Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
1. Introduction: Concrete is basically a compressive material, with its strength in tension being relatively low. Prestressing applies a precompression to the member that reduces or eliminates undesirable tensile stresses that would otherwise be present, fig 1a and 1b. Cracking under service loads can be minimized or even avoided entirely. Deflections may be limited to an acceptable value; in fact, members can be designed to have zero deflection under the combined effects of service load and prestress force. Deflection and crack control, achieved through prestressing, permit the engineer to make use of efficient and economical high-strength steels in the form of strands, wires, or bars, in conjunction with concretes of much higher strength than normal. Thus, prestressing results in the overall improvement in performance of structural concrete used for ordinary loads and spans and extends the range of application far beyond the limits for ordinary reinforced concrete, leading not only to much longer spans than previously thought possible, but permitting innovative new structural forms to be employed.
2. Principle of prestressing: Many important features of prestressed concrete can be demonstrated by simple examples. Consider first the plain, unreinforced concrete beam with a rectangular cross section shown in Fig. 1a. It carries a single concentrated load at the center of its span. (The self-weight of the member will be neglected here). As the load W is gradually applied, longitudinal flexural stresses are induced. If the concrete is stressed only within its elastic range, the flexural stress distribution at mid span will be linear (Fig 1a). At a relatively low load, the tensile stress in the concrete at the bottom of the beam will reach the tensile strength of the concrete fr, and a crack will form. Because no restraint is provided against upward extension of the crack, the beam will collapse without further increase of load. Now consider an otherwise identical beam, shown in Fig. 1b, in which a longitudinal axial force P is introduced prior to the vertical loading. The longitudinal prestressing force will produce a uniform axial compression fc = P/Ac, where Ac is the cross-sectional area of the concrete. The force can be adjusted in magnitude so that, when the transverse load Q is applied, the Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
superposition of stresses due to P and Q will result in zero tensile stress at the bottom of the beam as shown (fig 1b). Tensile stress in the concrete may be eliminated in this way or reduced to a specified amount. It would be more logical to apply the prestressing force near the bottom of the beam, to compensate more effectively for the load-induced tension. A possible design specification, for example, might be to introduce the maximum compression at the bottom of the beam without causing tension at the top, when only the prestressing force acts. It is easily shown that, for a beam with a rectangular cross section, the point of application of the prestressing force should be at the lower third point of the section depth to achieve this. The force P, with the same value as before, but applied with eccentricity e = h/6 relative to the concrete centroid, will produce a longitudinal compressive stress distribution varying linearly from zero at the top surface to a maximum of 2fc = P/Ac + Pec2/Ic at the bottom, where fc is the concrete stress at the concrete centroid, c2 is the distance from the concrete centroid to the bottom of the beam, and Ic is the moment of inertia of the cross section. The stress at the bottom will be exactly twice the value produced before by axial prestressing. This is shown in fig. 1c and further explained, mathematically, below. Refer fig. 1c. For zero tension, in upper fibers, fc must be equal to ft1 and hence: fc = fc1. We know that, fc = P/A and, fc1 = ft1 = Mc2/I Where, M = Pe, I = bh3/12, and A = bh Now, when,
h
c2 b
fc = ft1 = P/A = Mc2/I; c2 = h/2 Therefore, P/A = {Pe Ă— h/2}/ (bh3/12) Or P/bh = peh(h/2) Ă— 12/(2bh3) = 6e/h = 1 e = h/6 Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
Page 2 of 19
Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
