Pre Stressed Concrete

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1 BASIC CONCEPTS

1.1 INTRODUCTION Concrete is strong in compression, but weak in tension: its tensile strength varies from 8 to 14 percent 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 called a prestressing force, i.e., 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 and slenderness desired. Since the prestressing force is applied longitudinally along or parallel to The Diamond Baseball Stadium, Richmond, Virginia. Situ cast and precast post-tensioned prestressed structure. (Courtesy, Prestressed Concrete Institute.) 1

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Figure 1.1 Prestressing principle in linear and circular prestressing. (a) Linear prestressing of a series of blocks to form a beam. (b) Compressive stress on midspan section C and end section A or B. (c) Circular prestressing of a wooden barrel by tensioning the metal bands. (d) Circular hoop prestress on one wooden stave. (e) Tensile force F on half of metal band due to internal pressure, to be balanced by circular hoop prestress.

the axis of the member, the prestressing principle involved is commonly known as linear prestressing. Circular prestressing, used in liquid containment tanks, pipes, and pressure reactor vessels, essentially follows the same basic principles as does linear prestressing. The circumferential hoop, or “hugging” stress on the cylindrical or spherical structure, neutralizes the tensile stresses at the outer fibers of the curvilinear surface caused by the internal contained pressure. Figure 1.1 illustrates, in a basic fashion, the prestressing action in both types of structural systems and the resulting stress response. In (a), the individual concrete blocks act together as a beam due to the large compressive prestressing force P. Although it might appear that the blocks will slip and vertically simulate shear slip failure, in fact they will not because of the longitudinal force P. Similarly, the wooden staves in (c) might appear to be capable of separating as a result of the high internal radial pressure exerted on them. But again, because of the compressive prestress imposed by the metal bands as a form of circular prestressing, they will remain in place. 1.1.1 Comparison with Reinforced Concrete From the preceding discussion, it is plain that 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

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Photo 1.1 Bay Area Rapid Transit (BART), San Francisco and Oakland, California. Guideways consist of prestressed precast simple-span box girders 70 ft long and 11 ft wide. (Courtesy, Bay Area Rapid Transit District, Oakland, California.)

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. The reinforcement in the reinforced concrete member does not exert any force of its own on the member, contrary to the action of prestressing steel. The steel required to produce the prestressing force in the prestressed member actively preloads the member, permitting a relatively high controlled recovery of cracking and deflection. Once the flex-

Photo 1.2 Chaco-Corrientes Bridge, Argentina. The longest precast prestressed concrete cable-stayed box girder bridge in South America. (Courtesy, Ammann & Whitney.)

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Photo 1.3 tute.)

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Park Towers, Tulsa, Oklahoma. (Courtesy, Prestressed Concrete Insti-

ural tensile strength of the concrete is exceeded, the prestressed member starts to act like a reinforced concrete element. By controlling the amount of prestress, a structural system can be made either flexible or rigid without influencing its strength. In reinforced concrete, such a flexibility in behavior is considerably more difficult to achieve if considerations of economy are to be observed in the design. Flexible structures such as fender piles in wharves have to be highly energy absorbent, and prestressed concrete can provide the required resiliency. Structures designed to withstand heavy vibrations, such as machine foundations, can easily be made rigid through the contribution of the prestressing force to the reduction of their otherwise flexible deformation behavior. 1.1.2 Economics of Prestressed Concrete Prestressed members are shallower in depth than their reinforced concrete counterparts for the same span and loading conditions. In general, the depth of a prestressed concrete member is usually about 65 to 80 percent of the depth of the equivalent reinforced concrete member. Hence, the prestressed member requires less concrete, and about 20 to 35 percent of the amount of reinforcement. Unfortunately, this saving in material weight is balanced by the higher cost of the higher quality materials needed in prestressing. Also, regardless of the system used, prestressing operations themselves result in an added cost: Formwork is more complex, since the geometry of prestressed sections is usually composed of flanged sections with thin webs. In spite of these additional costs, if a large enough number of precast units are manufactured, the difference between at least the initial costs of prestressed and reinforced concrete systems is usually not very large. And the indirect long-term savings are quite substantial, because less maintenance is needed, a longer working life is possible due to

