Engineering Structures 30 (2008) 13–21www.elsevier.com/locate/engstruct

Ductility analysis of prestressed concrete beams with unbonded tendons

J.S. Dua, Francis T.K. Aub,∗, Y.K. Cheungb, Albert K.H. Kwanb

a School of Civil Engineering and Architecture, Beijing Jiao Tong University, Beijing 100044, PR Chinab Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received 27 October 2006; received in revised form 21 February 2007; accepted 22 February 2007Available online 6 April 2007

Abstract

This paper describes a numerical method for the full-range analysis of prestressed concrete flexural members with unbonded tendons, takinginto account the stress-path dependence of materials. The numerical results compare favourably with experimental results. Parametric studies arecarried out to evaluate the influence of loading type, span–depth ratio, combined reinforcement index (CRI), partial prestressing ratio, concretecompressive strength, and ratio of compressive reinforcement, etc. on the ductility behaviour. The results indicate that the curvature ductility factorof prestressed concrete members with unbonded tendons decreases with the increase of CRI. The curvature ductility factors for members withbonded and unbonded tendons for given values of CRI are also analyzed and compared. It is generally observed that when the CRI is between0.15 and 0.20, the ductility factor of an unbonded member is close to that of the bonded one. Above this range of CRI the ductility factor of anunbonded member is higher than that of a bonded one, while below this range the ductility factor of an unbonded member is lower than that of abonded one.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Ductility; Post-tensioning; Prestressed concrete; Unbonded tendons

1. Introduction

The use of unbonded tendons is now common in prestressedconcrete structures, and they may be in the form of internal orexternal tendons. With the increasing use of unbonded tendonsin both new construction and retrofitting of existing structures,there is the need for a closer look at their design and analysis.As the tendons and the surrounding concrete generally movewith respect to each other, the stress increase in the tendons dueto external loading subsequent to prestressing depends on thedeformation of the whole member, and cannot be determinedfrom the analysis of the cross section alone as in the case ofbonded tendons. Many experimental and analytical studies havebeen carried out within the past five decades for predictionof the flexural resistance of prestressed concrete beams withunbonded tendons, which is closely related to the ultimatetendon stress at flexural failure. However, there have been fewinvestigations on the ductility of prestressed concrete memberswith unbonded tendons. In fact, ductility is as important as

∗ Corresponding author. Tel.: +852 2859 2650; fax: +852 2559 5337.E-mail address: francis.au@hku.hk (F.T.K. Au).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.02.015

flexural resistance for a prestressed concrete member withunbonded tendons in order to assess its safety realistically.The present study aims to evaluate the flexural ductility ofprestressed concrete members with unbonded tendons, takinginto account the stress-path dependence of materials. As theplastic hinges formed in a beam enter the post-peak stage, theother sections will be unloaded, and it is therefore necessary toconsider the stress-path dependence. Using a nonlinear methodof analysis, extensive parametric studies are carried out toevaluate the influence of various parameters such as the loadingtype, the span–depth ratio, the effective reinforcement index,the partial prestressing ratio, the concrete compressive strength,and the ratio of compressive reinforcement on the ductilitybehaviour.

2. Ductility factor

The ductility of a structural member is usually expressed interms of a dimensionless ductility factor, which is the ratio ofa kind of deformation at failure to that at yield. The curvatureductility factor µφ used here is defined in terms of the ultimatecurvature φu and the curvature at first yield φy as

14 J.S. Du et al. / Engineering Structures 30 (2008) 13–21

Fig. 1. Equivalent yield curvature φy and maximum curvature φu .

µφ = φu/φy . (1)

The definitions of equivalent yield curvature and ultimatecurvature as proposed by Park [1] for general cases ofmoment–curvature response are adopted in the present study.The ultimate curvature φu is taken as the curvature of thesection when the resisting moment has dropped to 85% of thepeak resisting moment after reaching the peak (Fig. 1). Theyield curvature φy is taken as the curvature at the hypotheticalyield point of an equivalent perfectly elasto-plastic system withan elastic stiffness equal to the secant stiffness of the sectionat 75% of the peak resisting moment and a yield momentequal to the peak resisting moment itself. This definition ofductility factor applies even if there is no obvious yield pointidentified as a kink on the moment–curvature curve. However,it is required to obtain the full-range behaviour at least coveringthe deformation up to the point where the moment drops to 85%of the peak moment.

