Partially PrestressedConcrete StructuresA Design Challenge

Prof. Dr. Ir.A. S. G. BruggelingProfessor of Civil EngineeringDelft University of TechnologyDelft, The Netherlands

The merits of partially prestressedconcrete as opposed to fully pre-

stressed concrete have been debatedsince the early days of prestressed con-crete construction. On the one hand, thefollowers of Eugene Freyssinet be-lieved very strongly that full prestress-ing produced a crack-free material inwhich flexural tensile stresses shouldnot be allowed under full working load.

On the other hand, some felt that withpartial prestressing, tensile stressescould be permitted, In this manner, par-tially prestressed concrete could be re-garded similar to conventional rein-forced concrete in which cracking ispermitted in the flexural tensile zone. Agood historical account of this con-troversy is given in Ref. 1.

The main advocate of partial prestress-ing was Dr. Paul Abeles who in the1940's conducted research with partiallyprestressed concrete and after World

War II designed several partially pre-stressed structures for British RaiI-ways.'- 3 During the next decade, therevas little activity in partial prestressing.

The major breakthrough in the im-plementation of partial prestressingcame in 1968. Switzerland became thefirst nation in Europe to adopt partialprestressing in its national code, SIAStandard 162. In the last 15 years Swissengineers have successfully designedand built numerous partially prestressedstructures following their code. There isgeneral satisfaction with the provisionsin the Swiss Code and the experiencesgained during the interim using it.

Unfortunately, the progress in de-signing and building partially pre-stressed structures in other parts of theworld has been very limited. Reportsdelivered at the FIP Symposia inBucharest (1980), the University ofWaterloo Conference (1983), 5 and the

140

recent NATO-sponsored Workshop inParis (June 1984) reveal that there arenot only many differing approaches tothe design of partially prestressed con-crete structures but also varying opin-ions on the definition of partial pre-stressing. In addition, experiences withpartially prestressed structures actuallybuilt were reported to be scarce.

There are three major reasons whypartial prestressing is not fully acceptedtoday.

I. Many existing design proceduresfor analyzing cracked prestressed sec-tions are considered either too compli-cated or the calculations are too long,

2. Most codes of practice do not en-courage the use of partial prestressing.

3. Experience with actually built par-tially prestressed structures is lacking.

The purpose of this paper is to showthat the design of partially prestressedstructures is relatively simple and topresent several design examples of ac-tual structures. It should be mentionedthat much of this work has been done incooperation with Dr. Hugo Bachmanns. 7.

in Switzerland. In this manner it hasbeen possible to unify the differingviews on partial prestressing and to comeup with a more general design procedurefor designing such structures.

Experiences gained so far show thatpartially prestressed concrete structurescan be designed and the structural di-mensions determined as easily as thosefor reinforced or fully prestressed con-crete structures. In many cases, partialprestressing offers a better solution withregard to simplification of the rein-forcement and shape of the members,control of deflection and crack width,and favorable redistribution of bendingmoment by stiffness reduction ofcracked sections.

The most obvious expression for thedegree of prestress, K, is the ratio of theapplied partial prestressing force, Pp.,,and the prestressing force, P,, tr, whichcauses full prestress under maximumload, i.e., zero stress at the extreme fiber

SynopsisThe author presents a simplified

approach for the design of partiallyprestressed concrete structures.

It is shown that partial prestressing,together with supplemental nonpre-stressed reinforcement, offers a bettersolution (as compared to fully pre-stressed sections) with regard to di-mensioning and shaping of members,simplification of reinforcement, de-flection and crack control, and redis-tribution of stresses.

The proposed design procedure isillustrated with examples of box girderbridges; beams, double tees, and flatslabs for buildings; river tunnels; stor-age tanks; and other structures. Someof the examples pertain to precastpretensioned members.

of a concrete member:

K= P"rt

Phil

The values of Parr and Pmt are finalforces after subtraction of all losses.

It should be noted that this definitionof x is only correct when both prestress-ing forces have the same centroid.When this is not the case, as for examplewhen fewer tendons are used in partialprestressing, the degree of prestressshould be defined as the ratio:

K _ ""Dec

^Mmax

whereML, = moment which produces zero

concrete stress at the extremefiber of a section (nearest tothe centroid of the prestress-ing force), when added to theaction of the effective pre-stress alone

PCI JOURNALJMarch-April 1985 141

M,,= maximum moment caused bythe total service load (deadload plus live load)

Note that this is the same definitionused by Bachmann.7 It should be ap-preciated, however, that other defini-tions of the degree of prestress may beuseful, as for example the ratio A r , theforce which can be resisted by the pre-stressing steel above to the force whichcan be resisted by the total amount ofsteel (prestressing steel plus nonpre-stressed steel):

Avf„ + Ajj

Note that the ratio A r is independentof the magnitude of the actual prestress-ing force and refers to the portion ofprestress in the ultimate limit state. Onthe other hand, the ratio K refers to theserviceability state and is approximatelythe portion of the actual prestressingforce to the full prestress,

APPLICATIONSAfter presenting the philosophy for

designing partially prestressed concretestructures, several different designexamples are given (1) box girderbridge, (2) double tees, (3) roof beams,(4) submerged tunnel, (5) connection oftank wall to floor slab; (6) flat slabs. Notethat some of the members in theseexamples are precast pretensioned.

Design PhilosophyIn general, it is not necessary to

choose the degree of prestress be-forehand. In fact, in some cases it iseven inadvisable. The majority of thefollowing examples have been designedby starting with sound structural princi-ples. The cross section and shape of themember have been chosen on the basisof simple formwork and reinforcementcage, as well as ease of casting, vibra-tion, demolding, and curing. The type of

tendons also has been chosen and thenumber of tendons that can be placed inthe section has been determined. In ad-dition, the profile of the tendons and thelocation of the nonprestressed rein-forcement are known.

With the member roughly designedand all loading provided, the bendingmoments for various load combinationscan be determined. Knowing the profileand number of tendons, the prestressingforce and the degree of prestress can becalculated. Next, the amount of non-prestressed reinforcement which has tobe added to fulfill serviceability re-quirements can be determined, basedon an ultimate limit design state. Then,from the crack width calculation, the bardiameters of the reinforcement are de-tennined and, when necessary, the areais increased. In the event a section doesnot follow the above design require-ments this design procedure must berepeated, but this is seldom necessary.

This method avoids choosing a degreeof prestress beforehand including suchimpractical results as fractional numbersof tendons, tendons which cannot becontained in the section, and disruptionof the regular stirrup arrangement.Therefore, the best approach is to firstdesign the section and then choose theprestressing tendons and determine theamount of additional nonprestressedreinforcement. In this way a soundstructure is assured.

1. Box Girder BridgeA statically indeterminate box girder

for an overpass designed to carry highwaytraffic is shown in Fig. 1. This structurehas a center span of 41.60 m (136 ft 5 in.)and two side spans of 32 m (105 ft). Thebox girders are prestressed with Type12/12.9 VSL cables with an effectiveprestressing force of between 1200 and1350 kN (276 and 310 kips).

