manuals Design of prestressed concrete

Design of prestressed concretebridgesN. R. Hewson Hewson Consulting Engineers Ltd

This chapter provides coverage on all aspects of the design and construction ofprestressed concrete bridges, from the fundamentals of prestressing and theequipment used, to insight into the problems that can occur and how they have beendealt with in different projects around the world.

It explains key issues for the designer to consider and provides guidance to the siteengineer on some of the techniques and temporary works used in their construction.Simple in situ and precast beam bridges are described along with the more advancedtechniques of precast segmental and incrementally launched decks. Both pre-tensioning and post-tensioning are included as well as precast and in situconstruction. Recent developments in the use of non-metallic materials to improve thedurability of prestressing systems and higher strength concrete are included to showhow prestressed concrete bridges are likely to develop in the future.

IntroductionPrestressed concrete decks constitute a large proportion ofthe current bridge stock around the world. Since pre-stressing was introduced into concrete bridges in the early1930s it has challenged the bridge engineer’s imaginationas new techniques developed, spans became longer andthe appearance of bridges more important. In situ or pre-cast concrete; simply supported, continuous or cable-stayed structures; beams, slabs or boxes; all feature inprestressed concrete bridges.Concrete is strong in compression but weak in tension;

however, prestressing of the concrete can be used toensure that it remains within its tensile and compressivecapacity. Prestressing is applied as an external force to theconcrete, by the use of wires, strands or bars and canincrease the capacity of concrete alone many times. Thedevelopment of prestressed concrete bridges has giventhe bridge engineer increased flexibility in the selection ofbridge form and in the construction techniques available,resulting in prestressed concrete frequently being thematerial of choice for bridges with spans ranging from25mwith precast beams up to 450mwhen cable-supported.This chapter describes the design and construction of

the different types of prestressed concrete bridges andassumes that the reader has a basic understanding of pre-stressed concrete design which can be applied to thespecific application of bridgeworks.

Principle of prestressingPrestressing of concrete structures is achieved by forcetransfer between the prestressing tendon and the concrete.The tendons are placed within the concrete member asinternal tendons; or alongside the concrete as external

tendons. For post-tensioning, the tendon is pulled using ajack and the force is transferred directly on to the hardenedconcrete, while for pretensioning the tendon is jackedagainst a temporary anchor frame before concreting andthe force released from the anchor frame to the concretewhen sufficient strength has developed. In these ways anexternal compressive force is applied to the concretewhich can be used to counter the tensile stresses generatedunder the bending moments and shear forces present.The concrete can be either fully prestressed, which ensures

that the longitudinal stresses are always in compression, orpartially prestressed which allows some tension to occurunder certain loading conditions.

MaterialsConcreteThe required strength of the concrete is determined by thecompressive stresses generated in the concrete by the pre-stress and applied forces. A minimum strength of fcu equalto 45N/mm2 is typical for prestressed concrete; however, itis becoming more common to use higher strengths with fcuup to 70N/mm2 not unusual, while even higher values havebeen achieved for some projects. The rate of gain of strengthof the concrete is important as this governs the time at whichthe prestress can be applied. At the time of transfer of theprestress force to the concrete it is normally required thatthe concrete strength be at least a minimum of 30N/mm2,although this can vary depending on the tendon andanchor arrangement and the magnitude of load applied tothe tendon. To minimise creep and shrinkage losses in theprestressing, the cement content and water/cement (w/c)ratio of the concrete should be kept to a minimum, com-patible with the high concrete strengths required.

ICE Manual of Bridge Engineering # 2008 Institution of Civil Engineers www.icemanuals.com 195

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doi: 10.1680/mobe.34525.0195

CONTENTS

Introduction 195

Principle of prestressing 195

Materials 195

Prestressing systems 196

Prestress design 199

Design of details 212

Bridge construction anddesign 216

References 231

Further reading 232

Appendix I. Definitions 232

Appendix II. Symbols andnotation used 232

Recent developments include special concretes such asDuctal1 where the cement paste and fine aggregate aremixed with steel fibres to achieve compressive strengthsup to 200N/mm2. These special high-strength concretesallow thinner, lighter structures and include other advan-tages for the prestressing such as higher elastic modulus,minimal shrinkage and lower creep. Ductal1 was used forthe Peace Footbridge, Korea which is shown in Figure 1.

Prestressing tendonsHigh-tensile steel is used aswire, strand or bars, with nominaltensile strengths varying between 1570 and 1860N/mm2 forwire or strand, and between 1000 and 1080N/mm2 for bars.After the load is applied to the prestressing steel, stress

relaxation occurs which results in a reduction of the forcein the tendon. The magnitude of relaxation varies depend-ing on the steel characteristics and the initial stress levelsapplied, with typical values of 2.5–3.5% when a stress of0.70fpu is applied, which reduces to below 1% when theinitial stress is 0.50fpu or less.Typical material properties for tendons are indicated in

Table 1.Fibre-reinforced polymer tendons have been used on

several projects in Europe and the USA to eliminate the

corrosion problems associated with steel tendons. Thematerials available include carbon, glass or aramid fibreswith epoxy, vinylester or polyester resins, althoughcarbon-fibre-reinforced polymers (CFRPs) within anepoxy resin have become the most common. The propertiesof the CFRP depend on the fibre and resin design and cangive tensile strengths up to 3000N/mm2 and E-values of140–300 kN/mm2. The use of CFRP for prestressing ten-dons appears promising but has yet to be widely adopteddue to their current high cost compared with the traditionalsteel alternatives.

Cement groutCement grout is used to fill the void around post-tensionedtendons and their ducts; a w/c ratio of between 0.35 and0.40 is typically used, with admixtures often added toimprove flow and to reduce shrinkage and the w/c ratio. Insome countries such as the UK there is a current trend tomove towards pre-bagged grout where the cement andadmixture are premixed and delivered to site in accuratelymeasured quantities with just the water needing to be added.

Grease or wax groutGrease or wax grout is used for some external tendons toenable destressing or replacement. The grease or petroleumwax is injected into the duct at 80 or 908C before cooling togive a soft, flexible filler.

Prestressing systemsWiresIndividual wires are sometimes used in pre-tensioned beamsbut have become less common in favour of strand, whichhas better bonding characteristics. Wire diameters are typi-cally between 5 and 7mm with a minimum tensile strengthof 1570N/mm2 and can carry forces up to 45 kN.

Figure 1 Peace Footbridge, Korea (courtesy of VSL International)

Nominaldiameter:

mm

Nominalarea:mm2

Nominalmass:kg/m

Yieldstrength:N/mm2

Tensilestrength:N/mm2

Minimumbreakingload: kN

Young’smodulus:kN/mm2

Relaxation after1000h at 208Cand 70% UTS

7-wire strand

13mm super 12.9 100 0.785 1580 1860 186.0 195 2.5%

15mm super 15.7 150 1.18 1500 1770 265.0 195 2.5%

Stress bars

20mm – 314 2.39 835 1030 325 170 3.5%

32mm – 804 6.66 835 1030 828 170 3.5%

50mm – 1963 16.02 835 1030 2022 170 3.5%

Cold-drawn wire

7mm 7mm 38.5 302 1300 1570 60.4 205 2.5%

5mm 5mm 19.6 154 1390 1670 32.7 205 2.5%

Reference should be made to the manufacturers’ literature for the properties of individual wires, strands or bars being used.

Table 1 Typical tendon properties

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Strands and tendonsA seven-wire strand with a tensile strength of 1860N/mm2

and either 13 or 15mm diameter is a very common form ofprestressing and can be used either singularly for pre-tensioning or in bundles to form multi-strand tendons forpost-tensioning as shown in Figure 2. The most commonpost-tensioned tendon sizes utilise 7, 12, 19 or 27 strands tosuit the standard anchor blocks available, but tendons canincorporate up to 55 strands for larger tendons. Stressing to75% ultimate tensile strength (UTS) gives a typical jackingforce of 140 or 199kN for the 13 or 15mm diameter strand,respectively, while the larger multi-strand tendons can carryforces up to 10 000kN. During stressing, jacks are placedover the tendon, gripping each strand and pulling it untilthe required force is generated. Wedges are then placedaround the strand and seated into the anchor block so thaton release of the jack the wedges grip the strand and transferthe force on to the anchor and into the concrete.

BarsIndividual bars can vary in diameter from 15 up to 75mmand are used in post-tensioned construction with jackingforces ranging from 135 to 3000kN. The bars are placedinto ducts which have been cast into the concrete betweentwo anchor blocks on the concrete surface. The bars arepulled from one end by a stressing jack and held in placeby a nut assembly which transfers the load from the bar tothe anchor block and then into the concrete (see Figure 3).

AnchoragesAt each end of the tendon the forces are transferred into theconcrete by an anchorage system. For pre-tensioned tendonsthe anchorage is by bond of the bare strand cast into theconcrete, while for post-tensioned tendons the anchoragecan be either by anchorage blocks or by bond for sometypes of cast-in dead-end anchors. Figures 2 and 3 show typi-cal live-end anchorage for strands and bars respectively,while Figure 4 is a typical cast-in dead-end anchorage forpost-tensioned strand.

When external, post-tensioned tendons are used theyshould be removable and replaceable and the detail ofthe anchorage should be arranged to allow this. Wherecement grout is used, this can be achieved by providing alining on the central hole of the anchor to ensure thegrout around the tendon does not bond to the anchor,allowing the tendon to be cut and pulled out if necessary.

Stressing jacksTypical jacks for stressing of single-strand and multi-strandtendons are shown in Figures 5 and 6, respectively, whileFigure 7 shows a typical jack for bar tendons. For singlestrands, smaller tendons or smaller bars, the jacks, weighingup to 250 kg, can usually be easily handled and manoeuvredinto position with readily available lifting equipment, whilefor the larger tendons special lifting frames or cranes arerequired to move the jacks which can weigh up to 2000 kg.Designs should always take into account the access

needed to set up and operate the jacks and associatedequipment.

Figure 2 Prestressing strand, anchor andduct (courtesy of Freyssinet International)

Figure 3 Prestressing bar and anchor(courtesy of McCalls Special Products Ltd)

Figure 4 Strand with cast-in dead-end anchorage


Design of prestressed concrete bridges ice | manuals

Tendon couplersIt is possible to couple tendons together to assist in thestage-by-stage construction method frequently adopted.Figure 8 shows the coupler arrangement for strands,where a special anchor block is used to enable the tendonto be extended in to the next stage of concrete after it hasbeen stressed against the previous stage. This arrangementcan simplify the tendon layout and save the cost of aseparate anchor. However, often the concrete section hasto be widened to accommodate the coupler arrangement.Some design codes prohibit the coupling of all tendons atthe same location in a deck to reduce the impact of all thetendons being anchored and then extended at the samepoint. Bars can also be joined together by a simple threadedcoupler as shown in Figure 9.

Corrosion protection and ductingWires, bars and strand are generally used uncoated;however, to protect the tendons during storage or to

reduce friction losses during stressing they may be coatedwith soluble oil which is washed off prior to grouting theduct. Galvanised bars and strand are also available, asare epoxy-coated strand in some countries, although these

Figure 5 Single strand stressing jack(courtesy of VSL International)

Figure 6 Multistrand stressing jack

Figure 7 Jack for prestressing bars

Figure 8 Tendon coupler (courtesy of VSL International)



have yet to be commonly adopted in the prestressing ofbridges.Post-tensioned tendons are normally placed inside duct-

ing to allow them to be stressed inside the hardened con-crete and to provide protection to the tendons. Forinternal tendons the ducts have traditionally been manufac-tured using galvanised mild steel strips; more recently plas-tic ducting systems have been adopted to provide awatertight barrier around the tendon, as protection againstcorrosion additional to the cement used to fill the ductsafter the tendons have been stressed.For external tendons high-density polyethylene (HDPE)

ducts are used with either cement grout or grease aroundthe strands. The HDPE provides long-term durabilitywhile the advantage of using grease is that it allows the ten-dons to be more easily destressed and restressed orreplaced, which is an important benefit of external tendons.Where cement grout is used, restressing of the tendon is notpossible, and removal of the external tendon involves cut-ting it up into short lengths and pulling it out of the devia-tors and anchor blocks, which requires careful detailing ofthe duct arrangements. The HDPE ducts need to be strongenough to prevent deformation when pressurised duringgrouting and to resist the strand punching through atdeviated positions, requiring a wall thickness generallygreater than 6mm. Figure 10 shows the different protectionsystems used.Grouting of the ducts is normally carried out on site,

although grease filling of external tendons can be done inthe factory for single strands in individual sheaths or forsmaller multi-strand tendons. For cement grout, themixing and pumping into the duct is carried out in a contin-uous operation with typical equipment shown in Figure 11.The grout is pumped in at the low end and steadily pushedthrough until the duct is full. Grout vents need to be placedat regular intervals along the duct and at all high and lowpoints to ensure that all the air is expelled and the ducts

filled. Discovery during the 1990s of poorly grouted ductsin a number of bridges in the UK led to a review ofprocedures and to The Concrete Society Technical ReportNo. 47 being issued to give guidance on all aspects of grout-ing of tendons. The second edition of TR 47 was publishedin 2002 (The Concrete Society, 2002).

