Theme 6 Shearing stress in bending

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Elasici and plasici Theme 6 Shearing sress in bending Basic relaionships and condiions o soluions Shearing sress in chosen cross-secions Dimension

1 Elasici and plasici Theme 6 Shearing sress in bending Basic relaionships and condiions o soluions Shearing sress in chosen cross-secions Dimension o members in shear Shear lux and shear cenre Composie beams Deparmen o Srucural Mechanics Facul o Civil Engineering, SB - Technical Universi Osrava

2 Shearing sress in beams In bending here come up usuall ogeher wih bending momens also shear orces, which causes he shearing sress. F + - a b R a R b Plane bending: exernal and inernal orces are in planes x or x he main planes. In he plane x: N M x M 0, M 0 In he plane x : N M x M 0, 0 M Basic relaions and condiions o soluion 2 / 74

3 Basic examples o shearing sress Cracks in bridges aer lood in 2002, Souh Cech, phoo: Pro. Ing. ladimír Tomica, CSc. Basic examples o shearing sress 3 / 74

4 Basic examples o shearing sress Cracks in bridges aer lood in 2002, Souh Cech, phoo: Pro. Ing. ladimír Tomica, CSc. Basic examples o shearing sress 4 / 74

5 Basic examples o shearing sress Basic examples o shearing sress Crack in he suppor o he concree beam phoo: Pro. Ing. Radim Čajka, CSc. 5 / 74

6 Basic examples o shearing sress Deail o he screw connecion Základní vah a předpoklad řešení 6 / 74

7 The ormula o reciproci o he shearing sresses d x x dx S x d x x x x similarl x x Tenor o sresses: ecor o sresses: [ σ ] σ x sm. σ x x σ { σ } { σ σ σ } T Basic relaions and condiions o soluion x x x Jus 6 sresses componens more opic n.8 7 / 74

8 + Basic relaions or derivaion o shearing sress A, S I b ( ) S A T + b( ) B... Shear orce in secion... Cross secional characerisics or shear sress in bending are: - S - I 3 S A [ m ] x saical momen o separae par o cross secion... momen o ineria o he whole cross secion... Widh o cross-secion in he assessing place par T Grasho ormula x I. S. b ( ) Shearing sress in chosen cross-secions 8 / 74

9 Shear sress in a recangular cross secion Cross secion T h Disribuion o o 2 max x S I 1 12 A čási průr b. h 3 T 3 [ m ] S I b ( ) b... Shear orce in secion... max. S b( ) b bh h bh b 12 3 Z 1 / 2 I. b saical momen o separae par o cross secion... momen o ineria o he whole cross secion... Widh o cross-secion in he assessing place 3 2 ( ) 3 2 bh A Z Shearing sress in chosen cross-secions 9 / 74

10 Design and assessmen o recangular cross secion in shear Ed To make bigger secion Rd Design, A, req Assessmen aer Ulimae Limi Sae Ed Rd d Ed Rd 1 A max req Dimensioning d 3. 2 A k γ M d d 3 Realisaion Dimensioning 10 / 74

11 Shearing sress in hin-walled members Thin-walled beam,, w << h Smmeric I b (lange) w h w (web) h w h Open cross secions: I, U, T, C, Z Hollow box-beam secions (pipe): Shearing sress in chosen cross-secions 11 / 74

12 Shear sress in I proile Secion Deail w b Wall Disribuion o o 2 x Deail x h w h x,max Flange Disribuion o o 1 x Condiion o soluions: shearing sress is consan in he verical cu o he ragmen o he wall (see Deail) is parallel o circumerence o he secion Shearing sress in chosen cross-secions 12 / 74

13 Shear sress in he wall o he I-proile w b Disribuion x o 2 Basic ormula:. S I.... Thickness in he assessing ragmen x h w h x,max S... saical momen o a separae parn o he secion x... Shearing sress in he plane perpendicular o he axis x x x verical par horional par S A par T [ ] 3 m Shearing sress in he wall x (quadraic uncion) Shearing sress in chosen cross-secions 13 / 74

14 Shear sress in he lange o I-proile b Basic ormula x I. S. w h w h Shearing sress in he lange (Linear uncion) x T Disribuion o x S A par T [ ] 3 m o 1 Shearing sress in chosen cross-secions 14 / 74

15 Maximum sress in he I-proile b Disribuion o x o 2 w h w h x,max 0 Disribuion o x o 1 2 [ 4..( b )(. h ) h ] max. w + w. 8. I. w Shearing sress in chosen cross-secions 15 / 74

16 Shear cenre A double smmeric cross secions he resuling orce goes hrough he cenre o gravi, a non smmeric secions i is dieren i a plane o he load is no a plane o smmer, han he load mus go hrough he shear cenre A, oherwise he member would be wised (sress rom orsion). We obain resuling shear orces Q in he proile b he inegraion o shearing sresses along he single walls o he open proile. The are equivalen o shear orce. x Q h 0 Q + A T o 45 Q + h 0 x Q Q 2 Shear lux and shear cenre 16 / 74

17 Shear cenre o he U-proile Proiles U, UE, UPE - sie a in ables Q x I. S. h0. 2. I. s x x A T h 0 Q w s Flange: a b 0 Q h 1 S. s. 0.. h0. s 2 2 x Shear lux and shear cenre 17 / 74

18 Shear cenre o U-proile Q M x x A T h 0 Q w s Web x 8. I. w a b 0 Q 2 2 [ 4.. b. h +.( h )]. x Q w 0 0 w likewise I proile 2 Q. h. b0. a 0 Saical momens o poin M. a Q. h0 4. I h 2 0 Shear lux and shear cenre 18 / 74

19 Composie members a b 0 Q * x b. b ( ) x ( ). I. S. b. S I ( ) a b [kn/m] Shear orce or one connecing member a a welds, screws, bols b. S * Qx Qx. a. a I [kn] Composie members 19 / 74

20 Quesions or he exam 1. Shearing sress in bending o he recangular cross secion 2. Shearing sress in bending o hin-walled members, shear cenre 3. Assessmen o he members under shear sress in bending 4. Composie members Quesions or he exam 20 / 74

At the end of this lesson, the students should be able to understand

At the end of this lesson, the students should be able to understand Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress concenraion facor; experimenal and heoreical mehods.