CURVATURE DUCTILITY OF REINFORCED CONCRETE BEAM

CURVATURE DUCTILITY OF REINFORCED CONCRETE BEAM Monita Olivia, Parthasarathi Mandal ABSTRACT The aim o this paper is to examine the inluene o three variables on urvature dutility o reinored onrete beams. A omputer program was developed to predit moment-urvature and available urvature dutility o reinored onrete beams with or without axial loads. Ten beams with dierent variables were analysed using the program. The variables measured are onrete strength, amount o longitudinal reinorement and spaing o transverse reinorement. The input onsists o beam geometry, material properties and loading. A onined stress-strain urve or onrete proposed by Saatioglu and Razvi (1992) is applied in the program, while, steel stress-strain model is adopted rom BS 8110 (British Standard Institution 1985). Computer analysis indiates that the urvature dutility inreases with the inrease o longitudinal reinorement and onrete strength. On the other hand, the spaing o transverse reinorement does not have any signiiant inluene on the urvature dutility. Key words : omputer analysis, urvature dutility, moment-urvature, reinored onrete beams 1. INTRODUCTION Dutility o reinored strutures is a desirable property where resistane to brittle ailure during lexure is required to ensure strutural integrity. Dutile behaviour in a struture an be ahieved through the use o plasti hinges positioned at appropriate loations throughout the strutural rame. These are designed to provide suiient dutility to resist strutural ollapse ater the yield strength o the material has been ahieved. The available dutility o plasti hinges in reinored onrete is determined based on the shape o the moment-urvature relations. Dutility may be deined as the ability to undergo deormations without a substantial redution in the lexural apaity o the member (Park & Ruitong 1988). Aording to Xie et al, (1994), this deormability is inluened by some ators suh as the tensile reinorement ratio, the amount o longitudinal ompressive reinorement, the amount o lateral tie and the strength o onrete. The dutility o reinored onrete setion ould be expressed in the orm o the urvature dutility (µ φ ): φu µ φ = φ where φ u is the urvature at ultimate when the onrete ompression strain reahes a speiied limiting value, φ y is the urvature when the tension reinorement irst reahes the yield strength. The deinition o φ y shows the inluene o the yield strength o reinorement steel y Curvature Dutility O Reinored Conrete Beam 1

on the alulation o µ φ, while the deinition o φ u relets the eet o ultimate strain o onrete in ompression. Park & Paulay (1975) have suggested that the yield urvature o a reinored onrete setion is taken when the tension steel irst yields. Assuming an under-reinored setion, irst yield will our in the steel, then the moment and urvature are: φ y M y = A s y d'' ε sy y = = ( 1 k) d E (1 k) d s where k = {(ρ + ρ') 2 n 2 + 2 [ρ + (ρ'd/d)]n} - (ρ + ρ')n, ρ = A s /bd is the tensile reinorement ratio, ρ' = A s /bd is the ompression steel ratio, n = E s /E = modular ratio, E s, E is the modulus elastiity o the steel and the onrete, d'' is the distane rom entroid o ompressive ores in the steel and onrete to the entroid o tension. The ultimate urvature o reinored onrete setion is deined as the maximum value o onrete strain at the extreme ompressive ibre. It an be written as: As y A' s y where a = 0.85 ' b M u = 0.85 ' ab(d-a/2) + A's y (d-d') ε εβ φ 1 u = = a where y is the steel yield strength, ' is the onrete ompressive strength, β 1 is the depth o the equivalent retangular stress blok. ACI 318-71 (Park & Paulay 1975) onservatively reommends a value o 0.003. In Euroode 8 ENV 1994-1-3 (European Committee or Standardization 1994) the nominal value o ultimate onrete strain (ε u ) or unonined onrete needed to alulate Conventional Curvature Dutility Fator (CCDF) is 0.0035, while impliitly or onined onrete it is larger than 0.0035. Theoretial moment-urvature analysis or reinored onrete strutural elements indiating the available lexural strength and dutility an be onstruted providing that the stress-strain relations or both onrete and steel are known. Moment-urvature relationship an be obtained rom urvature and the bending moment o the setion or a given load inreased to ailure. Moment (M) First yielding First raking Curvature (ϕ) Figure 1. A trilinear moment-urvature relationship (Park & Paulay 1975). 2 Volume 6 No. 1, Oktober 2005 : 1-13