Prestress designGeneral approachThe choice of prestress type and arrangement is influencedby the structural form of the deck and the constructionmethods employed. Pre-tensioned strands or wires areused in precast beams with typical arrangements as shown

Figure 9 Bar coupler

Galvanised steel orplastic duct

Cement grout

(i) Internal tendons

(ii) External tendons

Strand or bars

(a) (b) (c)

Cement grout

13 mm or 15 mm7-wire strand

13 mm or15 mm strand

Cement groutGrease or wax

Greaseor wax

Plastic sheathHigh density polyethylene ducts

Figure 10 Tendon ducts and protection systems

Figure 11 Grouting equipment



in Figure 12(i), where the economies from constructionrepetition can offset the cost of providing a casting bedwith anchor frames strong enough to resist the totalprestress force.Prestressing bars generally have a higher cost per kN of

force than strand but are usually considered to be easierto handle and install and are often used where they simplifyconstruction or in temporary works, such as during theerection of precast segmental decks shown in Figure 13 orduring the launching of box girder decks.Post-tensioned tendons have the advantages of being

installed on site with flexibility in the tendon layoutsthat can be achieved and in the forces that can be applied.Tendons can be profiled to achieve maximum benefit alongthe span and over the support, such as the arrangementsshown in Figure 12(ii) and Figure 14. Jacking systemsused normally require a tendon length of more than 5m,while for internal tendons above 120m long the frictionlosses, even when double-end stressed, become excessive.

External tendons have much lower friction losses andhave been installed with lengths over 300m.External tendons as shown in Figure 15 have been used

extensively for bridge strengthening as well as for new con-struction, and offer a solution where the tendons can beeasily inspected and replaced if necessary – somethingthat is difficult to do where internal tendons are used. Fornew construction, external tendons cost more per tonnethan the equivalent internal tendons due to the higherHDPE cost; however, this can be offset against savings else-where, as with the tendons outside the concrete section theconcrete sizes, particularly the webs, can be reduced inthickness, and reinforcement fixing and concrete placingis made easier. In segmental construction, using externaltendons with span-by-span erection has resulted in veryrapid erection of long viaducts to give overall cost savingsin the project.The prestress tendon layouts for most types of bridge

structures are now well established, and any new designwould normally start by considering a tendon arrangementsimilar to that used on previous structures of the same type,for example straight strand in the bottom of precast beams,cantilever tendons and continuity tendons for balanced

(i) Pre-tensioned beams

(ii) Post-tensioned beams

Post-tensioned tendons

Multi-strand tendons

Beam thickened at endfor tendon anchor

A–A

B–B

B

BPrestressanchors

A

A

13 mm or 15 mm strandStrand

debondingat ends

Pre-tensioned strand

Figure 12 Typical prestress layout in beams

Figure 13 Prestressing bars used for deck erection

P P

Prestressanchorage

Grout vent

Drain on duct

Strand inside duct

Friction alongtendon

Figure 14 Typical post-tensioned tendon layouts



cantilever construction or simple draped tendons for in situ,continuous decks.For most structures, the prestress quantity is governed by

the serviceability stress check, and the number of tendonsrequired at each critical section such as mid-span andover supports can be derived by calculating the stresseson the section due to the applied loads and constructioneffects and then estimating the prestress required to keepthese stresses within the allowable limits based on an initialestimate of the secondary moment. The problem is that atthis stage the prestress secondary moment is not known,and therefore after estimating the prestress layout required,it is necessary to calculate the actual secondary momentsgenerated and to compare these with the values used inthe initial estimate of prestress. It may take several itera-tions of estimating the secondary moment and adjustingthe prestress arrangements before the required prestressstresses are obtained and the actual secondary momentmatches the assumed secondary moment, to give the finaldesign. After this the ultimate moment should be checkedat the critical sections and, if necessary, additional tendonsadded to ensure adequate resistance can be mobilised,although this would require a recheck of the serviceabilitystresses to ensure that they are still within the acceptablelimits.For structures with external, unbonded tendons the quan-

tity of prestress can be governed by the moment capacity atthe ultimate limit state, especially where the increase instress in the tendon is small as described in the sectionentitled ‘Ultimate moment’ and when no non-prestressedreinforcement is available, as is the case for precast segmen-tal construction. In these situations the number of tendonsrequired at critical sections can be determined from the ulti-mate moment check, and the serviceability check then car-ried out to ensure the stresses are within their acceptablelimits.

The quantity of prestress needed is proportional to thespan length of the bridge. Spans of 30m can use as littleas 15 kg/m2 of deck while spans of 100m can use up to50 kg/m2 of deck. Quantities do vary depending on theshape of the deck and type of construction, and also onthe approach taken by the designer.The magnitude of the prestress secondary moments set up

can vary greatly for any type of structural form and willdepend on the prestress layout chosen; however, for bridge-works it is normally found that the secondary moment issagging, reducing the design hogging moments over thepiers and increasing the design sagging moments in mid-span. For concrete box girders with spans in the range35–40m and with typical tendon layouts, the prestress sec-ondary moments can be of the order of 5000–10 000 kNm atinternal piers, while for heavily prestressed structures andfor longer spans the secondary moments can be signifi-cantly higher.Although not specified in the UK, many countries now

recommend that when designing post-tensioned structuresallowance is made for the future installation of an addi-tional 10% or more of prestress, which can be easilyincorporated with external tendons. The need for this hascome about because of excessive loss of prestress or deflec-tions in some existing structures, or the need to upgrade orstrengthen the structure to take heavier loads, and it wouldseem a sensible provision to make on all new post-tensionedbridges.

Primary and secondary effectsWhen the prestressing tendons apply load to the structurethe resultant forces and moments generated can be con-sidered as a combination of primary and secondary orparasitic effects. For a theoretical beam with a concordanttendon arrangement no secondary moments are set up, butthis rarely occurs in practice. The stage-by-stage construc-tion sequence and envelope of loads applied to the structureinevitably result in a non-concordant prestress layout andsecondary moments can be significant. The secondarymoments can often be beneficial by reducing the hoggingmoments at the piers, but at the expense of increasing thesagging moments in the span.The primary effect is the moment, shear and axial force

generated by the direct application of the force in thetendon on the section being considered. Secondary effectsoccur when the structure is statically indeterminate andrestraints on the structure prevent the prestressed memberfrom freely deflecting when the prestress is applied.These secondary effects can be derived using a number of

different methods, the most common being the equivalentload method where the loads from the tendons areapplied directly to the structural model and the combinedprimary and secondary effects as shown in Figure 16 arederived directly from the analysis output. The forces and

Figure 15 External tendons



moments from the tendon Ph, Pv and Mp are applied ateach anchor position, while along the structure equivalentloads are applied to the model wherever the tendons havean angle change. Care needs to be taken in setting up themodel and applying the prestress load to ensure the struc-tural behaviour is accurately represented. The memberlayout should follow the neutral axis of the structure, andfriction losses in the tendons incorporated by applyingappropriate forces and moments on the members alongthe model.For simply supported decks, unrestrained at the ends, the

structure is free to deflect when prestressed and no second-ary effects are set up. For continuous decks, when theprestress is applied, the intermediate supports restrain the

deck from vertical movement, and secondary momentsand shears occur. The primary and secondary momentsboth set up longitudinal bending stresses on the section,as indicated in Figure 17, and the resultant stress is takeninto the serviceability stress analysis.An alternative approach to deriving the secondary effects

is the influence coefficient method which takes into accountchanges in section and friction losses and is readily adapta-ble for use with spreadsheets. With the prestress loading onthe structure as shown in Figure 18, and by using the theoryof least work, the following equations can be derived:

f11X1 þ f12X2 ¼ �U1

f21X1 þ f22X2 ¼ �U2

PP

Tendon eccentricity, e

(i) Typical prestress layout

Sagging

Sagging

(iii) + (iv)

Hogging

(ii) Equivalent loads Force in tendonLoad intensity = Radius of tendon

(iii) Primary moments, Mp = Pe

(iv) Secondary moments, Ms

(v) Resultant moment (primary and secondary)

Draped tendonprofile

Prestressanchorage

Sectionneutral axis

Pv

Ph

Mp

Pv

Ph

Mp

Figure 16 Prestress in continuous decks



N NA A

e

P P

P

A

P

A

Pe

zt

Ms

zt

Applied force Primary stresses Secondary stresses

+

–P

A

Pe

zt

–Ms

zt

+

P

A

Pe

zb

Pe

zb

+Ms

zb

–P

A

Ms

zb

+

=

Resultant stresses

Figure 17 Prestress stresses on section

(i) Typical structure layout

(ii) Mp = Pe (after prestress losses)

(iii) m1

(iv) m2

(v) Secondary moment, Ms

End support End supportSupport 1 Support 2

x2x1

0

0 0

00

1.0

1.0

s

Tendon eccentricity, ePrestressanchorage

Support 1

e

Strand insideduct

P P

Sectionneutral axis

Support 2

0

Sagging

Hoggings

s

Figure 18 Secondary moments using the influence coefficient method



where:

f11 ¼ð

sðm2

1=EIÞds

f12 ¼ð

sðm1m2=EIÞ ds ¼ f21

f22 ¼ð

sðm2

2=EIÞds

U1 ¼ð

sðm1Mp=EIÞ ds

U2 ¼ð

sðm2Mp=EIÞ ds

By solving the equations, X1 and X2 can be determined,which are the secondary moments generated at their respec-tive support. This approach can be extended to derivesecondary moments for structures with many more spans.The secondary shear forces can be determined by considera-tion of the changes in the secondary moments along thestructure.Where the deck is built integral with abutments or piers,

the restraints will again set up secondary effects fromthe prestressing which must be taken into account in thedesign. An example of this is shown in Figure 19. Theprestress tends to compress the portal beam, but thiscompression is resisted by the columns, which generatessecondary moment and reduces the prestress compressiveforces on the beam. Similarly the prestress tends to bendthe portal beam under its primary effects, and this bendingis again resisted by the columns.Where post-tensioned tendons are used in curved decks,

torsions are generated due to the secondary effects of theprestress. The intermediate supports along the structureresist the curved decks’ tendency to twist, and where theprestress generates secondary moments along the deckand vertical reactions in the supports a complementary tor-sion is also present. An estimate of the torsion can bederived by computing the secondary moment diagramassuming the bridge is straight and dividing this by thehorizontal radius of the deck. This M/R diagram is thenapplied as a load to the structure and the shears generatedalong the deck are the torsions that will be present in theoriginal structure. Provided the tendons are symmetricalabout the vertical axis of the section, the primary effectsof the tendons are not affected by deck curvature; however,where this is not the case, such as with external tendonsbetween anchor and deviator points, a transverse bendingand further torsions can be generated.

Prestress force and lossesTendons can be stressed up to 80% UTS, although somecodes limit the jacking load to 75% UTS, and after

the jack releases the strand to a maximum stress of 70or 75% UTS, for post-tensioning and pretensioning,respectively. Prestressing bars are normally used straightand encounter little friction losses. When a multi-strandtendon is pulled by the jack, the movement is resisted byfriction along the duct, and the force F at any point alongthe tendon, x metres from the stressing anchor, is givenby the formula:

F ¼ F0 e�ðkxþ��Þ

Typical values for the coefficients are shown in Table 2.For external tendons, or pretensioned strands, k ¼ 0

where the tendons are free from any restraint.

P

Pv Pv

Ph Ph

Mp Mp

Prestress tendon

(i) Prestressed portal

(ii) Deflected shape

(iii) Prestress moments diagram

Secondary momentgenerated in column

Primary ‘hogging’ momentreduced by restraint at column

P

Figure 19 Prestress affect with integral bridge decks



The equation for the force in the tendon along its length issometimes written as:

F ¼ F0 e��ðkxþ �Þ

in which case the value of k is adjusted accordingly.When the tendon is released by the jack a length of reverse

friction is set up as the strand pulls in and the wedges grip.Figure 20 gives a typical force profile along the tendon,while the typical pull-in at lock-off for multi-strand tendonscould be up to 7mm.The extension of the tendon during stressing can be

calculated from the force profile:

Extension ¼ �Fdx=AtEt

¼ ðArea under the force profile=AtEtÞThe extension is usually checked during the stressingprocess to ensure that the prestress is achieved as expected.Too low extensions can indicate excessive friction or ablockage in the duct while too high extensions can indicatestrand slippage at anchors or strand breakage, althoughthat is usually obvious from the loud noise made.The force loss along the tendon is greatly dependent on

the tendon profile, and for tendons longer than 40m exces-sive losses can occur; however, this can be compensated for

by double end stressing, where the tendon is jacked fromboth ends.With post-tensioning, when each tendon is installed the

elastic shortening of the concrete causes losses in the ten-dons already stressed. The losses can be calculated by thefollowing formula:

lE ¼ �cðEt=EcÞN/mm2

where �c is taken as the increase in stress in the concreteadjacent to the tendon, occurring after the new tendonhas been stressed.For pre-tensioning, the full prestress force will generate

elastic shortening of the concrete and give a constant lossin each tendon. For post-tensioning, each tendon at asection will suffer a different loss depending upon when itis stressed in relation to the other tendons. For mostdesigns it is sufficient to calculate an average loss andapply this to all the tendons, with �c often taken as halfthe total final stress in the concrete, averaged along thetendon length.Relaxation of the tendon will reduce the prestress force

over a period of time, as will creep and shrinkage of the con-crete. Typical relaxation percentages are given in Table 1.

