Flexural stresses aftercracking in partiallyprestressed beams

Arthur H. NilsonProfessor of Structural EngineeringCornell UniversityIthaca, New York

A method is presented to calculate elasticflexural stresses in partially prestressed concretebeams, in which cracking can be expected atservice load.The effect of mild steelused to supplement thein such construction, cathe analysis.

reinforcement, oftenmain prestressing steeli easily be included in

A numerical example is given to demonstratethe proposed method.

Early in the development of pre-stressed concrete, the goal of pre-

stressing was the complete eliminationof concrete tensile stress at serviceloads. The concept was that of an en-tirely new, homogeneous materialwhich would remain untracked and re-spond elastically up to the maximumanticipated loading.

This kind of design, where the limit-ing tensile stress in the concrete at fullservice load is zero, is generally knownas full prestressing, while an alternativeapproach, in which a certain amount oftensile stress is permitted in the con-crete at full service load, is called par-

tial prestressing.

Abeles, l Thurlimann,2 Leonhardt,3

and others have pioneered in the de-velopment of partially prestressed con-crete construction. They have con-vincingly demonstrated its advantagesover full prestressing, and have shownthat substantially improved perfor-mance, reduced cost, or both, may beobtained through partial prestressing.

Fully prestressed beams may exhibitan undesirable amount of upward cam-ber due to the eccentric prestressingforce, a displacement which is onlypartially counteracted by the gravityloads producing downward deflection.This tendency is aggravated by creepin the concrete, which magnifies theupward displacement due to the pre-stressing force, but has little influenceon the downward deflection due tolive loads, which may be only inter-mittently applied.

Should heavily prestressed membersbe overloaded and fail, they may do soin a brittle way, rather than in a ductilemanner as for beams with a smalleramount of prestress. Furthermore, ex-perience indicates that in many casesimproved economy results from use of acombination of unstressed bar steel andhigh strength prestressing steel ten-dons.

Present ACI CodeProvisions

The tensile stress f2 at the bottom ofa concrete beam may be found fromthe expression:

Pr / ec2 M,f2 =— A I 1 rL I S (1)

,\ 2

in which P e is the effective prestress-ing force after losses, A, is the area ofthe concrete section, e is the eccen-tricity of the prestressing force, r isthe radius of gyration of the concretese-tion, M, is the total moment due todead and live loads, and c2 and S2 are,re>pectively, the distance from the see-

tion centroid to the bottom surface andthe section modulus with reference tothe bottom surface.

The ACI Code4 permits concretetension of 6V7 : psi at full service load,slightly less than the usual modulusof rupture. If explicit calculation ofdeflection indicates that it is withinallowable limits, a tensile stress of12\/fe' psi is permitted.

In each case, the tensile stress is tobe calculated on the basis of propertiesof the untracked cross section. Sincethe higher stress limit is well above themodulus of rupture, this limit corre-sponds to a nominal stress only.

It is further stated in the ACI Codethat a nominal tensile stress higher than12V7„' psi is permitted when it isshown experimentally or analyticallythat performance will not be impaired.

The above provisions clearly permituse of partially prestressed concretemembers.

Flexural StressesAfter Cracking

At the full service load stage, partiallyprestressed beams are cracked, al-though generally both concrete andsteel stresses remain within the elasticrange. While service load stresses at acracked cross section are of secondaryimportance, compared with thestrength and safety of the membershould it be overloaded, calculation ofsuch stresses may be necessary for sev-eral reasons:

1. For prestressed members, crackwidths at service load are related to theincrease in steel stress past the stage ofconcrete decompression; consequently,the service load steel stress must beknown, as well as the stress at decom-pression.

2. An accurate calculation of bothelastic and creep deflection at serviceload requires that curvatures be based

PCI JOURNAL/July-August 1976ҟ 73

P

y Ci 3untracked

dP conc. centroid 2 -

ds J e c2 cracked rneutral axis

S AP - -

PQ alone

^Cs3+—'ES2_^^

"'^EPEP2EPt - EPe-1t'IDecompression 0

Q3 Pe + service load 0

(a)Cracked cross sectionҟ(b) Concrete and steel strains

R

c^ e y

untracked conc. centroid t dP

cl ds crackedneutral axis

L F F

e

AP fPA s fs

(c) Decompressionҟ(d) Forces onҟ(e) Resulting stressesforceҟcracked section

Fig. 1. Basis for analysis of cracked cross section.

on actual, not nominal, stress and straindistributions.

