Flexural Strength Design of RC Beams with Consideration of Strain Gradient Effect

Transcription

1 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 Flexural Strength Design of RC Beams with Consideration of Strain Gradient Effet Mantai Chen, Johnny Ching Ming Ho International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ Abstrat The stress-strain relationship of onrete under flexure is one of the essential parameters in assessing ultimate flexural strength apaity of RC beams. Currently, the onrete stress-strain urve in flexure is obtained by inorporating a onstant sale-down fator of 0.85 in the uniaxial stress-strain urve. However, it was revealed that strain gradient would improve the maximum onrete stress under flexure and onrete stress-strain urve is strain gradient dependent. Based on the strain-gradient-dependent onrete stress-strain urve, the investigation of the ombined effets of strain gradient and onrete strength on flexural strength of RC beams was extended to high strength onrete up to 00 MPa by theoretial analysis. As an extension and appliation of the authors previous study, a new flexural strength design method inorporating the ombined effets of strain gradient and onrete strength is developed. A set of equivalent retangular onrete stress blok parameters is proposed and applied to produe a series of design harts showing that the flexural strength of RC beams are improved with strain gradient effet onsidered. Keywords Beams, Equivalent onrete stress blok, Flexural strength, Strain gradient. I. INTRODUCTION N flexural strength design of reinfored onrete (RC) I members, it is important to determinate the onrete stress distribution within the ompression zone of RC members under flexure. Currently, the stress-strain relationship of onrete in flexure is obtained by inorporating a sale-down fator k 3, whih is the ratio of maximum flexural onrete stress to ylinder strength and is taken to be 0.85 as proposed by Hognestad [] to aount for the effets of size, shape and asting position of members, in the uniaxial onrete stress-strain urve. However, experimental investigation by Sturman et al. [] has revealed that a larger maximum onrete stress was developed in eentrially loaded RC olumns with strain gradient than that in onentrially loaded one without strain gradient due to retardation of the formation of miro-raking in onrete. Various researhers [3]-[5] have also reported the effet of strain gradient on maximum onrete stress in flexure. By onduting experimental tests on eentrially loaded RC olumns, the authors have reported that the maximum onrete stress is influened by the strain gradient under flexure [6], [7] and have reommended modeling the variation of maximum onrete stress in terms of M. T. Chen is a MPhil student of the Department of Civil Engineering, The University of Hong Kong, Hong Kong (phone: ; fax: ; mt@hku.hk). J. C. M. Ho is a senior leturer of the Shool of Civil Engineering, The University of Queensland, QLD 407, Australia ( johnny.ho@uq.edu.au). fator k 3 against strain gradient in tri-linear manner [8]. Through imposing a strain-gradient-dependent fator k 3 [8] in onrete stress-strain relationship to onsider the effet of strain gradient, the authors have extended the investigation of strain gradient effet on flexural strength to high strength onrete (HSC) up to 00 MPa by nonlinear moment-urvature analysis [9]. From the results of theoretial analysis, it was found that the values of equivalent onrete stress blok parameters α and β depend on both strain gradient and onrete strength while a onstant value of an be used reasonably well as ultimate onrete strain for flexural design of RC members with various onrete strengths inorporating strain gradient effet. As a ontinued study, in this paper, the proposed value of ultimate onrete strain together with equivalent retangular onrete stress blok parameters are applied to develop a new flexural strength design method inorporating the ombined effets of strain gradient and onrete strength. A set of design equations and a series of design harts are produed for evaluating the flexural strength of both singly- and doubly-reinfored onrete beams with various onrete strengths inorporating strain gradient effet. II. NONLINEAR MOMENT-CURVATURE ANALYSIS To study strain gradient effet on the flexural behavior of RC beams, nonlinear moment-urvature analysis, whih takes into aount the onstitutive stress-strain urve of onrete and steel, was employed. It has been revealed in authors previous study that the ratio of maximum flexural onrete stress to ylinder strength k 3, and that of onrete strain at maximum flexural stress to uni-axial strength k o, are strain-gradient-dependent [8]. The empirial formulas developed to orrelate k 3 and k o to strain gradient, whih adopts a non-dimensional form in the ratio of effetive to neutral axis depth d/ to eliminate size effet, are given in (a) and (b). The onrete stress-strain urve developed by Attard and Setunge [0], whih is appliable to onrete strength from 0 to 30MPa, was modified to inlude strain gradient effet on k 3 and k o for 0 d /.3 k3 = 0.93( d / ) 0.35 for.3 < d / <.0 (a).5 for.0 d /.0 for 0 d /.3 k o = 0.43( d / ) for.3 < d / <.0 (b). for.0 d / International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /

2 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 For longitudinal steel reinforement, an idealized linear elasti-perfetly plasti stress-strain urve with stress-path dependene was adopted. The adopted stress-strain urves of onrete and steel reinforement are shown in Fig.. b = 300 mm h = 600 mm d = 550 mm d = 50 mm f y = 460 MPa E s = 00,000 MPa f = 30 to 00 MPa ρ t = A st / bd = to 8% ρ = A s / bd = 0 to % Fig. Beam setions analyzed International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ (a) Strain-gradient-dependent onrete Stress-strain urve (b) Stress-strain urve of steel with stress-path dependene onsidered Fig. Stress-strain urves of onrete and steel reinforement The moment-urvature relation of beam setion was analyzed by applying presribed urvatures to the setion inrementally starting from zero. At a presribed urvature and assumed neutral axis depth, the stresses developed in onrete and steel reinforement an be determined from strain distribution in the setion and their respetive stress-strain urves. An iterative proedure of suessively adjusting the neutral axis depth was needed until the unbalaned axial fore was negligibly small. The resulted neutral axis depth and resisting moment an be therefore evaluated. Suh proedure was repeated until the resisting moment inreased to the peak and dereased to half of the peak value. Flexural strength of a beam setion is defined as the peak moment in the moment-urvature urve. A parametri study using non-linear moment-urvature analysis is onduted to evaluate effet of strain gradient on flexural strength of RC beams. The setions analyzed and the values of various fators are shown in Fig.. III. RESULTS OF ANALYSIS A. Strain Gradient Effet on Flexural Strength of RC Beams The strain gradient effet on flexural behavior of singly- and doubly-reinfored beam setions with various onrete strengths and steel ratios are shown in Fig. 3. Two values of onrete strength (40 MPa and 80 MPa) and two values of steel ratio (3% and 8%) are hosen in Figs. 3 (a) and (b) to inlude the effet of onrete strength and to inlude both under-reinfored and over-reinfored senarios. The improvement in flexural strength of both under- and over-reinfored beam setions an be observed from both figures. It is evident that the relative improvement in flexural strength is larger for over-reinfored setion beause of the larger ompression zone and the extent of flexural strength improvement of RC beams due to strain gradient effet is also onrete-strength-dependent. Moment-urvature urves for setions with two values of ompression steel ratio (0% and %) are hosen in Fig. 3 () to investigate strain gradient effet on doubly-reinfored setion. It is apparent from Fig. 3 () that the flexural strength improvement due to strain gradient dereases as the ompression steel inreases due to smaller ompression zone. (a) f = 40 MPa, ρ = 0% International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /

3 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 stress blok as the following equations: M n P = αβ f ' b + f A (a) i= n h β h = αβ f ' b + f A d si si i (b) i= si si International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ (b) f = 80 MPa, ρ = 0% () f = 60 MPa, ρ t = 4% Fig. 3 Complete moment-urvature urves of singly- and doubly-rc beams with and without strain gradient effet onsidered B. Derivation of Equivalent Stress Blok Parameters For pratial flexural strength design of RC members, the nonlinear onrete stress distribution is usually replaed by simplified equivalent retangular stress blok with the width of αf representing equivalent onrete stress in flexure and the height of β representing the depth of equivalent stress blok as shown in Fig. 4. where b is the breadth of beam setion; is the neutral axis depth; n is the total number of steel bars; f si and A si are respetively the stress and area of the i th steel bar; d i is the distane of the i th steel bar from the extreme onrete ompressive fiber; h is the total depth of the beam setion; P and M are the axial fore and moment apaity alulated using non-linear analysis. The derived equivalent retangular stress blok parameters α and β for three seleted onrete strengths are plotted against strain gradient fator d/ in Figs. 5 (a) and (b), respetively. It is apparent that both onrete stress blok parameters, α and β, are dependent on strain gradient and onrete strength. Aordingly, the ombined effet of strain gradient and onrete strength should be onsidered simultaneously in the flexural strength design of RC beams. (a) α plotted against d/ Fig. 4 Atual nonlinear onrete stress distribution and equivalent stress blok The values of equivalent stress blok parameters α and β are derived based on both axial fore and moment equilibrium onditions to math the fore and moment obtained from the non-linear onrete stress-strain urve and the retangular (b) β plotted against d/ Fig. 5 Graphs of α and β plotted against strain gradient d/ Note: Dotted lines indiate that neutral axis depth falls out of setion For the appliation to pratial design of RC beams with International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /

