1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 6 Ductility and Deflections

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1 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk aahett Intitte o Tehnolog Otline /1.541 ehani and Deign o Conrete Strtre (3--9) Otline 6 Dtilit and Deletion Dtilit o Toghne, deormabilit, energ aborption apait b d kd = > o For beam ailing in lexre the dtilit ratio an be deined a the ratio o the rvatre at ltimate moment to the rvatre at ield. Yield ondition: = E d 1 ( 1 k) β 1 Ultimate ondition: = = a Dtilit = ( 1 k) d = E a β 1 The ratio give a meare o the rvatre dtilit o the ro etion. o Dtilit haraterize the deormation apait o member (trtre) ater ielding, or their abilit to diipate energ. o In general, dtilit i a trtral propert whih i governed b ratre and depend on trtre ize. 1 / 9

Oral Bkoztrk aahett Intitte o Tehnolog Otline 6 541 ehani and Deign o Conrete Strtre (3--9) Otline 6 Dtilit and Deletion Dtilit o Toghne, deormabilit, energ aborption apait b d kd

2 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 o The deined dtilit (ratio) hown above doe not give inormation abot eet o a moderate nmber o le nor abot the hape o the deending branh in moment-rotation rve. o For dobl reinored ro etion, 2 2 k = ( ρ+ ρ ) n + 2 ρ+ = jd ρ d d n 1/2 a = b.85 and the rvatre dtilit ratio ator i given 2.85β1E 2 ρ d = ( ρ+ ρ ) n ( ρ+ ρ ) n + 2 ρ+ n ( ρ ρ ) d 1/2 o Eet o axial load on lexral dtilit Dtilit o nonined olmn etion: b = Given,, d d h σ σ ditribtion ditribtion =.3 2 / 9

the deending branh in moment-rotation rve. o For dobl reinored ro etion, 2 2 k = ( ρ+ ρ ) n + 2 ρ+ = jd ρ d d n 1/2 a = b.

3 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 1. Ultimate nonined Crve 1 n b Crve 2 ield Balaned oint Crve 2 Crve 1.85 bh 2.85 bh b 2 h Conined etion b (Balaned point) Crve 1: Combination o and and h that ae the olmn to reah (ltimate onrete train) withot oninement Crve 2: Combination o and and h at whih the tenion teel irt reahe the ield trength. The dierene between Crve 1 and 2 indiate the amont o inelati bending deormation (energ aborption), whih or one the ielding tart. ( = pre axial trength.) Dtilit i reded b the preene o axial load. 3 / 9

85 bh b 2 h Conined etion b (Balaned point) Crve 1: Combination o and and h that ae the olmn to reah (ltimate onrete train) withot oninement Crve

4 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 Beae o the brittle behavior o nonined olmn it i reommended that the end o the olmn in rame in earthqake region be onined b loel paed tranvere reinorement when, generall, >.4. o Eet o oninement I the ompreion zone i onined b tie, hoop, and piral, dtilit will be improved. dditional oninement de to loading and pport ondition. o Diplaement dtilit ator meare o the dtilit o a trtre ma be deined b the diplaement dtilit ator. µ = where = the lateral deletion at the end o the pot-elati range, and = the lateral deletion when ield i irt reahed. o For olmn member, ator aeting the dtilit, other than thoe related to oninement, are: rate o loading onrete trength bar diameter ontent o longitdinal teel ield trength o the tranvere teel Deletion o Tpe o deletion 1. Immediate deletion (hort-term) 4 / 9

o Eet o oninement I the ompreion zone i onined b tie, hoop, and piral, dtilit will be improved. dditional oninement de to loading and pport ondition.