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better quality control of the concrete, and lighter foundations are achieved due to the smaller cumulative weight of the superstructure. Once the beam span of reinforced concrete exceeds 70 to 90 feet, the dead weight of the beam becomes excessive, resulting in heavier members and, consequently, greater long-term deflection and cracking. Thus, for larger spans, prestressed concrete becomes mandatory since arches are expensive to construct and do not perform as well due to the severe long-term shrinkage and creep they undergo. Very large spans such as segmental bridges or cable-stayed bridges can only be constructed through the use of prestressing. 1.2 HISTORICAL DEVELOPMENT OF PRESTRESSING Prestressed concrete is not a new concept, dating back to 1872, when P. H. Jackson, an engineer from California, patented a prestressing system that used a tie rod to construct beams or arches from individual blocks. [See Figure 1.1(a).] In 1888, C. W. Doehring of Germany obtained a patent for prestressing slabs with metal wires. But these early attempts at prestressing were not really successful because of the loss of the prestress with time. J. Lund of Norway and G. R. Steiner of the United States tried early in the twentieth century to solve this problem, but to no avail. After a long lapse of time during which little progress was made because of the unavailability of high-strength steel to overcome prestress losses, R. E. Dill of Alexandria, Nebraska, recognized the effect of the shrinkage and creep (transverse material flow) of concrete on the loss of prestress. He subsequently developed the idea that successive post-tensioning of unbonded rods would compensate for the time-dependent loss of stress in the rods due to the decrease in the length of the member because of creep and shrinkage. In the early 1920s, W. H. Hewett of Minneapolis developed the principles of circular prestressing. He hoop-stressed horizontal reinforcement around walls of concrete tanks through the use of turnbuckles to prevent cracking due to internal liquid pres-

Photo 1.4

Wiscasset Bridge, Maine. (Courtesy, Post-Tensioning Institute.)

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Photo 1.5 tute.)

Basic Concepts

Executive Center, Honolulu, Hawaii. (Courtesy, Post-Tensioning Insti-

sure, thereby achieving watertighteness. Thereafter, prestressing of tanks and pipes developed at an accelerated pace in the United States, with thousands of tanks of water, liquid, and gas storage built and much mileage of prestressed pressure pipe laid in the two to three decades that followed. Linear prestressing continued to develop in Europe and in France, in particular through the ingenuity of Eugene Freyssinet, who proposed in 1926 through 1928 methods to overcome prestress losses through the use of high-strength and high-ductility steels. In 1940, he introduced the now well-known and well-accepted Freyssinet system comprising the conical wedge anchor for 12-wire tendons. During World War II and thereafter, it became necessary to reconstruct in a prompt manner many of the main bridges that were destroyed by war activities. G. Magnel of Ghent, Belgium, and Y. Guyon of Paris extensively developed and used the concept of prestressing for the design and construction of numerous bridges in western and central Europe. The Magnel system also used wedges to anchor the prestressing wires. They differed from the original Freyssinet wedges in that they were flat in shape, accommodating the prestressing of two wires at a time. P. W. Abeles of England introduced and developed the concept of partial prestressing between the 1930s and 1960s. F. Leonhardt of Germany, V. Mikhailov of Russia, and T. Y. Lin of the United States also contributed a great deal to the art and science of the design of prestressed concrete. Lin’s load-balancing method deserves particular mention in this regard, as it considerably simplified the design process, particularly in continuous structures. These twentieth-century developments have led to the extensive use of prestressing throughout the world, and in the United States in particular.

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Photo 1.6 Stratford “B” Condeep offshore oil drilling platform, Norway. (Courtesy, the late Ben C. Gerwick.)

Today, prestressed concrete is used in buildings, underground structures, TV towers, floating storage and offshore structures, power stations, nuclear reactor vessels, and numerous types of bridge systems including segmental and cable-stayed bridges. Note the variety of prestressed structures in the photos throughout the book; they demonstrate the versatility of the prestressing concept and its all-encompassing applications. The success in the development and construction of all these landmark structures has been due in no small measure to the advances in the technology of materials, particularly prestressing steel, and the accumulated knowledge in estimating the short- and long-term losses in the prestressing forces.

1.3 BASIC CONCEPTS OF PRESTRESSING 1.3.1 Introduction The prestressing force P that satisfies the particular conditions of geometry and loading of a given element (see Figure 1.2) is determined from the principles of mechanics and of stress-strain relationships. Sometimes simplification is necessary, as when a prestressed beam is assumed to be homogeneous and elastic. Consider, then, a simply supported rectangular beam subjected to a concentric prestressing force P as shown in Figure 1.2(a). The compressive stress on the beam cross section is uniform and has an intensity

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Photo 1.7 Sunshine Skyway Bridge, Tampa Bay, Florida. Designed by Figg and Muller Engineers, Inc., the bridge has a 1,200-ft cable-stayed main span with a single pylon, 175-ft vertical clearance, and total length of 21,878 ft. It has twin 40-ft roadways and has 135-ft spans in precast segmental sections and high approaches to elevation + 130 ft. (Courtesy, Figg and Muller Engineers, Inc., now FIGG Engineers)

f

P Ac

(1.1)

where Ac = bh is the cross-sectional area of a beam section of width b and total depth h. A minus sign is used for compression and a plus sign for tension throughout the text. Also, bending moments are drawn on the tensile side of the member. If external transverse loads are applied to the beam, causing a maximum moment M at midspan, the resulting stress becomes ft  

