PRESTRESSED CONCRETE STRUCTURE SUBJECT CODE: CE 2404

SYLLABUS UNIT I INTRODUCTION – THEORY AND BEHAVIOUR Basic concepts – Advantages – Materials required – Systems and methods of prestressing – Analysis of sections – Stress concept – Strength concept – Load balancing concept – Effect of loading on the tensile stresses in tendons – Effect of tendon profile on deflections – Factors influencing deflections – Calculation of deflections – Short term and long term deflections - Losses of prestress – Estimation of crack width.

UNIT II DESIGN CONCEPTS Flexural strength – Simplified procedures as per codes – strain compatibility method – Basic concepts in selection of cross section for bending – stress distribution in end block, Design of anchorage zone reinforcement – Limit state design criteria – Partial prestressing – Applications. UNIT III CIRCULAR PRESTRESSING Design of prestressed concrete tanks – Pipes.

UNIT IV COMPOSITE CONSTRUCTION Analysis for stresses – Estimate for deflections Flexural and shear strength of composite members UNIT V PRE-STRESSED CONCRETE BRIDGES General aspects – pretensioned prestressed bridge decks – Post tensioned prestressed bridge decks – Principles of design only.

TEXT BOOKS: 1. Krishna Raju N., Prestressed concrete, Tata McGraw Hill Company, New Delhi 1998. 2. Mallic S.K. and Gupta A.P., Prestressed concrete, Oxford and IBH publishing Co. Pvt. Ltd.1997. 3. Rajagopalan, N, “Prestressed Concrete”, Alpha Science, 2002.

REFERENCES: 1.Ramaswamy G.S., Modern prestressed concrete design, Arnold Heinimen, New Delhi, 1990 2.Lin T.Y. Design of prestressed concrete structures, Asia Publishing House, Bombay, 1995. 3.David A.Sheppard, William R. and Philips, Plant Cast precast and prestressed concrete – A design guide, McGraw Hill, New Delhi 1992.

INTRODUCTION – THEORY AND BEHAVIOUR UNIT I

Basic concepts Advantages Materials required Systems and methods of prestressing Analysis of sections Stress concept Strength concept Load balancing concept Effect of loading on the tensile stresses in tendons Effect of tendon profile on deflections Factors influencing deflections Calculation of deflections Short term and long term deflections

Basic Concept A prestressed concrete structure is different from a conventional reinforced concrete structure due to the application of an initial load on the structure prior to its use. The initial load or ‘prestress’ is applied to enable the structure to counteract the stresses arising during its service period.

Prestressing, itself means, an intentional application of a pre-determined force on a system, for resisting the internal stresses that may be developed in the system, due to external loads.

Development of prestressed Concrete The basic principle of prestressing was applied to construction perhaps centuries ago, when ropes or metal bands were wound around wooden staves to form barrels.

Principle of prestressing applied to barrel construction Compressive prestress Wooden Staves Metal bands Wooden stave as a free body Radial Pressure Tensile prestress Half of metal band as a free body A wooden Barrel Principle of prestressing applied to barrel construction

When the bands were tightened, they were under tensile prestress which in turn created compressive prestress between the staves and thus enabled them to resist hoop tension produced by internal liquid pressure. In other words, the bands and the staves were both prestressed before they were subjected to any service loads.

Brief History Before the development of prestressed concrete, two significant developments of reinforced concrete are the invention of Portland cement and introduction of steel in concrete.

1824 Aspdin, J., (England) Obtained a patent for the manufacture of Portland cement. 1857 Monier, J., (France) Introduced steel wires in concrete to make flower pots, pipes, arches and slabs.

The following events were significant in the development of prestressed concrete. 1886 Jackson, P. H., (USA) Introduced the concept of tightening steel tie rods in artificial stone and concrete arches. Steel tie rods in arches

1888 Doehring, C. E. W., (Germany) Manufactured concrete slabs and small beams with embedded tensioned steel. 1908 Stainer, C. R., (USA) Recognised losses due to shrinkage and creep, and suggested retightening the rods to recover lost prestress.

1923 Emperger, F., (Austria) Developed a method of winding and pre- tensioning high tensile steel wires around concrete pipes. 1924 Hewett, W. H., (USA) Introduced hoop-stressed horizontal reinforcement around walls of concrete tanks through the use of turnbuckles. Thousands of liquid storage tanks and concrete pipes were built in the two decades to follow.

1925 Dill, R. H. , (USA) Used high strength unbonded steel rods 1925 Dill, R. H., (USA) Used high strength unbonded steel rods. The rods were tensioned and anchored after hardening of the concrete. 1926 Eugene Freyssinet (France) Used high tensile steel wires, with ultimate strength as high as 1725 MPa and yield stress over 1240 MPa.

In 1939, he developed conical wedges for end anchorages for post-tensioning and developed double-acting jacks. He is often referred to as the Father of Prestressed concrete.

1938 Hoyer, E., (Germany) Developed ‘long line’ pre-tensioning method. 1940 Magnel, G., (Belgium) Developed an anchoring system for post-tensioning, using flat wedges.

Prestressed concrete was started to be used in building frames, parking structures, stadiums, railway sleepers, transmission line poles and other types of structures and elements.

In India, the applications of prestressed concrete diversified over the years. The first prestressed concrete bridge was built in 1948 under the Assam Rail Link Project. Among bridges, the Pamban Road Bridge at Rameshwaram, Tamilnadu, remains a classic example of the use of prestressed concrete girders.

Pamban Road Bridge at Rameshwaram, Tamilnadu

Definitions Prestressing wire is a single unit made of steel. Strands Wires Prestressing wire is a single unit made of steel. Strands Two, three or seven wires are wound to form a prestressing strand. Tendon A group of strands or wires are wound to form a prestressing tendon.

Cable A group of tendons form a prestressing cable. Bars A tendon can be made up of a single steel bar. The diameter of a bar is much larger than that of a wire.

Nature of Concrete-Steel Interface Bonded tendon When there is adequate bond between the prestressing tendon and concrete, it is called a bonded tendon. Pre-tensioned and grouted post-tensioned tendons are bonded tendons.

Unbonded tendon When there is no bond between the prestressing tendon and concrete, it is called unbonded tendon. When grout is not applied after post-tensioning, the tendon is an unbonded tendon.

