Deformations of statically determinate bar structures

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1 Statics of Buiding Structures I., ERASMUS Deformations of staticay determinate bar structures Department of Structura Mechanics Facuty of Civi Engineering, VŠB-Technica University of Ostrava

2 Outine of Lecture Term deformation Virtua works principe Deformation of bar aia oading Deformation of bar transversa oading Deformation of bar torsiona oading Deformation of indirect bar Deformation of curved bar Deformation of pane truss structure Outine of Lecture / 6

3 Deformation Deformation: a) Goba deformation of structure b) Loca component of deformation in some point (dispacement, rotation) Term deformation 3 / 6

4 Deformation Why to cacuate deformations?. Usabiity of structure. Soution of staticay indeterminate structures 3. Verifying the correctness of the cacuation by measurement Cacuation assumptions: Physica inearity (Hooke's aw appies) Geometric inearity (sma deformations theory) Consequence: Equiibrium conditions are formuated on the deformed structure The First order Theory Appy the principe of superposition and the principe of proportionaity Term deformation 4 / 6

5 Deformation Noninear mechanics: nd order theory equiibrium conditions formuated on deformed structure (sma deformations) Physica noninearity (nonineary eastic or permanent deformations) Theory of big deformations Structures with uniatera inks Cabe structures Term deformation 5 / 6

6 Work of eterna forces and moments Work (eterna) of a force at point: L e P P cos c Work - scaar, units are Joues (J = N.m), kj, MJ Work of a moment at point: L e M. Notice: It is assumption that () has other cause than P (M). The work is positive when there are same directions of: vectors of force and dispacement, moment and rotation. Work of point force and point moment Virtua works principe 6 / 6

7 Work of continuous force and moment oading Work of eterna forces and moments: b a L q( ) w( ) d L m( ) ( ) e e b a Assumption magnitude of oading is constant during movement. d Work of continuous oading Virtua works principe 7 / 6

8 Virtua work ) Rea oading state a) Deformationa virtua work ) Virtua oading state b) Force virtua work a) Deformationa virtua state L L e e P w c P w c b) Force virtua state rea oading state virtua defection curve rea defection curve force virtua oading state Deformationa virtua work described by Lagrange to study equiibrium of structures Virtua works principe 8 / 6

9 Work of interna forces Loaded bar in a space: N, M y, M z, V z, V y, T Coordinate system of the bar Virtua works principe 9 / 6

10 / 6 Work of interna forces Work of interna forces of bar y z z z y y i T v V w V M M u N L d d ˆ d ˆ d d d Positive directions of interna forces Work of interna forces: Interna forces restrain deformations, they have opposite direction compared to picture beow, that is reason for negative sign in cacuation of L i. Virtua works principe

11 Virtua works principe Aiom: Tota virtua work on soved structure (i.e. sum of works of eterna and interna forces) is equa to zero. L e L i A) Deformationa principe of virtua works (principe of virtua dispacements) B) Force principe of virtua works (principe of virtua forces) Virtua interna forces Rea interna forces, causes deformations N du d EA d y M EI y y d N, M y, M z, Vz, Vy, T d z M EI z z d wˆ V z d * GAz d vˆ V y d * GAy d d T GI t d Virtua works principe / 6

12 L e Deformationa oading caused by temperature Force principa of virtua works NN M ym EA EI y y ez t th ( td th) h du t d t M z EI M z t z d V zv VyV z y TT t t d N tt M y t M z t d * * GAz GAy GIt h b t y d t t t h d h Uniform therma oading and decomposition of ineary changing therma oading across cross-section Virtua works principe / 6

Betti's theorem (87) M y,im y,ii P P d EI M y,iim y,i P3 3 M 4 4 d EI P P P3 3 M 4 4 y y st oading state Enrico Betti (83-89) The work done by the st oading state through the

13 Betti's theorem (87) M y,im y,ii P P d EI M y,iim y,i P3 3 M 4 4 d EI P P P3 3 M 4 4 y y st oading state Enrico Betti (83-89) The work done by the st oading state through the dispacements produced by the nd oading state is equa to the work done by the nd oading state the dispacements produced by the nd oading state. nd oading state Virtua works principe 3 / 6