Mc P  A Ig

(1.2a)

fb  

P Mc  A Ig

(1.2b)

and

where f t = stress at the top fibers fb = stress at the bottom fibers c =  h for the rectangular section Ig = gross moment of inertia of the section (bh3/12 in this case) Equation 1.2b indicates that the presence of prestressing-compressive stress − P/A is reducing the tensile flexural stress Mc/I to the extent intended in the design, either elimi-

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Figure 1.2 Concrete fiber stress distribution in a rectangular beam with straight tendon. (a) Concentric tendon, prestress only. (b) Concentric tendon, self-weight added. (c) Eccentric tendon, prestress only. (d) Eccentric tendon, self-weight added.

nating tension totally (even inducing compression), or permitting a level of tensile stress within allowable code limits. The section is then considered uncracked and behaves elastically: the concrete’s inability to withstand tensile stresses is effectively compensated for by the compressive force of the prestressing tendon. The compressive stresses in Equation 1.2a at the top fibers of the beam due to prestressing are compounded by the application of the loading stress − Mc/I, as seen in Figure 1.2(b). Hence, the compressive stress capacity of the beam to take a substantial external load is reduced by the concentric prestressing force. In order to avoid this limitation, the prestressing tendon is placed eccentrically below the neutral axis at midspan, to induce tensile stresses at the top fibers due to prestressing. [See Figure 1.2(c), (d).] If the tendon is placed at eccentricity e from the center of gravity of the concrete, termed the cgc line, it creates a moment Pe, and the ensuing stresses at midspan become

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Photo 1.8 The new I-35W bridge, Minneapolis, Minnesota. Designed by FIGG, the 10-lane interstate bridge has a 504 precast, prestressed concrete span over the Mississippi River, with 75 vertical clearance. The span’s 120 precast concrete segments, each 45 wide, up to 25 deep and weighing 200 tons, achieved an average concrete strength of 7600 psi. These segments for the main span were assembled in one directional cantilever from the main piers in just 47 days. The bridge has 50,000 cubic yards of concrete and nearly 3 million pounds of post-tensioning steel that runs the length of the twin 1,214 bridges. This new interstate bridge was designed and built in 11 months, opening on September 18, 2008. (Courtesy of FIGG, the bridge designer)

ft 

P Pec Mc   Ac Ig Ig

(1.3a)

fb  

P Pec Mc   Ac Ig Ig

(1.3b)

Since the support section of a simply supported beam carries no moment from the external transverse load, high tensile fiber stresses at the top fibers are caused by the eccentric prestressing force. To limit such stresses, the eccentricity of the prestressing tendon profile, the cgs line, is made less at the support section than at the midspan section, or eliminated altogether, or else a negative eccentricity above the cgc line is used. 1.3.2 Basic Concept Method In the basic concept method of designing prestressed concrete elements, the concrete fiber stresses are directly computed from the external forces applied to the concrete by longitudinal prestressing and the external transverse load. Equations 1.3a and b can be modified and simplified for use in calculating stresses at the initial prestressing stage and at service load levels. If Pi is the initial prestressing force before stress losses, and Pe is the effective prestressing force after losses, then 

Pe Pi

(1.3c)

can be defined as the residual prestress factor. Substituting r2 for Ig/Ac in Equations 1.3, where r is the radius of gyration of the gross section, the expressions for stress can be rewritten as follows: (a) Prestressing Force Only f t 

Pi ect a1  2 b Ac r

(1.4a)

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Photo 1.9 Tianjin Yong-He cable-stayed prestressed concrete bridge, Tianjin, China, the largest span bridge in Asia, with a total length of 1,673 ft and a suspended length of 1,535 ft, was completed in 1988. (Credits Owner: Tianjin Municipal Engineering Bureau. General contractor: Major Bridge Engineering Bureau of Ministry of Railways of China. Engineer for project design and construction control guidance: Tianjin Municipal Engineering Survey and Design Institute, Chief Bridge Engineer, Bang-yan Yu.)

fb 

Pi ecb a1  2 b Ac r

(1.4b)

where ct and cb are the distances from the center of gravity of the section (the cgc line) to the extreme top and bottom fibers, respectively. (b) Prestressing Plus Self-weight If the beam self-weight causes a moment MD at the section under consideration, Equations 1.4a and b, respectively, become Pi ect MD a1  2 b  t Ac S r

(1.5a)

Pi ecb MD a1  2 b  Ac Sb r

(1.5b)

f t  and fb 

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Figure 1.3

Basic Concepts

Prestressing tendon profile. (a) Harped tendon. (b) Draped tendon.