Stages of Loading The analysis of prestressed members can be different for the different stages of loading. The stages of loading are as follows. 1) Initial : It can be subdivided into two stages. a) During tensioning of steel b) At transfer of prestress to concrete.

2) Intermediate : This includes the loads during transportation of the prestressed members. 3) Final : It can be subdivided into two stages. a) At service, during operation. b)At ultimate, during extreme events.

Advantages of Prestressing The prestressing of concrete has several advantages as compared to traditional reinforced concrete (RC) without prestressing. A fully prestressed concrete member is usually subjected to compression during service life. This rectifies several deficiencies of concrete.

1) Section remains uncracked under service loads Reduction of steel corrosion Increase in durability. Full section is utilised Higher moment of inertia (higher stiffness)

Less deformations (improved serviceability). Increase in shear capacity. Suitable for use in pressure vessels, liquid retaining structures. Improved performance (resilience) under dynamic and fatigue loading.

2) High span-to-depth ratios Larger spans possible with prestressing (bridges, buildings with large column-free spaces) Typical values of span-to-depth ratios in slabs are given below. Non-prestressed slab 28:1 Prestressed slab 45:1

For the same span, less depth compared to RC member. Reduction in self weight More aesthetic appeal due to slender sections More economical sections.

3) Suitable for precast construction 3) Suitable for precast construction. The advantages of precast construction are as follows. Rapid construction Better quality control Reduced maintenance Suitable for repetitive construction Multiple use of formwork ⇒ Reduction of formwork Availability of standard shapes.

Typical precast members The following figure shows the common types of precast sections. Typical precast members

Limitations of Prestressing Although prestressing has advantages, some aspects need to be carefully addressed. Prestressing needs skilled technology. Hence, it is not as common as reinforced concrete. The use of high strength materials is costly. There is additional cost in auxiliary equipments. There is need for quality control and inspection.

Types of Prestressing Source of prestressing force This classification is based on the method by which the prestressing force is generated. There are four sources of prestressing force: Mechanical Hydraulic Electrical and Chemical.

Hydraulic Prestressing This is the simplest type of prestressing, producing large prestressing forces. The hydraulic jack used for the tensioning of tendons, comprises of calibrated pressure gauges which directly indicate the magnitude of force developed during the tensioning.

Mechanical Prestressing In this type of prestressing, the devices includes weights with or without lever transmission, geared transmission in conjunction with pulley blocks, screw jacks with or without gear drives and wire-winding machines. This type of prestressing is adopted for mass scale production.

Electrical Prestressing In this type of prestressing, the steel wires are electrically heated and anchored before placing concrete in the moulds. This type of prestressing is also known as thermo-electric prestressing.

Chemical Prestressing In the chemical method, expanding cements are used and the degree of expansion is controlled by varying the curing conditions. Since the expansive action of cement while setting is restrained, it induces tensile forces in tendons and compressive stresses in concrete.

External or internal prestressing This classification is based on the location of the prestressing tendon with respect to the concrete section. Pre-tensioning or Post-tensioning This is the most important classification and is based on the sequence of casting the concrete and applying tension to the tendons.

Pre - tensioning A method of prestressing concrete in which the tendons are tensioned before the concrete is placed. In this method, the prestress is imparted to concrete by bond between steel and concrete.

Post – tensioning A method of prestressing concrete by tensioning the tendons against hardened concrete. In this method, the prestress is imparted to concrete by bearing.

Linear or circular prestressing This classification is based on the shape of the member prestressed. Full, limited or partial prestressing Based on the amount of prestressing force, three types of prestressing are defined. Uniaxial, biaxial or multi-axial prestressing As the names suggest, the classification is based on the directions of prestressing a member.

External or Internal Prestressing External Prestressing When the prestressing is achieved by elements located outside the concrete, it is called external prestressing. The tendons can lie outside the member (for example in I-girders or walls) or inside the hollow space of a box girder. This technique is adopted in bridges and strengthening of buildings.

In the following figure, the box girder of a bridge is prestressed with tendons that lie outside the concrete. External prestressing of a box girder

Internal Prestressing When the prestressing is achieved by elements located inside the concrete member (commonly, by embedded tendons), it is called internal prestressing. Most of the applications of prestressing are internal prestressing. In the following figure, concrete will be cast around the ducts for placing the tendons.

Internal prestressing of a box girder

Pre-tensioning or Post-tensioning The tension is applied to the tendons before casting of the concrete. The pre-compression is transmitted from steel to concrete through bond over the transmission length near the ends. The following figure shows manufactured pre-tensioned electric poles.

Pre-tensioned electric poles

Post-tensioning The tension is applied to the tendons (located in a duct) after hardening of the concrete. The pre-compression is transmitted from steel to concrete by the anchorage device (at the end blocks). The following figure shows a post-tensioned box girder of a bridge.

Post-tensioning of a box girder

Linear or Circular Prestressing Linear Prestressing When the prestressed members are straight or flat, in the direction of prestressing, the prestressing is called linear prestressing. For example, prestressing of beams, piles, poles and slabs. The profile of the prestressing tendon may be curved. The following figure shows linearly prestressed railway sleepers.

Linearly prestressed railway sleepers

Circular Prestressing When the prestressed members are curved, in the direction of prestressing, the prestressing is called circular prestressing. For example, circumferential prestressing of tanks, silos, pipes and similar structures. The following figure shows the containment structure for a nuclear reactor which is circularly prestressed.

Circularly prestressed containment structure, Kaiga Atomic Power Station, Karnataka

Full, Limited or Partial Prestressing Full Prestressing When the level of prestressing is such that no tensile stress is allowed in concrete under service loads, it is called Full Prestressing (Type 1, as per IS:1343 - 1980).

Limited Prestressing When the level of prestressing is such that the tensile stress under service loads is within the cracking stress of concrete, it is called Limited Prestressing (Type 2). Partial Prestressing When the level of prestressing is such that under tensile stresses due to service loads, the crack width is within the allowable limit, it is called Partial Prestressing (Type 3).

Uniaxial, Biaxial or Multiaxial Prestressing Uniaxial Prestressing When the prestressing tendons are parallel to one axis, it is called Uniaxial Prestressing. For example, longitudinal prestressing of beams.

