ACI 423.5R-99 Partially Prestressed Concrete

ACI 423.5R-99

State-of-the-Art Report on Partially Prestressed Concrete Reported by Joint ACI-ASCE ACI-ASCE Committee 423 Ward N. Marianos, Jr.* Chairman

Henry Cronin, Jr. Secretary

Sarah L. Billington Kenneth B. Bondy

William L. Gamble Hans R. Ganz

H. Kent Preston Denis C. Pu

Robert N. Bruce, Jr. * Dale Buckner

J. Weston Hall Mohammad Iqbal

Julio A. Ramirez * Ken B. Rear

Ned H. Burns *

Francis J. Jacques

Dave Rogowsky

*

Daniel P. Jenny

Bruce W. Russell

Jack Christiansen

Paul Johal

David H. Sanders

Todd Christopherson

Susan N. Lane

Thomas Schaeffer*

Steven R. Close

Les Martin*

Gregory P. Chacos

Morris Schupack *

Thomas E. Cousins

Alan H. Mattock

Kenneth W. Shushkewich

Charles W. Dolan*

Gerard J. McGuire

Khaled S. Soubra

Apostolos Fafitis Mark W. Fantozzi Martin J. Fradua

Mark Moore

*

Antoine E. Naaman Kenneth Napior

Richard W. Stone *

Patrick Sullivan Luc R. Taerwe

Catherine W. French*

Thomas E. Nehil

H. Carl Walker

Clifford Freyermuth

Mrutyunjaya Pani

Jim J. Zhao Paul Zia*

*

Subcommittee preparing report (Michael Barker contributed to writing Chapters 4 and 5 of this report).

Partially Partially prestressed concrete construction uses prestressed, or a combination of prestressed and nonprestressed, reinforcement. Partially prestressed concrete falls between the limiting cases of conventionally reinforced concrete and fully prestressed concrete, which allows no flexural tension under service loads. When flexural tensile stresses and cracking are allowed under service loads, the prestressed prestressed members have historically been called

ACI Committee Reports, Guides, Standard Practices, and Commentaries are intended for guidance in planning, designing, executing, and inspecting construction. This document is intended for the use of individuals who are competent to evaluate the significance and limitations of its content and recommendations and who will accept responsibility for the application of the material it contains. The American Concrete Institute disclaims any and all responsibility for the stated principles. The Institute shall not be liable for any loss or damage arising therefrom. Reference to this document shall not be made in contract documents. If items found in this document are desired by the Architect/Engineer to be a part of the contract documents, they shall be restated in mandatory language for incorporation by the Architect/Engineer. Architect/Engineer.

partially prestressed. prestressed. This report report is presented presented as an overview overview of the current current state of the art for partial prestressing of concrete structures. Research findings findings and design applications applications are presented. Specific topics discussed include the history of partial prestressing, prestressing, behavior of partially prestressed concrete members under static loads, time-dependent effects, fatigue, and the effects of cyclic loadings. Keywords: bridges; buildings; buildings; concrete construction; corrosion; cracking;

crack widths; cyclic loading; deflections; earthquake-resistant structures; fatigue; partially prestressed concrete; post-tensioning; prestressing; prestress losses; shear; stresses; structural analysis; structural design; timedependent effects; torsion.

CONTENTS Chapter 1—Introduction, p. 423.5R-2 1.1 —Historical perspective 1.2 —Definition 1.3 —Design philosophy of partial prestressing ACI 423.5R-99 became effective December 3, 1999. Copyright © 2000, American Concrete Institute. All rights reserved including rights of reproduction and use in any form or by any means, including the making of copies by any photo process, or by electronic or mechanical device, printed, written, or oral, or recording for sound or visual reproduction or for use in any knowledge or retrieval system or device, unless permission in writing is obtained from the copyright proprietors.

423.5R-2

ACI COMMITTEE REPORT

1.4 —Advantages and disadvantages of partial prestressing 1.5 —Partial prestressing and reinforcement indexes 1.6 —Report objective

Chapter 2—Partially prestressed members under static loading, p. 423.5R-5 2.1 —Behavior 2.2 —Methods of analysis 2.3 —Cracking 2.4 —Deflections 2.5 —Shear and torsion Chapter 3—Time-dependent behavior, p. 423.5R-12 3.1 —Prestress losses 3.2 —Cracking 3.3 —Deflections 3.4 —Corrosion Chapter 4—Effects of repeated loading (fatigue), p. 423.5R-15 4.1 —Background 4.2 —Material fatigue strength 4.3 —Fatigue in partially prestressed beams 4.4 —Prediction of fatigue strength 4.5 —Serviceability aspects 4.6 —Summary of serviceability Chapter 5—Effects of load reversals, p. 423.5R-20 5.1 —Introduction 5.2 —Design philosophy for seismic loadings 5.3 —Ductility 5.4 —Energy dissipation 5.5 —Dynamic analyses 5.6 —Connections 5.7 —Summary Chapter 6—Applications, p. 423.5R-28 6.1 —Early applications 6.2 —Pretensioned concrete components 6.3 —Post-tensioned building construction 6.4 —Bridges 6.5 —Other applications Chapter 7—References, p. 423.5R-30 7.1—Referenced standards and reports 7.2—Cited references

Appendix—Notations, p. 423.5R-36 CHAPTER 1—INTRODUCTION 1.1—Historical perspective Application of prestressing to concrete members imparts a compressive force of an appropriate magnitude at a suitable location to counteract the service-load effects and modifies the structural behavior of the members. Although the concept of prestressed concrete was introduced almost concurrently in the U.S. and in Germany before the turn of the 20th century (Lin and Burns 1981), its principle was not fully

(Freyssinet 1933). Freyssinet recognized that as the load on a prestressed member is increased, flexural cracks would appear in the tensile zones at a certain load level, which he referred to as the transformation load. Even though the cracks would close as the load was reduced and the structure would recover its original appearance, Freyssinet advocated avoiding cracks under service load so that the concrete would behave as a homogeneous material. A different design approach, however, was proposed by von Emperger (1939) and Abeles (1940). They suggested using a small amount of tensioned high-strength steel to control deflection and crack width while permitting higher working stresses in the main reinforcement of reinforced concrete. Most of the early work in support of this design concept was done by Abeles (1945) in England. Based on his studies, Abeles determined that eliminating the tensile stress and possible cracking in the concrete is unnecessary in many designs. Abeles also realized that prestress can be applied to counteract only part of the service load so that tensile stress, or even hairline cracks, occur in the concrete under full service load. Abeles did specify that under dead load only, no flexural tension stress should be allowed at any member face where large flexural tensile stresses occurred under maximum load, so as to ensure closure of any cracks that may have occurred at maximum load. Additional bonded and well-distributed nonprestressed reinforcement could be used to help control cracking and provide the required strength. Abeles termed this design approach as “partially prestressed concrete.” Therefore, the design approach advocated by Freyssinet was then termed as “fully prestressed concrete.” In actual practice, nearly all prestressed concrete components designed today would be “partially prestressed” as viewed by Freyssinet and Abeles. Interest in partial prestressing continued in Great Britain in the 1950s and early 1960s. Many structures were designed by Abeles based on the principle of partial prestressing, and examinations of most of these structures around 1970 revealed no evidence of distress or structural deterioration, as discussed in the technical report on Partial Prestressing published by the Concrete Society (1983). Partially prestressed concrete design was recognized in the First Report on Prestressed Concrete published by the Institution of Structural Engineers (1951). Provisions for partial Code prestressing were also included in the British Standard Code of Practice for Prestressed Prestressed Concrete (CP 115) in 1959. In that code, a permissible tensile stress in concrete as high as 750 psi (5.2 MPa) was accepted when the maximum working load was exceptionally high in comparison with the load normally carried by the structure. Presently, the British Code (BS 8110) as well as the Model Code for Concrete Structures (1978), published by CEB-FIP, defines three classes of prestressed concrete structures: Class 1—Structures in which no tensile stress is permitted in the concrete under full service load; Class 2—Structures in which a limited tensile stress is permitted in the concrete under full service load, but there is no

PARTIALLY PRESTRESSED CONCRETE

Class 3—Structures in which cracks of limited width permitted under full service load. (0.2 mm [0.008 in.]) are permitted Calculations for Class 3 structures would be based on the hypothetical tensile stress in the concrete assuming an uncracked section. The allowable values of the hypothetical tensile stress vary with the amount, type, and distribution of the prestressed and nonprestressed reinforcement. reinforcement. Elsewhere in Europe, interest in partial prestressing also developed in the 1950s and 1960s. In the mid-1950s, many prestressed concrete structures in Denmark, especially bridges, were designed using the partial prestressing concept. Their performance was reported as satisfactory after 25 years of service (Rostam and Pedersen 1980). In 1958, the first partially prestressed concrete bridge in Switzerland (Weinland Bridge) was completed near Zurich. Provisions for partial prestressing were introduced in SIA Standard 162, issued by the Swiss Society of Engineers and Architects (1968), and since 1960, more than 3000 bridges have been designed according to this concept with highly satisfactory results (Birkenmaier 1984). Unlike the British Code and CEP-FIP Model Code, the limit of partial prestressing in the Swiss Code was not defined by the hypothetical tensile stress. Instead, it was defined by the tensile stress in the prestressed and nonprestressed reinforcement, and calculated using the cracked section. Under full service load, the allowable stress in the nonprestressed reinforcement was 22,000 psi (150 MPa), and in railroad bridges, the stress increase in the prestressed reinforcement was not to exceed 1/20 of the tensile strength. This value was taken as 1/10 of the tensile strength in other structures. It was required, however, that the concrete be in compression when the structure supported only permanent load. In the U.S., the design of prestressed concrete in the early 1950s was largely based on the Criteria for Prestressed Concrete Bridges (1954) published by the Bureau of Public Roads, which did not permit tensile stress and cracking in concrete under service loads. The ACI-ASCE Joint Committee 323 report (1958), however, recognized that “complete freedom from cracking may or may not be necessary at any particular load stage.” For bridge members, tensile stress was not allowed in concrete subjected to full service load. For building members not exposed to weather or corrosive * atmosphere, a flexural tension stress limit of 6 √ f ′ c′ psi was specified with the provision that the limit may be exceeded if “it is shown by tests that the structure will behave properly under service load conditions and meet any necessary requirements for cracking load or temporary overload.” Thus, partial prestressing was permitted in that first definitive design guide for prestressed concrete, and designers were quick to embrace the idea. When the balanced load design concept was published by Lin (1963), it provided a convenient design tool and encouraged the practical application of partial prestressing. In 1971, the first edition of the PCI Design Handbook was was published. Design procedures allowing tension stresses are * In this report, when formulas or stress values are taken directly directly from U.S. codes

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illustrated in that guide. The second edition (1978) mentioned the term “partial prestressing,” and by the third edition (1985), design examples of members with combined prestressed and nonprestressed reinforcement were included. Presently, ACI 318 permits a tensile stress limit of 12 √ f c′ ′ psi with requirements for minimum cover and a deflection check. Section 18.4.3 of ACI 318 31 8 permits the limit to be exceeded on the basis of analysis or test results. Bridge B ridge design guidelines or recommendations, however, did not follow the development until the publication of the Final Draft LRFD Specifications for Highway Bridges Design and Commentary Commentary (1993), even though most bridge engineers had been allowing tension in their designs for many years. The concept of partial prestressing was developed half a century ago. Over the years, partial prestressing has been accepted by engineers to the extent that it is now the normal way to design prestressed concrete structures. Bennett’s work (1984) provides a valuable historical summary of the development of partially prestressed concrete.

1.2—Definition Despite a long history of recognition of the concept of partial prestressing, both in the U.S. and abroad, there has been a lack of a uniform and explicit definition of the term, “partial prestressing.” For example, Lin and Burns (1981) state: “When a member is designed so that under the working load there are no tensile stresses in it, then the concrete is said to be fully prestressed. If some tensile stresses will be produced in the member under working load, then it is termed partially prestressed.” On the other hand, Naaman (1982a) states: “Partial prestressing generally implies a combination of prestressed and nonprestressed reinforcement, both contributing to the resistance of the member. The aim is to allow tension and cracking under full service loads while ensuring adequate strength.” According to Nilson (1987), “Early designers of prestressed concrete focused on the complete elimination of tensile stresses in members at normal service load. This is defined as full prestressing. As experience has been gained with prestressed concrete construction, construction, it has become evident that a solution intermediate between full prestressed concrete and ordinary reinforced concrete offers many advantages. Such an intermediate solution, in which a controlled amount of concrete tension is permitted at full service, is termed partial prestressing.” A unified definition of the term “partial prestressing” should be based on the behavior of the prestressed member under a prescribed loading. Therefore, this report defines partial prestressing as: “An approach in design and construction in which prestressed reinforcement or a combination of prestressed and non-prestressed reinforcement is used such that tension and cracking in concrete due to flexure are allowed under service dead and live loads, while serviceability and strength requirements are satisfied.” For the purposes of this report, fully prestressed concrete is defined as concrete with prestressed reinforcement reinforcement and no flexural tension allowed in the concrete under service loads. Conventionally reinforced concrete is defined as concrete

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flexural tension in concrete under service loads. Partially prestressed concrete falls between these two limiting cases. Serviceability requirements requirements include criteria for crack widths, deformation, long-term effects (such as creep and shrinkage), and fatigue. By the previous definition, virtually all prestressed concrete that uses unbonded tendons is “partially prestressed,” as codes require that a certain amount of bonded reinforcement be provided to meet strength requirements. Most pretensioned members used in routine applications such as building decks and frames, and bridges spanning to approximately 100 ft (30 m) will allow flexural tension under full service load. The addition of nonprestressed reinforcement reinforcement is used only in special situations, such as unusually long spans or high service loads, or where camber and deflection control is particularly important.

1.3—Design philosophy of partial prestressing The basic design philosophy for partial prestressing is not different from that of conventionally reinforced concrete or fully prestressed concrete. The primary objective is to provide adequate strength and ductility under factored load and to achieve satisfactory serviceability under full service load. By permitting flexural tension and cracking in concrete, the designer has more latitude in deciding the amount of prestressing required to achieve the most desirable structural performance under a particular loading condition. Therefore, partial prestressing can be viewed as a means of providing adequate control of deformation and cracking of a prestressed member. If the amount of prestressed reinforcement used to provide such control is insufficient to develop the required strength, then additional nonprestressed reinforcement is used. In the production of precast, pretensioned concrete members, serviceability can be improved by placing additional strands, as this is more economical than placing reinforcing bars. When this technique is used, the level of initial prestress in some or all of the strands is lowered. This is also a useful technique to keep transfer stresses below the maximum values prescribed by codes. At least for purposes of shear design, the ACI Building Code treats any member with effective prestress force not less than 40% of the tensile strength of the flexural reinforcement as prestressed concrete.

multispan bridges, camber control is important in improving riding comfort as a vehicle passes from one span to the next. The relatively large mild steel bars used in partially prestressed members result in a transformed section that can be significantly stiffer than a comparable section that relies solely on prestressing strand, thus reducing both camber and deflection. Nonprestressed reinforcement used in partially prestressed members will enhance the strength and also control crack formation and crack width. Under ultimate load, a partially prestressed member usually demonstrates greater ductility than a fully prestressed member. Therefore, it will be able to absorb more energy under extreme dynamic loading such as an earthquake or explosion. Because mild steel does not lose strength as rapidly as prestressing strands at elevated temperature, it is sometimes added to prestressed members to improve their fire-resistance rating. See Chapter 9 of the PCI Design Handbook (1992) and Design for Fire Resistance of Precast Prestressed Concrete (1989) for more information. Partial prestressing is not without some disadvantages. Under repeated loading, the fatigue life of a partially prestressed member can be a concern. In addition, durability is a potential problem for partially prestressed members because they can be cracked under full service load. Recent studies (Harajli and Naaman 1985a; Naaman 1989; and Naaman and Founas 1991), however, have shown that fatigue strength depends on the range of stress variation of the strand (refer to Chapter 4) 4) and that durability is related more to cover and spacing of reinforcement than to crack width, so these concerns can be addressed with proper design and detailing of the reinforcement (Beeby 1978 and 1979).