Therefore, zero tension in beam occurs only when prestressing force acts at e = h/6. Consequently, the transverse load can now be twice as great as before, or 2Q, and still cause no tensile stress. In fact, the final stress distribution resulting from the superposition of load and prestressing force in fig. 1c is identical to that of fig. 1b, with the same prestressing force, although the load is twice as great. The advantage of eccentric prestressing is obvious. For present purposes, it is sufficient to know that one practical method of prestressing uses high-strength steel tendons passing through a conduit embedded in the concrete beam. The tendon is anchored, under high tension, at both ends of the beam, thereby causing a longitudinal compressive stress in the concrete. The prestress force of fig. 1b and 1c could easily have been applied in this way. A significant improvement can be made, however, by using a prestressing tendon with variable eccentricity with respect to the concrete centroid, as shown in fig. 1d. The load 2Q produces a bending moment that varies linearly along the span, from zero at the supports to maximum at mid span. Intuitively, one suspects that the best arrangement of prestressing would produce a counter moment that acts in the opposite sense to the load-induced moment and that would vary in the same way. This would be achieved by giving the tendon an eccentricity that varies linearly, from zero at the supports to maximum at mid span. This is shown in fig. 1d. The stresses at mid span are the same as those in fig. 1c, both when the load 2Q acts and when it does not. At the supports, where only the prestress force with zero eccentricity acts, a uniform compression stresses fc is obtained as shown (fig 1d). For each characteristic load distribution, there is a best tendon profile that produces a prestress moment diagram that corresponds to that of the applied load. If the prestress counter moment is made exactly equal and opposite to the load-induced moment, the result is a beam that is subject only to uniform axial compressive stress in the concrete all along the span. Such a beam would be free of flexural cracking, and theoretically it would not be deflected up or down when that particular load is in place, compared to its position as originally cast. Such a result would be obtained for a load on ½ × 2Q = Q, as shown in Fig. 1e, for example.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
W f c h
(a)
fr Q h/2 (b)
P
fc= P/A c
fc
2fc
P fc
ft = f c
fc= P/A c
-ft1
0
2Q 2h/3 (c)
P
0 Prestressing force only
P
+fc1= fc
fc
2fc
0
2f c
2Q
(d)
P
h/2
2fc 2fc
0
2f = 2f c t Due to 2Q load 2fc
0
Final stress distribution
2f c
P 2f c h/3
0
2f = 2f c t Midspan
fc
fc
0 fc
Q
P
(e) P
h/2
fc
End fc
0
2fc h/3
fc
ft = f c
f c
Midspan f c
fc
0 fc
fc
End
Figure 1: Alternative schemes for prestressing a rectangular concrete beam. (a) Plain concrete beam, (b) Axially prestressed beam, (c) Eccentrically prestressed beam, (d) Beam with variable eccentricity, (e) Balanced load stage for beam with variable eccentricity. Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
Some important conclusions can be drawn from these simple examples as follows: i.
Prestressing can control or even eliminate concrete tensile stress for specified loads.
ii.
Eccentric prestress is usually much more efficient than concentric prestress.
iii.
Variable eccentricity is usually preferable to constant eccentricity, from the viewpoints of both stress control and deflection control.
3. Pre-stressing steel: 3a. Importance of high strength steel: Consider a short concrete member that is to be axially pre-stressed using a steel tendon, fig 2a. In the unstressed state, the concrete has length lc and the unstressed steel has length ls. After tensioning of the steel and transfer of force to the concrete through the end anchorages, the length of the concrete is shortened to l′c and the length of the stretched steel is l′s. These values must, of course, be identical, as indicated by the figure. fs , ksi
ls = Unstressed length of steel
150 ∆fs
lc = unstressed length of concrete
124
ls ' = l c ' = Stressed length of steel and concrete 30
∆fs 5.17
4.27
1.03
3
0.13
4 0
(εsh + εcu )lc
Figure 2: Effect of shrinkage and creep of concrete in reducing prestress force. (a) Axially prestressed concrete member, (b) Stress in steel.