If this overpass were designed as afully prestressed structure, it would re-quire 18 tendons, nine in each lateral

142

C-)

0CJJz

CAt

C,

CI1

F^ia^IA^i^1C\1

2500 500 5000/2

11000

Fig. 1. Cross section of fully prestressed box girder in midspan with only nonprestressed reinforcement shown.

200

200

60

705070

if x=0,68

zoo

60"' i20 70 60

50 4

Fig. 2. Location of tendons in box girder at midspan (2) and oversupport (1). The threetendons per web, which are not necessary if partial prestressing (K = 0.68) is applied,are shown hatched.

wall (taking into account losses of pre-stress due to shrinkage and creep of theconcrete, relaxation of the prestressingsteel, and frictional losses during ten-sioning of the tendons).

Fig. 2 shows the location of the pre-stressing tendons at midspan and overthe support, In both parts of the struc-ture, three tendons from the web arespread apart in the bottom and topflanges, respectively. This means thatalong the length of the girder the ten-dons from the bottom flange first have tobe curved sideways and then upwardinto the web.

Three tendons from the web aresimilarly curved into the bridge deck

over the support. This layout of the ten-dons is fairly complex and, therefore,not easy to carry out in the actual struc-ture. In addition, the solution is un-economical. If the prestressing tendonswere not spread over the bottom flangeand deck but rather kept only within theweb zone, a total of20 tendons would berequired because of the smaller internallever arm of the prestressing force.

Bending moment in a box girdercauses almost pure tension (or compres-sion) in the top flange. As is generallyknown, the required reinforcement per-centage for pure tension is quite large(between 0.4 and 0.6 percent) since thereinforcement should not be allowed to

144

Table 1. Amount of prestressed and nonprestressedreinforcement in top and bottom flanges of box girder for twovalues of degree of prestress.

K =1.02 K =0.68

A, A, Aa A,mm2 mm' mmg mm2

Location (in') (in?) (in. 2 ) (in.2)

End span (bottom flange) 21870 5100 14580 5100(33.9) (7.9) (22.6) (7.9)

Intermediate support (top 21870 12000 14580 12000

flange) (33.9) (18.6) (22.6) (18.6)

Center span (bottom flange) 21870 5100 14580 12800(33.9) (7.9) (22.6) (19.8)

yield if a tensile crack appears:

A 8fev > Ae}euc

where A, is the area and f,,, is the yieldstrength of the nonprestressed rein-forcement, respectively; A. is the area ofthe deck or bottom flange and f,, is theallowable tensile stress of the con-crete (5 percent probability of having alower value).

For example f,, = 400 N/mm 2 (58 ksi);f,,k = 2 NImm2 (290 psi);A s > 0.5 percentofA,. The area of this reinforcement canbe accounted for in calculating the ulti-mate bending moment M,,. In addition,the nonprestressed reinforcement helpscontrol the crack width,

if the box girder is partially pre-stressed, the degree of prestressing forthe girder in the middle of the centerspan is found to be K = 0.68. Now 12tendons are needed instead of 18 ten-dons. The tendon profiles have beenkept as simple as possible, stayingwithin the web zones and not spreadinto the deck (at the support) or in-to the bottom flange (at midspan) (seeFig. 2).

The nonprestressed reinforcement inthe deck and bottom flange must nowsatisfy the following conditions:

1. Be at least equal to the minimumrequired cross-sectional area (top andbottom flange subjected to almost puretension). In this case, the area of the pre-stressed reinforcement is not taken intoaccount because it is situated only in ornear the webs.

2. Together with the prestressingsteel, provide adequate safety againststructural failure.

3. Limit the crack width at the rele-vant sections.

4. Limit the stress variation A rr, in theprestressing steel and in the nonpre-stressed reinforcement due to maximumlive load. Note that a value of A a-, { 140Nlmm5 (20 ksi) has been adopted forboth types of reinforcement.

The prestressed and nonprestressedreinforcement is shown in Figs. 1 and 2,and their areas are given in Table 1for a fully prestressed (K = 0.68) struc-ture.

Table 1 shows that in the partiallyprestressed bridge, additional nonpre-stressed reinforcement is necessary onlyin the middle part of the center span.Here, the added nonprestressed rein-forcement reduces the crack width andstress variations in the prestressedreinforcement and brings the safetyfactor to the required level (also see

PCI JOURNAL/March-April 1985 145

Table 2. Mean concrete stresses and safety factors for two degrees ofprestress for middle part of center span.

Degree ofprestress Percentage of reinforcement

Mean concretestress Q,.,,, , ,,

Safetyfactor

Nonprestressed PrestressedK A A,* PN N/mms (psi) Y

1.02 (0.95) 0.11 0,48 –4.4 (-638) 2.10.68 (0.83) 0.28 0.32 –2.9 (-420) 1.8

"Only the reinforcement in the tensile bottom flange is taken into account.

Table 2 for other pertinent data).Calculations show that in the fully

loaded, partially prestressed bridge no(or only very fine) cracks occur in theflexural tensile zones in the end spanand over the intermediate support. Inthe end span the mean value of the con-crete tensile stress in the bottom flangeis 1.9 N/mm2 (275 psi). The tensile stressabove the intermediate support also isvery low under maximum load.

The crack width in the center span isbetween 0.1 and 0.15 mm (0.004 and0.006 in.) and the stress variation in thenonprestressed reinforcement is about130 Nlmm2 (18.9 ksi). The degree of pre-stress (0.68) is related only to the bend-ing moment of decompression, M,, inthe tensile zone of the center span (seeTable 2).

Changing from fully prestressed topartially prestressed concrete offersseveral advantages including fewer ten-dons, simpler tendon profiles, and amore appropriate safety factor (1.7 in theDutch Code). Also, the area of the non-prestressed reinforcement in the centerspan increases locally (bars 4 'k 19-200instead of bars 0 12-200; or #6 bars at 8in. instead of #4 bars at 8 in). Thus,cracks occur only in the center spanbox girder when the structure is fullyloaded and close altogether tinder deadload.

2. Double Tees

Double tees are factory produced ona large scale in the precast prestressedconcrete industry. The usual production

2400

14oU _600

_135 135

^e0 yea

Fig. 3. Cross section of a standard double tee.

146

Table 3. Initial concrete stresses at bo ttom fiber and safety factor for twodegrees of prestress.

Degree ofprestress Percentage of reinforcement

Initial stress atbottom fiber Q,,

safetyfactor

Prestressed NonprestressedK a r pp P. N/in • '° (psi) Y

1.0 (1A) 0.56 — -23 (-3335) 2.1

0.65 (0.85) 0.45 0.38 -13 (-1885) 1.9

method uses a prestressing bed withstraight strands, although draped strandsare also used. A cross section of a typicaldouble tee (manufactured in theNetherlands) is given in Fig. 3.

The simply supported element has aspan of 18.30 m (60 ft) and a depth of0.60 m (2 ft). The double tee has drapedtendons and the concrete quality is B37.5 corresponding with a standardprism strength of 30 NImm2 (4.35 ksi) at28 days.