1.5—Partial prestressing and reinforcement indexes Several indexes have been proposed to describe the extent of prestressing in a structural member. These indexes are useful in comparing relative performances of members made with the same materials, but caution should be exercised in using them to determine absolute values of such things as deformation and crack width. Two of the most common indices are the degree of prestress λ, and the partial prestressing ratio (PPR). These indexes are defined as

1.4—Advantages and disadvantages of partial prestressing In the design of most building elements, the specified live load often exceeds the normally applied load. This is to account for exceptional loading such as those due to impact, extreme temperature and volume changes, or a peak live load substantially higher than the normal live loads. By using partial prestressing, and by allowing higher flexural tension for loading conditions rarely imposed, a more economical design is achieved with smaller sections and less reinforcement. Where uniformity of camber among different members of a structure is important, partial prestressing will enable the

λ

M de c = --------------------- M D + M L

(1-1)

where M dec =

decomp decompres ressio sion n moment moment (the (the moment moment that that produc produces es zero concrete stress at the extreme fiber of a section, nearest to the centroid of the prestressing force, when added to the action of the effective prestress alone);

M D

=

dead dead-l -loa oad d mome moment nt;; and and

M L

=

live live-l -loa oad d mome moment nt


M np PP R = -------- M n

(1-2)

where M np =

nomina nominall moment moment capac capacity ity provi provided ded by prest prestres ressed sed reinforcement; reinforcement; and M n = tota totall nomi nomina nall mome moment nt cap capac acit ity. y. In the previous expressions, all moments are computed at critical sections. This report will generally use the PPR to describe the extent of prestressing in flexural members. The tests, studies, and examples described in this report usually concern members with PPR < 1, and the members are pretensioned unless otherwise noted. Characterizing the total amount of flexural reinforcement in a member is also important. This will be done with the reinforcement index ω

ω

=

f

f

f

ps ρ -- y--- + ρ p ----– ρ′ --- y- f c′ f c′ f c′

(1-3)

where

and A ps = As

=

A′ s

=

b d

= =

d p

=

′c f ′

=

f ps

=

f y

=

ρ

A = ------s bd

ρ′

A ′ = -----s-bd

ρ p

A ps = -------bd p

area area of pres prestre tresse ssed d reinf reinforc orceme ement nt in in tensi tension on zone zone,, 2 2 in. (mm ); area area of nonp nonpres restre tresse ssed d tensi tension on reinf reinforc orceme ement, nt, in. in. 2 (mm2); area area of nonpre nonprestr stress essed ed comp compres ressio sion n reinf reinforc orceement, in.2 (mm2); width width of of compr compress ession ion face face of member member,, in. in. (mm); (mm); distan distance ce from from extrem extremee compr compress ession ion fiber fiber to to cencentroid of nonprestressed tension reinforcement, in. (mm); distan distance ce from from extrem extremee compr compress ession ion fiber fiber to to cencentroid of prestressed reinforcement, in. (mm); specif specified ied compre compressi ssive ve stre strengt ngth h of concre concrete, te, psi (MPa); stress stress in pres prestre tresse ssed d reinf reinforc orceme ement nt at at nomin nominal al strength, psi (MPa); and yield yield stren strength gth of of nonpre nonprestr stress essed ed reinf reinforc orceme ement, nt, psi psi (MPa).

1.6—Report objective The objective of this report is to summarize the state of the art of the current knowledge as well as recent developments in partial prestressing so that engineers who are not experienced in prestressed concrete design will have a better

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CHAPTER 2—PARTIALLY PRESTRESSED MEMBERS UNDER STATIC LOADING 2.1—Behavior There are a number of investigations on the behavior of partially prestressed concrete beams under static loading (Abeles 1968; Burns 1964; Cohn and Bartlett 1982; Harajli 1985; Harajli and Naaman 1985a; Shaikh and Branson 1970; Thompson and Park 1980a; and Watcharaumnuay 1984). The following results were observed for beams having the same ultimate resistance in flexure but reinforced with various combinations of prestressed and nonprestressed reinforcement: • Partially prestressed prestressed beams show larger larger ultimate deflections, higher ductility, and higher energy absorption than fully prestressed beams; • Partially prestressed prestressed beams tend to crack at lower lower load levels than fully prestressed beams. Average crack spacing and crack widths are smaller. The stiffness of partially prestressed beams after cracking is larger; • For a given given reinforcement reinforcement index ω, the moment-curvature relationship is almost independent of the ratio of the tensile reinforcement areas (prestressed versus nonprestressed); • Changing the effective effective prestress in the prestressing prestressing tendons does not lead to any significant change in the ultimate resistance and curvature of flexural members; and • A decrease in effective effective prestress prestress leads to an increase increase in yield curvature and a decrease in curvature ductility.

2.2—Methods of analysis Several methods can be followed to analyze partially prestressed concrete members subjected to bending. In terms of assumptions, purpose, and underlying principles, they are identical to those used for reinforced and prestressed concrete (Nilson 1976, Naaman and Siriaksorn 1979, Siriaksorn and Naaman 1979, Al-Zaid and Naaman 1986, and Tadros 1982). In the elastic range of 2.2.1 Linear elastic analysis— In behavior, the analysis must accommodate either a cracked or an uncracked section subjected to bending, with or without prestress in the steel. The usual assumptions of plane strain distribution across the section, linear stress-strain relations, and perfect bond between steel and concrete remain applicable. Linear elastic analysis under service loads assuming an uncracked section is used for prestressed concrete. In the U.S., the design of reinforced concrete is predominantly based on strength requirement, but a linear elastic analysis under service loads is also necessary to check serviceability limitations such as crack widths, deflections, and fatigue. Prestressed concrete beams can act as cracked or uncracked sections, depending on the level of loading. In contrast to reinforced concrete, the centroidal axis of the cracked section does not coincide with the neutral axis point of zero stress (Fig. 2.1). 2.1). Moreover, the point of zero stress does not remain fixed, but moves with a change in applied load. When the effective prestress tends toward zero, the point of zero stress and the centroidal axis tend to coincide. Generalized equations have been developed to determine the zero stress point based on satisfying equilibrium, equilibrium, strain compatibility, compatibility, and stress-strain relations (Nilson 1976; Naaman and Siriaksorn

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Fig. 2.1—Assumed stress or strain distribution in linear elastic analysis of cracked cracked and uncracked sections (Naaman 1985).

provide unified treatment for cracked reinforced, prestressed, and partially prestressed sections. At ultimate or nominal moment 2.2.2 Strength analysis— At resistance, the assumptions related to the stress and strain distributions in the concrete, such as the compression block in ACI 318, or the stress and strain in the steel (such as yielding of the reinforcing steel) are identical for reinforced, prestressed, and partially prestressed concrete (Fig. 2.2) 2.2). The corresponding analysis is the same and leads to the nominal moment resistance of the section. Numerous investigations have shown close correlation between the predicted (based on ACI 318) and experimental values of nominal moments. The ACI 318 analysis, however, resulted in conservative predictions of section curvatures at ultimate load, leading to erroneous estimates of deformations and deflections (Wang et al. 1978, Naaman et al. 1986). To improve the prediction of nominal moment and curvature, either a nonlinear or a simplified nonlinear analysis may be followed. Simplified nonlinear analysis—In the simplified nonlinear analysis procedure (also called pseudo-nonlinear analysis), the actual stress-strain curve of the steel reinforcement is considered while the concrete is represented by the ACI 318 compression block. A solution can be obtained by solving two nonlinear equations with two unknowns, namely the stress and the strain in the prestressing steel at nominal moment resistance (Naaman 1977, Naaman 1983b). The best accuracy in determining Nonlinear analysis— The nominal moments and corresponding curvatures is achieved through a nonlinear analysis procedure (Cohn and Bartlett 1982, Naaman et al. 1986, Harajli and Naaman 1985b, Moustafa 1986). Nonlinear analysis requires as input an accurate analytical representation of the actual stress-strain curves of the component materials (concrete, reinforcing steel, and prestressing steel). Typical examples can be found in two references (Naaman et al. 1986, Moustafa 1986).

2.3—Cracking

Fig. 2.2—Assumed strain distribution and forces in: (a) nonlinear analysis; (b) approximate nonlinear analysis; and (c) ultimate strength analysis by ACI Code (Naaman 1985).

1986). They usually are third-order equations with respect to member depth. Although they can be solved iteratively, charts, tables, and computer programs have been developed for their

Partially prestressed concrete permits cracking under service loads as a design assumption. To satisfy serviceability requirements, the maximum crack width should be equal to, or smaller than, the code-recommended limits on crack width. The maximum allowable crack widths recommended by ACI Committee 224 (1980) for reinforced concrete members can be used, preferably with a reduction factor for prestressed and partially prestressed concrete members. To select the reduction factor, consideration should be given to the small diameter of the reinforcing elements (bars or strands), the cover, and the exposure conditions. Only a few formulas are used in the U.S. practice to predict crack widths in concrete flexural members. Because the factors influencing crack widths are the same for reinforced and partially prestressed concrete members, existing formulas for reinforced concrete can be adapted to partially prestressed concrete. Five formulas (ACI 224 1980; Gergely and Lutz 1968; Nawy and Potyondy 1971; Nawy and Huang 1977; Nawy and Chiang 1980; Martino and Nilson 1979; and Meier and Gergely 1981) applicable to partially prestressed


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Table 2.1—Crack width prediction equations applicable to par tially prestressed beams (Naaman 1985) Source

W max = 7.6 × 10

Gergely and Lutz (1968) ACI Code (1971, 1977, and 1983) ACI Committee 224 (1980)

Nawy and Potyondy (1971)

–5

β f s

3 d A c b

Multiply expression by 0.1451

f s = tensile stress in reinforcing steel d c = concrete cover to center of closest bar layer Ab = concrete tensile area per bar β = ratio of distances from tension face and steel centroid to neutral axis W max = 1.44 × 10 W max =

Nawy and Huang (1977) Nawy and Chiang (1980)

Equation * with SI system, (mm., N/mm 2)

Equation* wi with U.S. system, (in., ksi)

–4

Note: ACI Committee 224 recommends multiplication multiplication factor of 1.5 when strands, rather than deformed bars, are used nearest to beam tensile face.

( f s – 8.3 )

= 5.31 × 10

–4

( f s – 57.2 )

A

α × 10 –5 β -------t - ∆ f ps ΣO

At = area of concrete tensile zone ΣO = sum of perimeters of bonded reinforcing elements ∆ f ps = net stress change in prestressing steel after decompression

Multiply expression by 0.1451

5.85 if pretensioning α =   6.51 if post-tensioning –5

Martino and Nilson (1979)

*

= 2 × 10 d ′c f s + 0.08

d ′c= concrete clear cover

 W ma x   W ma x Meier and Gergely (1981)

–5

W max = 14 × 10 d c′ f s + 0.0031

= C 1 ε ct d c

(1)

(1) Sa Same eq equa ti tion

= C 2 ε ct d c 3 A b

(2) (2)

(2) Multi ultip ply by 220 220

C 1, C 2 = bond coefficients For reinforcing bars: C 1 = 12; C 2 = 8.4 C 1 = 16; C 2 = 12 For strands: εct = nominal concrete tensile strain at tensile face

In the formulas shown, f shown, f s can be replaced by ∆ f ps when applied to partially prestressed concrete.

able tensile stress in the reinforcing steel f s should be replaced by the stress change in the prestressing steel after decompression ∆ f ps. The ACI 318 formula initially developed by Gergely and Lutz (1968) for reinforced concrete could be used as a first approximation for partially prestressed concrete. Meier and Gergely (1981), however, suggested a modified form (shown in Table 2.1) 2.1) for the case of prestressed concrete. This alternate formula uses the nominal strain at the tensile face of the concrete (instead of the stress in the steel), and the cover to the center of the steel d c. Both the stress in the steel and the clear concrete cover are found to be the controlling variables in the regression equation derived by Martino and Nilson (1979). The two prediction equations proposed by Nawy and Huang (1977) and Nawy and Chiang (1980) contain most of the important parameters found in the cracking behavior of concrete members except the concrete cover, which is accounted for indirectly. Moreover, they are based on actual experimental results on prestressed and partially prestressed beams. As pointed out by Siriaksorn and Naaman (1979), large differences can be observed in predicted crack widths depending on the prediction formula used. Harajli and Naaman (1989) compared predicted crack widths with observed crack widths from tests on twelve partially prestressed concrete beams. They considered the three prediction equations

ng (1977), and Meier and Gergely (1981). Although none of the three equations gave sufficiently good correlation with experimental data for all conditions, the following observations were made (Fig. 2.3) 2.3): • The Gergely and Lutz equation equation gave a lower prediction prediction in all cases (Fig. 2.3(a)); 2.3(a)); • The Meier and Gergely Gergely equation gave gave the worst correlation (Fig. 2.3(c)); 2.3(c)); and • The Nawy and Huang equation equation gave a higher prediction prediction in most cases (Fig. 2.3(b)) 2.3(b)). Although more experimental data are needed to improve the accuracy of crack-width prediction equations available in U.S. practice, there is sufficient information to judge if the serviceability, serviceability, with respect to cracking or crack width under short-term loading, is satisfactory for a partially prestressed member. The effects of long-term loading and repetitive loading (fatigue) on the crack widths of partially prestressed members need to be further clarified. A research investigation provided an analytical basis to deal with the problem (Harajli and Naaman 1989); however, the proposed methodology is not amenable to a simple prediction equation that can be easily implemented for design.

2.4—Deflections Fully prestressed concrete members are assumed to be uncracked and linearly linearly elastic under service service loads. Instantaneo Instan taneous us

423.5R-8


The widely accepted concept of the effective moment of inertia I eff , initially introduced by Branson (1977) for reinforced concrete, has been examined by several researchers and modified accordingly to compute the deflection in cracked prestressed and partially prestressed members. The modified effective moment of inertia is defined (Naaman 1982a) as I ef f = I cr + ( ( M cr – M de c ) ⁄ ( M a – M de c ) )

3

(2-1)

( I g – I cr ) ≤ I g where I g = gros grosss mom momen entt of of ine inert rtia ia,, in. in.4 (mm4); I cr = mome moment nt of of iner inerti tiaa of cra crack cked ed sec secti tion on,, in. in. 4 (mm4); M cr = crac cracki king ng mom momen ent, t, in. in.-k -k (mm (mm-N -N); ); M dec = deco decomp mpre ress ssio ion n mome moment nt,, in.in.-k k (m-N (m-N); ); and and M a = appl applie ied d mom momen ent, t, in.in.-k k (m(m-N) N).. Although there is general agreement for the use of the previous expression, substantial divergence of opinion exists as to the computation of I cr and M dec. The computation difference is whether the moment of inertia of the cracked section should be computed with respect to the neutral axis of bending or with respect to the zero-stress point, and whether the decompression moment should lead to decompression at the extreme concrete fiber or whether it should lead to a state of zero curvature in the section. The discussion of Tadros’ paper (1982) by several experts in the field is quite informative on these issues. A systematic comparison between the various approaches, combined with results from experimental tests, is given in work by Watcharaumnuay (1984), who observed that the use of I cr with respect to the neutral axis of bending is preferable, while the use of M dec as that causing decompression at the extreme concrete fiber, is easier and leads to results similar to those obtained using the zero curvature moment.

2.5—Shear and torsion 2.5.1

Fig. 2.3—Comparison of observed and theoretically predicted crack widths (Naaman 1985).

principles of mechanics. To compute short-term deflections, customary U.S. practice is to use the gross moment of inertia I g for pretensioned members, or the net moment of inertia I n for members with unbonded tendons, and the modulus of elasticity of concrete at time of loading or transfer E ci. Several approaches proposed by various researchers to compute short-term and long-term deflections in prestressed or partially prestressed uncracked members are summarized in Table 2.2 (Branson and Kripanarayanan 1971; Branson 1974; Branson 1977; Naaman 1982a; Naaman 1983a; Branson and Trost 1982a; Branson and Trost 1982b; Martin 1977; Tadros et al. 1975; Tadros et al. 1977; Dilger 1982; and Moustafa 1986). Although no systematic evaluation or comparison of these different approaches has been undertaken, for common cases they lead to results of the

General— Nonprestressed Nonprestressed and fully prestressed concrete (tensile stress in the concrete under full service load is zero) are the two limiting cases of steel-reinforced concrete systems. Partially prestressed concrete represents a continuous transition between the two limit cases. A unified approach in design to combined actions including partial prestressing would offer designers a sound basis to make the appropriate choice between the two limits (Thurlimann 1971). The equivalent load concept provides a simple and efficient design of prestressed concrete structures under combined actions (Nilson 1987). For example, this approach allows the designer to calculate the shear component of the prestress anywhere in the beam, simply by drawing the shear diagram due to the equivalent load resulting from a change in the vertical alignment of the tendon (Fig. 2.4) 2.4). That equivalent load, together with the prestressing forces acting at the ends of the member through the tendon anchorage, may be looked upon as just another system of external forces acting on the member. This procedure can be used for both statically determinate and indeterminate structures, and it


423.5R-9

Table 2.2—Deflection prediction equations for prestressed and partially prestressed beams (from Naaman 1985) Source ACI 435 (1963)

ACI Code Section 9.5 (1971, 1977, and 1983)

Short-term instantaneous deflection

Long-term or additional long-term deflection

∆t is obtained from elastic analysis using F t , E ct , and I g .

Long-term deflection obtained by integrating curvatures curvatures with due account for creep effects and prestress losses with time.

• Uncracked Uncracked section section;; and • No provisi provisions ons for for As and A′s.

∆add shall be computed, taking into account stresses under sustained load, including effects of creep, shrinkage, and relaxation.

