Structural Analysis I
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- Pamela Mosley
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1 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss I f y y θ θ z z f z z θ y y f Sprng Semester, 8 Hae Sung ee Dept. of Cv and Envronmenta Engneerng Seou Natona Unversty Structura Anayss ab.
2 Dept. of Cv and Envronmenta Eng., SNU Ths page s ntentonay eft bank. Structura Anayss ab.
3 Dept. of Cv and Envronmenta Eng., SNU Contents. Introducton. Reactons & Interna orces by ree Body Dagrams. rncpe of Vrtua Work. Anayss of Statcay Indetermnate Beams 5. Anayss of Statcay Indetermnate Trusses 6. Anayss of Statcay Indetermnate rames 7. Infuence nes for Determnate Structures 8. Infuence nes for Indetermnate Structures Structura Anayss ab.
4 Dept. of Cv and Envronmenta Eng., SNU Ths page s ntentonay eft bank. Structura Anayss ab.
5 Dept. of Cv and Envronmenta Eng., SNU Chapter Introducton Structura Anayss ab.
6 Dept. of Cv and Envronmenta Eng., SNU. echancs of atera - Structura echancs robem Cacuate the reacton force at each support and draw the moment and shear force dagram for the two-span beam shown n the fgure. q Souton q Ra Rb Rc Equbrum Equaton y Ra Rb Rc q a q Rb Rc Rb Rc q b q q R a Rc Ra Rc c q Ra Rb Ra Rb q Snce there are three unknowns n two ndependent equatons, we cannot determne a unque souton for the gven structure, and thus we need one more equaton to sove ths probem. The man ssue of ths cass s how to bud addtona equatons to anayze statcay ndetermnate structures. Structura Anayss ab.
7 Dept. of Cv and Envronmenta Eng., SNU. echancs of atera q w w Governng Equaton eft span q '''' w q w a b c d Rght Span q '''' w q w a b c d Boundary Condtons eft support w, w Center support w w, w w, w w Rght support w, w Snce there are eght unknowns wth eght condtons, we can sove ths probem. Determnaton of Integraton Constant eft Support q d, w b w a c w Rght Support q w d, w b w a c Structura Anayss ab.
8 Dept. of Cv and Envronmenta Eng., SNU Center Support q w a c q w a c q q a c a 6 6 q q 6a 6a a a c c c q 8 q 8 q w w 8 q q q w, V w q 8 8 oment Dagram 8 -.5q Shear Dagram 8.7q -.65q - Reactons.75q.75q.5q.75q Structura Anayss ab.
9 Dept. of Cv and Envronmenta Eng., SNU. echancs of atera α 5.. an dea Orgna robem q Case I Remova of the center support q Case II Appcaton of the reacton force R Orgna robem Case I Case II compatbty condton Rb R.. Cacuaton of Bendng oment q q / q q Structura Anayss ab.
10 Dept. of Cv and Envronmenta Eng., SNU Governng Equaton 6 Boundary support Condtons eft Support w Rght Support w Determnaton of Integraton Constant eft Support w b Rght Support q q w w a b 6 q q w a a 6 q Defecton q w q w 5q 8.. Cacuaton of R Bendng oment Rb/ Rb Rb Rb Rb Structura Anayss ab.
11 Dept. of Cv and Envronmenta Eng., SNU Governng Equaton 7 w R w R w w R R Rb a b Rb Rb a b Boundary support and md-span Condtons Support wr, wr d-span w w, w w θ R R θ R R R R Determnaton of Integraton Constant eft Support w R b Rght Support & d-span R b Rb wr a b a R b wr a b wr a R b b wr a a wr Defecton w w R R Rb Rb Rb Rb 6 R wr wr R 8 b Compatbty Condton 5q Rb R R b q Structura Anayss ab.
12 Dept. of Cv and Envronmenta Eng., SNU. Structura echancs 8 Orgna robem q Case I Remova of the center support q q / Case II Appcaton of the reacton force Rb/ Rb Orgna robem Case I Case II rncpe of Vrtua Work R R 5q d, 8 R R R d Rb 8 Souton 5q Rb R R b q Structura Anayss ab.
13 Dept. of Cv and Envronmenta Eng., SNU 9 oment q / 5q /8 Rb 8 -.5q.7q Shear Structura Anayss ab.
14 Dept. of Cv and Envronmenta Eng., SNU.5 지점 Supports 고정단 fed support Structura Anayss ab.
15 Dept. of Cv and Envronmenta Eng., SNU 회전단 hnge support 이동단 roer support Structura Anayss ab.
16 Dept. of Cv and Envronmenta Eng., SNU.6 구조물의 차원이상화 가로보 Cross Beam 세로보 Strnger 주구조물 an Structure Cross Bracng Wnd Bracng 지점 Support Structura Anayss ab.
17 Dept. of Cv and Envronmenta Eng., SNU Truss 절점 Jont Beam Structura Anayss ab.
18 Dept. of Cv and Envronmenta Eng., SNU rame Structura Anayss ab.
19 Dept. of Cv and Envronmenta Eng., SNU.7 orce and Dspacement 5 Rea -D Structures force components and moment components dspacement components and rotatona components f y y θ θ z z y f z z θ y y f z Beam Ideazaton Vertca force and oment on z-as Vertca dspacement and rotatona ange w.r.t. z-as V, w V, w V, w, θ, θ, θ ane Truss Ideazaton Vertca and horzonta force Vertca and horzonta dspacement Structura Anayss ab.
20 Dept. of Cv and Envronmenta Eng., SNU 6 f f y y ane rame Ideazaton Vertca, horzonta force and moment w.r.t. z-as Vertca, horzonta dspacement rotatona ange w.r.t. z-as f y y z θ z f Structura Anayss ab.
