Chapter 5 Post-Tensioned Flat Slab
GENERAL
Post-tensioned concrete has been used for more than 40 years in the United States in a wide
variety of construction projects. First used primarily in bridge construction, applications for
post-tensioning now extend far beyond bridges to include tanks, office buildings, hotels,
parking structures, pavement, masonry, seismic walls, single-family homes and more.
They can be designed as two-way spanning flat slabs, one-way spanning ribbed slabs, or as
banded beam and slab construction. Flat slabs are supported, without the use of beams, by
columns with or without column heads. They may be solid or may have recesses formed in
the soffit to create a series of ribs running in two directions (waffle or coffered slab). The
design principles of continuous flat slab floors are similar to those of two-way reinforced
concrete slabs. A strip of slab of unit width, continuous over supports, is analyzed as a
continuous beam. Prestressing of continuous slab results in secondary moments. If the cable
profile is concordant, secondary moments can be eliminated. Since 1955 a number of number
of continuous flat slab have been built in U.S.A, in which unbounded tendons are to be
preferred both from the point of view of ultimate strength requirements and easy
maintenance under adverse exposure conditions.
The design of a continuous flat slab floors involves the computation of maximum and
minimum moments for various load combinations and the determination of suitable cable
profiles so that the resulting stresses in concrete are within the safe allowable limits as per
codes. Shear stresses at the junction of the column and slab should be carefully controlled by
proper design and detailing of the critical shear zones.
5.1 POST-TENSIONING SYSTEMS
In the U.S. and Canada, post-tensioned buildings and parking garages are typically
constructed with seven-wire, 12.7 mm diameter, un-bonded single-strand (monostrand)
Design of Reinforced Concrete & Post-Tensioned flat slab using software 1
Chapter 5 post-tensioned flat slab
tendons. These tendons, with a typical strength of 1860 MPa, are also greased and sheathed.
One reason for the widespread use of the 12.7 mm diameter strand is the Code requirement
that the tendon spacing not be greater than eight times the slab thickness. The use of 12.7 mm
diameter, 1860 MPa strand permits 110 and 125 mm slabs to meet both the minimum 0.85
MPa average pre-compression and the maximum tendon spacing requirement. In addition,
the tendons and stressing equipment are light enough for workers to handle them efficiently
on site. Larger diameter 15.3 mm strands are primarily used in pre-tensioning and bridge
construction. Higher strength steels and smaller diameter strands are also available but are
not commonly used for new construction.
5.2 STRUCTURAL MODELING
Several methods of floor slab analysis and design
1. Direct Design Method (DDM)
2. Equivalent Frame Method (EFM)
3. Strip Method
4. Closed Form Solution and Approximations
5. Finite Difference Method (FDM)
6. Finite Element Method (FEM)
7. Yield Line Method (YLM)
8. Experimental Techniques
9. Strut-And-Tie Method
In both one- and two-way systems, specifying the structural model includes defining the
design strips, irrespective of whether an FEM or Equivalent Frame Method of analysis is
used. Column-supported floors generally qualify as two-way systems; beam- and wall-
supported slabs and beams generally qualify as one-way systems.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 2
Chapter 5 post-tensioned flat slab
The fixity of the connections must also be specified. In some instances, such as corner
columns in flat slabs, the assumption of full fixity does not yield a satisfactory design. For
structural analysis, such connections may be assigned partial fixity or may be assumed as
hinged connections (releases). Connections that are assumed to be hinged must be detailed in
the construction documents to allow rotation, while retaining the integrity of the joint by
limiting crack width and transfer of axial and shear forces through the joint. Another instance
where a hinge connection may be beneficial is for short gravity columns at split levels in
parking structures, which have a ramp on one side and a level floor on the other side.
5.3 DESIGN GUIDELINES
There is a major difference between the design of a post-tensioned member and the design of
a conventionally reinforced concrete member. Once the geometry, loading, support
conditions, and material properties of a conventionally reinforced member are established, a
unique solution of the required area of reinforcement, As, is given by a formula.
For a post tensioned member, there are a number of acceptable reinforcement designs
because there are several additional parameters that must be specified by the engineer. These
parameters may be grouped as follows:
Average pre-compression (prestressing force);
Percentage of load to balance (uplift due to tendon drape); and
Tendon profile (shape and drape).
