CE 429ST
PRESTRESSED CONCRETE DESIGN
PRE STRESSED DESIGN
Basic Principle of Prestressing Beams
Prestressed Concrete – is defined as a structural concrete in which internal
stresses have been introduced to reduce potential tensile stresses in concrete due
to the imposed loads.
Method of Prestressing:
1. Pre tensioning – is a method of prestressing in which tendons are
tensioned before concrete is placed.
2. Post tensioning – is a method of prestressing in which tendons are
tensioned after concrete has hardened. This can be applied to members
either pre cast or cast in place.
When post tensioned, the tendons are anchored at their ends by means of
mechanical devices (hydraulic jack) to transmit the prestress to the concrete, such
a member is termed as end anchored.
In a post tensioning, the tendons generally have their prestress transmitted to the
concrete by their bond action near the ends. The effectiveness of such stress
transmission is limited to wires of small size and to larger diameter strands which
possess better bond properties than smooth wires.
Tendons is a prestressing steel used in pre tensioned applications. In post-tension
applications the tendon is a complete, assembly consisting of anchorages,
prestressing steel, and sheathing with coating for unbonded applications or ducts
with grout for bonded application.
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Types of Tendons
1. Bonded Tendons – it is a tendon that is bonded throughout their length to
the surrounding concrete. Non end-anchored tendons are necessarily
bonded ones. End-anchored tendons maybe either bonded or unbonded to
the concrete. The prestressing tendons are bonded to the concrete directly
or through grouting.
2. Unbonded Tendons – it is a tendon in which the prestressing steel is
prevented from bonding to the concrete and is free to move relative to the
concrete. The prestressing force is permanently transferred to the concrete
at the tendon ends by anchorage only.
Common Method of Stressing the Tendons
For both pre tensioning and post tensioning the tendons are stress by jacking. In
post tensioning, jacks are used to pull the steel with the reaction acting against
the hardened concrete, in pre tensioning jacks pull the steel with the reaction
against end bulkheads. Hydraulic jacks are used because of their high capacity to
apply the pressure. Care must be taken to see that the jack can be properly
mounted at the end of the bearing plates and that there is enough room at the
tensioning ends to accommodate the jacks.
Loss of Prestress
To determine the effective prestress, allowance for the following sources of loss
of prestress shall be considered.
1. Tendon seating at transfer.
2. Elastic shortening of concrete.
3. Creep of concrete.
4. Shrinkage of concrete.
5. Relaxation of tendons stress.
6. Friction losses due to intended or unintended curvature in post-tensioning
tendons.
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Permissible Stresses in Concrete (Flexural members)
Ultimate Stress Design
A. Stresses in concrete immediately after prestress transfer (before time-
dependent prestress losses)
1. Extreme fiber stress in compression except as permitted by
NSCP shall not exceed……………………………………………………… 0.60fci’
2. Extreme fiber stress in compression at ends of simply supported
members shall not exceed ……………………………………………… 0.70fci’
3. Where computed concrete tensile strength, ft, exceeds 0.5√fc’
at ends of simply supported members, or 0.25√fc’ at other
locations, additional bonded reinforcement shall be
provided in the tensile zone to resist the total tensile force in
concrete computed with the assumption of an uncracked section.
B. Stresses in concrete at service loads based on uncracked section properties
and after allowance for all pre stress losses shall not exceed the ff:
1. Extreme fibers stress in compression due to prestress plus
sustained loads …………………………………………………………….. 0.45fc’
2. Extreme fiber stress in compression due to prestress plus
total load …………………………………………………………………….. 0.60fc’
Permissible Stress in Prestressing Tendons
Tensile stress in prestressing tendons shall not exceed the following:
1. Due to prestressing tendon jacking force ……………………. 0.94fpy
but not greater than the lesser of 0.80fpu and the maximum
value recommended by manufacturer of prestressing tendons
or anchorage devices.