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ε s x 10
Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
But the concrete experiences a shrinkage strain εsh with the passage of time and, in addition, if held under compression will suffer a creep strain εcu. The total length change in the member ∆lc = (εsh + εcu)lc……………………..(a) may be such that it exceeds the stretch in the steel that produced the initial stress, and complete loss of prestress force will result. The importance of shrinkage and creep strain can be minimized by using a very high initial strain and a high initial stress in the steel. The reduction in steel stress from these causes depends only on the unit strains in the concrete associated with shrinkage and creep, and the elastic modulus Es of the steel: ∆fs = (εsh + εcu)Es……………………(b) It is independent of the initial steel stress. It is informative to study the results of calculations of representative values of the various parameters. Suppose first that the member is prestressed using ordinary reinforcing steel at an initial stress fst of 30 ksi. The modulus of elasticity Es for all steels is about the same and will be taken here 29,000 ksi. The initial strain in the steel is: εsi = fsi/Es = 30/29000 = 1.03 x 10-3 And the steel elongation is: ∆s = εsls = 1.03 × 10-3ls………………………(c) But a conservative estimate of the sum of shrinkage and creep strain in the concrete is about 0.90 × 10-3, and the corresponding length change is: (εsh + εcu)lc = 0.90 × 10-3lc…………………...(d) Since ls and lc are nearly the same, it is clear by comparing (c) and (d) that the combined effects of shrinkage and creep of the concrete is almost a complete loss of the stress in steel. The effective steel stress remaining after time-dependent effects would be fse = (1.03 – 0.90) × 10-3 × 29 × 103 = 4 ksi Alternatively, suppose that the prestress were applied using high strength steel at an initial stress of 150 ksi. In this case the initial strain would be: εsi = 150/29000 = 5.17 × 10-3………………..(e) εsls = 5.17 × 10-3ls……………………………(f) Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
The length change resulting from the shrinkage and creep effects would be the same as before, (εsh + εcu)lc = 0.90 × 10-3lc And the effective steel stress fse after losses due to shrinkage and creep would be: fse = (5.17 – 0.90) × 10-3 × 29 × 103 = 124 ksi The loss is about 17 percent of the initial steel stress in this case compared with 87 percent loss when mild steel was used. The results of these calculations are shown graphically in fig. 2(b) and illustrate clearly the need in prestressing for using steel that is capable of a very high initial stress. Prestressing steel is used in three forms: round wires, stranded cable, and alloy steel bars. Prestressing wire ranges in diameter from 0.192 to 0.276 in. It is made by cold drawing high-carbon steel after which the wire is stress-relieved by heat treatment to produce the prescribed mechanical properties. Wires are normally bundled in groups of up to about 50 individual wires to produce prestressing tendons of the required strength. Stranded cable, more common than wire in U.S. practice, is fabricated with six wires wound around a seventh of slightly larger diameter. The pitch of the spiral winding is between 12 and 16 times the nominal diameter of the strand. Strand diameters range from 0.250 to 0.600 in. Alloy steel bars for prestressing are available in diameters from 0.750 to 1.375 in. as plain round bars and from 0.625 to 2.50 in. as deformed bars. The tensile strengths of prestressing steels range from about 2.5 to 6 times the yield strengths of commonly used reinforcing bars. The grade designations correspond to the minimum specified tensile strength in ksi. Round wires may be obtained in Grades 235, 240, and 250, depending on diameter. For the widely used seven-wire strand, three grades are available: Grade 250 (fpu = 250 ksi), Grade 270, and Grade 300, although the last is not yet recognized in ASTM A 421. Grade 270 strand is used most often. For alloy steel bars, two grades are used: the regular Grade 150 is most common, but special Grade 160 bars may be ordered.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
3b. Stress Strain curves: Figure 3 shows stress-strain curves for prestressing wires, strand, and alloy bars of various grades. For comparison, the stress-strain curve for a Grade 60 reinforcing bar is also shown. It is seen that, in contrast to reinforcing bars, prestressing steels do not show a sharp yield point or yield plateau; i.e., they do not yield at constant or nearly constant stress. Yielding develops gradually, and in the inelastic range the curve continues to rise smoothly until the tensile strength is reached. Because welldefined yielding is not observed in these steels, the yield strength is somewhat arbitrarily defined as the stress at a total elongation of 1 percent for strand and wire and at 0.7 percent for alloy steel bars. 300
2000
Grade 270 Strand Grade 250 Strand
Grade 250 Wire
1500 200
1000
Grade 150 Bar
Grade 60 Reinforcing Bar
100
500
1 % Extension 0.7 % Extension
0 0
50
Strain, 0.001
0 150
100
Figure 3: Typical stress strain curves for prestressing steel.
