Determination of Design Moments in Bridges Constructed by Balanced Cantilever Method

Engineering Structures 24 (2002) 639–648www.elsevier.com/locate/engstruct

Determination of design moments in bridgesconstructed by balanced cantilever method

H.-G. Kwak *, J.-K. SonDepartment of Civil Engineering, Korea Advanced Institute of Science and Technology,

373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

Received 26 March 2001; received in revised form 21 August 2001; accepted 30 October 2001

Abstract

This paper introduces an equation to calculate the design moments in reinforced concrete (RC) bridges constructed by the balancedcantilever method. Through time-dependent analyses of RC bridges, considering the construction sequence and creep deformationof concrete, structural responses related to the member forces are reviewed. On the basis of the compatibility condition at everyconstruction stage, a basic equation which can describe the moment variation with time in the balanced cantilever construction isderived. It is then extended to take into account the moment variation according to changes in construction steps. By using theintroduced relation, the design moment and its variation over time can easily be obtained with only the elastic analysis results, andwithout additional time-dependent analyses considering the construction sequences. In addition, the design moments determined bythe introduced equation are compared with the results from a rigorous numerical analysis with the objective of establishing therelative efficiencies of the introduced equation. Finally, a more reasonable guideline for the determination of design moments isproposed. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Balanced cantilever method; RC bridges; Construction sequence; Creep; Design moment

1. Introduction

In accordance with the development of industrialsociety and global economic expansion, the constructionof long-span bridges has increased. Moreover, the con-struction methods have undergone refinement, and theyhave been further developed to cover many specialcases, such as progressive construction of cantileverbridges and span-by-span construction of simply sup-ported or continuous spans. Currently, among these con-struction methods, the balanced cantilever constructionof reinforced concrete box-girder bridges has beenrecognized as one of the most efficient methods of build-ing bridges without the need for falsework. This methodhas great advantages over other kinds of construction,particularly in urban areas where temporary shoringwould disrupt traffic and service below, in deep gorges,

* Corresponding author. Tel.:+82-42-869-3621; fax:+82-42-869-3610.

E-mail address: [email protected] (H.-G. Kwak).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (01)00128-6

and over waterways where falsework would not only beexpensive but also a hazard.

However, the design and analysis of bridges con-structed by the balanced cantilever method (FCM)require the consideration of the internal moment redistri-bution which takes place over the service life of a struc-ture because of the time-dependent deformation of con-crete and changes in the structural system repeatedduring construction. This means that the analysis ofbridges considering the construction sequence must beperformed to preserve the safety and serviceability of thebridge. All the related bridge design codes [1,2] havealso mentioned the consideration of the internal momentredistribution due to creep and shrinkage of concretewhen the structural system is changed during construc-tion.

Several studies have dealt with the general topics ofdesign and analysis of segmentally erected bridges,while a few studies have been directed toward the analy-sis of the deflection and internal moment redistributionin segmental bridges [3–5]. Alfred and Nicholas [3]investigated the time-dependent deformation of cantil-

640 H.-G. Kwak, J.-K. Son / Engineering Structures 24 (2002) 639–648

ever construction bridges both before and after closure,and Cruz et al. [6] introduced a nonlinear analysismethod for the calculation of the ultimate strength ofbridges. Articles on the design, analysis and constructionof segmental bridges have been published by manyresearchers, and detailed comparisons have been madebetween analytical results and responses measured inactual structures [7,8].

Moreover, development of sophisticated computerprograms for the analysis of segmental bridges consider-ing the time-dependent deformation of concrete has beenfollowed [9]. Most analysis programs, however, havesome limitations in wide use because of complexities inpractical applications. Consequently, a simple formulafor estimating the internal moment redistribution due tocreep and shrinkage of concrete, which is appropriate foruse by a design engineer in the primary design ofbridges, has been continuously required. Trost and Wolff[5] introduced a simple formula which can simulateinternal moment redistribution with a superposition ofthe elastic moments occurring at each construction step.A similar approach has been presented by the PrestressedConcrete Institute (PCI) and the Post-Tensioning Insti-tute (PTI) on the basis of the force equilibrium and therotation compatibility at the connecting point [10]; how-ever, these formulas do not adequately address thechanging structural system because of several simplify-ing assumptions adopted.