21 Dept. of Cv and Envronmenta Eng., SNU.8 구조물의안정 Stabty of Structures 7 내적안정 Interna Stabty 어느한계내의크기의어떠한하중의작용을받더라도형상이허물어지지 않는구조물의상태 외적안정 Eterna Stabty 어느한계내의크기의어떠한하중의작용을받더라도구조물이움직이지않 는상태 Structura Anayss ab.
22 Dept. of Cv and Envronmenta Eng., SNU Ths page s ntentonay eft bank. 8 Structura Anayss ab.
23 Dept. of Cv and Envronmenta Eng., SNU 9 Chapter Reactons & Interna orces by ree Body Dagrams Structura Anayss ab.
24 Dept. of Cv and Envronmenta Eng., SNU. ree Body Dagram Structura Anayss ab.
25 Dept. of Cv and Envronmenta Eng., SNU It s mpossbe to draw too many free-body dagrams. Tme spent n dong so s never wasted - C. H. Norrs & J. B. Wbur & S. Utku - Structura Anayss ab.
26 Dept. of Cv and Envronmenta Eng., SNU. Reactons Beams a b RA RB V R R A B A a RB Cockwse a b R B, R A a RA RBa V RA RB A a RB Cockwse a a R B, R A Structura Anayss ab.
27 Dept. of Cv and Envronmenta Eng., SNU Truss HA RA RB H H A H V, R R A B A A a a RB a Cockwse R A, R B rame HA q HB V H R A H q RB H A B R h RB H B q RA H A RB H q RA H A q q q H A H B H A H B 8, h B RA RB H H A B q 6 q 6 R R A B q 8 q 8 Structura Anayss ab.
28 Dept. of Cv and Envronmenta Eng., SNU. Interna orces n ramed Structures Aa orce Shear orce Bendng oment Torson Structura Anayss ab.
29 Dept. of Cv and Envronmenta Eng., SNU. 들보 Beam 5 Reactons q RAq/ Rb q/ Structura Anayss ab.
30 Dept. of Cv and Envronmenta Eng., SNU q 6 RA RB RAq/ RB q/ ree Body Dagram for Shear and oment RA V RB q V RA q V V RA q q RA q q q Shear orce and oment Dagrams q/ q /8 Structura Anayss ab.
31 Dept. of Cv and Envronmenta Eng., SNU 7 Defected Shape.5 Gerber Systems Structura Anayss ab.
32 Dept. of Cv and Envronmenta Eng., SNU 8 Structura Anayss ab.
33 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab Interna orces n a Gerber Beam - I ree Body Dagram R R H H C R R R C C H v R R R B H B A R R R A B A v 6 / RH RA RB RC /6 5/6 /
34 Dept. of Cv and Envronmenta Eng., SNU Shear orce V -/6 /6 V / /6 5/6 V -/ /6 5/6 /6 - - / / Bendng oment /6 /6 /6 Structura Anayss ab.
35 Dept. of Cv and Envronmenta Eng., SNU, 6 /6 5/6 / /6 5/6 / / /6 - /6 Defected Shape d w q d Structura Anayss ab.
36 Dept. of Cv and Envronmenta Eng., SNU.5. Interna orces n a Gerber Beam - II q ree Body dagram q q q q q q q Shear q q oment q q 8 q Structura Anayss ab.
37 Dept. of Cv and Envronmenta Eng., SNU.6 트러스 Truss Structura Anayss ab.
38 Dept. of Cv and Envronmenta Eng., SNU Assumpton. A jonts are hnges.. A members are straght.. Sma deformatom. The eterna oads are apped ony at jonts. Characterstcs of truss By the nd, rd and th assumptons d d d q d a b By the st assumpton a b, V No bendng moment and shear force are nduced n a members n a truss structure. Ony aa forces are the nterna forces n a truss. Structura Anayss ab.
39 Dept. of Cv and Envronmenta Eng., SNU.6. Interna orces n Howe Truss 5 U U 8 U H 6 / / At U and U 8 9 At, / At 5 5, - 6 5, 6 6 Structura Anayss ab.
40 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 6 At U At Aa orce Dagram / / / / / /
41 Dept. of Cv and Envronmenta Eng., SNU 7 Equvaent Beam Acton V/ / / / Defected Shape.6. Interna orces n Warren Truss U U U Structura Anayss ab.
42 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 8 At, At U, At , - / 6 5
43 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 9 At U, At, At U
44 Dept. of Cv and Envronmenta Eng., SNU At / / OK Defected Shape Equvaent oment Structura Anayss ab.
45 Dept. of Cv and Envronmenta Eng., SNU.6. ethod of Sectons U U U Cut out U 6 6 V 5 5 Structura Anayss ab.
46 Dept. of Cv and Envronmenta Eng., SNU.7 프레임 rame Structura Anayss ab.
47 Dept. of Cv and Envronmenta Eng., SNU.7. Interna orces n a rame Reactons q H qh qh qh reebody Dagram qh qh qh qh qh qh qh qh qh Structura Anayss ab.
48 Dept. of Cv and Envronmenta Eng., SNU Aa, Shear and oment dagram - - Aa qh Shear qh qh qh qh oment Defected Shape Structura Anayss ab.
49 Dept. of Cv and Envronmenta Eng., SNU 5.7. Interna orces n a -hnged rame q h HA HB RA RB Reactons :Cockwse for mement q RA RB H A H B R : R h B H B RB H B q h : R H A RA H A A q q H A H H H A B B q H A H B q 8 H H A B q 6 q 6 R R A B q 8 q 8 Structura Anayss ab.
50 Dept. of Cv and Envronmenta Eng., SNU 6 reebody Dagram q 6 q 6 q 6 q 6 q 6 q 8 q 6 q 8 q 8 q 6 q 8 q 6 q 6 q 6 Aa, Shear and oment dagram q 8 - q 6 q 8 - q 8 - Aa - - Shear q 8 q 8 q 6 q 6 Structura Anayss ab.