A trilinear moment-urvature relationship shown in Figure 1 is deined by the points o raking and yielding. Unraked, raked and yielded behaviors are depited by straight lines. Park & Paulay (1975) ound that the urve is linear in its initial stage, and the relationship between moment M and urvature is given by the lassial elasti equation: EI = MR = M/ϕ where EI is the lexural rigidity o the setion. Based on the proedures proposed by Park & Ruitong (1988), the moment, orresponding to the hosen value (ε m ) and axial load (P) is obtained by taking moments o the internal ores. M = n m Σ i Ai d i + Σ i= 1 j= 1 where i is the onrete stress in the i-th layer, sj is the steel stress in the j-th layer, A i is the area o onrete in the i-th layer, A sj is the steel stress in the j-th layer, d i and d j are the the distane o the entroid o i-th layer or onrete and o j-th lamina or steel, rom reerene axis or moment alulation, n is the number o layer o onrete, m is the number o layer or steel. The urvature is given by: ε ϕ = m where ε m is the onrete strain in the extreme ompression ibre, is the neutral axis depth. The variables aeting urvature dutility may be lassiied under three groups (ater Dereho 1989), namely 1) loading variable suh as the level o axial load; 2) geometri variables suh as the amount o tension and ompression reinorement, amount o transverse reinorement and the shape o the setion; and 3) material variables suh as the yield strength o reinorement and harateristi strength o onrete. The objetive o this study is to analyse urvature dutility o 10 beams with three dierent variables. A omputer program was developed to establish dutility analysis or those reinored onrete beams. The sotware, reerred to CD Analysis, provides momenturvature analysis and urvature dutility analysis. sj A sj d j 2. METHODOLOGY Many variables inluene the urvature dutility o reinored beams and the presented numerial analysis is done on speimens that are designed to address some o them. The parametri study has been arried out or all the speimens. The eet o dierent variables is studied by varying one variable at a time, keeping the value o other variables ixed. The summary o the speimen properties is given in Table 1. The variables studied in the presented test program are as ollows 1) onrete ompressive strength ( u ). The test speimens ontained our types o ompressive strength. The ompressive strength ( u ) ranged between 20-35 MPa; 2) longitudinal reinorement ratio (ρ'/ρ). The longitudinal reinorement onsisted o tension and ompression reinorement and was varied between 0.25 to 1.00. The reinorement ratios were alulated as A s /bh; where A s is the area o reinoring steel; b and h are width and height o the onrete setion, respetively; 3) spaing o oninement reinorement. The stirrups spaing were taken between 50-150 mm. Ten beams with 77 x 130 mm in ross setion and 1320 mm length were investigated in this researh. Curvature Dutility O Reinored Conrete Beam 3

Stress ' Table 1. Speimen properties Diameter Diameter o Diameter u o tensile ompression o (MPa) ρ ρ' ρ'/ρ reinorement reinore- stirrups (mm) ment (mm) (mm) 1 25 2#6 0.00704 2#6 0.00704 1.0 3 75 Beam No. Spaing o stirrups (mm) 2 25 2#8 0.01263 2#6 0.00710 0.55 3 75 3 25 2#10 0.01993 2#6 0.00717 0.36 3 75 4 25 2#12 0.02898 2#6 0.00725 0.25 3 75 5 20 2#10 0.01993 2#6 0.00717 0.36 3 75 6 30 2#10 0.01993 2#6 0.00717 0.36 3 75 7 35 2#10 0.01993 2#6 0.00717 0.36 3 75 8 25 2#10 0.01993 2#6 0.00717 0.36 3 50 9 25 2#10 0.01993 2#6 0.00717 0.36 3 100 10 25 2#10 0.01993 2#6 0.00717 0.36 3 150 A omputer program was run to estimate moment-urvature relationship and urvature dutility o reinored onrete setion. The program inorporates eet o onrete oninement. In this researh, stress-strain urve o onined onrete was adopted rom Saatioglu and Razvi (Saatiouglu & Razvi 1992). The assumed stress-strain urve o steel reinorement rom BS 8110 (British Standard Institution 1985) is modiied to simpliy the analysis. Saatioglu & Razvi (1992) proposed a stress-strain relationship whih is appliable to any ross setional shapes and reinorement arrangement used in pratie. The stress-strain urve onsists o a paraboli asending branh ollowed by a linear desending segment as shown in Figure 2. This part is onstruted by deining the strain orresponding to 85% o the peak stress. This strain level is expressed in terms o oninement parameters. A onstant residual strength is assumed beyond the desending branh, at 20% strength level. 0.85' ' o 0.85' Unonine Conined 0.20' Strain ε 01 ε 1 ε 85 ε 20 ε 085 Figure 2. Stress-strain relationship or onrete onined by irular spirals (Saatioglu & Razvi 1992). The equations relating to the various segments o the stress-strain urve are shown below: The paraboli part: 4 Volume 6 No. 1, Oktober 2005 : 1-13