5 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 Deletion that or at one pon appliation o load. Time-independent Elati-plati 2. Long-term deletion Deletion that or de to time-dependent behavior o material, mainl reep and hrinkage. Creep (nder tained loading) Shrinkage (independent o loading) Short-term deletion o Eetive moment o inertia Conider a beam trtre bjeted to bending deletion, B The deletion angle between and B i B θ = dx where = rvatre De the exitene o rak, the eetive moment o inertia will be allated a 3 3 r r Ie = Ig + 1 I r max max where max = max moment apait o the ro etion, r = moment apait at irt raking, I g = moment o inertia o gro etion, and 5 / 9

Creep (nder tained loading) Shrinkage (independent o loading) Short-term deletion o Eetive moment o inertia Conider a beam trtre bjeted to bending deletion, B The deletion

6 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 I = moment o inertia o raked etion. r It i known that r = I r g where r = maximm tre at raking, = maximm ditane rom N.. to the oter-mot ibre on the ro etion. Long-term deletion o The member when loaded ndergoe an immediate deletion. The additional deletion are aed de to reep and hrinkage. The rate o thee additional deletion dereae b time. o Eet o reep on the lexre behavior o onrete member Conrete reep relt in a hortening o the ompreed part o the onrete ro etion, hene ae additional rvatre and tre reditribtion in the etion. Eetive modl o elatiit onidering reep: For tre le than.5 ( σ < ): = reep Ct E where E = modl o elatiit at the intant o loading. = + C = + ( 1 C ) total t t E E E The eetive modl inlding reep i E e E = = 1 + C total t o Eet o hrinkage on the lexre behavior o onrete member 6 / 9

The rate o thee additional deletion dereae b time.

7 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 Conrete etion reinored mmetriall ae niorm tre ditribtion. In nmmetrial etion a non-niorm tre ditribtion, and hene, a rvatre i relted. h a = atal train de to h retraint o teel Conrete will tend to hrink, bt annot de to the teel retraint, hene onrete ndergoe tenile tre and teel reinorement will be in ompreion. Eetive onrete tre onidering reep and hrinkage: a ( ) = E (tenile tre in onrete) h h e = E (ompreive tre in teel) a h = (eqilibrim) = = a h E ES = E = E where h h, eetive E 1+ Ct E = area o onrete, h = ree hrinkage o onrete a h = atal hrinkage o onrete = area o teel, E = Yong modl o teel, and 7 / 9

h a = atal train de to h retraint o teel Conrete will tend to hrink, bt annot de to the teel retraint, hene onrete ndergoe tenile tre and teel reinorement will be in

8 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 E = Yong modl o onrete. Thi tenile tre,, ma exeed the tenile trength o onrete aing raking. Note that or mmetrial reinorement the rvatre i zero, de to hrinkage. Otherwie, t h b φ h b t = h Simpliied approah in ompting long-term deletion: De to omplexitie, or traditional appliation impliied method are ed. Compter baed anali an be perormed with eetive E modl to predit long-term deletion, taking into aont the load hitor. Control o deletion o Exeive deletion an lead to raking o pported wall and partition, ill-itting door and window, poor roo drainage, mialignment o enitive mahiner and eqipment, or viall oenive ag, and interere with ervie ondition. o The need or deletion ontrol 1. Senor aeptabilit 2. Servieabilit o the trtre 3. Eet on nontrtral element 8 / 9

Otherwie, t h b φ h b t = h Simpliied approah in ompting long-term deletion: De to omplexitie, or traditional appliation impliied method are ed.

9 1.54/1.541 ehani and Deign o Conrete Strtre Spring 24 ro. Oral Bkoztrk Otline 6 4. Eet on trtral element o Control trategie o deletion 1. Limiting pan/thikne ratio Deletion are ontrolled b etting itable pper limit on the pan-depth ratio o member. 2. Limiting ompted deletion Deletion are ontrolled b allating predited deletion and omparing thoe with peii limitation that ma be impoed b ode or b peial reqirement. In an ae, maximm allowable deletion are ontrained b trtral and ntional limitation. o Some permiible deletion b CI Code inimm thikne o beam: Tpe o pport inimm thikne Simpl pported One end ontino Both end ontino Cantilever S: Clear pan l in inhe. l 16 l 18.5 o dditional long-term deletion mltiplier l 21 l 8 ξ λ = ρ where ξ = a time-dependent oeiient haraterizing material propertie, and ρ =. bd Long-term deletion total = λ immediate 9 / 9

Limiting ompted deletion Deletion are ontrolled b allating predited deletion and omparing thoe with peii limitation that ma be impoed b ode or b peial reqirement.

Reinforced Concrete Design RECTANGULAR R/C CONCRETE BEAMS: TENSION STEEL ONLY

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