Initial Stiffness of Reinforced Concrete Columns and Walls

Transcription

1 Initial tiness o Reinorced Concrete Columns and Walls Bing Li cool o Civil and Environmental Engineering, Nanang Tecnological Universit, ingapore UMMARY: Te estimation o te initial stiness o columns and alls subjected to seismic loadings as long been a matter o considerable uncertaint. Tis paper reports a stud tat is devoted to addressing tis uncertaint b developing a rational metod to determine te initial stiness o RC columns and alls en subjected to seismic loads. A compreensive parametric stud based on a proposed metod is initiall carried out to investigate te inluences o several critical parameters. To simple equations are ten proposed to estimate te initial stiness o RC columns and alls. Te applicabilit and accurac o te proposed metod and equation are ten veriied it te experimental data obtained rom literature studies. eords: Reinorced Concrete; Column; Wall; Initial tiness. INTRODUCTION Te past 5 ears ave seen major developments in seismic design provisions, it a paradigm sit rom a orce-based approac to a displacement-based one and an increasing in ocus on te deormation caracteristics o structures. tiness properties o reinorced concrete (RC) column and all structures can aect te estimation o te undamental period, displacements and te distributions o internal orce response. Te initial stiness depends on te intensit and distribution o stress on alls and columns cross-section, as ell as te extent o lexural and sear cracks. Flexural cracking causes reduction in te net cross sectional area and moment o inertia, ence a reduction in initial lexural rigidit o te all and column section. Tis leads to te increasing diicult in making accurate predictions o te initial stiness o RC members. Tus, stiness reduction actor is emploed in te analsis o RC members under lateral loads. In practice, te value o.35 and.7 te gross moment o inertia or cracked and un-cracked members, respectivel is idel emploed. Hoever, tis simpliication ma not be appropriate in man cases as te recommended moment o inertia or alls and columns is independent o te reinorcement content and axial load level.. REIEW OF EXITING INITIAL TIFFNE MODEL Tere are to metods as illustrated in Fig. tat are commonl utilized to determine te initial stiness o RC columns ( i ). In te irst metod, te initial stiness o RC columns are estimated b using te secant o te sear orce versus lateral displacement relationsip passing troug te point at ic te applied orce reaces 75% o te lexural strengt (Point A in Fig.). In te second metod, te column is loaded until eiter te irst ield occurs in te longitudinal reinorcement or te maximum compressive strain o concrete reaces. at a critical section o te column (point A in Fig.). Generall, te to approaces give similar values. In tis stud, te later approac as adopted. Assuming te column is ixed against rotation at bot ends and as a linear variation in curvature over te eigt o te column, te measured initial moment o inertia can be determined as:

2 I 3 L i e (.) Ec Te stiness ratio (κ ) is deined as ollos: I κ I e g % (.) ere I g is te moment o inertia o te gross section; i is te initial stiness o columns and L is te eigt o columns and E c is te elastic modulus o concrete. ACI 38-8 (8) recommends te olloing options or estimating member stiness or te determination o lateral delection o building sstems subjected to actored lateral loads: (a).35ei g or members it an axial load ratio o less tan. and.7 EI g or members it an axial load ratio o more tan or equal to.; or (b).5 EI g or all members. FEMA 356 () suggests te variation o initial stiness values it te applied axial load ratio. Te initial stiness is taken as.5 EI g or members it an axial load ratio o less tan.3, ile a value o.7 EI g is adopted or members it an axial load ratio o more tan.5. Tis value varies linearl or intermediate axial load ratios as illustrated in Fig.. As son in Fig., ACE 4 (7) recommends tat te initial stiness is taken as.3 EI g or members it an axial load ratio o less tan., as.7ei g or members it an axial load ratio o more tan.5 and varies linearl or intermediate axial load ratios.according to Paula and Priestle s recommendation (99), te initial stiness is taken as.4ei g or members it an axial load ratio o less tan -.5, as.8ei g or members it an axial load ratio o more tan.5 and varies linearl or intermediate axial load ratios as illustrated in Fig.. tiness Ratio k (%) Axial Load Ratio ca g ACI38-.8(a) ACI38-.8(b) FEMA 356 ACE 4 PP9 EE9 Figure. Metods to Determine Initial tiness Figure. Relationsips beteen tiness Ratio and Axial Load Ratio o Existing Models 3. INITIAL TIFFNE OF RC COLUMN 3.. Proposed Metod Yield Force ( ) Te initial stiness o columns is determined b appling te second metod as described in te previous section. Te ield orce ( ) corresponding to point A in Fig. is obtained rom te ield moment (M ) en te reinorcing bar closest to te tension edge o columns as reaced its ield strain. Moment-curvature analsis is adopted to determine tis moment.

