Bending stress strain of bar exposed to bending moment

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Elastiit and Plastiit Bending stress strain of ar eposed to ending moment Basi priniples and onditions of solution Calulation of ending (diret)

1 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 of ar Department of Strutural ehanis Fault of Civil Engineering, VSB - Tehnial Universit Ostrava

2 Bars under ending The ending moments and shear fores eome in the ar in the ourse of ending. Simple ending a R a l R a l V Plane ending: inner and eternal fores are situated in plane or plane prinipal plains. n plane hold true: N V 0 V, 0 n plane hold true: N V 0, 0 V Basi priniples and onditions of solution / 7

3 Simple ending Laorator test 3 / 7

4 Simple ending Testing of strutures 4 / 7

5 Basi onditions a) deformated ross-setions sta on plane figure and perpendiular to deformated ais (Bernoulli hpothesis) Charater of ondition is deformation-geometrial. ) aial fires are not mutuall in ompression Daniel Bernoulli ( ) 0 a Basi priniples and onditions of solution 5 / 7

6 Relations etween inner fores and stress in ross-setion dn. da N likewise N. A N.. d A A da A. d A Cross-setion Centre of gravit Central line Plaement of inner fores resultant + + N The resultant fore an e replaed the normal fore on the normal and the two shear fores in, V V + Calulation of ending (diret) stress 6 / 7

7 Normal stress in ending d ma. e Distriution of normal stress in ending is linear over the hight of eam and etreme values are in outer fires. Zerro value of is on neutral aes. ma r e - setion modulus for outer fires [m 3 ] - moment of inertia ma e = n C A d d D B E d Neutral aes is the same as the entral line onl at simple loadind the ending moment. Etrem of stress is on outer fires where = e. 7 / 7

8 Normal ( ending) stress at simple ending N da A Simple ending: suma N = 0 8 / 7

9 Etrem of normal stress in ending - smmetrial ross setion e an determine signe of stress aording to ending moment, after deformation in ending there are lear tensile or ompressed 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 - asmmetrial ross setion, e. e, e1. e1, e, e1 ompressed fires tensile fires e upper lower upper lower e 1,e1 e 1,e e Neutral aes in entre of gravit of setion Setion modullus for outer fires [m 3 ] 0 Distane of outer fires from aes of enter of gravit e 1, (or 1, ), ma, upper, ma, lower n farther fires from neutral aes there are with higher stress ( je,min ) 10 / 7

11 Comination of stresses σ N N A n setion stress is alulated superposition and it is possile to gain: R a a R a N V N l + F N = n R ovement of neutral aes 11 / 7

12 Limited validitation of derived relation ma (ompression). a R a (tension) l h R Relation is valid for ase of simple ending, onstant ross-setion and the height of eam h << l (span). Limited validation 1 / 7

13 Limited validation of derived relation. a h R a l R Relation is not valid in arupt hanges of ross-setion. Limited validation 13 / 7

14 Limited validation of derived relation. (ompression) h Relation is not valid in ase of earing walls, where l < 3h. a R a l (tension) R Limited validation 14 / 7

15 15 / 7 Cross setional harateristis Cross setional harateristis 1,,1 Neutral ais in enter of gravit 1, 1 1,.,,. Cross-setion modulus alulation in ase of simple shapes h 1 1,, h h h h h d d 3. 3 d d Cross-setion modulus to outer fires [m 3 ] 0

16 Design and reliailit assessment of ar eposed to ending moments Design of arring struture, Ed, min f d ma Ed d min f Ed d Adjusted design Rd Dimensioning Reliailit assessment of design Limit state of arring apait. f Ed Rd min d Ed Rd 1 f d fk Realiation Design of ar eposed to ending moment Assumption in design: The same strength of material in ase of tension and ompression (steel), no shear stresses influene 16 / 7

17 Vertial, horiontal and unsmmetrial ending a.. Vertial ending Horiontal ending.. and omined stress of ar (unsmmetrial ending) Comined stress of ar 17 / 7

18 Eentri tension and ompression - ating N + + or another epression is when the N fore whose position is plaed against the entrer of gravit on eentrities e and e. Positive N fore on positive eentrities auses moments: N. e Bending stress is the sume of individual stresses: N A.. is possile to modif sustitution: i i A A N e.. e into:. 1 A i i Segments of neutral ais: N e. A i Comined stress of ar N. e Centre of gravit Central line of eam + e (the epres. in rakets =0, intersetion is otained sustituting ero for -oordinate) n i 0 e n Tension + e Neutral ais n n + N i e 18 / 7

Bending stress strain of bar exposed to bending moment

Bending stress strain of bar exposed to bending moment 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