4 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ onrete strength from 30 to 00 MPa inorporating strain gradient effet, a set of equivalent retangular stress-blok parameters α and β with equations shown in (3) and (4) are reommended based on the results obtained from the parametri study. α for 0 d /.3 d /.3 α = ( α ) α + α for.3 < d / <.0 (3a) 0.7 for.0 d / α ( f '/00) 0.9( f '/00) ( f '/00) 0.06( f '/00). 395 α = (3b) α (3) = β for 0 d /.3 d /.3 β = ( β β ) for.3 < d / <.0 + β (4a) 0.7 for.0 d / β ( f '/00) 0.83( f '/00) 0. 9 ( f '/00) 0.36( f '/00) β (4b) = β (4) = C. Ultimate Conrete Strain Equivalent stress blok parameters should be adopted together with appropriate ultimate onrete strain ε u in pratial flexural strength design of RC beams inorporating strain gradient effet. Sine the determination of ε u is quite diffiult for under-reinfored beams, whih show long flat plateau when reahing the maximum moment apaity in the moment-urvature urves, the authors have proposed a method to minimize the sensitivity of ε u and at the same time maintaining the auray in moment apaity predition by studying the lower bound and upper bound values of ε u that may ause at most % error in evaluating the maximum moment apaity. It was found that a onstant value of 300 με an always be adopted as the design value of ultimate onrete strain for all f and ρ t studied in this paper [9]. D. Verifiation of Proposal To validate the proposed equivalent onrete stress blok together with the proposed design value of ultimate onrete strain, the proposed equations (3) and (4) and ultimate onrete strain with design value of are used to evaluate the flexural strength of over 00 beam speimens tested by other researhers []-[5]. The predited flexural strengths of beam speimens based on the new proposal are ompared with the measured strengths and those alulated as per various RC design odes [6]-[8]. The auray of flexural strength predition is improved by 6% on average [9]. IV. APPLICATION ON FLEXURAL STRENGTH DESIGN OF RC BEAMS A. Singly-Reinfored Beams The proposed equivalent retangular stress blok parameters with strain gradient effet onsidered an be applied to flexural strength design of singly-reinfored beams with various tension steel ratios and onrete strengths based on fore and moment equilibrium onditions as shown in the following equations: αβ f ' b = f st ρ tbd (5a) M = αβf ' b d 0. 5β (5b) ( ) where α and β are proposed by (3) and (4) respetively. To failitate pratial design appliation, the above formulas have been onverted into a series of design harts for singly-reinfored beams with different onrete strength varied from 40 to 00 MPa and tension steel ratio varied from 0 to 8% as shown in Fig. 6. The flexural strength is expressed in terms of M/(bd ) to eliminate the size effet. The flexural strength improvement when strain gradient effet is onsidered is investigated by omparing the predited strength using proposed method (M p ) with the theoretial strengths (M ACI, M EC, M NZS ) alulated as per various design odes [6]-[8] respetively. The dotted line is referred to the strength alulated with allowable neutral axis depth of 0.45d for f 50 MPa or 0.35d for f > 50 MPa as stipulated in Euroode [7]. From Fig. 6, the maximum differene in flexural strength with strain gradient effet onsidered is 34% when f = 80 MPa and ρ t = 8%. However, for pratial design where 0.45d or 0.35d and ρ t 4% to allow for nominal dutility requirement [7], the maximum differene redues to % (f = 40 MPa and ρ t = 4%). (a) f = 40 MPa and ρ = 0% (b) f = 60 MPa and ρ = 0% International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /

5 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 () f = 60 MPa and ρ = 0% (a) f = 40 MPa and ρ = % International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ (d) f = 00 MPa* and ρ = 0% Fig. 6 Design harts for flexural strength of singly-reinfored beams Note: *: onrete strength of 00 MPa is beyond the limit speified in Euroode, no data an be reported in (d) B. Doubly-Reinfored Beams Similarly to singly-reinfored beams, the flexural strength of doubly-reinfored beams with various tension steel ratios and onrete strengths an be evaluated using following equations: αβ f ' b + f ρ bd = f ρ bd (6a) s st t ' b d 0.5β + f bd( d d s ( ) ) M = αβ f ρ (6b) where α and β are proposed by (3) and (4) respetively. A series of design harts is produed for flexural strength design of doubly-reinfored beams with ompression steel ratio of %, onrete strengths from 40 to 00 MPa and tension steel ratios from 0 to 8% as shown in Fig. 7. From Fig. 7, the maximum differene in flexural strength with strain gradient effet onsidered for doubly-reinfored beam with ρ = % is % when f = 60 MPa and ρ t = 8%. However, for pratial design onsidering nominal dutility requirement, the maximum differene redues to only 3% (f = 40 MPa and ρ t = 4%). (b) f = 60 MPa and ρ = % () f = 80 MPa and ρ = % International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /

6 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 Hong Kong (HKU) for the work presented herein is gratefully aknowledged. International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ (d) f = 00 MPa* and ρ = % Fig. 7 Design harts for flexural strength of doubly-reinfored beams (ρ = %) Note: *: onrete strength of 00 MPa is beyond the limit speified in Euroode, no data an be reported in (d) V. CONCLUSION A strain-gradient-dependent onrete stress-strain urve is adopted in nonlinear moment-urvature analysis to extend the investigation of strain gradient effet on flexural strength of RC beams to high strength onrete up to 00 MPa. The improvement of flexural strength due to strain gradient effet an be observed from both under- and over-reinfored onrete beams with or without ompression steel. It is found that the strain gradient effet on flexural strength improvement of RC beams is also onrete-strength-dependent. Therefore, two equations of equivalent retangular stress blok parameters onsidering the ombined effets of strain gradient and onrete strength together with a onstant design value of 0.003, whih were verified by omparing the predited strengths to the measured strengths of over 00 beam speimens, are proposed. Based on that, a new flexural strength design method onsidering the ombined effets of strain gradient and onrete strength is proposed for both singly- and doubly-reinfored beams. The proposed flexural strength design equations are further onverted into a series of design harts for the purpose of failitation of pratial flexural design. By omparing the predited flexural strength of singly- and doubly-reinfored beams with the theoretial strength alulated as per various RC design odes, it is apparent that the improvement in flexural strength predition due to onsideration of strain gradient effet is more signifiant for singly-reinfored beams (with maximum of 34% improvement) than doubly-reinfored beams (with maximum of % improvement). However, for pratial design, by limiting the neutral axis depth to be within 0.45 or 0.35 times the effetive depth and tension steel ratio to be smaller than 4% to allow for the minimum dutility requirement, the flexural strength improvement redues to the range between 3 and %. ACKNOWLEDGMENT The researh grant from Seed Funding Programme for Basi Researh (Projet ode ) of The University of REFERENCES [] E. Hognestad., A study of ombined bending and axial load in reinfored onrete members. Bulletin No. 399, Engineering Experiment Station, University of Illinois, Urbana. 95. [] G. M. 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7 World Aademy of Siene, Engineering and Tehnology Vol:8, No:6, 04 [3] S. W. Shin, S. K. Ghosh, and J. Moreno, Flexural dutility of ultra-high-strength onrete members, ACI Strutural Journal, vol. 86, no. 4, pp , 989. [4] S. W. Shin, S. H. Yoo, J. M. Ahn, and K. S. Lee, The dutile behaviour inluding flexural strength of high-strength onrete members subjeted to flexure, ACI Speial Publiation, vol. 7, no., pp , 999. [5] W. J. Weiss, K. Guler, and S. P. Shah, Loalization and size-dependent response of reinfored onrete beams, ACI Strutural Journal, vol. 98, no. 5, pp , 00. [6] ACI Committee 38, Building ode requirements for strutural onrete and ommentary ACI 38M-08, Manual of Conrete Pratie, Amerian Conrete Institute, Mihigan, USA, 008. [7] European Committee for Standardization, Design of onrete strutures: Part -: General rules and rules for buildings Euroode, British Standards Institution, Brussels, UK, 004. [8] Standards New Zealand, Conrete strutures standard: Part : The design of onrete strutures NZS 30, Wellington, New Zealand, 006. International Siene Index, Civil and Environmental Engineering Vol:8, No:6, 04 waset.org/publiation/ Mr. Mantai Chen obtained his BEng degree in 0 in ivil engineering from The University of Hong Kong (HKU). He is now a MPhil student in the ivil engineering department in The University of Hong Kong. His researh interest is strain gradient effets of flexural strength and dutility of reinfored onrete beams and olumns. Ir. Dr. Johnny Ching Ming Ho is Senior Leturer of the Shool of Civil Engineering, The University of Queensland. Before joining the university in 03, Dr Ho has been working as an Assistant Professor in The University of Hong Kong, and as a Civil Engineers in both Hong Kong and Brisbane offies of Arup on some large sale infrastruture projets suh as The Stoneutters Bridge in Hong Kong and Ipswih Motorway Upgrade in Queensland, Australia. Dr Ho s researh interests are on dutility and deformability of high-strength onrete beams and olumns, plasti hinge analysis of reinfored onrete members and behaviour of single- and double-skinned onrete-filled-steel-tube olumns with external onfinement and internal onrete expansive agent. International Sholarly and Sientifi Researh & Innovation 8(6) sholar.waset.org/ /