at pane pints and. hat additina vertica ad must be appied at pane pint t increase the defectin at pane pint by 5 percent? A members have a crss-sectina area f in and a mduus f easticity f 9, kips/in k 8 8 Figure S. S.4 A cantiever circuar arch rib with a unifrm sectin is shwn in Figure S.4. A hriznta frce H is appied t the free end f the rib s that end can defect ny verticay when the vertica ad V is appied at. Determine the rati f H/V. H V Figure S.4

20 78 Structura Anaysis: In Thery and Practice S.5 The frame shwn in Figure S.5 has a vertica ad f kips appied at 5. Determine the resuting vertica defectin at nde 4. The crss-sectina areas f the members are in, 4 5 in, and 4 4 in. The secnd mment f areas f the members are 9 in 4 and 4 5 in 4. The mduus f easticity f a members is 9, kips/in k 4 4 Figure S.5 S.6 The tw-stry, singe-bay frame shwn in Figure S.6 has the reative secnd mments f area indicated. The crss-sectina area f the cumns is A. The mduus f easticity f a members is E, and the mduus f rigidity is G. Determine the bending mments and shear frces in the members and cacuate the hriznta defectin f nde due t the ad f 4 kips. 4 k 6 I I I 5 I I I 4 Figure S.6

21 Virtua wrk methds 79 S.7 The parabic arch shwn in Figure S.7 has a secnd mment f area that varies directy as the secant f the spe f the arch rib and a temperature cefficient f therma epansin f α per F. Negecting the effects f aia and shearing frces, determine the supprt reactins at the fied end prduced by a temperature rise f t F. a /4 Figure S.7 References. Di aggi, I. F. Principe f virtua wrk in structura anaysis. Prc. Am. Sc. Civi Eng. 86 ( ST ). Nv. 96. pp Ghse, S. The vume integratin methd f structura anaysis. The Structura Engineer. 4. arch 964. pp. 95.

Theoretical Evaluation on Effects of Opening on Ultimate Load-carrying Capacity of Square Slabs

Theoretical Evaluation on Effects of Opening on Ultimate Load-carrying Capacity of Square Slabs Eectrnic Jurna f Structura Engineering (8 008 Theretica Evauatin n Effects f Opening n Utimate ad-carrying Capacity f Square Sabs Chee Khn Ng Universiti Maaysia Sarawak, Maaysia Timthy Juius Edward Universiti