Biaxial Prestressing When there are prestressing tendons parallel to two axes, it is called Biaxial Prestressing. The following figure shows the biaxial prestressing of slabs. Multiaxial Prestressing When the prestressing tendons are parallel to more than two axes, it is called Multiaxial Prestressing. For example, prestressing of domes.

Biaxial prestressing of a slab

Other Terminologies Anchorage A device generally used to enable the tendon to impart and maintain prestress in the concrete. The commonly used anchorages are the Freyssinet, Magnel Blaton, Gifford-Udall, Leonhardt-Baur, LeeMcCall, Dywidag, Roebling and B.B.R.V. systems.

Concordant Prestressing Prestressing of members in which the cables follow a concordant profile. In the case of statically indeterminate structures, concordant prestressing does not cause any change in the support reactions.

Non-distortional prestressing In this type, the combined effect of the degree of prestress and the dead-weight stresses is such that the deflection of the axis of the member is prevented. In such cases, the moments due to prestress and dead-weight exactly balance resulting only in an axial force in the member.

Eccentric Prestressing A section at which the tendons are eccentric to the centroid, resulting in a triangular or trapezoidal compressive stress distribution.

Transfer The stage corresponding to the transfer of prestress to concrete. For pretensioned members, transfer takes place at the release of prestress from the bulk-heads. For post-tensioned members, it takes place after the completion of the tensioning process.

Supplementary or untensioned reinforcement Reinforcement in prestressed members not tensioned with respect to the surrounding concrete before the application of loads. These are generally used in partially prestressed members. Transmission Length The length of the bond anchorage of the prestressing wire from the end of a pre-tensioned member to the point of full steel stress.

Cracking load The load on the structural element corresponding to the first visible crack. Creep in concrete Progressive increase in the inelastic deformation of concrete under sustained stress component. Shrinkage of concrete Contraction of concrete on drying.

Relaxation in steel Decrease of stress in steel at constant strain Proof stress The tensile stress in steel which produces a residual strain of 0.2 percent of the original gauge length on unloading. Creep coefficient The ratio of the total creep strain to elastic strain in concrete.

Cap cable A short curved tendon arranged at the interior supports of a continuous beam. The anchors are in the compression zone, while the curved portion is in the tensile zone.

Degree of prestressing A measure of the magnitude of the prestressing force related to the resultant stress ocurring in the structural member at working load. Debonding Prevention of bond between the steel wire and the surrounding concrete.

Materials Required High Strength Concrete Prestressed concrete requires concrete which has a high compressive strength at a reasonably early age, with comparatively higher tensile strength than ordinary concrete.

Low shrinkage, minimum creep characteristics and a high value of Young’s modulus are generally deemed necessary for concrete used for prestressed members. Many desirable properties, such as durability, impermeability and abrasion resistance are highly influenced by the strength of concrete.

The minimum 28-day cube compressive strength prescribed in the IS: 1343-1980 is 40N/mm2 for pre-tensioned members and 30N/mm2 for post-tensioned members.

According to IS:1343-1980, The compressive stress varies linearly from 0.54 to 0.37 fci for post tensioned work and from 0.51 to 0.44 fci for pre-tensioned work depending on the strength of concrete. (fci Compressive strength of concrete at initial transfer of prestress ) At transfer, there is no tensile stress

At service load, the compressive stress varies linearly from 0.41 to 0.35 fck depending upon the strength of concrete.(fck-Characteristic cube strength of concrete) At service load, the tensile stresses are Type 1 members: none Type2 members: tensile stresses not to exceed 3N/mm2 Type 3 members: hypothetical tensile stresses vary from 3.2N/mm2 for M30 to a maximum of 7.3N/mm2 for M50 grade concrete depending upon the limiting crack-width.

Shrinkage of concrete The shrinkage of concrete in pre-stressed members is due to the gradual loss of moisture which results in changes in volume. The drying shrinkage depends on the aggregate type and quantity, relative humidity, water/cement ratio in the mix and the time of exposure.

The values of total residual shrinkage strain recommended in the IS code for the purpose of design are 3.0 X 10-4 for pre-tensioned members and (2.0 X 10-4)/log(t+2) for post-tensioned members, where t is the age in days of the concrete at transfer.

Creep of concrete The loss of prestress due to creep of concrete can be estimated by the creep coefficient method, as recommended in the IS code IS:1343-1980. The values of creep coefficient, which is the ratio of the ultimate creep strain to the elastic strain, is 2.2 at 7 days of loading, 1.6 at 28 days and 1.1 when the age at loading is 1 year.

Deformation characteristics of concrete The deformation characteristics of concrete under short-term and sustained loads is necessary for determining the flexural strength of beams and for evaluating the modulus of elasticity of concrete, which is required for the computation of deflections of prestressed members.

High tensile steel For prestressed concrete members, the high tensile steel used generally consists of wires, bars or strands. The higher tensile strength is generally achieved by marginally increasing the carbon content in steel in comparison with mild steel

Permissible stresses in steel At the time of initial tensioning, the initial stress not to exceed 80% of the characteristic tensile strength of tendons. Immediately after prestress transfer, no stress in steel. Final stress after allowing for all losses of prestress should not be less the 45% of the characteristic tensile strength of tendons.

Stress Corrosion The phenomenon of stress corrosion in steel is particularly dangerous, as it results from the combined action of corrosion and static tensile stress, which may be either residual or externally applied. If the ducts of post-tensioned members are not grouted, there is a possibility of stress corrosion leading to a catastrophic of the structure.

Other common types of corrosion frequently encountered in prestressed concrete constructions are pitting corrosion and chloride corrosion. Some of the important protective measures against stress corrosion include protection from chemical contamination, protective coatings for high-tensile steel and grouting of ducts immediately after prestressing operations.

Hydrogen Embrittlement Atomic hydrogen is liberated as a result of the action of acids on high tensile steels. This penetrates into the steel surface, making it brittle and fracture prone on being subjected to tensile stress. Even small amounts of hydrogen can cause considerable damage to the tensile strength of high-tensile steel wires.