3. Numerical analysis

3.1. Numerical model and material properties

The numerical model adopted is based on the incrementaldeformation method as described by Au et al. [2] to evaluate thenonlinear response of partially prestressed concrete memberswith unbonded tendons. In addition, the present model cansimulate the post-peak behaviour of the member and takeinto account the strain reversal in non-prestressed steel andconcrete at the descending branches of the stress–strain curves.Analytical models of the stress–strain curves of constituentmaterials are used. An iteration procedure is implementedto satisfy compatibility and equilibrium at every step in thesolution process. The assumptions adopted are: (1) planesections remain plane after bending; (2) the constitutiverelations for prestressing tendons, non-prestressed steel andconcrete are known; (3) the friction between the concrete andprestressing tendons is neglected; and (4) the member hasadequate shear strength.

The stress–strain relationship for concrete in compressionas proposed by Attard and Setunge [3] has been shown to beapplicable to concrete strengths from 20 to 130 MPa, and isused in the study. The parameters to establish the equationinclude the modulus of elasticity of concrete Ec, in situ concrete

strength f ′c , strain at peak stress εco, and the stress fci and

strain εci at the inflection point on the descending branch ofthe stress–strain curve. The in situ concrete strength f ′

c canbe obtained from the cylinder strength or cube strength bysuitable conversion factors. The concrete stress σc is related tothe concrete strain εc by

σc

f ′c

=A(εc/εco) + B(εc/εco)

2

1 + (A − 2)(εc/εco) + (B + 1)(εc/εco)2 (2a)

in terms of parameters A and B. These parameters have beenobtained by Attard and Stewart [4] as follows.

(a) For the ascending branch of the stress–strain curve:

A = Ecεco/ f ′c; (2b)

B =(A − 1)2

0.55− 1. (2c)

(b) For the descending branch of the stress–strain curve:

A =fci (εci − εco)

2

εcoεci ( fc − fci ); (2d)

B = 0. (2e)

In the above equations, the values of Ec, εco, fci and εci maybe determined from

Ec = 4370( f ′c)

0.52; (2f)

εco = 4.11( f ′c)

0.75/Ec (2g)

fci/ f ′c = 1.41 − 0.17 ln( f ′

c); (2h)

εci/εc = 2.50 − 0.30 ln( f ′c) (2i)

where Ec and f ′c are in MPa and εco is dimensionless. The

stress–strain relationship for concrete in tension is assumedto be linear with a slope equal to the elastic modulus incompression at zero stress. The tensile strength of concrete istaken as 0.1 f ′

c . The contribution of concrete in tension aftercracking is neglected. The model of Bahn and Hsu [5] for theunloading stress–strain relationship of concrete is adopted:

σunlo

f ′c

= 0.95σc

f ′c

(εunlo/εco − εcp/εco

εc/εco − εcp/εco

)1+√

εcp/εco

; (3a)

εcp

εco= 0.30

(εc

εco

)2

(3b)

where σunlo and εunlo are respectively the stress and strain inconcrete on unloading; σc and εc are respectively the stress andstrain in concrete in Eq. (2a); εcp is the residual plastic strainin concrete corresponding to unloading from point (εc, σc); andεco is concrete strain at peak stress f ′

c . Fig. 2 shows typicalloading and unloading curves adopted in the study.

The non-prestressed steel is assumed to be perfectly elasto-plastic as shown in Fig. 3. To cater for strain reversal, thestress-path dependence of the stress–strain relation is taken intoaccount by assuming that the unloading path follows the initialelastic slope. When the steel strain εs increases, the steel stress

J.S. Du et al. / Engineering Structures 30 (2008) 13–21 15

Fig. 2. Stress–strain curve for concrete.

