Prestressing Optimizationand Its Implications

for Design

M. Z. CohnProfessorDepartment of Civil EngineeringUniversity of WaterlooWaterloo, Ontario, Canada

A. J. MacRaeResearch Assistant

Department of Civil EngineeringUniversity of Waterloo

Waterloo, Ontario, Canada

SynopsisResults of a comprehensive study

on the optimization of some 240beams of reinforced, prestressed andpartially prestressed concrete usingnonlinear programming techniquesare presented.

Evaluation of these results providessome tentative answers to the desir-ability of partial prestressing, its ex-tent, its economic feasibility vis-a-visreinforced or fully prestressed con-crete and the possible benefits of aunified design approach between re-inforced, prestressed, and partiallyprestressed concrete.

Note: This article is based on a paper presented at theInternational Symposium on Nonlinearity and Con-tinuity in Prestressed Concrete, University of Waterloo,Waterloo, Canada, July 4-6, 1983.

E xtensive theoretical and experi-mental work on partially pre-

stressed concrete has led to its ac-ceptance in structural design applica-tions. Explicitly or not, partial pre-stressing also has found a place in stan-dard codes of the "unified" type, i.e.,those which present the design practicefor both reinforced and prestressed con-crete in the same document.1.5

Despite this acceptance, there still is awide range of opinions on the very def-inition of partial prestressing, its rec-ommended amount, practical applica-tion and best design procedures. Someof the related questions faced in theirpractice by structural concrete designersmay be summarized as follows:

1. What is the desirable extent of par-tial prestressing?

2. When is partial prestressing to bepreferred over normally reinforcedconcrete or prestressed concrete?

68

3. Is a unified procedure for the de-sign of RC, PC, PPC possible, de-sirable, practical?

4. Which is the preferred method forpartial prestressing design?

A comprehensive study on the op-timization of RC, PC, and PPC memberswas undertaken to provide some spe-cific answers to the above questions.Details of the mathematical formulationare found elsewhere. 6- 7 This paper islimited to a discussion of the results andfocuses on some practical lessons de-rived from formal optimization tech-niques.

PREMISESThis section discusses some of the

underlying theoretical considerationsused in this investigation includingfeasible design space, optimizationtechniques, and parametric study of op-timal solutions.

Feasible Design SpaceRecent studies e- 9 have shown that the

entire spectrum of flexural design solu-tions for RC, PC, and PPC sections maybe defined by three variables:

o_ (Apfp8 + ABf8 — A g f,)lbdf (1)

[= WP + w $ - Wg]

K = M d/M. (2)

y = M v /M . I' A ,.fv l(A ,.fns (3)+AJ. ,)]

where to is the net reinforcing index, y isthe partial prestressing ratio, and K is theratio of decompression moment to theservice moment. The effective depth dis taken to the centroid of the total ten-sion force and other symbols are stan-dard and can be found in the Notationsection (see Appendix).

The variable w governs the ultimatebehavior of flexural sections and haspractical lower and upper bounds thatprevent premature brittle failures of the

tension and compression zones, respec-tively.

The variable K characterizes the ser-vice behavior and practically varies fromzero (for RC) to 1 (for PC). Note that foroverdesigned PC sections K> 1, since itis conceivable (but uneconomical) todesign sections such that the servicemoment, M 8, be less than the decom-pression moment, M d . The historicaldefinition of partial prestressing (al-lowing cracking at service' o- 11 ) impliesthat 0 < K< 1.

The variable y describes the con-tribution of the prestressed reinforce-ment to the ultimate flexural strength. Itvaries from zero for sections with onlypassive nonprestressed steel,A B , to 1 forsections with only active prestressedsteel, Ap.

For the definitions and assumptions ofthe American codes,' ,4 it has beenshown 8- 9 that the primary variables a,, K,

and y satisfy the following equation:

(K/y) (1 – 0.59u,) (1 – 0.5w) _ (4)F 1 (a) F2(13)

where a3 is the dead to live load mo-ments ratio, a is a parameter dependingon section geometry, and F 1 (a) and F2(f3)are functions depending on a and /3.8

For a given data set of section designF,(a) F2 (f3) = constant.

