## Prestressed Concrete

8.1 Introduction 8-2

### 8.2 Concrete for Prestressed Elements 8-2

Compressive Strength • Tensile Strength • Shear Strength • High-Strength Concrete • Initial Compressive Strength and Modulus • Creep • Shrinkage

8.3 Steel Reinforcement Properties 8-6

Non-Prestressing Reinforcement • Prestressing Reinforcement

8.4 Maximum Permissible Stresses 8-8

Concrete Stresses in Flexure • Prestressing Steel Stresses

8.5 Partial Loss of Prestress 8-8

Steel Stress Relaxation (R) • Creep Loss (CR) • Shrinkage Loss

(SH) • Losses Due to Friction (F) • Example 1: Prestress Losses in Beams • Example 2: Prestressing Losses Evaluation Using SI Units

8.6 Flexural Design of Prestressed Concrete Elements 8-18

Minimum Section Modulus • Example 3: Flexural Design of Prestressed Beams at Service Load Level • Development and Transfer Length in Pretensioned Members and Design of their Anchorage Reinforcement • Posttensioned Anchorage Zones: Strut-and-Tie Design Method • Example 4: End Anchorage Design by the Strut-and-Tie Method • Ultimate-Strength Flexural Design • Limit States in Bonded Members from Decompression to Ultimate Load • Example 5: Ultimate Limit State Design of Prestressed Concrete Beams • Example 6: Ultimate Limit State Design of Prestressed Beams in SI Units

### 8.7 Shear and Torsional Strength Design 8-42

Composite-Action Dowel Reinforcement • Example 7: Design of Web Reinforcement for Shear • SI Expressions for Shear in Prestressed Concrete Beams • Design of Prestressed Concrete Beams Subjected to Combined Torsion, Shear, and Bending in Accordance with the ACI 318-02 Code

8.8 Camber, Deflection, and Crack Control 8-52

Serviceability considerations • Long-Term Effects on Deflection and Camber • Permissible Limits of Calculated Deflection • Long-Term Deflection of Composite Double-Tee Cracked Beam • Cracking Behavior and Crack Control in Prestressed Beams • ACI Expression for Cracking Mitigation • Long-Term Effects on Edward G. Nawy Crack-Width Development • Tolerable Crack Widths • Example 9:

Department of Civil and Environmental Crack Control Check • SI Deflection and Cracking Expressions

Engineering Acknowledgments 8-70

Rutgers University — The State

University of New Jersey, Qo^aty 8-71

Piscataway, NJ References 8-75

*This chapter is a condensation from several chapters of Prestressed Concrete — A Fundamental Approach, 4th edition, 2003, 944 pp., by E. G. Nawy, with permission of the publishers, Prentice Hall, Upper Saddle River, NJ.

8.1 Introduction

Concrete is strong in compression, but weak in tension: its tensile strength varies from 8 to 14% of its compressive strength. Due to such a low tensile capacity, flexural cracks develop at early stages of loading. In order to reduce or prevent such cracks from developing, a concentric or eccentric force is imposed in the longitudinal direction of the structural element. This force prevents the cracks from developing by eliminating or considerably reducing the tensile stresses at the critical midspan and support sections at service load, thereby raising the bending, shear, and torsional capacities of the sections. The sections are then able to behave elastically, and almost the full capacity of the concrete in compression can be efficiently utilized across the entire depth of the concrete sections when all loads act on the structure.

Such an imposed longitudinal force is termed a prestressing force, that is, a compressive force that prestresses the sections along the span of the structural element prior to the application of the transverse gravity dead and live loads or transient horizontal live loads. The type of prestressing force involved, together with its magnitude, are determined mainly on the basis of the type of system to be constructed and the span length. As a result, permanent stresses in the prestressed structural member are created before the full dead and live loads are applied, in order to eliminate or considerably reduce the net tensile stresses caused by these loads.

With reinforced concrete, it is assumed that the tensile strength of the concrete is negligible and disregarded. This is because the tensile forces resulting from the bending moments are resisted by the bond created in the reinforcement process. Cracking and deflection are therefore essentially irrecoverable in reinforced concrete once the member has reached its limit state at service load. In prestressed concrete elements, cracking can be controlled or totally eliminated at the service load level. The reinforcement required to produce the prestressing force in the prestressed member actively preloads the member, permitting a relatively high controlled recovery of cracking and deflection.

### 8.2 Concrete for Prestressed Elements

Concrete, particularly high-strength concrete, is a major constituent of all prestressed concrete elements. Hence, its strength and long-term endurance have to be achieved through proper quality control and quality assurance at the production stage. The mechanical properties of hardened concrete can be classified into two categories: short-term or instantaneous properties, and long-term properties. The short-term properties are strength in compression, tension, and shear; and stiffness, as measured by the modulus of elasticity. The long-term properties can be classified in terms of creep and shrinkage. The following subsections present some details on these properties.

### 8.2.1 Compressive Strength

Depending on the type of mix, the properties of aggregate, and the time and quality of the curing, compressive strengths of concrete can be obtained up to 20,000 psi or more. Commercial production of concrete with ordinary aggregate is usually in the range 4,000 to 12,000 psi, with the most common concrete strengths being in the 6,000 psi level.