• No provisio provisions ns for partial partial prestressing (cracking, As and A′s ).

∆t shall be obtained from elastic analysis using I g for uncracked sections.

Remarks

1 + η - k r C cu ∆ad d = η – 1 +  ----------- 2

Branson et al. (1971, 1974, and 1977)

∆t is obtained from elastic analysis using E ct and I g.

( ∆ t ) F t + k r C CU ( ∆ t ) G + K CA k r C CU ( ∆t ) SD where C CU = ultimate creep coefficient of concrete; η = F / F Ft ; k r = 1/(1 + As / A ps); and K CA = age at loading factor for creep

The long-term deflection is estimated from:

Naaman (1982 and 1983)

Bronson and Trost (1982)

∆t is obtained using I g and the predicted elastic modulus at time of loading E c (t ). ).

For cracked members, the short-term deflection is computed using I eff modified for partial prestressing.

∆ ( t )

2

=

φ 1 ( t ) l--- + [ φ 2 ( t ) – φ 1 ( t ) ]

2

l -----48

8 where φ1(t ) = midspan curvature at time t ; φ2(t ) = support curvature at time t ; ]; and φ(t ) = M /[ E E ce(t ) × I ]; E ce(t ) = equivalent equivalent modulus

Long-term deflection is not addressed but it is assumed that for a given ∆ , t the earlier method is applicable.

∆ad d = λ1 ( ∆i ) G + λ2 ( ∆i )F i Martin (1977)

Tadros et al. (1975 and 1977)

Dilger (1982)

Moustafa (1986)

∆t is obtained from elastic analysis using E ct and I g.

∆t is obtained from elastic analysis using E c(t ) and I g.

∆t is obtained from long-term deflection expression at initial loading time. The age adjusted effective effective modulus and a creep transformed moment of inertia are used.

∆t is obtained from nonlinear analysis using actual material properties.

• Uncracked Uncracked section; section; • k r is applicable only when (∆t ) F t + G is a camber; and • F t = initial prestressing force immediately after transfer. transfer.

• Uncracked Uncracked section; section; • The pressure pressure line line is assumed assumed resulting from the sustained loadings; • The profile profile of the pressur pressuree line is assumed parabolic; • Prestress Prestress losses losses must must be estimated a priori; • Design Design chart is provide provided d for the equivalent modulus; and • As and A′s are accounted for through I t and neutral axis of bending. • Cracked Cracked members. members.

α = ( 2 – 1.2 As ′ ⁄ As ) ≥ 0.6 λ2 = ηλ 1

• k r = same as Branson; • Uncracked Uncracked section; section; • Design Design values values of λ1 and λ2 were recommended; and • The method method is adopt adopted ed in PCI Design Handbook .

The long-term deflection is obtained by integrating the curvatures modified by a creep recovery parameter and a relaxation reduction factor that are time-dependent.

• Uncracked Uncracked section sections; s; and • For common common loadin loading g cases, only the curveatures at the support and midspan sections are needed.

λ1

E ci α = k r ----- E c

The long-term deflection is obtained by integrating the curvature along the member. The time-dependent curvature is modified by the effect of an equivalent force acting at the centroid of the prestressing steel due to creep and shrinkage strain. φ ( t ) = φ i C c ( t ) – ------ M ---------c----- I tr E ca ( t ) I tr = transformed moment of inertia; M c = moment due to equivalent transformed force; and E ca(t ) = age adjusted modulus The nonlinear analysis takes both creep and shrinkage into account, using ACI creep and shrinkage functions and a time step method.

• Uncracked Uncracked section sections; s; and • A relaxation relaxation reduction reduction factor factor is used.

• A computer computer program program is available from PCI to perform the nonlinear analysis.

423.5R-10


Fig. 2.4—Equivalent loads and moments produced by prestressing prestressing tendons (Nilson 1987); P = prestressing force.

prestressing, as well. This approach allows the designer to treat a prestressed concrete member as if it was a nonprestressed concrete member. The prestressing steel is treated as mild (passive) reinforcement for ultimate conditions, with a remaining tensile capacity of ( f ps – f pe), where f ps is the stress in the reinforcement at nominal strength, and f pe is the effective stress in the prestressed reinforcement (after allowance for all losses). Most codes of practice (ACI 318; AASHTO Bridge Design Specifications, Eurocode 2; and CSA Design of Concrete Structures for Buildings) use sectional methods for design of conventional beams under bending, shear, and torsion. Truss models provide the basis for these sectional design procedures that often include a term for the concrete contribution (Ramirez and Breen 1991). The concrete contribution supplements the sectional truss model to reflect test results in beams and slabs with little or no shear reinforcement and to ensure economy in the practical design of such members. In design specifications, the concrete contribution has

cracking, or as the capacity of an equivalent member without transverse reinforcement. Therefore, detailed expressions have been developed in terms of parameters relevant to the strength of members without transverse reinforcement. These parameters include the influence of axial compression, member geometry, support conditions, axial tension, and prestress. The following behavioral changes occur in 2.5.2 Shear— The partially prestressed members at nominal shear levels, as some of the longitudinal prestressing prestressing steel in the tension face of the member is replaced by mild reinforcement, but the same total flexural strength is maintained: • Due to the lower effective effective prestress, the external external load required to produce inclined cracking is reduced. This results in an earlier mobilization of the shear reinforcement; and • After inclined cracking, cracking, there is a reduction in the concrete contribution. The reduction is less significant significant as the degree of prestressing decreases. This can be explained as follows: •The addition of mild reinforcement results in an


tension reinforcement and the reinforcement stiffness; and •The increase in stiffness of the longitudinal tension reinforcement delays the development of the cracking pattern, so that the cracks are narrower and the flexural compression zone is larger than in fully prestressed members of comparable flexural strength. These behavioral changes are well documented in a series of shear tests carried out by Caflisch et al. (1971). In this series of tests, the only variable was the degree of prestressing. The cross sections of the prestressing steel and the reinforcing steel were selected so that all the beams had the same flexural strength. These tests also showed that for the same external load, a higher degree of prestressing delays the onset of diagonal cracking and results in a decrease in the stirrup forces. The decrease in stirrup forces can be explained by the fact that a higher degree of prestressing in the web of the member results in a lower angle of inclination of the diagonal cracks. The lower angle of inclination of the cracks leads to the mobilization of a larger number of stirrups. In ACI 318, a cursory review of the design approach for shear indicates that partially prestressed members can be designed following the same procedure as for fully prestressed members. In ACI 318, it is assumed that flexure and shear can be handled separately for the worst combination of flexure and shear at a given section. The analysis of a beam under bending and shear using the truss approach clearly indicates that, to resist shear, the member needs both stirrups and longitudinal reinforcement. The additional longitudinal tension force due to shear can be determined from equilibriequilibrium conditions of the truss model as ( V cot cot θ), where V is is the shear force at the section, and θ is the angle of inclination of the inclined struts with respect to the longitudinal axis of the member. In the shear provisions of ACI 318, no explicit check of the shear-induced force in the longitudinal reinforcement is performed (Ramirez 1994). The difference between the flexural strength requirements for the prestress reinforcement and the ultimate tensile capacity of the reinforcement can be used to satisfy the longitudinal tension requirement. The 1994 AASHTO LRFD Bridge Design Specifications, in the section for shear design, includes a check for longitudinal reinforcement. These recommendations are based on a modified compression field theory (Vecchio and Collins 1986). ACI 318 includes design recommenda2.5.3 Torsion— ACI tion for the case of torsion or combined shear and torsion in prestressed concrete members. These provisions model the behavior of a prestressed concrete member before cracking as a thin-walled tube and after cracking using a space-truss model with compression diagonals inclined at an angle θ around all faces of the member. For prestressed members, θ can be taken equal to 37.5 degrees if the effective prestressing force is not less than 40% of the tensile strength of the prestressed reinforcement. For other cases, θ can be taken equal to 45 degrees. This approach is based on the work carried out in the 1960s and 1970s by European investigators

423.5R-11

supported by the theory of plasticity, in which a space truss with variable inclination of compression diagonals provides a lower-bound (static) solution. This procedure is representative of the behavior of thinwalled tubes in torsion. For these members, the shear stresses induced by torsion can be determined using only equilibrium relationships. Because the wall of the tube is thin, a constant shear shear stress can be a ssumed across its thickness. In the longitudinal direction, equilibrium conditions dictate that the torsion-induced shear stresses be resisted by a constant shear flow around the perimeter of the section. For other sections before cracking, the strength in torsion can be computed from the elastic theory (de Saint-Venant 1956) or from the plastic theory (Nadai 1950). Rather than using these more complex approaches, an approximate procedure is used in ACI 318 based on the concept that most torsion is resisted by the high shear stresses near the outer perimeter of the section. In this approach, the actual cross section before cracking is represented by an equivalent thinwalled tube with a wall thickness t of of A --t = 0.75 -----cp P cp

(2-2)

where Acp = area enclosed by outside perimeter of concrete cross section, and Pcp = outside perimeter of the concrete cross section. While the area enclosed by the shear flow path, Ao, could be calculated from the external dimensions and wall thickness of the equivalent tube, it is reasonable to approximate it as equal to 2 Acp /3. / 3. Cracking is assumed to occur when the principal tensile stress reaches 4 √ f ′ c′ . For prestressed members, the cracking torque is increased by the prestress. A Mohr’s Circle analysis based on average stresses indicates that the torque required to cause a principal tensile stress equal to 4 √ f ′ c′ is the corresponding cracking torque of a nonprestressed beam times f 1 + ------ pc ------4 f c′

(2-3)

where f pc in psi is the average precompression due to prestress at the centroid of the cross section resisting the externally applied loads or at the junction of web and flange if the centroid lies within the flange. In ACI 318, the design approach for combined actions does not explicitly consider the change in conditions from one side of the beam to the other. Instead, it considers the side of the beam where shear and torsional effects are additive. After diagonal cracking, the concrete contribution of the shear strength V c remains constant at the value it has when there is no torsion, and the torsion carried by the concrete is taken as zero. The approach in ACI 318 has been compared with test results by MacGregor and Ghoneim (1995). In the AASHTO LRFD Specifications, the modified com-

423.5R-12


Fig. 3.1—Typical stress change in prestressing steel at end of service life for sections cracked and uncracked under permanent loads (Watcharaumnuay (Watcharaumnuay and Naaman 1985).

been extended to include the effects of torsion. Similar to the ACI 318 procedure, the AASHTO approach concentrates on the design of the side of the beam where the shear and torsional stresses are additive. As in the case of shear, torsion leads to an increase in the tensile force on the longitudinal reinforcement. The longitudinal reinforcement requirement requirement for torsion should be superimposed with the longitudinal reinforcement requirement for bending that acts simultaneously with the torsion. In ACI 318, the longitudinal tension due to torsion can be reduced by the compressive force in the flexural compression zone of the member. Furthermore, in prestressed beams, the total longitudinal reinforcement, including tendons at each section, can be used to resist the factored bending moment plus the additional tension induced by torsion at that section. ACI 318 and AASHTO Specifications recognize that, in many statically indeterminate structures, the magnitude of the torsional moment in a given member will depend on its torsional stiffness. stiffness. Tests have shown (Hsu 1968) that when a member cracks in torsion, its torsional stiffness immediately after cracking drops to approximately 1/5 of the value before cracking, and at failure can be as low as 1/16 of the value before cracking. This drastic drop in torsional stiffness allows a significant redistribution of torsion in certain indeterminate beam systems. In recognizing the reduction of torsional moment that will take place after cracking in the case of indeterminate members subjected to compatibilityinduced torsion, ACI 318R states that a maximum factored torsional moment equal to the cracking torque can be assumed to occur at the critical sections near the faces of supports. This limit has been established to control the width of the torsional cracks at service loads.

components, which are generally grouped into instantaneous and time-dependent losses. Instantaneous losses are due to elastic shortening, anchorage seating, and friction. They can be computed for partially prestressed concrete in a manner similar to that for fully prestressed concrete. Time-dependent losses are due to shrinkage and creep of concrete and relaxation of prestressing steel. Several methods are available to determine time-dependent losses in prestressed concrete members: lump-sum estimate of total losses, lump-sum estimates of separate losses (such as loss due to shrinkage or creep), and calculation of losses by the timestep method. Because the numerical expressions— described, for instance, in the AASHTO’s Standard Specifications for Highway Bridges or the PCI Design Handbook (1992)—for the first two methods were developed assuming full prestressing, they are not applicable to partially prestressed members. The combined presence of nonprestressed reinforcing bars and a lower level of prestress in partially prestressed concrete should lead to smaller time-dependent prestress losses than for fully prestressed concrete. Another significant factor is that a partially prestressed member can be designed as a cracked member under sustained loading. A computerized time-step analysis of the concrete and steel stresses along partially prestressed sections shows that the effect of creep of concrete on the stress redistribution between the concrete and steel tends to counteract the effect of prestress losses over time. This is particularly significant for members that are cracked under permanent loads. The result is illustrated in Fig. 3.1 (Watcharaumnuay and Naaman 1985), which was derived from the analysis of 132 beams with various values of the partially prestressed ratio ( PPR) and the reinforcement index ω. While time-dep endent prestress losses can be 14% for uncracked fully prestressed sections, they remain low for cracked sections up to relatively high values of the 3.1 also shows that for uncracked sections, timePPR. Fig. 3.1 also dependent prestress losses decrease with a decrease in PPR. Creep redistribution of force to reinforcing steel may reduce the precompression in the concrete. An investigation under NCHRP Project 12-33 has addressed prestress losses in partially prestressed normal and high-strength concrete beams. This work was described by Naaman and Hamza (1993) and was adopted in the Final Draft LRFD Specifications for Highway Bridge Design and Commentary (Transportation Research Board 1993). It led to lump-sum estimates of time-dependent losses for partially prestressed beams that are assumed uncracked under the design sustained loading. These are summarized in Table 3.1 and can be used as a first approximation in design.

3.2—Cracking CHAPTER 3—TIME-DEPENDENT BEHAVIOR 3.1—Prestress losses The stress in the tendons of prestressed concrete structures structures decreases continuously with time. The total reduction in stress during the life span of the structure is termed “total

Crack widths in reinforced and cracked partially prestressed concrete members subjected to sustained loads are known to increase with time. Bennett and Lee (1985) reported that crack widths increase at a fast rate during the early stages of loading then tend toward a slow steady rate of


423.5R-13

Table 3.1—Time-dependent losses, ksi (LRFD Bridge Design Specifications 1994) Type of beam section

Level

Rectangular beams Upper bound and solid slab Average Box girder rder

I-girder

Single-T, double-T, hollow core, and voided slab

For wires and strands with f pu = 235, 250, or 270 ksi 29.0 + 4.0 PPR 26.0 + 4.0 PPR

Uppe Upperr bound ound

21.0 1.0 + 4.0 PPR

Average

19.0 + 4.0 PPR

Average

f c′ – 6.0 - + 6.0 PP R 33.0 1.0 – 0.15 -----------------6.0

Upper bound

f c′ – 6.0 - + 6.0 PP R 39.0 1.0 – 0.15 -----------------6.0

Average

f c′ – 6.0 33.0 1.0 – 0.15 ------------------ + 6.0 PP R 6.0

For bars with f pu = 145 or 160 ksi 19.0 + 6.0 PPR 15.0 19.0 + 6.0 PPR

f c′ – 6.0 - + 6.0 PP R 31.0 1.0 – 0.15 -----------------6.0

camber increase with time. Crack widths, however, represent only localized effects, and their relative increase with time is not proportional to the increase of deflections. No studies have been reported where an analytical model of crack width increase with time was developed. An investigation by Harajli and Naaman (1989), however, discussed in Chapter 4 of 4 of this report, has led to the development of a model to predict the increase in crack width under cyclic fatigue loading. The model accounts for the effect of change in steel stress due to cyclic creep of concrete in compression, the increase in slip due to bond redistribution, and concrete shrinkage.