For 0 ε ε 1 = ' ε ε 2 ε1 ε1 2 (1+ 2K ) The linear part: For ε 1 ε ε 20 = ' ε ε ε ε 85 1 0.15 ' 1 0.20 ' For ε ε 20 = 0.20' where K = k 1 le / k 1 = 6.7( le ) -0.17 le = k 2 l ' = ' + k 1 1e or a retangular setion : le lex bx = b x + leyby + b y l ΣA = s yh s b sinα b 1 b k2 = 0.26 1.0 s s1 1 ' = ' + k 1 1e ε o = 0.002 ε 1 = ε o (1+5K) ε 85 = 260ρε 1 + ε 85 ΣAs ρ = s b + b ( ) x y where ' is the unonined strength o onrete; ε, is the strain and orresponding stress rom stress-strain urve; ε 1, ε 85 is the strain orresponding to the peak stress and 85% o the peak stress, or onined onrete; ε o, ε o85 is the strain orresponding to the peak stress and 85% o the peak stress, or unonined onrete; ε 20 is the strain at 0.20 o maximum stress on the alling branh o stress-strain urve or unonined onrete; 1 is the uniorm onining pressure (MPa); le is the equivalent uniorm pressure (MPa); lex is the eetive lateral pressures ating perpendiular to ore dimension b x ; ley is the eetive lateral pressures ating perpendiular to ore dimension b y ; A s, yh is the area and yield strength o transverse reinorement; b x, b y is the ore dimensions o retangular setion; s is the entre to entre Curvature Dutility O Reinored Conrete Beam 5

distane o tie spaing; s is the spaing between laterally supported longitudinal reinorement. Figure 3. Design stress-strain urve rom BS 8110 (British Standard Institution 1985) In this researh, a simple bilinear idealisation o the steel stress-strain relationship is adopted in whih no strain hardening o the material is taken into aount (Figure 3) rom BS 8110 (British Standard 1985). BS 8110 idealises an idential behaviour o the steel in tension and ompression, as being linear in the elasti range up to the design yield stress o y /γ m where, y is the harateristi yield stress, the partial saety ator γ m = 1.15. Inputs or the program are beam geometry, material properties and axial loading. Geometri data or a beam onsist o width (b = 77 mm), depth (h = 130 mm), onrete over = 9 mm, the ratio o the amount o longitudinal reinorement (ρ'/ρ = 0.025-1.0) and the spaing o transverse reinorement (s = 50-100 mm). While, material properties inlude Young's Modulus (200 MPa), onrete ompressive strength ( u = 20-35 MPa), yield strength o reinoring steel (250 MPa), modulus elastiity o steel (E s = 200,000 Mpa) and maximum elongation o steel reinorement grade 250 is 0.22 (BS EN 10002 1992). The omputational proedure or obtaining the urvature dutility rom the momenturvature behaviour o ross setion is as ollows (Saatioglu & Yalin 1999). Firstly, alulate the ultimate axial load (P o ) that the setion an arry using P o = (A - A s )(maximum stress o onrete) + A s y where A is the gross area o ore onrete or onined setion, A s is the area o longitudinal steel, y is the yield strength o longitudinal steel. I the given axial load is less than the ultimate axial load (P o ), the proess will ontinue to the next step. Then, analysis is onduted or the strain at the extreme ompressive iber as i the setion is loaded under one axial load without any moment. The strain proile is established or the value o ibre strain. It is assumed that strain has a linear variation over the beam ross setion. The setion is divided into retangular strips (lamina) or the purpose o alulating ompressive ores in onrete as shown in material models desribed in Figure 4. Figure 4 shows a retangular setion with stress and strain diagram, and the ores ating on the ross setion. Corresponding stresses in onrete and steel are determined rom its appropriate stress-strain models. Internal ores in reinoring steel are alulated. One internal ores are omputed, the axial ore is alulated. The momenturvature urve is plotted rom the values o moment and urvature. Curvature at eah setion is obtained rom the moment-urvature relationship. The setional analysis ontinues until either the yield ondition o steel is being satisied at this partiular iteration or the ultimate ondition has reahed. I the yield ondition is satisatory, the present urvature is set as 6 Volume 6 No. 1, Oktober 2005 : 1-13