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MPa
Stress, ksi
Grade 160 Bar
Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
Figure 3 shows that the yield strengths so defined represent a good limit below which stress and strain are fairly proportional, and above which strain increases much more rapidly with increasing stress. It is also seen that the spread between tensile strength and yield strength is smaller in prestressing steels than in reinforcing steels. It may further be noted that prestressing steels have significantly less ductility. While the modulus of elasticity Es, for bar reinforcement is taken as 29,000,000 psi, the effective modulus of prestressing steel varies, depending on the type of steel (e.g., strand vs. wire or bars) and type of use, and is best determined by test or supplied by the manufacturer. For unbonded strand (i.e., strand not embedded in concrete), the modulus may be as low as 26,000,000 psi. For bonded strand, Es, is usually about 27,000,000 psi, while for smooth round wires Es, is about 29,000,000 psi, the same as for reinforcing bars. The elastic modulus of alloy steel bars is usually taken as Es, = 27,000,000 psi.
3c. Relaxation When prestressing steel is stressed to the levels that are customary during initial tensioning and at service loads, it exhibits a property known as relaxation. Relaxation is defined as the loss of stress in stressed material held at constant length. (The same basic phenomenon is known as creep when defined in terms of change in strain of a material under constant stress). To be specific, if a length of prestressing steel is stressed to a sizable fraction of its yield strength fpy (say 80 to 90 percent) and held at a constant strain between fixed points such as the ends of a beam, the steel stress fp will gradually decrease from its initial value fpi. In prestressed concrete members this stress relaxation is important because it modifies the internal stresses in the concrete and changes the deflections of the beam some time after initial prestress was applied. The amount of relaxation varies, depending on the type and grade of steel, the time under load, and the initial stress level. A satisfactory estimate for ordinary stress relieved strand and wires can be obtained from eq. (1), which was derived from more than 400 relaxation tests of up to 9 years duration:
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Department of Civil Engineering, N-W.F.P UET, Peshawar
fp fpi
1
log t 10
Prestressed Concrete
fp fpy
0.55 ....................(1)
Where, fp= final stress after t hours, fpi = initial stress, and fpy = nominal yield stress. In eq. (1), log t is to the base 10 and fpi/fpy not less than 0.55; below that value essentially no relaxation occurs. The tests on which eq. (1) is based were carried out on round, stress-relieved wires and are equally applicable to stress-relieved strand. In the absence of other information, results may be applied to alloy steel bars as well. Low-relaxation strand has replaced stress-relieved strand as the industry standard. According to ASTM A 416, such steel must exhibit relaxation after 1000 hours of not more than 2.5 percent when initially stressed to 70 percent of specified tensile strength and not more than 3.5 percent when loaded to 80 percent of tensile strength. For low-relaxation strand, eq. (1) is replaced by:
fp fpi
1
log t 45
fp fpy
0.55 ....................(2)
4. Concrete used for prestressed construction: For several reasons the concrete used for prestressed construction is characterized by a higher strength than that used for ordinary reinforced concrete. It is usually subjected to higher forces, and an increase in quality generally leads to more economical results. Use of high strength concrete permits the dimensions of member cross sections to be reduced to the minimum. Significant saving in dead load results in longer spans become technically and economically possible. Objectionable deflection and cracking, which would otherwise be associated with the use of slender members at high stress, are easily controlled by prestressing. There are other advantages. High strength concrete has a higher elastic modulus than the low strength concrete, so that loss of prestress force resulting from elastic Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