3.3—Deflections Several experimental investigations have dealt with the time-dependent deflection of partially prestressed concrete members (Bennett and Lee 1985; Bruggeling 1977; Jittawait and Tadros 1979; Lambotte and Van Nieuwenburg 1986; Abeles 1965; and Watcharaumnuay and Naaman 1985). Deflections and cambers in partially prestressed concrete members are expected to vary with time similarly to reinforced or fully prestressed concrete. When positive deflection (opposite to camber) is present, in all cases the deflection in partially prestressed beams falls between those of reinforced concrete and fully prestressed concrete beams (Fig. 3.2). 3.2). A limited experimental study by Jittawait and Tadros (1979) also seems to confirm this observation. A study of the long-term behavior of partially prestressed beams has been conducted at the Magnel laboratory in Belgium (Lambotte and Van Nieuwenburg 1986). Twelve partially prestressed beams with PPR of 0.8, 0.65, and 0.5 were either kept unloaded or were loaded with an equivalent full service load. For the unloaded beams, increased camber with time was generally observed for PPR = 0.8; for PPR = 0.65, the camber reached a peak value and then decreased with time, resulting in a practically level beam; and for PPR = 0.5, the initial camber decreased with time, resulting in a final downward deflection. For the loaded beams, deflections were observed in all cases and increased with time. These observations are illustrated in Fig. 3.3. 3.3. After 2 years of load-

Fig. 3.2—Long-term deflections of fully prestressed and partially prestressed prestressed (cracked and uncracked) beams (Naaman 1982); 1 in. = 25.4 mm.

was about 1.25 for pretensioned members and 1.5 for posttensioned members. Several analytical investigations investigations have dealt with the timedependent deflections of prestressed and partially prestressed beams assumed to be uncracked (Table 2.2) 2.2). The evaluation of deflections for cracked, partially prestressed members has been conducted by several researchers (Branson and Shaikh 1985; Ghali and Tadros 1985; Tadros et al. 1985; Ghali and Favre 1986; Al-Zaid et al. 1988; Elbadry and Ghali 1989; Ghali 1989; and Founas 1989). Watcharaumnuay and Naaman (1985) proposed a method to determine time- and cyclic-dependent deflections in simply supported, partially prestressed beams in both the cracked and uncracked state. The time-dependent deflection is treated as a special case of cyclic deflection. Compared with the time-step method where the deflection is obtained from

423.5R-14


K D , K F =

cons consta tant ntss dep depen endi ding ng on type type of load loadin ing g and and steel profile;

M

=

sust sustai aine ned d ext exter erna nall mome moment nt at mids midspa pan; n;

F(t)

=

pres prestr tres esssing ing for force ce at time t ;

I eff (t)

=

effe effect ctiv ivee mome moment nt of of iner inerti tiaa of cra crack cked ed sec secti tion on at time t ; and

E ce (t)

=

equiva equivalen lentt elasti elasticc modulu moduluss of concre concrete te at time time t .

The eqivalent elastic modulus of concrete can be approximated by the following equation E c ( t ) E ce ( t ) = -------------------------------1 + C c ( t – t A )

Fig. 3.3(a)—Variation with time of midspan deflection for unloaded specimens (Lambotte and Van Nieuwenburg 1986); 1 in. = 25.4 mm.

(3-2)

where t

= time time or or age age of con concr cret ete; e;

t A

= age of of concre concrete te at time time of of loadin loading; g;

E c (t)

= instanta instantaneous neous elastic elastic modulus modulus of concre concrete te at time t ;

and C c (t - t A) = creep creep coeffi coefficient cient of concre concrete te at at time time t when loaded at time t A.

Several expressions are available to predict the creep coefficient of concrete. The recommendations of ACI Committee 209 (1982) can be followed in most common applications. Fig. 3.3(b)—Variation with time of midspan deflection for specimens under service loading (Lambotte and Van Nieuwenburg Nieuwenburg 1986); 1 in. = 25.4 mm.

intervals, this method leads to the deflection at any time t and and cycle N directly. directly. The method satisfies equilibrium and strain compatibility. Using a slightly different approach, the method proposed by Watcharaumnuay and Naaman (1985) was generalized by Al-Zaid et al. (1988) and Founas (1989), and was extended to include composite beams as well as noncomposite beams. In dealing with time-dependent deflections, several timedependent variables should be determined. These include the prestressing force, the moment of inertia of the section, and the equivalent modulus of elasticity of the concrete. The expression for the effective moment of inertia described in Eq. (2-1) can be used. In this expression, however, M cr and I cr are time-dependent variables because they depend on the value of the prestressing force and the location of the neutral axis (zero stress point along the section), both of which vary with time. The equivalent modulus of elasticity of the concrete depends on the variation of creep strain with time. The following method can be used to estimate deflection at any time t in in a cracked, simply supported beam K D M K F F ( t ) ∆ ( t ) = ----------------------------+ ---------------------------- E ce ( t ) I ef f ( t ) E ce ( t ) I ef f ( t )

(3-1)

3.4—Corrosion The reinforcement in a fully prestressed member is better protected against corrosion than the reinforcement in a partially prestressed member. Cracks in partially prestressed beams are potential paths for the passage of corrosive agents. Although corrosion also occurs along uncracked sections, cracking can facilitate corrosion. Abeles (1945) suggested that corrosion of the prestressing steel in partially prestressed members can be mitigated by requiring that the member faces that are cracked under full service load be in compression under permanent (dead) loads. He demonstrated the effectiveness of this strategy in the behavior of beams partially prestressed using small-diameter wires that were used in the roof of an engine shed for steam locomotives. These beams successfully resisted a very corrosive atmosphere caused by the mixture of smoke and steam ejected onto them from the funnels of the locomotives. Limiting the size of crack widths to reduce the probability of corrosion has been common practice in design. Later studies (ACI 222R-89), however, move away from this approach by pointing out that corrosion is due to many causes, most of which can proceed with or without cracking to be activated. Corrosion in prestressing steels is much more serious than corrosion in nonprestressed reinforcing steels. Prestressing steel is generally stressed to over 50% of its strength, making it susceptible to stress corrosion, and the diameter of individual prestressing steel wires is relatively small. Even a small, uniform corrosive layer or a corroded spot can progressively reduce the cross-sectional area of the steel and lead to wire


Corrosion is mostly an electrochemical problem and should be treated accordingly. Precautions should be taken to prevent or to reduce prestressing steel corrosion. ACI 423.3R addresses the historical causes of corrosion in unbonded tendons. The Post-Tensioning Manual (Post-Tensioning Institute 1990) provides guidance for corrosion protection for bonded and unbonded tendons. Occasionally, for pretensioned concrete, epoxy-coated prestressing strands have been specified for corrosive environments. High curing temperatures, however, could adversely affect the bond of epoxy-coated strand. Guidelines for the Use of Epoxy-Coated Strand (PCI (PCI Ad Hoc Committee 1993) contains recommendations for its use. Lenschow (1986) reported that crack widths less than 0.004 to 0.006 in. (0.1 to 0.15 mm), which develop under maximum load, will heal under long-term compression. Crack widths that increase to less than 0.01 in. (0.3 mm) under rare overload (every 1 to 3 years) can reduce to 0. 004 to 0.006 in. (0.1 to 0.15 mm) under sustained compression. Keeping crack widths under such limits should avoid problems with corrosion.

CHAPTER 4—EFFECTS OF REPEATED REPEATED LOADING (FATIGUE) 4.1—Background Two major requirements should be considered when designing members subjected to repeated loads: member strength and serviceability. The static strength and fatigue strength of the member should exceed loads imposed and adequate serviceability requirements (deflection and crack control) should be provided. The fatigue strength of a member is affected primarily by the stress range (difference between maximum and minimum stress), the number of load applications or cycles, and the applied stress levels. The fatigue life of a member is defined as the number of load cycles before failure. The higher the stress range imposed on the member, the shorter the fatigue life. Reliability analyses indicate that the probability of fatigue failure of reinforcement in partially prestressed beams is higher on average than failure by any other common serviceability or ultimate limit state criterion (Naaman 1985, Naaman and Siriaksorn 1982). Fatigue can be a critical loading condition for partially prestressed concrete beams because high stress ranges can be imposed on the member in the service-load range (Naaman and Siriaksorn 1979). Partially prestressed concrete beams generally crack upon first application of live load (Naaman 1982b). Subsequent applications of live load cause the cracks to reopen at the decompression load (when the stress at the extreme tensile face is zero), which is less than the load that caused first cracking. To maintain equilibrium in the section after cracking, the neutral axis shifts toward the extreme compression fiber. This shift generates higher strains (stresses) in the tensile reinforcement. Under repeated loads, the larger stress changes (created by opening and closing of the cracks) cause fatigue damage

423.5R-15

increased crack widths and deflections under service loads (Naaman 1982b; Shakawi and Batchelor 1986; and ACI Committee 215 1974). The increase in crack widths in partially prestressed beams, however, has been smaller than that generated in similarly loaded, precracked, fully prestressed beams (Harajli and Naaman 1984). Because the proportion of dead to total load often increases as the span length increases, the significance of fatigue as a critical limit state tends to diminish as span lengths increase (Freyermuth 1985). Abeles demonstrated the practicability of using partially prestressed concrete members when fatigue resistance is a serious consideration (Abeles 1954). He persuaded British Railways to consider the use of partial prestressing in the reconstruction of highway bridges over the London to Manchester line, when it was electrified around 1950. British Railways financed extensive cyclic loading tests of fullscale members, which were designed to allow 550 psi (approximately 8 √ f ′ c′ or 3.8 MPa) tension under full service load and 50 psi (0.34 MPa) compression under dead load only. These members were cracked under static load and were then subjected to 3 million cycles of load producing the design range of stress, 50 psi (0.34 MPa) compression to 550 psi (3.8 MPa) tension at the flexural tension face. Behavior was satisfactory, with essentially complete closure of cracks and recovery of deflection after 3 million cycles of load. The strength under static loading was not decreased by the cyclic loading. Many relatively short-span bridges were constructed using such partially prestressed members and they performed satisfactorily. satisfactorily. Recently, Roller et al. (1995) conducted an experimental program including four full-size, pretensioned, bulb-tee girders made with high-strength concrete and pretensioned. The girders were 70 ft (21.3 m) long and 54 in. (1.4 m) deep with a concrete compressive strength of 10,000 psi (69 MPa). One of the four test girders with a simple span of 69 ft (21.0 m) was subjected to cyclic (fatigue) flexural loading using two point loads spaced 12 ft (3.66 m) apart at midspan. A concrete deck 10 ft (3.05 m) wide and 9.5 in. (250 mm) thick had been cast on the girder to represent the effective flange of the composite girder in a bridge. During the cyclic flexural loading, the upper limit of the load produced a midspan tensile stress at the extreme fiber of the lower flange equal to 6 √ f c′ ′ . The lower limit of the load was selected such that a steel stress range of 10,000 psi (69 MPa) would be produced. After each million cycles of loading, the girder was tested statically to determine its stiffness. Slight reductions in stiffness and camber were observed, but there was no significant change in prestress loss. The girder performed satisfactorily satisfactorily for 5 million cycles of fatigue loading. After completion of the long-term fatigue load test, the girder was tested under static load to determine its ultimate flexural strength. It developed an ultimate moment equal to 94% of the ultimate moment capacity of a companion girder that had been under long-term sustained load. The measured moment capacity also exceeded the calculated moment capacity by 7.5% based on the AASHTO Standard Specifi-

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Tests have been conducted on ordinary reinforcement, both in-air and embedded in concrete, to determine its fatigue properties. These tests have yielded varying results (Rehm 1960, Soretz 1965). For straight deformed bars, ACI Committee 215 (1974), Model Code for Concrete Concrete Structur Structures es (CEB-FIP 1978), FIP Commission on Model Code (1984), and Ontario Highway Bridge Design Code (Ministry of Transportation and Communications 1983) recommend stress range limits of 20, 22, and 18 ksi (138, 152, and 124 MPa), respectively. The lowest stress range found to cause fatigue failure in a hotrolled bar is 21 ksi (145 MPa) (ACI Committee 215 1974). ACI 343R recommends limiting the reinforcement stress range in terms of the minimum stress and reinforcement deformation geometry f f = 21 – 0.33 f mi n + 8 ( r ⁄ h )

(4-2)

where f = f f min = r/h =

Fig. 4.1—Comparison of observed fatigue life of prestressing strands with existing data (Harajli and Naaman 1985a).

4.2—Material fatigue strength The fatigue resistance of a structural concrete member is directly related to the fatigue properties of its component materials (Naaman 1982b). Therefore, the fatigue behavior of the constituent materials should be investigated first. Fatigue of concrete—The applied stress range limit for concrete recommended by ACI 215 (1974) is given by the formula cr

= 0.4 f c′ – f mi n ⁄ 2

(4-1)

where f cr = maximu maximum m reco recomme mmende nded d stre stress ss range range for concre concrete; te; f ′ = specif specified ied concre concrete te comp compres ressiv sivee stren strength gth;; and and c′ f min = mini minimu mum m app appli lied ed stre stress ss.. Concrete can sustain a fluctuating stress between zero and 50% of its static strength for approximately 10 million cycles in direct compression, tension, or flexure without failure (Norby 1958; Gylltoft 1978; McCall 1958; Stelson and Cernica 1958; and Hilsdorf and Kesler 1966). Concrete stresses resulting from service loads are generally smaller than this magnitude (Shahawi and Batchelor 1986). Consequently, concrete fatigue failure generally will not control in the case of repetitively loaded partially prestressed beams (Naaman 1982b; Harajli and Naaman 1984; and Bennett 1986). Fatigue of reinforcement reinforcement —Fatigue —Fatigue failure of underreinforced prestressed concrete beams is believed to be governed by the fatigue failure of the steel reinforcement (Warner and Huls-

safe safe stres tresss rang ange, ksi; ksi; mini minimu mum m appl applie ied d stre stress ss,, ksi ksi;; and and ratio ratio of base base radius radius-to -to-he -heigh ightt of rolled rolled-on -on transtransverse deformation (a value of 0.3 can be used in the absence of specific data). The fatigue strength of prestressing reinforcement depends upon the steel type (bar, wire, strand), anchorage (unbonded post-tensioned reinforcement), reinforcement), extent of bond (ACI Committee 215 1974), and steel treatment. Paulson et al. (1983) conducted fatigue tests (in-air) of 50 seven-wire strand samples obtained from six different manufacturers. All of the strands conformed with ASTM A 416 requirements. The minimum stresses applied in the tests ranged from 75 to 165 ksi (517 to 1138 MPa), and the stress ranges varied from 22 to 81 ksi (152 to 559 MPa). A significant variation was observed in results from even two samples of the same product produced by the same manufacturer. The effect of the end grips dominated the fatigue curves in the region of long-life, low-stress-range. The following relationship was found to lie above 95 to 97.5% of the failure points log N = 11 – 3.5 log f sr

where N = f sr =

(4-3)

num number of cycl ycles; es; and maxim maximum um stress stress range range for a fati fatigue gue life life of of N cy cycles, ksi. The researchers did not find the effect of minimum stress on fatigue life great enough to warrant inclusion in the equation. The FIP Commission on Prestressing Steel (1976) recommends a stress range of 15% of f pu with a minimum applied stress not greater than 75% of f pu for a fatigue life of 2 million cycles. For the same fatigue life of two million cycles, however, Naaman (1982b) recommends a reduced stress range of 10% f pu with a minimum applied stress not greater than 60% of f pu to better correlate with test results (Fig. 4.1). 4.1). The following equation can be used to predict other maxi-


f sr ⁄ f pu = – 0.123 log N f + 0.87

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(4-4)

where f sr =

maximu maximum m safe safe stress stress range range for for a fatigu fatiguee life life of of N cycles; f specif specified ied tensil tensilee stren strength gth of the the pres prestre tressi ssing ng pu = strand; and = numb number er of cycl cycles es to fail failur ure. e. N f The endurance limit (stress range for which the reinforcement will not fail for an infinite number of cycles) has not been found for prestressing steel (Naaman 1982b); however, a fatigue life of 2 million cycles is considered to be sufficient for most applications. The previous discussion applies to pretensioned strands. For post-tensioned tendons, two more levels of fatigue strength have to be considered: the strand/duct assembly and the tendon anchorages. For the strand/duct assembly, fretting fatigue may govern if high contact stresses between strand and corrugated steel duct are combined with small relative movements at cracks. Under such circumstances, the fatigue strength of the strand/duct assembly can drop to as low as 14,300 psi (100 MPa). Fatigue strengths of anchorages are in the order of 14,300 psi (100 MPa), according to FIP Commission on Prestressing Steel and Systems (1992). Designers typically place tendon anchorages away from areas with high stress variations and avoid fatigue problems at the anchorages. A similar approach normally will not work to avoid fretting fatigue because maximum stresses often occur at sections with maximum tendon curvature and maximum contact stresses between strand and duct. Fretting fatigue between strand and duct, however, can be avoided by using thick-walled plastic ducts rather than corrugated steel ducts (Oertle 1988). With a thick-walled plastic duct, the strand reaches fatigue strengths comparable to those of strand in air. Fig. 4.2 shows the fatigue performance of tendons with steel and plastic ducts in simply supported beams under four-point loading. In the specimen with a steel duct, 50% of the tendons failed at a fatigue amplitude of 25,000 psi (175 MPa); in contrast, only 18% of the tendons in the specimen with a plastic duct failed at a fatigue amplitude of 39,400 psi (275 MPa).