51 Dept. of Cv and Envronmenta Eng., SNU 7 q oment q 6 q 6 q 8 V q 6 q 6 q q 8 q q 8 Defected Shape Structura Anayss ab.
52 Dept. of Cv and Envronmenta Eng., SNU 8.8 Arches Structura Anayss ab.
53 Dept. of Cv and Envronmenta Eng., SNU 9.8. Three Hnged Arch y h HA HB RA RB h Arch Curve : y Reactons reebody Dagram HA y V R R A B H, H H A B h h V H h H A y RA y h h h ma 8 RA Structura Anayss ab.
54 Dept. of Cv and Envronmenta Eng., SNU Aa force and Shear orce 5 S V A θ H H Acosθ S sn θ H cosθ V Asn θ S cosθ V sn θ sn θ A cosθ S A cosθ S sn θ sn θ H cosθ V h tan θ y cosθ h, sn θ h h A cosθ sn θ h S sn θ cosθ h h h h h Defected Shape Structura Anayss ab.
55 Dept. of Cv and Envronmenta Eng., SNU 5.8. Zero oment Arch I y h HA HB RA RB Reactons reebody Dagram R R A B, H H A B h h HA y V H V H h H A y RA y h h y RA y Structura Anayss ab.
56 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab Zero oment Arch II Reactons q R q R B A, h q H h q H B A reebody Dagram q R y H A A h y q q y h q HA RA HB RB q h y HA RA V H y q
57 Dept. of Cv and Envronmenta Eng., SNU 5 Chapter rncpe of Vrtua Work The prncpe of vrtua work s the most mportant subject n the area of the structura anayss!!!! Structura Anayss ab.
58 Dept. of Cv and Envronmenta Eng., SNU. Beam robems 5 q d V VdV.. Governng Equatons Equbrum for vertca force V dv V qd dv d q Equbrum for moment d d d Vd qd V d Emnaton of shear force d d q Stran-dspacement reaton d w ε y d Stress-stran reaton Hooke aw σ Defnton of oment d w ε E y d E d w d w σyda EεydA E y da d d A A A Beam Equaton d w q d Structura Anayss ab.
59 Dept. of Cv and Envronmenta Eng., SNU odeng of Concentrate oads - Drac deta functons 55 ε m ε ξ ε ξ ξ ξ d m ε ξε d ξε ξε ε d ξε d m ε ε ε f ξ d m ε m ε ξε ε f d ξε ξε ξε ξε f ε d ξε f d ξ ε ξ ε f d m ξ ε ε f ξ.. rncpe of Vrtua Work Beam Equbrum equaton n an ntegra form d w d q d Integraton by part twce d w d d w d dw d dw d d d d wqd d w d d wqd d w d wqd wv d θ In case w s a dspacement fed of the same structure caused by another oad case q, then the boundary terms vansh snce ether dspacement or reacton shoud be zero at a boundary support. Structura Anayss ab.
60 Dept. of Cv and Envronmenta Eng., SNU rncpe of vrtua work 56 d w d d d d wqd Equbrum equaton for oad case q nt W W et d w d q d Vrtua work epresson d wqd Bett-awe s Recproca Theorem d d wqd Cacuaton of dspacement for the oad case q d wqd wqd In case q system represents a snge unt concentrated oad apped at the poston where you want to cacuate the dspacement for q system... Eampe d w d w w d A smpe beam subject to an unform oad oment of oad case q q q /8 Structura Anayss ab.
61 Dept. of Cv and Envronmenta Eng., SNU 57 oment of oad case q / Defecton at the center of the span w d / q / q q d or from the ntegraton tabe, q q q d / q q 96 / q d q q d q ab w d Vaues of roduct Integras U d q 8 5q 8 Structura Anayss ab.
62 Dept. of Cv and Envronmenta Eng., SNU.. Conservaton of Energy 58 q q w Equbrum and Conservaton of Energy Equbrum Equaton w Eterna work W et [ [ wqd d w q d d w w d d d w d w d w d w d d d d w d w d wv d d d w d w d d d θ d W dw d w ] d d ] nt Conservaton of Energy n each oad case d wqd, d wqd Two oad cases are apped smutaneousy. d d d d wq wq d w w q q d d wq wq wq wq d Structura Anayss ab.
63 Dept. of Cv and Envronmenta Eng., SNU Eterna work for sequenta oadng q frst 59 q q w w w wqd wqd wqd qwd rncpe of Vrtua work w qwd wqd wqd wq wq d w w q q d wq wq wq wq d qwd d wq wq d wqd..5 Genera Conservaton and Equbrum qwd wqd ds v n Conservaton n Genera By dvergence theorem, S v fdv S nds V v nds vdv where,,,,. y z V Structura Anayss ab.
64 Dept. of Cv and Envronmenta Eng., SNU S fdv vdv fdv v f dv v nds V V V V 6 Snce the ntegra equaton shoud hod for a systems, v f In a potenta probem, the vector fed of a system s defned by a gradent of a scaar functon referred to as a potenta functon v k Φ The famous apace equaton for a conservatve system. v f Φ f Φ f or Φ Φ Φ y z f Equbrum n Genera orce Equbrum: y z or T ds bdv or ds bdv S V T for,, S V Suppose T σ n or T σ jn j j σ n σ σ σ σ n σ σ σ σ σ, n n σ σ σ σ n Dvergence Theorem S T ds V bdv S V σ nds σ b dv V b dv V σ dv V b dv for,, Snce the ntegra equaton shoud hod for a systems n equbrum, σ σ σ σ j σ b b b for,, or y z σ σ σ σ y σ y σ y j σ z σ z σ z j b b b Structura Anayss ab.