In order to prevent hydrogen embrittlement, it is essential that the steel is properly protected from the action of acids. Protective coverings like bituminous crepe-paper covering during transport, reduces the chances of contamination. The wires should be protected from rain water and excessive humidity by storing them in dry conditions.

Durability, Fire Resistance and Cover Requirements for PSC members The IS code IS1343-1980 provides for a minimum clear cover of 20mm for protected pre-tensioned members, while it is 30mm or the size of the cable(whichever is bigger) in the case of protected post-tensioned members. If the prestressed members are exposed to an aggressive environment, these cover requirements are increased by 10mm.

The IS 1343-1980 also prescribes minimum cement content and maximum water cement ratio in concrete to ensure durability under specified conditions of exposure. Fire resistance of structural concrete elements is influenced by the following parameters: Size and shape of the element Detailing, type and quality of reinforcement or prestressing tendons The level of load supported and pattern of loading Type of concrete and aggregate Conditions at end bearing Protective cover to reinforcement

Protection of prestressing steel, sheathing and anchorages To prevent deterioration due to corrosion, unbonded tendons should be coated by non-reactive materials like epoxy or zinc or zinc aluminium. Non-corroding sheathing material like high density polyethylene(HDPE) is beneficial.

The space between sheathing and duct can be filled with corrosion inhibiting materials like grease, wax or petroleum jelly. External parts of anchorages and projecting cables should be covered by suitable casing. The prestressing steel stored at site should also be protected by proper packaging films to guard against corrosion.

Prestressing Systems The various methods by which pre-compression is imparted to concrete are classified as follows: Generation of compressive force between the structural element and its abutments using flat jacks. Development of hoop compression in cylindrically shaped structures by circumferential wire winding.

Use of longitudinally tensioned steel embedded in concrete or housed in ducts. Use of the principle of distortion of a statically indeterminate structure either by displacement or by rotation of one part relative to the remainder. Use of deflected structural steel sections embedded in concrete until the hardening of the latter. Development of limited tension in steel and compression in concrete by using expanding cements.

Post-tensioning Systems Introduction Stages of Post-tensioning Advantages of Post-tensioning Disadvantages of Post-tensioning Devices Manufacturing of a Post-tensioned Bridge Girder

Stages of Post-tensioning In post-tensioning systems, the ducts for the tendons (or strands) are placed along with the reinforcement before the casting of concrete. The tendons are placed in the ducts after the casting of concrete. The duct prevents contact between concrete and the tendons during the tensioning operation.

Unlike pre-tensioning, the tendons are pulled with the reaction acting against the hardened concrete. If the ducts are filled with grout, then it is known as bonded post-tensioning. The grout is a neat cement paste or a sand-cement mortar containing suitable admixture.

In unbounded post-tensioning, as the name suggests, the ducts are never grouted and the tendon is held in tension solely by the end anchorages. The following sketch shows a schematic representation of a grouted post-tensioned member. The profile of the duct depends on the support conditions.

For a simply supported member, the duct has a sagging profile between the ends. For a continuous member, the duct sags in the span and hogs over the support.

Post-tensioning

Among the following figures, the first photograph shows the placement of ducts in a box girder of a simply supported bridge. The second photograph shows the end of the box girder after the post-tensioning of some tendons.

Post-tensioning ducts in a box girder

Post-tensioning of a box girder

The various stages of the post-tensioning operation are summarised as follows. 1) Casting of concrete. 2) Placement of the tendons. 3) Placement of the anchorage block and jack. 4) Applying tension to the tendons. 5) Seating of the wedges. 6) Cutting of the tendons.

The stages are shown schematically in the following figures. After anchoring a tendon at one end, the tension is applied at the other end by a jack. The tensioning of tendons and pre-compression of concrete occur simultaneously. A system of self-equilibrating forces develops after the stretching of the tendons.

Advantages of Post-tensioning The relative advantages of post-tensioning as compared to pre-tensioning are as follows. Post-tensioning is suitable for heavy cast-in-place members. The waiting period in the casting bed is less. The transfer of prestress is independent of transmission length.

Disadvantage of Post-tensioning The relative disadvantage of post-tensioning as compared to pre-tensioning is the requirement of anchorage device and grouting equipment.

Devices The essential devices for post-tensioning are as follows Devices The essential devices for post-tensioning are as follows. 1) Casting bed 2) Mould/Shuttering 3) Ducts 4) Anchoring devices 5) Jacks 6) Couplers (optional) 7) Grouting equipment (optional).

Anchoring Devices In post-tensioned members the anchoring devices transfer the prestress to the concrete. The devices are based on the following principles of anchoring the tendons. 1) Wedge action 2) Direct bearing 3) Looping the wires

Wedge action The anchoring device based on wedge action consists of an anchorage block and wedges. The strands are held by frictional grip of the wedges in the anchorage block. Some examples of systems based on the wedge-action are Freyssinet, Gifford-Udall, Anderson and Magnel-Blaton anchorages. The following figures show some patented anchoring devices.

Freyssinet “T” system anchorage cones

Direct bearing The rivet or bolt heads or button heads formed at the end of the wires directly bear against a block. The B.B.R.V post-tensioning system and the Prescon system are based on this principle. The following figure shows the anchoring by direct bearing.

Anchoring with button heads

Looping the wires The Baur-Leonhardt system, Leoba system and also the Dywidag single-bar anchorage system, work on this principle where the wires are looped around the concrete. The wires are looped to make a bulb. The following photo shows the anchorage by looping of the wires in a post-tensioned slab.

Anchorage by looping the wires in a slab

Sequence of Anchoring The following figures show the sequence of stressing and anchoring the strands. The photo of an anchoring device is also provided.

Couplers The couplers are used to connect strands or bars. They are located at the junction of the members, for example at or near columns in post-tensioned slabs, on piers in post-tensioned bridge decks. The couplers are tested to transmit the full capacity of the strands or bars. A few types of couplers are shown in the figure below.

Grouting Grouting can be defined as the filling of duct, with a material that provides an anti-corrosive alkaline environment to the prestressing steel and also a strong bond between the tendon and the surrounding grout.

The major part of grout comprises of water and cement, with a water-to-cement ratio of about 0.5, together with some water-reducing admixtures, expansion agent and pozzolans. The following figure shows a grouting equipment, where the ingredients are mixed and the grout is pumped.