Eq. (4) defines the feasible designspace for all conceivable structural con-crete solutions, as illustrated in Fig. 1:Line AB (w > 0, K = y = 0) correspondsto RC designs; Curve EF (a,> 0, K(Kmax= 1, y = 1) defines the set of full pre-stressing designs; any point (a,, K, y)within the solid ABC DE F (w > 0, K> 0,y > 0) corresponds to a partially pre-stressed concrete solution.

Fig. 1 gives a complete and continu-ous geometric illustration of all theoreti-cal solutions in reinforced, prestressed,and partially prestressed concrete.Identifying a design by the three pri-mary variables (co, K, y) (rather than byone only variable) completely defines

PCI JOURNAL/July-August 1984 69

its general strength-deformation re-sponse under service and ultimateloads.

Optimization of Structural ConcreteMembers

Let us assume that the simply sup-ported beam in Fig. 2(a) with the crosssection in Fig. 3 is to be designed in turnas a RC, PC or PPC beam. In the lattertwo cases either pretensioning or post-tensioning may be used, according tothe layouts shown in Figs. 2(b) and (c).

The optimization problem consists offinding the design variables in Fig. 3that minimize the total cost of the beam,while satisfying all the required perfor-mance constraints. The adopted meritfunction is the total cost C, expressed(per unit beam length L) as:

CIL = w,(A.–A 8–A8–Ap)c, (5)

+ w8(A 8+A3)c8 + w.Apcp

+ Pfc f + C„ IL

where w, A, and c are unit weights, areasand unit costs in general and subscriptsc, s, p, andf refer to concrete, reinforc-ing steel, prestressing steel, and form-work, respectively. The variable Pf is thesection perimeter of the form and C„ isthe total cost of the web (stirrup) rein-forcement.

A variety of merit functions may beadopted in the optimization process byproperly adjusting the unit costs c., c8,

c p, and c f. For example, an optimal de-sign with the minimum steel consump-tion may be obtained by settingc, = cr=0 in Eq. (5); similarly, for an optimalheight design it suffices to setc, = c p = 0.

The performance constraints must en-sure satisfactory response with regard tothe serviceability and ultimate limitstates, code limitations on steel rein-forcements, as well as some additionalrequirements related to fatigue, ductil-ity and minimum construction rein-forcement. Along with the criterion ofallowable stresses under service load,the additional requirement of zero ten-

70

W

(d) L

cgcP P

(b) -- --^

\_cgc

cgcP—•j_•— —Pe

(c) cgs

Fig. 2. Structural concrete beam: (a) loading; (b) pretensioning;(c) post-tensioning.

b(x4) ds

h f (x 2 ) QA's (xg) T

L b t /2d(x11)

h( x1 ) bw,(x6) dp(x10)

Ap(xg)ti (su PPO

I2

^'_ (MID SPAN)h} (x3)

1:5. ')) d ^s

b(x5)

Fig. 3. Design variables for structural concrete sections.

PCI JOURNAUJuly-August 1984 71

Table 1. Types of constraints used in the optimization process.

Design solution

RC PC PPCLimit state Constraint type Particulars Load level

Transfer • •

ConcreteCOperating • Service • •Fatigue •

SL(service)(

Effective stresses Steel A,ServiceFatigue

••

Transfer • •

SteelA D Service • •Fatigue •

Camber Erection • •

Deflection Instantaneous live load Service • • •

Additional long term Service • • •

Crack width — Service •

Flexural strength • • •ULS Ductility factor Ultimate • •

(ultimate) Reinforcement index • • •

Top (A') Service • •Bottom (A s) Transfer • •

Minimum

TopAB) Construction • •reinforcementBottom (A.) • •

sile stress under operating (i.e., deadplus sustained live load) conditions is

imposed.A summary of the types of design con-

straints considered in the optimizationprocess is given in Table 1.

The optimal member design was con-ducted for the support and midspansections only, but further sections maybe considered, as desired. Time effects(creep and shrinkage effects on pre-stressing losses and deflections) and thesatisfaction of shear requirements viathe merit function were automaticallyincluded as described in Ref. 7.

Because for every PPC section thereare up to 11 variables (Fig. 3) and at least20 constraints (Table 1), some of whichare nonlinear, the optimization ofstructural concrete members as formu-

lated is a nonlinear programming prob-lem. It was solved using an appropriatecomputer code, FCDPAK, 12 for which aspecialized program, OSCON, 7 wasadded to generate all the required con-straints. The program produces optimalsolutions for any RC, PC, or PPC designparameters.