The compressive strength f' is based on standard 6 in. by 12 in. cylinders cured under standard laboratory conditions and tested at a specified rate of loading at 28 days of age. The standard specifications used in the United States are usually taken from American Society for Testing and Materials (ASTM) C-39. The strength of concrete in the actual structure may not be the same as that of the cylinder because of the difference in compaction and curing conditions.

### 8.2.2 Tensile Strength

The tensile strength of concrete is relatively low. A good approximation for the tensile strength fct is 0.10f' < ft < 0.20f'. It is more difficult to measure tensile strength than compressive strength because of the gripping problems with testing machines. A number of methods are available for tension testing, the most commonly used method being the cylinder splitting, or Brazilian, test.

For members subjected to bending, the value of the modulus of rupture fr rather than the tensile splitting strength ft' is used in design. The modulus of rupture is measured by testing to failure plain concrete beams 6 in.2 in cross-section, having a span of 18 in., and loaded at their third points (ASTM C-78). The modulus of rupture has a higher value than the tensile splitting strength. The American Concrete Institute (ACI) specifies a value of 7.5 for the modulus of rupture of normal-weight concrete.

In most cases, lightweight concrete has a lower tensile strength than does normal-weight concrete. The following are the code stipulations for lightweight concrete:

1. If the splitting tensile strength fct is specified fr = 1.09ft < 7.5-Pf (8.1)

2. If fct is not specified, use a factor of 0.75 for all-lightweight concrete and 0.85 for sand-lightweight concrete. Linear interpolation may be used for mixtures of natural sand and lightweight fine aggregate. For high-strength concrete, the modulus of rupture can be as high as 11-12yf.

### 8.2.3 Shear Strength

Shear strength is more difficult to determine experimentally than the tests discussed previously because of the difficulty in isolating shear from other stresses. This is one of the reasons for the large variation in shear-strength test values reported in the literature, varying from 20% of the compressive strength in normal loading to a considerably higher percentage of up to 85% of the compressive strength in cases where direct shear exists in combination with compression. Control of a structural design by shear strength is significant only in rare cases, since shear stresses must ordinarily be limited to continually lower values in order to protect the concrete from the abrupt and brittle failure in diagonal tension

8.2.4 High-Strength Concrete

High-strength concrete is termed as such by the ACI 318 Code when the cylinder compressive strength exceeds 6,000 psi (41.4MPa). For concrete having compressive strengths 6,000 to 12,000 psi (42 to 84 MPa), the expressions for the modulus of concrete are [1-3]

Today, concrete strength up to 20,000 psi (138 MPa) is easily achieved using a maximum stone aggregate size of | in. (9.5 mm) and pozzolamic cementitious partial replacements for the cement such as silica fume. Such strengths can be obtained in the field under strict quality control and quality assurance conditions. For strengths in the range of 20,000 to 30,000 (138 to 206 MPa), other constituents such as steel or carbon fibers have to be added to the mixture. In all these cases, mixture design has to be made by several field trial batches (five or more), modifying the mixture components for the workability needed in concrete placement. Steel cylinder molds size 4 in. (diameter) x 8 in. length have to be used, applying the appropriate dimensional correction.

PHOTO 8.1 A rendering of the new Maumee River Bridge, Toledo, Ohio. This cable-stayed bridge spans the Maumee River in downtown Toledo as a monument icon for the city. The design includes single pylon, single plane of stays, and a main span with a horizontal clearance of 612 ft in both directions. The main pylon is clad on four of its eight sides with a glass curtain wall system, symbolizing the glass industry and heritage of Toledo. This glass prismatic system and stainless steel clad cables create a sleek and industrial look during the day. At night, the glass becomes very dynamic with the use of LED arrays back-lighting the window wall. Owner: Ohio Department of Transportation (courtesy of the Designer, Figg Engineering Group, Linda Figg, President, Tallahassee, Florida).

### 8.2.5 Initial Compressive Strength and Modulus

Since prestressing is performed in most cases prior to concrete's achieving its 28-day strength, it is important to determine the concrete compressive strength ft at the prestressing stage as well as the concrete modulus Ec at various stages in the loading history of the element. The general expression for the compressive strength as a function of time [4] is

where fc = 28-day compressive strength t = time in days a = factor depending on type of cement and curing conditions = 4.00 for moist-cured type-I cement and 2.30 for moist-cured type-III cement = 1.00 for steam-cured type-I cement and 0.70 for steam-cured type-III cement b = factor depending on the same parameters for a giving corresponding values of 0.85, 0.92, 0.95, and 0.98, respectively

Hence, for a typical moist-cured type-I cement concrete fi =-t-f (8.4b)

8.2.6 Creep

Creep, or lateral material flow, is the increase in strain with time due to a sustained load. The initial deformation due to load is the elastic strain, while the additional strain due to the same sustained load is the creep strain. This practical assumption is quite acceptable, since the initial recorded deformation includes few time-dependent effects. The ultimate creep coefficient, Cu, is given by

or average Cu ffi 2.35.

Branson's model, verified by extensive tests, relates the creep coefficient Ct at any time to the ultimate creep coefficient (for standard conditions) as t0.6

or, alternatively,

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