The initial steel stress in the prestress-ing steel at transfer is 1300 N/mm2 (189ksi). The dead load of the double tee is8.9 kN/m (0.62 kips/ft) and the live loadis 10.6 kN/m (0.74 kips/ft), or 55 percentof the total load. In the case of full pre-stressing (K = 1) in each rib of the dou-ble tee, ten seven-wire strands with ac ross section of 100 mm 2 (0.155 in. ․ )each are needed.

After transfer of the prestress, thecompressive stress in the bottom fiber atmidspan is 23 N/mm 2 (3335 psi). Thecompressive stresses in the anchoragezones are also lower due to the use ofdraped tendons. Note that the DutchCode allows only 18 N/mm 2 (2610 psi)permissible initial compressive stress inthe bottom fiber for the given concretequality.

From this example, it is clear that withthe given span and depth, a fully pre-stressed double tee cannot be used.

Therefore, one of the following so-lutions must be chosen: (1) increase the

depth of the double tee or (2) decreasethe initial prestressing force by eitherreducing the number of prestressingstrands and adding mild steel rein-forcement in the middle part of the span,or decreasing the initial stress in thestrands in such a way that the initialcompressive stress in the bottom fiber atmidspan is 18 Nlmm2 (2610 psi) or less.In both cases, partially prestressed con-crete is used, permitting cracks in thetensile zone at midspan.

Fig. 4 shows the distribution ofstrands and bars 'h k 12 over the sectionsat midspan. The use of mild steel rein-

bent up ® bars 0kl2near support 0 strands 7 0 kr:

135

Fig. 4. Solution with partial prestressing.

12

30

i2

30

12

30

12

30

12815

PCI JOURNAL^March-April 1985 147

roof slab

joist

1504'k10- 350 L ^ 3 41 1 1-m l

75f I ' I

mklo-3_i40 3 of 14in)

I I

_

925 I I i 21O(2#31

I I 0k 10-350

2I(t)E231

I©1 2mk1012*31

125 o

t a 2 C325

I 1 L4k1214N4 1

400

Fig. 5. End section (left) and midspan section (right) of fully prestressed roof beam.

forcement in the production of doubletees might appear uneconomical inwhich case using a lower initial stress inthe strands can be considered. How-ever, one should also consider theproblem of durability and the increasein steel stress due to cracking of the con-crete.

Therefore, the preferred solutionwould be to reduce the number of pre-stressing strands and to add nonpre-stressed reinforcement at midspan.

The deflection of the double teeunder permanent load is calculated tobe 14 mm (0.55 in.), Under full load(taking into account loss of stiffness dueto cracking of the concrete), the total

deflection load is computed to be 37 mm(1.46 in.).

Table 3 shows that the initial com-pressive stress in the bottom fiber in thedouble tee is much less than in the caseof x = 1. The effect of reducing the ini-tial prestressing force and adding mildsteel reinforcement can now be seen.This illustrates one advantage of partialprestressing in factory produced pre-stressed concrete members.

In general, partial prestressing is ad-vantageous if the ratio of the dead loadto live load is high and some elements ina multistory building carry higher loads(e.g., archives) with no increase of con-struction depth allowed in that area.

148

o longitudsnal reinforcement

1(2#3)

DIi ^^ I lh& 3 at 14 in)

DI Io 1B12 (8-#4)

. 7\______ I 3tendons

BBR -B20

150

75

1375

800

220

45 45 5

220

Fig. 6. Midspan section of partially prestressed roof beam.

3. Roof Beams

The roof beam considered here has aspan of 30.4 m (100 ft) and is supportedon 7 m (23 ft) high columns. The beamcarries the actual roof structure consist-ing of roof slabs and joists. The loadingon the beam consists of dead load q,, =11.7 kNlm (0.80 kips/ft) and live load qt= 6.9 kN/m (0.47 kips/ft). The deadweight of the beam is 12.8 kNlm (0.88kips/f). The cross section of the beam,designed as a fully prestressed member,is shown in Fig. 5.

The beam is prestressed by four ten-dons, BBR-B20, with 20 wires 4 P 7 mmeach. In addition, the beam also has

nonprestressed longitudinal and stirrupreinforcement. The roof beam, designedas a partially prestressed concretemember, is shown in Fig. 6.

As a result of partial prestressing, thecross-sectional shape of the beam can bemade simpler than with full prestress-ing. This can be explained as follows.With fully prestressed concrete thebottom flange has to be of considerablewidth (due to the large prestressingforce) because the initial compressivestress in the bottom fiber is not allowedto exceed a certain maximum value, de-pending on the specified strength ofconcrete employed.

However, with partially prestressed

PC3 JOURNALIMarch-April 1985 149

concrete the compressive force is sig-nificantly smaller because less prestressis needed. Therefore, a concrete sectionwith a smaller bottom flange can beused (or in this case a section without abottom flange). The beam is also easierto extract from its formwork because theside walls do not have to be removed.

In addition, partial prestressing re-duces the number of prestressing ten-dons to three, as compared with fourtendons in the case of full prestressing.Supplementary nonprestressed rein-forcement has to he installed only at thetensile face in the midspan region of thebeam. This reinforcement does not ex-tend over the whole length of the beam.Detailing of the main longitudinalreinforcement is shown in Fig. 7.

When the beam is fully loaded, thecalculated maximum crack width is lessthan 0.05 mm (0.002 in.). Some pertinentdata of the two solutions are given inTable 4.

Although it will not be furtheranalyzed here, it should be mentionedthat this roof can also be produced as aprecast concrete member with preten-sioned steel. In that case, 22 strands of12.4 mm (0.5 in.) nominal diameter steelgrade FeP 1860 (f, = 298 ksi) will berequired together with 14 reinforcingbars of 12 mm nominal diameter (13-#4bars) in the midspan region.

4. Submerged TunnelFig. 8 shows a typical cross section of

a European submerged tunnel. Themaximum load is 220 kNIne (4.3 kips/fl2)or approximately 20 m (66 ft) waterdepth. An abnormal load of 265 kN/m$(5.2 kips/ft) or about 24.5 in) waterdepth is also considered, bu prob-ability of this load occurring is only oncein 10,000 years based on the water con-tinuing to flow over existing river dikes.

Generally, submerged tunnels are

00co

7x2 s5k12(7x2#4)

—4012(4.#4)

31ooa12

Fig. 7. Main longitudinal reinforcement (prestressed and nonprestressed) near undersideof partially prestressed roof beam (note that vertical scale is five times horizontal scale).

150

Table 4. Mean concrete stresses and safety factor for two values ofdegree of prestress.

Percentage o Mean concreteDegree of atonprestresse stress, after losses Safety

prestress reinforcement

7Percentage

a«.m factor

K A: P. a Nfmm` (psi) Y

0.90 (0.95) 0.15 0.60 —6.1 (-885) 2.0

0.68 (0.84) 0.42 0.47 —4.9 (-710) 1.7

Pertains to midspan section.

composed of precast segments 100 to150 m (325 to 500 ft) in length concretedin excavated dry dock. The completedsegments are floated to their destinationwhere they are sunk to the designateddepth (in a trench dredged in advance)and joined to one another. The wallthickness of the segments is governedmore by transporting and sinking re-quirements than by considerations ofstructural strength. Such tunnels arebuilt mainly with reinforced concrete.