where St and Sb are the moduli of the sections for the top and bottom fibers, respectively. The change in eccentricity from the midspan to the support section is obtained by raising the prestressing tendon either abruptly from the midspan to the support, a process called harping, or gradually in a parabolic form, a process called draping. Figure 1.3(a) shows a harped profile usually used for pretensioned beams and for concentrated transverse loads. Figure 1.3(b) shows a draped tendon usually used in post-tensioning. Subsequent to erection and installation of the floor or deck, live loads act on the structure, causing a superimposed moment Ms. The full intensity of such loads normally occurs after the building is completed and some time-dependent losses in prestress have already taken place. Hence, the prestressing force used in the stress equations would have to be the effective prestressing force Pe. If the total moment due to gravity loads is MT, then MT  MD  MSD  ML where

(1.6)

MD = moment due to self-weight MSD = moment due to superimposed dead load, such as flooring ML = moment due to live load, including impact and seismic loads if any

Equations 1.5 then become ft 

Pe ect MT a1  2 b  t Ac S r

(1.7a)

fb  

Pe ecb MT a1  2 b  Ac Sb r

(1.7b)

Some typical elastic concrete stress distributions at the critical section of a prestressed flanged section are shown in Figure 1.4. The tensile stress in the concrete in part (c) permitted at the extreme fibers of the section cannot exceed the maximum permissible in the code, e.g., ft = 6 2f c¿ at midspan in the ACI code. If it is exceeded, bonded nonprestressed reinforcement proportioned to resist the total tensile force has to be provided to control cracking at service loads. 1.3.3 C-Line Method In this line-of-pressure or thrust concept, the beam is analyzed as if it were a plain concrete elastic beam using the basic principles of statics. The prestressing force is considered an ex-

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Figure 1.4 Elastic fiber stresses due to the various loads in a prestressed beam. (a) Initial prestress before losses. (b) Addition of self-weight. (c) Service load at effective prestress.

ternal compressive force, with a constant tensile force T in the tendon throughout the span. In this manner, the effects of external gravity loads are disregarded. Equilibrium equations H = 0 and M = 0 are applied to maintain equilibrium in the section. Figure 1.5 shows the relative line of action of the compressive force C and the tensile force T in a reinforced concrete beam as compared to that in a prestressed concrete beam. It is plain that in a reinforced concrete beam, T can have a finite value only when transverse and other external loads act. The moment arm a remains basically constant throughout the elastic loading history of the reinforced concrete beam while it changes from a value a = 0 at prestressing to a maximum at full superimposed load. Taking a free-body diagram of a segment of a beam as in Figure 1.6, it is evident that the C-line, or center-of-pressure line, is at a varying distance a from the T-line. The moment is given by

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Figure 1.5 Comparative free-body diagrams of a reinforced concrete (R.C.) beam and a prestressed concrete (P.C.) beam. (a) R.C. beam with no load. (b) P.C. beam with no load. (c) R.C. beam with load w1. (d) P.C. beam with load w1. (e) R.C. beam with typical load w. (f) P.C. beam with typical load w.

M  Ca  Ta

(1.8)

and the eccentricity e is known or predetermined, so that in Figure 1.6, e¿  a  e

(1.9a)

M e T

(1.9b)

ft 

Ce¿ct C  Ac Ic

(1.10a)

fb  

Ce¿cb C  Ac Ic

(1.10b)

Since C = T, a = M/T, giving e¿  From the figure,

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Figure 1.6

Free-body diagram for the C-line (center of pressure).

But in the tendon the force T equals the prestressing force Pe; so Pe Pee¿ct  Ac Ic

(1.11a)

Pe Pee¿cb  Ac Ic

(1.11b)

ft   fb  

Since Ic = Acr2, Equations 1.11a and b can be rewritten as ft 

Pe e¿ct a1  2 b Ac r

(1.12a)

fb  

Pe e¿cb a1  2 b Ac r

(1.12b)

Equations 1.12a and b and Equations 1.7a and b should yield identical values for the fiber stresses. 1.3.4 Load-Balancing Method A third useful approach in the design (analysis) of continuous prestressed beams is the load-balancing method developed by Lin and mentioned earlier. This technique is based on utilizing the vertical force of the draped or harped prestressing tendon to counteract or balance the imposed gravity loading to which a beam is subjected. Hence, it is applicable to nonstraight prestressing tendons.

Photo 1.10 East Huntington Bridge over Ohio River. A segmentally assembled precast prestressed concrete cable-stayed bridge spanning 200–900–608 ft. (Courtesy, Arvid Grant and Associates, Inc.)

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Figure 1.7

Basic Concepts

Load-balancing forces. (a) Harped tendon. (b) Draped tendon.