3. If fatigue is a factor in the design,it is necessary to determine actual stressranges in both concrete and steel.

4. Finally, it may be necessary tocompute stresses in the cracked sectionto demonstrate compliance with designcodes.

For an ordinary reinforced concretebeam, calculation of stresses at acracked section is a simple matter. Thetransformed section concept permitsuse of the familiar equations ofmechanics for homogeneous elasticbeams to locate neutral axis, determinesection properties, and calculatestresses. Alternately, explicit equationsmay be derived for nonhomogeneousreinforced concrete sections.5

For cracked prestressed concretebeams, matters are more complicated.The neutral axis location and effec-tive section properties depend not onlyon the geometry of the cross sectionand the material properties, as for rein-forced concrete beams, but also on theaxial prestressing force and the loading.The axial force is not constant aftercracking, but depends on the loadingand on the section properties.

The effective cross section of a typi-cal partially prestressed beam at ser-vice load is shown in Fig. 1(a). Themember shown includes both prestress-ing steel of area AP and nonprestressingsteel bar reinforcement of area A, asis commonly the case. It is assumed thatthe member has cracked, that bothconcrete and steel are stressed only

74

within their elastic ranges, and that thecontribution of the tensile concrete canbe disregarded.

The strains and stresses in the con-crete and steel will be considered atseveral load stages, certain of which arenot actually experienced by the mem-ber, but are considered only as a com-putational convenience.2

Load Stage (1), Fig. 1(b), corre-sponds to application of effective pre-stress Pe alone. At this stage, the stressin the tendon is

fpi = fpe = Pe/Ap (2)

The compressive strain in the barreinforcement at this stage, assumingperfect bond between the two mate-rials, is the same as that in the con-crete at the same level. Consequently,the bar reinforcement is initially sub-jected to a compressive stress:

f8l = — E,EB (3)

Next, it is useful to consider a ficti-tious Load Stage (2) corresponding tocomplete decompression of the con-crete, at which there is zero concretestrain through the entire depth asshown in Fig. 1(b).

Compatibility of deformation of theconcrete and steel requires that thechanges of stress in the tendon and thebar reinforcement as the beam passesfrom Stage (1) to Stage (2) are, re-spectively:

fp2 = EpEp2 (4)

fs2 = E 8Es2 (5)

At this hypothetical load stage, thestress in the bar reinforcement, neglect-ing the effects of shrinkage and creep,is

fs = Es (— Es2 + E82) = 0 (6)

The change in strain in the tendon isthe same as that in the concrete at thatlevel, and can be calculated on the ba-sis of the uncracked concrete sectionproperties:

/ z

Ep2 = A,EO I 1 + r2 (7)

after which f2 can be found fromEq. (4).

The bar reinforcement is unstressedat Stage (2), as noted, but in order toproduce the zero stress states in theconcrete, the tendon must be pulledwith a ficticious external force:

F Ap (fp1 + fp2) (8)

as shown in Fig. 1(c).The effect of this fictitious decom-

pressing force is now cancelled by ap-plying an equal and opposite force Fas shown in Fig. 1(d). This force, to-gether with the external moment Mtdue to self-weight and superimposedloads, can be represented by a resultantforce R applied with eccentricity eabove the uncracked concrete centroid,where R = F and

e = (Mt — Fe)/R (9)

The beam can now be analyzed asan ordinary reinforced concrete mem-ber subjected to an eccentric compres-sion force. The resulting strain distri-bution (3) in the concrete is shown inFig. 1(b). The incremental strains inthe tendon and bar reinforcement,Ep3 and Es3, respectively, together withtheir corresponding stresses fp3 andfs3, are superimposed on the strainsand stresses already present in the ten-don and bar.

These incremental steel stresses, aswell as the stress in the concrete, canbe found using the transformed sectionconcept. 5 The tendon is replaced by anequivalent area of tensile concretenpAp and the bar reinforcement is re-placed by the area n8A,, where np =Ep/Ee and n8 = E3/E0, as shown inFig. 2(a).