3 Displacement at ield orce ( ) Te displacement o a column at ield orce ( ) can be considered as te sum o te displacement due to lexure, bar slip and sear. + (3.) lex sear ere is te displacement o a column at ield orce; bar slip at ield orce; and lex sear Flexure Deormations ( ) lex is te displacement due to sear at ield orce is te displacement due to lexure and In tis proposed metod, te simpliied concept o an initial lengt o te member suggested b Priestle et al. (996) as used to account or te displacement due to bar slip in lexure deormations. Assuming a linear variation in curvature over te eigt o te column, te contribution o lexural deormations and bar slips to te displacement at te ield orce or RC columns it a ixed condition at bot ends can be estimated as ollos: lex φ ( L + L ) 6 sp ere φ is te curvature at te ield orce determined b using moment-curvature analsis and L is te clear eigt o columns. Te strain penetration lengt ( L ) is given b: sp (3.) L. d (3.3) sp l b ere is te ield strengt o longitudinal reinorcing bars; and d b is te diameter o longitudinal l reinorcing bars. ear Deormations ( ) sear Park and Paula (975) derived a metod to determine te sear stiness b appling te truss analog or sort or deep rectangular beams o unit lengt. Te sear stiness is te magnitude o te sear orce, en applied to a beam o unit lengt tat ill cause unit sear displacement at one end o te beam relative to te oter. Tis model is reliable in estimating sear deormations o sort or deep beams in ic te inluences o lexure are negligible. Te beaviors o RC columns under seismic loading are muc more complex because o te interaction beteen sear and lexure. Te inluences o axial strain due to lexure in estimating sear deormations o RC columns sould be considered to accuratel predict te initial stiness o RC columns. B appling a metod tat is similar to Park and Paula s analogous truss model (975), te sear stiness o RC columns is derived in tis part o te paper. Te eects o lexure in sear deormations are incorporated in te proposed model troug te axial strains at te center o columns ( ε )., CL Assuming tat transverse reinorcing bars start resisting te applied sear orce en te sear cracking starts occurring, te stress in transverse reinorcing bars at te ield orce is calculated as: s ( ) cr s (3.4) A d tanθ st

4 ere d is te distance rom te extreme compression iber to centroid o tension reinorcement; s is te spacing o transverse reinorcement; Ast is te total transverse steel area itin spacing s ; and θ is te angle o diagonal compression strut. Hence te strain in transverse reinorcing bars is: s ε x ε t (3.5) E s ere ε t is te ield strain o transverse reinorcing bars; E s is te elastic modulus o steel. imilar to Park and Paula s model (975), te concrete compression stress at te ield orce is given as: (3.6) bl cosθ cs ere b is te idt o columns; L cs d sinθ is te initial dept o te diagonal strut as son in Fig. 3a. Hence te strain in te concrete compression strut is given as: ere E c ε (3.7) Ec is te elastic modulus o concrete given as: Ec 5 c (3.8) Based on eccio and Collins s model (986), te initial compressive strengt o concrete is calculated as ollos: ce c.8 + 7ε c B appling Mor s circle transormation or te mean strains at te center o ection C-C as son in Fig. 3b, it gives: ε x + ε, CL ε x ε, CL γ x ε + + (3.) ε x + ε, CL ε x ε, CL γ x ε + (3.) γ x tan θ (3.) ε ε x, CL For te axial mean strains, compatibilit requires tat te plain sections remain plane. Hence te mean strain at te center o section C-C is given as: ε, CL ε, top + ε, bot (3.3) (3.9)