Figure 1.7 demonstrates the balancing forces for both harped- and draped-tendon prestressed beams. The load balancing reaction R is equal to the vertical component of the prestressing force P. The horizontal component of P, as an approximation in longspan beams, is taken to be equal to the full force P in computing the concrete fiber stresses at midspan of the simply supported beam. At other sections, the actual horizontal component of P is used. 1.3.4.1 Load-Balancing Distributed Loads and Parabolic Tendon Profile. Consider a parabolic tendon as shown in Figure 1.8. Let the parabolic function Ax2  Bx  C  y

(1.13)

represent the tendon drape; the force T denotes the pull to which the tendon is subjected. Then for x = 0, we have y0

C0

dy 0 dx

B0

and for x = l/2, ya

A

4a l2

But from calculus, the load intensity is qT

0 2y 0x2

(1.14)

Finding ∂2y/∂x2 in Equation 1.13 and substituting into Equation 1.14 yields qT

Figure 1.8

8Ta 4a 2 2 2 l l

Sketched tendon subjected to transverse load intensity q.

(1.15a)

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or T

ql 2 8a

(1.15b)

Ta 

ql 2 8

(1.15c)

Hence, if the tendon has a parabolic profile in the prestressed beam and the prestressing force is denoted by P, the balanced-load intensity, from Equation 1.15a, is wb 

8Pa l2

(1.16)

Figure 1.9 gives a free-body diagram of the forces acting on a prestressed beam with a parabolic tendon profile. Clearly, the two sets of equal and opposite transverse loads wb cancel each other, and no bending stress is produced. This is reasonable to expect in the load-balancing method, since it is always the case that T = C, and C has to cancel T to satisfy the equilibrium requirement that ΣH = 0. As there is no bending, the beam remains straight, without having a convex shape, or camber, at the top face. The concrete fiber stress across the depth of the section at midspan becomes fbt 

P¿ C  A A

(1.17)

This stress, which is constant, is due to the force P = P cos . Figure 1.10 shows the superposition of stresses to yield the net stress. Note that the prestressing force in the loadbalancing method has to act at the center of gravity (cgc) of the support section in simply supported beams and at the cgc of the free end in the case of a cantilever beam. This condition is necessary in order to prevent any eccentric unbalanced moments. When the imposed load exceeds the balancing load wb such that an additional unbalanced load wub is applied, a moment Mub = wubl2/8 results at midspan. The corresponding fiber stresses at midspan become fbt  

Mubc P¿ < Ac Ic

(1.18)

Equation 1.18 can be rewritten as the two equations ft 

Figure 1.9

Mub P¿  t Ac S

Load-balancing force on free-body diagram.

(1.19a)

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Photo 1.11 Walt Disney World Monorail, Orlando, Florida. A series of hollow precast prestressed concrete 100-foot box girders individually post-tensioned to provide six-span continuous structure. Designed by ABAM Engineers, Tacoma, Washington. (Courtesy, Walt Disney World Corporation.)

and fb  

Mub P¿  Ac Sb

(1.19b)

Equations 1.19 will yield the same values of fiber stresses as Equations 1.7 and 1.12. Keep in mind that P is taken to be equal to P at the midspan section because the prestressing force is horizontal at this section, i.e., = 0.

Figure 1.10 Load-balancing stresses. (a) Prestress stresses. (b) Imposed-load stresses. (c) Balanced-load stresses. (d) Net stress.

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1.4 COMPUTATION OF FIBER STRESSES IN A PRESTRESSED BEAM BY THE BASIC METHOD Example 1.1 A pretensioned simply supported 10LDT24 double T-beam without topping has a span of 64 ft (19.51 m) and the geometry shown in Figure 1.11. It is subjected to a uniform superimposed gravity dead-load intensity WSD and live-load intensity WL summing to 420 plf (6.13 kN/m). The initial prestress before losses is fpi ≅ 0.70 fpu = 189,000 psi (1,303 MPa), and the effective prestress after losses is fpe = 150,000 psi (1,034 MPa). Compute the extreme fiber stresses at the midspan due to (a) the initial full prestress and no external gravity load (b) the final service load conditions when prestress losses have taken place. Allowable stress data are as follows: f c = 6,000 psi, lightweight (41.4 MPa) fpu = 270,000 psi, stress relieved (1.862 MPa) = specified tensile strength of the tendons fpy = 220,000 psi (1.517 MPa) = specified yield strength of the tendons fpe = 150,000 psi (1,034 MPa) ft = 12 2f c¿ = 930 psi (6.4 MPa) = maximum allowable tensile stress in concrete f ci = 4,800 psi (33.1 MPa) = concrete compressive strength at time of initial prestress fci = 0.6 f ci = 2,880 psi (19.9 MPa) = maximum allowable stress in concrete at initial prestress fc = 0.45 f c = maximum allowable compressive stress in concrete at service Assume that ten -in.-dia. Seven-wire-strand (ten 12.7-mm-dia strand) tendons with a 108-D1 strand pattern are used to prestress the beam. Ac  449 in.2 12,915 cm2 2

Ic  22,469 in.4 1935,347 cm4 2

r 2  Ic>Ac  50.04 in.2

10'-0"