Creep losses; lc ¼ �cðEt=EcÞ�N/mm2

Shrinkage losses; ls ¼ �csEt N/mm2

Derivations of � and �cs are given in Appendix C of BS5400, Part 4 (BSI, 1990) and �c is the total stress in the con-crete adjacent to the tendon. Typical values for shrinkageand creep losses of standard bridge construction are givenin Clauses 6.7.2.4 and 6.7.2.5 of BS 5400, Part 4 (BSI, 1990).External tendons are not bonded to the concrete and the

elastic shortening, creep and shrinkage losses should beestimated by considering the total deformation between

k mm

Bare strand in metal ducts 0.001–0.002 0.2–0.3

Bare strand in UPVC ducts 0.001 0.14

Bare strand in HDPE ducts 0.001 0.15

Greased and plastic sheathed strand inpolyethylene ducts

0.001 0.05–0.07

Table 2 Typical values for the coefficients

F

x

Length of tendon

Tendon force, F/kNProfile after lock-off

Jacking force, Fo

F = Fo e–(kx + µθ)

Distance fromanchor, x/m

Dead end

dx

Figure 20 Typical tendon force profile



the anchorages or other fixed points as being uniformlydistributed along the complete tendon length.

Serviceability limit state stress checkLongitudinal stresses in the prestressed deck are checked ateach stage of construction, and throughout the structure’sdesign life the stresses in the concrete under service loadsmust be within allowable limits. Typical allowable stressesin the concrete under service load are as follows:

Compressive stresses:

0.5fci < 0:4fcu with triangular stress distribution, or

0.4fci < 0:3fcu with uniform stress distribution.

Tensile stresses:

0N/mm2 with permanent loads and normal live load

0.45ffiffiffiffiffiffi

fcup

full loading and other effects on pre-tensionedmembers

0.36ffiffiffiffiffiffi

fcup

full loading and other effects on post-tensionedmembers.

Tensile stresses during construction:

1N/mm2 during erection with prestress and coexistentdead and temporary loads

0.45fcuc, for pretensioned members under any loading

0.36fcu, for post-tensioned members under any loading.

With precast segmental construction, no tension is allowedat the joints during either the construction stages or in thecompleted structure.The stress levels should be checked at all the critical stages

in the structure’s life, including:

i at transfer of prestress to the concreteii during construction, with temporary loads appliediii at bridge opening, with full live loadiv after long-term losses in the prestress, and full creep

redistribution of moments have occurred.

With external tendons, to allow for tendon replacement, thedesign of the deck should be checked for the situation whenany two tendons are removed. The live loading applied maybe reduced to reflect traffic management systems that couldbe implemented.

DeflectionsFor normally proportioned prestressed concrete decks andstresses within the allowable limits, deflections are usuallynot critical and do not need to be checked, other than toconfirm the precamber values to be catered for during theconstruction.

Vibrations and fatigueThe vibration from traffic or wind seldom creates a problemand for most prestressed concrete decks it is not necessary

to consider vibrations further; however, if external tendonsare used the vibrations of the individual tendons can giverise to fluctuations in stress levels in certain circumstancesand lead to fatigue failures of the wires.For either internal or external tendons, the fluctuation in

the direct stress in the tendons due to live and other loadingis very small and fatigue is usually not critical; however,with external tendons it is necessary to check that the ten-dons are not subjected to vibrations that are likely to giverise to fretting of the strand or bending stresses that couldcause fatigue problems. This is normally achieved by ensur-ing the frequency of the free length of tendons, betweenanchors or deviator points, is not the same as the naturalfrequency of the deck or the traffic using the bridge.Approximate values for frequencies are as follows:

Frequency of the external tendon ¼ 1=2LT

pðF=mtÞHz

Frequency of the deck ¼ ðk2f =2�L2spÞ

pðEI=mdÞHz

where kf depends on span continuity and is � for simplysupported spans and between 3–4.5 for a continuous deck.Vibration frequency of live load is often taken as between

1–3Hz for highway traffic, while for rail traffic thebehaviour of track supports, track stiffness and naturalfrequencies of the train bogey leads to a more complexbehaviour.To prevent vibrations in highway bridges, it is normally

sufficient to limit the free length of the tendon to 12m orless, which should result in the tendon frequency beingdifferent from that of the deck and traffic.

Ultimate momentAs the applied bending moment on a prestressed beamincreases, the compression on one side goes up while onthe other side the concrete goes into tension. When thetensile strength of the concrete is exceeded, crackingoccurs and the load is transferred either to the prestressingtendons or to any non-prestressed reinforcement present.As the applied moment further increases, the crack in theconcrete opens and propagates across the section with apure couple set up between the compression in the concreteand tension in the tendon and non-prestressed reinforce-ment, with the maximum moment reached when eitherthe concrete or tendon and reinforcement fail. To ensurethat there is a sufficient factor of safety against failure theultimate moment of resistance of any section alongthe deck must exceed the bending moment generated bythe ultimate loads. It is normally sufficient to only checkkey points along the deck, such as over the pier and atmid-span.The ultimate moment of resistance is derived by consider-

ing a balance of tensile force in the tendon and compressiveforce in the concrete as shown in Figure 21 for internaltendons.



In the case of rectangular sections, or flanged sectionswith the neutral axis in the flange, the compressive stressin the concrete is taken as an average of 0.4fcu over thedepth of compression, while the concrete strain is limitedto 0.0035. The increases in stress in the tendon, fpi, andnon-prestressed reinforcement, fsi, are derived by consider-ing the increase in strain caused by the bending within thebeam and can be obtained by reference to the tendonstress–strain curves similar to Figures 2 and 3 in BS 5400Part 4 (BSI, 1990), which are used with �m ¼ 1:15.To determine the ultimate moment of resistance, an

iterative approach is often used, with a first estimate of dctaken, from which C and T can then be calculated.

C ¼ 0:4fcudcbs

T ¼ At½ð fp þ fpiÞ=1:15� þ Ap½ð fs þ fsiÞ=1:15�d is then adjusted and C and T recalculated until C ¼ T ,giving the ultimate moment of resistance ¼ TL.The American bridge design code (AASHTO) requires

that the factored moment at a section, M4�Mr, where� ¼ 1:0 for conventional concrete construction, reducedwith precast segmental construction to 0.95 where fullybonded tendons are used and 0.9 where external, unbondedtendons are used.For external tendons unbonded from the concrete over

long lengths, the change in stress under ultimate loads is

StrainStress

Neutral axis

Applied loading

(i) Loaded beam

(ii) Initial stress and strain distribution

(iii) Stress and strain distribution at ultimate moment of resistance

0.4fcu

fs + fsi

1.15

dtεp + (0.0035 – εct) dc

0.0035

Tendon

L

C

d

εct

εs

εcb

εp

fct

fs

fcb

dc

dt

fp

fp + fpi

1.15

T

dtεs + (0.0035 – εct) dc

Figure 21 Ultimate moment of resistance



derived by considering the increase in tendon lengthbetween its fixed points as the deck deflects and thenusing this to estimate an average strain change. The increasein stress in the tendon is thus limited and ultimate failure isnormally governed by the compression in the concretewhich can give rise to a brittle failure mechanism. The esti-mate of the ultimate moment of resistance follows the sameprocedure as for internal tendon above, although forlong tendons installed over several spans fpi ¼ 0N/mm2 isnormally taken. For short tendons anchored within onespan, fpi can be significant depending on the tendonlayout and deck arrangement.When the ultimate moment of resistance of a section with

external tendons is being considered, as the deck deflects thetendon will remain a straight element between the nearestrestraints, and at a point away from the anchors or devia-tors the tendon moves closer to the compression face withthe lever arm, L, being reduced. This can become significantif the free length of external tendons is greater than a quar-ter of the span length and the tendons are not restrainednear the critical points.To allow the future replacement of external tendons

the section should have adequate capacity to carry areduced live loading with any two of the external tendonsremoved.

Ultimate shearShear need only be considered at the ultimate limit state,when the shear resistance must exceed the shear forces gen-erated by the ultimate loads applied. The shear in the deckis checked adjacent to each pier and at regular intervalsalong the spans, and reinforcement, normally in the formof links, provided as required to give adequate resistancewhen combined with the concrete resistance. It is usual toassume that all the shear acts on the webs of a sectionand only the web resistance taken into account in thedesign. For haunched decks, such as with long-span boxgirders, the longitudinal bending in the deck gives rise tocompression in the bottom slab which acts parallel to thesoffit as shown in Figure 22. This compression has a verticalcomponent that acts against the shear forces and can bededucted from the shear force, V , in the design.The shear capacity at any section is the sum of Vc the

ultimate shear resistance of the concrete andVs the ultimateshear resistance of the reinforcement present.With internal tendons, designing to BS 5400 Part 4 (BSI,

1990),Vc is the lesser ofVco, the ultimate shear resistance ofa section uncracked in flexure and Vcr, the ultimate shearresistance of a section cracked in flexure.

Vco ¼ 0:67bhpð f 2t þ fcp ftÞ þ Vp

Vcr ¼ 0:037bdpfcu þ ðMcr=MÞV

Mcr ¼ ð0:37pfcu þ fptÞðI=yÞ

The values for Vp, fcp and fpt are usually based on theprestressing forces after all losses have occurred and aremultiplied by the partial safety factor, �fl ¼ 0:87. Wherethe vertical component of prestress, Vp, resists shear atthe section being considered, this is usually ignored whencalculating Vcr.BS 5400 Part 4 (BSI, 1990) requires a minimum shear

reinforcement to be provided such that:

ðAsv=SvÞ5 ð0:87fyv=bÞ0:4N=mm2

When V exceeds Vc, shear reinforcement is required suchthat:

ðAsv=SvÞ ¼ ðV þ 0:4bdt � VcÞ=ð0:87fyvdtÞLongitudinal reinforcement should be provided in the ten-sile zone such that:

Asl 5V=2ð0:87fylÞThis quantity can include both the non-prestressedreinforcement and any bonded prestressing tendon notbeing utilised at the ultimate limit state for other purposes.The maximum shear force V, is limited to 0.75

pfcubd

to prevent excessive principal stresses occurring in theconcrete.Where internal post-tensioned tendons are placed within

ducts in the webs of the concrete, allowance has to be madefor the reduction in effective web width. Prior to groutingthe tendons, the full duct width is deducted from b, whileafter grouting, two-thirds of the duct diameter is deducted.When checking the ultimate shear capacity at the ends of

pretensioned beams, allowance must be made for the lossof prestress over its anchorage length, Lt, as defined inthe section entitled ‘Anchorage’. It is normally sufficientto consider the prestress force as varying linearly over thislength and, by applying the formula above, derive Vc. Ifthe shear resistance based on the beam being simply

V M

Stress in bottom slabgenerated by moment, M

Vertical componentof force

θ

Figure 22 Resistance to shear in haunched decks



reinforced, and ignoring the prestress, is greater than thatcalculated assuming a prestressed beam, then the highervalue can be taken.AASHTO’s approach to load factor design differs from

BS 5400 Part 4 in that for shear, V 4�ðVc þ Vs þ VpÞ inwhich Vc is 0.083�

pf 0cbd. In this equation, � is a factor

indicating the ability of diagonally cracked concrete totransmit tension and where at least the minimum shearreinforcement is provided varies between 1.5 and 6.32depending on the applied shear stress and the longitudinalstrain in the section.Vs ¼ asv fyvdðcot �þ cot�Þ sin�=Sv where � and � are the

inclination of the diagonal compressive stress and inclina-tion of shear reinforcement respectively.