Grouting equipment

Manufacturing of Post-tensioned Bridge Girders The following photographs show some steps in the manufacturing of a post-tensioned I-girder for a bridge. The first photo shows the fabricated steel reinforcement with the ducts for the tendons placed inside. Note the parabolic profiles of the duct for the simply supported girder.

After the concrete is cast and cured to gain sufficient strength, the tendons are passed through the ducts, as shown in the second photo.

The tendons are anchored at one end and stretched at the other end by a hydraulic jack. This can be observed from the third photo.

The following photos show the construction of post-tensioned box girders for a bridge. The photograph shows the fabricated steel reinforcement with the ducts for the tendons placed inside. The top flange will be constructed later.

The photo below shows the formwork in the pre-casting yard. The formwork for the inner sides of the webs and the flanges is yet to be placed.

In this photo a girder is being post-tensioned after adequate curing.

This photograph shows a crane on a barge that transports a girder to the bridge site.

The completed bridge can be seen in the last photo.

Analysis of Prestress and Bending Stresses Basic Assumptions The analysis of stresses developed in a prestressed concrete structural element is based on the following assumptions: Concrete is a homogeneous elastic material

Within the range of working stresses, both concrete and steel behave elastically, notwithstanding the small amount of creep which occurs in both the materials under sustained loading, and A plane section before bending is assumed to remain plane even after bending, which implies a linear strain distribution across the depth of the member.

Analysis of prestress The stresses due to prestressing alone are generally combined stresses due to the action of direct load and bending resulting from an eccentrically applied load. The stresses in concrete are evaluated by using the well known relationship for combined stresses used in the case of columns.

The following notations and sign conventions are used for the analysis of prestress: P = Prestressing force e = eccentricity of prestressing force M = P . e = Moment A = cross-sectional area of the concrete member I = second moment of area of section about its centroid.

Zt and Zb = section modulus of the top and bottom fibres fsup and finf = prestress in concrete developed at the top and bottom fibres (positive when compressive and negative when tensile in nature) yt and yb = distance of the top and bottom fibres from the centroid of the section i = radius of gyration

Concentric Tendon Consider a concrete beam with a concentric tendon as shown in the figure P P Stress = P/A Uniform prestress in concrete = P/A, which is compressive across the depth of the beam.

Eccentric Tendon The figure shows a concrete beam subjected to an eccentric prestressing force of magnitude P located at an eccentricity e. The stresses developed at the top and bottom fibres of the beam are obtained by the relations:

Eccentric Prestressing

Resultant stresses at a section The concrete beam shown in figure, supports uniformly distributed live and dead loads of intensity q and g. The beam is prestressed by a straight tendon carrying a prestressing force P at an eccentricity e.

The resultant stresses in concrete at any section are obtained by superposing the effect of prestress and the flexural stresses developed due to the loads. If Mq and Mg are the live and dead load moments at the central span section,

In the case of prestressed members, the cross-sectional area of high tensile steel being a very small percentage of the total concrete area, the stress computations are generally based on the nominal concrete cross-sectional properties.

Pressure line or Thrust line and internal resisting couple At any given section of a prestressed concrete beam, the combined effect of the prestressing force and the externally applied load will result in a distribution of concrete stresses that can be resolved into a single force. The locus of the points of application of this resultant force in any structure is termed as the ‘pressure or thrust line’.

The concept of pressure line is very useful in understanding the load-carrying mechanism of a prestressed concrete section. In the case of prestressed concrete members, the location of the pressure line depends upon the magnitude and direction of the moments applied at the cross-section and the magnitude and distribution of stress due to the prestressing force.

Consider a concrete beam shown in the figure below, which is prestressed by force P acting at an eccentricity e. The beam supports a uniformly distributed load (including self-weight) of intensity q per unit length.

The load is of such magnitude that the bottom-fibre stress at the central span section of the beam is zero. Figure below shows the resultant stress distribution at the support, centre and quarter span sections of the beam.

At the support section, since there are no flexural stresses resulting from the external loads, the pressure line coincides with that of the centroid of steel, located at an eccentricity of h/6.

At the centre of the span section, the external loading is such that the resultant stress developed is maximum at the top fibre and zero at the bottom fibre. It ca easily be seen that for this section the pressure line has shifted towards the top fibre by an amount equal to h/3 from its initial position.

The external moment at the quarter span section being smaller in magnitude, the shift in the pressure line also is correspondingly smaller, being equal to h/4 from the initial position.

In a similar manner, it can be shown that a larger uniformly distributed load on the beam would result in the pressure line being shifted even higher at the centre and quarter span sections.

The pressure line location in the beam is shown in the figure below.

These observations lead to the following important principle: “A change in the external moments in the elastic range of a prestressed concrete beam results in a shift of the pressure line rather than in an increase in the resultant force in the beam.”

This in contrast to a reinforced concrete beam section, where an increase in the external moment results in a corresponding increase in the tensile force and the compressive force. The increase in the resultant forces are due to a more or less constant lever arm between the forces, characterised by the properties of the composite section.

Basically, the load-carrying mechanism is comprised of a constant force with a changing lever-arm, as in the case of prestressed concrete sections, and a changing force with a constant lever arm prevailing in reinforced concrete sections as shown in the figure below.

However, if a prestressed concrete member is cracked, it behaves in a manner similar to that of a reinforced concrete section. The pressure line or thrust line method is generally referred to as the internal resisting couple method or the C-line method, the prestressed beam is analysed as a plain concrete elastic beam using the basic principles of statics.

The prestressing force is considered as an external compressive force with a constant tensile force T in the tendon throughout the span. Consequently, at any section of a loaded prestressed beam, equilibrium is maintained satisfying the equations, H= 0 and M = 0.

The figure below shows the free-body diagram of a segment of a beam without and with transverse loads respectively.