From the many possible design solutions for a post-tensioned member, the one that meets
the Code requirements for serviceability and strength and is the least expensive to build is
usually the preferred solution. Generally, for a given slab dimension, loading, and
construction method, less material means a more economical design. Values for the three
parameters listed above must be established before the required amount of post-tensioning
can be determined. The amount of supplemental reinforcement as required for strength
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 3
Chapter 5 post-tensioned flat slab
design of the post-tensioning member is determined by the amount of the post-tensioning
reinforcement and the reinforcement profile. The typical ranges of spans for the
various forms of construction and their corresponding slab and beam depths
are shown here.
Fig. 5.1 Typical economical spans with different types of slab
Table 1 Typical slab and beam depths
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 4
Chapter 5 post-tensioned flat slab
5.3.1 AVERAGE PRE-COMPRESSION
The average pre-compression is the total post tensioning force divided by the gross cross-
sectional area normal to the force. ACI 318 requires a minimum 0.85 MPa effective pre-
compression (pre-compression after all prestress losses). In general, 0.85 MPa should be used
for the initial average pre-compression. For roofs and parking structures, use 1.0 to 1.4 MPa
if water tightness or cracking is a concern. however, an increase in pre-compression does not
guarantee water tightness and may not completely eliminate cracking. To avoid leakage, the
increased post-tensioning must be supplemented by other measures, such as a membrane
overlay. In one-way slab and beam construction, the member is defined as the beam and its
tributary slab area. Maximum pre-compression should be 2.0 MPa for slabs and 2.50 MPa for
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 5
Chapter 5 post-tensioned flat slab
beams; although the Code’s limit of maximum compressive stress is much higher, values
higher than these typically mean the design will be less economical.
Fig. 5.2 Tributaries used when computing average pre-compression
5.3.2 PERCENTAGE OF LOAD TO BALANCE
This is expressed as the ratio (percentage) of the dead load that is balanced. For slabs, it is
customary to balance between 60 and 80% of the dead load. For beams, this is usually
increased to between 80 and 110%. One reason for higher balanced loading for the beams is
that beam deflection is more critical to service performance of a floor system. To determine
the required post-tensioning force, start with the critical span. For the spans adjacent to the
critical span, a lower percentage of the dead load should generally be balanced because less
upward force in an adjacent span helps to reduce the design values of the critical span.
Balancing all the spans of a continuous member to the same percentage of dead load is not
always economical. In practice, tendon profiles are reversed parabolas; such example is
shown in Figure 2. Tendons thus exert both upward and downward forces in the same span.
In such cases and for the purpose of design, the percentage of dead load balanced is
considered as the sum of the upward forces divided by the total dead load (DL) on the span.
For the design shown in Fig. 4, this becomes: % of DL balanced = 100[(W2+W3)/DL]
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 6
Chapter 5 post-tensioned flat slab
Fig. 5.3 Post-tensioning tendon of reversed parabola shape, exert both an
upward and a downward force in the same span
5.3.3 TENDON PROFILE: SHAPE
For beam tendons and slab tendons in the distributed direction, a reversed parabola tendon
profile with inflection points at one-tenth of the span length (Fig. 5.4) is typically used. For
an exterior span, the tendon is at the mid depth of the slab at the slab edge and at its high
point (typically somewhat higher than mid depth) at the other end. Moving the tendon low-
point to 0.4L, results in a more uniform uplift over the exterior span.
Fig. 5.4 The tendon profile, reversed parabola with inflection
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 7
Chapter 5 post-tensioned flat slab
5.3.4 TENDON PROFILE: DRAPE
The high point of the tendon profile should be as close to the top surface of the member as
practical, allowing for clearance and reinforcement in the orthogonal direction, if necessary.
At the low point of the profile, it is best to place the tendons as close to the soffit of the
member as allowable, to take full advantage of the uplift and contribution to strength that the
tendon can provide. This arrangement is possible for the critical spans in a continuous
member, but may need to be adjusted for other spans. Maximum drape results in excessive
uplift in a span other than the critical span, the first choice should be to reduce the
prestressing force. Tendons should be anchored at the centroid of the slab even if there is a
transverse beam or drop cap/panel at the slab edge (fig. 5.5)
Fig. 5.5 Anchorage at exterior support
5.4 ANCHOR LOCATIONS
Tendons in stand-alone beams (beams not cast monolithically with the slab) should be
anchored at the centroid of the beam. Tendons in flanged beams such as in one-way slab and
beam structures should be anchored at the centroid of the combined beam stem and its
tributary. In the traditional load- balancing analysis used by most designers, the force in the
tendons, generally considered constant, is represented by an axial force, P, at a location that
results in “uniform pre-compression”, if a force is acting at the centroid of a member, it will
disperse in to a uniform compression at a distance “sufficiently far” from the point of
application of the force.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 8
Chapter 5 post-tensioned flat slab
5.5 ADDITIONAL CONSIDERATIONS
5.5.1 COVER FOR FIRE RESISTANCE
When determining fire ratings, designers typically consider the end spans in column-
supported structures unrestrained. To achieve fire resistance equal to that of interior spans,
provide a larger cover for tendons at the low point of exterior spans unless the end support is
a wall or transverse edge beam. Only the first and last spans of tendons along a slab edge are
considered as “end spans”.