Parametric Study ofOptimal Solutions

The OSCON program was used to op-timize a set of beam designs in whichspecial cases and discrete values are as-signed to the variables as follows:

(a) Structural concrete type: threecases (RC, PC, PPC).

(b) Prestressing type: two cases (pre,post-tensioning).

72

"i- 'r SIT ib

—i b M M-

SectionNumber

h(mm)

h,(mm)

b(mm)

b^(mm)

d (mca)mm)

1 1200 125 2400 300 1025

2 1000 125 2400 300 11503 900 125 1200 350 7504 S00 300 400 400 650

Fig. 4. Concrete section designs: (a) shapes; (b) geometry; (c) dimensions.

(b)

(C )

(c) Section type: four cases as shownin Fig. 4.

(d) Span/height ratio: four values persection in the range 5-35.

(e) Live load intensity: three values,for light (w), normal (2w) andheavy loading (3w).

The range of design solutions isshown in Fig. 5, in terms of the span toheight(L/h) ratios considered for varioussection types and load intensities.

The design data set for all solutions isas follows:

(a) Serviceability requirements:crack width limit = 0.25 mm de-flection limit, according to Ref. 4.

(b) Member layout: cable hold-downat midspan; stirrup diameter =10 mm; concrete cover = 40 mm.

(c) Loading: fraction of live load sus-

tained = 0.25; superimposeddead load = 0 kN/m.

(d) Time effects: end of creep period= 4400 hours ( 1/2 year).

(e) Prestressing parameters: k 3 = 1.6(pretensioning), 1.0 (post-tension-ing), creep factor = 2.3; k,, = k r =

1, esh = 0.0275 percent, jacking

stress = 0.7f,, = 1302 MPa.(f) Material properties: E 8 = E p =

200,000 MPa; f = 1860 MPa; fp,,= 1640 MPa; f ff = 30 MPa (at 3days); f f = 40 MPa (at 28 days).

(g) Merit function parameters: w, _23.6 kN/m 3 ; w 8 = w,, = 76.4 kN/m 3 ; c^ = $100/m 3 ; c 1 = $ 10/m2;

Metric (SI) conversion factors: 1 in. = 25.4 mm; 1 sqin. = 645.2 mm'; 1 psi = 0.006895 MPa; 1 cu ft =0.02832 m'; 1 kip per cu ft = 157 kN/m'.

PCI JOURNAL/July-August 1984 73

RC - PC,PPC

Fig. 5. Ranges of span to height ratios for optimal design solutions.

PC/PPC

RC0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35

L/h L/h

(a) (b)

Fig. 6. Variations of reinforcement indices with span to height ratio (Section 3, normalload 2w).

0.15w PRETENSIONING PC

W

Cp/ C l - 8/9 PPC0.10 USECTION 3

0.05

RC•.•

c8 /c C = c„ /cc = 9; cr /cc = 8 (pre-tensioning); c9 lc, = 36 (post-ten-sioning).

(h) Minimum reinforcement (con-struction): A 8 = 200 mm 2 ; A8 =600 mm 2 (Sections 1, 2, 3); A8 =200 m 2 (Section 4).

RECOMMENDED EXTENTOF PRESTRESSING

The total amount of reinforcement(prestressed and nonprestressed), the

74

degree of prestress, and the relativeamount of prestressing reinforcementare defined by the primary variables w,K, y. The particular values of these vari-ables for the optimal solutions obtainedin the parametric study are presentedbelow.

Variable wA typical plot of c versus L/h for Sec-

tion 3 and the case of normal loading isshown for pre and post-tensioned mem-bers in Figs. 6(a) and (b), respectively.

Table 2. Critical span-to-height (L/h) ratios for which the required

reinforcing index t, is the same for RC, PC, and PPC.

Section

Live 1 2 3 4load

Pre Post Pre Post Pre Post Pre Post

w 22 15 17 15 18 15 18 162w 16 11 14 9 16 10 16 113w 11 8 12 9 14 10 18 —

As expected, the total steel requiredincreases with the larger L/h ratios. The

increase is steeper for Section 4, whichhas the smallest compression zoneamong the sections analyzed (Fig. 3).Note that the values of w are smaller forRC than for PPC or PC designs for L/hratios below a certain limit, and largerabove it. The limiting ratios where RCdesign curves intersect the PC and PPCcurves are given in Table 2 for the vari-ous section types and loadings analyzed.