4.3—Fatigue in partially prestressed beams To illustrate the relative importance of fatigue for partially prestressed beams compared with that for ordinary reinforced or fully prestressed beams, Naaman (1982b) analyzed three concrete concrete beams, beams, identical identical except for the partially prestressed reinforcement ratio ( PPR = 0, 0.72 and 1.0). Note that PPR = 0 represents an ordinary reinforced beam; PPR = 1.0 represents a fully prestressed beam; and PPR = 0.72 represents a partially prestressed beam. All of the beams were designed to provide the same ultimate moment capacity. Material properties and relevant data are given by Naaman and Siriaksorn (1979). For each beam, computed stress ranges in ordinary and pre stressed steel were plotted with respect to the applied load (in

Fig. 4.2—Fatigue resistance of post-tensioned tendon in steel duct and in thick-walled plastic duct (Oertle 1988); 1 ksi = 6.9 MPa.

live load (Fig. 4.3). 4.3). For the same type of beam section, the effect of the PPR was plotted with respect to the reinforcement stress range due to the application of live loads (Fig. 4.4). 4.4 ). The discontinuity in the plots corresponds with first cracking of the concrete in the beams. It is evident from the figures that higher stress ranges are associated with partially prestressed sections. Thus, fatigue problems are more significant in partially prestressed sections than in their ordinary reinforced or fully prestressed counterparts.

4.4—Prediction of fatigue strength The studies described have a common conclusion summarized by Naaman (1982b) and Warner and Hulsbos (1966b). The critical limit state (fatigue failure) of partially prestressed concrete beams is generally due to failure of the reinforcement. reinforcement. The fatigue life of the member can be predicted from the smaller of the fatigue lives of the reinforcing steel or the prestressing steel. Many of these investigations have indicated that in-air test results of reinforcement provide a good indication of the member fatigue life. Naaman therefore recommends using Eq. (4-4) or Fig. 4.1 to estimate the fatigue life of stress-relieved seven-wire strand for the appropriate stress range. A strand subjected to a minimum stress less than 60% of its tensile strength with a stress range of 10% of the tensile strength should provide a fatigue life of approximately two million cycles. For ordinary reinforcement, Naaman recommends using Eq. (4-2) to determine safe stress ranges that provide fatigue lives in excess of 2 million cycles. ACI Committee 215 (1974) and Venuti (1965) recommend conducting a statistical investigation of at least six to 12 reinforcement samples at appropriate stress levels to establish the fatigue characteristics of the material. At least three stress levels are required to establish the finite-life portion of the S - N diagram: diagram: one stress level near the static strength, one near the fatigue limit, and one in between. The choice of the PPR and relative placement of the reinforcement have a significant effect on the fatigue response of the members. Naaman (1982b) states that proper selection of these variables can maintain the stress ranges in the reinforcement to

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Fig. 4.3—Typical comparison of stress changes in steel for reinforced, prestressed, and partially prestressed prestressed beams (Naaman 1982a).

Balaguru (1981) and Balaguru and Shah (1982) have presented a method and a numerical example for predicting the fatigue serviceability of partially prestressed members. The method compares the stress ranges in the beam constituents to the fatigue limits of each individual component (concrete, prestressing steel and nonprestressing steel) using the equations derived by Naaman and Siriaksorn (1979) to calculate stresses for both uncracked and cracked sections. Naaman and Founas (1991) also presented models to calculate the structural responses that account for shrinkage, static and cyclic creep, and relaxation of prestressing steel. For any time t and and cycle N , the models can be used to compute stresses, strains, curvatures, and deflections.

4.5—Serviceability aspects

Fig. 4.4—Typical 4.4—Typical stress changes in steel at different levels of

In a cracked concrete member, whether nonprestressed, partially prestressed, or fully prestressed, the crack widths and deflections generally increase under repeated loadings (Naaman 1982b). The increase in crack widths and deflections in concrete members is mostly attributed to the cyclic creep of concrete and bond deterioration accompanied by slip between the reinforcement and concrete on either side of existing cracks. ACI Committee 224 (1980) notes that 1 million cycles of


Harajli and Naaman (1989) developed an analytical model for cyclic slip and stresses to compute crack widths in partially prestressed concrete beams. In the analysis, equilibrium was assumed between the reinforcement bond stresses and the concrete tensile stresses in a concrete tensile prism joining two successive cracks. In addition, a local bond stress-slip relationship relationship was assumed for the reinforcement. The results of the analyses were in agreement compared with the experimental results shown in Fig. 4.5 and 4.6 for monotonic and cyclic loading, respectively. Agreement was also obtained with tests conducted by Lovegrove and El Din (1982) for which the steel stresses ranged from 7 to 43 ksi (48 to 297 MPa). The analytical models were also useful in predicting trends in the crack patterns in terms of growth and mechanisms of growth. From the analyses, Harajli and Naaman (1989) attributed increased crack widths to increases in steel stresses caused by cyclic creep of concrete in compression and bond redistribution between cracks. They indicated that partially prestressed beams have smaller crack spacings and less crack growth than fully prestressed beams subjected to fatigue loads. The presence of ordinary reinforcement in prestressed beams helps to control cracking in static and fatigue tests. Ordinary reinforcement reduces the increased steel stresses attributed to cyclic creep of concrete. In addition, it minimizes bond redistribution. Crack widths were observed to be a function of reinforcement stress and crack spacing. In tests that generated similar crack spacings (3 to 4.5 in., [76 to 114 mm]), a nearly linear relationship relationship was observed between crack widths and reinforcement stress (Fig. 4.7). 4.7). To estimate the crack spacing acs and crack width W the the following equations were given a cs = 1.20 A t ⁄ P

(4-5)

W = 2 S o B

(4-6)

W N = 2 S o, N B N + [ ( f so , N – f so ) a cs B N ⁄ E s ]

(4-7)

where acs = At =

P

=

W W N S o S o, N N B B N f so f so, N N N

= = = = = = = = =

crac crack k spac spacin ing, g, in. in. (mm (mm); ); effect effective ive area area of concre concrete te tensi tension on zone zone surroun surroundding all reinforcementhaving the same centroid as the reinforcement, in. 2 (mm2); sum of the the perim perimeter eterss of all tension tension reinforcem reinforcement, ent, in. (mm); crack crack widt width h at the first first load loading ing cycle; cycle; crack rack widt width h at at the the N th th loading cycle; slip slip of the the reinfo reinforce rcemen mentt at the the first first cycle cycle;; slip slip of of the the reinfo reinforce rcemen mentt at at the the N th th cycle; distan distance ce ratio ratio at the first first cycle; cycle; dist distan ance ce rati ratio o at at the the N th th cycle; steel steel stress stress at the the prima primary ry crack crack at the the first first cycle; cycle; steel steel stres stresss at the the prim primary ary crack crack at at cycle cycle N ; numb number er of load load cycl cycles es;; and and

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Fig. 4.5—Comparison of observed and predicted crack widths for monotonically tested beams (Harajli and Naaman 1989); 1 ksi = 6.9 MPa; MPa; 1 in. = 25.4 mm.

Tests by Shahawi and Batchelor (1986) indicated a similar linear relationship between crack width and reinforcement stress. They recommend the use of the FIP-CEB equation (Eq. 4-8) for predicting the maximum crack widths in partially prestressed beams subjected to fatigue W ma x = f s × 10

–3

(4-8)

where W max = maxi maximu mum m cra crack ck widt width, h, mm; mm; and and f s = change change in steel steel stre stress ss from from decomp decompres ressio sion n at the the extreme tensile fiber, MPa. The previous FIP-CEB equation accounts for partial debonding, but does not consider cyclic creep of the concrete. Balaguru (1981) and Balaguru and Shah (1982) developed an equation to estimate maximum crack widths that incorporates the effects of cyclic creep and progressive deterioration deterioration of flexural stiffness. Comparing the results of this equation equa tion with those of the FIP-CEB crack width formula using data by Bhuvasorakul (1974), Balaguru determined the following ratios of maximum crack width after N cycles cycles to initial maximum crack width: W max,N / W max = 2.45 (experimental) = 1.003 (FIP-CEB) = 2.15 (Balaguru). He attributed the disparity in the result obtained by the FIP-CEB equation to the lack of consideration of cyclic creep of concrete.

4.6—Summary of serviceability In all types of concrete members, crack widths and deflections generally increase under repeated loadings (Naaman 1982b). These increases occur during early load stages and then generally stabilize until approximately 90%

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Fig. 4.6—Comparison of observed and predicted crack width increases with number of loading cycles (Harajli and Naaman 1989); 1 in. = 25.4 mm.

Shahawi and Batchelor 1986) show a nearly linear relationship between maximum crack width and reinforcement stress. Deflections can increase to twice the static deflection of the first load cycle. The importance of correct reinforcement detailing for crack control should be emphasized. Pretensioned wires or nonprestressed bars can be placed near the tensile face to help limit crack widths according to the report Partial Prestressing (Concrete Society 1983). The presence of ordinary reinforcement in prestressed beams helps to control cracking in static and fatigue tests. Ordinary reinforcement reduces the increased steel stresses attributed to cyclic creep of concrete. In addition, it minimizes bond redistribution. Balaguru (1981) and Balaguru and Shah (1982) have

cyclic creep of the concrete and tension stiffening effects to estimate maximum crack widths and deflections at a given load cycle. Naaman and Founas (1991) also presented models to compute stresses, strains, and deflections in partially prestressed beams for static and cyclic loading incorporating effects of shrinkage, creep, and relaxation.

CHAPTER 5—EFFECTS OF LOAD REVERSALS 5.1—Introduction Cyclic load reversals can result from dynamic effects of earthquake, shock, wind, or blast loadings. Earthquake loadings are imparted to the base of the structure through ground motion, whereas wind loadings transmit forces to the aboveground portions of the structure. Shock or blast loadings can


423.5R-21

Fig. 4.7—Crack width variation versus reinforcing steel stress at different PPR (Harajli and Naaman 1985); 1 ksi = 6.9 MPa, 1 in. = 25.4 mm.

to wind loadings and ground motions accompanying either buried, air, or contact blasts.

5.2—Design philosophy for seismic loadings In design for seismic effects, the primary concern is safety of the occupants (Bertero 1986), and a secondary concern is economics. Even for an earthquake with reasonable probability of occurrence; however, it is not possible to guarantee with absolute certainty complete life safety and no structural damage. The Federal Emergency Management Agency, in their document NEHRP 1997 Recommended Provisions (1998), give minimum requirements to provide “reasonable and prudent life safety.” To achieve these objectives, structures should be proportioned to resist design forces with adequate strength and stiffness to limit damage to acceptable levels. Design forces are determined by using a responsemodification coefficient R to reduce the forces obtained, assuming that the structure responds linear elastically to the ground motion. The reduction factor accounts for the reduced strength demand due to a number of effects, including energy dissipation through hysteretic damping and lengthening of the structural period as the structure becomes inelastic. In simplistic terms, there are two basic options in seismic design: (1) make the structure strong enough so that it will respond elastically; or (2) permit the structure to deform inelastically while ensuring adequate ductility and energy

to be designed for considerably lower forces than required for the first option. Designing partially prestressed concrete structures in accordance with the second approach, raises the same concerns as for the case of designing with conventional reinforced concrete; that is, the designer must ensure: • Ductility at plastic hinges; hinges; and • Adequate energy energy dissipation through through damping and and hysteresis. This chapter concentrates on these characteristics as they relate to partially prestressed concrete.

5.3—Ductility Ductility allows the redistribution of forces in indeterminate systems and ensures gradual rather than brittle failure, providing a warning to the occupants before collapse. If adequate ductility is not provided at the critical regions (plastic hinge zones), the member will be unable to develop the required inelastic rotation. Ductility is usually expressed for structural members and systems in terms of a deformation ductility ratio, where the deformation is described in terms of displacement, rotation, or curvature (Naaman et al. 1986, Thompson and Park 1980a, Giannini et al. 1986). Deflection and rotation ductilities give an indication of drift (ratio of lateral displacement to height) and are used in structural analysis. Curvature ductility is used to define member or section behavior at plastic hinges. The ductility ratio is defined as the ratio of the ultimate

423.5R-22


Fig. 5.1—Definitions of yield for determination of ductility (Naaman et al. 1986).

prestressed concrete structures, the definition of yield deformation can be quite arbitrary (Naaman et al. 1986; Thompson and Park 1980a; Giannini et al. 1986). The section contains both prestressed and nonprestressed reinforcement that can have different yield stresses; prestressing steel does not have a definite yield point, and reinforcement located in different layers will yield at different member deformations. The yield deformation, therefore, can be defined in many ways. Among numerous investigations on partially prestressed concrete, there has not been a consistent definition of yield deformation ( deformation (Fig. Fig. 5.1) 5.1). Cohn and Bartlett’s study (1982) on partially prestressed concrete assumed that the yield curvature corresponded to yielding of ordinary reinforcement. This approach gives a relatively high value of ductility because the prestressing reinforcement yields at a much larger deformation. Thompson and Park (1980a) defined first yield as the intersection of the tangent to the elastic portion of the load-deformation curve and a horizontal line at ultimate load. In Park and Falconer’s study (1983) on prestressed piles, the yield deformation was taken at the intersection of the secant from zero to 75% of the ultimate moment capacity and the postelastic slope. Naaman et al. (1986) defined the yield deformation as the intersection of the secant from zero to the proportional limit of the prestressing steel and the postelastic slope. Although these latter definitions give slightly different results, they are all based on the ductility of the member section rather than yielding of a particular component in the cross section. Local section ductility demands can be much more severe than overall deflection ductilit ies of frames. Tests by

frames indicated that the ratios of section curvature ductilities to deflection ductilities of the frame were 1.24 and 1. 53 for partially prestressed concrete systems with bonded and unbonded tendons, respectively. Numerous investiga5.3.1 Factors affecting ductility— Numerous tions have been conducted to determine the effect of different parameters on ductility. Generally, factors that affect the ductility of ordinary-reinforced concrete sections affect the ductility of partially prestressed concrete sections as well. Effects of anchorage, bond, transfer lengths, grouting, characteristics of high-strength prestressing steels, and level of prestressing are additional parameters uniquely important to prestressed and partially prestressed concrete members (Bertero 1986). Investigations were conducted to determine the effects of the reinforcement ratio (Cohn and Ghosh 1972; MacGregor 1974; Scott et al. 1982; and Park and Thompson 1977), concrete confinement (Park and Falconer 1983; Scott et al. 1982; Burns 1979; Kent and Park 1966; Wight and Sozen 1975; Sheikh and Uzumeri 1982; and Sheikh 1982), material strengths and stress-strain properties (Wight and Sozen 1975, Wang 1977), section geometry, ratio of compression reinforcement (Park and Thompson 1977, Burns and Seiss 1962), PPR, and axial load (Lin and Burns 1981). Another investigation by Thompson and Park (1980a) was conducted to determine the effects of the content and distribution of longitudinal reinforcement, transverse reinforcement, and concrete cover. Naaman et al. (1986) analyzed beams with four types of sections (rectangular, T, I, and box sections), five different concrete strengths, three types of prestressing steel, four levels of PPR, and seven reinforcing indexes ω. For the rectangular and T-sections, five levels of effective prestress f pe /f pu were analyzed, and in selected cases, four levels of compression reinforcement ratios and the effect of confinement were investigated. The following observations were made from the parametric evaluation: • Reinforcement index—Section ductility decreases with increasing ω because of the significant reduction in ultimate curvatures associated with increased reinforcement reinforcement ratios (Fig. 5.2 and 5.3 5.3)). prestress—When the effective prestress was • Effective prestress decreased, it led to an increase in ultimate curvature for reinforcing indices ω greater than approximately 0.12. It also led to an increase in yield curvature irrespective of the reinforcement index. Fig. 5.2 illustrates the influence of effective prestress on ductility factor or the ratio of these two quantities. The increase in yield curvature tended to dominate the results and caused a reduction in ductility with decreased effective prestress irrespective of the reinforcing indices. This effect was more significant for fully prestressed beams. Partially prestressed ratio—A decrease in PPR led to • an increase in both ultimate and yield curvatures, but in terms of ductility, there was no consistent trend. The effect of PPR on ductility became negligible with increased concrete compressive compressive strengths. • —Increases in concrete Transverse reinforcement —Increases confinement resulted in corresponding increases in sectional ductility irrespective of the reinforcement


423.5R-23

Fig. 5.2—Typical 5.2—Typical variation of curvature ductility as function of effective prestress in steel from analytical study ( Naaman et al. 1986).