Vs ¼ ðasv fyvd=SvÞ5 0:644pf 0cbd and d5 0:8h

The strength reduction factor, �, is taken as 0.9 for internaltendons and 0.85 for external tendons.For external tendons, AASHTO treats shear design in the

same way as for internal tendons when calculating Vc andthe shear reinforcement to be provided, while in the UK,BD 58/94 requires that sections with unbonded externaltendons are designed as reinforced-concrete columns, sub-jected to an externally applied load, with:

v ¼ ðV � 0:87VpÞ=bdwhere v should not exceed 5.8N/mm2.When: v4 svc;Asv 5 0:4bsv=0:87fyvWhen: v > svc;

Asv 5 bsvðvþ 0:4� f1þ ½0:05ð0:87PÞ=Ac�gsvcÞ=0:87fyvwhere vc and s are given in BS 5400 Part 4 (BSI, 1990),Tables 8 and 9, respectively.With external tendons the shear capacity should be ade-

quate to carry a reduced live loading with any two tendonsremoved.

Ultimate torsionTorsion does not normally govern the dimensions of theconcrete members, but will require additional reinforce-ment in the section and is considered at the ultimate limitstate.In determining the torsional capacity it is necessary to

calculate the torsional shear stresses generated by the ulti-mate loads and where these stresses exceed Vtmin as givenin Table 10 of BS 5400 Part 4 (BSI, 1990), then reinforce-ment is provided by means of transverse links and longitu-dinal bars. The stresses generated by torsion and shear areadded together and the sum must not exceed 0.75

pfcu, and

be less than 5.8N/mm2.For rectangular sections, the torsional shear stress, vt, is

given by:

vt ¼ ½2Tu=h2minðhmax � hmin=3Þ�

and torsional reinforcement provided such that:

Ast=Sv 5Tu=½1:6xi yið0:87fyvÞ�

Asl=Sl5Astð fyv=fylÞ=Sv

where T and I sections are used, the torsion is considered asacting on the individual rectangular elements, with thesection divided up to maximise the sum of (hmaxh

3min) of

each rectangle. Each rectangle is then designed to carry aproportion of the torsion based on its value of (hmaxh

3min)

in relation to the sum of the values for all the rectangles,reinforcement being determined as for normal rectangularsections and detailed to tie the individual rectanglestogether.For box sections:

vt ¼ Tu=2hwoAo

and reinforcement provided such that:

Ast=Sv 5 ½Tu=2Aoð0:87fyvÞ�

Asl=Sl5Astð fyv=fylÞ=Sv

Where a part of the section is in compression, this com-pressive force may be used to reduce the longitudinalreinforcement required. This reduction in Asl=SL is givenby fchwo=0:87fyl.For precast segmental decks where no reinforcement

passes through the joints, and where the joints should bein compression at all times, the longitudinal torsionalstresses need to be countered by a residual longitudinalcompression from the prestress equal to Tu=2hwoAo.

Longitudinal shearAs the bending moments change, the flow of stress throughthe section gives rise to longitudinal shear which is checkedat the slab–web interfaces, as indicated in Figure 23. Theultimate longitudinal shear force per unit length is given by:

VL ¼ VðAL y=IÞ

when AL and y refer to the area of concrete outside of thesection being considered.The concrete is able to provide some resistance to this

shear, with reinforcement required to ensure that VL isless than both:

k1 fcuLS ðaÞ

and:

vLLS þ 0:7Ar fy ðbÞ

Values for vL and k1 are given in BS 5400 Part 4, Table 31.For precast beam-and-slab construction, a minimum areaof reinforcement of 0.15% of the contact area should be



provided across the interface between the beam and slab.BS 5400 Part 4 (BSI, 1990) allows reinforcement that isprovided for other purposes to be utilised to resist thelongitudinal shear.

Partial prestressingClassified in BS 5400 Part 4 (BSI, 1990) as Class 3, partialprestress is not currently adopted for bridgeworks in theUK; however, several other countries, including Denmarkand Australia, have successfully adopted this approachinto their bridge designs. For fully prestressed structuresthe design is based on the concrete being uncrackedunder service loading, either as Class 1 where the stressesare in compression under all loading conditions, or asClass 2 where under transient loading small tensions areallowed which are kept below the tensile strength of theconcrete. For partially prestressed structures the philoso-phy is to allow the concrete to crack under serviceloading and to limit the crack widths to the normal allow-able limit for reinforced concrete. This is often related toa hypothetical allowable tension in the concrete of up to 6or 7N/mm2 when carrying out the longitudinal stresscheck.The ultimate moment and torsion design is carried out in

the normal manner, while for ultimate shear the concreteresistance is calculated as cracked in flexure, with:

Vcr ¼ ½1� 0:55ð fpe=fpuÞ�vcbd þMoðV=MÞ

Mo ¼ fptðI=yÞ

and any vertical component of prestress ignored.The values of fpe and fpt are based on the prestressing

force after all losses have occurred and multiplied by thepartial safety factor, �fl ¼ 0:87, and for the purpose ofthis equation fpe should be 50:6fpu while vc is given inBS 5400 Part 4 (BSI, 1990), Table 8 in which the value ofAs is the total area of prestressed or non-prestressedreinforcement in the tensile zone.A fatigue check of the prestress is needed to ensure that

the service stress fluctuation in the strand is not criticalfor the level of stress present. Normally checking to

ensure the stress range in the prestressis below 120N/mm2 is sufficient.The advantages of partial prestressing

include reduction in the quantity of pre-stress, full utilisation of non-prestressedreinforcement for ultimate strength,smaller deflections than reinforced con-crete due to the presence of the prestressand reduced creep in the concrete whencompared to a fully prestressed section.The disadvantages include the need formore non-prestressed reinforcementand a concern that the durability of the

structure will be reduced in harsh environments, althoughthis concern has not been borne out where partially pre-stressed structures have been used.

PrecamberDeflections of the concrete deck occur under the self-weightand from the weight of the permanently applied loadsfollowed by further movements due to long-term creep ofthe concrete and losses in prestress. The deck must becast and erected so that the theoretical profile is achievedupon completion. The adjustment made to the profileduring casting to achieve the desired shape is called theprecamber and this will be affected by the constructionsequence and concrete properties.The creep effects will cause the deck to change its profile

over its life and it is usual to aim to achieve the desiredalignment at time of bridge opening, although the long-term changes in deflections should be checked to ensurethey are not excessive.

Construction sequence and creepanalysisThe way the bridge is built affects the moments and shearsgenerated in the structure and this needs to be fully takeninto account during the design. The structure should bechecked for strength and stability, and serviceability stres-ses assessed at each stage of construction with the finalmoments and shears derived to reflect the constructionsequence. For example, Figure 24 shows the dead loadbending moments in a four-span deck constructed instages, with the final moments after creep being betweenthe as-built moments and the moment if the deck werebuilt instantaneously.When a concrete structure’s statical system is changed

during construction, creep of the concrete will modify theas-built bending moments and shear forces towardsthe ‘instantaneous’ moment and shear distribution, theamount of the change being dependent on the creepfactor, �, of the concrete.

� ¼ creep strain/elastic strain

AAF(at interface)Area AL

for section A

Centre ofarea AL Critical sections

C L

Area AL

for section E

C

D EA B

NN

y

Figure 23 Longitudinal shear



� varies depending on the concrete constituents and details,environmental conditions and age of concrete, and thisvariation can be from 1.3 to 3 or more. For precast con-struction, � is normally around 1.6, while for in situconstruction it would normally be between 2.0 and 2.5.BS 5400 Part 4 (BSI, 2000), Appendix C.2 gives the detailedderivation of �.Where the change to the statical system is sudden, such as

connecting cantilevers with a mid-span stitch, the modifica-tion to the moments is:

Mfinal ¼ Mas-built þ ð1� e��ÞðMinst �Mas-builtÞWhere the change is gradual, such as the differentialshrinkage between precast beams and in situ top slab, themodification to the moments becomes:

Mfinal ¼ Ma½ð1� e��Þ=��Shears are modified in a similar way to the moments.

Temperature effectsChanges in effective temperature of the deck will cause it toexpand or contract, while differential temperature gradientsthrough the concrete result in stresses that need to beconsidered in the prestress design.Under a uniform temperature change, the change in deck

length is �L ¼ �t�L.The coefficient of thermal expansion, �, is typically

12� 10�6/8C for normal concrete or 9� 10�6/8C withlimestone aggregates.Where a deck is free to expand or contract, this overall

change in effective temperature will not give rise to anyforces in the structure, although the movement does needto be allowed for in the bearing and expansion jointdesign. When a restraint exists that restricts free movement,

such as integral decks or multiple fixed piers, forces andstresses are generated throughout the structure whichmust be taken into account in the prestress design.Differential temperature effects are generated through the

concrete section as the outer surface heats or cools morequickly than the rest. BS 5400 Part 2 (BSI, 2000) Figure 9defines a series of simplistic temperature gradients thatcan be applied to different structural types. Figure 25 illus-trates how the temperature gradients generate stresses andforces within the deck. The equivalent forces generated bythe temperature differential give rise to an out-of-balanceaxial force and moment. Provided the deck is notrestrained, the force and moment will relieve at the endsand the stresses adjusted accordingly to give the resultantstress distribution across the section. These stresses arethen catered for in the design when considering the service-ability stress check of the prestressed concrete.

In situ stitchDeck built as

balanced cantilever

(i) Structure arrangement

(ii) Dead load bending moment diagram

Instantaneousmoment

Momentrange Moment

after creep

As-builtmoment

Figure 24 Creep redistribution of moments

Stress from (ii) Relieving ofaxial forcefrom (iv)(a)

Relieving ofmomentfrom (v)

Relieving momentdistributed down structure

(v) Relieving effect

(iv) Balanced stresses

(i) Typical temperaturegradient

(ii) Equivalent stresss = TaEc

(iii) Equivalent forces

(a) Axial force

0

0

0

0

(b) Moment

Resultantstress

distribution

+ + =

S (Fe)

T2

h

T3

T4

T1

F2e

N A

F1

F4

F3

s2

s3

s4

s1h1

h2

h3

h4

S (Fe)

SF SF

S (Fe)

zb

SF

A

S (Fe)

zt

Figure 25 Differential temperature stresses



The moment effect gives rise to shear forces in the spansand these should be considered during the ultimate sheardesign.

AnalysisModern computer techniques are increasingly being used toanalyse and design prestressed concrete bridges. The latestgeneration of software takes into account the 3D natureof bridges and combines the prestressing tendons with thestructural analysis as well as simulating the stage-by-stageconstruction and long-term effects that are essential whenconsidering the design of prestressed concrete bridges.Analysis using these software packages includes all theprestress losses, creep and shrinkage of the concrete andcombines it with the self-weight, superimposed deadloads, live loads and temperature loading to simplify thedesign process. A typical model for a precast segmentalbox girder deck built by the balanced cantilever techniqueis shown in Figure 26 as generated using the MIDAS soft-ware package. Several stages of the analysis are shown asthe deck is built up and the prestress installed.

Design of detailsAnchorageWith pretensioned strand, the force transfer into the con-crete is achieved through bond between the two materials.At the end of the strand it slips into the concrete as thebond gradually builds up the force transferred, until thetotal force in the strand has been taken up by the concreteover a transmission length, Lt. With Fo 4 75% UTS andfci 5 30N/mm2, then this length is defined as:

Lt ¼ ktDt=fci

kt can be taken as 600 for plain, indented or crimped wirewith a wave height of less than 0.15Dt, 400 for crimpedwire with a total wave height greater than 0.15Dt, 240 for7-wire standard and super-strand or 360 for 7-wire drawnor compacted strand.When post-tensioned tendons are anchored, they apply a

large concentrated force to the concrete which needs to becontained. CIRIA Guide No. 1 (Clark, 1976) describesthe behaviour and design of anchor blocks with a typicalarrangement as shown in Figure 27. Behind each anchor,splitting forces and tensile stresses occur and reinforcementneeds to be provided as follows.

Bursting reinforcementThis is provided as a spiral or series of links around eachindividual anchor, with the area of reinforcement:

A ¼ Fbst=0:87fy

where Fbst depends on the end block arrangement and isgiven by:

ypo=yo 0.3 0.4 0.5 0.6 0.7

Fbst=Fo 0.23 0.20 0.17 0.14 0.11

0.87fy should be replaced by a stress of 200N/mm2 or less tocontrol cracking when the concrete cover to the reinforce-ment is less than 50mm.Bursting reinforcement is placed between 0.2yo and 2yo

from the anchor face and encloses a cylinder or prismwith dimensions 50mm larger than the face of the anchorblock. The bursting forces in the two principal directionsshould be determined and sufficient reinforcement providedto suit. The loaded area used for determining yo is taken assymmetrical about the anchor, extending to the nearestedge of the concrete or to the mid-point of any adjacentanchors.