14 I I Mawe s theorem A specia case of Betti's theorem. In each oading state acts ony one force P or moment M. P P P P P II II I II I II James Cerk Mawe (83-879) Dispacement done by the first force in the pace and direction of second force is equa to dispacement done by second force in the pace and direction of the first force. st state Zváštní případ Bettiho věty, kdy v každém z obou zatěžovacích stavů působí na konstrukci ediná sía P nebo ediný moment M. nd state Virtua works principe 4 / 6

15 Unit force method NN M ym y M zm z VzV VyV z y TT Le. * EA EI y EI z GAz GAy GI t * d Force oading t t Ntt M yt M zt d h b Therma oading Unit force method Virtua works principe 5 / 6

16 Deformation of bar aia oading Force oading u e E NN A d Therma oading Constant cross-section Variabe cross-section Simpson s rue u f e EA A NNd EA ( ) d f 4( f f3 ) f f4 3 N d ue tt Nd tt A N Deformation of bar eposed to aia oading Deformation of bar aia oading 6 / 6

17 Eampe. Cacuate horizonta dispacement u c for force and therma oading state A = 64 mm, E =,. 8 kpa, t =,. -5 K - Force oading state: R R a a 8,4.,5 8 3kN N 3 8,4. u c,. NN AN d EA EA 9, 8.6,4. 5,685 m Probem definition and soution of eampe. Deformation of bar aia oading 7 / 6

18 Eampe. Dispacement caused by therma oading: u u c c N t,. t 5 d t t Nd t.( ).,48 m,48mm t A N Probem definition and soution of eampe. Deformation of bar aia oading 8 / 6

19 Eampe. Cacuate vertica dispacement w b of top of coumn for oading by sef-weight Concrete r = 4 kg.m -3 E =. 7 kpa Probem definition and soution of eampe. Deformation of bar aia oading 9 / 6

20 Eampe. z A ( z).(,8,8. ),8,. z Nm 3 4kNm 3 n( z) A (,8,. z).4 9, 4,8. z z N( z) (9, 9, 4,8. z). 9,. z,4. z w b E 4 in i NN dz EA E 9,. z i,4. z,8,. z i 4 N dz A i z i Z E Probem definition and soution of eampe. Deformation of bar aia oading / 6

21 Eampe. Soution using: ) Simpson s rue 4 i z A N N/A m m kn knm -,8,. -,6 -,.6, -48-4, 3 3,4-79, -56, ,6-5, -7, d 4 f ( )d ( f 4( f f3) f f4) d N A dz ( 4.(,6 56,57).4 7) 3 54,895kNm w b 54, ,745. m. 7,7745 mm Deformation of bar aia oading / 6

22 Eampe. Soution using: ) Rectanguar method (numerica integration) n z w b z i Z E 4,4 n 54,9756 7, m i z i N i /A i m knm -,,87486,6 5, , 8,64 4,4, ,8 4,5986 6, 7,379 7,6, ,, ,4 5,46 3,8 7,59385 S N i /A i 54,9756 Deformation of bar aia oading / 6

23 Aiay oaded bar coumn Structure with aiay oaded bar Graded cross-section of coumn on ta buiding, Chicago, USA 3 / 6

24 Deformation of bar transversa oading Force oading MM VV d E I G * A d Constant cross-section MMd VVd * EI GA Therma oading t t M d h Types of transversay oaded direct beams Deformation of direct beam transversa oading 4 / 6

25 Vereshchagin s rue Aid for computation of the integra M Md A. M M T Deformation of direct beam transversa oading 5 / 6

26 Vereshchagin s rue Paraboic parts of moment diagrams for use within Vereshchagin s rue Deformation of direct beam transversa oading 6 / 6

27 Eampe.3 Cacuate vertica defection = w a. Use Vereshchagin s rue. Reinforced concrete cantiever E =,. 7 kpa Negect the work of shear forces. Probem definition and soution of eampe.3 Deformation of direct beam transversa oading 7 / 6

28 Eampe.3 3 bh EI E,. 3 7,446 knm w 3 d EI,5 5, ,446. S M 3,333.(,75),5kNm S M.(,5) 5kNm S a MM EI MMd A MMd A MMd A M 3 3.(,667),667 knm 7,8.,8. ( S ,547 m M M M S S ) Probem definition and soution of eampe.3 Deformation of direct beam transversa oading 8 / 6

Eampe.4 The same probem as in Eampe.3 but the work of shear forces is taken into account w A c * GA S S w w c c w * 6 9,4..,4 3,888. V,. V, 6 3,888. 5 * ( S bh,8.,8,4m, A A VV d * GA GA. V.( ) knm V 6.