yield urvature. I the ultimate ondition has reahed, urvature dutility o the setion an be determined. The results are presented in a tabulation orm in term o moment-urvature values and a value o urvature dutility o a setion. Program lowhart is in Figure 5. b h φ ε Atual strain proile Assumed strain proile 0.85' Neutral axis s1 1 2 3 4 S 1 P h/2 M M ε s s2 S 2 Setion Idealised setion Strain Stress Internal Fores External Fores Figure 4. Setion with strain, stress and ore distribution. One internal ores are omputed, the axial ore is alulated. The momenturvature urve is plotted rom the values o moment and urvature. Curvature at eah setion is obtained rom the moment-urvature relationship. The setional analysis ontinues until either the yield ondition o steel is being satisied at this partiular iteration or the ultimate ondition has reahed. I the yield ondition is satisatory, the present urvature is set as yield urvature. I the ultimate ondition has reahed, urvature dutility o the setion an be determined. The results are presented in a tabulation orm in term o moment-urvature values and a value o urvature dutility o a setion. Program lowhart is in Figure 5. 3. RESULTS AND DISCUSSION The numerial model was employed to analyse the ten beams. The program output onsists o numerial results and urvature dutility values. The parameters onsidered were inluded in the dutility omputation. The eets o the major variables on moment-urvature urves are disussed in the ollowing paragraphs. Coninement reinorement spaing. Figure 6 shows omparison o moment-urvature relationship or our beams having the same onrete strength and the same amount o longitudinal reinorement but dierent oninement reinorement spaing. With onining the setion, the ultimate ompressive strain and dutility is inreased. The yield and maximum moment apaity o the setion remain unaltered beause the stress-strain model used in the numerial analysis assumes that shape o the initial asending segment o stress-strain urve is unhanged with the amount o transverse steel. The urvature at yield does not show any signiiant hange with the amount o transverse steel reinorement. On the other hand, the ultimate urvature inreases beause the ompressive strain also inreases. Shin et al, (1989) have reported their test results on ultra high strength onrete beams or speimens having the same onrete strength and the same amount o longitudinal reinorement but dierent oninement spaing. They ound that a loser spaing has a ontribution to postpone the bukling o the ompressive reinorement, and ailure takes plae in tension steel, hene there is no eet on dutility. Thereore, it is assumed in the urrent study that there is no eetiveness o a loser oninement spaing. Curvature Dutility O Reinored Conrete Beam 7

Start i) Geometry geometry (width, depth, over, et.) o the setion, ii) material speiiation (grade o onrete and steel, Young's modulus or steel), iii) axial load on the setion (P), and iv) strain inrement in the extreme ompression ibre ( ε). Calulate the strain ( p) on the setion under axial load Set strain at the top most ibre = p + A i) Divide the setion into laminae ii) Depending on and, and assuming linear variation in strain, alulate strain at the middle o eah laminae iii) Calulate the stress on eah lamina, using the stress-strain model or onrete iv) Also, alulate the stress in steel Calulate axial ore on the setion (Pal) Calulate moment (M) on the setion Is y > 0.0? NO Is the yield ondition o steel ahieved? YES NO y = Set the ultimate onditions Continue 8 Volume 6 No. 1, Oktober 2005 : 1-13

Figure 5. The program low hart or the moment-urvature alulation 4.50E+06 4.00E+06 3.50E+06 Moment (Nmm) 3.00E+06 2.50E+06 2.00E+06 1.50E+06 1.00E+06 5.00E+05 Beam 3 Beam 8 Beam 9 Beam 10 (s = 75 mm) (s = 50 mm) (s = 100 mm) (s = 150 mm) 0.00E+00 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04 Curvature (rad/mm) Figure 6. Computed moment-urvature urves or dierent spaing o oninement reinorement. Curvature Dutility O Reinored Conrete Beam 9