When the gravity loads are zero, the C and T lines-coincide since there is no moment at the section. Under transverse loads, the C-line, or the centre of pressure or thrust line, is at a varying distance a from the T-line. If M = bending moment at the section due to dead and live loads. e = eccentricity of the tendon T = P = Prestressing force in the tendon

Moment equilibrium yields the relation, The shift of pressure line e measured from the centroidal axis is obtained as

The resultant stresses at the top and bottom fibres of the section are expresses as

Concept of Load Balancing It is possible to select suitable cable profiles in a prestressed concrete member such that the transverse component of the cable force balances the given type of external loads. This can be readily illustrated by considering the free-body of concrete, with the tendon replaced by forces acting on the concrete beam as shown in the figure and table below.

The various types of reactions of a cable upon a concrete member depend upon the shape of the cable profile. Straight portions of the cable do not induce any reactions except at the ends, while curved cables result in uniformly distributed loads. Sharp angles in a cable induce concentrated loads. The concept of loading – balancing is useful in selecting the tendon profile, which can supply the most desirable system of forces in concrete.

In general, this requirement will be satisfied if the cable profile in a prestressed member corresponds to the shape of the bending moment diagram resulting from the external loads. Thus, if the beam supports two concentrated loads, the cable should follow a trapezoidal profile. If the beam supports uniformly distributed loads, the corresponding tendon should follow a parabolic profile.

Losses of Prestress Introduction In prestressed concrete applications, the most important variable is the prestressing force. In the early days, it was observed that the prestressing force does not stay constant, but reduces with time.

Even during prestressing of the tendons and the transfer of prestress to the concrete member, there is a drop of the prestressing force from the recorded value in the jack gauge. The various reductions of the prestressing force are termed as the losses in prestress.

The losses are broadly classified into two groups, immediate and time-dependent. The immediate losses occur during prestressing of the tendons and the transfer of prestress to the concrete member. The time-dependent losses occur during the service life of the prestressed member.

The losses due to elastic shortening of the member, friction at the tendon-concrete interface and slip of the anchorage are the immediate losses. The losses due to the shrinkage and creep of the concrete and relaxation of the steel are the time-dependent losses.

Loss due to Elastic deformation of concrete Pretensioned members When the tendons are cut and the prestressing force is transferred to the member, the concrete undergoes immediate shortening due to the prestress. The tendon also shortens by the same amount, which leads to the loss of prestress.

Post tensioned members If there is only one tendon, there is no loss because the applied prestress is recorded after the elastic shortening of the member. For more than one tendon, if the tendons are stretched sequentially, there is loss in a tendon during subsequent stretching of the other tendons.

The elastic shortening loss is quantified by the drop in prestress (Δfp) in a tendon due to the change in strain in the tendon (Δep) It is assumed that the change in strain in the tendon is equal to the strain in concrete (ec) at the level of the tendon due to prestressing force. This assumption is called strain compatibility (Δep= ec) between concrete and steel.

The loss of prestress due to elastic deformation of concrete depends on the modular ratio and the average stress in concrete at the level of steel. If fc = prestress in concrete at the level of steel Es = modulus of elasticity of steel Ec = modulus of elasticity of concrete

Strain in concrete at the level of steel = Stress in steel corresponding to this strain = Therefore, loss of stress in steel = If the initial stress in steel is known, the percentage loss of stress in steel due to the elastic deformation of concrete can be computed.

A pretensioned concrete beam, 100mm wide and 300mm deep, is prestressed by straight wires carrying an initial force of 150kN at an eccentricity of 50mm. The modulus of elasticity of steel and concrete are 210 and 35 kN/mm2 respectively. Estimate the percentage loss of stress in steel due to elastic deformation of concrete if the area of steel wires is 188mm2.

P = 150kN e = 50mm A = 100 X 300 = 3 x 104 mm2 I = 225 x 106mm4 ae = Es/Ec = 210/35 =6

Initial stress in steel = 150 x 103 / 188 = 800 N/mm2 Stress in concrete, fc = (150 x 103 /3 x 104 ) + ((150 x 103 x 50 x50)/225x106) = 6.66 N/mm2 Loss of stress due to elastic deformation of concrete = ae x fc = 6 x 6.66 = 40N/mm2 Percentage loss of stress in steel =(40 / 800)x 100 = 5%

Loss due to shrinkage of concrete The shrinkage of concrete in prestressed members results in a shortening of tensioned wires and hence contributes to the loss of stress. The shrinkage of concrete is influenced by the type of cement and aggregates and the method of curing used. Use of high-strength concrete with low water cement ratios results in a reduction in shrinkage and consequent loss of prestress.

The primary cause of drying shrinkage is the progressive loss of water from concrete. The rate of shrinkage is higher at the surface of the members. The differential shrinkage between the interior and surface of large members may result in strain gradients leading to surface cracking. Hence, proper curing is essential to prevent shrinkage cracks in prestressed members.

As per IS 1343-1980, the loss of prestress due to the shrinkage of concrete is

Loss due creep of concrete The sustained prestress in the concrete of a prestressed member results in creep of concrete which effectively reduces the stress in high-tensile steel. The loss of stress in steel due to creep of concrete can be estimated if the magnitude of ultimate creep strain or creep coefficient is known.

Hence, loss of stress in steel = ec x Es = f (fc /Ec) X Es = f. fc Hence, loss of stress in steel = ec x Es = f (fc /Ec) X Es = f. fc . (Es / Ec) =

The magnitude of the creep coefficient, f, varies depending upon the humidity, concrete quality, duration of applied loading and the age of the concrete when loaded. The general values recommended for the creep coefficient vary from 1.5 for watery situations to 4.0 for dry conditions with a relative humidity of 35 percent.

=15.33N/mm2 15.33 =131.99N/mm2

15.33 =147.168N/mm2

Loss due to Relaxation of stress in steel When a high-tensile steel wire is stretched and maintained at a constant strain, the initial force in the wire does not remain constant but decreases with time. The decrease of stress in steel at constant strain is termed relaxation.

Most of the codes provide the loss of stress due to relaxation of steel as a percentage of the initial stress in steel. The Indian standard code recommends a value varying from 0 to 90 N/mm2 for stress in wires varying from 0.5 fpu to 0.8 fpu. The loss of prestress due to relaxation of steel recommended in Indian code IS 1343-1980 is shown below.

Loss of stress due to friction In the case of post tensioned members, the tendons are housed in ducts preformed in concrete. The ducts are either straight or follow a curved profile depending upon the design requirements. Consequently, on tensioning the curved tendons, loss of stress occurs in the post-tensioned members due to friction between the tendons and the surrounding concrete ducts.