shortening of the concrete is reduced. Creep losses, which are roughly proportioned to elastic losses, are lower also. High bearing stresses in the vicinity of tendon anchorages for post tensioned members are more easily accommodated, and the size of expensive anchorage hardware can be reduced. In the case of pretensioned elements, higher bond strength results in a reduction in the development length required to transfer prestress force from the cables to the concrete. Finally, concrete of higher compressive strength also has a higher tensile strength so that the formation of flexural and diagonal tension crack is delayed. Figure 4 shows typical set of compressive stress strain curve for normal density concrete, obtained from uniaxial compressive test performed at normal, moderate testing speeds on concretes that are 28 days old. In present practice, compressive strength between 4, 000 and 8, 000 psi (28 and 55 MPa) is commonly specified for prestressed concrete members, although strengths as high as 12,000 psi (83 MPa) have been used. It should be emphasized, however, that the concrete strength assumed in the design calculations and specified must be attained with certainty, because the calculated high stresses resulting from prestress force really do occur. In recent years there has been a rapid growth of interest in high-strength concrete. Although the exact definition is arbitrary, the term generally refers to concrete having uniaxial compressive strength in the range of about 8000 to 15,000 psi or higher. Such concretes can be made using carefully selected but widely available cements, sands, and stone; certain admixtures including high-range water-reducing super plasticizers, fly ash, and silica fume; plus very careful quality control during production. In addition to higher strength in compression, most other engineering properties are improved, leading to use of the alternative term high-performance concrete. For bridges, too, smaller cross sections bring significant advantages, and the resulting reduction in dead load permits longer spans. The higher elastic modulus and lower creep coefficient result in reduced initial and long-term deflections, and in the case of prestressed concrete bridges, initial and time-dependent losses of prestress force are less. Other recent applications of high-strength concrete include offshore oil structures, parking garages, bridge deck overlays, dam spillways, warehouses, and heavy industrial slabs. Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
12
80
10
8
6
40
MPa
Compressive Stress fc', ksi
60
4 20 2
0
0.001
0.002
0.003
0 0.004
Strain Îľc
Figure 4: Typical compressive stress strain curves for normal density concrete with wc = 145 pcf.
5. Methods of prestressing: Although many methods have been used to produce the desired state of precompression in concrete members, all prestressed concrete members can be placed in one of two categories: pretensioned or post-tensioned. Pretensioned prestressed concrete members are produced by stretching the tendons between external anchorages before the concrete is placed. As the fresh concrete hardens, it bonds to the steel. When the concrete has reached the required strength, the jacking force is released, and the force is transferred by bond from steel to concrete. In the case of post-tensioned prestressed concrete members, the tendons are stressed after the concrete has hardened and achieved sufficient strength, by jacking against the concrete member itself.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
A. Pretensioning: The tendons, usually in the form of multiple-wire stranded cables, are stretched between abutments that are a permanent part of the plant facility, as shown in fig. 5a. The extension of the strands is measured, as well as the jacking force. Tendon Anchorage
Beam
Casting Bed
Support force Anchorage
Beam 1
Jack
(a) Tendon
Jack
Hold Down (b) Force Beam 2
Jack
(c)
Casting Bed
Figure 5: Method of pretensioning. (a) Beam with straight tendon, (b) beam with variable tendon eccentricity, (c) Long-line stressing and casting. With the forms in place, the concrete is cast around the stressed tendon. High early strength concrete is often used, together with steam curing to accelerate the hardening of the concrete. After sufficient strength is attained, the jacking pressure is released. The strands tend to shorten, but are prevented from doing so because they are bonded to the concrete. In this way, the prestress force is transferred to the concrete by bond, mostly near the ends of the beam, and no special anchorage is needed. It is often advantageous to vary the tendon eccentricity along a beam span. This can be done when pretensioning by holding down the strands at intermediate points and holding them up at the ends of the span, as shown in fig. 5b. One, two or three intermediate cable depressors are often used to obtain the desired profile. These hold-down devices remain embedded in the membrane. To minimize the Prof Dr. Qaisar Ali (http://www.eec.edu.pk)
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