2"

2"

53/4" topping

24"

5'-0" Figure 1.11

33/4" Example 1.1

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Photo 1.12 Prestressed lightweight concrete mid-body for Arctic offshore drilling platform. Global Marine Development. (Courtesy, the late Ben C. Gerwick.) cb  17.77 in. 1452 mm2 ct  6.23 in. 1158 mm2

ec  14.77 in. 1375 mm2

ee  7.77 in. 1197 mm2

Sb  1,264 in.3 120,714 cm3 2 St  3,607 in.3 159,108 cm3 2

WD  359 plf 14.45 kN>m2

Solution: (i) Initial Conditions at Prestressing Aps  10  0.153  1.53 in.2

Pi  Aps fpi  1.53  189,000  289,170 lb 11,287 kN2

Pe  1.53  150,000  229,500 lb 11,020 kN2 The midpan self-weight dead-load moment is MD 

359 1642 2 wl 2   12  2,205,696 in.-lb. 1249 kN-m2 8 8

From Equations 1.5 and 1.7, ft  

Pi ect MD a1  2 b  t Ac S r 2,205,696 289,170 14.77  6.23 a1  b 449 50.04 3,607  540.3  611.5  70 psi 1C2

fb   

Pi ecb MD a1  2 b  Ac Sb r 289,170 2,205,696 14.77  17.77 a1  b 449 50.04 1,264

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1.5 C-Line Computation of Fiber Stresses

21 4,022.1  1,745.0  2,277 psi 1C2

fci 2,880 psi allowed, O.K.

(ii) Final Condition at Service Load Midspan moment due to superimposed dead and live load is MSD  ML 

420 1642 2 8

 12  2,580,480 in.-lb

Total Moment MT  2,205,696  2,580,480

 4,786,176 in.-lb. 1541 kN-m2

f t  

Pe ect MT a1  2 b  t Ac S r 229,500 4,786,176 14.77  6.23 a1  b 449 50.04 3,607

429  1,327  898 psi 1C2 17 MPa2 6 fc  0.45  6,000  2,700 psi, O.K.

fb  

Pe ecb MT a1  2 b  Ac Sb r 229,500 4,786,176 14.77  17.77 a1  b 449 50.04 1,264

3,192  3,786  594 psi 1T2 15.2 MPa2 6 ft  122f c¿  930 psi, O.K.

1.5 C-LINE COMPUTATION OF FIBER STRESSES Example 1.2 Solve example 1.1 for the final service-load condition by the line-of-thrust, C-line method. Solution: Pe  229,500 lb MT  4,786,176 in.-lb a

MT 4,786,176   20.85 in. Pe 229,500

e¿  a  e  20.85  14.77  6.08 in. From Equations 1.12, Pe e¿ct f t  a1  2 b Ac r 

229,500 6.08  6.23 a1  b  898 psi 1C2 449 50.04

fb   

e¿cb Pe a1  2 b Ac r 229,500 6.08  17.77 a1  b  594 psi 1T2 449 50.04

Notice how the C-line method is shorter than the basic method used in Example 1.1.

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Photo 1.13 Dauphin Island Bridge, Mobile County, Alabama. (Courtesy, PostTensioning Institute.)

1.6 LOAD-BALANCING COMPUTATION OF FIBER STRESSES Example 1.3 Solve Example 1.1 for the final service load condition after losses using the load-balancing method. Solution: P¿  Pe  229,500 lb at midspan At midspan, a  ec  14.77–  1.231 ft For the balancing load, we have Wb  8

8  229,500  1.231 Pa  l2 1642 2

 552 plf 18.1 kN>m2

Thus, if the total gravity load would have been 552 plf, only the axial load P / A would act if the beam had a parabolically draped tendon with no eccentricity at the supports. This is because the gravity load is balanced by the tendon at the midpan. Hence, Total load to which the beam is subjected  WD  WSD  WL  359  420  779 plf Unbalanced load Wub  779  552  227 plf Unbalanced moment Mub 

Wub 1l2 2 8



2271642 2 8

 1,394,688 in.-lb From Equations 1.19, f t 

Mub 1,394,688 229,500 P¿   t  Ac S 449 3,607

 12

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Photo 1.14 Heidrun Offshore Oil Drilling Platform in the North Sea weighing 2.9 million Kg. It measures 110 m on each side; has four slip-form constructed hulls and module-support 50-ft. span beams (Courtesy, C. E. Morrison, CONOCO Inc., Houston, Texas.) 511  387  898 psi 1C2 fb 

Mub 1,394,688 229,500 P¿    Ac Sb 449 1,264

 511  1,104  594 psi 1T2

ft  930 psi allowed, O.K.