In this paper, a simple, but effective, formula is intro-duced to calculate the internal moment redistribution insegmental bridges after completion of construction. Withpreviously developed computer programs [8,12,12],many parametric studies for bridges erected by the bal-anced cantilever method are conducted, and correlationstudies between the numerical results obtained withthose obtained by the introduced formula are includedto verify the applicability of the formula. Finally, reason-able guidelines to determine the internal designmoments, which are essential in selecting a proper initialsection, are proposed.

2. Construction sequence analysis

Every nonlinear analysis algorithm consists of fourbasic steps: the formulation of the current stiffnessmatrix, the solution of the equilibrium equations for thedisplacement increments, the stress determination of allelements in the model, and the convergence check. Pre-vious papers [8,11,12] presented an analytical model topredict the time-dependent behavior of bridge structures.Experimental verification and correlation studiesbetween analytical and field testing results were conduc-ted to verify the efficiency of the proposed numericalmodel. The rigorous time-dependent analyses in thispaper are performed with the analytical model intro-

duced. Details to the analytical model can be found inprevious papers [8,11,12].

Balanced cantilever construction is the term used forwhen a phased construction of a bridge superstructurestarts from previously constructed piers cantilevering outto both sides. Each cantilevered part of the superstruc-ture is tied to a previous one by concreting a key segmentand post-tensioning tendons. It is thus incorporated intothe permanent continuous structure; consequently theinternal moment is continuously changed according tothe construction sequence and the changing structuralsystem. To review the structural response due to thechange in the construction sequence, three differentcases of FCM 1, FCM 2 and FCM 3, shown in Fig. 1,are selected in this paper.

For the time-dependent analysis of bridges consider-ing the construction sequence, a five-span continuousbridge is selected as an example structure. This bridgehas a total length of 150 m with an equal span lengthof 30 m, and maintains a prismatic box-girder sectionalong the span length. The assumed material and sec-tional properties are taken from a real bridge and aresummarized in Table 1. The creep deformation of con-crete is considered on the basis of the ACI creep withan ultimate creep coefficient of f�

cr=2.35 [13].As shown in Fig. 1, the time interval between each

construction step is assumed to be 50 days. FCM 1 isdesigned to describe the construction sequence in whichconstruction of all the cantilever parts of the superstruc-ture is finished first at the reference time t=0 day. Thecontinuity of the far end spans and center span followsat t=50 days, and then the construction of the superstruc-ture is finally finished at t=100 days by concreting thekey segments at the midspans of the second and fourthspans. FCM 2 describes the continuity process from thefar end spans to the center span, and FCM 3 the step-by-step continuity of the proceeding spans from a farend span. The corresponding bending moments at typicalconstruction steps are shown in Figs. 2–4, where TS(total structure) means that all the spans are constructedat once at the reference time t=0 day.

After construction of each cantilever part, the negativemoment at each pier reaches M=wl2/8=1160 (tonFm)(l=30 m), and this value is maintained until the structuralsystem changes by the connection of an adjacent span.The connection of an adjacent span, however, causes anelastic moment redistribution because the structural sys-tem moves from the cantilevered state to the over-hanging simply supported structure (see Fig. 1a). Never-theless, there is no internal moment redistribution bycreep deformation of concrete in a span if the structuralsystem maintains the statically determinate structure. Asshown in Fig. 1, the statically indeterminate structurebegins at t=100 days in all the structural systems (FCM1–3). Therefore, it is expected that the dead load bending

641H.-G. Kwak, J.-K. Son / Engineering Structures 24 (2002) 639–648

Fig. 1. Construction sequences in balanced cantilever bridges: (a) FCM 1; (b) FCM 2; (c) FCM 3.

Table 1Material and sectional properties used in application

AC rsc=rst WD f�c fsy ES

4.5 m2 0.62% 10.3 t/m 400 kg/cm2 4000 kg/cm2 2.1×106 kg/cm2

moments in the structures start the time-dependentmoment redistribution after t=100 days.