Fig. 3. Stress–strain relationship of non-prestressed steel allowing for stress-path dependence.

σs is given by

At elastic stage: σs = Esεs (4a)

After yielding: σs = fy (4b)

where Es is the modulus of elasticity and fy is the yieldstrength. On unloading after yielding, the steel stress σs canbe written in terms of the steel strain εs and the residual strainεsp as

σs = Es(εs − εsp) (5)

where the residual strain εsp can be evaluated from the laststrain increment as

εsp = εs − σs/Es . (6)

The stress–strain formula for prestressing steel proposedby Menegotto and Pinto [6] was shown by Naaman [7] to berealistic, and it is adopted here. The stress σps is related to thestrain εp by

σps = E pεp{Q + (1 − Q)/[1 + (E pεp/(K f py))N]1/N

} (7a)

Q = ( f pu − K f py)/(E pεpu − K f py) (7b)

as shown in Fig. 4 where E p is the modulus of elasticity ofprestressing steel; f py is the yield stress of prestressing steel;f pu and εpu are the ultimate stress and strain of prestressingsteel, respectively; and the empirical parameters N , K and Qare respectively 7.344, 1.0618 and 0.01174 for 7-wire strandsof Grade 270 with ultimate tensile strength of 1863 MPa.

Fig. 4. Stress–strain relationship of prestressing tendons.

Fig. 5. Arrangement of elements along the beam for analysis.

3.2. Numerical procedures

It is assumed that the symmetrically loaded beam also failssymmetrically with a plastic hinge formed at mid-span. For theanalysis, half of the beam is subdivided into m beam elements(k = 1, 2, . . . , m) as shown in Fig. 5. For convenience, thefirst element (k = 1) at the centre of the beam is taken asthe control element where the concrete strain at the top fibreis increased by increments to simulate the applied loading orimposed displacement. The length of the control element istaken as half of the plastic hinge length lp, while the lengthof the other elements is about 0.5dp where dp is the depth tocentroid of the prestressing tendon. Corley’s expression for theplastic hinge length lp as modified by Mattock [8] is adopted,namely,

lp = 0.5dp + 0.05Z (8)

where Z is the shear span or the distance between the point ofmaximum moment and point of contra-flexure.

For each value of concrete strain at the top fibre of thecontrol element, a three-level iteration procedure is carried outto satisfy the following criteria: (1) equilibrium of forces acrossthe depth of all beam elements; (2) equilibrium between theexternally applied load and the internal moment resistance ateach beam element; and (3) compatibility of the average strainand average elongation between the end anchorages of theunbonded prestressing tendons. The concrete strain at the topfibre of the control element is increased monotonically until themoment at which the control element has experienced its peakmoment and dropped to 85% of the peak value.

The computer program written can cope with variouscases including central point load, third-point loading anduniform loading as well as straight, harped and parabolictendon profiles. It can also conduct a ductility analysisof the corresponding bonded beam containing the samenumber of tendons as that of the unbonded one. It requires

16 J.S. Du et al. / Engineering Structures 30 (2008) 13–21

Fig. 6. Simplified load–deflection curve for unbonded partially prestressedconcrete beams.

the input of member and section dimensions, pattern ofloading, reinforcement areas, tendon profile, material strengthand stress–strain characteristics. The output includes thestress increment in tendons, equivalent yielding and ultimatecurvatures, ductility factor, mid-span deflection of the beam,etc. Unless otherwise stated, the span–depth ratio is the ratioof the span L to the effective depth dp at mid-span.