Spalling reinforcementThis is provided to prevent concrete spalling off the end facearound the anchor with the area of reinforcement:

A ¼ 0:04Fo=0:87fy

Construct cantilevers

2nd cantilever extended out

Complete construction

Prestress tendon layout

Figure 26 MIDAS analysis model for box girder deck



The stress in the reinforcement (0.87fy) should be kept tobelow 200N/mm2 to control cracking while the barsshould be placed as near to the end face as possible andanchored around the concrete edges.Where the anchor is positioned non-symmetrically on an

end face, additional spalling stresses are set up and addi-tional reinforcement is provided in the unsymmetricaldirection such that:

A1 ¼ 0:2½ðd1 � d2Þ=ðd1 þ d2Þ�ðFo=0:87fyÞ

Equilibrium reinforcementThis is provided to maintain the overall equilibrium of theend block within the bridge deck arrangement. The forcefrom the anchor block is assumed to have fully spreadout at a distance h from the anchor face. By considering

the concrete block with the anchor force on one side andthe stress distribution on the other, the equilibrium of anyhorizontal plane, such as A–A in Figure 27(ii), is checkedand vertical reinforcement provided to resist the out-of-bal-ance moment, using a lever arm of h/2 to calculate thequantity of reinforcement needed. The reinforcement isthen placed over the distance 0.25h or 0.5h as indicated inFigure 27(iii) depending on the direction of the out-of-bal-ance moment.The shear on each horizontal plane should also be

checked to ensure that the shear stress does not exceed:

ð2:25þ 0:65pfyÞN=mm2

where p is the ratio of the reinforcement crossing the planeand is taken as As=bh.

b

h

Loadedarea

Stress distribution‘h’ from anchor

Anti-spallingreinforcement

Equilibriumreinforcement


Anti-burstingspirals

A A

Prestress anchorageend block

(i) Anchor layout

(iii) Reinforcement design

(ii) Stress distribution

0.25h 0.25h

–ve +ve

0.5h

d2

d1

Fo

Fo

bγpo

γpo

γo

γo

σt

σb

Figure 27 End block design



Anchor blistersFigure 28(i) shows the additional reinforcement needed totie the tendon into the main body of concrete where anchorsare placed on blisters, or concrete blocks cast on the side ofthe concrete member.The bursting and spalling reinforcement quantities are

calculated as for standard anchors, but the additional tie-back reinforcement is required to prevent cracks occurringbehind the anchor due to the tensile forces generated toachieve strain compatibility in the concrete around the blis-ter or anchor block. It has been traditional to provide tie-back reinforcement to cater for 50% of Fo, althoughfinite-element analysis can show that significantly lessreinforcement than this is needed in some cases.Equilibrium effects occur as the force distributes across

the section and into the webs and slabs, requiring reinforce-ment to be provided as described above.

Anchor pocketsFigure 28(ii) shows an arrangement where recesses or pocketsare provided in the concrete to anchor the tendons. Theanchor forces tend to cause cracking of the concrete behindthe recesses and similar tie-back reinforcement should be pro-vided as for the anchor blisters, as well as the normal bursting

and spalling reinforcement, while equilibrium reinforcementis again needed in the webs and slabs to restrain the forceas it spreads out. Theuse of anchor pockets on the top surfaceof decks is discouraged in some countries due to concernsover the ingress of water through the resultant constructionjoint and if used they should be well constructed and fullyprotected with additional waterproofing measures.

DuctsDuct sizes are governed by the practicalities of having spaceto thread the tendon through and to allow the grout to flowfreely around the strands or bars, with the area of a ductnormally not less than twice the tendon area.With internal tendons, the ducts have to be supported and

held in place during the concreting operation. This is nor-mally done by installing additional reinforcement aroundthe duct and fixing this to the normal reinforcement cage.The spacing of those supports will depend on the ductsize and stiffness and is normally between 0.5 and 1.0m.To ease concrete placing and prevent delaminating of the

adjacent concrete, ducts cast into the concrete should havea minimum clear spacing of not less than the greater of thefollowing:

n duct internal diameter

n aggregate size þ 5mm

n 35mm.

The concrete cover to the duct should not be less than50mm although this depth may need to be increasedwhen the ducts are curved, to prevent the tendon burstingout of its concrete.With curved tendons, the strand presses against the edge

of the duct and exerts a pressure on the concrete whichcauses splitting forces that need to be considered. The split-ting force on the concrete depends on the force in thetendon and the radius of the curve. When the radius islarge enough no additional reinforcement is needed as thetensile strength of the concrete is sufficient to counter theforces generated. Where the radius of the duct is signifi-cantly less than the value given in Tables 36 and 37 of BS5400 Part 4 (BSI, 2000) for the tendon force and cover orduct spacing, reinforcement should be provided to restrainthe tendon as shown in Figure 29.Where a curved tendon exerts a force on the deck section,

it is sometimes necessary to provide additional reinforce-ment to counter the bursting and bending effects, such aswhere the tendons placed inside the bottom slab of ahaunched box girder give a resulting downward force asindicated in Figure 30. The tendon will tend to pull out ofthe concrete section and needs to be tied back while italso generates transverse bending and shears in thebottom slab and tension in the webs which need to bereinforced against.

Equilibriumreinforcement tosuit force flow intodeck section

Equilibriumreinforcement

Anti-burstingspiral

Anti-burstingspiral

Anti-burstingspiral


(i) Blister reinforcement

Equilibrium reinforcementto suit force flow into

deck section

(ii) Anchor pocket reinforcement

Tie-backreinforcement

Tie-backreinforcement


Fo

Fo


Figure 28 Anchor reinforcement



Where the ducts are curved, the tendon will move tothe edge of the duct, resulting in the tendon centrelinebeing offset from the duct centreline. This offset will varydepending on tendon and duct sizes and arrangement,and needs to be taken into account when considering thetendon eccentricity in the longitudinal prestress design.

Typical design offsets and duct radii for tendons areindicated in Table 3. With strands, the tendons can followa profile with fairly tight curves, the minimum radii oftenbeing governed by the ability of the duct to bend. Individualstrands can be bent to radii as little as 0.5m.

DiaphragmsDiaphragms are generally used in the deck at points ofsupport to transfer the load from the webs into the sub-structure below. Typical diaphragm arrangements areshown in Figure 31. For the arrangement shown in Figure31(i), a truss analogy is normally used to model the forcetransfer from the webs into the bearings. To ensure thatall the load is picked up and transferred on to the truss atthe top of the webs, ‘hanging reinforcement’ needs to beprovided in the form of vertical bars in the web and dia-phragm. The vertical force is then held by a ‘tie’ acrossthe top of the diaphragm and a strut towards the bearing.Where significant torsion exists in the deck, it is also neces-sary to consider the horizontal forces present in the top andbottom slabs, and reinforcement provided to tie thesetogether.In Figure 31(ii) and 31(iii) the diaphragm will behave as a

beam and bending moments and shears can be determinedand reinforced against in the normal way. Where a web isdirectly over a bearing, then the load will go straight intothe support; elsewhere ‘hanging reinforcement’ will beneeded to take the force from the web up to the top ofthe diaphragm beam.The concrete above the bearings needs to contain the high

loads applied and can be designed in a similar manner to ananchor end block, with spalling and bursting reinforcementbeing provided accordingly.

DeviatorsWhere deviators are used to deflect external tendons to givethe desired profile they can be subject to large forces thatneed to be tied into the deck section. Figure 32 showsseveral different types of deviator arrangements that canbe used. The force applied to the deviator is equal to theforce in the tendon multiplied by its ‘angle change’ (inradians) and the deviator should be designed to cater forthis load. In the UK, where BD 58/94 (Highways Agency,

P/kN

P/kN

Restrainingreinforcement

A

A

Typicalforce system

PR/4

PRPR/2 PR/2

1

2

ReinforcementSection A–A

Restrainingreinforcement

PR PPR = kN per m run R

Figure 29 Restraint of curved tendon

F/R F/R

F, force in tendon

R, radius of soffit

Force on bottomslab = F/R m run

Figure 30 Force on curved soffit

Typicaltendonsize:mm

Duct sizeinternaldiameter:

mm

Typicaltendonoffset:mm

Minimumradii forinternalducts: m

Minimumradii forexternalducts: m

7� 15 strand 65 10 3 2.5

19� 15 strand 90 17 5 3.0

27� 15 strand 110 20 7 5.0

Table 3 Typical duct details



1994) is used, the design is carried out at the ultimate limitstate with the applied ultimate force based on the charac-teristic strength of the tendon. Elsewhere the design isoften based on the direct load applied by the tendons atthe serviceability limit state.

Bridge construction and designDeck formMany factors affect the choice of the bridge type, the spanarrangement and general layout, while prestressed concretedecks cover a wide range of construction forms with spanlengths ranging from 25m for single spans to over 450min cable-stayed bridges. For spans below 25m, reinforcedconcrete is likely to be preferred, while for spans above450m steel or composite cable-stayed decks are used; how-ever, between these span lengths prestressed concrete oftengives an economic, aesthetic and simple solution.

The choice for the deck form depends largely on theindividual site constraints and the advantages and dis-advantages of each option need to be carefully consideredto arrive at the optimum solution. Figure 33 indicates thetypical span range for different deck types.For decks with an overall length of 60m or less, it is

common to build the deck continuous and integral withthe piers and abutments to reduce the number of expansionjoints and bearings and eliminate the maintenance prob-lems that can occur with these elements. For longerstructures, the deck should still be made continuous for aslong as practical depending on the structural form, withconcrete box girders having been constructed up to1.7 km between expansion joints. The need for bearingsbetween the deck and substructure depends on the decktype and stiffness of the substructure. Precast segmentalconstruction usually utilises bearings to simplify erectionwhile in situ deck construction can either have bearings,

Pier

Pier

(i) Concrete box girder

(ii) Multicell box girder

Pier

(iii) Precast beams Equivalent beam

Equivalent beam

Equivalent truss

Bearing Bearing

Access holethrough diaphragm

Bearings

Soffit drainage pipe

Tie across top

Lower strut

Bearing

Loadfromweb

LoadfromwebBearing

TorsionalbracingDiaphragm wall

typically 1.0 m thick

Diaphragm wall

Access holethrough diaphragm

Bearings

Soffit drainage pipe

Force from webs

Diaphragm wallIn situ top slab

Precast beam

Force from webs

Bearings

Bearing Bearing

Figure 31 Diaphragm arrangements



or easily be built into the substructure provided that deckmovements do not generate excessive forces in the piersand foundations.Span arrangements are frequently governed by the nature

of the obstruction being bridged, but if possible the spansshould be arranged to suit the type of bridge being con-structed. With precast beams, it is preferred to keep astandard beam length throughout to standardise castingand erection equipment, while with in situ, precast segmen-tal or launched continuous decks, the end spans shouldbe arranged to be approximately 70% of the length ofthe internal spans to balance out the design bendingmoments. Where the deck is built by balanced cantilever,the end spans should be reduced to approximately 60%

(i) Concrete beam deviator

Galvanised steel tubewith bellmouth

Galvanised steel tube

Tendon inHDPE duct

A–A

A

A

(ii) Concrete block A–A alternative

A

A

(iii) Steel frame B–B

Steel strut

B

B

HDPE duct

Steel frameSteel deviator tube

Coupler

HDPE ductCoupler

Figure 32 Deviator arrangements

20 40 60 80 100 200 500

Span length/metre

In situ concrete slab

Precast concrete beams

Concrete box girder (launched)

In situ or precast segmental box girder (constant depth)

Precast segmental box girder (variable depth)

In situ segmental box girder (variable depth)

Concrete cable-stayed bridges

Figure 33 Typical span range for different deck types



of the length of the internal spans to minimise any out-of-balance effects of the end balanced cantilever, althoughcare should be taken to ensure that no uplift occurs at theabutment bearings, otherwise a tie-down arrangement isneeded.

Solid-slab bridgesIn situ, solid-slab decks are often used for shorter concretespans where easy access is available, with the advantagesincluding simple construction and formwork layout, whilethe disadvantages are that prestressing a solid concrete ele-ment is inefficient and dead load becomes excessive as spansget longer. For these reasons prestressing of solid-slabs isnot commonly seen for bridgeworks.

Voided-slab bridgesIn situ voided-slab decks were commonly used during theearly period of prestressed concrete bridge developmentwithin the span range 25–35m and depth/span ratios of upto 1:20. The voids are often shaped from polystyrene voidforms, either round as shown in Figure 34 or rectangularwith bevelled corners leaving a 150mm thick concretesection above and below, while the concrete is cast supportedby falsework from the ground. The advantages includesimple construction and formwork arrangement, while thedisadvantages include a fairly heavy deck section and theneed to hold down the voids during concreting, and thishas more recently resulted in beam-and-slab or multi-cellboxes being used in preference.Analysis of the voided-slab is easily carried out with suf-

ficient accuracy using a grillage model. It is convenient toplace the longitudinal grillage members to match thelayout of the prestress tendons or groups of tendons, withtransverse members placed over the supports and at suit-able spacing between to give an evenly balanced gridpattern. The section properties of both the longitudinaland transverse members are based on the deck width tomid-way between the members, with the resultant voidedsection taken longitudinally and the section at the centreof the void taken for the full width transversely. This is

normally found to give sufficiently accurate results forcarrying out the design. When assigning the torsionalstiffness to the grillage members it is sufficient to giveboth longitudinal and transverse members the same valueof twice the ‘I ’ value of the longitudinal members.The longitudinal prestress design is carried out by con-

sidering the bending moments and shear forces from theimposed loads on the longitudinal grillage members anddesigning the prestress to balance the serviceability limitstate stresses, with checks to ensure adequate ultimatelimit state strength in the normal way.