Comparing the numerical results obtained in Figs. 2–4, the following can be inferred: (1) the time-dependentmoment redistribution causes a reduction of negativemoments near the supports and an increase of positivemoments at the points of closure at the midspans; (2)the final moment at an arbitrary time t after completingthe construction converges to a value within the regionbounded by two moment envelopes for the final stati-

cally determinate stage at t=100 days and for the initiallycompleted five-span continuous structure (TS in Figs. 2–4); and (3) the final moments in the structure depend onthe order that the joints are closed in the structures,which means that the magnitude of the moment redistri-bution due to concrete creep may depend on the con-struction sequence, even in balanced cantilever bridges.

Under dead load as originally built, elastic displace-ment and rotation at the cantilever tips occur. If the mid-span is not closed, these deformations increase over time


Fig. 2. Moment redistribution in FCM 1.



due to concrete creep without any increase in the internalmoment. On the other hand, as the central joints areclosed, the rotations at the cantilever tips are restrainedwhile introducing the restraint moments. Moreover, thisrestraint moment causes a time-dependent shift or redis-tribution of the internal force distribution in a span. Ifthe closure of the central joints is made at the referencetime t=0 day, then the final moments Mt will convergewith the elastic moment of the total structure (TS in Figs.2–4). However, the example structure maintains thestatically determinate structure which does not causeinternal moment redistribution until t=100 days, so thatonly the creep deformation after t=100 days, which is arelatively small quantity of time, affects the time-depen-dent redistribution of the internal moment. Therefore, themoment distribution at time t represents a differencefrom that of the total structure. On particular, as shownin Figs. 2–4, the difference is relatively large at theinternal spans. This means that the moment redistri-bution caused in proportion to the elastic moment differ-ence between the statically determinate state and thefive-span continuous structure will be concentrated at theinternal spans. Figure 5, which represents the creepmoment distribution of the FCM 1 bridge, shows thatthe creep moments at the center span are about 3.5 timeslarger for the negative moment and about 7.0 timeslarger for the positive moment than those of the endspans.

Figure 6 shows the final moment distribution of theexample structure constructed by FCM 1, FCM 2, andFCM 3 at t=100 years. As this figure shows, the differ-ence in construction steps does not have a great influenceon the final moment distributions, but there is remark-able difference in the final moments between the initiallycompleted continuous bridge (TS in Figs. 2–4 and 6)and the balanced cantilever bridges. Balanced cantileverbridges represent relatively smaller values for the posi-tive moments and larger values for the negative moments

Fig. 5. Creep moment distribution of FCM 1 bridge.


Fig. 6. Internal moment distribution at t=100 years.

than those of a five-span continuous structure (see Figs.2–4 and 6). This difference is induced from no contri-bution of the creep deformation of concrete up to t=100days at which the structural system is changed to thestatically indeterminate state. From the results obtainedfor the time-dependent behavior of balanced cantileverbridges, it can be concluded that the prediction of moreexact positive and negative design moments requires theuse of sophisticated time-dependent analysis programs[8,9,11,14], which can consider the moment variationaccording to the construction sequence. To be familiarwith those programs in practice, however, is time-con-suming and involves many restrictions caused by com-plexity and difficulty in use because the adopted algor-ithms, theoretical backgrounds and the styles of inputfiles are different from each other. Accordingly, theintroduction of a simple but effective relation, which canestimate design moments on the basis of elastic analysisresults without any time-dependent analysis, is in greatdemand in the preliminary design stage of balanced can-tilever bridges.

3. Determination of design moments

3.1. Calculation of creep moment

Unlike temporary loads such as live loads, impactloads and seismic loads, permanent loads such as thedead load and prestressing force are strongly related tothe long-term behavior of a concrete structure, so thatthese are classified by the load which governs the time-dependent behavior of a structure. Of these loads, thedead load includes the self-weight continuously actingon a structure during construction. Thus the moment anddeflection variations arising from changes in the struc-tural system are heavily influenced by the dead load. Thedesign moments of a structure can finally be calculated

by the linear combination of the factored dead and liveload moments. Since the dead load moment depends onthe construction method because of the creep defor-mation of concrete, determination of the dead loadmoment through time-dependent analysis consideringthe construction sequence must be accomplished toobtain an exact design moment.