3.3. Verification of numerical results

The numerical results are verified by comparison of theload–deflection curves with some experimental results. Theload–deflection curves of most unbonded and bonded partiallyprestressed concrete beams clearly exhibit three stages asshown in Fig. 6, namely (a) elastic, (b) cracked-elastic and (c)plastic. The transition from the first stage to the second stageis caused by cracking, while the transition to the third stageis caused by yielding of the non-prestressed steel. In the workof Du and Tao [9] on simply supported unbonded and bondedpartially prestressed concrete beams, all the test beams were160 mm × 280 mm in cross section and were tested with third-point loading over a 4200 mm span. The span–depth ratio was19.1. The unbonded prestressed concrete beams were dividedinto three categories and each beam was designed for the non-prestressed steel to carry about 30%, 50% and 70% of the

total ultimate load. The reinforcement was characterized by thecombined reinforcement index (CRI) q0 at mid-span defined as

q0 =Ap f pe + As fy

bdp f ′c

(9)

where f pe is the effective stress in prestressed steel; fy isthe yield strength of non-prestressed tension steel; f ′

c is thecompressive strength of concrete; Ap is the cross-sectional areaof prestressing steel; As is the cross-sectional areas of non-prestressed tension steel; b is the width of compression faceof the beam; dp is the distance from extreme compression fibreto centroid of prestressed steel.

The 26 partially prestressed concrete beams were dividedinto four groups, namely Groups A, B, C and D, which wereunbonded except for those in Group D. The beams in GroupsA, B and C were classified according to the CRI q0, namelylow (q0 < 0.15), medium (0.15 < q0 < 0.25) and high(q0 > 0.25). The nine beams in Group A were reclassified intothree categories, each containing three beams corresponding tothe three different levels of q0 as stated above. Groups A and Bwere similar in that each beam in Group B was identical to onein Group A, except that the strengths of the prestressing steeland concrete in the former were higher. Group C consisted offour beams which were identical to Beams A-1, A-3, A-7 andA-9, except that cold stretched bars of higher strength were usedinstead of ordinary non-prestressed steel. The four beams inGroup D were largely duplicates of their counterparts in GroupA, except that those in Group D were bonded. Beam-D0 was anordinary reinforced concrete beam. Beams D-1 and D-3 werebonded partially prestressed concrete beams while Beam D-10was a fully prestressed beam. The load–deflection curves forBeams A-1, A-2 and A-3 in Group A and Beams D-1 and D-3 in Group D are plotted in Fig. 7(a) and (b), respectively. Itcan be seen that the numerical results agree quite well with theexperimental results.

In the study by Campbell and Chouinard [10], six partiallyprestressed concrete beams with unbonded tendons having asection of 160 mm × 280 mm and overall length of 3600 mmwere tested under third-point loading. The span length was3300 mm, and the span–depth ratio was 15. The study focused

(a) Unbonded beams. (b) Bonded beams.

Fig. 7. Comparison of numerical results with experimental results of Du and Tao [9].

J.S. Du et al. / Engineering Structures 30 (2008) 13–21 17

Table 1Comparison of experimental and calculated values of ultimate tendon stress f ps and ultimate flexural moment Mu of Campbell and Chouinard [10]

Beam no. Ultimate tendon stress f ps Ultimate flexural moment MuTested (MPa) Calculated (MPa) Error (%) Tested (kN m) Calculated (kN m) Error (%)

1 1476 1521 3.0 45.5 46.7 2.62 1467 1461 −0.4 63.3 58.5 −7.63 1381 1325 −4.1 81.1 71.8 −11.54 1348 1382 2.5 98.0 91.5 −6.65 1274 1315 3.2 105.5 102 −3.36 1269 1272 0.2 120.0 117 −2.5Mean 0.8 −4.8Standard deviation 2.8 4.9

on the influence of non-prestressed steel on the strength.Table 1 shows the experimental and computed results ofultimate tendon stress f ps and ultimate flexural moment Mu .Good agreement is observed for the six beams tested. Theaverage discrepancy of the computed values with respect to themeasured values of ultimate tendon stress is 0.8% while thestandard deviation of discrepancy is 2.8%. The correspondingdiscrepancies for ultimate flexural strength have an average of−4.8% and a standard deviation of 4.9%.