Beam-and-slab bridgesBeam-and-slab bridges utilise either precast or in situ con-crete beams, with the deck slab cast in situ (see Figure35). Spans normally range from 25m up to 40m althoughprecast beams have been used on spans up to 50m.Depth/span ratios are usually of the order of 1:16 witheither post-tensioned or pre-tensioned strand positionedto take the sagging moments as indicated in Figure 12.Figure 36 shows some typical arrangements for the beams

at piers, where the beams may be connected or the top slabmade continuous over the piers to minimise the need forexpansion joints in the road surface. The beams arenormally connected together at the support location by atransverse in situ concrete beam, or diaphragm, which pro-vides rigidity to the arrangement and can be used to transferload from the beams on to the supports. The advantages ofbeams include economies in repetitive precasting and rapidconstruction while the disadvantages include the need forgood access and heavy lifting equipment for the longerspans.Where the beam-and-slab deck is cast in situ, traditional

formwork and falsework arrangements may be used tosupport the concrete while it hardens. Post-tensioningtendons are then threaded through the ducts cast into theconcrete and stressed, lifting the deck off the formworkand allowing the falsework and formwork to be removed.Precast beams are usually made in special casting yards

which have a series of casting beds to form the beams.For pre-tensioned beams, jacking or anchor frames ateither end of the casting bed provide anchorages for thestrands to be stressed against. The reinforcement and shut-ters are assembled around the stressed strand, and theconcrete poured. Although the side shutters can be strippedaway after one or two days, the concrete is normallyrequired to reach at least 35N/mm2 before the strands arereleased from the jacking frames and the force transferredinto the beam. For post-tensioning, the beams are madewith ducts and anchorages cast in to allow the tendon tobe threaded through and stressed after the concrete hassufficient strength. Figures 37 and 38 show a typical pre-tensioning and post-tensioning casting bed arrangement,respectively.

900 c/cService trench

Post-tensionedtendons

600 dia. polystyrenevoid formers

Varies with span typically900 for 20 m span

C bridge L

Figure 34 Typical voided deck slab arrangement



For simply supported beams the applied moment reducesat the end of beams to zero at the supports. If a constantprestress force with uniform eccentricity is applied alongthe beam’s length this can generate high compression inthe bottom and tension in the top over a significantlength of the ends of the beams. To prevent this being aproblem for the design, in post-tensioned beams the ten-dons may be draped with the anchors raised up at theends while with pretensioned beams some of the strandsmay be debonded by placing in a plastic tube over a

length at each ends, or alternatively some strands may bedeflected up by means of an anchor frame at 1

4or 1

3points

along the beam.The placing of the beams into position is normally done

by crane; however, for long, multi-span bridges or whereground access is poor, an erection gantry may be used tolift the beams and carry them into place. Figures 39 and40 show crane and gantry erection, respectively. Afterpositioning the beams, the formwork is placed to allowthe top slab to be cast.

Post-tensioningtendons


Pre-tensionedstrand

Pre-tensionedstrand

Pre-tensionedstrand

Pre-tensionedstrand Pre-tensioned

strandPost-tensioned

tendons


In situ top slab

(i) In situ beams (ii) ‘M’ beams

(iii) Box beams (iv) ‘T’ beams

(v) ‘U’ beams

(vii) ‘Y’ beams

(vi) ‘T’ beams

In situ top slab

In situ top slabIn situ top slab

In situ top slabIn situ top slab

Pre-tensionedstrand

In situ top slab

Figure 35 Typical prestressed beam-and-slab arrangements



Design of beam-and-slab decks involves the superimpos-ing of many different effects to build up the overall designstate. Dead load, superimposed dead load and live loadeffects, both longitudinally in the beams and transverselyin the deck slab, may be derived from a grillage analysiswhich will give satisfactory results for most standardbeam-and-slab arrangements. For the grillage analysis,the deck is modelled as a series of discrete members bothlongitudinally and transversely. Longitudinally it is sim-plest to provide a grillage member along the line of eachbeam web, with section properties to include a portion ofthe top slab up to midway to adjacent webs and similarlya portion of the bottom slab where U-beams or box-beams are used. Where box structures are present and

are represented by individual grillagemembers for each web, the torsionalstiffness of the box is calculated and aquarter of this is assigned to each web,with the remaining half of the torsionalstiffness assigned to the transversemembers. Transversely, grillage mem-bers are positioned at each diaphragmlocation with section properties tomatch the diaphragm beam, and also atregular intervals between with sectionproperties to match the length of thetop and bottom slabs between, whereappropriate. The location of the trans-verse members should result in a ratioof the spacing of longitudinal membersto transverse member of between 1:1and 1:2 to achieve reasonably accurateload distribution.Where unusual beam-and-slab arrange-

ments are used or secondary effects such

Pier

Pier

PierPier

(ii) Continuous over support

(iii) Continuous top slab

(i) Non-continuous

Beam Beam

Beam Beam

Beam BeamBeam Beam

Bearing

Movement joint

Bearing

Bearing Bearing

In situ connection

In situ slab

Figure 36 Beam continuity

Figure 37 Casting bed for pre-tensioned beam

Figure 38 Casting bed for post-tensioned beam (courtesy of HyderConsulting Ltd)



as distortions or transverse bending in the beams become sig-nificant then three-dimensional finite element models shouldbe used for deriving the load effects. Shell elements may beused to build up the webs, slabs and diaphragms of thethree-dimensional structure, and the dead, superimposedand live load applied to the model to give the forces andmoments at the critical sections for the beam design.Full account of the construction and stressing sequence

needs to be allowed for in the design. When the beamsare cast they are supported by the formwork along theirtotal length; however, on applying the prestress they tendto lift up over the length of the beam and when removedto the storage yard they are usually supported at the endsof the beams only, giving a typical stress profile at mid-span as indicated in Figure 41(i). At this stage the beamsare subject to their own dead weight and the appliedprestress and the resulting stresses at the top and bottomof the beam may be accurately derived. The common

problem at this stage is to ensure that the compression inthe bottom and tension in the top of the beam do notexceed allowable limits. Once erected, the top slab is castwith the weight being carried by the beam section aloneas indicated in Figure 41(ii). When the top slab concretehas hardened, creep on the concrete will redistribute thedead load stresses and prestress forces across the section.The superimposed and live loads are carried by the com-bined section of the beam and top slab.The creep redistribution of the dead load and prestress

stresses may be estimated by considering the as-built

Figure 39 Crane erection of precast beams

Figure 40 Gantry for erection of precast beams

A

ATension Compressionσbt

σbbStress

Stress

A–A

(i) Beam cast and prestress installed

B

B

B–B

(ii) Top slab cast

(iii) Differential shrinkage

σbt

σst

σbb

Instantaneous stress

(iv) Creep redistribution of dead load

As-built stress

Stress after creep

Instantaneous stress

(v) Creep redistribution of prestress

As-built stress

Stress after creep

σbb

σbt

σst

(iii) + (iv) + (v)after creep

(vi) Resultant stress distribution

Figure 41 Longitudinal stress distribution in beams



condition, where all the load is carried on the beam and thestresses are a combination of Figure 41(i) and 41(ii), and thetheoretical ‘instantaneous’ condition where the full deadload and prestress is applied to the composite section.The stresses will creep from the as-built conditiontowards the instantaneous condition as indicated inFigures 41(iv) and 41(v) with the final stress at any level inthe section being given by:

�final ¼ �as-built þ ð1� e��Þð�inst � �as-builtÞThe loss of prestress forces due to relaxation in the strand,and concrete creep and shrinkage that occur after castingof the top slab will affect the stresses in both the slab andbeam, and these long-term prestress losses should be con-sidered in two phases. For the first phase before the deckslab is cast, the prestress losses should be estimated andthe prestress force in the beam reduced accordingly. Inthe second phase, the remaining losses that occur aftercasting the slab should be estimated and then these appliedas a ‘tensile’ force to the composite section at the position ofthe centroid of the strand or tendons.When the top slab is cast, the concrete in the beam will

already have completed a large proportion of its shrinkage,resulting in differential shrinkage occurring between theslab and beam. The deck slab tries to shrink more thanthe beam and is restrained by it, creating tension in theslab and a combined compressive force and saggingmoment in the beam. The equivalent differential shrinkageforce, assuming the top slab was fully restrained, may beestimated from:

force;Fs ¼ �sAsEc½ð1� e��Þ=�Þ�with ð1� e��Þ=� reducing the effect due to creep of theconcrete.The stresses in the section may be derived by adding the

‘fully restrained’ stress to the ‘released’ stress in thecomposite section, i.e:

(i) tensile stress from force Fs acting on slab alone at the centroidof slab, plus

(ii) stresses generated from a compressive force Fs acting on thecomposite section, applied at the centroid of the slab giving astress profile similar to Figure 41(iii).

The stresses generated at the top of the slab, top of beamand bottom of beam as indicated in Figure 41(vi) arechecked at each stage of construction as well as in thepermanent condition after opening and again after thelong-term effects have occurred.When the beams are fully connected longitudinally over

the piers to make the deck continuous, the dead load andprestress effects will be redistributed due to creep of theconcrete. The creep from the prestress load on the concretemay cause a hogging deflection of the beam at mid-spanand a sagging restraint over the pier, which may lead to

excessive cracks in the soffit in this region if the effect isnot fully considered and suitable reinforcement provided.The final moment at the pier due to the crept dead loadand prestress may be derived by a similar approach tothat described in the section entitled ‘Constructionsequence and creep analysis’, with:

Mfinal ¼ Mas-built þ ð1� e��ÞðMinst �Mas-builtÞ� is based on the residual creep left in the beam at the timeof casting the connection andMas-built is equal to zero wherethe connections are made after erecting the beam andcasting the top slab. The moment calculated as if the deckwere built ‘instantaneously’ as continuous, Minst, shouldinclude the prestress secondary moment as well as the pre-stress primary moment and dead load. The differentialshrinkage between the top slab and beam will generate asecondary hogging moment in the deck at the pier afterthe connection has been made, which must also be consid-ered in the design.The design for ultimate moment, shear and torsion is

carried out in the normal manner, while the check of thelongitudinal shear along the interface between the slaband beam must include the forces generated by the differen-tial shrinkage and creep effects.During the lifting and transporting of the beams they may

be subjected to additional loads resulting in a change to thebending moments which must be considered in the design.Care must be taken during handling to minimise anyimpact or dynamic loading, while lifting or temporarysupport positions should be located as near to the perma-nent support position as possible. Where long, slenderbeams are used, transverse instability may occur eitherduring lifting, or when placing the concrete for the topslab, and suitable restraint will need to be provided.

In situ multi-cell box-girder decksMulti-cell box-girder bridges are similar to voided slabswith the voids occupying a larger proportion of the decksection. A typical arrangement is shown in Figure 42,which would have transverse diaphragms at each pier posi-tion. Concreting and formwork construction normally

150 × 150chamfer

Access holesthrough diaphragm

Depth varieswith span length

C deck L

Footpath Footpath200

200

150

2

1

300

Figure 42 Typical multi-cell box girder arrangement



dictate a minimum depth of 1200mm, and with depth/spanratios of up to 1:25, the span lengths are normally greaterthan 30m and may extend up to 50m. Above 50m otherforms of construction are usually preferred. The advantagesinclude efficient use of concrete and prestress and simpleconstruction where easy access is available. The disadvan-tages include the need for extensive labour activities onsite, and the falsework may become complex if access isdifficult.The formwork for concreting may be supported either

from falsework off the ground or from a gantry spanningbetween the piers. It usually simplifies the formwork tocast the deck section in several stages, with the bottomslab, outer webs and diaphragms being cast first, followedby the interior webs and top slab soon after. Figure 43shows a multi-cell box girder being prepared to cast thefirst stage.Where thin webs and slabs are being used, the box section

may significantly deform and distort under the imposedloading, and the deck analysis is usually most accuratelycarried out by using a three-dimensional finite-elementmodel with shell elements used for each web and slabsection. The moments and shears in each element in eachdirection may be taken directly from the model and usedfor both the longitudinal prestress design and the transverse

reinforcement design. For the longitudinal design thedeck section is divided up into a series of I-beams consistingof a web and associated top and bottom slabs up to themid-point of each cell. Prestress layouts are then designedfor each web to balance the stresses derived from thefinite-element analysis and to give adequate ultimatestrength.