•

•

other investigators (Watanabe et al. 1980; Muguruma et al. 1982b; Iyengar et al. 1970; Okamoto 1980; and Hawkins 1977). Concrete strength—For high values of reinforcement index (ω > 0.25), the concrete compressive compressive strength did not have a significant effect on ductility. For low values of ω (< 0.25), however, the use of higher-strength concrete reduced section ductility as much as 30%, particularly in cases of low PPR (ordinary reinforced concrete). This trend was also more pronounced for rectangular sections as compared with box, T- and Isections. Tests by Muguruma et al. (1983) indicate that high-strength concrete is effective in improving ductility, but that the consequences of high-strength tendon fracture and brittle compressive failure of highstrength concrete should be considered. Section geometry—The section geometry did not have an appreciable effect on ductility, provided that the

Fig. 5.3—Comparison of analytical results with prediction equations (Naaman et al. 1986); γ = ratio of compressionto-tension steel force at ultimate; and 1 ksi = 6.9 MPa.

reinforcement index ω (computed using the web for Tsections). Naaman et al. (1986) found the reinforcement index ω to be an excellent independent variable for describing section ductility, as it is proportional to c/d ctf (c = distance from extreme compression fiber to neutral axis; and d ctf = depth to centroid of tensile steel). Based on the reinforcement ratio and concrete compressive strength, it can be used to describe all three types of systems: reinforced concrete, prestressed concrete, and partially prestressed concrete. Based on the results of analysis and experimental tests on twelve partially prestressed concrete beams (Harajli and

423.5R-24


where ω = rein reinfforce orcem ment ent ind index; ex; depth to to centr centroid oid of tens tensile ile force force in in steel steel;; and and d ctf = depth L p = equi equiva vale lent nt pla plast stic ic hin hinge ge len lengt gth. h. In Fig. 5.4, 5.4, the equations are compared with the results of several investigations (Kent and Park 1966; Corley 1966; Mattock 1964; Mattock 1967; Harajli and Naaman 1985c; Bishara and Brar 1974; and Baker and Amarakone 1964). The equations show fairly good correlation with the results (assuming a plastic hinge length L p of d ctf /2), / 2), and are recomrecommended by Naaman et al. for use as a first approximation in the design and detailing of ordinary reinforced, fully prestressed or partially prestressed members. In a discussion of this work, Loov et al. (1987) recommend specifying c/d rather rather than ω to ensure ductility, as is done in the Standards Association of Australia’s Draft Unified Concrete Structure for Buildings (1984) and the CSA’s Design of Concrete Structures for Buildings. The use of c/d ensures consistent ductility demands from sections independent of geometry and concrete compressive strength. To accommodate this, Naaman et al. also provided the expressions in terms of c/d ctf , valid for c/d ctf between 0.08 and 0.42: Curv Curvat atur uree duct ductil ilit ity y rati ratio o Upper bound

1 -------------------------------------------0.73 c ⁄ d ct f – 0.053

1.86 – 0.73 c ⁄ d ct f L P ----------------------------------------- -------------621 c ⁄ d ct f – 42 d ct f ⁄ 2

(5-4)

Lower bound

1 -------------------------------------------1.42 c ⁄ d ct f – 0.102


(5-5)

Average

1 ----------------------------------------------1.095 c ⁄ d ct f – 0.087


(5-6)

where c =

Fig. 5.4—Comparison of prediction equations experimental results results (Naaman et al. 1986).

with

et al. developed prediction equations for sectional ductility and plastic rotation as a function of the reinforcement index ω. Three equations were derived for both the sectional ductility and plastic rotation to give upper limit, lower limit, and average values for ranges of reinforcement index of 0.05 to 0.3 (Fig. 5.3). 5.3). The The upper and lower limits accommodate the effects of parameters described previously, other than reinforcement index ω which were shown to have an effect on ductility. For example, Naaman et al. suggest the lowerbound equation be used to describe cases with high-strength concrete, low effective prestress, and high PPR. The upperbound equations are suggested for cases with normal strength concrete, high effective prestress, and low PPR. The equations are given as follows Curva Curvatur turee duct ductili ility ty ratio ratio

Plasti Plasticc rotat rotation ion (radia (radians) ns)

Upper bound

1 ----------------------ω – 0.045

L 1.05 – ω ------------------------- ---------P----850 ω – 35 d ct f ⁄ 2

(5-1)

Lower bound

1 ---------------------------------1.94 ω – 0.086

1.05 – 1.65 ω L P ------------------------------- -------------1300 ω – 40 d ct f ⁄ 2

(5-2)

Average

1 ----------------------------1.5 ω 0.075

1.07 – 1.58 ω L P ------------------------------- -------------1050 45 d ⁄ 2

(5-3)

Plas Plasti ticc rota rotati tion on (ra (radi dian ans) s)

depth depth to to neutra neutrall axis axis from from extre extreme me comp compres ressio sion n fiber; d ctf = depth depth to to centr centroid oid of tens tensile ile force force in in steel steel;; and and L p = equi equiva vale lent nt pla plast stic ic hin hinge ge len lengt gth. h. Thompson and Park (1980a) agreed with a form of c/d as as a measure of the ductility in an investigation in which a moment-curvature relation was developed applicable to ordinary reinforced, prestressed, and partially prestressed concrete beams, and verified with experimental tests on symmetrically reinforced (prestressed and ordinary reinforced) rectangular beam-column beam-column assemblies subjected to cyclic load reversals. The model was subsequently used to conduct a parametric study. Their recommendations are as follows: • Effect of prestressing steel content —ACI —ACI 318 limits the prestressing steel content by requiring ω p ≤ 0.36 β1, where β1 is a parameter used to modify the neutral axis depth c to determine the depth of the equivalent rectangular stress block a. Although the code limits the prestressing steel content, ACI 318 allows this value to be exceeded if the additional reinforcement is not considered in the flexural strength calculation. Although this can result in sufficient sufficient ductility for gravity loading, it can be insufficient for seismic loading. To ensure adequate ductility, one can limit ω p to less than 0.2 (rather than 0.3 for the case of β1 = 0.85). Alternatively, rather than using the previous equation, one can also limit a/h ≤ 0.2 (a = depth of rectangular stress block; and


ductility by specifying the location of the neutral axis independent of tendon positions. This relationship has been adopted by FIP (Cement and Concrete Association of England 1977) and the Standards Association of New Zealand’s Code of Practice for General Structural Design and Design Loading for Buildings (1980). prestressing ssing steel distribution distribution on ductility— • Effect of prestre Sections with only one tendon have reduced moment capacity as the concrete in compression deteriorates. Thompson and Park (1980a) recommend the use of at least two grouted tendons (one near the top and one near the bottom of the section) for seismic loading, and suggested locating a third tendon towards the center of the section. The use of unbonded tendons was generally not recommended for primary earthquake-resistant members because of lack of information on the performance of the anchorages (Muguruma 1986). In cases where partially prestressed beams are designed with nonprestressed reinforcement in the extreme fibers providing at least 80% of the seismic resistance, however, they indicated that the prestress can be provided by one or more grouted or unbonded tendons in the middle third of the beam depth. The centrally located prestressing steel is recommended because it can delay cracking and enhance strength without much reduction in ductility. Studies at the National Institute of Standards and Technology (NIST) and as part of the National Science Foundation Precast Seismic Structural Systems (PRESSS) program (Cheok and Lew 1990) have explored the use of unbonded prestressing reinforcement through the connections. In the studies at NIST, central post-tensioning was used in conjunction with mild steel reinforcement at the top and bottom of the beam-column interface. The central post-tensioning maintained integrity of the joint, while energy dissipation was accommodated with the mild steel reinforcement. A study by Priestley and Tao (1993) proposed the use of lightly stressed, unbonded prestressing steel through the joint (nonlinear (nonlinear-elas -elastic tic concept) concept).. The system system offers offers ductil ductilee behavior through crack opening at the interface, while maintaining a “self-restoring” force. Because the strands do not yield, they work to bring the connection back to its originally undeformed position. This system has been studied further with other proposed connection systems as part of the PRESSS program (Cheok and Lew 1990). Effect of transver transverse se reinfor reinforceme cement nt on ductility ductility—The • Effect degree of confinement provided by the transverse reinforcement to the compression regions significantly increases ductility. The confinement enhances the performance of the concrete within the core and also inhibits buckling of the compression reinforcement (Park and Thompson 1977; Park and Paulay 1975; and Thompson 1975). Thompson and Park recommend that the stirrups be spaced less than d /4 or 6 in. (150 mm). • Effect of cover thickness—For ductility, Thompson and Park suggest that the cover be made as small as possible, because loss of cover under cyclic loads causes a reduction in capacity. The effect of cover spalling on member strength is not as great for deeper members where cover is a smaller percentage of member depth. Holding all other parameters constant, ductility decreases with an increase in reinforcement index. ACI 318 gives a maximum value for the reinforcement index of ω ≤ 0.36 β1.

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order of 1.5 to three. With Park’s suggested limit of 0.2, ductilities in excess of four or five can be obtained. Tests by Muguruma et al. (1982a) indicate that if the sections do not contain compression reinforcement, ω < 0.2 can give ductilities of only about three. If the reinforcement index is maintained at less than 0.1, ductilities in excess of 10 can be achieved (Naaman et al. 1986). Tests by Nakano and Okamoto (1978) on beam-column subassemblages indicated that even when tension reinforcement is adequate (for example, ω < 0.15 to provide ductility of five), hoop reinforcement (> 0.5%) is required to provide the required deflection ductility. Okamoto (1980) carried out similar tests on eleven simply supported beams and nine cantilever beams and reached similar conclusions. 5.3.2 Special cases Compression members— Relatively Relatively little information is available on partially prestressed concrete columns (Bertero 1986). Tertea and Onet (1983) analyzed 20 columns with different PPR ranging from zero to 100%. As observed in the beam tests, curvature ductility of columns improved with the addition of nonprestressed reinforcement. Curvature ductility factors of 3.9, 6.1, and 9.8 corresponded with PPR of 100, 50, and 0%. Irrespective of the degree of prestressing, column ductilities improved with increased confinement. Tests by Park and Falconer (1983) on prestressed piles found that the curvature ductility increased with reduced axial load levels and increased confinement by spiral reinforcement. They discourage the use of hard-drawn reinforcing wire for spirals as it commenced fracture at displacement ductility factors of four to six. They recommend requiring the wire to have high fracture strains. Shear— Beams Beams should be proportioned to ensure ductile flexural behavior and to avoid brittle shear behavior (Bertero 1986). To accomplish this objective, the FIP Commission (Cement and Concrete Association of England 1977) recommends assuming material overstrengths on the order of 15% in calculating the design shear force from the plastic hinge moments. The New Zealand Code of Practice for General Structural Design and Design Loading for Buildings (Standards Association of New Zealand 1980) recommend neglecting the concrete contribution to shear resistance when the axial compression is less than 0.1 f ′ c′ . Muguruma et al. (1983) determined that concrete resistance to shear can be improved by prestressing. Experiments were conducted on 7 x 10 in. (180 x 250 mm) simply supported beams with span lengths of 7.22 ft. (2.20 m) to investigate shear in prestressed beams. The flexural shear cracking load was found to increase in direct proportion to the increase in flexural cracking load (due to an increase in prestressing). Muguruma (1986) suggests in the design of partially prestressed concrete beams that the concrete contribution to shear resistance can be increased by the shear force corresponding to the decompression moment at the critical section. This effect is taken into consideration in ACI 318. Beam-column joints should be Beam-column joints— Beam-column designed and detailed so that the plastic-hinge zones in the beams can develop their full plastic capacities. Muguruma

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for ordinary reinforced or partially prestressed systems, except that the prestressing tendons do not bond to the concrete as well as ordinary deformed reinforcement. Research on the performance of bond between tendons and concrete in joints (Muguruma and Tominaga 1972) has indicated that tendons undergo bond deterioration at early load stages. Tendon slip causes a reduction in energy dissipation capacity by pinching the hysteresis loops. Park’s investigation of beam-column subassemblages with grouted post-tensioned tendons (Applied Technology Council 1981) recommended minimizing the possibility of failure in the joint. To avoid this type of failure, the hinge zones should be displaced away from the beam-column interface. This can be accomplished by haunching the beams, adding additional reinforcement at the interface, or using cruciform shapes (Bertero 1986). Displacing the hinge from the interface has the added advantage of locating the failure in a region in which nonprestressed reinforcement is located, increasing ductility and energy dissipation. Displacing the hinge further into the beam results in increased shear at the interface when the hinge reaches its capacity and produces increased ductility demands relative to hinges at the interface when subjected to similar drift levels. Using post-tensioned tendons through the joint core at middepth can reduce the amount of required joint shear reinforcement as it serves to maintain integrity of the joint (Park and Thompson 1977).

5.4—Energy dissipation Energy dissipation can be attributed to viscous and hysteretic damping described separately below. Energy dissipation of prestressed concrete is as little as 0.15 times that of conventional reinforced concrete of comparable size and strength (Bertero 1986). In a 1978 state-of-the-art report 5.4.1 Viscous damping— In by Hawkins (1977) on seismic resistance of prestressed and precast structures, a thorough review of the research on damping was presented. Tests by Penzien (1964) indicated that the level of prestress and concrete strength affects damping by affecting cracking. Decreases in cracking cause a reduction in damping. Tests by Spencer (1969) on centrally prestressed beams indicated that damping ratios were not frequency dependent. Damping ratios tended to increase with end rotations and prestress. Spencer also found damping to be higher for beams loaded in uniform shear rather than uniform moment and attributed this result to the higher effects of interface shear and bond slip in the case of uniform shear loading. Tests by Brondum-Nielsen (1973) on centrally prestressed beams indicated damping decreased with increased prestress levels, which is the opposite of what Spencer observed. Values of damping obtained by Nakano (1965) from tests of 1/3 scale four-story frames were 1% before cracking, 3% at cracking, and 7% before inelastic behavior. The energy dissipation (area 5.4.2 Hysteretic behavior— The enclosed within the force-deformation hysteresis loops) of prestressed concrete members is lower than that of

Fig. 5.5—Moment-curvature idealizations for prestressed, reinforced, and partially prestressed concrete (Thompson and Park 1980b); α = ratio of strength of reinforced concrete component to total strength of partially prestressed concrete system; and β = ratio of strength of prestressed concrete component to total strength of partially prestressed concrete system.

strengths (Thompson and Park 1980b). Prestressed concrete members can achieve significant elastic recoveries even at large deformations (Schoeder 1977) because of the initial tensile stress in the strand due to prestressing. In elastic recoveries, the strand unloads its tensile force and is capable of achieving large recoveries. The member will not achieve significant energy dissipation unless the prestressing steel yields, the concrete crushes, or the nonprestressed deformed bar yields (Nakano 1965; Blakeley and Park 1973b; Guyon 1965; Blakeley and Park 1971; and Bertero 1973). The addition of nonprestressed reinforcement in partially prestressed concrete members has been shown to provide a marked improvement in energy dissipation characteristics (Thompson and Park 1980b, Inomata 1986). This is evident in Fig. 5.5, 5.5, which shows typical hysteresis loops for cyclically loaded fully prestressed, ordinary reinforced, and partially prestressed concrete members. In addition, by reducing the strength degradation, the presence of longitudinal nonprestressed reinforcing steel improves ductility by acting as compression reinforcement (Park and Thompson 1977, Inomata 1986). As mentioned previously, Cheok and Lew (1990) have investigated the use of lightly prestressed unbonded reinforcement. The performance of the PRESSS connections has the advantage of a selfrestoring force (nonlinear-elastic system). The stiffness reduction compared with that of a monolithic counterpart can compensate for the reduced energy dissipation. Hybrid connections (Cheok and Lew 1991) have been investigated that combine the nonlinear elastic system with elements that yield or dissipate energy. Thompson and Park (1980a, 1980b) developed idealized moment-curvature relations for partially prestressed concrete sections by combining relations obtained by Blakeley (1971) and Blakeley and Park (1973a) for prestressed concrete and by Ramberg and Osgood for


Fig. 5.6—Comparison of idealized and experimental moment-curvature relations for prestressed concrete section (Thompson and Park 1980b).

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Fig. 5.7—Comparison of idealized and experimental moment-curvature relations for reinforced concrete section (Thompson and Park 1980b).

curves agree with those obtained experimentally from tests on beam-column subassemblages. subassemblages. Fig. 5.6 through 5.8 show the idealized and experimental moment-curvature relationships for fully prestressed, reinforced, and partially prestressed members subjected to cyclic loading. In the fully prestressed member (Fig. 5.6) 5.6), the energy dissipation capacity of the member increased (although still with considerable elastic recovery) after the beam-moment capacity had been reached and crushing of the concrete had commenced. The compression steel in the beam was nominal and the significant degradation in the strength and stiffness was attributed to the reduction in the sectional area when the cover concrete crushed. After the beam-moment capacity had been reached and crushing of the concrete had commenced for the partially prestressed member ( member (Fig. Fig. 5.8) 5.8), the energy dissipation capacity of the partially prestressed member was considerably greater than that observed for the fully prestressed member. For this particular test, the beneficial effect of the nonprestressed longitudinal steel in the compression zone of the beam plastic hinge region was evident from the stable moment-curvature relations. The strength degradation was not large because after concrete crushing commenced, the nonprestressed steel was able to carry some of the compression previously carried by the cover concrete and therefore helped maintain the internal lever arm (Thompson and Park 1980b). Similar findings were obtained by Muguruma et al. (1982a, 1982b), who tested 24 partially prestressed (eccentric) beams with various levels of prestress, and also by Nakano and Okamoto (1978), who tested 21 partially prestressed (symmetric) beams subjected to cyclic-load reversals. Confining the concrete has a marked benefit on the ductility of partially prestressed concrete and also improves the hysteretic behavior. Closely spaced stirrup ties or spirals in the hinging regions enhance ductility and energy dissipation by confining the concrete compression zone, reducing stiffness

Fig. 5.8—Comparison of idealized and experimental moment-curvature relations for partially prestressed concrete section (Thompson and Park 1980b).