4. Parametric studies

Parametric studies were conducted on simply supportedrectangular beams as shown in Fig. 8. Emphasis is placed on theparameters that may influence the stress increase in prestressingtendons together with the curvature ductility factor at mid-span.They include a wide range of span–depth ratios (L/dp = 5–45),three different load applications (central point, third-point anduniform), three tendon profiles (straight, harped and parabolic),and numerous reinforcement and strength parameters. Thepartial prestressing ratio (PPR) and the compression steel indexγc defined as follows were adopted:

PPR =Ap f pe

Ap f pe + As fy(10)

γc =A′

s f ′y

Ap f pe + As fy(11)

where A′s and f ′

y are the cross-sectional areas and the yieldstrengths of non-prestressed compression steel, respectively.

4.1. Influence of concrete compressive strength

To study the influence of concrete compressive strength onthe curvature ductility factor of prestressed concrete memberswith unbonded tendons, a beam of span–depth ratio equal to 10,with straight tendon profile and loaded with third-point loadingis chosen. Fig. 9(a)–(d) illustrate the variation of curvatureductility factor with the CRI q0 for concrete compressivestrengths of 30, 60 and 90 MPa. It is generally observed thatat a given CRI q0, the curvature ductility factor µφ decreasesas the concrete compressive strength increases. However, thetrend is less significant for the higher range of CRI and concretecompressive strength, such as when the CRI is greater than 0.15and the concrete compressive strength is greater than 60 MPa.Taking as an example the case of PPR = 0.3 and f pe = 0.5 f puas shown in Fig. 9(a), at CRI of 0.05, the curvature ductilityfactor of the beam with concrete strength 60 MPa is about49% of that with concrete strength 30 MPa, while the curvatureductility factor of the beam with concrete strength 90 MPais about 66% of that with concrete strength 60 MPa. Whenthe CRI is increased to 0.15, the curvature ductility factor ofthe beam with concrete strength 60 MPa is about 54% of thatwith concrete strength 30 MPa, while the curvature ductilityfactor of the beam with concrete strength 90 MPa is about

Fig. 8. Prestressed concrete members studied: (a) tendon profiles, (b) section.

18 J.S. Du et al. / Engineering Structures 30 (2008) 13–21

Fig. 9. Influence of concrete compressive strength on ductility of unbonded members with straight tendons and L/dp = 10 under third-point loading: (a) PPR = 0.3,f pe = 0.5 f pu ; (b) PPR = 0.3, f pe = 0.6 f pu ; (c) PPR = 0.7, f pe = 0.6 f pu ; (d) PPR = 0.7, f pe = 0.7 f pu .

76% of that with concrete strength 60 MPa. It can therefore beconcluded that at high values of CRI, the concrete compressivestrength has limited influence on the curvature ductility factorof prestressed concrete members with unbonded tendons.

4.2. Effect of effective prestress and PPR on ductility

A beam of span–depth ratio equal to 10, with straight tendonprofile and loaded with third-point loading, is again chosen.Figs. 10 and 11 show the variation of curvature ductility factorwith CRI at different values of effective prestress f pe and PPR.It can be seen from Fig. 10 that the ductility factor increasesslightly at given CRI as the effective prestress f pe increases.However, when the CRI is above 0.15, the variation of ductilityfactor with the change of effective prestress f pe is minimal.Fig. 11 demonstrates that the ductility factor decreases with theincrease of PPR at given CRI. It is generally observed that thehigher the CRI, the smaller the influence of PPR.

4.3. Influence of span–depth ratio, type of loading and tendonprofiles on ductility

To illustrate the influence of member span–depth ratio onthe curvature ductility factor, members with a wide rangeof span–depth ratios are studied. While keeping the depthto straight prestressing tendons dp constant, the span lengthis varied to give span–depth ratios in the range of 5–45. Inaddition, the concrete compressive strength is taken as 60 MPa,with PPR = 0.7 and f pe = 0.7 f pu . Fig. 12(a)–(c) show thevariation of curvature ductility factor with span–depth ratio

under three different loading types. It can be observed fromFig. 12(a) that under third-point loading, for a given CRI,the span–depth ratio has almost no influence on the curvatureductility factor. On the other hand, under central point loadingand uniform loading as shown in Fig. 12(b) and (c) respectively,at the lower CRI value of q0 = 0.10, the curvature ductilityfactor increases by about 27% when the span–depth ratioincreases from 5 to 15, and then remains constant thereafter.However, at higher CRI values of q0 = 0.20 and q0 = 0.30, thespan–depth ratio has no significant influence on ductility.