In situ single-cell box-girder bridgesIn situ single-cell box girders such as the typical sectionshown in Figure 44 are used for a wide range of spansfrom 40 up to 300m when haunched. The advantages arethe efficient use of concrete and prestress and the flexibilityin span arrangements, while the disadvantages include thelabour-intensive activities on site and the long constructiontimes needed for the larger structures.For spans up to 60m a constant depth section with depth/

span ratio of 1:20 would typically be used, with theconcrete cast in sections span by span supported by false-work from the ground as seen in Figure 45 or with a trussspanning between the piers.Above 60m spans it is common to use a haunched section

cast within a travelling form as a balanced cantilever abouta pier as shown in Figure 46, with segments between 3 and5m long and a typical casting cycle resulting in a segmentbeing cast every seven or eight days. At mid-span, thetwo opposing cantilevers are joined to make the deckcontinuous. Depth-to-span ratios are typically 1:16 at thepier, reducing to 1:45 at mid-span.Traditionally, for a single-cell box girder the longitudinal

and transverse designs have been considered separatelyalthough it is now becoming more commonplace to estab-lish three-dimensional finite-element models that combinethe effects.The deck may be modelled longitudinally as a two-

dimensional or three-dimensional frame by standardstructural analysis software, and the moments and shearsderived for dead load, superimposed dead load and liveload as well as for the secondary effects such as support

Figure 43 Multi-cell box girder deck under construction (courtesy ofHyder Consulting Ltd)

350 mid-span500 near support

400

200

200

250

250

100

200

1500

200 2000

1000Varies

(2500 typical)

Vari

es (

span/2

0)

C deck L

Figure 44 Typical single-cell box girder section



settlement. Creep effects of the dead load and the prestress,and the temperature effects may be determined as describedearlier in the chapter.Transversely, the box is subjected to bending and shear

due to the dead and superimposed dead loads and the liveloads on the top slab. A three-dimensional finite-elementmodel with shell elements should be set up, similar to thatshown in Figure 47, covering sufficient length of the boxto allow the longitudinal distribution of the live load andthe three-dimensional behaviour of the box to be accuratelymodelled. Once the moments and shears around the boxhave been calculated, reinforcement is determined asnecessary.

Under torsional loading, concrete boxgirders warp with their cross-sectionsundergoing out-of-plane displacements.The sections undergo torsional and dis-tortional warping while the appliedloads may also cause distortion to thecross-section. These effects are fullydescribed in the C&CA technical reporton concrete box beams (Maisel andRoll, 1974). If a full three-dimensionalfinite-element model is set up over thefull length of the deck, the warpingeffects may be directly modelled, alongwith both the longitudinal and trans-verse effects.For normal-proportioned box girders,

the prestress tends to balance the deadand live load moments, and shear lag isnot a problem and need not be consid-ered further. For wider or slimmerdecks with unusual proportions, shearlag may be significant and needs to beallowed for in the design. In these

cases, the shear lag will modify the stress distributionacross the section and peak stresses will need to bedetermined for use in the serviceability longitudinal stresscheck; however, shear lag need not be considered in theultimate limit state. The shear lag effect may be determinedfrom a full three-dimensional finite-element model of thestructure, although care is needed in modelling the prestressforces to give an accurate representation of their effect.

Precast segmental box girdersPrecast single-cell box girders are found to be very eco-nomic for long bridge lengths due to the savings associatedwith maximising repetition in factory conditions. Spanlengths vary typically from 40m up to 150m, abovewhich the segment weights become excessive. Up to spans

Figure 45 In situ box girder, construction on falsework (courtesy of Hyder Consulting Ltd)

Figure 46 In situ balanced cantilever construction (courtesy of HyderConsulting Ltd)

zxy

Figure 47 Typical finite element model of box girder



of 70m a constant depth section would be used with adepth/span ratio of approximately 1:20, with segmentsbeing positioned by crane, where access is suitable, or byerection gantry as shown in Figures 48 and 49, respectively.Above 60m spans the deck is more likely to be haunchedwith depth-to-span ratios of 1:16 at the pier, reducing to1:35 at mid-span. The advantages of precast segmentalconstruction include rapid construction with minimal on-site work, while the disadvantages include the costs of set-ting up the casting yard and the special erection equipmentneeded.The segments are made in purpose-built casting yards in

specially designed formwork, most commonly using theshort bed method shown in Figure 50. Each segment iscast against its neighbour and when hardened it is movedinto the counter-cast position to be cast against, afterwhich it is moved to a storage area. In the alternative

long bed method, the segments are cast against each otherin a long line, the formwork moving down the line foreach segment casting. After casting a series of segmentsthey are all removed from the bed to the storage area.By casting directly against each other either in the short

bed or long bed method, a perfect fit is achieved when thesegments are erected, and by carefully controlling therelative position of the segments when they are beingcast, complex horizontal and vertical alignments may beachieved, especially with the short bed casting method.Before the segment being placed is positioned against the

already erected portion of the deck the end faces are lightlysandblasted or cleaned to remove any deleterious materialand to provide a good key for the epoxy glue which isspread evenly over the matching faces. Temporary prestressis then applied to hold the segment in place as indicated inFigure 51. The temporary prestress should apply an averagecompressive stress of between 0.2N/mm2 and 0.3N/mm2

with a minimum local stress of 0.15N/mm2 and a flexuralstress difference less than 0.5N/mm2 over the segmentjoint being epoxied, to ensure that the joint thickness isconstant throughout, normally between 1 and 3mm. Theepoxy helps to lubricate the joint during erection, allowingthe segment to slide into position, and once hardened theepoxy seals the joint and provides structural continuitybetween the segments.Several bridges have been built with dry joints, where no

epoxy is used between the segments and the design relies onshear friction to transfer the loads in both the permanentand temporary conditions. To date dry joints have beenused only in tropical climates where freeze–thaw cycles donot occur and their effectiveness in more aggressive climateshas yet to be proved.Figure 52 indicates three alternative systems for segment

erection. The most common method of erecting the seg-ments is by the balanced cantilever technique, either witha gantry as shown in Figure 49, a crane as shown in

Figure 48 Precast segmental erection by crane

Figure 49 Precast segmental erection by gantry

Figure 50 Precast segment casting cell



Figure 48, or by a special lifting frame fixed to the deck. Thefirst segment over the pier is lifted into position and placedon temporary supports to allow it to be correctly aligned.Subsequent segments are transported into position and

lifted either side of the pier segment, being fixed to it byepoxy and temporary prestress. Further segments arepositioned with erection progressing out from both sidesof the pier in a balanced manner and with permanent

Section

Internal bottomslab tendons

Internal cantilevertendons

C Deck L

C Segment L

C Segment L

Temporary blister

Temporary blisterfor temporarystressing bars

End anchorage

Plan

Directionof erection Temporary prestressing

bars for next segment

Symmetrical aboutsegment chord

Temporary prestressing barsto hold segment during erection

and until permanent prestressingtendons installed

Bulkhead face

Matchcast face

Figure 51 Typical temporary prestress for precast segmental erection

C Pier L C Pier L C Pier L C Pier L C Pier L C Pier L

Launch girder

(i) Span-by-span erection

(ii) Balanced cantilever erection

Erect segments

Direction ofconstruction

GL▼ GL

▼

C Pier L C Pier L C Pier L

Launch girder


GL▼

C Pier L C Pier L C Pier L C Pier L C Pier L C Pier L C Pier L

Erect segments Erect next pier segment

GL▼

GL▼

(iii) Progressive placing erection

C Pier L C Pier L C Pier L

Launch girder


GL▼

C Pier L C Pier L C Pier L C Pier L C Pier L C Pier L

Erect segments and temporary props

Temporary props

Build out over next pier

GL▼

GL▼


Figure 52 Precast segmental deck erection methods



prestress cantilever tendons being installed as required.When the cantilever reaches mid-span it is connected upto the opposing cantilever from the adjacent pier by an insitu concrete stitch and the remaining prestressing tendonsare installed along the length of the span and across thestitch to complete the connection.For spans of less than 50m, an alternative erection

method is to place the segments span by span. In this tech-nique a gantry, either overhead, as shown in Figure 53, orunderslung, is used to support a complete span of segmentswhich are then pulled together by applying the permanentprestress, allowing the gantry to release the span on to thebearings and to launch itself forward ready to erect thenext span. Above 50m and the span weight usuallybecomes too large for this method to be economical.A third method is progressive placing, where erection is

started at one end of the bridge and continued out progres-sively until the other end is reached. This method usuallyneeds temporary props along the span to support thedeck until the next pier is reached, such as is shown inFigure 54. This method is the least used for precast segmen-tal decks due to its slower erection rate when compared tothe other methods available.With match cast segments erected with epoxy in the joints

the design of the concrete box in its completed condition issimilar to that for an in situ box girder. The presence of thejoints creates a discontinuity in the longitudinal reinforce-ment and the joints should be kept in compression underall loading conditions. The epoxy creates a bond of greaterstrength than the concrete itself and the ultimate shearchecks may be carried out as if the deck is monolithic;however, where wide, cast-in-situ concrete joints are incor-porated into the structure, the shear force under ultimateloads should not be greater than:

0:7ðtan�Þ0:87P

where tan� depends on the interface and for roughenedfaces may be taken as 0.7 during erection and 1.4 aftercompletion of the bridge, and P is the horizontal com-ponent of the prestress force after all losses.Shear keys are provided on the surface of the joint to help

align the segments during erection and to transfer the shearforces before the epoxy has set. Either large or smallmultiple shear keys may be used as shown in Figure 55,and both are designed to prevent a shear failure under thetemporary erection loads.

Figure 53 Span-by-span segments erection Figure 54 Progressive placing of segments

50

50

50

2525

340

340

340

20°

340

340

340

Reinforcement

(i) Small shear keys

(i) Large shear keys

Figure 55 Shear key arrangement



Considering the shear friction concept as given inAASHTO, the capacity of the shear key is estimatedusing a coefficient of friction of 1.4, resulting in the shearresistance being:

Vk ¼ 1:4½Ask fpk þ Arð0:87fyÞ�

For the large shear keys these may be designed as areinforced-concrete corbel where appropriate reinforce-ment is provided, while for the small shear key, reinforce-ment is not normally required.

Incrementally launched box-girderbridgesWhere the bridge alignment is straight or on a constantradius curve, either vertical or horizontal, launchedsingle-cell box girders may be used to overcome accessproblems or to avoid obstructions at ground level. Com-monly used for spans up to 60m, the technique hasoccasionally been used for longer spans up to 100m withthe help of temporary piers placed to reduce the effectivespan length during launching. Deck depth must be constantfor launching purposes, with the ratio to the launched spangenerally 1:16 or less.Casting segments behind the abutment, the deck is

pushed or pulled out over the piers as shown in Figure 56.A specially prepared casting area is located behind the

abutment with sections to assemble the reinforcement, toconcrete, and to launch with a typical layout shown inFigure 57. Segment lengths are normally standardised fora bridge and are usually 20–30m long.The first segment is cast and moved forward on tem-

porary bearings. The second segment is then cast againstthe first and both are moved forward by a furtherincrement, with subsequent segments cast and the deckmoved until it reaches the opposite abutments and itsfinal position.The area immediately behind the casting bay is utilised

for steel fixing and placing the prestressing ducts, whichmay progress at the same time as other operations. Whenthe deck is launched the steel cage is attached to the con-crete and pulled into position ready for concreting. Theformwork system needs to be specifically designed toallow it to be lowered, leaving the deck on temporary sup-ports ready for launching. Temporary bearings are used oneach pier and in the casting area to launch the deck and con-sist of a steel plate, surfaced with stainless steel on lami-nated rubber pads, with a Teflon pad fed in between thebearing and concrete deck to provide a low-friction slidingsurface. The launching devices may be fixed to the abut-ments which provide the resistance against the thrustneeded. A typical launching device as shown in Figure 58would jack up the deck a small amount to grip the structureand then push or pull the deck forward a small amount

Figure 56 Incrementally launched box girder



before dropping down to release the structure and to moveback ready to start another stroke.As the deck moves out the pushing force has to increase to

overcome the frictional force on the temporary bearing,which may be between 2–6% of the vertical load. Greaterpushing forces are needed where a deck is being launched

up a slope, while if going down a slope a braking deviceis needed. The launching force is normally resisted by theabutment which has to be designed to resist sliding oroverturning. Additional resistance may be mobilised byproviding the casting area with a ground slab as aworking platform and by connecting this slab withthe abutment. To keep the deck correctly aligned duringthe launching operations, guides are fixed to the piers.When the launching is complete, the deck is jacked upand the temporary bearings replaced by the permanentbearings.As the deck is launched over a pier, large cantilever

moments occur until the next pier is reached and toreduce these moments a temporary lightweight steel launch-ing nose is fixed to the front of the box. The effectiveness ofthe launching nose is governed by its length and stiffness,and the optimum arrangement needs to be chosen tobalance the cost of the nose against the cost of cateringfor additional moments in the deck. Typically the launchingnose length would be about 60% of the span length, andwould have a stiffness (EI) of about 10–15% of the concretedeck. An alternative to a launching nose is to utilise a tem-porary tower and stay-cables over the front portion of the

Figure 57 Casting area for launched box girder

Figure 58 Launching device (under launch nose)



deck in order to reduce bending moments. The tension inthe stays is adjusted as the deck passes over a pier tocontrol the moments and forces imposed on the structure.As the deck moves over the piers, each section is subject

to a change in moment and shear. At some stage, eachsection of deck is over a pier and subject to hoggingmoments, or in mid-span and subject to sagging moments,and the prestress design needs to cater for the full range ofmoments as indicated in Figure 59. During launching, afirst-stage prestress is usually arranged to give a constantcompressive stress across the section, normally of theorder of 5N/mm2. This prestress would often consist ofstraight tendons placed in the top and bottom slabs,anchored on the construction joint between segments andcoupled or lapped to extend through each new length ofdeck cast. After completion of launching, a second-stageprestress is installed with a draped profile along the spansto balance the bending moment profile generated by thedeck in its final position.The full length of box section also needs to be strong

enough to resist the shear forces and the temporary bearingload under the webs, as the deck passes over the piers. Thewebs are normally kept a constant thickness and theweb/bot-toms slab corners stiffened throughout to distribute the localloads from the temporary bearings during the launch.Unevenness of the concrete surface and differential

settlement of the piers and temporary supports generateadditional moments and shears in the deck during the

launching operation and these need to be allowed for inthe design.As the deck is launched, the friction in the temporary

bearings applies a load to the top of the piers. The tempor-ary bearings are aligned parallel to the deck rather thanhorizontal like the permanent bearings and this induces afurther horizontal load on to the pier. The piers have tobe designed to resist these horizontal loads in combinationwith the coexistent vertical loads, although the effects maybe reduced by providing stays or guys to the top of the piers.

