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(c) Creep of concrete;
ation of initial stress level (0.7fpu or higher), type of steel
(stress-relieved or low-relaxation wire, strand, or bar), exposure
conditions, and type of construction (pretensioned, bonded
post-tensioned, or unbonded post-tensioned).
(d) Shrinkage of concrete;
(e) Relaxation of prestressing steel stress;
(f) Friction loss due to intended or unintended
curvature in post-tensioning tendons.
18.6.2 — Friction loss in post-tensioning tendons
R18.6.2 — Friction loss in post-tensioning tendons
18.6.2.1 — Ppx , force in post-tensioning tendons a
distance lpx from the jacking end shall be computed by
The coefficients tabulated in Table R18.6.2 give a range that
generally can be expected. Due to the many types of
prestressing steel ducts and sheathing available, these values
can only serve as a guide. Where rigid conduit is used, the
wobble coefficient K can be considered as zero. For largediameter prestressing steel in semirigid type conduit, the
wobble factor can also be considered zero. Values of the
coefficients to be used for the particular types of
prestressing steel and particular types of ducts should be
obtained from the manufacturers of the tendons. An unrealistically low evaluation of the friction loss can lead to
improper camber of the member and inadequate prestress.
Overestimation of the friction may result in extra
prestressing force. This could lead to excessive camber and
excessive shortening of a member. If the friction factors are
determined to be less than those assumed in the design, the
tendon stressing should be adjusted to give only that
prestressing force in the critical portions of the structure
required by the design.
– ( Kl px + μ p α px )
(18-1)
Where (Klpx + μpαpx) is not greater than 0.3, Ppx shall
be permitted to be computed by
Ppx = Ppj (1 + Klpx + μpαpx)–1
(18-2)
18.6.2.2 — Friction loss shall be based on experimentally determined wobble K and curvature μp friction
coefficients, and shall be verified during tendon
stressing operations.
TABLE R18.6.2 — FRICTION COEFFICIENTS
FOR POST-TENSIONED TENDONS FOR USE
IN EQ. (18-1) OR (18-2)
Mastic
coated
Unbonded tendons
Grouted tendons in
metal sheathing
Wobble
Curvature
coefficient, K per coefficient, μp per
radian
meter
Pregreased
Ppx = Ppj e
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Actual losses, greater or smaller than the computed values,
have little effect on the design strength of the member, but
affect service load behavior (deflections, camber, cracking
load) and connections. At service loads, overestimation of
prestress losses can be almost as detrimental as underestimation, since the former can result in excessive camber and
horizontal movement.
Wire tendons
0.0033-0.0049
0.15-0.25
High-strength bars
0.0003-0.0020
0.08-0.30
7-wire strand
0.0016-0.0066
0.15-0.25
Wire tendons
0.0033-0.0066
0.05-0.15
7-wire strand
0.0033-0.0066
0.05-0.15
Wire tendons
0.0010-0.0066
0.05-0.15
7-wire strand
0.0010-0.0066
0.05-0.15
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18.6.2.3 — Values of K and μp used in design shall
be shown on design drawings.
R18.6.2.3 — When the safety or serviceability of the
structure may be involved, the acceptable range of
prestressing steel jacking forces or other limiting requirements
should either be given or approved by the licensed design
professional in conformance with the permissible stresses of
18.4 and 18.5.
18.6.3 — Where loss of prestress in a member occurs
due to connection of the member to adjoining
construction, such loss of prestress shall be allowed
for in design.
18.7 — Flexural strength
R18.7 — Flexural strength
18.7.1 — Design moment strength of flexural
members shall be computed by the strength design
methods of the Code. For prestressing steel, fps shall
be substituted for fy in strength computations.
R18.7.1 — Design moment strength of prestressed flexural
members may be computed using strength equations similar
to those for nonprestressed concrete members. The 1983
Code provided strength equations for rectangular and
flanged sections, with tension reinforcement only and with
tension and compression reinforcement. When part of the
prestressing steel is in the compression zone, a method
based on applicable conditions of equilibrium and compatibility of strains at a factored load condition should be used.
For other cross sections, the design moment strength φMn is
computed by an analysis based on stress and strain
compatibility, using the stress-strain properties of the
prestressing steel and the assumptions given in 10.2.
18.7.2 — As an alternative to a more accurate
determination of fps based on strain compatibility, the
following approximate values of fps shall be permitted
to be used if fse is not less than 0.5fpu .