Fig. 12(d) shows the effect of three different loading typeson the ductility factor for a member with straight tendon profileof span–depth ratio 15, f ′

c = 60 MPa, PPR = 0.7 andf pe = 0.7 f pu . Everything else being equal, at a given CRI q0,the curvature ductility factor of a prestressed concrete memberwith unbonded tendons under central point loading is higherthan that under third-point loading by up to 30%, while thecorresponding value of ductility factor for uniform loading isslightly lower than that for central point loading.

Three tendon profiles, namely straight, harped and parabolictendons as shown in Fig. 8(a), are adopted to study the effectof different tendon profiles on the curvature ductility factor.The results indicate that, at given CRI and load patterns, thethree tendon profiles considered have very little influence onthe curvature ductility factors.

4.4. Influence of compression steel content on ductility

To analyze the influence of compression steel content onductility, three compression steel indices, namely γc = 0.0,

J.S. Du et al. / Engineering Structures 30 (2008) 13–21 19

Fig. 10. Effect of f pe on ductility of unbonded members with straight tendons and L/dp = 10 under third-point loading: (a) PPR = 0.3, f ′c = 60 MPa; (b)

PPR = 0.3, f ′c = 90 MPa.

Fig. 11. Effect of PPR on ductility of unbonded members with straight tendons and L/dp = 10 under third-point loading: (a) f pe = 0.5 f pu , f ′c = 30 MPa; (b)

f pe = 0.5 f pu , f ′c = 90 MPa.

Fig. 12. Effect of span–depth ratio and loading type on ductility of unbonded members with straight tendons: (a) third-point loading; (b) central point load; (c)uniform loading; (d) span–depth ratio = 15.

20 J.S. Du et al. / Engineering Structures 30 (2008) 13–21

Fig. 13. Influence of compression steel content on ductility of unbonded members with straight tendons under third-point loading: (a) PPR = 0.3; (b) PPR = 0.7.

0.15 and 0.25, are used. Fig. 13(a) and (b) present the changeof curvature ductility factor with CRI for the three compressionsteel indices. Neither a clear nor a consistent trend can beobserved from Fig. 13 when γc varies from 0.0 to 0.25. It isobserved that when CRI is above 0.15, the compression steelindex has almost no influence on the ductility factor. However,a decrease in ductility factor of up to 28% can be observed forthe CRI value of q0 = 0.05 when γc increases from zero to0.25.

5. Effect of bonding on ductility of members

The effect of bonding on ductility of prestressed concretemembers with straight tendon profiles under third-point loadingis evaluated. For a given value of CRI q0, the correspondingcurvature ductility factors for unbonded and bonded members

are analyzed and compared. Some of the typical results areplotted in Fig. 14(a)–(d). It is found that, everything else beingequal, the curvature ductility factors of both the unbonded andbonded members decrease with the increase of CRI q0. It isgenerally observed that when CRI q0 is between 0.20 and 0.30,the ductility factor of the unbonded member is higher than thatof the bonded one by about 10%–15%. For CRI q0 between0.15 and 0.20, the ductility factor of the unbonded member isclose to that of the bonded one. For CRI q0 below 0.15, theductility factor of the unbonded member is usually lower thanthat of the bonded one by about 10%–15%. In the case of CRIq0 = 0.05, the ductility factor of the unbonded member cansometimes be lower than that of the bonded one by up to 30%.However, at such a low level of CRI, the ductility factor is wellabove what is normally considered adequate.

Fig. 14. Effect of bonding on ductility of members with straight tendons under third-point loading: (a) PPR = 0.3; (b) PPR = 0.7; (c) γc = 0.15; (d) γc = 0.25.