1.7 SI WORKING LOAD STRESS CONCEPTS Example 1.4 Solve Example 1.1 using SI units Given Stress Data f c¿  41.4MPa fpu  1,860 MPa fpi  0.70 fpu  0.70  1860  1300 MPa fpy  1,520 MPa

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Photo 1.15 Hibernia Platform, Grand Banks, Newfoundland, 1997: 80 m water depth, 165,000 m3, 70 MPa strength concrete and 8,000 tons of prestressing tendons; first platform designed for iceberg impacts (Courtesy, Dr. George C. Hoff, Mobile Research and Development Corp.) fpe  1,034 MPa ft  2f c¿ is deflection check O.K.  6.4 MPa, use 1 ft  2f c¿ if no check made on deflection 2 f c¿i  0.8 f ¿c  33.1 MPa fc i  0.6 f c¿i  19.9 MPa fc  0.45 f c¿  18.6 MPa Aps  10 tendons 17.7 mm diameter  10  99mm2  990 mm2 Section Geometry Try 108-D1 Strand Pattern Ac  2897 cm2 Ic  935,346 cm2

r2  Ic>Ac  323 cm2

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Photo 1.16 Pier 37 Rebuild, Seattle Washington: a 1200-ft-long by 40-ft wide pier consisting of 20-ft-long prestressed concrete deck panel supported by situ-cast concrete pile caps and prestressed concrete piles (Designed by BERGER/ABAM Engineers, Federal Way, Washington, courtesy Robert Mast, Chairman) cb  45.1 cm

ct  15.8 cm

ec  37.5 cm

ee  19.74 cm

S  59,108 cm t

3

Sb  20,713 cm3 WD  5.24 kN>m WSD  WL  6.13 kN>m l  19.51 m Solution: 1. Initial conditions at prestressing Aps  990 mm2 Pi  Apsfpi  990  1,303  1,290 kN Pe  Apsfpe  900  1034  1,024 kN Midspan self-weight dead-load moment MD 

5.24 119.512 2 wl 2   249 kN-m 8 8

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From Equation 1.1a, f t 



Pi Md ect a1  2 b  t Ac S r 37.5  15.8 249  102 kN-cm 1290 a1  b 2897 323 59,108 cm2

0.37  0.42 kN>cm2 0.5  106 N>m2

 0.5 MPa 1C2 6 fti in tension 6 fci, O.K. From Equation 1.1b, fb   

Pi ecb MD a1  2 b  Ac Sb r 1290 37.5  45.1 249  102 kN-cm a1  b 2897 323 20,713 cm2

Photo 1.17 Paramount Apartments, San Francisco, CA: the first hybrid precast prestressed concrete moment-resistant 39 floor high-rise frame building in high seismicity zone, 2002. (Courtesy Charles Pankow Ltd., Design/Build contractors. Structural Engineers: Robert Englekirk Inc.; Design Architects: Elkus/Manfredi; Executive Architect: Kwan Henmi, Inc.; Owner: The Related Companies)

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Photo 1.18 Rendering of the New Maumee River Bridge, Toledo. The design includes a unique single pylon clad with glass emitting LED arrays at night, single plane of stays, and a main span of 612 feet in both directions. Courtesy of the designer, FIGG Engineers of Tallahassee, Florida.

2.33  1.23 kN>cm2 11.0  106 N>m2

 11.0 MPa 1C2 6 allowable fci  19.9 MPa, O.K.

2. Final condition at service load WSDL  6.13 kN>m MSDL 

6.13119.512 2 8

 292 kN-m

Total Moment MT = 343 + 292 = 635 kN − m. From Eq. 1.7a, f t  

Pe MT ect a1  2 b  t Ac S r 1024 37.5  15.8 541  102 kN-cm a1  b 2897 323 59,108 cm3

0.29  0.92 kN>cm2 6.3  106 N>m2

6.3 MPa 1C2 6 allow fc  18.6 MPa, O.K.

From Equation 1.7b, fb  

Pe ecb MT a1  2 b  Ac Sb r 1024 37.5  45.1 541  102 kN-cm a1  b 2897 323 20,713 cm3