Variable yThe values of y for optimal solutions

depend significantly on the relativecosts of prestressing and nonprestress-

ing steels, c, /c8 . If pretensioning is used,these unit costs are comparable and, as aresult, there is no advantage in replacinghigh strength steel by mild steel rein-

forcement.In this study, with c,, lc, = 8/9, y values

of at least 0.9 are found for all cases, ex-cept for Section 4 at the maximum L/hratio and under heavy loading. If nolower bounds are placed on the rein-forcements for detailing purposes (e.g.,

A, > 200 mm 2 ; A, -_ 600 mm 2 for Sec-

tions 1, 2, 3, and A > 200 for Section 4)

y tends toward unity.If post-tensioning is used, for a steel

cost ratio of c9 /c8 = 4, the range of y val-ues is wider than in the pretensioningcase and a minimum value of y = 0.65 is

found.

Variable aOptimal solutions for all cases inves-

tigated confirm the expectation that K

values are larger for PC than for PPCbeams. This is due to the allowable ten-sile stress constraint for concrete, whichresults in higher prestress of A D , andcorrespondingly larger K values. Thevalue of K remains a designer's choice; ifprestressing steel is present it should beprestressed as much as allowable.

In practical prestressed concrete (PCor PPC) design, the governing criteriaare the service concrete stresses, de-flection limitation and/or ultimatestrength. However, for Llh ratios belowa critical value for pretensioned mem-bers, the active constraint is alwaysfound to be the limitation on the servicestress ofA9.

This is quite undesirable, becauselarge amounts of prestressing steel arerequired mostly to sustain the prestresslosses. For the post-tensioned membersthe critical L/h ratios tend, in general, tobe smaller than for pretensioning andthe K values corresponding to the optimavary between K -= 0.4 to 1. For varioussection designs and loadings critical L/hvalues are given in Table 3.

PRACTICAL USE OFPARTIAL PRESTRESSINGContrary to some misconceptions, the

theoretical and practical potential of

PCI JOURNALJJuly-August 1984 75

200C ($

I50

50

Table 3. Critical span-to-height (L/h) ratios below which theeffective prestress constraint becomes critical.

SectionLivelload 1 2 3 4

Pre Post Pre Post Pre Post Pre Post

w 25 29 29 28 27 28 25 192w 19 17 25 15 22 22 19 133w 15 13 20 15 22 17 19 10

partial prestressing were recognized fora long time.'°.".13 In particular, Guyonhas pointed out the special benefits ofpartial prestressing to hyperstatic de-sign, where it combines the best fea-tures of RC (continuity) and PC (crack-free, stiffer members).13

For isostatic structures, however,these benefits are not obvious. Theparametric study of optimal solutionsintends to provide some practical guid-ance, by identifying the ranges of designvariables for which PPC is more costeffective than corresponding RC and PCdesigns. The relevant parameters arethe prestressing type (pre and post-ten-

sioning), the section type (Fig. 4), theL/h ratios and the live load intensity, w.

Optimization results are illustrated bythe typical plots in Fig. 7, where thetotal member costs for RC, PC and PPCsolutions are shown against the L/h ra-tios for a moderate loading 2w. The preand post-tensioning cases are shown inFigs. 7(a) and (b), respectively.

The diagrams in Fig. 7, and similarones for other design cases investigated,show that within the ranges ofL/h ratioswhere both members with or withoutprestressing are feasible (i.e., satisfy alldesign constraints) RC solutions aremore expensive. However, there is a

0 5 10 15 20 25 30 35L/h

PRE TENSIONEDcp/cs • 8/9

PCWSECN3PPC

RC

PCPOST-TENSIONED

cp/cs • 4 PPC

SECTION 3

PCRC

0 S 10 15 20 25 30 35L/h

(a) (b)

Fig. 7. Variation of member cost with span to height ratio (Section 3, normal load 2w).

76

Table 4. Critical span-to-height (L/h) ratios below which RC is

more economical than PC or PPC.

Section

Live 1 2 3 4load

Pre Post Pre Post Pre Post Pre Post

w 16 18 15 — 17 18 13 162w 10 16 10 16 12 17 10 13

3w 10 14 9 15 10 16 10 13

critical ratio below which RC is cheaperand prestressing brings no economicbenefit, as shown in Table 4.