On the other hand, post-tensioning tendons (cantilevertendons) may be installed to connect each segment dur-ing construction, and the prestressing forces introducedwill also be redistributed from the cantilevered structuralsystem to the completed structural system due to con-crete creep and the relaxation of tendons. However,unlike the dead load from the self-weight of a structureand the continuity tendons installed after completion ofconstruction, the cantilever tendons have a minor effecton the internal moment redistribution, which is directlyrelated to the construction sequence [7]. Thus the influ-ence by cantilever tendons has been excluded in thispaper while determining the dead load moment consider-ing the construction sequence.

The time-dependent behavior of a balanced cantileverbridge can be described using a double cantilever withan open joint at the point B, as in Fig. 7. When theuniformly distributed load of q is applied on the struc-ture, the elastic deflection of d=ql4/8EI and the rotationangle of a=ql3/6EI occur at the ends of the cantilevers(see Fig. 7b), where l and EI refer to the length of thecantilever and the bending stiffness, respectively. If the

Fig. 7. Deformation of cantilevers before and after closure: (a) con-figuration of cantilever; (b) elastic deformations in a cantilever; (c)restraint moment Mt after closure.


joint remains open, then the deflection at time t willincrease to d·(1+ft) and the rotation angle to a·(1+ft),where ft is the creep factor at time t. However, if thejoint at the point B is closed after application of the load,an increase in the rotation angle a·ft is restrained, andthis restraint will develop the moment Mt, as shown inFig. 7c. The moment Mt, if acting in the cantilever,causes the elastic rotation at the point B, defined asb=Mtl/EI, and also accompanies the creep deformation.Since the creep factor increases by dft during a timeinterval dt, the variations in the angles of rotation willbe a·dft and db (the elastic deformation) +b·dft (thecreep deformation) for a and b, respectively.

From these relations and the fact that there is no netincrease in discontinuity after the joint is closed, thecompatibility condition for the angular deformation(a·dft=db+b·dft) can be constructed. The integration ofthis relation with respect to ft gives the restraint momentMt [10]:

Mt�ql2(1−e−ft)

6�qL2

(1−e−ft)

24(1)

where ft means the creep factor at time t, and L=2l.From Eq. (1), it can be found that for a large value

of ft, the restraint moment converges to Mt=qL2/24,which is the same moment that would have beenobtained if the joint at the point B had been closed beforethe load q was applied. This illustrates the fact thatmoment redistribution due to concrete creep following achange in the structural system tends to approach themoment distribution that relates to the structural systemobtained after the change.

Referring to Fig. 8, which shows the moment distri-bution over time, the following general relationship maybe stated [10]:

Mcr�MIII�MI�(MII�MI)(1�e−ft) (2)

where Mcr=the creep moment resulting from change ofstructural system, MI=the moment due to loads before achange of structural system, MII=the moment due to the

Fig. 8. Moment distribution over time.

same loads applied on the changed structural system, andMIII=the restraint moment Mt.

The derivation of Eq. (2) is possible under the basicassumption that the creep deformation of concrete startsfrom the reference time, t=0 day. If it is assumed thatthe joint is closed after a certain time, t=C days, whilemaintaining the same assumptions adopted in the deri-vation of Eq. (2), then the structure can be analyzed bymeans of the rate-of-creep method (RCM) [15], and thecreep moments obtained in Fig. 8 can be represented bythe following expression [16]:

Mcr�(MII�MI)(1�e−(ft−fC)) (3)

Namely, in balanced cantilever bridges, the restraintmoment grows continuously from the time at which thestructural system is changed (t=C days), and its magni-tude is proportional to (1�e−(ft−fC)) [10,15,16].

Generally, construction of a multispan continuousbridge starts at one end and proceeds continuously to theother end. Therefore, change in the structural system isrepeated whenever each cantilever part is tied byconcreting a key segment at the midspan. Moreover, theinfluence by the newly connected span will be deliveredinto the previously connected spans so that there aresome limitations in direct applications of Eq. (3) to cal-culate the restraint moment at each span because of themany different connecting times of t=C days. To solvethis problem and for a sufficiently exact calculation ofthe final time-dependent moments, Trost and Wolff [5]proposed a relation on the basis of the combination ofelastic moments (SMS,i; equivalent to MI in Eq. (3))occurred at each construction step (see Fig. 9), and themoment obtained by assuming that the entire structure

Fig. 9. Combination of MS,i.


is constructed at the same point in time (ME; equivalentto MII in Eq. (3)):

MT��MS,i�(ME��MS,i)ft

1+rft

(4)

where ft and r represent the creep factor and corre-sponding relaxation factor, respectively.