The magnitude of this loss is of the following types: Loss of stress due to the curvature effect, which depends upon the tendon form or alignment which generally follows a curved profile along the length of the beam. Loss of stress due to the wobble effect, which depends upon the local deviations in the alignment of the cable. The wobble or wave effect is the result of accidental or unavoidable misalignment, since ducts or sheath cannot be perfectly located to follow a predetermined profile throughout the length of the beam.

From the figure, it is seen that the magnitude of the prestressing force, Px, at a distance x from the tensioning end follows an exponential function of the type, Px = Po.e -(ma+kx)

A concrete beam of 10m span, 100mm wide and 300mm deep, is prestressed by 3 cables. The area of each cable is 200mm2 and the initial stress in the cable is 1200 N/mm2. Cable 1 is parabolic with an eccentricity of 50mm above the centroid at the supports and 50mm below at the centre of span. Cable 2 is also parabolic with zero eccentricity at supports and 50mm below the centroid at the centre of span. Cable 3 is straight with uniform eccentricity of 50mm below the centroid. If the cables are tensioned from one end only, estimate the percentage loss of stress in each cable due to friction. Assume µ = 0.35 and K = 0.0015 per m.

= (4 x 100 )/ (10 x 1000)

(4 x 50) / (10 x 1000)

Loss due to Anchorage Slip In most post-tensioning systems, when the cable is tensioned and the jack is released to transfer prestress to concrete, the friction wedges, employed to grip the wires, slip over a small distance before the wires are firmly housed between the wedges. The magnitude of slip depends upon the type of wedge and the stress in the wires.

The magnitude of the loss of stress due to the slip in anchorage is computed as follows: If Δ = slip of anchorage, mm L = length of the cable, mm A = cross-sectional area of the cable, mm2 Es = modulus of elasticity of steel, N/mm2 P = Prestressing force in the cable, N Then,

Since the loss of stress is caused by a definite total amount of shortening, the percentage loss is higher for short members than for long ones.

A concrete beam is post tensioned by a cable carrying an initial stress of 1000N/mm2. The slip at the jacking end was observed to be 5mm. The modulus of elasticity of steel is 210 kN/mm2. Estimate the percentage loss of stress due to anchorage slip if the length of the beam is (a) 30m and (b) 3m.

A post-tensioned cable of beam 10m long is initially tensioned to a stress of 1000N/mm2 at one end. If the tendons are curved so that the slope is 1 in 24 at each end, with an area of 600mm2, calculate the loss of prestress due to friction given the following data. coefficient of friction between duct and cable = 0.55; friction coefficient for ‘wave’ effect = 0.0015 per m. During anchoring, if there is a slip of 3mm at the jacking end, calculate the final force in the cable and the percentage loss of prestress due to friction and slip. Es = 210kN/mm2.

Total losses allowed for in Design It is normal practice in the design of prestressed concrete members to assume the total loss of stress as a percentage of the initial stress and provide for this in the design computations. Since the loss of prestress depends on several factors, such as the properties of concrete and steel, method of curing, degrees of prestress and the method of prestressing, it is difficult to generalize the exact amount of the total loss of prestress.

Typical values of the total losses of stress that could be encountered under normal conditions of work are outlined as below:

In these recommendations it is assumed that temporary overstressing is done to reduce relaxation, and to compensate for friction and anchorage losses. The value of η is generally taken as 0.75 for pre-tensioning and 0.80 for post-tensioned members.

A pretensioned beam, 200mm wide and 300mm deep is prestressed by 10 wires of 7mm diameter initially stressed to 1200 N/mm2, with their centroids located 100mm from the soffit. Find the maximum stress in concrete immediately after transfer, allowing only for elastic shortening of concrete. If the concrete undergoes a further shortening due to creep and shrinkage while there is a relaxation of 5 percent of steel stress, estimate the final percentage loss of stress in the wires using the Indian standard code (IS: 1343-1980) regulations, and the following data:

Ac = 200 X 300 = 6 x 104 mm2 Ec = 5700 (42)1/2 = 36940 N/mm2 I = 200 x 3003/12 =45 x 107 mm4 αe = Es/Ec = 210x103/36900 = 5.7 P = 1200 x 10 x π x 72/4 = 461.8 x 103 N = 462 kN

Stress in concrete at the level of steel is given by fc = ((462 x 103/6 x 104) + ((462 x 103 x 50)50/45 x 107)) = 10.3 N/mm2 Loss of stress due to elastic deformation of concrete = 5.7 x 10.3 = 58.8 N/mm2 Force in wires immediately after transfer = (1200 – 58.8)x10 x π x 72/4 = 440 x 103 N = 440 kN

Stress in concrete at the level of steel is given by fc = ((428 x 103 / 6 x 104) + (428 x 103 x 50)50/45 x 107)) = 9.78 N/mm2 Type of losses of prestress Elastic deformation = αe x fc = 5.7 x 10.3 = 58.8 N/mm2 Creep of concrete = ø x fc x αe = 1.6 x 9.78 x 5.7 = 89.2 N/mm2 Shrinkage of concrete = Ɛcs x Es = 3 x 10-4 x 210 x 103 = 63 N/mm2 Relaxation of steel stress = 5/100 x 1200 = 60 N/mm2 Total loss = 271.0 N/mm2

Final stress in wires = 1200 – 271.0 = 929 N/mm2 Percentage loss = (271.0 / 1200) x 100 = 22.58 %

A prestressed concrete beam, 200mm wide and 300 mm deep, is prestressed with wires (area = 320 mm2) located at a constant eccentricity of 50 mm and carrying an initial stress of 1000 N/mm2. The span of the beam is 10m. Calculate the percentage loss of stress in wires if (a) the beam is pre-tensioned and (b) the beam is post-tensioned, using the following data: Es = 210 kN/mm2 and Ec = 35 kN/mm2; relaxation of stress in steel = 5 percent of the initial stress; shrinkage of concrete = 300 x 10-6 for pretensioning and 200 x 10-6 for post tensioning; creep coefficient = 1.6; slip at anchorage = 1 mm; frictional coefficient for wave effect = 0.0015 per m.