29 Eampe.4 The same probem as in Eampe.3 but the work of shear forces is taken into account w A c * GA S S w w c c w * 6 9,4..,4 3,888. V,. V, 6 3, * ( S bh,8.,8,4m, A A VV d * GA GA. V.( ) knm V 6..( ) 6kNm S 9,76.,93.,7 5,47 ) 5 5 kn m,93mm Reinforced concrete cantiever G = 9,4. 6 kpa Rea and virtua shear forces in eampe.4 Deformation of direct beam transversa oading 9 / 6

30 Tabe. Equations for cacuation of integra MMd Deformation of direct beam transversa oading 3 / 6

31 Eampe.5 Cacuate vertica dispacement w c and rotation a Wood E = 7 kpa Probem definition and soution of eampe.5 Deformation of direct beam transversa oading 3 / 6

32 Tabe.3 Loca deformations of a cantiever beam and simpy supported beam Deformation of direct beam transversa oading 3 / 6

33 Eampe.6 Therma oading the change of temperature is inear aong height of the cross-section Cacuate defections w c a w s. Stee t =,. -5 K - h =,4 m t h t tt Md h A M Probem definition and soution of eampe.6 Deformation of direct beam transversa oading 33 / 6

34 Eampe ttm tt tt wc d Md A Mc h 9. 9 h t h t ws A Ms h AMs A Mc w c 7.,75 5,..6.( 9),7 m 7,mm,4 6,5 w s 5,..6.6,5,49 m,4 4,9mm Probem definition and soution of eampe.6 Deformation of direct beam transversa oading 34 / 6

35 Eampe.6 Shape, oading see Eampe.3 Variabe cross-section Reinforced concrete cantiever E =,. 7 kpa Probem definition and soution of eampe.7 Deformation of direct beam transversa oading 35 / 6

36 Research energetic center, VŠB-TU Ostrava Cantiever beam: Stee weded and roed I- profie Trapeze meta pate Concrete foor Eampe of the structure with variabe cross-section 36 / 6

37 Research energetic center, VŠB-TU Ostrava Cantiever beam: Stee weded and roed I- profie Trapeze meta pate Concrete foor Eampe of the structure with variabe cross-section 37 / 6

38 Deformation of bar torsiona oading Force virtua state Torsiona rotation c TT GI t d GI t TTd A GI T t Deformation of direct beam torsiona oading 38 / 6

Eampe.8 Determine torsiona rotation of the right end b. Use unit force method. Stee - G = 8,. 7 kpa 4 4 It I p ( r r ) 4 4 5 4.(3 4 ) 7,59.

39 Eampe.8 Determine torsiona rotation of the right end b. Use unit force method. Stee - G = 8,. 7 kpa 4 4 It I p ( r r ) (3 4 ) 7,59. mm 7 7 GI 8,..7,59. 6,8466 knm t TT GI t d GI t AT TTd GI AT.(,7,5)..,7.,6,336 knm,336 o b,384 rad, 6,8466 t Probem definition and soution of eampe.7 Deformation of direct beam torsiona oading 39 / 6

40 Deformation of poyine beam pane oading m NN d E A E I G MM d VV A * d At most of staticay determinate cases the work of shear and norma forces is negected m E Constant MM I d cross-section m E I MMd 3 oca components of deformation: u, v a At the point c c w u c c tan u w c c Therma oading t m M t, Nd t, d h Deformation of poyine beam pane oading 4 / 6

41 Cacuate u d, w d,, d Eampe.9 Stee I = m 4 I = 3,8. -5 m 4 I 3 = 9,. -5 m 4 E =,. 8 kpa Probem definition and soution of eampe.9 Deformation of poyine beam pane oading 4 / 6

42 Stee frame structure of industria ha Span,5 m Eampes of structures of poyine beams 4 / 6

43 Industria ha, Vítkovice Foor pan dimensions 3 3 m Cranes capacity 8 a t Undermined region Eampes of structures of poyine beams 43 / 6

44 Mutipurpose ha, Frýdek - Místek Eampes of structures of poyine beams 44 / 6

45 Mutipurpose ha, Frýdek - Místek Rámová oceová konstrukce Eampes of structures of poyine beams 45 / 6

46 Patform of footba stadium, Ostrava Bazay Undermined region Eampes of structures of poyine beams 46 / 6

47 Patform of footba stadium, Ostrava Bazay Detai of moment hinge Eampes of structures of poyine beams 47 / 6

48 Deformation of curved beam pane oading Span, defection f, reative defection F Φ f Shape and supporting of a pane curved beam Deformation of curved beam pane oading 48 / 6