(a) For members with bonded tendons
⎧
f pu d
γp
- ρ p -------- + ------ ( ω – ω′ )
fps = fpu ⎪⎨ 1 – ----fc ′ dp
β1
⎪
⎩
⎫
⎪
⎬
⎪
⎭
(18-3)
where ω is ρfy /fc′ , ω′ is ρ′fy /fc′ , and γp is 0.55 for
fpy /fpu not less than 0.80; 0.40 for fpy /fpu not less
than 0.85; and 0.28 for fpy /fpu not less than 0.90.
If any compression reinforcement is taken into
account when calculating fps by Eq. (18-3), the term
f pu d
- + ------ ( ω – ω′ )
ρ p ------fc ′ dp
shall be taken not less than 0.17 and d ′ shall be no
greater than 0.15dp .
R18.7.2 — Equation (18-3) may underestimate the strength
of beams with high percentages of reinforcement and, for
more accurate evaluations of their strength, the strain
compatibility and equilibrium method should be used. Use
of Eq. (18-3) is appropriate when all of the prestressed
reinforcement is in the tension zone. When part of the
prestressed reinforcement is in the compression zone, a
strain compatibility and equilibrium method should be used.
By inclusion of the ω′ term, Eq. (18-3) reflects the increased
value of fps obtained when compression reinforcement is
provided in a beam with a large reinforcement index. When
the term [ρp (fpu /fc′ ) + (d/dp)(ω – ω′)] in Eq. (18-3) is small,
the neutral axis depth is small, the compressive reinforcement does not develop its yield strength, and Eq. (18-3)
becomes unconservative. This is the reason why the term
[ρp (fpu /fc′ ) + (d/dp)(ω – ω′)] in Eq. (18-3) may not be taken
less than 0.17 if compression reinforcement is taken into
account when computing fps. If the compression reinforcement is neglected when using Eq. (18-3), ω′ is taken as zero,
then the term [ρp(fpu /fc′ ) + (d/dp)ω] may be less than 0.17
and an increased and correct value of fps is obtained.
When d′ is large, the strain in compression reinforcement
can be considerably less than its yield strain. In such a case,
the compression reinforcement does not influence fps as
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(b) For members with unbonded tendons and with a
span-to-depth ratio of 35 or less:
f c′
f ps = f se + 70 + ---------------100ρ p
(18-4)
but fps in Eq. (18-4) shall not be taken greater than
the lesser of fpy and (fse + 420).
(c) For members with unbonded tendons and with a
span-to-depth ratio greater than 35:
f c′
f ps = f se + 70 + ---------------300ρ p
(18-5)
but fps in Eq. (18-5) shall not be taken greater than
the lesser of fpy and (fse + 210).
favorably as implied by Eq. (18-3). For this reason, the
applicability of Eq. (18-3) is limited to beams in which d′ is
less than or equal to 0.15dp.
The term [ρp (fpu /fc′ ) + (d/dp)(ω _ ω′)] in Eq. (18-3) may
also be written [ρp (fpu /fc′ ) + As fy /(bdp fc′) _ As′fy /(bdp fc′)].
This form may be more convenient, such as when there is no
unprestressed tension reinforcement.
Equation (18-5) reflects results of tests on members with
unbonded tendons and span-to-depth ratios greater than 35
(one-way slabs, flat plates, and flat slabs).18.10 These tests also
indicate that Eq. (18-4), formerly used for all span-depth ratios,
overestimates the amount of stress increase in such members.
Although these same tests indicate that the moment strength of
those shallow members designed using Eq. (18-4) meets the
factored load strength requirements, this reflects the effect of
the Code requirements for minimum bonded reinforcement, as
well as the limitation on concrete tensile stress that often
controls the amount of prestressing force provided.
18.7.3 — Nonprestressed reinforcement conforming to
3.5.3, if used with prestressing steel, shall be permitted
to be considered to contribute to the tensile force and
to be included in moment strength computations at a
stress equal to fy . Other nonprestressed reinforcement
shall be permitted to be included in strength computations only if a strain compatibility analysis is performed
to determine stresses in such reinforcement.
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18.8 — Limits for reinforcement of flexural
members
R18.8 — Limits for reinforcement of flexural
members
18.8.1 — Prestressed concrete sections shall be
classified as either tension-controlled, transition, or
compression-controlled sections, in accordance with
10.3.3 and 10.3.4. The appropriate strength reduction
factors, φ, from 9.3.2 shall apply.