This relation has been broadly used in practicebecause of its simplicity. In particular, the exactness andefficiency of this relation can be expected in a bridgeconstructed by the incremental launching method (ILM)or the movable scaffolding system (MSS), that is, in aspan-by-span constructed bridge. However, there are stilllimitations in direct applications of Eq. (4) to balancedcantilever bridges because this equation excludes theproportional ratio, (1�e−(ft−fC)) in Eq. (3), which rep-resents the characteristic of the balanced cantilevermethod.

The difference in the internal moments (ME�ΣMS,i inEq. (4) which is equivalent to MII�MI in Eq. (3)) is notrecovered immediately after connection of all the spansbut gradually over time, and the internal restraintmoments occurring at time t also decrease with timebecause of relaxation accompanied by creep defor-mation. From this fact, it may be inferred that Eq. (4)considers the variation of the internal restraint momentson the basis of a relaxation phenomenon. When a con-stant stress s0 is applied at time t0, this stress will bedecreased to s(t) at time t (see Fig. 10). Considering thestress variation with the effective modulus method(EMM), the strain e(t) corresponding to the stress s(t)can be expressed by e(t)=s0/E0·(1+ft). Moreover, thestress ratio, which denotes the relaxation ratio, becomesR(t,t0)=s(t)/s0=1/(1+ft), and the stress variation�s(t)=ft/(1+ft)·s0. That is, the stress variation is pro-portional to ft/(1+ft). If the age-adjusted effective modu-lus method (AEMM) is based on calculation to allowthe influence of aging due to change of stress, the stressvariation can be expressed by �s(t)=cft/(1+cft)·s0,where c is the aging coefficient [17].

Fig. 10. Stress variation due to relaxation.

3.2. A proposed relation

With the background for the time-dependent behaviorof a cantilever beam effectively describing the internalmoment variation in balanced cantilever bridges, and bymaintaining the basic form of Eq. (4) suggested by Trostand Wolff [5], considering the construction sequencewhile calculating the internal moments at an arbitrarytime t, the following relation is introduced:

MT��MS,i�(ME��MS,i)(1�e−(ft−fc))·f(ft) (5)

where f(ft)=cft/(1+cft). c is the concrete aging coef-ficient which accounts for the effect of aging on the ulti-mate value of creep for stress increments or decrementsoccurring gradually after application of the original load.It was found that in previous studies [11,12,14] an aver-age value of c=0.82 can be used for most practical prob-lems where the creep coefficient lies between 1.5 and3.0. An approximate value of c=0.82 is adopted in thispaper. In addition, if the creep factor ft is calculated onthe basis of the ACI model [13], f(ft)=cft/(1+cft) hasthe values of 0.62, 0.64, and 0.65 at 1 year, 10 years,and 100 years, respectively.

Comparing this equation (Eq. (5)) with Eq. (4), thefollowing differences can be found: (1) to simulate thecantilevered construction, a term, (1�e−(ft−fC)) describ-ing the creep behavior of a cantilevered beam is addedin Eq. (5) (see Eq. (3)); and (2) the term ft(1+rft) inEq. (4) is replaced by f(ft)=cft/(1+cft) in Eq. (5) on thebasis of the relaxation phenomenon.

To verify the effectiveness of the introduced relation,the internal moment variations in FCM 1, FCM 2, andFCM 3 bridges (see Fig. 1), which were obtainedthrough rigorous time-dependent analyses, are comparedwith those by the introduced relation. The effect of creepin the rigorous numerical model was studied in accord-ance with the first-order algorithm based on the expan-sion of a degenerate kernel of compliance function[8,11,12]. Figures 11–13, representing the resultsobtained at t=1 year, t=10 years, and t=100 years aftercompletion of construction, show that the relation of Eq.(4) proposed by Trost and Wolff gives slightly conserva-tive positive moments even though they are still accept-able in the preliminary design stage. On the other hand,the introduced relation of Eq. (5) effectively simulatesthe internal moment variation over time regardless of theconstruction sequence and gives slightly larger positivemoments than those obtained by the rigorous analysisalong the spans. Hence the use of Eq. (5) in determiningthe positive design moments will lead to more reason-able designs of balanced cantilever bridges. In addition,the underestimation of the negative moments, which rep-resents the equivalent magnitudes with overestimation ofthe positive moments, will be induced. The negativedesign moments, however, must be determined on the