Because of the heavy water load onthe structure, the bending moments to

which it is subjected are very large,whereas the compressive stresses in thewalls, roof, and bottom slab are rela-tively small. These conditions necessi-tate a large quantity of nonprestressedreinforcement, which can be accommo-dated only by the use of large diameterbars, often arranged in several layers. As

a result, it is difficult to limit the crackwidths in the tensile zones to low valuesand calculated crack widths of 0.4 mm(0.016 in.) or more are actually quitecommon.

Why, then, is prestressing not consid-

ELA

r

to.L5

Fig. 8. Idealized cross section of submerged tunnel.

PCI JOURNALJMarch-April 1985 151

ered? Curved prestressing cables canexert pressures of the same magnitudeas the water pressure on the tunnel sothat no tensile stresses need occur.However, additional prestressing mustbe applied while the tunnel segmentsare still in their construction dock andthe loads acting on them are due only totheir dead load weight and the reduc-tion of the subgrade, which are bothsmall in comparison with water pres-sure. Thus, full prestressing of the tun-nel segments would cause large tensilestresses and failure of the structurewould he very probable,

Nevertheless, prestressing of tunnelsegments remains an attractive proposi-tion. It will be apparent, however, that itwill not be possible to apply full pre-stressing, only partial prestressing to adegree that no problems arise in stress-ing the tendons in the constructiondock. Tensile stresses are allowed tooccur, both in the dock and when thesegment has been sunk into position, i.e.,

the combination of nonprestressed andprestressed reinforcement must ensurethat the crack widths are suitably lim-ited.

If partial prestressing is accepted, theapplication of a symmetrical nonpre-stressed reinforcement is attractive be-cause of its simplicity. This means thatthe top and bottom reinforcement (e.g.,in a roof or bottom section of the tunnel)have the same cross-sectional area overthe entire width of the tunnel, Thestructure must then be designed so thatthe crack widths will be very nearlyequal in the following cases: in the con-struction dock under initial prestressplus dead load, and under service con-ditions with effective prestress plusdead and live load.

In this context, note that a particularcrack width in the dock is less seriousthan the same crack width under serviceconditions, with regard to the temporarycharacter of the stay in the dock. Crack-ing also can be expected to occur in the

concreting after prestressing D k 25-200 k16-200(#t8at8"^ /{#5at8")

B.R.Y. –U R- 500stirrups

(t16 (f^ 5),

ON 25-200 i,15-200(;# 8 at 8 -) (#s at 8"?

Fig. 9. Prestressed and nonprestressed reinforcement of tunnel roof.

152

tunnel roof. This will occur in the con-struction dock on both the upper face atmidspan and on the lower (inner) facenear the center support, as well as underservice conditions on the lower face atmispan and on the upper face near thecenter support. The stress situation isjust the opposite for the bottom slab ofthe tunnel.

The reinforcement and prestress ar-rangements for the tunnel roof areshown in Fig. 9. The area of the non-prestressed reinforcement at the upperas well as the lower face is approxi-mately 0.27 percent of the concretecross-sectional area.

The prestress is located about 0.11 m(3.6 ft) from both the upper face of theroof near the center support and atmidspan from the lower face. The pre-stressing tendons are entirely within thereinforcement installed at the upper andlower faces. The prestressing systemchosen for this tunnel is BBR-U3 with12 strands of 12.9 mm diameter and ini-tial prestressing force of 1680 kN (386kips).

The calculations show that crackingcan be expected only in that part of thetunnel roof which is adjacent to thecenter support. The crack widths (givenin mm) were calculated with the formulagiven in the Dutch Code of Practice forconcrete construction:

w = (8) (10') r ( 2c + ^-) (1)

whereo•, = steel stress (N/mm2)c = concrete cover (mm)(b k = bar diameter (mm)p = As IAA

The calculated crack widths near theinner support are:During stressing of the tendons

.......... 0.11 mm on inner faceUnder dead plus live load

............. 0.16 mm on outer faceUnder abnormal load

.......... 0.39 mm on outer face

A further analysis of cracking showsthat the crack widths calculated in thisway are sensitive to the amount of cal-culated effective prestress. If this pre-stress decreases to 90 percent of thevalue adopted, the crack width underdead plus live load will increase from0.16 to 0.26 mm. This underlies the ex-treme importance of taking into accountrealistic prestress losses.

The tunnel roof at the above men-tioned center support can be charac-terized as follows:

• Degree of prestress – x = 0.73 (A,= 0.82)

• Percentage of nonprestressed re-inforcement – p, = 0.27 percent

• Percentage of prestressed rein-forcement – p,, = 0.27 percent

• Mean concrete stress:a,,^,a = –3.3 N/mm2 (-480 psi)(prestress only)

= –4.1 Nlmm2 (-595 psi)(stresses by horizontal water pres-sure included)

It can be concluded that the tunnelroof and the bottom slab can be effi-ciently constructed with partially pre-stressed concrete. This will result in asimplified reinforcement arrangementand a relatively simple prestress profile.Moreover, crack widths can be very ef-fectively controlled.

5. Connection of Tank Wall toFloor Slab

In many storage tank structures withhorizontally prestressed concrete walls,the circular curved wall is designed tomove freely in relation to the bottom orfloor slab. The joint between wall andfloor is detailed in such a way that theanticipated horizontal displacementscan be properly accommodated and thejoint remains liquid tight. A joint of thistype is shown in Fig. 10.

A special case is the joint between thebase slab and the safety wall around atank containing liquified natural gas(LNG). This joint has to remain liquid

PCI JOURNALJ March-April 1985 153

ou,

me

Fig. 10. Joint under safety wall around LNG tank.

tight at very low temperatures to ensurethat, in the event of an inner tank failure,the cold liquid will not flow out over alarge area." Such a joint is complicatedand expensive because the nonpre-stressed reinforcement, the horizontaltendons, and the anchorages of the ver-tical tendons are situated in the jointarea. In addition, the joint has to be in-spected regularly and maintained.

Separate connection details areneeded for transmitting to the base slabthe horizontal forces that may act uponthe safety wall in the event of an exter-nal explosion or earthquake. Fur-thermore, in the occurrence of differ-ential settlement of the relatively flexi-ble slab, the joint will have to allow de-formation in the vertical direction.Therefore, from a technical viewpoint,the construction of a wall-to-floor slabconnection is quite complex.

In the tanks of sewage treatmentplants and other such installations, a"free" joint between wall and base slabis a construction detail which requires

much care and attention and which,after being put into service, will con-tinue to need maintenance. If thewall-to-floor slab connection is ofmonolithic construction, many problemsand difficulties are eliminated.

Such a joint does not require extramaintenance and the tank can be em-bedded easily in backfill soil. However,at the time of horizontally prestressingthe wall, the deformation in the hori-zontal direction is restrained by themonolithic joint. This means that part ofthe prestress is transmitted to the floorslab, resulting in shear forces andbending moments (with respect to thehorizontal plane) occurring in that partof the wall adjacent to the floor slab.This can result in local curvature of thefloor slab (see Fig. 11).