The neutral axis for the equivalenthomogeneous transformed section, adistance y from the top surface, can befound from the equilibrium condition

PCI JOURNAL/July-August 1976ҟ 75

*ҟ* ҟRI -" fc3 —''1

^1ҟ_ _

centroidҟeҟeecrackedconcrete

P uncrackes _ _ __ҟds• concrete

C2 neutral axis=*N ._._.._.—.ҟng As.^--

(a) Transformed crackedҟ(b) Stressescross section

Fig. 2. Transformed cracked cross section and stress diagram of partiallyprestressed beam.

that the moment of all internal forcesabout the line of action of R must bezero. These internal forces are based onthe concrete stresses and the stressesacting on the transformed steel areasas shown in Fig. 2(b).

The moment equation for the inter-nal forces about the external resultantR results in . a cubic equation for ywhich can be solved by successivetrials. Once y is known, the effectivetransformed area Apt and moment ofinertia 'at of the cracked section, aboutits own centroid cl° from the top sur-face, can be found. The incrementalstresses sought, as loading passes fromStage (2) to Stage (3) are

R Re*ci*fc3 — T

" Ict(10)

Rc Re°(d, — cl#) -1 )fp3 = y

At + lot (11

n R + Re*(dt,B — cl°) (12s3 — af [ - Act Ict )

where geometric terms are as definedin Fig. 2.

The final stress in the tendon is nowfound by superimposing the stresses ofEqs. (2), (4), and (11). That in the barreinforcement is given by Eq. (12).

The concrete stress- at the top surface ofthe beam is given by Eq. (10). Specific-ally:

fp = fp1 + fp2 +f53.

(13)

f8 = f83 (14)

10=103 (15)

Summary of StressReview Procedure

The procedure for calculating elasticstresses in cracked prestressed concretebeams is- summarized briefly as follows:

1. Calculate the effective stress in thetendon after losses, fpl = fie, usingEq. (2).

2. Find f,2, the increase in stress inthe tendon as the member passes to ahypothetical decompression stage, withthe aid of Eq. (7) and Eq. (4).

3. Use Eq. (8) to determine the fic-titious force F needed to produce thedecompression stage.

4. Apply an equal and opposite forceF to the member, in combination withthe moments due to dead and liveloads. The resultant force R = F hasan equivalent eccentricity given byEq. (9). Find the neutral axis of the

76

cracked section and the section prop-erties by the usual methods of me-chanics.

5. Determine 10 = f C3, the maximumcompression in the concrete at serviceload using Eq. (10), the service loadtension in the nonprestressing steel re-inforcement, f8 = fs3, from Eq. (12),and the incremental tensile stress fp3

in the tendon from Eq. (11). The totaltension f, in the tendon is given byEq. (13).

Design Example

DataThe partially prestressed T-beamshown in cross section in Fig. 3(a) issubjected to superimposed dead andservice live load moments of 38 and191 'ft-kips (52 and 259 kN •m) in addi-tion to a moment of 83 ft-kips (113kN•m) due to its own weight.

An effective prestressing force of 123kips (547 kN) is applied using sixGrade 250 I/2-in. (12.7 mm) diameterstrands. Two nonprestressing steelGrade 60 No. 8 bars are located closeto the tension face of the beam.

The elastic moduli for the concrete,tendon steel, and bar steel are, respec-tively, 3.61 x 106, 27 x 106, and 29 x106 psi (24,900, 186,000, and 200,000N/mm2). The modulus of rupture ofthe concrete is 500 psi (3.5 N/mm2).

RequiredFind the stresses in the concrete, pre-stressing steel, and bar reinforcementat the full service load.

Solution'First, the tensile stress in the concreteat the bottom of the beam will bechecked, assuming the member is un-cracked. The properties of the un-cracked cross section are

Ae =212in2 c1 = 13.1 in.

PCI JOURNAL/July-August 1976

S1 = 1664 in.3 c2 = 16.9 in.

S 2 = 1290 in. r2 = 103 in .2

Then using Eq. (1):

Pc( eG2) Mtf2=—A`1 F 22 J { S

_ _ 123,000 11.9 x 16.9212 (1+ 103 +

312,000 X 121290

+1186 psi

This stress greatly exceeds the modu-lus of rupture, indicating that the sec-tion has indeed cracked. Analysis willproceed according to the method de-scribed above.