For pretensioning, PPC brings negli-gible cost savings versus PC in all casesinvestigated. For such cases, the sim-plicity of PC (uncracked section) designcoupled with the marginal cost differ-ence favor PC over PPC solutions.

For post-tensioning, the cost savingsof PPC are slightly larger. Hence, inrepetitive applications the additionalcomplexity of PPC design versus PC de-

sign is warranted.To summarize:(a) RC or PC solutions are cost effec-

tive at low or high L/h and live load in-

tensities, PPC is the most economicalsolution for members with moderate L/hand live loads.

(b) Prestressing (PC or PPC) is un-economical below the critical L/h values

given in Table 4.(c) Unit costs of prestressed and non-

prestressed steel have a decisive influ-ence on the overall economy of PC andPPC designs; as a result PPC is moreefficient for post-tensioning, while PC ispreferable for pretensioning.

UNIFYING DESIGNMETHODS FOR RC, PC,

AND PPC DESIGNStructural concrete design has

evolved considerably since Freyssinet's

old claim that RC and PC are two com-pletely different materials to the presentuse of "unified" codes for RC, PC, and

PPC. 2.3 Unified design methodologiesfor the three classes of structural con-crete may be considered from either abroader or a narrower viewpoint.

The broader concept of "unification"refers to defining a common set of per-formance and engineering criteria (e.g.,Table 1) to be satisfied by any design ofRC, PC, and PPC members. The generalacceptance of this broad unifying con-cept is reflected by many structural con-crete standards.'-5

The narrower viewpoint of "unifica-tion" refers to the design of RC, PC, andPPC members by the same subset ofcriteria that is suited for a particularclass only. Currently, the main trend isto extend the prevailing ultimatestrength method for RC to PC and PPCmembers. These attempts, motivated bythe legitimate quest for simplicity andrapidity of design, demand an objectiveapproach to the problem of unification.Such an approach consists of identifyingand comparing the active constraints ofthe optimal solutions for RC, PC, andPPC members. Essentially, the gov-erning criteria (which do not includefatigue) fall into the following catego-ries:

(a) Service stresses such as allowabletensile stress under service load-ing (PC) or zero tensile stress un-der operating loading (PPC);

PCI JOURNAUJuly-August 1984 77

MS

0 .7

Os

0.4

• x

RC• x

0-

• x• •

0-, •• •

- • x

0

5

0 0 rl oo 0 pL PC: POST -TENSIONING

0 0 O

O O 0 0 0

0 0 0

• 0 00 Q

0 i1

o p zt

0 0

• O jj

1

w

0 0 q 00 0 q 0

PPC: POST-TENSIONING0 n •

• 0 • (

O • C

0 0 •>p•• x

O • x

O •

O •

5 10

20 30 10 20 30 35

IO 20 30 35L/h L/h

L/h

(a) (b)

(c )

LEGEND:• ULTIMATE. o SERVICE, • ULTIMATE + SERVICE, it DEFLECTION

Fig. 8. Active design criteria for optimal solutions: (a) reinforced concrete;(b) prestressed concrete; and (c) partially prestressed concrete.

Table 5. Summary of ranges of span-to-height ratios (Llh) for which

particular constraints control the design.

No. Criterion R

Pretensioned Post-tensioned

PPC PC PPC

1 Service stresses —f30-31

20-35 12-35 10-352 Deflections 16-32 30-35 18-30 20-353 Ultimate flexural strength 6-2 15-35 10-15 12-334 Reinforcement index

limitation — 25-31 25-31 — —

(b) Deflections under long-term con-

ditions;(c) Ultimate flexural strength, and

(d) Reinforcement index (w) limita-

tions.The results obtained from some 240

optimal design solutions are illustratedin Figs. 8(a), (b), and (c) for RC, PC, andPPC members (post-tensioned steel),respectively. Each point in these plotsidentifies an active design constraint in

the plane M. /M g versus L/h.Plots in Fig. 8 and similar ones for the

pretensioning case show that:

1. For the vast majority of RC beamsultimate strength governs, with the de-flection being critical only for Llh > 20and high M. /M8 values.