65 Dept. of Cv and Envronmenta Eng., SNU 6 oment Equbrum: for,, or S vds fdv mdv V V σ σ, σ σ, σ σ What s σ, and how s σ reated to a potenta functon? : out of scope of ths cass!..6 Dspacement on boundares d d wv wqd wv θ θ wqd wv θ w V w V θ θ By concdng the postve drecton of forces and dspacement w V w V θ θ Defecton of a cantever beam subject to an end oad V,, w, θ d q rea system q vrtua system - w d - Or, you can obtan the same answer by assumng the unt concentrate oad s apped at just eft sde of the boundary. Rotaton of a cantever beam subject to an end oad V,, w, θ q rea system q vrtua system - Structura Anayss ab.
66 Dept. of Cv and Envronmenta Eng., SNU 6 θ d Rotaton n the a body or a structure odeng of a unt moment apped at /ε ε /ε d m ε w[ ε m [ w ε ε ε ε w ε ε ] dw d ε ] d θ by concdng the postve drecton of the rotatona ange wth that of the apped moment. θ d Structura Anayss ab.
67 Dept. of Cv and Envronmenta Eng., SNU. rncpe of Vrtua Work n Genera 6.. -Dmensona Eastc Body Rgd Body If a rea q-force system s actng on a rgd body s n equbrum and remans n equbrum as the body s gven any sma dspacement, the vrtua work done by the q-force system s equa to zero. W et w vds w fdv w vds fdv S V S V Deformabe Body q-orce System q -orce System If a deformabe body s n equbrum under a rea q force system whe t s subjected to sma and compatbe dspacement caused by a vrtua q force system, the eterna vrtua work done by the rea q force system s equa to nterna vrtua work done by the nterna q stress!!! W et S w vds w fdv V S w vds S j V w v ds j S j w σ j w dv σ j n ds j j S w σ j σj w j V j j w σ n j j ds dv Structura Anayss ab.
68 Dept. of Cv and Envronmenta Eng., SNU W et j σ j w j V j j j V j V j V j V w w σ jdv j w σ jdv j w σ jdv j w j w j V j σ w j V j V j dv σ dv V σ j j j w σ jdv w j σ jdv j V w f dv f j dv j j V ε σ dv W w j σ jdv nt 6.. ramed Structures j V ε j σ w vds w fdv S V j V j dv V εσdv V γτdv e ε j e V σ j dv εσdv Interna vrtua work by norma stress bendng moment e V εσdv e V e d w d d w d e A d w y E y dv d d w Ey da d d e e V d w d w E y d d d w d w d d d Interna vrtua work by norma stress Aa orce dv e εσdv dv da d V e V e EA A Interna vrtua work by shear stress e EA A e A e e e V e A EA γτdv d w d w E y d d V V τ Q and γ Q where Q Ib y GIb y yda γτdv Q QdV dad V Tota dspacement e V e V GIb y V Ib y e VV G A e Q I b y e e e VV w d fs d d GA EA e a y d f s GA e d VVd dad e d Structura Anayss ab.
69 Dept. of Cv and Envronmenta Eng., SNU.. Effect of Shear Deformaton 65 or smpe beam wth a unform oad case Shear Effect 5q w 8 Shear force of oad case q q/ Shear force of oad case q / Defecton by shear force w S e / f s f s f s VVd VVd GA GA GA w w s fsq /8GA 5q / 8 f s VV 8 GA f s q 8GA for a rectange secton of h b wth stee w w bh s 5 bh h.5 or sma h/, the effect of shear deformaton can be negected. Structura Anayss ab.
70 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 66. Truss robems.. rncpe of Vrtua Work rom Genera prncpe α α e s S EA d EA d GA VV f d v u w ds e e e cos cos w q rom equbrum equaton, m j j m j j Y V H for njn,, where m, j H and j V are the number of member connected to jont, the horzonta component and the vertca component of the bar force of j-th member connected to jont, respectvey. ] [ njn m j j m j j v Y V u H θ θ θ θ θ θ njn nmb e e njn njn m j j j m j j j njn m j j j m j j j Y v u v v u u v Y u v u v Y u sn cos sn cos sn cos θ θ n nmb e nmb e nmb e nmb e nmb e v Y u EA EA EA v v u u sn cos Y j m Y
71 Dept. of Cv and Envronmenta Eng., SNU 67 u u cosθ v v sn θ v v θ θ u u or, by appyng Bett-awe recproca theorem njn nmb e u Y v EA The dspacement of a jont k n a truss s obtaned by appyng a unt oad at a jont k n an arbtrary drecton. nmb e k k k k k k u Y v u cosα u cosα EA k u cosα k u Snce α represnts the ange between the apped unt oad and the dspacement vector, k u cosα are the dspacement of the jont k n the drecton of the apped unt oad. or vertca dspacement or Horzonta dspacement Structura Anayss ab.
72 Dept. of Cv and Envronmenta Eng., SNU.. Eampe 68 Rea System Vrtua System ember Tabe for cacuaton of the defecton of a truss EA EA EA EA Structura Anayss ab.
73 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab Conservaton of Energy Equbrum and Conservaton of Energy Equbrum Equaton, m j j m j j Y V H for njn,, Eterna work njn m j j m j j njn njn et v V u H Y v u W nt nmb e e nmb e nmb e e njn m j j j m j j j njn m j j j m j j j njn m j j m j j et W EA v v u u v u v u v V u H W θ θ θ θ θ θ sn cos sn cos sn cos et W nt W Conservaton of Energy n each oad case njn nmb EA, njn nmb EA Two oad cases are apped smutaneousy njn nmb njn nmb EA EA Δ Δ Δ Δ Eterna work for sequenta oadng frst njn njn njn njn njn njn njn njn njn njn nmb EA njn njn nmb EA
74 Dept. of Cv and Envronmenta Eng., SNU. rame robems 7 e e d s e VV f d GA where s the dspacement n the drecton of apped unt concentrate oad n the vrtua system. e EA d HA HB/ RA/ RB/ / - / - / - / / oment Shear Aa d / d / / / / 6 6 Structura Anayss ab.