5.5—Dynamic analyses Giannini et al. (1986) presented an analytical investigation on the seismic behavior of fully prestressed and partially prestressed concrete. The influences of the fundamental period, PPR, and seismic intensity were considered. The response was obtained in the form of time histories and peak values in terms of displacements, curvatures, local strains (in the constituents), forces, and dissipated energy. Partially prestressed concrete seems to couple the advantages of fully prestressed concrete with those of ordinary reinforced concrete in resisting loads. Fully prestressed concrete has a tendency to greater strain and deformation demands than ordinary reinforced concrete. After a strong impulse, the displacements of the reinforced concrete and partially prestressed concrete systems tended to damp out, whereas those of the fully prestressed concrete system did not. The analysis showed that fully prestressed concrete resists the design seismic intensity, but for greater intensities, it rapidly approaches collapse. Fully prestressed concrete structures, however, can behave better for low

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Similar conclusions were obtained from a study by Thompson and Park (1980b). They used the idealized moment-curvature relations (discussed previously) to conduct a nonlinear dynamic analysis of ordinary reinforced, fully prestressed, and partially prestressed concrete singledegree-of-freedom systems. Displacements of fully prestressed concrete systems are larger than those of ordinary reinforced and partially prestressed concrete systems because of the reduced energy dissipation associated with prestressing. Fully prestressed systems also undergo a significant reduction in stiffness after cracking that counteracts the increased displacements displacements under dynamic loading. The reduction in stiffness causes a reduction in the effective period of vibration that can reduce the displacement response. Under the combination of both of these effects, the analyses indicate that fully prestressed frames will have, on average, 30% greater maximum lateral displacements than reinforced concrete systems designed to have the same strength, viscous damping ratio, and initial stiffness (Thompson and Park 1980b). Similar analytical studies have been conducted as part of the NSF PRESSS program (Palmieri et al. 1996, El Sheikh et al. 1999) that indicated systems incorporating unbonded post-tensioned connections experienced moderate increases in displacements in comparison with those expected from monolithic concrete frame systems. An advantage of the unbonded post-tensioned connections was the reduction in residual drift associated with the nonlinear elastic behavior of the system in comparison with the permanent system deformations associated with the inelastic response of the ordinary reinforced frames. Fully prestressed concrete members can be detailed to accommodate increased displacements in a ductile manner; however, these higher displacements can result in greater nonstructural damage and P-delta effects. An advantage of using full prestressing and partial prestressing is the reduction in residual displacements due to the self-restoring nature of these systems. The addition of nonprestressed reinforcement can reduce these deflections.

5.6—Connections Results of tests (Berg and Degenkolb 1973; Elliott 1972; Kunze et al. 1965; Sutherland 1965; Ozaka 1968; FIP 1970; FIP 1974; Arya 1970; and Architectural Institute of Japan 1970) and observations on the past performance of prestressed concrete in earthquakes indicate that most prestressed concrete members perform well. Most failures have been found to occur in connections between elements. Several investigations have been conducted to study the performance of connections between fully prestressed and partially prestressed concrete precast elements. As mentioned previously, ductile connection concepts have been tested as part of the PRESSS program. French et al. (1989a, 1989b), Amu and French (1985), Hafner and French (1986), Jayashankar and French (1987), and Tarzikhan and French (1987) investigated post-tensioned, welded, bolted,

connections. The most viable alternatives were continuous post-tensioned connections displacing the plastic-hinge region from the beam-column interface, and taperedthreaded splice connections that emulate the performance of monolithically cast beam-column connections. It was also suggested that advantage be taken of composite action between the beam and the slab.

5.7—Summary If members are detailed properly, it is possible to ensure ductile behavior in fully prestressed and partially prestressed concrete beams (Applied Technology Council 1981). Ordinary reinforced concrete has an advantage in energydissipation characteristics. Prestressed concrete has an advantage in its ability to reach large displacements with only a small loss in load-carrying capacity and almost complete recovery of displacements. Joint performance is improved by the use of continuous prestressed strand through the joint. Partially prestressed concrete couples the advantages of prestressed concrete with those of ordinary reinforced concrete (Giannini et al 1986). The sections can be detailed so that energy dissipation and deformation demands are on the same order as those of ordinary reinforced concrete. Partially prestressed concrete also offers the possibility of calibrating the subsequent yielding of both prestressed and nonprestressed of reinforcement to optimize stiffness and energydissipation properties (Giannini et al 1986). A recommended design concept is to balance all or part of the gravity loads, satisfy serviceability criteria with draped or blanketed prestressing strands, and place sufficient nonprestressed reinforcement in the section to meet the additional requirements for seismic design strength, ductility, and energy dissipation (Hawkins 1977; Inomata 1986; and Thompson and Park 1980b). Stirrup ties should be located in hinging regions to prevent bar buckling, confine concrete, and increase shear strength (Park and Thompson 1977, Park and Paulay 1975).

CHAPTER 6—APPLICATIONS 6—APPLICATIONS 6.1— Early applications Partial prestressing has been used in the design of structures since the early 1950s. In 1952, partially prestressed beams with post-tensioned cables were first used in the roof of a freight depot at Bury St. Edmunds, England (Bennet 1984). Lower concrete tensile stresses were specified initially, but with successful experience and an increasing amount of test data, the permissible fictitious tensile stresses were increased to 750 psi (5.2 N/mm 2) for pretensioning and 650 psi (4.5 N/mm2) for post-tensioning (Bennet 1984). Other early applications of partial prestressing included the development of a concrete mast for overhead power lines and structures designed to withstand mining collapse; with both applications, the most severe design load was usually a


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6.2—Pretensioned concrete components In the early 1980s, a survey was conducted among the pretensioned, precast-concrete plants in the U.S. and Canada (Jenny 1985) concerning the use of partial prestressing in the industry. Sixty-five plants responded to the survey. Four of the plants manufactured only bridge girders, and partial prestressing was rarely considered in their production. The survey revealed, however, that there was a general familiarity with partial prestressing and the concept was used for meeting special design situations rather than as a routine production standard. Only seven of the 61 remaining plants responding to the survey indicated that they did not allow ′ psi any product to exceed a nominal tensile stress of 6 √ f c′ MPa) for certain members, such as stemmed units, (√ f ′ c′ /2 inverted tees, spandrel beams, hollow-core slabs, or columns (Jenny 1985). The nominal stress levels reporte d by the plants were: 7.5 to 12√ f ′ c psi ( √ f ′ c / 1.6 to √ f ′ c MPa) for stemmed units; 7.5 to 12√ f ′ for beams; beams; 7.5 to 9√ f ′ c psi (√ f ′ c /1.6 to √ f ′ c MPa) for c psi (√ f ′ / 1.6 to √ /1.33 MPa) for hollow-core slabs; and ′ ′ f c c ′ psi (√ f ′ ′c /1.6 6 MPa) for columns. 7.5√ f c′ /1. columns. Reasons for designing these members with a nominal tensile stress greater than 6√ f ′ MPa) were to meet special design loads, c′ psi (√ f ′ c′ /2 reduce camber, save steel and lower the cost, reduce the required concrete strength at release, and achieve competitive span-depth ratios (Jenny 1985).

6.3—Post-tensioned building construction The majority of post-tensioned building construction in the U.S. is based on the use of unbonded tendons in conjunction with code-specified minimum amounts of bonded nonprestressed reinforcement. Maximum design tensile stresses ′ /2 to √ f ′ range from 6 to 12√ f ′ c′ psi (√ f c′ c′ MPa) (Feyermuth 1985). For designs where durability considerations do not govern, service-load nominal tensile stress levels of 6 to ′ /2 2 to √ f c′ ′ MPa) are now routine for one-way, 12√ f ′ / c′ psi (√ f c′ post-tensioned systems. According to Feyermuth, “The use of partial prestressing, especially as applied to flat slabs, has reduced the incidence of significant cracking problems in post-tensioning buildings. This results from lower prestress levels, inducing proportionately smaller elastic and creep shortening movements, and the presence of bonded nonprestressed reinforcement to control crack widths” (1985). In the design of office and hotel structures, the use of partial prestressing is often cost-effective. Post-tensioned beams, as well as one-way slabs, designed to an extreme nominal fiber ′ psi (√ f c′ ′ /2 and √ f ′ stress in tension between 6 and 12√ f c′ c′ MPa) have performed well in practice. The use of partially prestressed design is especially effective when certain designated areas of a building are required to have a higher design live load than the majority of the floor area. The 1100 Peachtree Tower (shown in Fig. 6.1) 6.1) is a 27-story, cast-inplace concrete office building located in Atlanta, Ga. Th e slab is 5 in. (127 mm) thick, conventionally reinforced, and supported by post-tensioned beams 14 in. (356 mm) wide and 22 in. (559 mm) deep. By allowing the nominal tensile stress ′ psi (√ f ′ in the beams to reach 10 √ f c′ MPa) under service service c′ /1.2

Fig. 6.1—Structural floor plan layout of 1100 Peachtree Office Tower, Atlanta, Ga.

of tendons and by reducing the overall height of the structure, owing to the shallower beam depths. For two-way flat-plate designs, ACI 318 limits the stress at column locations as calculated by the equivalent frame 2 method or other approximate methods to 6 √ f ′ / c′ ( √ f ′ c′ /2 MPa). This is in recognition of the gross approximation of the actual strains in the vicinity of the column. Higher allowable stresses, however, are permitted in conjunction with more precise analytical procedures.

6.4—Bridges Prestressed concrete bridges designed to permit cracking under service loads are still rare in the U.S., even though it is common to allow flexural tension in extreme fibers under service load. The primary concerns about bridge members being cracked under service loads have to do with fatigue and durability. Partially prestressed concrete bridges have been built in Europe and Japan. Taerwe (1990) has described a number of partially prestressed concrete bridges constructed in Belgium. These highway bridges were designed according to the Class 2 conditions described in Chapter 1 of this report. As of 1990, the bridges described by Taerwe (built between 1965 and 1975) performed satisfactorily. satisfactorily. The Kannonji Viaduct in Japan was completed in 1991 (Soda and Kazuo 1994). The bridge has a total length of 2464 ft (751 m), with an average span length of 92 ft (28 m). The bridge superstructure units are continuous for three or four spans each. It was designed to allow flexural cracking under service loads at midspan but not over interior supports. Four other partially prestressed concrete highway bridges also have been constructed in Japan (Soda and Kazuo 1994). AASHTO’s Standard Specifications for Highway Bridges does not currently allow service load cracking in prestressed

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and resistance-factor design specification for highway bridges (AASHTO LRFD Bridge Design Specifications) that does specifically address partially prestressed concrete. This can lead to a wider use of partially prestressed concrete for highway bridges in the U.S.

The reports and standards listed above were the latest editions at the time this document was prepared. Because these documents are revised frequently, the reader is advised to contact the proper sponsoring group if it is desired to refer to the latest version.

6.5—Other applications

AASHTO 444 N. Capitol Street NW Suite 249 Washington D.C. 20001

*

Dolan reports that the prismatic beams on the Walt Disney World Monorail were designed for tensile stresses of nearly 12√ f ′ certain load combinations. combinations. c′ psi (√ f ′ c′ MPa) for certain The beams were precast and post-tensioned to provide continuity. Reinforcing bars were used to control cracking at the continuity joint. Partial prestressing also has applications in nuclear-containment structures, cylindrical shells, and rectangular water vessels. In the 1960s, the first prestressed concrete containment structures for nuclear reactors were partially prestressed in the vertical direction only with nonprestressed reinforcement in the circumferential (hoop) direction of the cylinder and in the dome. Two examples are the containment structures at R.E. Ginna and H.B. Robinson Unit 2 nuclear stations (Ashar et al. 1994). Similarly, partial prestressing is used vertically in the walls of cylindrical shell and rectangular water-storage structures. If these structures are designed with restrained bases, high negative or positive vertical bending moments, or both, will be present. An economical design can be produced when a small amount of vertical prestressing in the range of 200 psi (1.4 MPa) is provided and nonprestressed reinforcement reinforcement is added to resist any net flexural tensile stresses. These high bending moments occur only at one point in the wall. Therefore, the economy is achieved by not overreinforcing the entire height of wall. This method of design has been used since the 1970s. †

CHAPTER 7—REFERENCES 7.1—Referenced standards and reports American Concrete Institute 222R Corrosion of Metals in Concrete 318 318 Building Co Code Re Requirements fo for St Structural Co Concrete and Commentary 343R 343R Ana Analysis sis and and Des Design of Rei Rein nforced Con Concre crete Bridge Structures 423.3R 423.3R Reco Recomm mmen enda dati tion onss for for Conc Concre rete te Memb Member erss PrePrestressed with Unbonded Tendons ASTM A 416 416 Sta Standar ndard d Spec Specif ific icaation tion for for Unc Uncoa oatted Sev SeveenWire Stress-Relieved Strand for Prestressed Concrete American Association of State Highway and Transportation Officials (AASHTO) AASHTO AASHTO// Bridge Bridge Design Design Specif Specifica icatio tions ns LRFD AASHTO AASHTO Standa Standard rd Specif Specifica icatio tion n for Highwa Highway y Bridges Bridges

*

April 16, 1991 corresponence from Charles W. Dolan to Paul Zia.

American Concrete Institute P.O. Box 9094 Farmington Hills, Mich. 48333-9094 ASTM 100 Barr Harbor Drive West Conshohocken, PA 19428

7.2—Cited references Abeles, P. W., 1940, “Saving Reinforcement by Prestressing,” Concrete and Constructional Engineering, V. 35, 3 pp. Abeles, P. W., 1945, “Fully and Partially Prestressed Reinforced Concrete,” ACI J OURNAL , Proceedings V. 41, No. 3, Jan., pp. 181-214. Abeles, P. W., 1954, “Static and Fatigue Tests on Partially Prestressed Concrete Constructions,” ACI J OURNAL, V. 51, No. 12, Dec., 361 pp. Abeles, P. W., 1965, “Studies of Crack Widths and Deformation Under Sustained and Fatigue Loading,” PCI Journal, V. 10, No. 6, Dec., pp. 43-52. Abeles, P. W., 1968, “Preliminary Report on Static and Sustained Loading Tests,” PCI Journal, V. 13, No. 4, Aug., pp. 12-32. ACI Committee 209, 1982, “Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures,” Designing for Creep and Shrinkage, Shrinkag e, SP-76, B. B. Goyal, ed., American Concrete Institute, Farmington Hills, Mich., pp. 193-300. ACI Committee 215, 1974, “Considerations for Design of Concrete Structures Subjected to Fatigue Loading,” ACI JOURNAL , V. 71, No. 3, Mar., pp. 97-121. ACI Committee 224, 1980, “Control of Cracking in Concrete Structures,” Concrete International, V. 2, No. 10, Oct., pp. 35-76. ACI-ASCE Joint Committee 323, 1958, “Tentative Recommendations for Prestressed Concrete,” ACI J OURNAL , Proceedings V. V. 54, No. 7, Jan., pp. 545-578. Al-Zaid, R. Z., and Naaman, A. E., 1986, “Analysis of Partially Prestressed Composite Beams,” Journal of Structural Engineering, ASCE, V. 112, No. 4, Apr., pp. 709-725. Al-Zaid, R. Z.; Naaman, A. E.; and Nowak, A. S., 1988, “Analysis of Partially Prestressed Composite Beams under Sustained and Cyclic Fatigue Loading,” Journal of Structural Engineering, ASCE, V. 114, No. 2, Feb., pp. 269-291. Amu, O. O., and French, C. W., 1985, “Moment Resistant Connections in Precast Structures Subjected to Cyclic Lateral Loads,” Structural Research Report No. 87-07, Deptartment of Civil and Mineral Engineering, University of Minnesota, Minneapolis, Minn.