R18.8.1 — The net tensile strain limits for compression- and
tension-controlled sections given in 10.3.3 and 10.3.4 apply
to prestressed sections. These provisions take the place of
maximum reinforcement limits used in the 1999 Code.
The net tensile strain limits for tension-controlled sections
given in 10.3.4 may also be stated in terms of ωp as defined
in the 1999 and earlier editions of the Code. The net tensile
strain limit of 0.005 corresponds to ωp = 0.32β1 for
prestressed rectangular sections.
18.8.2 — Total amount of prestressed and nonprestressed reinforcement in members with bonded
prestressed reinforcement shall be adequate to
develop a factored load at least 1.2 times the cracking
load computed on the basis of the modulus of rupture
fr specified in 9.5.2.3. This provision shall be permitted
to be waived for flexural members with shear and
flexural strength at least twice that required by 9.2.
R18.8.2 — This provision is a precaution against abrupt flexural failure developing immediately after cracking. A flexural
member designed according to Code provisions requires
considerable additional load beyond cracking to reach its
flexural strength. Thus, considerable deflection would warn
that the member strength is approaching. If the flexural
strength were reached shortly after cracking, the warning
deflection would not occur. Transfer of force between the
concrete and the prestressing steel, and abrupt flexural failure
immediately after cracking, does not occur when the
prestressing steel is unbonded18.11; therefore, this requirement does not apply to members with unbonded tendons.
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18.8.3 — Part or all of the bonded reinforcement
consisting of bars or tendons shall be provided as
close as practicable to the tension face in prestressed
flexural members. In members prestressed with
unbonded tendons, the minimum bonded reinforcement consisting of bars or tendons shall be as
required by 18.9.
R18.8.3 — Some bonded steel is required to be placed near
the tension face of prestressed flexural members. The
purpose of this bonded steel is to control cracking under full
service loads or overloads.
18.9 — Minimum bonded reinforcement
R18.9 — Minimum bonded reinforcement
18.9.1 — A minimum area of bonded reinforcement
shall be provided in all flexural members with unbonded
tendons as required by 18.9.2 and 18.9.3.
R18.9.1 — Some bonded reinforcement is required by the
Code in members prestressed with unbonded tendons to ensure
flexural performance at ultimate member strength, rather than
as a tied arch, and to limit crack width and spacing at service
load when concrete tensile stresses exceed the modulus of
rupture. Providing the minimum bonded reinforcement as
stipulated in 18.9 helps to ensure adequate performance.
Research has shown that unbonded post-tensioned members
do not inherently provide large capacity for energy dissipation
under severe earthquake loadings because the member
response is primarily elastic. For this reason, unbonded
post-tensioned structural elements reinforced in accordance
with the provisions of this section should be assumed to
carry only vertical loads and to act as horizontal diaphragms
between energy dissipating elements under earthquake loadings of the magnitude defined in 21.1.1. The minimum
bonded reinforcement areas required by Eq. (18-6) and (18-8)
are absolute minimum areas independent of grade of steel or
design yield strength.
18.9.2 — Except as provided in 18.9.3, minimum area
of bonded reinforcement shall be computed by
As = 0.004Act
(18-6)
where Act is area of that part of cross section between
the flexural tension face and center of gravity of gross
section.
R18.9.2 — The minimum amount of bonded reinforcement
for members other than two-way flat slab systems is based
on research comparing the behavior of bonded and
unbonded post-tensioned beams.18.12 Based on this
research, it is advisable to apply the provisions of 18.9.2
also to one-way slab systems.
18.9.2.1 — Bonded reinforcement required by Eq. (186) shall be uniformly distributed over precompressed
tensile zone as close as practicable to extreme tension
fiber.
18.9.2.2 — Bonded reinforcement shall be required
regardless of service load stress conditions.
18.9.3 — For two-way flat slab systems, minimum area
and distribution of bonded reinforcement shall be as
required in 18.9.3.1, 18.9.3.2, and 18.9.3.3.
R18.9.3 — The minimum amount of bonded reinforcement
in two-way flat slab systems is based on reports by Joint
ACI-ASCE Committee 423.18.6,18.11 Limited research available for two-way flat slabs with drop panels18.13 indicates
that behavior of these particular systems is similar to the
behavior of flat plates. Reference 18.11 was revised by
Committee 423 in 1983 to clarify that Section 18.9.3 applies
to two-way flat slab systems.
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