The bending stiffness and bendingmoment at the wall-floor slab connec-tion is reduced by cracking. Therefore,when partial prestressing is applied andhorizontal cracks in the wall occur, thebending stiffness and the vertical

154

Ld med

med wall

formed slab

hin

—pile (deformed)

a

rigid connection

Fig. 11. Deformation of wall and bottom slab due to horizontal prestressing of wall.

bending moment in the wall are re-duced (Fig. 12). Research has shownthat the bending moment at the wall-floor connection can he nearly halvedand the shear forces reduced.

The mild steel reinforcement and pre-stressing steel arrangements for a con-nection of this type are shown in Fig. 13.This particular example concerns asafety tank for liquified petroleum gas(LPG) according to the C-IS system.This is an isolated steel tank, sur-rounded by a prestressed concretesafety tank. The circular space betweenthe walls is filled with nitrogen gas. Thetank wall has been designed to resist awater pressure corresponding to a depthof 26 m (85.5 ft) in the water test. Thewall-to-floor connection was found tomeet expectations entirely.

The following data are relevant to thepattern of forces acting at the connec-tion:(1) Ring prestress (tangential stresses) oftank wall:

(a) Free movement of wall in relationto the bottom: o- = –11.7 N/mm2(-1695 psi).

(b) Monolithic connection, as in Fig.13, analyzed on nonlinear elasticassumption: o = –2.6 Nlmm2(-375 psi).

(2) Bending moment at connection (Fig.12a):

(a) Analyzed on linear elastic as-sumption: M = –500 kNm/m(-112 kip-ft/ft)

(b) Analyzed on nonlinear elastic as-sumption: M = –370 kNm/m (-85kip-ft/ft)

PCI JOURNAL/March-April 1985 155

r=25m

25akNm/m

(ALE)

Mxx

2.QQrn J i 90mpJ

x

8m7

654 d w =O.6Orr

32 EQ1oTI

E

I

01rn

Fig. 12. Moment diagrams at wall-to4loor connection of tank. Left: after initial prestress. Right: water test and effective prestress.LE = linear elastic, NLE = nonlinear elastic.

(3) Water test, 26 m (86.5 ft):(a) Tangential compressive stress at

base of wall: o = –0.9 N/mm2(-130 psi)

(b) Bending moment at connection,analyzed on nonlinear elastic as-sumption (Fig. 12b): M = +250kNm/m (+52 kip-ft/ft)

It can be seen that the cracking of apartially prestressed structure has beenused to reduce the bending moment to avalue which can he resisted by the sec-tion.

6. Flat Slab Floors

In Europe, flat slab floors are usuallydesigned by the support strip method.Such slabs are extremely suitable forpartially prestressed concrete construc-tion. The design should be based on asimple bottom reinforcement mesh,combined with simple mesh reinforce-ment at the upper face in the columnstrips. This mesh reinforcement must besupplemented with extra mild steel barsover the support. The prestress is

600vertical prestressing

a (U-shaped)

sv - 0 PSV = 0.67

Psh = 0.41 % ° Psh = 0.43a

ring prestressing construction-joint0

0

o i) o o foundation slab

0 0

concrete piles0.45 x 0.45 m

Fig. 13. Monolithic wall-to-floor connection of LPG safety tank.

PCI JOUHNALJMarch-April 1985 157

CDm` 5 x 7, 20

0 0N N

zone where unbondedtendons are concentrated

EIIi 1111111 — o

1110L 7,20

Fig. 14. Plan of middle panel, Unbonded tendons are concentrated in a zone of 31 in. (800 mm) aroundcenterlines of columns.

applied in the column strips and func-tions as an artificially applied loadwhich counteracts part of the load in themiddle panel between the column strips(see Fig. 14).

As shown in Fig. 14, a load of 4.7kN/m2 (0.093 kips/ft) has been adopted,excluding the dead weight of the con-crete slab which is 0.22 m (9 in.) thick.The column strips contain ten prestress-ing tendons, each consisting of one 12.9mm diameter strand. These tendons areinstalled in the two mutually perpen-dicular column strips which act as sup-port strips.

Fig. 15 shows the reinforcement andprestressing arrangements at the slab-to-column connection. Studies indicatethat in flat slab floors the effect of thenormal force due to prestress upon thecrack width is often negligible.

The middle panel under considera-tion has been analyzed for two cases,namely, prestress without bond and pre-stress with bond, both for p, = 0.27 per-cent and a degree of prestress, K = 0.56.The difference in p, of nonprestressedreinforcement is small: p, = 0.44 percentfor prestress without bond and pr = 0.41percent for prestress with bond. In bothcases the safety factor y = 1.7 and themean concrete stress o-,,„ , ,, = —0.8NImm2 (- 116 psi).

At the center of the panel, the ap-proximate crack width under full load is0.26 mm without bond and 0.11 mmwith bond. It is apparent from these val-ues that in the latter case the prestress-ing steel acts also as reinforcement andthus helps to reduce the crack width.

INFLUENCE OFTIME-DEPENDENT EFFECTS

In a reinforced concrete column sub-ject to a large axial compressive loadover a long period of time, a redistribu-tion of stresses occurs. This stress redis-tribution is due to shrinkage and creepof the concrete and elastic shortening of

the reinforcing steel. As a result of thistime-dependent shortening of the con-crete, the longitudinal reinforcementalso shortens and carries a larger pro-portion of the column's load, thus reliev-ing the concrete of part of the load.

For an axially loaded column the con-crete stress at time t = x can be calcu-lated by various methods. 1° Two of theseare:(1)Effective modulus method:

17

r9 l + Tep(l+^m) (aco )pEt€n,,,

(2)

(2) Dischinger's method:

ac^ _ aco Ivey + Es^E^p^1(1–ems)4),,(ijl

+n

(3)

where

= np 4xI +tip

Although there are more advancedanalytical methods than Dischinger'sequations to determine the concretestresses, it appears from the calculationsin Ref. 10 that this method gives the bestresults when compared to test data.

Although the preceding discussionrelates to a column, the theory alsoapplies to the tensile zone of a pre-stressed concrete beam, except that thebeam calculations are usually morecomplicated. The general behaviormode of a partially prestressed beamsubject to sustained load is that thecompressive stress in the extreme fiberof the concrete section decreases whilethe compressive stress in the reinforcingsteel increases, i.e., the tensile stress inthe prestressing steel decreases.

Using Dischinger's method, the rela-tion between the concrete stress o-, inthe bottom fiber, combinations of time-dependent effects p ; €, ), and thereinforcement cross section (p„ ; p,) havebeen plotted in Fig. 16.

The example chosen is a partially pre-

PCI JOURNALJMarch-April 1985 159

column strip 1700 mm

Ok12-200

220

180

200 400 200

Fig. 15. Slab-to-column connection.

'l'

600 mmv0C- (Tc20CXZ

-6

Ow ^EE

? 5

C 4

t ^d-4

U,

mE^

n oE

-3

-2

U -

-- 2

150 m

150640

A s100

A

P5=4

methodsimple

Mme= 169 kNmP. = 573 kN'Qp i = 1100 N/mm2Ap = 521 mm2q c = 180 x 103mm 2E c = 35,000 N Ircm2

Ps'ZPp E =205,000N/mm2

1-150■10 6

2- 550■10 6

3-350xifl 6

creep coefficient IQmfree shrinkage Ecsm

_PS=6PpFig. 16. Relation between concrete stress cr cz in bottom fiber and variouscombinations of creep factor 4. and shrinkage value € , for three values of ps.

stressed concrete tee beam. Relaxationlosses of the steel stresses were nottaken into account. The calculation wasbased on the assumption that the bend-ingmomentMo due to the dead load doesnot change with time and at t - 0 theprestressing force is 573 kN (129 kips)regardless of the cross-sectional area ofthe nonprestressed steel.