5 ere ε, top, ε, bot are te axial strains at te extreme tension and compression ibers, respectivel as son in Fig. 3(d). Tere are six variables, namel ε x, ε, CL, γ x, ε, ε and θ ; and six independent equations (3.5), (3.7), (3.), (3.), (3.) and (3.3). B solving tese six independent equations, te sear strain γ ) at te center o section C-C could be determined. ( x Te column is divided into several segments along its eigt o te column to determine te total sear deormation at te top o te column. Te mean axial strain at te center o te section is determined based on te moment-curvature analsis. Te sear strains at te loer and upper section o te segment are calculated using te above equations. Hence, te total sear displacement caused b te ield orce can be calculated as ollos: sear n i i γ + x γ i+ x i (3.4) i i+ ere γ x and γ x are te sear strains at te loer and upper section o te segment i ; i is te eigt o segment i and n is te number o segments. Initial tiness Once te lexural and sear deormations at te top o columns under ield orce are obtained, te initial stiness o columns can be determined as: i (3.5) + lex sear (a) (b) (c) (d) Figure 3. Inluences o Flexure in Estimating ear Deormations 3.. alidation o te Proposed Metod Te proposed metod is validated b comparing its results to te initial stiness o six columns obtained rom te experimental stud previousl conducted b Tran and Li. (). It as ound tat te average ratio o experimental to predicted initial stiness b te proposed metod as.735 as tabulated in Table. It sos a relativel good correlation beteen te analtical and experimental results. Te mean ratio o te experimental to predict initial stiness and its coeicient o variation ere.4 and.6,.3 and.76,.6 and.54,.3 and.84,.3 and.46, and.588 and.4 or ACI 38-8(a) (8), ACI 38-8(b) (8), FEMA 356 (), ACE 4 (7), Paula and Priestle (99) respectivel. Comparison o available models it experimental data indicated tat te proposed metod produced a better mean ratio o te experimental to predicted

6 initial stiness tan oter models. Te proposed metod ma be suitable as an assessment tool to calculate te initial stiness o RC columns. Table. Experimental eriication o te Proposed Metod pecimen i exp i exp i exp i exp i exp i exp i exp i exp (kn/mm) i p i ACI ( a) i ACI ( b) i FEMA i ACE i PP i EE C C C C C C Mean Coeicient o Parametric tud or Initial tiness o Columns A parametric stud conducted to improve te understanding o te eects o various parameters on te initial stiness o RC columns is presented itin tis section. Te parameters investigated are concrete compressive strengt ( c ), aspect ratio (a/d) and axial load ratio (P/ c A g ). In te parametric stud, te eects o te parameters tat ere investigated on te initial stiness o RC columns are presented b te dimensionless stiness ratio (k). pecimen C-.4-. it an aspect ratio o.4 (Tran & Li ) is considered as te reerence specimen in te parametric stud. An axial load o. c A g as applied to te specimen. Te concrete compressive strengt o te specimen ( c ) at 8 das as 5. MPa. Te longitudinal reinorcement consisted o 8-T ( mm diameter). Tis resulted in te ratio o longitudinal steel area to te gross area o column to be.5%. Te transverse reinorcement consisted o R6 bars (6 mm diameter) it 35 bent spaced at 5 mm, corresponding to a transverse reinorcement ratio o.9%. Inluence o Concrete Compressive trengt Fig. 4a illustrates te inluence o concrete compressive strengt on stiness ratios or to dierent axial loads o.5 c A g and. c A g. Te concrete compressive strengts investigated ere 5MPa, 35MPa, 45MPa, and 55MPa. For bot axial loads, it an increase in concrete compressive strengt, no signiicant canges on stiness ratios ere observed. Inluence o Longitudinal Reinorcement Ratio Te inluence o longitudinal reinorcement ratios on stiness ratios is presented in Fig. 4b or to dierent column axial loads o.5 c A g and. c A g. Four tpes o longitudinal reinorcement, 8T6, 8T, 8T and 8T5 corresponding to longitudinal reinorcement ratios ρ l o.66%,.5%,.48% and 3.% respectivel, ere considered. As son in Fig. 4b, te stiness ratios or columns under an axial load o.5 c A g ere observed to rise sligtl it an increase in longitudinal reinorcement ratio; ile or columns under an axial load o. c A g te stiness ratios almost remained te same. Tis suggested tat or simplicit te inluence o longitudinal reinorcement ratio on te initial stiness o RC columns could be ignored. Inluence o Aspect Ratio Fig. 4c so te inluence o aspect ratio on stiness ratios o RC columns. ix aspect ratios o.5,.8,.,.43,.7, and 3. ere investigated. In general, te stiness ratio increased it an increase in aspect ratio. It can be seen tat it an increase in aspect ratio rom.5 to.8,.,.43,.7, and 3.; te stiness ratios o columns itout axial loads rose b approximatel 8.5%, 39.8%, 6.8%, 83.6%, 9.4%, respectivel. imilar trends ere observed or te columns it an axial load ratio o.. Te stiness ratios increased b approximatel 5.6%, 7.4%, 37.8%, 45.% and 5.3% or columns under an axial load o.6 c A g it an increase in aspect ratio rom.5 to