2. For the large majority of PC beamsservice stresses govern the design (iso-lated cases of the reinforcement limita-tion constraint are active for Llh > 28 in

pretensioning).3. For the PPC beams both the ulti-

mate flexural strength and the servicestresses are active in a wide range of thedesign parameters.

A summary of the trends in constraintactivity is given in Table 5.

A "unified" method (e.g., based onultimate strength) is valid only to theextent that all other criteria are satisfiedimplicitly when the primary criterion(ultimate strength) is met within thespecified margins. Fig. 8 and Table 5show that in general, the optimal PPCsolutions correspond to a variety of gov-

erning criteria, which does not alwaysinclude the ultimate strength. There-fore, the postulated assumption of theultimate strength criterion governingany design solution is not proved.

A design method may be consideredpractical when it is both convenient andreliable; also it is desirable when simplein application. However, if a method issimple and hence convenient, but ofunproved reliability, it cannot be con-

sidered practical.In summary, unified criteria for RC,

PC, and PPC design are codified andgenerally acceptable. Unified methods,based on subsets of these criteria, havenot been shown to always result in validsolutions and thus, however desirable,are not always practical.

PRACTICAL METHODSFOR PPC DESIGN

A method recommended for design

practice should be feasible, economical,and simple. For PPC the design task isnot too easy because:

(a) An initial solution (prestressingforce and eccentricity) must beguessed;

(b) Cracked sections must be used;

(c) All relevant criteria (e.g., Table 1)should be satisfied.

Hand calculations that overcome inturn the above difficulties can only beiterative and, except for experienced de-

PCI JOURNAL/July-August 1984 79

Table 6. Summary of nonlinear constraints.

Index Description of nonlinear constraints

Value of nonlinearconstraint

Actual Allowable

Critical section number = 11* Support concrete stress-Top/compression (MPa) 0.67 16.552* Support concrete stress-Top/tension (MPa) 0.67 -2.633* Support concrete stress-Bot/compression (MPa) 4.96 16.554* Support concrete stress-Bot/tension (MPa) 4.96 -2.635* Minimum top mild flexural steel (mm2) 1.00 1.00

Critical section number = 26 Transfer concrete stress-Top/tension (MPa) 2.68 -1.317 Transfer concrete stress-Bot-compression (MPa) 1.02 16.558* Operating concrete stress-Top/compression (MPa) 3.13 9.399* Operating concrete stress-Bot/tension (MPa) -0.45 0.0

10 Service concrete stress-Top/compression (MPa) 11.39 15.5211 Service mild steel stress (MPa) 173.09 400.0012 Service prestressing steel stress (MPa) 1283.52 1640.0013 Crack width (mm) 0.1331 0.2514* Minimum top mild flexural steel (mm2) 1.00 1.0015* Minimum bottom mild flexural steel (mm2) 1167.49 420.0316 Flexural strength capacity (kN-m) 1264.97 1277.9317* Ductility factor 36.63 2.0018* Maximum reinforcement index 0.0419 0.3019 Camber + creep due to prestressing (mm) -4.37 44.4620 Long term + instantaneous deflection (mm) 25.26 44.46

*Represents constraints not considered in Ref 20.Metric (SI) conversion factors: 1 in. = 25.4 mm; 1 sq in. = 645.2 mm; 1 psi = 0.006895 MPa; 1 ft-kip1.356 kN-m.

signers, require some analytical effort.Feasibility and accuracy may be ob-tained, but not in a simple, direct proce-dure. Simplifications are possible withsome sacrifice of accuracy (assumptionson initial P and e values, cracked sectionproperties and governing criteria).

Eventually, however, the final designmust be feasible, even if not necessarilyeconomic or simple to obtain. Designprocedures developed by Swiss engi-neers d5.17d 18 are among the most appeal-ing, but their practicality is best left forjudgment by the profession and thepassing of time.

On the other hand, optimal solutionsusing computers by definition meet thecriteria of feasibility and economy, butmay be deficient only on the require-ments of simplicity for the designer.

Better programs and easy manuals forthe users may be the answer to a direct,reliable, and optimal design of PPCmembers. Optimal solutions for PC19and PPC 20 are available and a versatileoptimization method for RC, PC, andPPC is provided by the OSCON pro-gram.'

One useful feature of this program isthe capability of comparing optimal de-signs with (feasible) solutions obtainedfor specific problems by various authors.For valid comparisons the optimizationshould be conducted only with the crite-ria used by such authors. However, inextreme cases, neglect of some con-straints may lead to optima of incorrector irrelevant formulations.