Actually, the ultimate load carryingcapacity of the member is increasedwhen supplemental reinforcing steel isplaced in the tensile zone. Directly aftertendon stressing, the compressive stressin the concrete is reduced from -6.1N/mm2 (-885 psi) for p, = 0 to -5.3N/mm2 (-770 psi) forp, = 2 p p or to -4.2N/mm2 (-610 psi) for p, = 6 p,. This isdue to the influence of the nonpre-stressed steel.

In the case where 0- = 2 and e cd,a =-250 x 10-8, the following values of aare found: -4.7, -2.4, and -0.3 N/mm2(-680, -350, and -45 psi), for p $ = 0, pg= 2 p n, and p a = 6 p p, respectively.

Fig. 16 shows that when increasingthe ratio of nonprestressed reinforce-ment, p, (for an equal initial prestressingforce A„ a-,,), the compressive stress 0c2

in the concrete on the flexural tensileface of the beam decreases. This com-pressive stress continues to decreasewith tine-dependent effects, 0. andE es. A

Most codes of practice for fully pre-stressed concrete members allow asimplified method to calculate prestressloss due to shrinkage and creep. Thiscalculation neglects the nonprestressedsteel in the flexural tensile zone andleads to the decrease of o- 2 , as indicatedby the dashed line in Fig. 16. The fol-lowing example illustrates the designapproach.

ExampleSuppose that immediately after

stressing the tendons, the concretestress at the level of the steel centroid is-6.1 N/mm2 (- 885 psi). With 0„ = 2 and

-250 x 10-", the magnitude of thetime-dependent shortening in the ten-sile zone is:

Ec+g 2(- 5.) -- 250 x 101

= -559x10

Hence, the decrease in the steel stressis 559 x 10- e x 205,000 = 114 N/mm2(16.5 ksi).

For a bending moment:Md = 169kNmandPe = 521 (1100 -

114) N, we now obtain rr, g = -4.5N/mm2 ( - 650 psi). For the case where p,= 0, Discbinger's method gives a valueof -4.7 N/mm2 (-680 psi),

This calculation turns out to be verysimple and as Fig. 16 shows, reasonablyaccurate when p, = 0. However, if sup-plemental reinforcing steel is present,the concrete compressive stress o•,swhich develops over time is consid-erably smaller. Therefore, the compres-sive stress in the flexural tensile zone ofa member can be significantly reducedby time-dependent effects when non-prestressed steel is added in the tensilezone of an originally fully prestressedsection.

However, in a partially prestressedconcrete member, part of the prestress-ing steel used to produce full prestressis replaced by nonprestressed steel. Inthis case, the time-dependent effectsonly moderately reduce (which canoften be neglected) the compressivestress. Therefore, a question arises as tohow the simplified analysis affects themagnitude of the decompression mo-ment, MB,,, and the crack width, w, as-sociated with full load, Md+t .

To answer this question, a study wasconducted at the Delft University ofTechnology" using mathematical mod-els in which the stress distributions inthe cross sections of partially pre-stressed structures were determined.Using the simplified design method,several structures (double tees and boxbeam) were analyzed with different de-grees of prestress.

162

Ez

cdE0Ec0NU)C)I-

0E0uC)

degree of prestress x

Fig. 17. Relation between decompression moment M. and degree of prestress K formidspan section of double tee with span of 66 ft (20 m).

This simplified calculation procedureresulted in an almost linear relationshipbetween the decompression momentMme, and the degree of prestress K. Themathematical models also showed whatthe actual M,, versus K relationship is,based on a particular variation of creepand shrinkage of the concrete)' The re-sult is presented in Fig. 17.

It appears from Fig. 17 that M cal-culated for various degrees of prestressby the simplified method differs onlyslightly from M, computed by Dis-chinger's method. Also, variations inshrinkage and creep of the concretehave little effect on the magnitude ofM. The line terminating at 1175 kNmin the diagram was calculated by thesimplified method. The line terminatingat 1106 kNm relates to e,. , ,, = 200 x 10and 4 = 1.5. The next line relates to

= 250 x 10-1 and ¢. – 2.0, and thebottom line to e^ g, m = 300 x 10- 1 and ¢. _2.5.

Although this study appears to con-tradict the results of the calculations

presented in Fig. 16, this is not the case.To clearly show the effect of the steelcross section in the flexural tensile zoneon the stress redistribution due totime-dependent effects (see Fig. 16),reinforcement was added to the sectionwithout the cross-sectional area of theprestressing steel being reduced inproportion. However, in Fig. 17, such areduction was made in the calculations.

The magnitude of the prestressingforce is reduced when the steel cross-sectional area is increased so that theultimate moment Mu of the section re-mains constant. Therefore, reduction ofthe prestressing force means a decreasein the magnitude of the compressivestress o-, in addition to a decrease inthe influence of time-dependent effects.

As a result, the structure becomesstiffer and the concrete stresses undergosmaller increases per unit of bendingmoment than when there is little steel inthe flexural tensile zone. This appliesmore particularly to the stage where thetensile zone is uncracked.

PCI JOURNALJMarch-April 1985 163

DEGREE OF PRESTRESSUNDER FULL LOAD

The Delft University of Technologystudy" investigated the magnitudethe steel stress attains if a structure isdesigned and built with partially pre-stressed concrete. In the study, it wasassumed that the initial stress in the pre-stressing steel was o = 1395 N/mm2(202 ksi). This value is 75 percent of thecharacteristic value of the tensilestrength (1860 N/mm2 or 298 ksi).

The analysis was based on completebond of the prestressing steel to the con-crete, such as occurs more particularlyin prestressed concrete with preten-sioned steel. As an example, the resultsof the calculations for a double tee witha span of 66 ft (20 m) are given in Fig. 18.Other calculations using other beamsgave similar results."

The steel stresses occurring underthree bending moments — Mi, = 695kNin (dead load), MD+L = 1175 kNm (fullload), and M = 940 kNm (half live load)— are shown in the diagram. Thebending moment at which the tensilezone cracks is indicated asM^r. It appearsfrom the figure that the variation inshrinkage and creep of the concrete haslittle effect on the steel stress where theflexural tensile zone has cracked.

The diagram also shows that for lowdegrees of prestress, the steel stressunder full load can become substantiallyhigher than the initial steel stress, due tothe effect ofcracking in the tensile zone.Thus, for K = 0.6 the value attained isapproximately equal to the initial stress,and for x = 0.4 it is higher, namely, 1450N/mm2 (210 ksi). This means that themagnitude of the stress in the prestress-ing steel should he properly checkedwhen low degrees of prestress areemployed.