Conrete ompressive strength. The omparison o the moment-urvature urve or speimens having the same oninement reinorement spaing and amounts o longitudinal reinorement but dierent onrete strength is shown in Figure 7. The igure shows that or a member with lower strength exhibits less urvature at ultimate than a member with higher strength does. It is evident that the urvature at yield dereases and the urvature at ultimate inreases with high harateristi strength o onrete. The urvature orresponding to moment appears to inrease slightly or the higher members. The higher strength onrete members are stier than lower strength onrete members, beause the lexural rigidity (EI) o onrete inreases with strength (Xie et al, 1994). Mandal (1993) also reported that inrease in the harateristi strength o onrete inreases the neutral axis depth, hene inreases the moment apaity o the setion. It an be assumed that there is signiiant hange with an inrease in the onrete strength. Moment (Nmm) 4.50E+06 4.00E+06 3.50E+06 3.00E+06 2.50E+06 2.00E+06 1.50E+06 1.00E+06 5.00E+05 0.00E+00 Beam 3 Beam 5 Beam 6 Beam 7 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04 Curvature (rad/mm) ( u = 25 MPa) ( u = 20 MPa) ( u = 30 MPa) ( u = 35 MPa) Figure 7. Computed moment-urvature relationship urve or dierent onrete strength Longitudinal reinorement ratio. Figure 8 shows omparison o moment-urvature urve or our beams with the same onrete strength and oninement reinorement spaing but dierent amounts o longitudinal reinorement. The parameter ρ'/ρ varying rom 0.25 to 1.0, was ound to be the important ator in determining the shape o moment-urvature urves. Beam 4 in Figure 8 shows the lowest urvature dutility, reers to over reinored ondition. With inrease in the amount o tension steel, the depth o neutral axis inreases. At yield, in longitudinal steel is ixed stress and neutral axis depth inrease with the urvature. At ultimate ondition, the strain at the maximum ompressive ibre o onrete is ixed, so the urvature at ultimate dereases. As a result, the urvature dutility dereases. Shin et al, (1989); Xie et al, (1994), reported that member with high values o ρ'/ρ undertaking large urvature at relatively onstant level o moment beore the ultimate load was attained. On the other hand, although the beam with low values o ρ'/ρ was able to sustain inreasing moments, but only a small urvature an be ahieved beore the ultimate ondition. On the other hand, beam 1 shows the highest urvature dutility reers to under reinored ondition. For the very low amount o tension steel, the ultimate ondition may arrive due to raturing o tension steel. In this ase the strain at tension steel is ixed at ultimate ondition, hene, the urvature at ultimate inreases. As a result, the urvature dutility inreases with derease in the amount o tension steel. 10 Volume 6 No. 1, Oktober 2005 : 1-13

6,00E+06 Moment (Nmm) 5,00E+06 4,00E+06 3,00E+06 2,00E+06 Beam 1 Beam 2 Beam 3 Beam 4 (ρ'/ρ = 1.0) (ρ'/ρ = 0.55) (ρ'/ρ = 0.36) (ρ'/ρ = 0.25) 1,00E+06 0,00E+00 0,00E+00 2,00E-04 4,00E-04 6,00E-04 8,00E-04 1,00E-03 1,20E-03 1,40E-03 1,60E-03 1,80E-03 Curvature (rad/mm) Figure 8. Computed moment-urvature urves or dierent longitudinal reinorement ratios. Available urvature dutility o the beams that alulated using the CD program is listed in the Table 2. The table generally shows that or the same amounts o longitudinal and oninement reinorement. Curvature dutility rise gradually as the onrete strength inreases rom 20-35 MPa. Thus, the dutility o high strength onrete beams was generally higher than those o beams with moderate onrete strength. For the same onrete strength, the urvature dutility inrease drastially as the ratio o ρ'/ρ inreases. Finally, the results did not show the expeted eet o derease spaing in oninement reinorement on urvature dutility. In seismi design it would appear to be reasonable to aim at an available urvature dutility ator o at least 10 when ε u = 0.004 is reahed in the potential plasti hinge regions o beams (Park & Ruitong 1988). Aording to Dowrik (1987), during a severe earthquake the urvature dutility (µ φ ) available at the reinored onrete beams setion may be in the range o 10 to 20. Table 2. Properties and urvature dutility rom experimental and omputation results o reinored onrete beams (steel reinorement with Y = 250 MPa). Beams u (MPa) Longitudinal bars dia. (mm) Curvature Dutility O Reinored Conrete Beam ρ'/ρ Stirrups spaing (mm) Numerial Curvature Dutility (φ) 1 25 6 1.0 75 108.83 2 25 8 0.55 75 53.65 3 25 10 0.36 75 21.84 4 25 12 0.25 75 11.597 5 20 10 0.36 75 17.023 6 30 10 0.36 75 26.99 7 35 10 0.36 75 32.057 8 25 10 0.36 50 27.295 9 25 10 0.36 100 19.138 10 25 10 0.36 150 17.833 11