 12.20  2.162  107 N>m2 4.1  106 N>m2

 4.1 MPa 1T2 6 allow. ft  2f c¿  6.4 MPa, O.K.

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SELECTED REFERENCES 1.1 Freyssinet, E. The Birth of Prestressing. London: Public Translation, Cement and Concrete Association, 1954. 1.2 Guyon, Y. Limit State Design of Prestressed Concrete, vol. 1. Halsted-Wiley, New York, 1972. 1.3 Gerwick, B. C., Jr. Construction of Prestressed Concrete Structures. Wiley-Interscience, New York, 1993, 591 p. 1.4 Lin, T. Y., and Burns, N. H. Design of Prestressed Concrete Structures. 3d ed. John Wiley & Sons, New York, 1981. 1.5 Nawy, E. G. Reinforced Concrete—A Fundamental Approach. 6th ed., Prentice-Hall, Upper Saddle River, NJ: 2009, 936 p. 1.6 Dobell, C. “Patents and Code Relating to Prestressed Concrete.” Journal of the American Concrete Institute 46, 1950, 713–724. 1.7 Naaman, A. E. Prestressed Concrete Analysis and Design. McGraw Hill, New York, 1982. 1.8 Dill, R. E. “Some Experience with Prestressed Steel in Small Concrete Units.” Journal of the American Concrete Institute 38, 1942, 165–168. 1.9 Institution of Structural Engineers. “First Report on Prestressed Concrete.” Journal of the Institution of Structural Engineers, September 1951. 1.10 Magnel, G. Prestressed Concrete. London: Cement and Concrete Association, 1948. 1.11 Abeles, P. W., and Bardhan-Roy, B. K. Prestressed Concrete Designer’s Handbook. 3d ed. Viewpoint Publications, London, 1981. 1.12 Nawy, E. G., Fundamentals of High Performance Concrete, 2nd ed. John Wiley & Sons, New York, 2001, pp. 460. 1.13 Nawy, E. G., editor-in-chief, Concrete Construction Engineering Handbook, 2nd ed., CRC Press, Boca Raton, FL, 2008, 1560 p.

PROBLEMS 1.1 An AASHTO prestressed simply supported I-beam has a span of 34 ft (10.4 m) and is 36 in. (91.4 cm) deep. Its cross section is shown in Figure P1.1. It is subjected to a live-load intensity WL = 3,600 plf (52.6 kN/m). Determine the required  -in.-dia stress-relieved seven-wire strands to resist the applied gravity load and the self-weight of the beam, assuming that the tendon eccentricity at midspan is ec = 13.12 in. (333 mm). Maximum permissible stresses are as follows: f c¿  6,000 psi 141.4 MPa2 fc  0.45f c¿

Figure P1.1

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29  2,700 psi 119.7 MPa2

ft  12 2f c¿  930 psi 16.4 MPa2

fpu  270,000 psi 11,862 MPa2 fpi  189,000 psi 11,303 MPa2

fpe  145,000 psi 11,000 MPa2 The section properties, are: Ac  369 in2 Ig  50,979 in4

r2  Ic>Ac  138 in2 cb  15.83 in. Sb  3,220 in3 St  2,527 in3 WD  384 plf WL  3,600 plf Solve the problem by each of the following methods: (a) Basic concept (b) C-line (c) Load balancing 1.2 Solve problem 1.1 for a 45 ft (13.7 m) span and a superimposed live load WL = 2,000 plf (29.2 kN/m). 1.3 A simply supported pretensioned pretopped double T-beam for a floor has a span of 70 ft (21.3 m) and the geometrical dimensions shown in Figure P1.3. It is subjected to a gravity live-load intensity WL = 480 plf (7 kN/m), and the prestressing tendon has an eccentricity at midspan of ec = 19.96 in. (494 mm). Compute the concrete extreme fiber stresses in this beam at transfer and at service load, and verify whether they are within the permissible limits. Assume that all permissible stresses and materials used are the same as in example 1.1. The section properties are: Section Properties Ac = 1185 in.2 Ig = 109,621 in.4 cb = 25.65 in. ct = 8.35 in.

Sb = St = WD = V/S =

4274 in.3 13,128 in.3 1234 plf 82 psf 2.45 in.

14'-113/4" 4"

3'-87/8"

7'-6"

2'-10" 2'-6"

Figure P1.3

3'-87/8"

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Design the prestressing steel needed using  -in.-dia stress-relieved seven-wire strands. Use the three methods of analysis discussed in this chapter in your solution. 1.4 A T-shaped simply supported beam has the cross section shown in Figure P1.4. It has a span of 36 ft (11 m), is loaded with a gravity live-load unit intensity WL = 2,500 plf (36.5 kN/), and is prestressed with twelve  -in.-dia (twelve 12.7-mm-dia) seven-wire stress-relieved strands. Compute the concrete fiber stresses at service load by each of the following methods: (a) Basic concept (b) C-line (c) Load balancing Assume that the tendon eccentricity at midspan is ec = 9.6 in. (244 mm). Then given that f c¿  5,000 psi 134.5 MPa2

ft  122f c¿  849 psi 15.9 MPa2

fpe  165,000 psi 11,138 MPa2 the section properties are as follows: Ac  504 in2 Ic  37,059 in.4

r2  Ic>Ac  73.5 in.2 cb  12.43 in. Sb  2,981 in.3 St  2,109 in.3 WD  525 plf ec  9.6 in. 1 Aps  twelve -in.-dia, seven-wire stress-relieved strands 2

Figure P1.4 1.5 Solve problem 1.4 if fc = 7,000 psi (48.3 MPa) and fpe = 160,000 psi (1,103 MPa).