J.S. Du et al. / Engineering Structures 30 (2008) 13–21 21

6. Discussions and conclusions

Based on the parametric studies carried out using thenumerical method developed for full-range analysis, thefollowing conclusions can be drawn concerning the ductility ofprestressed concrete beams with unbonded tendons:

(1) As the tendon stress f ps at nominal moment resistanceof unbonded members is member-dependent, it cannot bedetermined directly from the analysis of a cross sectionby strain compatibility. The CRI q0, which makes use ofthe effective prestress f pe instead, is more convenient fordescribing the flexural ductility of unbonded members. Itis observed that the curvature ductility factor of unbondedmembers decreases with the increase of CRI q0.

(2) For equal values of CRI q0, the curvature ductilityfactor of an unbonded member decreases as the concretecompressive strength increases. However, the trend isless notable for the higher range of CRI and concretecompressive strength. At a given value of CRI q0, thecurvature ductility factor of an unbonded member decreaseswith the increase of PPR and effective prestress f pe. Itis observed that the higher the CRI q0 is, the smaller isthe influence of PPR, the effective prestress f pe and thecompression steel content.

(3) Under third-point loading and at a given CRI q0, thespan–depth ratio has almost no effect on the curvatureductility factor of an unbonded member. On the other hand,under central point and uniform loading, the span–depthratio has some effect on ductility only at low CRI and lowspan–depth ratio.

(4) Everything else being equal, at a given CRI q0, thecurvature ductility factor of an unbonded member undercentral point load is higher than that under third-pointloading by up to 30%, while the corresponding value ofductility factor under uniform loading is slightly lower thanthat under a central point load.

(5) It is generally observed that when the CRI q0 is between0.15 and 0.20, the ductility factor of an unbonded memberis close to that of the bonded one. Above this range of CRI

q0, the ductility factor of an unbonded member is higher;while below this range, the ductility factor of an unbondedmember is lower but at such a low level of CRI the ductilityfactor is well above what is normally considered adequate.

(6) The method presented enables the designer to assess theductility of unbonded members reasonably accurately.

Acknowledgements

The work described in this paper has been partiallysupported by the Internal Award for CAS Membership of TheUniversity of Hong Kong and the Research Grants Council ofthe Hong Kong Special Administrative Region, China (RGCproject no. HKU 7101/04E).

References

[1] Park R. Ductility evaluation from laboratory and analytical testing. In:Proceedings of the 9th world conference on earthquake engineering, vol.VIII. 1988. p. 605–16.

[2] Au FTK, Du JS, Cheung YK. Service load analysis of unbonded partiallyprestressed concrete members. Magazine of Concrete Research 2005;57(4):199–209.

[3] Attard MM, Setunge S. The stress–strain relationship of confined andunconfined concrete. ACI Materials Journal 1996;93(5):433–42.

[4] Attard MM, Stewart MG. A two parameter stress block for high-strengthconcrete. ACI Structural Journal 1998;95(3):305–17.

[5] Bahn BY, Hsu CT. Stress–strain behavior of concrete under cyclicloading. ACI Materials Journal 1998;95(2):178–93.

[6] Menegotto M, Pinto PE. Method of analysis for cyclically loadedRC plane frames, including changes in geometry and non-elasticbehavior or elements under combined normal force and bending. In:IABSE preliminary report for symposium on resistance and ultimatedeformability of structures acted on by well-defined repeated loads. 1973.p. 15–22.

[7] Naaman AE. Partially prestressed concrete: Review and recommenda-tions. PCI Journal 1985;30(6):31–71.

[8] Mattock AH. Discussion of ‘rotational capacity of concrete beams’ byCorley, W. Journal of the Structural Division ASCE 1967;93(2):519–22.

[9] Du GC, Tao XK. Ultimate stress of unbonded tendons in partiallyprestressed concrete beams. PCI Journal 1985;30(6):72–91.

[10] Campbell TI, Chouinard KL. Influence of non-prestressed reinforcementon the strength of unbonded partially prestressed concrete members. ACIStructural Journal 1991;88(5):546–51.