Beams have a higher urvature value ompared to olumns beause they are designed to ail in a dutile manner with yielding o the tension steel. Although some odes o pratie require the available dutility or beams in seismi design, the values will generally be exeeded during a severe earthquake (Park & Paulay 1975). The available urvature dutility values arising rom CD Analysis are ranging rom 11-108. It is evident that the available urvature dutility rom CD analysis its the ranges o dutility values aording to some indings rom previous researh. 4. CONCLUSION The eet o geometri and material variables on the available urvature dutility o reinored onrete beam an be readily assessed using Curvature Dutility Program. As expeted it was ound that, with other variables held onstant, the available urvature dutility ator is inreased i the longitudinal reinorement ratio is inreased and the onrete ompressive strength is inreased. While, there is no signiiant inrease i the onined reinorement spaing is dereased. REFERENCES British Standard Institution, BS 8110: Part 1: 1985. Strutural Use o Conrete, London: BSI. 1985. Dowrik, D.J., 1987. Earthquake Resistant Design, Great Britain: John Wiley & Sons. European Committee or Standardization, Euroode 8: 1994.Design provisions or earthquake resistane o strutures Part 3: Speii rules or various materials and elements. Mandal, P., 1993. Curvature dutility o reinored onrete setions with and without oninement, Master Thesis Department o Civil Engineering, Kanpur: Indian Institute o Tehnology Kanpur. Park, R. & Paulay, T., 1975. Reinored Conrete Strutures, Canada: John Wiley & Sons. Park, R. & Ruitong, D., 1988. Dutility o doubly reinored onrete beam setion, ACI Strutural Journal 85: 217-225. Saatioglu, M. & Razvi, S.R., 1992. Strength and dutility o onine onrete olumns, ASCE Journal Strutural 106: 1079-1102. Saatioglu, M. & Yalin, C., 2000, Inelasti analysis o reinored onrete olumns, Computer and Strutures 77 [online], London: Elsevier Siene Ltd. Available at: <http://www.sienediret.om/siene/journal/00457949> [Aessed 29 August 2000]. Shin, S., Ghosh, S.K. & Moreno, J., 1989. Flexural dutility o ultra high strength onrete members, ACI Strutural Journal 86: 394-400. Xie, Y., Ahmad, S., Yu, T., Hino, S. & Chung, W., 1994. Shear dutility o reinored onrete beams o normal and high strength onrete, ACI Strutural Journal 91: 140-149. ACKNOWLEDGEMENTS This artile is a part o MS Dissertation o Strutural Engineering. The author wish to aknowledge the inanial support given by Engineering Eduation Development Projet ADB-INO 1432. Speial thanks are due to the tehnial sta o the Strutural Laboratory o 12 Volume 6 No. 1, Oktober 2005 : 1-13

Civil & Strutural Department, UMIST, Manhester, i.e. John Mason, Steve Edwards, Paul Nedwell, and John Wall or their kind assistane throughout the projet. RIWAYAT PENULIS Monita Olivia MS, adalah sta pengajar pada Jurusan Teknik Sipil Fakultas Teknik Universitas Riau, Pekanbaru. Parthasarathi Mandal PhD adalah sta pengajar pada Civil & Strutural Engineering Department, UMIST, Manhester, United Kingdom. Curvature Dutility O Reinored Conrete Beam 13