From Eq. (2), the effective stress inthe tendon when Pe acts alone is

fpl = Jpe = Pe/`gyp

= 123,000/0.863

= 143,000 psi

Then, with reference to Fig. 1(b)and using Eq. (7) the change in strainin the tendon as the section is decom-pressed is

Pe e2ep2 A\1 + r2

e

_ 123,00011.92212x3.61x106 (1+ 103

= 0.0004

Thus, the corresponding increase instress in the tendon is found from Eq.(4) to be

/p2 = EPEp2

= 27 x 106 x 0.0004

= 10,800 psi

To obtain decompression of the con-crete, the fictitious external tensiongiven by Eq. (8):

F = AA(fpl + fp2)

77

r4T1^— 5

13.1

f r 1j 1ҟuncracked17 30 conc. centroid

16.9ҟ11.9

— Ap = 0.863 in25 3

ҟ

ҟA5= 1.57 in2

I^-8(a) Member cross section

1

r'4_:--jҟ R

3.15ҟ fc35ҟ7.62ҟ10.77

ro

L_ y= 13.4ҟcrackedҟ_- conc. centroid

25ҟ cracked .neutral axis

f(25)^= np Ap=6.46 in 2 c3 yns As= 12.61 in2

fc3(2y )

(b) Transformed crackedcross section

(c) Concrete stresses

Fig. 3. T-beam design example using cracked section analysis.

= 0.863(143 + 10.8)

= 133 kips

must have been applied to the tendon.This is now cancelled by applying anequal and opposite force F. This force,acting together with the total momentof 312 ft-kips, is equivalent to a com-pressive force R = 139 kips appliedwith eccentricity [Eq. (9) 1:

e = (Mt — Fe)/R

= (312 x 12 — 133 x 11.9)/133

= 16.25 in.

above the centroid of the uncrackedconcrete, or 3.15 in. above the top sur-face of the member as shown in Fig. 3.

With n,= 27/3.61 = 7.48 and ns= 29/3.61 = 8.03 the transformedareas of the tendon and the bars are,respectively, 6.46 and 12.61 in. 2 Theeffective cross section of the crackedbeam, wire neutral axis dimension ystill unknown, is shown in Fig. 3(b).

The stresses in the concrete andtransformed steel, as the loads passfrom Stage (2) to Stage (3), are shownin Fig. 3(c). Taking moments of the re-sulting forces about the force R gives acubic equation in y which is solved bysuccessive trials to obtain y = 13.4 in.as shown.

With y known, the location of thecentroid of the cracked transformedsection is a routine matter. Taking mo-

78

ments of the partial areas about the topsurface locates the centroid c 1 = 7.62in. from the top of the section. Sectionproperties are

A0t = 133 in.2 ICt = 9232 in .4

The eccentricity of the force R withrespect to the centroid of the crackedtransformed section is

e°=16.25-13.1-}-7.62

= 10.77 in.

Now the incremental stress in theconcrete and steel can be found fromEqs. (10), (11), and (12):

fp = fp1 + fp2 ±1p3

= 143,000 + 10,800 + 12,700

= 166,500 psi (1148 N/mm2)

while the stress in the bar reinforce-ment

orce-ment is

I8 = f83 = 16,100 psi (111 N/mm2)

and that at the top surface of the con-crete is

fc = fc3 = —2180 psi (-15 N/mm2)

Additional CommentsR Re"ci°

fc3 = — — —Aet lot

_ 133,000 _133

133,000xlO.77x7.629232

= —2180 psi

— n r — R Re°(dp — cl*)

lfp3 L Act + jet

=7.48[-131330+

133, 000x 10.77x 17.38]9232

= 12,700 psi

— n — R -f- Re'(d$ — ci*) JAot jot

= 8.03 r — 133,000133 +

133,000x10.77x19.38]9232

= 16,100 psi

The final stress in the tendon at fullservice load is found by summing thethree parts:

1. The stress increase in the tendonas the beam is brought to full serviceload is about 17 percent of the effec-tive prestressing force. In calculatingservice load stresses in partially pre-stressed beams, this increase clearlycannot be neglected.