shop on Design of Prefabricated Concrete Buildings for Earthquake Loads, Los Angeles, Calif., Apr., Architectural Institute of Japan, 1970, Design Essentials in Earthquake Resistant Buildings, Elsevier Co., New York, pp. 186-200. Arya, A. S., 1970, “Use of Prestressed Concrete in Earthquake Resistance Structures,” Proceedings, Fourth Symposium on Earthquake Engineering, Roorkee, India, Nov. Nov. Ashar, H.; Naus, D.; and Tan, C.P., 1994, “Prestressed Concrete in U.S. Nuclear Power Plants (Part 1),” Concrete International V. 16, No. 5, May, pp. 30-34. Baker, A. L. L., and Amarakone, A. M. N., 1964,, “Inelastic Hyperstatic Frame Analysis,” Proceedings, International Symposium on the Flexural Mechanics of Reinforced Concrete, SP-12, American Concrete Institute, Farmington Hills, Mich., pp. 85-142. Balaguru, P. N., 1981, “Analysis of Prestressed Concrete Beams for Fatigue Loading,” PCI Journal, V. 26, No. 3, May-July, pp. 70-94. Balaguru, P. N., and Shah, S. P., 1982, “Method of Predicting Crack Widths and Deflections for Fatigue Loading,” Fatigue of Concrete Structures, SP-75, S. P. Shah, ed., American Concrete Institute, Farmington Hills, Mich., pp. 153-175. Beeby, A. W., 1978, “Corrosion of Reinforcing Steel in Concrete and its Relation to Cracking,” Structural Engineer , London, V. V. 56A, No. 3, Mar., pp. 77-81. Beeby, A. W., 1979, “The Prediction of Crack Widths in Hardened Concrete,” Structural Engineer, London, V. 57A, No. 1, Jan., pp. 9-17. Bennet, E. W., 1984, “Partial Prestressing—An Historical Overview,” PCI Journal, V. 29, No. 5, Sept.-Oct., pp. 104-117. Bennett, E. W., 1986, “Partially Prestressed Concrete Members: Repeated Loading,” Partial Prestressing, From Theory to Practice, NATO ASI Series, V. 1, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, pp. 135-149. Bennett, E. W., and Lee, K. H., 1985, “Deflection of Partially Prestressed Beams under a Combination of LongTime and Short-Time Loading”, Deflections of Concrete Structures, SP-86, G. M. Sabnis, Ed., American Concrete Institute, Farmington Hills, Mich., pp. 185-214. Berg, G. V., and Degenkolb, H. J., 1973, “Engineering Lessons from the Managua Earthquake,” Managua, Managua, Nicaragua Nicaragua Earthquake, Earthquake, Dec. 1972, Earthquake Engineering Research Institute, Oakland, Calif., Nov., pp. 761-763. Bertero, V. V., 1973, “Experimental Studies Concerning Reinforced, Prestressed, and Partially Prestressed Concrete Structures and their Elements,” Resistance and Ultimate Deformability of Structures Acted on by Well Defined Repeated Loads, IABSE, Lisbon. Bertero, V. V., 1986, “Partially Prestressed Concrete Members for Earthquake-Resistant Design and Construction,” Partial Prestressing, from Theory to Practice, NATO ASI Series, V. 1, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, pp. 151-188. Bhuvasorakul, T., 1974, “Performance of Members Reinforced with Smooth Welded Wire Fabric Under Static and Repeated Loads,” PhD thesis, Oklahoma State University. Birkenmaier, M., 1984, “Developments in the Design of Partial Prestressing,” Partial Prestressing, from Theory to Practice, NATO ASI Series, Martinus Nijhoff Publishers, Dordrecht, The Netherlands.

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ity of Prestressed Concrete Beams,” Journal of the Structural Division, ASCE, V. 100, No. ST9, Sept., pp. 18831895. Blakeley, R. W. G., 1971, “Ductility of Prestressed Concrete Frames under Seismic Loading,” PhD thesis, University of Canterbury, Canterbury, Christchurch, New Zealand. Blakeley, R. W. G., and Park, R., 1971, “Seismic Resistance of Prestressed Concrete Beam-Column Assemblies,” ACI JOURNAL , V. V. 68, No. 9, Sept., pp. 677-692. Blakeley, R. W. G., and Park, R., 1973a, “Response of Prestressed Concrete Structures to Earthquake Motions,” New Zealand Engineering, V. 28, No. 2, Feb., pp. 42-54. Blakeley, R. W. G., and Park, R., 1973b, “Prestressed Con Journal of the Struccrete Sections With Seismic Loadings,” Journal tural Division, ASCE, V. 99, No. ST8, Aug., pp. 1717-1742. Branson, D. E., 1974, “The Deformation of Non-Composite and Composite Prestressed Concrete Members,” Deflections of Concrete Structures, SP-43, American Concrete Institute, Farmington Hills, Mich., pp. 83-128. Branson, D. E., 1977, Deformation of Concrete Structures, McGraw-Hill, New York. Branson, D. E., and Kripanarayanan, K. M., 1971, “Loss of Prestress, Camber, and Deflections of Non-Composite and Composite Prestressed Concrete Structures,” PCI Journal, V. V. 16, No. N o. 5, Sept.-Oct. pp. 22-52. Branson, D. E., and Shaikh, A. F., 1985, “Deflection of Partially Prestressed Members,” Deflections of Concrete Structures, SP-86, G. M. Sabnis, Ed., American Concrete Institute, Farmington Hills, Mich., pp. 323-363. Branson, D. E., and Trost, H., 1982a “Unified Procedures for Predicting the Deflection and Centroidal Axis Location of Partially Cracked Nonprestressed and Prestressed Concrete Members,” ACI JOURNAL , V. 79, No. 2, Mar.-Apr., pp. 119-130. Branson, D. E., and Trost, H., 1982b “Application of the IEffective Method in Calculating Deflections of Partially Prestressed Members,” PCI Journal, V. 27, No. 5, Sept.-Oct. Brondum-Nielsen, T., 1973, “Effect of Prestress on the Damping of Concrete,” Resistance and Ultimate Deformability of Structures Acted on by Well Defined Repeated Loads, IABSE, Lisbon. Bruggeling, A. S. G., 1977, “Time Dependent Deflection of Partially Prestressed Beams,” Report No. 5-77-1, Department of Civil Engineering, Delft Technical University, Mar., 16 pp. Bureau of Public Roads, 1954, Criteria for Prestressed Concrete Bridges, U.S. Department of Commerce, Washington, D.C. Burns, N. H., 1964, “Moment Curvature Relationship for Partially Prestressed Concrete Beams,” PCI Journal, V. 9, No. 1, Feb., pp. 52-63. Burns, N. H., and Siess, C. P., 1962, “Loading Deformation Characteristics of Beam-Column Connections in Reinforced Concrete,” Civil Engineering Studies, Structural Research Series, Bulletin No. 234, University of Illinois, Urbana, 261 pp. Burns, V. V., 1979, “Inelastic Behavior of Structural Elements and Structures,” Proceedings, NSF Workshop (Dec. 2-4), University of Illinois at Chicago, pp. 96-167. Caflisch, R.; Krauss, R.; and Thurlimann, B., 1971, “Beige-und Schubversuche and Teilweise Vorgespannten Betonbalken, Serie C.,” Bericht 6504-3, Institut fur Bausta-

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Flexural Members,” PCI Journal, V. 22, No. 3, May-June. Moustafa, S. E., 1986, “Nonlinear Analysis of Reinforced and Prestressed Concrete Members,” PCI Journal, V. 31, No. 5, Sept-Oct. Muguruma, H., 1986 “Seismic Problems of PPC Building Structures with Special Reference to Basic Research in Japan,” Partial Prestressing, From Theory to Practice, NATO ASI Series, V. 2, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, pp. 231-254. Muguruma, H., and Tominaga, H., 1972, “Deformation Characteristics of Prestressed Lightweight Concrete BeamColumn Assemblies under Seismic Force,” Proceedings , 18th National Symposium on Bridge and Structural Engineering, Japan, pp. 123-136. Muguruma, H.; Watanabe, Watanabe, F.; and Fujii, M., 1983, “Shear Behavior of Prestressed Reinforced Concrete Beams,” Transactions , Japan Concrete Institute, V. V. 5, pp. 225-230. Muguruma, H.; Watanabe, F.; and Fukai, S., 1980, “Behavioor of Class 3 Partially Prestressed Concrete Beam under Reversed Cyclic High Over-Load,” Proceedings of FIP Symposium on Partial Prestressing (Bucharest), Federation Internationale de la Precontrainte (FIP), London, Sept., pp. 118-127. Muguruma, H.; Watanabe, F.; and Fukai, S., 1983, “On the Class 3 Prestressed Concrete Member with High Strength Concrete,” (in Japanese), Proceedings, Annual Meeting of Japan Concrete Institute, Tokyo, June, pp. 449-452. Muguruma, H.; Watanabe, F.; and Nasu, T., 1982a, “On the Fundamental Behaviors of a PPC Beam under Monotonic and Cyclic High Overload,” Journal, Japanese Prestressed Concrete Engineering Association, V. V. 24, Sept., pp. 101-108. Muguruma, H.; Watanabe, F.; and Tanaka, H., 1982b, “Improving the Flexural Ductility of Prestressed Concrete by Using High Yield Strength Lateral Hoop Reinforcement,” Journal, Japanese Prestressed Concrete Engineering Association , V. V. 24, Sept., pp. 109-123. Naaman, A. E., 1977, “Ultimate Analysis of Prestressed and Partially Prestressed Sections by Strain Compatibility,” Compatibility,” PCI Journal, V. V. 22, No. 1, Jan.-Feb., pp. 32-51. Prestressed Concrete Analysis and Naaman, A. E., 1982a, Prestressed Design, McGraw Hill, N.Y, 670 pp. Naaman, A. E., 1982b, “Fatigue in Partially Prestressed Beams”, Fatigue of Concrete Structures, SP-75, S. P. Shah, ed., American Concrete Institute, Farmington Hills, Mich., pp. 25-46. Naaman, A. E., 1983a, “Time-Dependent Deflection of Prestressed Beams by the Pressure-Line Method,” PCI Journal, V. 28, No. 2, Mar.-Apr., pp. 98-119. Naaman, A. E., 1983b, “An Approximate Nonlinear Design Procedure for Partially Prestressed Concrete Beams,” Com puters and Structur Structures es, V. 17, No. 2, pp. 287-293. Naaman, A. E., 1985, “Partially Prestressed Concrete: Review and Recommendations,” Recommendations,” PCI Journal, V. 30, No. 6, Nov-Dec, pp. 30-71. Naaman, A. E., 1989, “Fatigue of Reinforcement in Partially Prestressed Beams,” Proceedings, ASCE Structures Congress, San Francisco, Volume on Structural Materials, J. F. Orofino, ed., May, pp. 377-381. Naaman, A. E., and Founas, M., 1991, “Partially Prestressed Beams under Random Amplitude Fatigue Loading,” Journal of Structural Engineering, ASCE V. 117, No.

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APPENDIX—NOTATIONS a

= depth of equiv equivalen alentt rectangular rectangular compre compressi ssive ve stress block

Acp

= area enclos enclosed ed by outside outside perime perimeter ter of concret concretee cross section = area of prest prestresse ressed d reinforcem reinforcement ent in tension tension A ps zone As = area of nonpre nonprestres stressed sed tension tension reinforc reinforcemen ementt = area of nonprestre nonprestressed ssed compress compression ion reinforce reinforcement ment A′s At = effectiv effectivee area of concrete concrete tension zone sursurrounding all reinforcementhaving the same centroid as reinforcement = width width of compr compressio ession n face face of member member b B = distance distance ratio ratio at at first first loading loading cycle cycle B N = distan distance ce rati ratio o at the N th th loading cycle c = depth depth to neutral neutral axis from from extrem extremee compressi compression on fiber c N = depth depth of neutr neutral al axis axis after after N th th cycle of loading c2 = constant C c(t–t ) = creep creep coef coeffi ficie cient nt of conc concret retee at time time t when A loaded at time t A = distance distance from from extreme extreme compres compression sion fiber fiber to cend troid of nonprestressedtension nonprestressedtension reinforcement d c = cover cover to center center of reinforc reinforcing ing steel steel ′c d ′ = concre concrete te clear clear cover cover d ctf = depth depth to centroi centroid d of tensile tensile force force in steel steel d e = depth depth to centroid centroid of nonprest nonprestress ressed ed tension tension reinforcement d max,N = maxim maximum um deflec deflectio tion n after after N th th cycle d = distance dist ance from fr om extreme extr eme compres com pression sion fiber fiber to cen p troid of prestressed reinforcement er,n = average average strain strain in in concrete concrete between between the the cracks cracks after N cycles cycles of loading es,n = steel steel strain strain at crack calcul calculated ated using using cracked cracked section elastic analysis after N cycles cycles of loading E c(t ) = inst instan anta tane neou ouss ela elast stic ic modu modulu luss of of con concr cret etee at at time t E ce(t ) = equi equiva vale lent nt elas elasti ticc mod modul ulus us of conc concre rete te at time time t E ci = elastic elastic modulus modulus of of concrete concrete at time time of loading loading or transfer E i(t ) = inst instan anta tane neou ouss ela elast stic ic modu modulu luss of of con concr cret etee at at time t E = cyclic cyclic modulus modulus of elastic elasticity ity after after N cycles cycles N E s = modulus modulus of elast elasticit icity y of steel steel f c′ = concrete concrete compressi compressive ve strength strength = maximum maximum recommend recommended ed stress stress range range for for concrete concrete f cr f = safe sa fe str s tres ess s rang ra nge e f f min = minimum minimum applied applied stress stress (in (in concrete concrete or reinforcing reinforcing steel, as applicable) f = average average precom precompress pression ion in concrete concrete due due to pre pc stress at centroid of cross-section f = effecti effective ve stress stress in prestr prestressin essing g steel steel pe f = stress stress in prestress prestressed ed reinforcem reinforcement ent at nominal nominal ps bending strength f = ultimate ultimate stren strength gth of the the prestressi prestressing ng steel steel pu f s = tensile tensile stress stress in reinf reinforcin orcing g steel, steel, or change change in steel stress from decompression f so = steel steel stress stress at primar primary y crack crack at first first cycle cycle f so N N = steel stee l stress str ess at primary prim ary crack cr ack at cycle cycl e , N f sr = maximum maximum recomm recommende ended d stress stress range in reinforc reinforcing ing steel (prestressedor nonprestressed, as applicable) f = yield yield strength strength of nonprestres nonprestressed sed reinforce reinforcement ment y F = prestr prestress essing ing force force = prestressing force at time t F (t )


h1 h2 I cr I eff (t ) I eN I g K D, K F L p M M a M D M dec

M cr M L M n M np N N f P Pcp PPR Q

= distance distance from from neutral neutral axis axis to tensio tension n face = distance distance from from neutral neutral axis axis to centroid centroid of reinforcement = moment moment of inert inertia ia of crack cracked ed section section = effe effect ctiive mome moment nt of iner inerti tiaa of of cra crack cked ed sect sectio ion n at at time t = effecti effective ve moment moment of inertia inertia after N cycles cycles = gross gross mome moment nt of inerti inertiaa = constants constants depend depending ing on type type of loadin loading g and steel profile = equival equivalent ent plast plastic ic hinge hinge lengt length h = sustained sustained external external moment moment at midsp midspan an = appl applie ied d mome moment nt = dead dead load load mome moment nt = decompress decompression ion moment moment (mome (moment nt which which produces zero concrete stressat extreme fiber of a section, nearest to centroid of the prestressing force, when added to action of effective prestress alone) = crac cracki king ng mom momen entt = live live-lo -load ad moment moment = total total nomina nominall momen momentt capacit capacity y = moment capacity capacity provided provided by prestres prestressed sed reinforcement = number number of load load cycle cycless = number number of of cycles cycles to to failur failuree = sum of perimet perimeters ers of all tension tension reinfor reinforceme cement nt = outside outside perimete perimeterr of concrete concrete cross cross sectio section n = partia partiall prestre prestressi ssing ng ratio ratio = appl applie ied d loa load d

Qser r / h

S o S o N , N t t A V c W W N W max W max N , N Z

β1 ∆ f ps ∆(t ) λ ε ρ ρ′ ρ p Σo ω ω p

423.5R-37

= applie applied d serv service ice load load = ratio ratio of base radius radius to height height of of rolled-on rolled-on transtransverse deformation (value of 0.3 may be used in absence of specific data) = slip of of reinforc reinforcement ement at first first cycle cycle = slip slip of reinf reinforc orceme ement nt at N th th cycle = time time or age of conc concret retee = age of of concrete concrete at time time of loading loading = concrete concrete contri contributi bution on of shear shear strength strength = crack width at the first first loading loading cycle cycle = crac crack k widt width h at the the N th th loading cycle = maximu maximum m crac crack k widt width h = maximu maximum m crack crack width width afte afterr N cycles cycles = slope of of descending descending portio portion n of concret concretee compression curve = parameter parameter relati relating ng neutral neutral axis axis depth to depth depth of equivalent equivalent rectangular compression block = change change in stress stress in prestre prestressing ssing steel steel = deflection at time t = degree degree of prestr prestress ess = strain strain (in genera general) l) = reinforcem reinforcement ent ratio ratio of tensile tensile nonprestr nonprestressed essed reinforcement = reinforcement reinforcement ratio ratio of compress compressive ive nonprenonprestressed reinforcement = reinforcement reinforcement ratio of tensile tensile prestressed prestressed reinforcement = sum of perimet perimeters ers of all tensio tension n reinforcem reinforcement ent = rein reinfo forc rcin ing g index index = reinforci reinforcing ng index index for prestre prestressing ssing steel steel

ACI 423.5R-99 Partially Prestressed Concrete

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