Size effect in behavior of lightly reinforced concrete beams

Transcription

1 Size effect in behavior of lightly reinforced concrete beam M.L.Zak Ariel Univerity Ariel 47 Irael Abtract The aim of thi work i to ae the effect of ize (depth of ection in the behavior of reinforced concrete beam. A computation cheme and tre-train relation are propoed for analyi of reinforced concrete ection with regard to thi effect for the cae of hort-term loading. The propoed model include two ditinct tre-train relation for tenile concrete. One relation repreent the behavior of concrete in axially-loaded reinforced concrete member (the tenion tiffening effect. It i intended for tenile concrete cloe to the reinforcing bar. The other relation i intended to repreent the behavior of plain concrete in flexural tenion. Reported in literature experimental data in particular data obtained in tet on beam with depth of ection from 2 to 8 cm are ued for calibration of parameter in thi work. Analye baed on the propoed model how that the hape of moment-curvature diagram for beam with low teel ratio i markedly influenced by the ize effect. Keyword: beam modeling reinforced concrete tenile propertie. Introduction The unatifactory performance of modern reinforced concrete building in the 21/211 Canterbury New Zealand earthquake i attributed among other factor to the unexpected by the deigner rocking rather than flexural mode of behavior of laterally loaded lightly reinforced concrete wall [1]. In principle the tranition from the ductile flexural behavior typical for relatively mall laboratory pecimen to brittle rocking behavior of reinforced concrete wall in multi-torey building can be identified a a ize effect. owever there i no conenu with regard to thi effect even in behavior of reinforced concrete beam the mot common tructural element. The ize effect in their behavior i recognized by the fib MC [2] and not recognized by the ACI code [3]. The aim of thi work i to conider thi iue and re-examine available tet data. Two different approache to modeling of the behavior of reinforced concrete beam are found in literature. According to one approach the cracked concrete i modeled a a gradually oftening material and the averaged tre ditribution in a ection i a hown in Fig. 1a. Calculation baed on thi approach in different verion (ee e.g. [4-6] yield reult (curvature deflection in reaonable agreement with tet data for member with depth of ection 4 cm or le. According to the other approach (ee e.g. [2 7 8] which i baed in particular on experimental tudie [9 1] of large reinforced concrete beam the cracked concrete bear the tenile tre within ome area cloe to reinforcing bar (the tenion tiffening effect but behave in a brittle manner in the remain of tenile concrete area; the averaged tre ditribution i a hown in Fig. 1b. In thi paper the above approache are combined and a model of concrete behavior i preented which i equally uitable for analye of reinforced concrete beam of any ize. The paper i organized a follow: 3-33

2 1. A computation cheme i given for nonlinear analyi of reinforced concrete ection in flexure for the cae of hort-term loading. 2. Uing thi cheme and experimental moment-curvature diagram for reinforced concrete beam with depth of ection from 2 to 8 cm aement i made of the effective (averaged tre-train relation of plain concrete in flexural tenion. 3. The effect of ize on moment-curvature diagram i illutrated for ection with different teel ratio. Analyi of reinforced concrete ection in flexure Conider a reinforced concrete ection (Fig. 2 acted upon by the bending moment M. A uual crack in the beam are meared and linear ditribution of the train over the ection i aumed = + φ x. (1 ere φ i the curvature the train of the extreme compreive fiber (at which the x-axi origin i located. The concrete ection comprie three zone: A cc - compreive concrete with tre-train relation σ c =σ cc ( A rct reinforced tenile concrete with averaged tre-train relation σ c =σ rct ( A pct - plain tenile concrete with averaged tre-train relation σ c =σ pct (. Following [11] the depth d rct =2(-D of zone A rct i adopted where D are the total and effective depth of the ection repectively. The olution ( φ of the equilibrium equation bxdx + A D = M c bdx + A = (2a (2b where b=b(x σ A are the width of the ection and the teel tre and area repectively i found uing the following iteration cheme ( ( i+ 1 Ec bx dx + E A D + ( E c 2 2 ( i bx dx + E A D φ + 1 = M (3a ( ( i+ 1 Ec b dx + E A + ( E c ( i bx dx + E A D φ + 1 = (3b i = (3c where cc ( / if < Ec if = Ec = (4 pct ( / if > and x < d rct rct ( / if > and x d rct E c - initial Young modulu of the concrete E = 2 GPa - Young modulu of the teel 3-34

3 = + φ x ( ( = φ = with the convergence criterion ( i+ 1 ( i+.1 φ 1 φ ( i.1φ. (5a (5b (6 Stre-train relation of concrete For the tenile concrete in zone A pct the tre-train relation can be taken in form imilar to that adopted in Ref. [12] Ec if ft = f ct / Ec pct ( = (7 f ctϕ( ft γ if > ft where f ct i the tenile trength of the concrete and γ ( / f if γ > 1 ϕ( γ = γ /( γ 1 f 1 + ( γ 1( / (8 f if γ = 1. ere γ i a parameter that depend on the teepne of the decending branch of the tre-train diagram (ee Fig.3. Fitting moment-curvature diagram obtained with ue of the above computation cheme to experimental data preented in Fig. 4-6 that i finding for each group of data uch a value γ that φmax 2 [ M th ( γ M exp ] dφ = min (9 where M th (γ= M th (φγ i the theoretical bending moment correponding to the pecific value of γ M exp = M exp (φ - the experimental bending moment and φ max - the curvature in the cracking tage correponding to the end point of the diagram we obtain (ee Fig. 7 γ = 3.2.7( / 1 = 2cm (1 Other relation adopted for contructing the diagram in Fig. 4-6 are a follow: - for the tenile concrete in zone A rct a per Ref. [16 17] Ec if fct rct ( = ϕ ( fct 2.5 if > fct (11 - for the compreive concrete a per Ref. [18] cc ( = f cϕ( fc Ec fc / f c (12 where f c.8f cu i the compreive trength of the concrete f cu the cube trength and fc the train correponding to f c ; - for E c a per Ref.[19] 515 ( fc / fc Ec = 2 + ( fc / fc [ MPa] (13 where f c =-1MPa; - for fc a per Ref. [221] 3-35