75 Dept. of Cv and Envronmenta Eng., SNU 7 S f s GA d / d / / / / S V f svv GA f s d GA 8 f s GA A / s d EA d / / / / A V AA EA d EA 9 6 EA f s S A 6 6 GA h h EA In most cases, the deformaton caused by the shear force and the aa force neggby sma compared to that caused by the bendng moment. If ths s the case, the dspacement of a frame can be appromated by consderng ony the bendng moment. e d e Structura Anayss ab.
76 Dept. of Cv and Envronmenta Eng., SNU 7 Ths page s ntentonay eft bank. Structura Anayss ab.
77 Dept. of Cv and Envronmenta Eng., SNU 7 Chapter Anayss of Statcay Indetermnate Beams θ R θ Structura Anayss ab.
78 Dept. of Cv and Envronmenta Eng., SNU. ropped Cantever Beam 7 q A Equbrum equaton... The frst dea RA R A A R B q q R B RB R B Compatbty condton R B The end dspacements of the cantever beam for two oads cases are cacuated by the prncpe of vrtua work. -q / - q d d q 8 d d Structura Anayss ab.
79 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton and the fna souton 75 oment Dagram q RB RB q up 8 8 R A 5 q, 8 A q 8 q /8 -q / Defected shape -q /8 - /8 9 q 8... The second dea A θ Compatbty condton θ A θ Aθ Structura Anayss ab.
80 Dept. of Cv and Envronmenta Eng., SNU Rotona Ange at the fed end 76 q /8 q θ d d 8 q θ d d Compatbty condton and the fna souton q A A q 8 Other reactons by a free body dagram q /8 q/ q/ q/8 q/8 q /8 5q/8 q/8 Structura Anayss ab.
81 Dept. of Cv and Envronmenta Eng., SNU. Cantever Beam wth Sprng Support 77 Robn BC The thrd type BC V w kw rmary structure Ra R B Compatbty Condton wbeamsprng R B sprng q 8 R B R k B R B q 8 k q 8 k k As k, RB q, and As k, R B 8 Defected Shape for k Structura Anayss ab.
82 Dept. of Cv and Envronmenta Eng., SNU. Support Settement 78 rmary structure RB Compatbty condton R B R R B B Defected Shape Structura Anayss ab.
83 Dept. of Cv and Envronmenta Eng., SNU. Temperature Change 79 T T rmary structure A θ θ A Curvature due to temperature change hd θ α T T d α T T d dθ α T d h T d w d α T T w a b h α T T or smpe beam, w w a h Comaptbty condton α T T θ Aθ T T A A α h h Structura Anayss ab.
84 Dept. of Cv and Envronmenta Eng., SNU.5 Shear Effect 8 rmary Structures Shear force dagram q Compatbty condton S S R B RB S VV f f f d d q GA GA GA fq GA S VV f f f d d GA GA GA f GA q / 8 q / fq / GA f / GA q f / GA f / GA R B 8 8 q. h /. 78 h / or retanguar secton f GA 6 Ebh / 5 Ehb / v. h h or h R B q. h / q. 6 q. 7.7 % error h / 8. 8 Structura Anayss ab.
85 Dept. of Cv and Envronmenta Eng., SNU or h 8 R B or R B q. h / q. q. 6.6 % error h / h 5 q. h / q. 6 q.. % error h / 8. 8 You may negect the effect of the shear deformaton n most cases!!.6 -Span Contnuous Beam q q rmary structure q q θ R θ Compatbty θ R θ R R R θ θ θ θ θ θ θ θ R B B B q Structura Anayss ab. q 8
86 Dept. of Cv and Envronmenta Eng., SNU 8 θ θ R d 6 q q 8 d q 6 q 5 8 q d θ θ R d d d B θ θ θ θ R R 5 q Defected shape Structura Anayss ab.
87 Dept. of Cv and Envronmenta Eng., SNU.7 ed-ed End Beam 8 q A B RA RB.7.. rmary Structure type I -q / - q d d q d d q 8 q 6 d Structura Anayss ab.
88 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton ebty equaton 8 q 8 q 6 q, q.7.. rmary Structure type II q 8 q d 8 q d 8 q q d d 6 d 6 Structura Anayss ab.
89 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton ebty equaton 85 q q 6 6 q Reactons and oment Dagrams q q q q Defected Shape Structura Anayss ab.
90 Dept. of Cv and Envronmenta Eng., SNU.8 -Span Contnuous Beam Unform oad case q rmary structure d q 8 q 8 q I I d q 8 q 8 q I I d I I d 6 6 d I I Structura Anayss ab.
91 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton ebty equaton 87 q q I I I I 6 I I 6 I I q 8 I I I.5 I In case I I, q.8.. Compcated oad Case q rmary structure Structura Anayss ab.
92 Dept. of Cv and Envronmenta Eng., SNU 88 d q 8 q q q q q d q 8 Compatbty condton ebty equaton q 5q q 6 q, Compatbty Condton ebty Equaton n Genera n j j j Structura Anayss ab.
93 Dept. of Cv and Envronmenta Eng., SNU 89 Chapter 5 Anayss of Statcay Indetermnate Trusses Structura Anayss ab.
94 Dept. of Cv and Envronmenta Eng., SNU 5. Varous Types of Trusses 9 Determnate Truss Eternay Indetermnate Truss Internay Indetermnate Truss ed Indetermnate Truss Structura Anayss ab.
95 Dept. of Cv and Envronmenta Eng., SNU 5. A Smpe Truss ethod - I Structura Anayss ab.
96 Dept. of Cv and Envronmenta Eng., SNU 9 U U,,:.5A,5:A rmary structure At Structura Anayss ab.
97 Dept. of Cv and Envronmenta Eng., SNU At ,.75 At U , 5 Structura Anayss ab.
98 Dept. of Cv and Envronmenta Eng., SNU Aa force tabe for prmary structure 9 em A EA EA EA EA.5 EA.75.5EA.5EA.5 EA.75.5EA EA.79 EA 7.59 EA Compatbty Condton. 6 na Souton em Structura Anayss ab.