49 Deformation of curved beam pane oading Span, defection f, reative defection F Φ f Defection and reative defection on different curved beams. Deformation of curved beam pane oading 49 / 6

50 Deformation of curved beam pane oading Use of Unit force method Force oading Therma oading L NN EA MM VV d s ds EI * GA L L ds t t L Nds tt L M h ds Soution ds d cos After modification: Force oading E b a NN d Acos E b a MM d I cos G b a VV d * A cos Therma oading t t b a N d tt cos b a M d hcos Deformation of curved beam pane oading 5 / 6

51 5 / 6 Deformation of curved beam pane oading Cacuation of deformation Numerica integration Simpson s rue Rectanguar method 3 ) )... ( )... 4( ( )d ( 4 3 d f f f f f f f f f n n n n i i i i i n i i i i i n i i i i i i n i i i i i i s I M M E s A N N E I M M E A N N E I MM E A NN E b a b a cos cos d cos d cos Deformation of curved beam pane oading

52 Eampe. Cacuate u b Paraboic centra ine z ( ) k. k z a a z b b tg dz d k.. k. cos sin tg tg tg EI = 6,7. 4 knm Probem definition and soution of eampe. Deformation of curved beam pane oading 5 / 6

53 Eampe. i [m] tg ψ cos ψ M [knm] [m] M/ cos ψ [knm ] - 5, - 3,75 -,,5 3 -,5 -,8 -,6 -,4 -,,788 7,8574 9,984 8, ,,, 5,5,, ,5,4, ,75,6, ,,8,788 7,,, 8,4375,875 9,8 47,5,5 76,739 57,875,875 9,35 57,5, 5, 43,5,875 8,46 8,75,5 46,447 4,375,875 4,668,,, Probem definition and soution of eampe. Deformation of curved beam pane oading 53 / 6

54 Curved beam Gateway Arch, span of the stee arch from the year 966 is 9,5 m, Saint Louis, Missouri. Eampes of panary oaded curved structures 54 / 6

55 Curved beam Gateway Arch, rozpětí a vzepětí oceového obouku z roku 966 9,5 m, Saint Louis, Missouri. Eampes of panary oaded curved structures 55 / 6

56 Curved beam Rovinně zakřivený vazník, Výzkumné energetické centrum VŠB-TU Ostrava Eampes of panary oaded curved structures 56 / 6

57 57 / 6 Deformation of pane truss structure Deformation of pane truss p p p A N N E A N N E EA N N. d d Therma oading Virtua work of Norma forces ony p t p t p t t N t N t N,,, d d

58 Eampe. Cacuate w c A = m 4 A =. -4 m 4 A 3 = m 4 A 4 = m 4 A 5 =. -4 m 4 A 6 =. -4 m 4 A 7 = m 4 = 3 = 6 =,36 m Cacuation in tabe Probem definition and soution of eampe. Deformation of pane truss 58 / 6

59 Eampe. Cacuation in tabe A [m ] [m] N [kn] Ñ [] (N Ñ /A ). -3 [kn/m],4, -9, -, 75,,,36 34,64,36 559,7 3,8,36-67,8,, 4,8, -6, -, 33,333 5,,,,, 6,,36 67,8,36 79,58 7,8, -6, -, 33,333 8,9 w c 7 3 N N 8,9 3 5,6 m 8 E A, 5,6mm Deformation of pane truss 59 / 6

60 Raiway bridge, Poanka s conunction Eampes of pane truss structures from / 6

61 Raiway bridge, Poanka s conunction Eampes of pane truss structures 6 / 6

62 Road bridge, Ostrava - Hrabová Truss bridge over Ostravice river Ukázky koubových příhradových konstrukcí 6 / 6

Work and energy method. Exercise 1 : Beam with a couple. Exercise 1 : Non-linear loaddisplacement. Exercise 2 : Horizontally loaded frame

Work and energy method. Exercise 1 : Beam with a couple. Exercise 1 : Non-linear loaddisplacement. Exercise 2 : Horizontally loaded frame Work and energy method EI EI T x-axis Exercise 1 : Beam with a coupe Determine the rotation at the right support of the construction dispayed on the right, caused by the coupe T using Castigiano s nd theorem.