It is not possible to make definite as-sertions as to the maximum permissiblesteel stress under full load. However,this maximum should not be higher thanthe initial stress o , and preferably, it

should be lower. The problem of thestress in the prestressing steel under fullload and with a low degree of prestressis, however, not as serious as might beassumed.

Complete bond between the prestress-ing steel and the concrete only occurswhen pretensioned tendons are used:partial prestressing with a low degree ofprestress is seldom used. Should a lowdegree of prestress be considered, it willbe necessary to limit the initial stress inthe prestressing steel so that aftercracking, a high value does not occurunder full load. The steel stress at thecrack can he calculated according to themethod described in Ref. 11.

In addition, with prestressed concreteusing post-tensioned tendons, there isnever complete bond between the pre-stressing steel and the concrete. This isobvious when unbonded tendons areused. If the tendons are enclosed insheaths which are grouted after ten-sioning, the role played by the tendon isdifferent than in the case where eachtendon is directly embedded in con-crete. At a flexural tensile crack, thestress in the steel of a post-tensionedtendon does not increase as much as thestress in a fully bonded (embedded) barof the reinforcement or a pretensionedtendon at such a crack. Due to the poorbond, the elongation is distributed overa greater length of the tendon.

In cases with bonded nonprestressedreinforcement in the tensile zone, theeffect of the steel in grouted cables(post-tensioned tendons) can be re-stricted by the introduction of a reduc-tion factor c for the area Ap of the pre-stressing steel:e

1, 1 (4)

J n ^P

wheren = number of wires in a tendon;

each wire in a strand to becounted separately

= diameter of a bar of the non-

164

prestressed reinforcementdi p = diameter of prestressing steel,

i.e., diameter of individualwires or bars or the wires ofwhich a strand is composed

To calculate the steel -stress at thecrack, it is necessary to take into accountthe same proportion of the cross sectionof prestressing steel, namely, the crosssectional area of the steel to be adoptedas tensile reinforcement is equal to A,plus cAp . For example, consider a pre-stressing cable comprising 20 strands (7wires 0 p = 4 mm) in conjunction withreinforcing bars 0, = 12 mm. Then:

c= i^ x 4 =0.25

Therefore, when cracking occurs inthe tensile zone of a member with onlypost-tensioned, grouted cables, therewill be wide cracks at great distances.To distribute the cracks and limit the

crack width, at least a minimum amountof nonprestressed reinforcement shouldbe used in the tensile zone of post-tensioned beams or slabs in whichcracking is not fully excluded. This alsois advocated by Bachmann"-' andothers.

In prestressed concrete with post-tensioned tendons, the tendons are usu-ally located at somewhat greater dis-tances from the underside of the beamthan the reinforcing bars are. For thisreason, too, the increase in stress in theprestressing steel due to cracking willbe less than that in the nonprestressedreinforcement.

The research at Delft Universityshows that the stress increase in the pre-stressing steel after cracking may be-come quite significant when low de-gress of prestressing are employed. Thisis, however, particularly true of pre-stressed concrete with pretensionedsteel.

1500 °p1175 initial. stress Opi =1395 (N /mm2)940

695M

(kNm)

Mcr

Mae

_06 0.8 1.0

degree of prestress x

Fig. 18. Relation between degree of prestress K and stress (r, in prestressing steel underthree different loads.

E 1400E

1300a,

1200e-N

^' 1100LA

N

c 1000

1175940695

U)U,vL

0 0.2 0.4

PCI JOURNALMarch-April 1985 165

LIMITING CRACK WIDTHUNDER FULL LOAD

The magnitude of the crack width isgoverned by the following factors:

1. The steel cross-sectional area that isjoined to the concrete by bond in thetensile zone.

2. The quality (specified strength) ofthe concrete.

3. The diameter of the reinforcing barsand their surface profile (ribs, etc.).

4. The depth of the concrete cover andthe location of the tendons within thesection of the member.

5. The increase of the stresses in theprestressed and nonprestressed steelwhen the bending moment increasesfrom Move to Mmax

In applying the simplified method ofanalysis, the stress in the reinforcingsteel or the increase of stress in the pre-stressing steel can be determined byconsidering the critical section as an ec-centrically loaded section. The princi-ple of this analysis is shown in Fig. 19.

The increase of strain in the nonpre-stressed reinforcement and in the pre-stressing steel will be calculated, asshown by Bachmann" using the princi-ple "bending with compressive forcePe ." The section is conceived as beingloaded by the effective prestressingforce P e which acts at the centroid of theprestressing steel, and by a bendingmoment M1J+L .

The eccentrically acting compressiveforce is resisted by the reinforced con-crete section containing steel with across-sectional area A, 1 in the compres-sive zone and A, + cA„ in the tensilezone. The compressive zone is sub-jected to a total compressive force:

N –Pe+A,u,+Ap4Qpwhere A o- denotes the increase in ten-sile stress (over and above the effectivestress) in the prestressing steel due tothe cracking of the concrete.

For prestressing cables (post-tensioned tendons), a reduction factor cis introduced. This compressive force is

resisted by the concrete in the compres-sive zone (which has a depth h,.) and bythe reinforcing steel A„ . The conditionsof compatibility must be satisfied:

vc,. 1 _ L . 1 . 1E, h.. Ep c (d„ – hi.)

= a12 • 1 1 (5)E, (d3 —hr

In the analysis, the magnitude of h;,.has to be determined by trial and errorso this condition is satisfied. This ap-proach is comparable to the one adoptedin the modular ratio method for rein-forced concrete except that in purebending hr changes when the bendingmoment changes in magnitude. Thetensile stress o-c in the reinforcing steelis obtained from this analysis. The crackwidth w can then be determined withthe aid of governing codes of practice forreinforced concrete.

Fig. 20 shows the relation betweenstress increase a-,,.,. in the prestressingsteel due to cracking in the tensile zoneand the action of both full load (M =1175 kNm) and dead load plus half thelive Ioad (940 kNm). It can now be seenthat by decreasing the degree of pre-stress the tensile zone increase becomesmore pronounced and the effects of con-crete shrinkage and creep become neg-ligible.

In order to show the effects which gov-ern crack widths, a reinforced concretetension member subjected to elongationwill briefly be considered. Such amember is comparable to the tensileflange of a box beam just after thecracking moment has been exceeded.The relation between crack width, bardiameter, and steel stress at the crack(here indicated as a- ,.r) is shown in Fig.21. The reinforcing steel consideredconsists of ordinary deformed bars witha ratio fs between rib height and ribspacing of 0.065.12

It is apparent from this diagram that inthe case of elongation and an associatedsteel stress of, for example, 130 Nlmm2,

166

L0cMz

C

0

deformations

cri

ds I dp

e

us2Fig. 19. Derivation of steel stresses under M. ES

3DC S.Cr

kN

1175

940 .

0.2 04 0.6 Q8 1.0

degree of prestress x

Fig. 20. Relation between degree of prestress K and 'increase of steel

stress A Qs, G , due to cracking.

EE

z

CDL.

da00

a

C

UC

25C

20(

15C

1DC

St

C

_55C

the crack width can be limited to 0.05mm by using bars with 0 k = 6 mm. Thisis, of course, not a suitable diameter forpractical use. If bars with rt k = 10 mmnare used, the crack width is limitedto about 0.08 mm, while 16 mm and 25mm bars limit the crack widths to about0.10 mm and 0.15 mm, respective-lv.