99 Dept. of Cv and Envronmenta Eng., SNU 5.. ethod - II 95 Structura Anayss ab.
100 Dept. of Cv and Envronmenta Eng., SNU 96 AE Compatbty condton AE rmary structure Structura Anayss ab.
101 Dept. of Cv and Envronmenta Eng., SNU Aa force tabe for prmary structure 97 em A EA EA EA ` EA.56 EA.68 EA.8 EA.5 EA.5 EA EA.6 EA AE AE AE AE H.8. 6 Structura Anayss ab.
102 Dept. of Cv and Envronmenta Eng., SNU A Truss wth Roer Support rmary Structures Structura Anayss ab.
103 Dept. of Cv and Envronmenta Eng., SNU Aa force tabe for prmary structure 99 em EA EA EA Compatbty condton EA Structura Anayss ab.
104 Dept. of Cv and Envronmenta Eng., SNU Temperature change and fabrcaton error EA α T f In case of no eterna oads EA.8 α T f. α T f EA 5. Truss wth Two Hnge Supports rmary structure and compatbty condton EA Structura Anayss ab.
105 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab.
106 Dept. of Cv and Envronmenta Eng., SNU Aa force tabe for the prmary structure em EA EA EA EA EA EA Structura Anayss ab.
107 Dept. of Cv and Envronmenta Eng., SNU Chapter 6 Anayss of Statcay Indetermnate rames, w Structura Anayss ab.
108 Dept. of Cv and Envronmenta Eng., SNU 6. Γ-shaped rame-i.75 RC A RA Equbrum equaton R A R A C R C 6.. rmary Structure type I Rc Structura Anayss ab.
109 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton 5 R C End Dspacements - - / - - / d d { } d d { } d d Compatbty condton and the fna souton 9 R C R C. 5, R A. 55, A Structura Anayss ab.
110 Dept. of Cv and Envronmenta Eng., SNU oment Dagram /.5.5 / Defecton Shape Structura Anayss ab.
111 Dept. of Cv and Envronmenta Eng., SNU 6... rmary Structure type II 7 B θ θ B Compatbty Condton θ Bθ Rotaton Ange / / / / / d d 6 6 d d Structura Anayss ab.
112 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton and the fna souton 8 6 B B 6 6. Γ-shaped rame-ii w A, RA, HB 6.. rmary Structure type I RB - -w / Structura Anayss ab.
113 Dept. of Cv and Envronmenta Eng., SNU 9 w d d w d d w 6 w 8 d d { } d d d d Compatbty condton ebty Equaton w 6 w 8 w, 8 w 7 Reactons w /8 w w/8 w/7 w/8 w/7 Structura Anayss ab.
114 Dept. of Cv and Envronmenta Eng., SNU oment Dagram -w /8 -w /8 w /96 Defected Shape 6... rmary Structure type II w /8 Structura Anayss ab.
115 Dept. of Cv and Envronmenta Eng., SNU d d w d d d d 6 d d Compatbty condton ebty Equaton w w 6 6 w, 8 w 8 Structura Anayss ab.
116 Dept. of Cv and Envronmenta Eng., SNU 6. orta rame subject to Horzonta oad, w 6... rmary Structure type I w w / Structura Anayss ab.
117 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. w d d w d d 6 w d d 6 d d 5 d d d d d d d d d d Compatbty condton ebty Equaton w w w atr form w w w 6 6 5
118 Dept. of Cv and Envronmenta Eng., SNU 5 w w w w 6 7 w w 6 5 Reactons w 9w/ 5w/ 59w /5 w /5 oment Dagram w /7 w /7 Defected Shape Structura Anayss ab.
119 Dept. of Cv and Envronmenta Eng., SNU 6... rmary Structure type II 5 w -w / - d d d d d d w 6 w 8 w 8 d d Structura Anayss ab.
120 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 6 d d d d d d 6 d d d d Compatbty condton ebty Equaton atr orm w w w w w w w w w w
121 Dept. of Cv and Envronmenta Eng., SNU 6. orta rame subject to Vertca oad 7 a, ab - Structura Anayss ab.
122 Dept. of Cv and Envronmenta Eng., SNU 8 d d d d 6 6 b a ab ab ab a b 6 ab a b 6 d atr orm ab a b a 7b a b ab 7a b a b ab Sdesway : b a 8 Defected Shape a Structura Anayss ab.
123 Dept. of Cv and Envronmenta Eng., SNU 6.5 Order of Indetermnacy 9 # of unknowns # of member # of nterna force per member # of reactons - # of known quanttes # of equatons # of member # of E.E. per member # of jonts # of E.E. per jont - # of used equatons # of Indetermnacy # of unknowns - # of equatons Order of Indetermnacy of the frame shown above Order of Indetermnacy Beam Number of Interna orces n a ember : Number of Equbrum Equatons n a ember : Number of Equbrum Equatons at a Jont : Smpe Beam - # of unknowns : 6 or 6 - # of equatons : 6 Structura Anayss ab.
124 Dept. of Cv and Envronmenta Eng., SNU - Order of Indetermnacy : 6 6 Overhanged Beam - # of unknowns : 8 - # of equatons : 8 - Order of Indetermnacy : 8 8 Gerber Beam - # of unknowns : - # of equatons : - Order of Indetermnacy : Contnuous Beam - # of unknowns : 6 - # of equatons : - Order of Indetermnacy : Order of Indetermnacy - Truss Number of Interna orces n a ember : Number of Equbrum Equatons n a ember : Number of Equbrum Equatons at a Jont : Structura Anayss ab.
125 Dept. of Cv and Envronmenta Eng., SNU Determnate Truss - # of unknowns : - # of equatons : - Order of Indetermnacy : Internay Indetermnate Truss - # of unknowns : # of equatons : - Order of Indetermnacy : 8 Internay Indetermnate Truss - # of unknowns : # of equatons : - Order of Indetermnacy : 6 Structura Anayss ab.