7 .8,.,.43,.7, and 3., respectivel. Tis suggested tat te aspect ratio signiicantl inluences te stiness ratio. Inluence o Axial Load It is generall recognized tat te presence o column axial load can initiall increase te lexural strengt o columns and tus lead to larger initial lexural stiness, ic results in a iger stiness ratio. Te analses as illustrated in Fig. 4d ere carried out to assess te inluence o axial load ratio on stiness ratio Te axial load ratio as varied rom to.6. In general, te stiness ratio increased it an increase in axial load ratio. Figure 4d soed tat it an increase in axial load ratio rom to.,.4, and.6; te stiness ratios or specimens it an aspect ratio o.5 rose b approximatel 35.%, 98.7% and 67.9%, respectivel. imilar trends ere observed or oter aspect ratios. It can tus be concluded tat te axial load ratio signiicantl aects te stiness ratio. 5 5 tiness Ratio k (%) 5 tiness Ratio k (%) 5. c A g 5.5 c A g 5.5 c A g tiness Ratio k (%) Concrete Compressive trengt. c A g c (MPa) Longitudinal Reinorcement Ratioρ l (%) a. eect o concrete strengt b. eect o longitudinal reinorcement ratio a/.5 5 a/.8 a/.. c A g.5 c A g. c A g a/.43.5 c A g. c A g.5 c A g.3 c A g.35 c A g.4 c A a/.7 g 5.45 c A g.5 c A g.55 c A g a/3..6 c A g Axial Load Ratio c A g Aspect Ratio a/ tiness Ratio k (%) c. eect o aspect ratio d. eect o axial load Figure 4. Parametric studies or Initial tiness o Columns 3.4. Proposed Equation or Eective Moment o Inertia o RC Columns It is observed tat te stiness ratio apparentl increased it an increase in aspect ratios (R a ) and axial load ratio (R n ). Te transverse and longitudinal reinorcement ratios and concrete compressive strengt insigniicantl inluenced te stiness ratio o RC columns. For simplicit, te inluences o tese actors ere ignored. Based on te results o te parametric stud, te stiness ratio (κ ) is given b te olloing equation: (.43R +.96R +.739)( 3.3R +.573) n n a κ (3.6)

8 Berr et al. (4) collected a database o 4 tests o RC columns, ic contained te steretic response, geometr, column axial load and material properties o test specimens. Tis database provided te data needed to evaluate te accurac o te proposed equation or te stiness ratio. Te veriication as limited to te range o te parametric stud. Te axial load as limited rom to.6 c A g, and te aspect ratio as limited rom.5 to 3.. Onl rectangular columns tested in te double-curvature coniguration under unidirectional quasi-static cclic lateral loading ere cosen. It as ound tat te average ratio o te experimental to predicted stiness ratio b te proposed equation is.945 as son in Fig. 5, soing a good correlation beteen te proposed equation and experimental data. Tereore, te proposed equation ma be suitable as an assessment tool to calculate te stiness ratio o RC columns itin te range o te parametric stud. Comparison o available models it experimental data (son in Tran&Li ) indicated tat te proposed equation produced a better mean ratio o te experimental to predicted stiness ratio tan oter models 4 35 Proposed tiness Ratio (%) Experimental tiness Ratio (%) Figure 5. Comparisons beteen experimental and proposed stiness ratio or columns Figure 6. Comparison o eective stiness beteen te analtical results and tested data or alls 4. INITIAL TIFFNE OF RC WALL 4.. Proposed Metod Anoter approac can be proposed to estimate initial stiness o RC Walls b emploing direct calculation o crack angle to calculate sear deormation. im and Mander (7) provided Eqn. 4. b considering te energ minimization on te virtual ork done b te sear and lexural components. 4 ρ Av ρ n +.57 (4.) ρ v Ag α tan + ρ n Te total sear distortion can be reritten includeing to components: elongation o te orizontal reinorcements,, and te sortening o te compression strut, C. Te sear distortion, can be deined b + R + C / sinα (4.) ere α is te inclination o compression strut ( α 9 θ in Fig. 3) Assuming tat te sear orce taken b te all panel is, te stress o orizontal reinorcement can be expressed as: s d cotα A (4.3)