2. The service load stress of only16,100 psi (111 N/mm2) in the barreinforcement indicates that require-ments of strength, not service loadstress, probably controlled the choiceof bar area.

3. Although the allowable concretestress was not given, the stress of 2180psi (15 N/mm2) appears reasonable forconcrete having a compressive strengthof about 5 ksi (35 N/mm2).

4. The strain and stress informationdeveloped provides a rational basis forjudging the serviceability of the beam.For example, an estimate of crackwidth could be made based on thestress in the bar reinforcement, usingstandard methods, or could be basedon the increase in stress in the tendonas the . member passes from the de-compression stage to the full serviceload stage.

PCI JOURNAL/July-August 1976ҟ 79

Conclusion

Nominal concrete tensile stresses maybe calculated in partially prestressedbeams based on the properties of theuncracked cross section. According topresent American practice, dimension-ing of the section may proceed on thebasis of such nominal stresses eventhough they may exceed the modulus ofrupture.

Circumstances may exist, however, inwhich it is necessary to obtain a more

realistic appraisal of service load con-ditions. Specific reference is made toconsideration of cracking, deflection,and fatigue, as well as satisfaction ofspecified limits on stresses.

A method is given for the calculationof stresses in the concrete, prestressingsteel, and bar reinforcement in partiallyprestressed beams after cracking, per-mitting a more satisfactory assessmentof serviceability.

Notation

AG = area of concrete cross section,in.2

A8t = area of transformed cracked con-crete section, in.2

Ar = area of prestressing steel, in.2A, = area of nonprestressing steel re-

inforcement, in.2of = distance to centroid of cracked

transformed concrete sectionfrom top of concrete, in.

C2 = distance to bottom of memberfrom centroid of uncracked con-crete section in.

E8 = modulus of elasticity of con-crete, psi.

EP = modulus of elasticity of pre-stressing tendon, psi

E3 = modulus of elasticity of non-prestressing steel reinforcement,psi

= eccentricity of prestressing forcewith respect to centroid of un-cracked concrete section, in.

= eccentricity of force R with ref-erence to centroid of uncrackedconcrete member, in.

IC = stress in concrete, psi

fo = compressive strength of con-crete, psi

fp = stress in prestressing steel rein-forcement, psi

fpe = effective prestressing force in

tendons, psifk= stress in nonprestressing steel

reinforcement, psifyl, f12, f13 = incremental stresses in pre-

stressing steel reinforcement, psifsr, f8 2, fu = incremental stresses in non-

prestressing steel reinforcement,psi

12 = stress in concrete at bottom ofmember, psi

F = fictitious decompressing force,lb

I,t= moment of inertia of trans-formed cracked concrete section,in.4

Mt = total moment due to superim-posed loads, in-lb

n. = modular ratio EP/EC

ns = modular ratio ES/ECP e = effective prestress force after

losses, lbR = resultant eccentric force on con-

crete, lbr = radius of gyration of concrete

section, in.S2 = section modulus of concrete

section with reference to bottomsurface, in.3

Epl , Ep2, Ep3 = incremental strains inprestressing steel reinforcement

E, 1, e,2, e83 , = incremental strains in non-prestressing steel reinforcement

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References

1. Abeles, P. W., Introduction to Pre-stressed Concrete, V. I and II, Con-crete Publications Ltd., London, En-gland, 1964 and 1966.

2. Thurlimann, B., "A Case for PartialPrestressing," Structural ConcreteSymposium Proceedings, Universityof Toronto, May 1971, pp. 253-301.

3. Leonhardt, F., "To New Frontiersfor Prestressed Concrete Design andConstruction," PCI JOURNAL, V.

19, No. 5, September-October 1974,pp. 54-69.

4. ACI Committee 318, `BuildingCode Requirements for ReinforcedConcrete (ACI 318-71)," AmericanConcrete Institute, Detroit, 1971.

5. Winter, G., and Nilson, A. H., De-sign of Concrete Structures, 8th Edi-tion, McGraw-Hill Book Co., NewYork, 1972.

Discussion of this paper is invited.Please forward your discussion toPCI Headquarters by December 31,1976.

PCI JOURNAL/July-August 1976ҟ 81