2. Immediately after prestress transfer…………………………… 0.82fpy
but not greater than ………………………………………………….. 0.74fpu
3. Post-tensioning tendons, at anchorage devices and
couplers, immediately after force transfer ………………… 0.70fpu
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Notations in Prestressed Concrete
Aps = area of prestressed reinforcement in tension zone, mm2
dp = distance from extreme compression fiber to centroid of prestressed
reinforcement, mm
fci’ = compressive strength of concrete at time of initial prestress, MPa
fpc = average compressive stress in concrete due to effective prestress force only
(after allowance for all prestress losses), MPa
fps = stress in prestressed reinforcement at nominal strength, MPa
fpu = specified tensile strength of prestressing tendons, MPa
fpy = specified yield strength of prestressing tendons, MPa
fse = effective stress in prestressed reinforcement (after allowance for all
prestress losses), MPa
lx = length of prestressing tendon element from jacking end to any point x
n = number of monostrand anchorage at jacking end
Ps = prestressing tendon force at jacking end
Px = prestressing tendon force at any point x
Psu = factored post-tensioned tendon force at the anchorage device
α = total angular change of prestressing tendon profile in radians from tendon
jacking end to any point x
γp = factor for type of prestressing tendon
= 0.55 for fpy/fpu not less than 0.80
= 0.40 for fpy/fpu not less than 0.85
= 0.28 for fpy/fpu not less than 0.90
μ = curvature friction coefficient
ω = рfy/fc’
ω’ = р’fy/fc’
ωp = рpfps/fc’
ωw, ωpw,ω’w = reinforcement indices for flanged sections computed as for ω,ωp,
and ω’ except that b shall be the web width, and reinforcement are shall be
that required to develop compressive strength of web only.
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Ultimate Stress Design
Determination of stress in prestressed reinforcement at nominal strength (fps) as
an alternative to a more accurate determination based on strain compatibility,
the following approximate values shall be permitted to be used if fse is not less
than 0.5fpu:
1. For members with bonded tendons:
fps = fpu{1- γp/β1[рp(fpu/fc’) + (d/dp)(ω –ω’)]}
if any combination reinforcement is taken into account when calculating fps,
the term:
[pp(fpu/fc’) + (d/dp)(ω –ω’)]
shall be taken not less than 0.17 and d’ shall be no greater than 0.15dp
where:
β1 = 0.85 if fc’ is less than 28 MPa
β = 0.85 – (0.05/7)(fc’ – 28) but not less than 0.65
γp = factor for type of prestressing tendon
γp = 0.55 for fpy/fpu not less than 0.80
γp = 0.40 for fpy/fpu not less than 0.85
γp = 0.28 for fpy/fpu not less than 0.90
рp = Aps/bd
dp = distance from extreme compression fiber to centroid of prestressed
reinforcement, mm
d = distance from extreme compression fiber to centroid of nonprestressed
tension reinforcement, mm.
ω = рfy/fc’
ω’ = р’fy/fc’
2. For members with unbonded tendons:
a. For members with unbonded tendons and with a span-to-depth ratio of
35 or less:
fps = fse + 70 + fc’/100рp
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but fps shall not be taken greater than fpy, nor greater than (fse + 415)
b. For members with unbonded prestressing tendons and with a span-to-
depth ratio greater than 35:
fps = fse + 70 + fc’/300рp
but fps shall not be taken greater than fpy, nor greater than (fse + 210)
Minimum Bonded Reinforcement
1. A minimum area of bonded reinforcement shall be taken as:
As = 0.004Act
Where : Act = area of that part of cross section between flexural tension
face and center of gravity of gross section.
2. Bonded reinforcement shall not be required in positive moment areas
where ft, the extreme fiber stress in tension in the pre-compressed
tensile zone at service load (after allowance for prestress losses) does
not exceed 0.17√fci’
3. In positive moment areas where computed tensile stress in concrete at
service load exceed 0.17√fci’ minimum area bonded reinforcement shall
be computed by:
As = Nc/0.5fy
Where: fy should be less than 415 MPa
Nc = tensile force in concrete due to unfactored dead load plus live load
4. In negative moment areas at column supports, minimum area of bonded
reinforcement As in the top of the slab in each direction shall be
computed by:
As = 0.00075Act
Where : Act = is the larger gross cross-sectional area of the slab-beam strips
in two orthogonal equivalent frames intersecting at a column in a two-way
slab.
Bonded reinforcement required shall be distributed between lines that are
1.5h outside opposite faces of the column support. At least four bars or
wires shall be provided in each direction. Spacing of bonded reinforcement
shall not exceed 300 mm.
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5. In positive moment areas, minimum length of bonded reinforcement
shall one-third the clear span length ln, and centered in positive moment
area.
6. In negative moment areas, bonded reinforcement shall extend one-sixth
the clear span, ln, on each side of support.
Reinforcement Index (ωp)
1) When ωp ,< 0.36β
C=T
0.85fc’ab = Apsfps
a = Apsfps/0.85fc’b
Mu = φT(d – a/2)
Mu = φApsfps(d – a/2)
2) Use ωp > 0.36β
a = Apsfps/0.85fc’b
a = ρpbdfps/0.85fc’b
a = ρpfpsd/0.85fc’
a = ωp/0.85
a = 0.36βd/0.85
a = 0.423β1d
Mu = φC(d – a/2)
Mu = φ0.85fc’ab(d – a/2)