Of more practical interest is the use ofthe OSCON program as an automated

80

means for the analysis of existing solu-tions and procedures, by exploring theactivity of all possible constraints for agiven design. This has proved an in-structive approach in testing some of theavailable solutions in the literature.'7 2°

For the purpose of illustration, the ex-ample used by Naaman and Siriaksorn20is briefly presented. The authors haveused a computer-iterative approach tominimize the amount of prestressing

steel, as reflected by the "partial pre-stressing ratio" (PPR), which is analo-gous to the variable y of this paper.

The example is a simply supportedT-beam of 21.34 m (70 ft) span, thatcontains specified dead and live loads of8.94 and 5.84 kN/m (620 and 400 lb perft), respectively, with f f = 34.5 and fp„ _1860 MPa (5000 and 270,000 psi).

Using the OSCON program the actualconstraint activities and the corre-sponding allowable levels are obtained,as shown in Table 6. Note that in thetable the asterisked indices representconstraints not considered in Ref. 20.

It is noted that all constraints are sat-isfied, except the tension stress at thebottom fiber of the midspan section (No.2) under operating loads (sustained loadis only the self weight), a criterionwhich the authors did not consider in

their design.To summarize:

(a) Practical hand methods must be,to a smaller or larger extent, iterative,limited in scope and skill-dependent;some recent procedures 14, 15.17.2° should

prove attractive, at least as an initial de-

sign.(b) Optimization programs such as

OSCON can produce direct optimal so-lutions for RC, PC, and PPC membersunder a variety of merit functions. Theanalysis capability of such optimizationprograms may be of considerable help inautomatically checking design solutions

of large-scale projects.

(c) The most practical method for aparticular problem may be at either ofthe above extremes (of hand or com-

puter design), or at an intermediateplace, depending on the nature of theproblem and the designer's experience.

CONCLUSIONSThe following answers are offered to

the four questions on partial prestress-ing outlined in the Introduction.

1. In sections with mixed reinforce-ment optimal y values of 0.8 to 0.9 forpretensioning and 0.65 to 0.80 forpost-tensioning are found; high strengthreinforcement should be prestressed asmuch as allowable.

2. Full is more economical for pre-tensioning; partial prestressing is moreeconomical for post-tensioning; partialprestressing is more economical thanboth reinforced and fully prestressedconcrete for moderate span to height ra-tios and live loadings.

3. A direct, unified design method forreinforced, partially, and fully pre-stressed concrete members is desirablebut not always practical, as its generalreliability is not established.

4. Some of the manual design meth-ods based on the ultimate strength crite-rion have an inherent practical appealthat could be validated only by time andexperience; with the increasing avail-ability of computers, optimal designsoftware, such as that referred to in thepaper, may prove to be just as practicaland common a professional tool.

ACKNOWLEDGMENTSResults presented in the paper are

part of a MASc thesis prepared in theDepartment of Civil Engineering, Uni-versity of Waterloo, by the second au-thor, under the direction of the first au-

thor.The financial assistance of the Natural

Sciences and Engineering ResearchCouncil (NSERC) of Canada underGrant A-4759 is gratefully acknowl-

edged.

PCI JOURNAL/July-August 1984 81

REFERENCES

1. ACI Committee 318, "Building CodeRequirements for Reinforced Concrete(ACI 318-77)," Detroit, Michigan, 1978,103 p.

2. BS, Code of Practice for the StructuralUse of Concrete Unified Code, CP110-72, London, 1972.

3. CEB-FIP, Model Code for ConcreteStructures, International Recommen-dations, Third Edition, 1978.

4. CSA/NBC Joint Committee on Rein-forced Concrete Design, Code for theDesign of Concrete Structures forBuildings, Can 3-A23.3-M77, CanadianStandards Association, Rexdale, Ontario,Canada, 1971, 131 p.

5. FIP, Recommendations on Practical De-sign of Reinforced and Prestressed Con-crete Structures Based on CEB-FIPModel Code, MC 78, June 1982.

6. Cohn, M. Z., and MacRae, A. J., "Opti-mization of Structural Concrete Beams,"SM Paper Series, Solid Mechanics Divi-sion, University of Waterloo, Ontario,Canada, to be published, May 1983.