Concrete cable-stayed bridgesProviding stay cables to support the deck results in aslimmer deck section or in longer spans being achieved.The deck elements may be either a concrete box or abeam-and-slab form with typical arrangements beingshown in Figure 60. The deck may be made up of precastelements, transferred to site and lifted into position, orin situ segments cast inside a travelling form. Eithermethod normally involves building the deck out from thepylons in a balanced manner, installing the stay cables asthe construction proceeds. The advantages of a concretedeck are that the horizontal component of the stay forcecauses compression in the deck which is readily taken bythe concrete, while the mass and stiffness of the deckreduces its susceptibility to vibrations or aerodynamicmovements.The box girder in Figure 60(i) has inherent torsional

strength which makes it suitable for use with a singleplane of stays, while the concrete cross-section is well

(i) Before reaching pier

Bending moment

Bending moment(ii) After reaching pier

Figure 59 Bending moment range in deck during launching

Prestress tendons

Cross-girder

C L

Cross-girder

Prestress tendons

Prestress tendonsin top slab

In situ concretetop slab

Prestress tendons

Stay cable

Stay cable

Stay cableBeam

section

Prestressingtendons

(i) Box girder with single plane of stays

(ii) Semi-box girder with twin plane of stays

(iii) Beam-and-slab with twin plane of stays

Figure 60 Types of cable-stayed decks



suited to precasting to enable fast and simple erection. Thisform of deck was used for the Brotonne Bridge in France,which had a main span of 320m and box depth of 3.8m,and also for the Sunshine Skyway Bridge in Florida witha main span of 366m and box depth of 4.47m.Where twin planes of stays are used then either a ‘semi-

box’ as shown in Figure 60(ii) or a beam-and-slab asshown in Figure 60(iii) are commonly used. A semi-boxarrangement was used for the Pasco-Kennewick Bridge,USA with a main span of 299m and deck depth of2.13m, while typical examples of the beam-and-slab typeare the Vasco da Gama Bridge, Portugal shown in Figure61, with a main span of 420m and deck depth of 2.6mand the River Dee Crossing, UKwith a main span of 194m.

Extradosed bridgesA variation on the cable-stayed arrangement, the ‘extra-dosed’ bridge combines a stiff concrete deck with ‘shallow’cable stays anchored to a reduced-height pylon and hasproven economic in the 100–200m span range. The relativestiffness of the deck gives a structural behaviour more likean externally prestressed deck than a cable-stayed structureand the design is carried out on that basis. A typical arrange-ment is shown in Figure 62(i).

Fin-backed bridgesThese are similar to the extradosed bridge, with the staysencased in a concrete wall or ‘fin’ as shown in Figure62(ii). The fin is an extension of the deck section and stiffensthe deck adjacent to the pier while also providing protectionto the stay arrangement.

Truss bridgesAlthough not commonly used because ofthe complexity in casting and joining thestruts and chords of the truss, severalprecast segmental viaducts and cable-stayed decks have utilised trusses forthe ‘webs’ in order to reduce weight,with a typical arrangement such as thatused on Bubiyan Bridge, Kuwaitshown in Figure 63.

Stressed-ribbon bridgesSeveral footbridges have been con-structed utilising prestressed concrete asa ‘stressed-ribbon’ deck where the ten-dons are laid out between the abutmentsand a concrete deck constructed aroundthem before additional prestressis imposed to compress the deck. Figure64 shows the Kilmacanogue footbridgein Ireland where this technique wasused to good effect.

Figure 61 Vasco da Gama Bridge under construction

Typical cable-stayed arrangement

(i) Extradosed bridge

(ii) Fin-backed bridge

AA

Section A–A

Prestress tendonsConcrete ‘fin’

Extradosed stay arrangement

Concrete ‘fin’

Figure 62 Extradosed and fin-back bridge arrangements



References

America Association of State Highway and Transportation Offi-

cials. (2007) LRFD Bridge Design Specifications, 4th edn.

AASHTO, Washington, DC.

British Standards Institution. (2006) Steel, Concrete and Com-

posite Bridges, Part 2: Specification for Loads. BSI, London,

BS 5400.

British Standards Institution. (1990) Steel, Concrete and Com-

posite Bridges, Part. 4: Code of Practice for the Design of Con-

crete Bridges. BSI, London, BS 5400.

Clark J. L. (1976)AGuide to the Design of Anchor Blocks for Post-

tensioned Prestress Concrete. CIRIA, London.

Highways Agency. (1994) Design Manual for Roads and Bridges.

The Design of Concrete Highway Bridges and Structures with

External and Unbonded Prestressing. HMSO, London, BD 58/

94.

Maisel B. I. and Roll F. (1974)Methods of Analysis and Design of

Concrete Box Beams with Side Cantilevers. Cement and Con-

crete Association, London.

The Concrete Society. (2002)Durable Bonded Post-tensioned Con-

crete Bridges. The Concrete Society, Camberley, Technical

Report No. 47, 2nd edition.

Further reading

Hambly E. C. (1979) Bridge Deck Behaviour. Wiley, New York.

Hewson N. R. (2003) Prestressed Concrete Bridges: Design and

Construction. Thomas Telford, London.

Podolny W. and Muller J. M. (1982) Construction and Design of

Prestressed Concrete Segmental Bridges. Wiley, New York.

Appendix I. DefinitionsBonded Where the tendon is bonded to thetendons concrete member, by being cast-in as for

pre-tensioning or through grout for post-tensioning.

External Where the tendon is placed outside thetendons concrete section.

Internal Where the tendon is placed within thetendons concrete section.

Post-tensioning Where the tendon is stressed after theconcrete has hardened, with load transferduring the stressing operation.

Pre-stressing A wire, strand, a bar or bundle of strandstendons used to prestress the concrete.

Pre-tensioning Where the tendon is stressed beforeplacing the concrete, with load transferoccurring after the concrete has set.

Unbonded Where the tendon is not connected to thetendons concrete other than at the anchor

positions, allowing the tendons andconcrete to act independently along thetendon length.

Appendix II. Symbols and notationusedA or A1 area of reinforcement to be providedAc area of concrete sectionAL area of concrete considered for longitudinal

shear checkAo area enclosed by median wall lines around the

boxAp area of non-prestressed reinforcementAr area of reinforcement across failure planeAs area of slabAsk area of shear keyAsl area of longitudinal reinforcementAst area of leg of link around sectionAsv area of shear reinforcementAt cross-sectional area of tendonb breadth of member or webbs width of slabC compressive force generated in concrete at

ultimate moment capacityDt nominal diameter of tendon

Figure 64 Kilmacanogue stressed-ribbon footbridge, Ireland (courtesyof Roughan and O’Donovan, Dublin)

Figure 63 Bubiyan Bridge under construction (courtesy of Bouygues)



d distance from tendons to compression facedc depth of compression in concrete at ultimate

moment capacitydt distance from reinforcement to compression faced1 larger dimension from line of action of anchor

force to the boundary on non-symmetrical prismd2 smaller dimension from line of action of anchor

force to the boundary on non-symmetrical prismEc 28-day secant modulus of elasticity of concreteEt modulus of elasticity of tendonfc stress in the concrete at point consideredf 0c 28-day cylinder strength of concretefci cube strength of concrete at age being consideredfcp compressive stress from prestress at centroidfcu 28-day cube strength of concretefd stress due to unfactored dead load at tensile face

of section subject to Mcr

fp initial stress in tendonfpc stress due to prestress only at the centroid of the

tendonsfpe effective prestress after all lossesfpi stress increase in tendonfpk average compressive stress over shear keysfpt stress due to prestress only at the tensile facefpu characteristic strength of tendonfs initial stress in non-prestressed reinforcementfsi increase in stress in non-prestressed

reinforcementft tensile strength of concrete, taken as 0.24

pfcu

fy characteristic strength of reinforcementfyl characteristic strength of longitudinal

reinforcementfyv characteristic strength of link reinforcementF force in tendon at point being consideredFbst anchor bursting forceFo force applied by the jack at the anchorFs force generated in top slab due to differential

shrinkageh overall depth of memberhmax larger dimension of the sectionhmin smaller dimension of the sectionhwo web or slab thicknessI second moment of area of sectionKL longitudinal shear coefficientk wobble coefficientki concrete bond coefficientkt coefficient dependent on type of tendonl lever arm at ultimate momentlc losses of stress in tendon due to creep of concretelE losses of stress in tendon due to elastic

shortening of concretels losses of stress in tendon due to shrinkage of

concreteL length of deck

Ls width of longitudinal shear failure planeLsp length of span of deckLT free length of tendonLt transmission length for anchorage of

pretensioned strandM moment at section due to ultimate loadsMa moment generated by the change applied

‘instantaneously’Mas-built moment as constructedMcr cracking moment of sectionMfinal final moment after creep effectsMinst: moment if the structure is built instantaneouslyMo moment necessary to produce zero stress in the

concrete at the tensile faceMp primary moment from prestress on sectionMr ultimate moment of resistance at a sectionMs prestress secondary momentmt mass per metre run of tendonmd mass per metre run of deckm1 moment due to unit restraint moment applied at

pier 1m2 moment due to unit restraint moment applied at

pier 2N–A neutral axis of sectionP total, unfactored prestress force acting on sectionPh horizontal force from prestress tendonPv vertical force from prestress tendonp ratio of reinforcement, Ar=bhSl spacing of longitudinal reinforcementSv spacing of link reinforcementT tensile force generated at ultimate momentTs torsional moment due to serviceability loadsTu torsional moment due to ultimate loadsV shear force due to ultimate loadsVc ultimate shear resistance of concrete at sectionVco ultimate shear resistance of concrete uncracked

in flexureVcr ultimate shear resistance of concrete cracked in

flexureVd shear force at section due to unfactored dead loadVk ultimate shear resistance of shear keyVL longitudinal shear force per unit lengthVp vertical component of prestressVs ultimate shear resistance provided by the

reinforcementv shear stress in the concrete due to ultimate loadsvc ultimate shear stress allowed in concretevL ultimate longitudinal shear stress in the concretevt torsional shear stressx distance of point being considered from the

tendon anchorxi smaller centreline dimension of torsion linkY or y0 distance in section from N–A to point being

considered



yi larger centreline dimension of torsion linkyo half length of side of anchor blockypo half length of side of loaded areayt distance in section from N–A to tensile facezt elastic sectional modulus referred to top facezb elastic sectional modulus referred to bottom face� coefficient of thermal expansion of the concrete/8C�m partial safety factor for strength�cs shrinkage strain deformation of the concrete�l change in deck length due to �t

�s differential shrinkage strain between in situ slaband precast beam

�t change in effective temperature"cb strain in concrete at bottom fibre

"ct strain in concrete at top fibre"p strain in prestress tendon"s strain in non-prestressed reinforcement� total angle change (in radians) in the tendon

over distance x� friction coefficients depth factor for shear, given in Table 9 of BS

5400 Part 4�as-built stress in section due to construction sequence�c stress in concrete adjacent to prestress tendon�final final stresses in section�inst: stress in section if built instantaneously� creep factor� AASHTO strength reduction factor



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manuals Design of prestressed concrete