AbstractIn recent years, external prestressing has become a primary methodfor strengthening existing concrete structures andhas beenincreasingly usedin the construction of newly erectedones, particularly segmental bridges. Analysis of externally prestressedmembers is more difficult than that of members with internal bonded tendons. 30574
This is because external tendons are unbonded tothe concrete and the stress in such tendons depends on the deformations of the whole member and is assumed uniform at all sec-tions. In this paper, a simple analytical model is outlined for predicting the flexural behavior of reinforced concrete members withexternal tendons under service loads. The analysis accounts for various factors that affect the flexural behavior including eccen-tricity variations of external tendons and span-to-depth ratios. Good agreement has been found between the predicted results andthose in the literature.# 2004 Publishedby Elsevier Ltd.Keywords: External tendons; Span-to-depth ratio; Eccentricity variations; Flexure 1. IntroductionThe use of external prestressing technique has beengrowing rapidly in recent years. Analysis and design ofstructures with external tendons is conceptually differ-ent from that of structures with internal tendons and isstill not fully understood. The main difference in beha-vior between members with internal tendons and thosewith external tendons lies in the deflected shape of thebeam andthe tendons andin the strain incompatibilitybetween the concrete andexternal tendons. This makesthe deformation and hence the stress in the externaltendon member dependent rather than section depen-dent. Many investigations have been carried out onexternally prestressedmembers. Muller andGauthier[1] developed a finite element computer program forthe ultimate response of simply supportedandcontinu-ous beams with external tendons. Their model requiresinformation regarding the moment versus curvature ormoment versus rotation relationship. Alkkairi andNaaman [2] have proposeda simplifiedmethodology tocompute the stress in unbonded internal/external steel tendons in the elastic range as well as the ultimateresistance. The methodology introduces strainreduction coefficients to convert the analysis of a beamwith unbonded tendons to analysis of a beam withbonded tendons, hence allowing a conventional sec-tional (fictitious section) analysis to be performed. Thismethodology requires the calculations of differentstrain reduction coefficients for different spans, loads,andtendon profiles. Wu andLu [3] have proposedamodel for non-linear analysis of externally prestressedbeams andcapable of simulating the slip of the steeltendons at the deviators but under-predicts the nom-inal strength of the beams. Their study did not accountfor the effects of span-to-depth ratios on the flexuralbehavior of the beam.This study is one of the continuing efforts to betterunderstand the flexural behavior of beams with exter-nal prestressing tendons.2. Analytical modelIn this model, a reinforced concrete member withexternal tendons is modeled as an assemblage of straightbeam elements simulating the reinforcedconcrete member, an assemblage of truss elements simulatingthe external tendons, and rigid arms as shown in Fig. 1.A computer program was developed for the presentinvestigation to verify the analysis.2.1. Strains and stressesIf a normal force increment DN anda momentincrement DM are appliedat an arbitrary referencepoint, an immediate change in the strains and stresseswill occur. Two parameters Deo (strain) and Dr (stress)are used to define the strain and stress distributions:De ¼ Deo þ Dwy ð1ÞDr ¼ EðDeo þ DwyÞð2ÞThese two parameters are obtainedfrom equilibriumrequirements as follows:DN ¼ðDr dA ¼ EDeoðdA þ EDwðy dA ð3ÞDM ¼ðDry dA ¼ EDeoðy dA þ EDwðy2dA ð4Þwhich can be written in the following equations:DN ¼ ADro þ BDc ð5ÞDM ¼ BDro þ IDc ð6Þwhere w ð¼ de=dyÞ, curvature or slope of the straindiagram; c ð¼ dr=dy ¼ EwÞ, slope of the stress dia- gram; Deo and Dro ð¼ EDeoÞ, strain andstressincrement, respectively, at a reference point;A; B; and I, cross-section area, its first, andsecondmoment about the reference axis of the transformedsection, respectively.Solving Eqs. (5) and(6) yields the strain increment atthe reference point andthe curvature, respectively, asfollows [4]:Deo ¼ IDN   BDMEðAI   B2Þ; Dw ¼  BDN þ ADMEðAI   B2Þð7Þor, when put in a matrix form:DeoDw  ¼ 1EðAI   B2ÞI  B BA  DNDM  ð8Þ2.2. Stiffness matrix of beam elementIn this study, flexibility matrix is evaluated first atone endof the member, ½f  3    3, using the unit-loadthe-orem andfixing the other:f ½ ¼lEA 000l33EIl22EI0l22EIlEI26 6 6 6 6 437 7 7 7 7 5ð9ÞThe inversion of ½f  3    3will yieldthe stiffness matrix½S  3    3 ð½f   1ð3    3ÞÞ. The forces at the fixedendareobtainedby equilibrium andthus the stiffness matrixfor the six degrees of freedom, [S ], is obtainedas fol-lows:½S  ¼ H ½ Tf ½  1H ½  ð10Þ½H ¼100  10 00100  1 l0010 0  12435 ð11Þwhere l is the length of the member.Taking the reference point at the centroidof thecross-section, the stiffness matrix of the beam elementwill be [5]: 2.3. Stiffness matrix of external tendonThe external tendon is treated as an assemblage oftruss elements. The conventional local stiffness matrixfor a truss element with two degrees of freedom isSt ½ ¼ EAl1  1 11  ð13ÞThe truss element contributes to the stiffness of thebeam element in which it is defined as shown in Fig. 1.This contribution can be evaluatedby transforming thelocal stiffness matrix of the truss element [St] to thelocal stiffness of the beam element. This transformationis done by geometrical relations of displacements asshown in Fig. 1:St ½ transf¼ T ½ TSt ½  T ½  ð14Þwhere [T] is as follows:T ½ ¼ cosðaÞ sinðaÞ e1cosðaÞ 00 00 0 0 cosðaÞ sinðaÞ e2cosðaÞ  ð15Þwhere a, slope of the external tendon, refers to Fig. 1;e1 and e2, tendon eccentricities at the end of the con-crete members, refer to Fig. 1. Therefore, the transformedstiffness matrix of thetruss element ½St transfis given as follows:St ½ transf¼C2 SC  e1C2 C2SC e2C2 SC S2e1SC SC  S2 e2SC e1C2e1SC e21C2e1C2 e1SC  e1e2C2 C2SC e1C2C2 SC  e2C2SC  S2 e1SC  SC S2e2SCe2C2 e2SC  e1e2C2 e2C2e2SC e22C226 6 6 6 6 6 437 7 7 7 7 7 5ð16Þwhere C¼cosðaÞ; S¼sinðaÞ.The transformedstiffness matrix of the truss element½St  transfis then to be added to the beam local stiffnessmatrix [S ].3. AnalysisThe beam shown in Fig. 2 was usedin the analysis.The span andthe cross-section of the beam, the tendonprofile, andall the dimensions are shown. A live loadof 14 kN/m is appliedon the beam. The prestressingforce usedis 2900 kN. The moduli of elasticity of theconcrete, normal steel, andprestressing steel are 25,200, and186 GPa, respectively. 3.1. Eccentricity variationsThe deflection of the external tendons is restricted tothe deflection of the beam at the deviator locations,and is different from the deflection at any otherlocation. The eccentricity variations were foundto beaffectedby the span-to-depth ratios as shown in Fig. 3.In Fig. 3, the ratio of the eccentricity at the initial tothe corresponding eccentricity after loading for thesame section, e(i)/e(f), was plottedversus the ratio ofdistance between the support and the section to thebeam span, x/S, for different span-to-depth ratios, S/d.It can be seen that the ratio e(i)/e(f) is equal to unityover the support andat the deviators. At all other loca-tions the eccentricities after loading become smallerthan the initial eccentricities.3.2. Span-to-depth ratiosFig. 3 shows that the ratio e(i)/e(f) decreases with theincrease in span-to-depth ratio. It can be noticed alsoin Fig. 3 that the change in eccentricity of beams withspan-to-depth ratios less than 20 can be safely neglec-ted. The change in eccentricity for beams with span-to-depth ratios higher than 20 is significant and has to betaken into consideration. These findings agree well withAlkkairi andNaaman [2]. The reduction in the eccen-tricity leads to a reduction in the flexural rigidity of thebeam.3.3. Compressive and tensile stresses in concreteIn Fig. 4, the distribution of the top and bottomfiber stresses are plottedagainst one-half the span ofthe beam. It can be seen from Fig. 4 that sudden change in the stresses occurs at the location of externaltendon deviator. It can be noticed also that the largestchange in top stress occurs at the locations of thedeviators. This is due to the loss in prestressing takingplace between the initial loading and after loading. Itcan be seen also from Fig. 4 that there is a significantincrease in the bottom stresses. This may leadto crack-ing of the beam before the application of the live loads.This is due to the non-existence of bond between exter-nal tendons.3.4. Flexural behaviorFig. 5 shows the moment–deflection relation. It canbe seen that a reduction in beam flexural stiffnessoccurs due to the loss of prestressing force. It can alsobe seen that beams with high span-to-depth ratios exhi- bit large deflections associated with relatively smallincrease in the bending moments, which agrees withHarajli [6]. This is due to a reduction in the flexuralstiffness which is a result of a reduced prestressing forceandcontinuous change in eccentricity between theexternal cables andthe beam at the section ofmaximum moment.4. ConclusionsA simple model of an assemblage of straight beamandtruss elements is usedfor modeling a concretemember with external tendons. The presented modeldefines all the elements in prestressed concrete memberandit can be usedfor a generalizedcase of reinforce-ment and any external tendon profiles. The model eval-uates the flexural behavior of beams with unbondedexternal tendons. In the current analytical study, thestress in the external tendons is found to vary linearlywith the appliedloadregardless of the cracks that maydevelop in the beam. Also the strain in external ten-dons varies linearly with the deflection of the beam atthe deviator locations. High span-to-depth ratios resultin a significant decrease in the eccentricity. High span-to-depth ratios tend to decrease the eccentricitybetween the external tendons and the beam. This leadsto large deflections and deformations and thereforeleads to a reduced rigidity and lower flexural capacityeven before the application of live loads.References[1] Muller J, Gauthier Y. Ultimate behavior of precast segmentalbox-girders with external tendons. In: Naaman AE, Breen JE,editors. External Prestressing in Bridges. Proceedings of the Inter-national Symposium. ACI SP 120-17. Detroit (MI, USA):American Concrete Institute (ACI); 1989, p. 355–73.[2] Alkkairi FM, Naaman AE. Analysis of beams prestressedwithunbonded internal or external tendons. Journal of StructuralEngineering 1993;119(9).[3] Wu Xiao-Han, Lu Xilin. Tendon model for nonlinear analysis ofexternally prestressedconcrete structures. Journal of StructuralEngineering 2003;129(1):96–104.[4] El-Ariss B, Elbadry M. Serviceability and strength of externallyprestressedconcrete structures. Structural Specialty. CSCE 1996Annual Conference, May 29–June 1, Edmonton, Alta., Canada.1996.[5] Ghali A, Neville AM. Structural Analysis: a UnifiedClassical andMatrix Approach, 3rd ed. London, New York: Chapman andHall; 1989.[6] Harajli MH. Strengthening of concrete beams by external pre-stressing. PCI Journal 1993;38(6). 体外预应力钢筋混凝土梁的刚度英文文献:http://www.youerw.com/fanyi/lunwen_26357.html