7. MacRae, A. J., "Optimal Design of Par-tially Prestressed Concrete Beams,"MASc Thesis, Department of Civil Engi-neering, University of Waterloo, Ontario,Canada, February, 1983, 207 p.

8. Bartlett, M., "Computer Analysis of Par-tially Prestressed Concrete," MASc The-sis, Department of Civil Engineering,University of Waterloo, Ontario, Canada,March, 1982, 285 p.

9. Cohn, M. Z., and Bartlett, M., "NonlinearFlexural Response of Partially Pre-stressed Concrete Sections," SM PaperNo. 168, Solid Mechanics Division, Uni-versity of Waterloo, Ontario, Canada,October, 1981, 55 p; also published inJournal of the Structural Division,ASCE, V. 108, No. ST12, December1982, pp. 2747-2765.

10. Abeles, P. W., "Philosophy of Design ofPartial Prestressing," Proceedings, In-ternational Symposium on Concrete De-sign: U.S. and European Practices,

(ACl/CEB), Philadelphia, Pennsylvania,1976, pp. 287-304.

11. Lebelle, P., "Partially PrestressedStructures," (in French), Travaux (Paris),No. 375-376, April-May, 1966, pp. 848-858.

12. Best, M. J., "FCDPAK: A Fortran IVSubroutine to Solve DifferentiableMathematical Programmes—User'sGuide—Level 3.1," Research ReportCORP 75-24, Department of Combina-torics and Optimization, University ofWaterloo, Ontario, Canada, 1973, 85 p.

13. Guyon, Y., "The Possibilities and theFuture of Prestressing," (in French),Travaux (Paris), No. 375-376, April-May1966, pp. 867-870.

14. Huber, A., "Practical Design of PartiallyPrestressed Concrete Beams," ConcreteInternational, V. 5, No. 4, April 1983, pp.49-54.

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16. Naaman, A. E., "A Proposal to ExtendSome Code Provisions on Reinforcementto Partial Prestressing," PCI JOURNAL,V. 26, No. 2, March-April 1981, pp. 74-91.

17. Bachmann, H., "Partial Prestressing ofConcrete Structures," IABSE Surveys5-11, Acta Periodica, November 1979,pp. 1-20.

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20. Naaman, A. E., and Siriaksorn, A., "Ser-viceability Based Design of PartiallyPrestressed Concrete Beams," Part 1,PCI JOURNAL, V. 24, No. 2, March-April 1979, pp. 64-69, Part 2, PCI JOUR-NAL, V. 24, No. 3, May-June 1979, pp.40-60.

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APPENDIX - NOTATION

C = total cost of member ff = allowable service tensile

C„ = cost of web (stirrup) rein- concrete stress (PC)

forcement f = allowable transfer tensile

Cc,f,p.s = unit cost of concrete, form- concrete stress (PC)

work, prestressing steel and fro = allowable tensile concrete

mild steel reinforcement, stress under sustained loads

respectively (=0 for PPC)

d = depth of tensile force from fv = yield strength of bottom

top concrete fiber, at ulti- mild steel reinforcement

mate limit state h = height of beam

E p,E s = modulus of elasticity of pre- k b = water-cement coefficientstressing reinforcement and k, = fraction of ultimate loss co-mild steel reinforcement, efficientrespectively kd = age of loading coefficient

f = allowable service compres- L = length of beamsive stress in concrete (PC, Md = decompression momentPPC)

= allowable transfer compres- M. = nominal (ultimate) section

sive stress in concrete (PC,capacity

PPC) Mo = operating load moment

fro = allowable compressive MP„ = nominal (ultimate) capacity

stress in concrete under of section reinforced with

sustained loads (PPC) = op- prestressing steel only

erating stress M8 = service load moment

f = standard cylinder strength P = prestressing force

of concrete Pf = formwork perimeter

fp8 = stress in prestressing rein- w = live load

forcement at ultimate limit w,,wp,w,= unit weights of concrete,

state prestressing steel, and mild

fpv = yield strength of prestress- steel reinforcement, respec-

ing steel tively

fp„ = ultimate strength of pre- y = partial prestressing ratio

stressing steel a„h = shrinkage strain of concrete

= stress in prestressing steel, K = degree of prestressing

after all losses w = net reinforcement index

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

PCI JOURNAL/July-August 1984 83