frictional loss of tension, it is common practice to stretch the straight cable, then to depress it to the final profile by using auxiliary jacks. Allowance must be made, in this case, for the increase in tension as the cable is forced out of straight alignment. Pretensioning is well suited to the mass production of beams using the long-line method of prestressing as suggested by fig. 5c. In present practice anchorage and jacking abutments may be as much as 800 ft apart. The strands are tensioned over the full length of the casting bed at one time, after which a number of individual members are cast along the stressed tendon. When the jacking force is released, the prestress force is transferred to each member by bond, and the strands are cut free between members. Although a straight tendon is shown in the sketch, cable depressors are often used with long-line prestressing, just as with individual members. Pretensioning is a particularly economical method of prestressing, not only because the standardization of design permits reusable steel or fiberglass forms, but also because the simultaneous prestressing of many members at once results in great saving of labor. In addition, expensive end-anchorage hardware is eliminated. B. Post tensioning: While prestressing is applied by post-tensioning, usually hollow conduits containing the unstressed tendons are placed in the beam forms, to the desired profile, before pouring the concrete, as shown in figure 6a. The tendons may be bundled parallel wires, stranded cable, or solid steel rods. The conduit is wired to auxiliary beam reinforcement (unstressed stirrups) to prevent accidental displacement, and the concrete is poured. When it has gained sufficient strength, the concrete beam itself is used to provide the reaction for the stressing jack, as shown in the sketch. With the tendon anchored by special fittings at the far end of the member, it is stretched, and then anchored at the jacking end by similar fittings, and the jack removed. The tension is gauged by measuring both the jacking pressure and the elongation of the steel. The tendons are normally
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
tensioned one at a time, although each tendon may consist of many strands or wires. Beam
Anchorage
(a) Anchorage
Tendon in conduit Intermediate diaphragms
(b) Anchorage
Jack
Jack
Beam Jack
Slab
(c)
Wrapped tendon
Figure 6: Methods of post-tensioning. (a) Beam with hollow conduit embedded in concrete, (b) Hollow cellular beam with intermediate diaphragms, (c) Continuous slab with plastic-sheathed tendons. Tendons are normally grouted in their conduits after they are stressed. A cement paste grout is forced into the conduit at one end under high pressure, and pumping is continued until the grout appears at the far end of the tube. When it hardens, the grout bonds to the tendon and to the inner wall of the conduit, permitting transfer of force. Although the anchorage fittings remain in place to transfer the main prestressing force to the concrete, grouting improves the performance of the member should it be overloaded and increases its ultimate flexural strength. An alternative method of post-tensioning is illustrated in figure 6b. A hollow cellular concrete beam with solid end blocks and intermediate diaphragms is shown. Anchorage fittings are provided as before, but the tendons pass through the void spaces in the member. The desired cable profile is maintained by passing the steel through sleeves positioned in the intermediate diaphragms.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
In many cases, particularly in relatively thin slabs, post-tensioning tendons are wrapped with asphalt-impregnated paper or encased in plastic sheathing, as shown in figure 6c. Anchorage and jacking hardware is provided. The wrapping prevents the concrete from bonding to the steel. When the concrete has hardened, the tendons are stretched and anchored, and the jack removed. Obviously, bonding of the tendons by grouting is impossible with such an arrangement. Countless patented systems of post-tensioning are available, along with all necessary hardware. A significant advantage of all post-tensioning schemes is the ease with which the tendon eccentricity can be varied along the span to provide the desired counter moment.
Related Images
Figure 7: Anchor blocks and wedges.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
Figure 8: Post tensioning under progress.
Figure 9: Post tensioning under progress
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
Figure 10: Post tensioning.
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Department of Civil Engineering, N-W.F.P UET, Peshawar
Prestressed Concrete
References 他 Design of Concrete Structures by Nilson, Darwin and Dolan (13th Ed.) 他 Design of Prestressed Concrete (2nd ed. by Arthur H. Nilson.)
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