Thus, crack width associated withelongations can he controlled by a suit-able choice of deformed bar diameter,reinforcement cross-sectional area forthe given concrete quality (specified

strength) and location of the bars in thestructural section. Raising the loadabove the value corresponding to thecracking moment causes the steel to in-crease, a fact that must he considered.This can be done by assuming that, athigher steel stress values, the crackwidth increases proportionately to thesteel stress.

In the case of members loaded inbending, it is usually possible to applythe rules for crack width calculationwhich are given in various codes ofpractice.

CONCLUSION

This article briefly reviews researchcarried out at the Delft University ofTechnology with regard to the pos-sibilities offered by partially prestressedconcrete. On the basis of this research it

is shown that a simplified approach tocrack width calculation is permissible.This means that the stresses in thereinforcing steel under full load can becalculated by a procedure similar to the

168

3rOk

2E

a 2a,E

D1L01D

B-22.5

■■MMMENNE ONNUM■■■

• ■►^^\►N\►\ M\NZ

■■■■\i■■■■^1i■■■■rte■■■■■■■■■■■■■■■■■■

steel stress O s.cr( N/mm2)

Fig. 21. Relation between bar diameter d., steel stress o$, cr , and crack width w.Concrete cube strength is 22.5 Nlmm 2 (cylinder strength is 18 NImm2 or 2.61 ksi).

one commonly applied to reinforcedconcrete structures eccentrically loadedin compression. The compressive forceexternally applied to the section is theeffective prestressing force which canbe determined from the initial prestress-ing force in the same way as with fullyprestressed designs.

If the stress in the reinforcing steel isknown, the bar diameters of that steelcan he chosen, within the limits used inactual practice, so that crack width limitrequirements can be fulfilled. However,to do this, it may be necessary to adaptthe reinforcement cross-sectional area tosuit individual cases.

As a result of cracking in the flexural

tensile zone, the stresses in the prestress-ing steel crossing the crack increase. Byusing the simplified method of analysisand design, this effect can be taken intoaccount as well as bond behavior differ-ences between post-tensioned prestress-ing cables and reinforcing steel an-chored by bond.

In conclusion, it has been shown thatthe procedure for designing partiallyprestressed concrete structures is nomore difficult than the design of fullyprestressed or conventionally reinforcedmembers. Partial prestressing offersmany distinct advantages which can beused to produce more efficient and im-aginative concrete structures.

ACKNOWLEDGMENTThe author wishes to express his ap- Technology, The Netherlands, for re-

preciation to Mr. J. Brakel, senior scien- viewing and checking the original man-tific officer of Delft University of uscript.

PCI JOURNAUMarch-April 1985 169

REFERENCES1. Bennett, E. W., "Partial Prestressing—A

Historical Overview," PCI JOURNAL,V. 29, No. 5, September-October 1984,pp. 104-117.

2. Abeles, P. W., "Partially PrestressedConstructions, Built in the Eastern Re-gion of British Railways," IABSE Publi-cation, V. 12, 1952, pp. 1 -14.

3. Abeles, P. W., "Static and Fatigue Testson Partially Prestressed Concrete Con-struction," ACI Journal, V. 53, No. 12,December 1954, pp. 261-276.

4. Reports by Inomata, Kavyrchine et al.,FIP Notes 90 and 91, Fl? Symposium onPartial Prestressing, Bucharest, Romania,September 15-16, 1981.

5. International Symposium on Nonlinear-ity and Continuity in Prestressed Con-crete, V. 1, 2, and 3, Preliminary Publi-cation, University of Waterloo, Waterloo,Ontario, Canada, July 4-6, 1983.

6. Bachmann, Hugo, "Partially PrestressedConcrete: Simplified Design Based onSwiss Practice Since 1968," InternationalSymposium on Nonlinearity and Con-tinuity in Prestressed Concrete, Prelimi-nary Publication, V. 1, University of Wa-terloo, Waterloo, Ontario, Canada, July4-6, 1983, pp. 29-45.

7. Bachmann, Hugo, "Design of PartiallyPrestressed Concrete Structures Basedon Swiss Experiences," PCI JOURNAL,V. 29, No. 4, July-August 1984, pp. 84-105.

8. Bruggeling, A. S. G., "Theory and Prac-tice of Prestressed Concrete" (in Dutch),Stichting Professor Bakkerfonds, Delft,The Netherlands, 1982, Two Volumes,1029 pp.

9. Bruggeling, A. S. G., "Prestressed Con-crete for the Storage of Liquified Gases,"Viewpoint Publication, Cement andConcrete Association, London, England,1981.

10. Bruggeling, A. S. G., Brunekreef, S. H.,and Walraven, J. C., "Partially Pre-stressed Concrete: Theory and Experi-ments," Heron (Delft, The Netherlands),V. 23, No. 1, 1978, pp. 36-62.

11. Bruggeling, A. S. G., "Towards a SimpleMethod of Analysis of Partially Pre-stressed Concrete," Research Report5-82-D16, Delft University of Technol-ogy, May 1983.

12. Bruggeling, A. S. G., "Imposed Defor-mations and Crack Width," Research Re-port 5-80-D13, Delft University ofTechnology, December 1980.

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by November 1, 1985.

170

APPENDIX - NOTATION

= cross-sectional area of con-crete

= cross-sectional area of pre-stressing steelcross-sectional area of non-prestressed reinforcing steel

= modulus of elasticity of con-crete

= E, = modulus of elasticity ofsteel

= bending moment due to deadload

= bending moment of decom-pression

= maximum bending moment= ultimate bending moment= normal force= initial prestressing force= effective prestressing force

(1/ n) (b k l(^ p ) = factor thattakes into account the reducedbond properties of groutedtendons; (n = number ofwiresof the tendon)

= yield strength of nonpre-stressed reinforcement

= permissible tensile strength(upper value) of concrete

= internal lever arm= modular ratio= time= crack width= safety factor for ultimate load

= shrinkage strain from t = 0 tot=00

r1 = coefficient = n p 0, l(1 + n p)K = degree of prestress K =

MDer I Mn+L

p = A:+p /A,p ^ = ABA,p v = An/A,a cz = concrete stress at top fiber

(compressive zone)o cz = concrete stress at bottom fiberCrro = concrete stress at t = 06^ a – concrete stress at t = x

= concrete stress, mean value att=-(P„+Ma)

CrV, = initial stress in prestressingsteel (t = 0)

U„ = steel stress in reinforcementin compressive zone

Q z = steel stress in reinforcementin tensile zone

6„,cr = steel stress in reinforcementat cracking

= creep factor from t = 0 to t =O k = bar diameter of reinforcement

= bar diameter of prestressingsteel

A cr, = stress increase in prestressingsteel due to cracking of con-crete

A U,,« = stress increase in normalreinforcement due to crackingof concrete

A,

At,

A,

E„

MA

Mme'

'WD+LM„

N

P!

FeC

^` xu

I tk

.1

n

t

U)

7

PCI JOURNALJMarch-April 1985 171