126 Dept. of Cv and Envronmenta Eng., SNU Order of Indetermnacy - rame Number of Interna orces n a ember : 6 Number of Equbrum Equatons n a ember : Number of Equbrum Equatons at a Jont : Internay Indetermnate rame - # of unknowns : 6 - # of equatons : - Order of Indetermnacy : rame wth Hnges - # of unknowns : # of equatons : Order of Indetermnacy : 6 6 orta rame wth ed Supports - # of unknowns : # of equatons : - Order of Indetermnacy : Structura Anayss ab.
127 Dept. of Cv and Envronmenta Eng., SNU Seectng a rmary Structure for a Compcated rame Order of Indetermnacy of the frame - # of unknowns : # of equatons : Order of Indetermnacy : By Repacng a Rgd Jont wth a Hnge, we can reduce - # of unknowns by the number of members at the jont - # of equatons by one In the rmary Structure - Reducton n unknowns : 9 - Reducton n equatons : 5 - Reducton n Indetermnacy : 7 5 Structura Anayss ab.
128 Dept. of Cv and Envronmenta Eng., SNU 6.6 Genera rame q q / rmary Structure q /8 q,, Structura Anayss ab.
129 Dept. of Cv and Envronmenta Eng., SNU 5 q /8 q - q d 8 q q d 6 d d, d, d, 6 q 6 Structura Anayss ab.
130 Dept. of Cv and Envronmenta Eng., SNU Compatbty Condton 6 q q 6 6 One Addtona Equbrum Equaton na Compatbty Condton q q 6 6 q 6 6 q q 6, q, q 6 6 In case n members are connected to a jont, and a hnge s used to reease moment at the jont you, have n- compatbty equatons and one equbrum equaton, whch eads to tota of n- compatbty equatons wth n- unknowns. Structura Anayss ab.
131 Dept. of Cv and Envronmenta Eng., SNU Genera Jont Compatbty n Jont Compatbty Condton n n k j j j j for n j j n One Addtona Equbrum Equaton n Structura Anayss ab.
132 Dept. of Cv and Envronmenta Eng., SNU 8 Ths page s ntentonay eft bank. Structura Anayss ab.
133 Dept. of Cv and Envronmenta Eng., SNU 9 Chapter 7 Infuence nes for Determnate Structures Structura Anayss ab.
134 Dept. of Cv and Envronmenta Eng., SNU 7. Infuence uncton Infuence functon p ξ Ip,ξ Convouton ntegra Superposton ξ qξdξ q dr I, ξ q ξ dξ p p Rp Drac deta functons R I, ξ q ξ dξ p p ε m ε ξ ε ξ ξ ξ d m ε ξε d ξε ξε ε d ξε d m ε ε ε f ξ d m ε m ε ξε ε f d ξε ξε ξε ξε f ε d ξε f d ξ ε ξ ε f d m ξ ε ε f ξ Structura Anayss ab.
135 Dept. of Cv and Envronmenta Eng., SNU Concentrated oads of ntensty at ξ ξ Responses by severa concentrated oads ξn ξ n Ip,ξ R p I n p I, ξ q ξ dξ, ξ p I, ξ n p 7. Infuence ne for Smpe Beams 7.. oment ξ ξ dξ n I p, ξ ξ ξ dξ ξ RA - ξ/ RBξ/ ξ ξ RA - ξ/ ξ ξ ξ Structura Anayss ab.
136 Dept. of Cv and Envronmenta Eng., SNU ξ RA - ξ/ ξ ξ Infuence ne / 7... Shear orce ξ RA - ξ/ RBξ/ ξ ξ V RA - ξ/ ξ V V ξ Structura Anayss ab.
137 Dept. of Cv and Envronmenta Eng., SNU ξ V RA - ξ/ ξ V V ξ Infuence ne / 7.. amum oment n a Smpe Beam / ξ / ξ ξ I / I / ξ ξ p ma p ξ ξ ξ ξ 5 ξ 7 p ma p ξ Structura Anayss ab. at
138 Dept. of Cv and Envronmenta Eng., SNU ξ ξ 9 7 p ξ p at ξ ma 5 9 ξ ξ ξ ξ ma p p ξ at 5 7. Infuence ne of a Gerber Beam / 7... Shear orce at / ξ ξ V RA ξ V RA 5 ξ. ξ ξ ξ ξ Infuence ne Structura Anayss ab.
139 Dept. of Cv and Envronmenta Eng., SNU 7... oment at the fed end 5 ξ ξ ξ ξ. 5 ξ ξ ξ ξ Infuence ne - ξ ξ ξ 7.. amum oment n the Gerber Beam / - ξ / ξ qi d q ξ / ξ I d ξ q 7 q ξ ξ ξ ma q. 88q 8 Structura Anayss ab.
140 Dept. of Cv and Envronmenta Eng., SNU ξ ξ ξ ξ 6 ξ 5 ξ q q 5 ξ ξ ξ ξ q 7 q ξ ξ ξ ξ 8 8 5ξ ξ q 5 5 6ξ ξ 6 ma q 8.9q 5 5 q q q ξ. 5 q q ξ ξ 8 ξ ξ ξ 6 ma q. 875 q or.5 ξ or.5 ξ. 5 The mamum moment shoud be smaer than any of the above cases. Therefore, the mamum moment s. at ξ ma 9q 5 6 Structura Anayss ab.
141 Dept. of Cv and Envronmenta Eng., SNU 7 7. Indrect oad / Ra Equvaent to Rb -/ / Ra Rb v R R a a b Rb R b, R a R b In case the unt oad s apped drecty on the smpe beam, -/ / Ra Rb v Ra Rb a Rb Structura Anayss ab.