42.00

Deflections of Prestressed Concrete Members Suitable control on deflection is very essential for the following reasons: Excessive, sagging of principal structural members is not only unsightly, but at times, also renders the floor unsuitable for the intended use.

Large deflections under dynamic effects and under the influence of variable loads may cause discomfort to the users. Excessive deflections are likely to cause damage to finishes, partitions and associated structures.

Factors influencing Deflections The deflections of prestressed concrete members are influenced by the following salient factors: Imposed load and self-weight Magnitude of the prestressing force

Cable profile Second moment of area of cross-section Modulus of elasticity of concrete Shrinkage, creep and relaxation of steel stress Span of the member Fixity conditions

Short term deflections of uncracked members Short term or instantaneous deflections of prestressed members are governed by the bending moment distribution along the span and the flexural rigidity of the members. Mohr’s moment area theorems are readily applicable for the estimation of deflections due to the prestressing force, self-weight and imposed loads.

Consider the figure above in which the beam AB is subjected to a bending moment distribution due to the prestressing force or self-weight or imposed loads. ACB is the centre line of the deformed structure under the system of given loads.

If θ = slope of the elastic curve at A AD = intercept between the tangent at C and the vertical at A a = deflection at the centre for symmetrically loaded, simply supported beam (since the tangent is horizontal for such cases) A = area of the B.M.D. between A and C x = distance of the centroid of the B.M.D. between A and C from the left support EI = flexural rigidity of the beam

Then, according to Mohr’s first theorem, Mohr’s second theorem states that

Effect of tendon profile on deflections In most of the cases of prestressed beams, tendons are located with eccentricities towards the soffit of beams to counteract the sagging bending moments due to transverse loads. Consequently, the concrete beams deflect upwards (camber) on the application or transfer of prestress. Since the bending moment at every section is the product of the prestressing force and eccentricity, the tendon profile itself will represent the shape of the B.M.D.

Straight tendons: The figure shows a beam with a straight tendon at a uniform eccentricity below the centroidal axis.

Trapezoidal tendons: A draped tendon with trapezoidal profile is shown in the figure below. Considering the B.M.D., the deflection at the centre of the beam is obtained by taking the moment of the area of the B.M.D. over one-half of the span. Thus,

Parabolic tendons (central anchors): The deflection of a beam with parabolic tendons having an eccentricity e at the centre and zero at the supports is given by,

Consequently, the resultant deflection becomes, Parabolic tendons (eccentric anchors): The figure below shows a beam, with a parabolic tendon having an eccentricity e1 at the centre of the span and e2 at the support sections. The resultant deflection at the centre is obtained as the sum of the upward deflection of a beam with a parabolic tendon of eccentricity (e1 + e2) at the centre and zero at the supports and the downward deflection of a beam subjected to a uniform sagging bending moment of intensity pe2 throughout the length. Consequently, the resultant deflection becomes,

Sloping tendons (eccentric tendons): From the figure below, the deflection is calculated as

Parabolic and straight tendons: the deflection at the centre of the beam is obtained as,

Parabolic and straight tendons (eccentric anchors): the maximum central deflection is obtained by superposition,

Deflections due to self-weight and imposed loads At the time of transfer of prestress, the beam hogs up due to the effect of prestressing. At this stage, the self-weight of the beam induces downward deflections, which further increase due to the effect of imposed loads on the beam.

If g = self-weight of the beam/m q = imposed load/m (uniformly distributed) The downward deflection is computed as,

A concrete beam with a rectangular section 300 mm wide and 500 mm deep is prestressed by 2 post-tensioned cables of area 600 mm2 each, Initially stressed to 1600 N/mm2. The cables are located at a constant eccentricity of 100 mm throughout the length of the beam having a span of 10 m. The modulus of elasticity of steel and concrete is 210 and 38 kN/mm2 respectively. (a) Neglecting all losses, find the deflection at the centre of span when it is supporting its own weight. (b) Allowing for 20 percent loss in prestress, find the final deflection at the centre of span when it carries an imposed load of 18 kN/m. Dc = 24 kN/m3.

Prediction of Long Time Deflection The deformations of prestressed members change with time as a result of creep and shrinkage of concrete and relaxation of stress in steel. The prestressed concrete member develops deformations under the influence of two usually opposing effects, which are the prestress and transverse loads. The net curvature øt at a section at any given stage is obtained as

Under the section of sustained transverse loads, the compressive stress distribution in the concrete changes with time. However, in practical cases, the change of stress being small, it may be assumed that the concrete creeps under constant stress. The creep strain due to transverse loads is directly computed as a function of the creep coefficient so that the change of curvature can be estimated by the expression,

The change of curvature due to the sustained prestress (øpt)depends upon the cumulative effects of creep and shrinkage of concrete and relaxation of stress in steel. According to Neville and ACI committee report, the creep curvature due to prestress is obtained on the simplified assumption that creep is induced by the average prestress acting over the given time. Using this approach, if Pi = initial prestress Pt = prestress after a time, t

Loss of prestressing force due to relaxation, shrinkage and creep, Lp = (Pi-Pt) e = eccentricity of the prestressing force at the section EI = flexural rigidity The curvature due to prestress after time t can be expressed as

If ai1 = initial deflection due to transverse loads aip = initial deflection due to prestress Then, the total long time deflection after time t is obtained from the expression, In this expression, the –ve sign refers to deflections in the upward direction(camber).

A much simplified but an approximate procedure is suggested by Lin for computing long time deflections. In this method, the initial deflection due to prestress and transverse loads is modified to account for the loss of prestress which tends to decrease the deflection, and the creep effect which tends to increase the deflection. According to this method, the final long time deflection is expressed as

A concrete beam having a rectangular section 100mm wide and 300mm deep is prestressed by a parabolic cable carrying an initial force of 240kN. The cable has an eccentricity of 50 mm at the centre of span and is concentric at the supports. If the span of the beam is 10 m and the live load is 2kN/m, estimate the short time deflection at the centre of span. Assuming E=38 kN/mm2 and creep coefficient P=2.0, loss of prestress=20 percent of the initial stress after 6 months. Estimate the long time deflection at the centre of span at this stage, assuming that the dead and live loads are simultaneously applied after the release of prestress.