Bending stress strain of bar exposed to bending moment

Transcription

1 Elasticit an Plasticit Bening stress strain of ar epose to ening moment Basic principles an conitions of solution Calculation of ening (irect) stress Design of ar epose to ening moment Comine stress of ar Department of Structural ecanics Facult of Civil Engineering, VSB - Tecnical Universit Ostrava

2 Bars uner ening Te ening moments an sear forces ecome in te ar in te course of ening. Simple ening a a l V R a l R + Plane ening: inner an eternal forces are situate in plane or plane principal plains. n plane ol true: N V 0 V, 0 n plane ol true: N V 0, 0 V Basic principles an conitions of solution / 7

3 Simple ening Laorator test 3 / 7

4 Simple ening Testing of structures 4 / 7

5 Basic conitions a) eformate cross-sections sta on plane figure an perpenicular to eformate ais (Bernoulli potesis) Caracter of conition is eformation-geometrical. ) aial fires are not mutuall in compression Daniel Bernoulli ( ) 0 a Basic principles an conitions of solution 5 / 7

6 Relations etween inner forces an stress in cross-section N. A N A likewise N. A (. ) A N. A A (. ) A Cross-section Centre of gravit Central line Placement of inner forces resultant + τ τ + N V V + Calculation of ening (irect) stress 6 / 7

7 Normal stress in ening ϕ ma. e Distriution of normal stress in ening is linear over te igt of eam an etreme values are in outer fires. Zerro value of is on neutral aes. ma r e - section moulus for outer fires [m 3 ] - moment of inertia ma e n C A D B E Neutral aes is te same as te central line onl at simple loain te ening moment. Etrem of stress is on outer fires were e. 7 / 7

8 Normal ( ening) stress at simple ening N A A Simple ening:suma N 0 Více vi přenáška 8 / 7

9 Etrem of normal stress in ening - smmetrical cross section Signe of stress we can etermine accoring ening moment, after eformation in ening tere are clear tensile or compresse fires.,upper inus stress Positive stress,lower ( ) Upper fires: upper, upper, ma, upper, ma, lower! Lower fires:, lower, lower 9 / 7

10 Etrem of normal stress in ening - asmmetrical cross section, e. e, e1. e1, e, e1 compresse tensile fires fires e upper lower upper lower e 1,e1 e 1,e e Neutral aes in centre of gravit of section Section moullus for outer fires [m 3 ] 0 Distance of outer fires from aes of center of gravit e 1, (or c 1, ), ma, upper, ma, lower n farter fires from neutral aes tere are wit iger stress ( je,min ) 10 / 7

11 Design an assessment of ening memers in elastic state Ultimate Limit State Get iger te section R E Design, min, f R carring E ma capacit min Dimensioning in f ening E E assessment R min. f Realisation f fk γ nitial conitions for assessment: At material were tere is te same strengt in tension an compression. 11 / 7

12 Comination of stresses N N A n section c stress is calculate superposition an it is possile to gain: R a a R a N V N c - l + F N n R ovement of neutral aes 1 / 7

13 Limite valiitation of erive relation. a ma (tension) (compression) R a l R Relation is vali for case of simple ening, constant cross-section an te eigt of eam << l (span). Relation is approimate onl if V leas 0 to sear stresses an loss of cross-section plainness. t is sufficientl accurate if l > 5. Limite valiation 13 / 7

14 Limite valiation of erive relation. a R a l R Relation is not vali in arupt canges of cross-section. Limite valiation 14 / 7

15 Limite valiation of erive relation. (compression) Relation is not vali in case of earing walls, were l < 3. a R a l (tension) R Limite valiation 15 / 7

16 Cross sectional caracteristics, c1. c 1, c. c, c1, c,c1,c c c 1, c1 c 1, c c Neutral ais in center of gravit Cross-section moulus to outer fires [m 3 ] 0 Cross-section moulus calculation in case of simple sapes π. 3 π Cross sectional caracteristics 16 / 7

17 Design an reliailit assessment of ar epose to ening moments Design of carring structure, E, min f ma E min f E Ajuste esign R Dimensioning Reliailit assessment of esign Limit state of carring capacit. f E R min E R 1 f fk γ Realiation Design of ar epose to ening moment Assumption in esign: Te same strengt of material in case of tension an compression (steel), no sear stresses influence 17 / 7

18 Vertical, oriontal an unsmmetrical ening a.. Vertical ening Horiontal ening.. common action of te an comine stress of ar (unsmmetrical ening) Comine stress of ar 18 / 7

19 . Eccentric tension an compression Bening moments come into eing N e N. e Neutral ais Direct stress N +.. A is possile to moif: i A i A Centre of gravit Central line of eam + n Tension n + N N e. 1+ A. e. + i i e + Segments of neutral ais: e N e A i n i e 0 n i e Comine stress of ar 19 / 7

20 Te core of section t is neee to assign in case of materials wit f t < fc Te core of te section is area closeness to centre of gravit, were te resultant of inner forces ave to e in action to ave irect stress wit te same sign in all cross-section. Solution: Let te neutral ais is te tangent line to cross-section 3. i i E.g. : A n n i e i e a) n ) n c) n ) n Comine stress of ar e e e e e n 3 Neutral ais a) 3 3 N / 7