142 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 8 R b, R R b a Statcay equvaent to the ndrect oad ξ ξ ξ ξ ξ ξ 7.. Infuence ne at the md-span 7.. Truss Case Ra Rb -/ / ξ Ra Rb -/ /
143 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. 9 Ra a 6 Rb a Infuence ne of Truss 7.5. Dagona ember a a a 6 6 a a a a a a a a a a a a a a
144 Dept. of Cv and Envronmenta Eng., SNU Structura Anayss ab. Infuence ne for the dagona member a a a a a 6 6 a a a Bottom ember Take moment about pont A cockewse. a a a a a a a a a a a a a R a 6 A 6 5
145 Dept. of Cv and Envronmenta Eng., SNU Chapter 8 Infuence nes for Indetermnate Beams Structura Anayss ab.
146 Dept. of Cv and Envronmenta Eng., SNU 8. Infuence nes at Supports 8... Reacton orce ξ By the ebty ethod ξ Rb?? dbξ dξ dbb Compatbty Condton R d d R b bb bξ Bett-awe s Recproca Theorem b d d bξ bb db ξ d b d d d d d ξ ξb ξ d bξ Infuence ne : R b d d bξ bb d d ξb bb oment Dagram / / / / d bb 6 8 Structura Anayss ab.
147 Dept. of Cv and Envronmenta Eng., SNU Cacuaton of Defecton w w a b Boundary condtons w b Defecton of the Beam w a a Infuence ne R b d d ξb bb d b w / 6 [ ] 8... oment ξ By the force method ξ θbξ dξ, θξ db θbb Structura Anayss ab.
148 Dept. of Cv and Envronmenta Eng., SNU Compatbty Condton b θ θ Bett-awe s Recproca Theorem bb bξ b θ θ bξ bb ξ d b d θ ξ d d ξb θ ξ θ bξ Infuence ne : b θ θ bξ bb d θ ξb bb Cacuaton of Defecton db eft span Boundary condtons w w a 6 w b w a a 6 6 Defecton of the eft span θbb b d b w 6 θ bb countercockwse Anayss of Center and Rght Span db Structura Anayss ab.
149 Dept. of Cv and Envronmenta Eng., SNU Compatbty condton: θ cc, θ cb 6 θcb θ θ θ cb c cc c cc 5 oment Dagram Defecton of Center span 5 5 w c wc a b w b c 5 wc w 7 b c 5 R 7 θ bb w c Cockwse θ bb θ bb θ R bb a a v Defecton of Rght Span w R wr a b 8 na Infuence ne eft span : b w θ bb w b R wr a a 8 6 w b R 5 / 8 5 Structura Anayss ab.
150 Dept. of Cv and Envronmenta Eng., SNU 6 b b Center Span : b w θ C bb / b b Rght Span: b w θ R bb / 8 6. b 5. b Structura Anayss ab.
151 Dept. of Cv and Envronmenta Eng., SNU 8.. Infence nes n embers oment ξ b?? By the ebty ethod ξ dξ, θξ θbb θbξ db Compatbty Condton: b θ θ bb bξ b θ θ bξ bb Bett-awe s Recproca Theorem ξ d b d / θ ξ d d ξb θ ξ θ bξ Infuence ne : b θ θ bξ bb d θ ξb bb Cacuaton of Defecton oment Dagram θ bb 8 Structura Anayss ab.
152 Dept. of Cv and Envronmenta Eng., SNU Suspended span 8 Boundary condtons w S ws a b w b S ws w Defecton of the suspended span O a?? w S a Overhanged span w wo c O Boundary condtons wo ws e a wo c e 6 v Rght span Boundary condtons Defecton v Determnaton of a, c, e w R wr f g w g R wr f w R f 7 θ O θr c c 7 c e e e 6 6 a a Structura Anayss ab. e
153 Dept. of Cv and Envronmenta Eng., SNU v Defecton of the eft span 9 Suspended span: w S a Overhanged span: w O 8 na Infuence ne Suspended span ws 8 b / θ 8 bb Overhanged span Rght span b. 8 6 w O 8 b θbb 6 w R b θbb 8 b 6., b Structura Anayss ab.
154 Dept. of Cv and Envronmenta Eng., SNU 8... Infuence ne of Shear orce usng the Infuence ne of oment 5 ξ Vb?? ξ χ b V b b Vb V b b 5 ξ Overhanged span b b Vb b Vb 8 ξ Rght span b Vb b Vb Structura Anayss ab.
155 Dept. of Cv and Envronmenta Eng., SNU 8... Infuence ne of Shear orce by üer Bresau s rncpe 5 ξ Vb?? Remove Redunduncy and Appy an Unt oad dbb db V b d d b bb ree Body Dgram and oment Dagram / d bb 6 Structura Anayss ab.
156 Dept. of Cv and Envronmenta Eng., SNU Defecton of the Beam Suspended span 5 w S ws a b 6 Boundary condtons w b S θs θ O 8 a?? Defecton of the suspended span w S a 6 Overhanged span w wo c 6 O Boundary condtons θo θs c a 8 wo c e Rght span w R wr f g 6 Boundary condtons Defecton v Determnaton of a, c, e w g R wr f f 6 w R 6 7 θ O θr c c 8 7 c e e e a a 8 6 e Structura Anayss ab.
157 Dept. of Cv and Envronmenta Eng., SNU v Defecton of the eft span 5 Suspended span : w S a Overhanged span : w O 8 8 Rght span : w R 6 na Infuence ne Suspended span : ws V 5 / 5 b d 6 6 bb w Overhanged span : O V 8 b d wr Rght span : Vb d bb bb Structura Anayss ab.
Strain Energy in Linear Elastic Solids
Duke Unverst Department of Cv and Envronmenta Engneerng CEE 41L. Matr Structura Anass Fa, Henr P. Gavn Stran Energ n Lnear Eastc Sods Consder a force, F, apped gradua to a structure. Let D be the resutng
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