9 ere d is te lengt o all panel, s is te distance beteen orizontal reinorcements, and te area o orizontal reinorcement spaced at a distance s. Hence te elongation o te orizontal reinorcement becomes A is E d E s cot α A (4.4) Te concrete compression stress is obtained cd b L sinα C (4.5) ere b is te dept o all panel and L C is initial dept o te compression strut as son in Fig. 3. Hence te sortening o te concrete strut is C E cd C / cosα E b C L C sinα cosα (4.6) ere is te eigt o te all panel. B making te appropriate substitution or eb orizontal steel content, ρ A sd, and modular ratio, n E EC, te sear distortion in te all panel can be expressed as θ + R E b cotα ρ + L C n sin α cosα (4.7) en θ and L C d cosα, te sear stiness o te all panel can be deined b te olloing expression: ρ sin α cos α (4.8) Ebd sin α + nρ 4 Eqn. (4.8) indicates tat te unit sear stiness o te all panel is mainl dependent on te extent o te crack angles Hence te sear displacement caused b te ield lateral orce F ould be v F (4.9) Combination o ear and Flexure Response Ater te lexural and sear deormation at te top o all under ield lateral load are obtained, te initial stiness o alls can be determined as: F i + (4.) v

10 4.3. Proposed Equation or Moment o Inertia o tructural Walls Based on te similar parametric studies conducted on RC columns, Eqn. (4.3) ic considers tree parameters investigated: ield tensile strengt o steel bars in all boundaries, axial loads, and aspect ratios is proposed to properl evaluate te eective stiness o squat structural alls (details o tose calculations and illustrations are presented in Li and Xiang ). For simplicit, te inluence o longitudinal reinorcement content in all boundaries on all eective stiness is conservativel disregarded. I I e g.9 + N c Ag L L (4.) 4.4. Comparison o te Proposed Approac it oter test results on RC Walls Results rom RC structural alls (Li and Wang ) are compared it analtical results using te proposed model, Eqn. (4.) and oter provisions previousl revieed. Experimental eective stiness values, EI e rom te tests are back calculated b dividing te displacements at te ield point b te tested ield strengt it an elastic model. All tested alls ave aspect ratios not larger tan to and axial load ratios ic range rom zero to., ic covers almost all conditions likel to be encountered in practice. Yield strengts o outermost longitudinal bars or all specimens range rom 3 MPa to 585 MPa. It is believed tat te proposed stiness model is applicable or all values o ield strengts studied. Te longitudinal and transverse reinorcement content in te all eb is limited and remains at a lo level or all alls selected. Fig 6 presents te comparison beteen te experimental and calculated stiness (EI e /EI g ) or te proposed model and tat presented b oter proposals. O te tree equations proposed, te currentl proposed equation it a standard deviation o.4 appears to be more accurate in eective stiness evaluation. Tis can be explained b te act tat te previous models (bot Fenick and Bull s design equation and te NZ 3 code (995)) ignored te sear deormation in calculating te eective stiness o squat alls; ereas, te sear deormation as considered in te proposed model. 5. CONCLUION Tis paper presents an analtical metod to estimate te initial stiness o RC columns and alls. Compreensive parametric studies are carried out based on te proposed metod to investigate te inluences o several critical parameters. To simple equations to estimate te initial stiness o RC columns and alls are also proposed. eriications o te proposed models it experiment data on RC columns test and RC alls tests soing good agreement. REFERENCE ACI Committee 38 (8) Building Code Requirements or tructural Concrete (ACI 38-8) and Commentar (38R-8), American Concrete Institute, Farmington Hills, Mic., 465 pp. ACE 4 (7) eismic Reabilitation o Existing Buildings, American ociet o Civil Engineers, Reston, A. FEMA 356 () Prestandard and Commentar or te eismic Reabilitation o Buildings, Federal Emergenc Management Agenc, Wasington D.C., UA. Li, B and Xiang, W.Z. (). Initial tiness o quat tructural Walls. ACE Journal o tructural Engineering 37:, Tran C T N. and Li, B (). Initial tiness o Reinorced Concrete Columns it Moderate Aspect Ratios. Advances in tructural Engineering 5:, Park, R., and Paula, T. (975) Reinorced Concrete tructures, Jon Wile & ons, Ne York, 769 pp.