Fig. 11. Moment variations of FCM 1 bridge after; (a) 1 year; (b)10 years; (c) 100 years.

basis of the cantilevered state because it has themaximum value in all the construction steps, as notedin Fig. 2. This means that the negative design momenthas a constant value of M=1160 t m in this examplestructure and is calculated directly from the elasticmoment of a cantilevered beam.

Fig. 12. Moment variations of FCM 2 bridge after: (a) 1 year; (b)10 years; (c) 100 years.

4. Application to segmental bridges

A time-dependent analysis of balanced cantileverbridges was conducted by assuming that the cantileversare constructed simultaneously while maintaining a con-stant time interval (see Fig. 1). The cantilevers in realbridges are usually constructed by sequential connectionof segments 3 to 6 m long. These segments may be cast


Fig. 13. Moment variations of FCM 3 bridge after: (a) 1 year; (b)10 years; (c) 100 years.

in place or transported to the specific piers after pre-casting in a nearby construction yard. Accordingly, asegmental concrete bridge has been taken as an examplestructure to review the applicability and effectiveness ofthe introduced relation of Eq. (5). The example structureis shown in Fig. 14, and each segment with a length of2.7 m is assumed to be cast-in-place with a time intervalof 8 days. All the sectional dimensions and material

Fig. 14. Casting sequence in a segmental concrete bridge.

properties used are the same as those used previously.The results obtained at t=100 years in FCM 1, FCM 2,and FCM 3 bridges (see Fig. 1) are given in Fig. 15.

Comparing the obtained results in Fig. 15 and in Figs.11–13, the positive moments in the segmental bridgeshow slightly larger values than those obtained when theentire length of the cantilever is cast at the same time.This difference in the numerical results seems to arisenot from the difference in the construction method ofthe cantilever part but from the difference in time whenthe structural system is changed. From the resultsobtained, it can be inferred that the most influential fac-tors on the internal moment variation in balanced cantil-ever bridges are the magnitude of the ultimate creep fac-tor and the time when the structural system is changedto a statically indeterminate state. This is because thetime-dependent deformations of concrete become veryimportant as a result of early loading to the young con-crete. Moreover, it can be concluded that the introducedrelation of Eq. (5) can be used effectively even in seg-mental bridges, and by using this relation, the designmoment required to determine the concrete dimensionsin the preliminary design stage can easily be calculatedwithout any rigorous time-dependent analysis.

5. Conclusions

A simple, but effective, relation which can simulatethe internal moment variation due to the creep defor-mation of concrete and the changes in the structural sys-tem during construction is proposed, and a new guidelineto determine the design moments is introduced in thispaper. The positive design moment for a dead load canbe determined by the introduced relation, while the nega-tive design moment for a dead load must be calculateddirectly from the elastic moment of a cantilevered beamin balanced cantilever bridges.

Moreover, since the internal moments by other loads,except the dead load, are not affected by the construction


Fig. 15. Moment distribution in segmental bridges at t=100 years:(a) FCM 1; (b) FCM 2; (c) FCM 3.

sequence, the calculation of the final factored designmoment can be followed by the linear combination ofmoments for each load. If the cantilever tendons, whichmay affect the internal moment redistribution duringconstruction, need to be considered in calculating theinternal moments and the corresponding normal stressesat an arbitrary section, it may be achieved on the basis ofthe final continuous structure even though the calculated

results represent slightly conservative values [7]. Inaddition, if a rigorous time-dependent analysis is con-ducted with the initial section determined on the basisof the initial design moments obtained by using Eq. (5),then a more effective design of balanced cantileverbridges can be expected.

Acknowledgements

The research presented in this paper was sponsoredpartly by the Samsung Engineering and Construction.Their support is greatly appreciated.

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Determination of Design Moments in Bridges Constructed by Balanced Cantilever Method

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