Elastic-plastic stress analysis of prismatic bar under bending
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1 Easticit and Pasticit Eastic-astic stress anasis o prismatic ar under ending Department o Structura ecanics Facut o Civi Engineering, VSB - Tecnica Universit Ostrava
2 Idea eastic-astic materia section - - Hook s aw Pastic state ree increase o deormations σ,c Compression -B B-C Unoading Re-increasing o strain α arctan E B ε ε p ε e Tension ε p astic (permanent) deormation ε e eastic deormation - Easto-astic stress anasis o prismatic ar under ending / 7
3 Eastic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section σ stress eavior σ σ,ma outer ires o crosssection ε state I. σ Ed,ma Ed Bending (direct) stress in outer ires σ <,ma Ed Rd, e γ k Easto-astic stress anasis o prismatic ar under ending 3 / 7
4 Eastic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section σ stress eavior σ astied Rd, e ε e outer ires o crosssection ε state II. Bending (direct) stress in outer ires σ,ma Rd, e γ k Easto-astic stress anasis o prismatic ar under ending 4 / 7
5 Easto-astic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section astied σ stress eavior Rd, e, σ ε e, outer ires o crosssection ε Load increases permanent, Ed > Rd, e, astic ones come state III. into eing: under, under. Cross-section residuum as sti eastic eavior. Bernoui potesis is sti vaid, ε is continuous inear. In point is ε E Easto-astic stress anasis o prismatic ar under ending 5 / 7
6 state IV. Easto-astic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section astied σ stress eavior N d Rd, σ astic joint σ d.( ) + ε d, cross-section areas in astic state. Neutra ais is in te midde o cross-section area. (in case o nonsmetric cross-sections is moved under astiation) Cross-section is astied, so-caed astic joint comes into eing, carring capacit under ending is eausted. ε Easto-astic stress anasis o prismatic ar under ending 6 / 7
7 Easto-astic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section state IV. astied Rd,..d..d +..d. S S S + S Rd,.. S.,,. S σ stress eavior Rd, σ astic moduus o cross-section [m 3 ] astic joint Cross-section is astied, so-caed astic joint comes into eing, carring capacit under ending is eausted. σ ( S S ) ε ε Easto-astic stress anasis o prismatic ar under ending 7 / 7
8 Easto-astic stress anasis Idea easto-astic materia, prismatic ar, rectanguar cross-section Cross-section state IV. astied Speciica: σ stress eavior Rd,. S ,. S..... σ astic joint ε Pastic reserve o rectanguar cross-section,, e ,5 ε 5% Easto-astic stress anasis o prismatic ar under ending 8 / 7
9 Eame Idea easto-astic materia, prismatic ar, rectanguar cross-section ssign o eame: Determine,e, or cross-section wit astied outer quarters 3 Soution:, e,, e +, , e, Resut: (vaid or astied outer quarters o cross-section on!!!) 48, e, astied Rd, e, state [m 3 ] II. III. IV ,6..,96..,5.. Easto-astic stress anasis o prismatic ar under ending 9 / 7
10 Eame. Idea easto-astic materia, prismatic ar, rectanguar cross-section ssign o eame: a Determine maima oad-carring capacit o eam q d [kn/m] or condition o soution: a) maima ending (direct) stress σ q d? R a Input vaues: Soution: 6m 8 Ed, e,ma. qd, e. mm 8mm R 5 3,..,3. e m 6 35Pa γ, 5 k γ k 4,35Pa σ, ma Sd, e, e q. d, e 8.., e qd, e,97kn/m 8., e Easto-astic stress anasis o prismatic ar under ending / 7
11 Eame. Idea easto-astic materia, prismatic ar, rectanguar cross-section ssign o eame: a Determine maima oad-carring capacit o eam q d [kn/m] or condition o soution: ) outer quarters o cross-section are astied q d? R a Input vaues: Soution: 8 Ed, e,. qd, e,. 6m q mm 8mm R , e,..,93. m. 35Pa γ, 5 Sd, e, d, e, σ,ma 8.., e, q,33kn/m, e, 8. d, e,, e, k γ k 4,35Pa Easto-astic stress anasis o prismatic ar under ending / 7
12 Eame.3 Idea easto-astic materia, prismatic ar, rectanguar cross-section ssign o eame: a Determine maima oad-carring capacit o eam q d [kn/m] or condition o soution: c) astic joint comes into eing q d? R a Input vaues: Soution: 8 Ed,. qd,. 6m mm 8mm R 5 3,.. 3,. m 4 Sd, qd,. 8.., q,45kn/m, 8. d,, 35Pa γ, 5 k γ k 4,35Pa Easto-astic stress anasis o prismatic ar under ending / 7
13 Eame - summar Idea easto-astic materia, prismatic ar, rectanguar cross-section a) σ Rd, e ) astied c) Rd, e, Rd, Pastic reserve o rectanguar cross-section q q d, d, e,45,97,5 5% a) ) c) ε state [m 3 ] q d [kn/m] a),3. -5,97 ),93. -5,33 c) 3,. -5,45 Easto-astic stress anasis o prismatic ar under ending 3 / 7
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