4 fc = ( fc / fc - for f ct a per Ref. [22] 2/3 2 f =.52( f [ kgf / cm ] ct cu (14 (15 Effect of ize on moment-curvature diagram See the diagram in Fig. 8 contructed uing the computation cheme and tre-train relation given above. Concluion Unlike beam with teel ratio.1 and more the behavior of lightly reinforced concrete beam i markedly influenced by the ize effect (depth of ection. A computation cheme and tre-train relation are given for analyi of reinforced concrete ection with regard to thi effect for the cae of hort-term loading. Reference 1. enry RS Lu Y Ingham JM: Experimental teting and modeling to addre the performance of RC wall during the 21/211 Canterbury earthquake Proc. of the Tenth U.S. National Conf. on Earthquake Engineering Anchorage Alaka fib (The International Federation for Structural Concrete: Model Code for Concrete Structure 21 Ernt & Sohn Wiley Berlin ACI Committee 318: Building Code Requirement for Structural Concrete (ACI Farmington ill MI Lam JYK Ng PL Kwan AK: Tenion tiffening in concrete beam. Part 2: Member analyi Proceeding of the Intitution of Civil Engineer: Structure and Building ( Kaklauka G.: Flexural layered deformational model of reinforced concrete member Magazine of Concrete Reearch 24 56( Bazant ZP Oh B: Deformation of progreively cracking reinforced concrete beam ACI J ( Boronyoi A Balaz GL: Model for flexural cracking in concrete: the tate of the art Structural Concrete 25 6( Marti P Alvarez M Kaufmann W Sigrit V: Tenion chord model for tructural concrete Structural Engineering International ( Leonhardt F: Crack and crack control in concrete tructure PCI J Breen JE Frantz GC: Cracking on the ide face of large reinforced concrete beam ACI J ( Schiel P Wolfel E: Kontruktionregebn zur Bechraukung der Ribreite Beton- und Stahlbetonbau ( Prakhya GKV Morley CT: Tenion-tiffening and moment-curvature relation of reinforced concrete element ACI Struct. J ( Bach C Graf O: Veruche mit Eienbetonbalken zur Ermittlung der Beziehung Zwichen Formanderungwinkel und Biegungmoment Deutcher Auchu fur Eienbeton Ruch Rehm G: Veruche mit Betonformtahlen Deutcher Auchu fur Stahlbeton

5 15. Braam CT: Control of crack width in deep reinforced concrete beam eron ( Belarbi A u TTC: Contitutive law of concrete in tenion and reinforced bar tiffened by concrete ACI Struct. J ( wang LS Rizkalla S: Effective tenile tre-train characteritic for reinforced concrete Proceeding of the Canadian Society of Civil Engineering Annual Conference Ottawa Carreira DJ Chu K: Stre-train relationhip for plain concrete in compreion ACI Journal ( Gvozdev AA Yahin AV: Elatic deformation and creep of concrete Strength Structural Change and Deformation of Concrete (ed. Gvozdev AA Mocow (In Ruian. 2. Popovic S: A numerical approach to the complete tre-train curve of concrete Cement and Concrete Reearch ( Yahin AV: Some data on deformation and tructural change of concrete in axial compreion New Data on Strength of Reinforced Concrete (ed. Mikhailov KV Mocow (In Ruian. 22. eilmann G: Beziehungen zwichen Zug- und Druckfetigkeit de Beton Beton ( Figure and figure caption Fig. 1. Effective area (hatched of tenile concrete and concrete tre ditribution (f ct concrete tenile trength in reinforced concrete ection according to concrete model a a oftening material (a and a a tenile-tiffening material (b 3-37

6 Fig. 2. Reinforced concrete ection; train ditribution Fig. 3. Averaged tre-train diagram for concrete in flexural tenion 3-38

7 Fig. 4. Etimate of γ for =2 cm and =4 cm; moment-curvature diagram fitted to experimental data from Ref. [13] (f cu concrete cube trength ρ=a /(bd 3-39

8 Fig. 5. Etimate of γ for =62.5 cm; moment-curvature diagram fitted to experimental data from Ref. [14] Fig. 6. Etimate of γ for =8 cm; moment-curvature diagram fitted to experimental data from Ref. [15]; cro-ection of beam (dimenion in cm 3-4

9 Fig. 7. γ veru Fig. 8. Comparion of moment-curvature diagram for reinforced concrete rectangular ection of different depth (f c =-3 MPa D=