Bending stress strain of bar exposed to bending moment
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1 Elasticit and Plasticit Bending stress strain of ar eposed to ending moment Basic principles and conditions of solution Calculation of ending (direct) stress Design of ar eposed to ending moment Comined stress of ar Department of Structural ecanics Facult of Civil Engineering, VSB - Tecnical Universit Ostrava
2 Bars under ending Te ending moments and sear forces ecome in te ar in te course of ending. Simple ending a a l V R a l R + Plane ending: inner and eternal forces are situated in plane or plane principal plains. n plane old true: N V 0 V, 0 n plane old true: N V 0, 0 V Basic principles and conditions of solution / 7
3 Simple ending Laorator test / 7
4 Simple ending Testing of structures 4 / 7
5 Basic conditions a) deformated cross-sections sta on plane figure and perpendicular to deformated ais (Bernoulli potesis) Caracter of condition is deformation-geometrical. ) aial fires are not mutuall in compression Daniel Bernoulli ( ) 0 a Basic principles and conditions of solution 5 / 7
6 Relations etween inner forces and stress in cross-section dn. da N A likewise N. A (. ) da N. A A (. ) da d Cross-section Centre of gravit Central line Placement of inner forces resultant + τ τ + N V V + Calculation of ending (direct) stress 6 / 7
7 Normal stress in ending dϕ ma. e Distriution of normal stress in ending is linear over te igt of eam and etreme values are in outer fires. Zerro value of is on neutral aes. ma r e - section modulus for outer fires [m ] - moment of inertia ma e n C A D B E Neutral aes is te same as te central line onl at simple loadind te ending moment. d d d Etrem of stress is on outer fires were e. 7 / 7
8 Bending stress at simple ending N A d A Simple ending:suma N 0 Více vi přednáška 8 / 7
9 Etrem of normal stress in ending - smmetrical cross section e can determine te sign of stress according to distriution of ending moment, after deformation in ending tere are clear tensile or compressed 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 ending - asmmetrical cross section, e. e, e1. e1, e, e1 compressed tensile fires fires e upper lower upper lower e 1,e1 e 1,e e Neutral aes in centre of gravit of section Section modullus for outer fires [m ] 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 Comination of stresses N N A n section c stress is calculated superposition and it is possile to gain: R a a R a N V N c - l + F N n R ovement of neutral aes 11 / 7
12 Limited validitation of derived relation ma (compression). a (tension) R a l R Relation is valid for case of simple ending, constant cross-section and te eigt of eam << l (span). Limited validation 1 / 7
13 Limited validation of derived relation. a R a l R Relation is not valid in arupt canges of cross-section. Limited validation 1 / 7
14 Limited validation of derived relation. (compression) Relation is not valid in case of earing walls, were l <. a R a l (tension) R Limited validation 14 / 7
15 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 modulus to outer fires [m ] 0 Cross-section modulus calculation in case of simple sapes d d π. d π. d Cross sectional caracteristics 15 / 7
16 Design and reliailit assessment of ar eposed to ending moments Design of carring structure, Ed, min f d ma Ed d min f Ed d Adjusted design Rd Dimensioning Reliailit assessment of design Limit state of carring capacit. f Ed Rd min d Ed Rd 1 f d fk γ Realiation Design of ar eposed to ending moment Assumption in design: Te same strengt of material in case of tension and compression (steel), no sear stresses influence 16 / 7
17 Vertical, oriontal and unsmmetrical ending a.. Vertical ending Horiontal ending.. common action of te and comined stress of ar (unsmmetrical ending) Comined stress of ar 17 / 7
18 Eccentric tension and compression For eccentric tension and compression, in te cross-section tere is te normal force N and te ending moments and. Anoter epression of te same prolem is te normal force, wose position is placed against te center of gravit on eccentricities e a e. Positive normal force on positive eccentricities causes moments:. N e N. e Normal stress is te sum of stresses from individual internal forces: N +.. A is possile to modif sustitution: into: i N. 1+ A i A A e.. e + i i Segments of neutral ais: (equation 0, te epression in rackets must e equal to ero) and intersection is otained sustituting ero for -coordinate: N e A i Comined stress of ar n Central line of eam + i e e Centre of gravit 0 n Tension + e and similarl wit te ais Neutral ais n n + N i e 18 / 7
19 Te core of section Te neutral ais divides te cross section into te pulling and pusing part. f te neutral ais is outside te cross-section, te entire cross-section is pulled or pused. Te oundar etween tese two cases is te neutral ais toucing te cross section. f we set te neutral ais to te edges of te cross section so tat it does not cross te cross section, te area corresponding to tese neutral aes defines te so-called core of te crosssection: Te core of te section is an area closeness to te centre of gravit, were te resultant of inner forces is placed and stress is wit te same sign in te wole cross-section. t is needed to assign in case of materials wit Te neutral ais is te tangent line to cross-section f < t f c e n N Neutral ais a) Comined stress of ar 19 / 7
20 Te core of section t is needed to assign in case of materials wit f < t f c Solution: Let te neutral ais is te tangent line to cross-section E.g. : i. 1.. A 1 i 1 n n i e i e a) n ) n c) n d) n e e e e e n Neutral ais a) N Comined stress of ar 0 / 7
Bending stress strain of bar exposed to bending moment
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
Bending stress strain of bar exposed to bending moment
Elastiit and Plastiit Bending stress strain of ar eposed to ending moment Basi priniples and onditions of solution Calulation of ending (diret) stress Design of ar eposed to ending moment Comined stress
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