5.5.2 TENDON LAYOUT
The preferred tendon layout for two-way slabs is to concentrate the tendons over the supports
in one direction (the banded tendons) and distribute them uniformly in the other direction.
Place the distributed tendons in the orthogonal direction, parallel to one another, making sure
that a minimum of two tendons pass over each support as required by ACI 318-02.
5.5.3 TENDON STRESSING
Most engineers in North America design with final effective forces—the post-tensioning
forces after all prestress losses. The post-tensioning supplier determines the number of
tendons required to provide the force shown on the structural drawings, based on the
effective force of a tendon. The effective force of a tendon is a function of a number of
parameters, including the tendon profile, certain properties of the concrete, and the
environment. For typical designs, however, a constant force of 120 kN may be assumed for
12.7 mm. Un-bonded tendons shall meet the following conditions for stressing.
Tendon length (length between anchorages) shall less than 72 m.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 9
Chapter 5 post-tensioned flat slab
Tendons less than 36 m long are stressed at one end; and
Tendons longer than 36 m but less than 72 m are stressed at both ends.
Tendons that do not meet these conditions may be used, as long as the assumed effective
force is lowered to account for the higher friction losses.
5.6 OTHER DESIGN CONSIDERATION
5.6.1 SEISMIC DESIGN
Although seismic design technology is not currently as well developed for post-tensioned
structures as for some other structural systems, a number of experimental research projects
have been completed which indicate that energy dissipation characteristics conforming with
accepted standards can be achieved by appropriate combinations of prestressed and non-
prestressed reinforcement. The preliminary report on the moment transfer tests at the
University of Washington sponsored by the Post- Tensioning Institute and the Reinforced
Concrete Research Council states: "It is apparent that prestressing could provide an excellent
means for tieing a slab together and ensuring ductile behavior for seismic loading. However,
reversed cyclic large edge deflection tests are necessary to validate that potential and develop
the rules necessary to ensure ductile behavior" (Fig. 5.6).
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 10
Chapter 5 post-tensioned flat slab
Fig. 5.6 Comparison of Lateral Load-Edge Deflection Relationships for Reinforced and
Prestressed Concrete Slab - Interior Column Specimens
Graph shows that after low intensity reversed cyclic loading one of the post-tensioned slab
specimens could still develop a lateral load - edge deflection relationship for unidirectional
loading paring very favorably with the same relationship for a similarly loaded reinforced
concrete slab with integral beam stirrups. For the prestressed slab, the cracking that
developed at the column face during the first cycle between maximum lateral loads of two
kips resulted in a permanent edge deflection for zero lateral load. However, for the second
and third cycles to the same maximum lateral loads, and even when the lateral loads were
increased to three kips, the hysteresis loops remained stable and spindle shaped. Further,
there was no degradation in stiffness with cycling. Use of post-tensioned flexural systems
with un-bonded tendons in areas of high seismicity has sometimes been questioned because
the tendons may not be stressed beyond the elastic range, even in a severe earthquake, and,
for this reason, the tendons do not dissipate much energy. This objection may be overcome
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 11
Chapter 5 post-tensioned flat slab
when the slabs are an integral part of the moment resisting frame by use of a combination of
un-bonded tendons and non-prestressed bonded reinforcement as discussed above. However,
in most post-tensioned structures in seismic zones the post-tensioned elements are not part of
the ductile moment resisting space frame, or the seismic resistance is provided by shear walls
or some other force resisting system. For such cases, Uniform Building Code 1976 Edition,
Section 2314 states: " framing elements not required by design to be part of the lateral force
resisting system shall be investigated and shown adequate for vertical load carrying capacity
and induced moment due to 3/K times the distortions resulting from the Code required lateral
forces." K is the horizontal force factor. It has been shown that under reinforced prestressed
concrete members can meet the ductility requirements of the above Uniform Building Code
provision. Also suggests" that in the range of seismic disturbances slightly greater than
moderate, prestressed concrete can provide the necessary deformability and still respond
almost elastically." The elastic response results from the fact that it is not possible for the
stress in an un-bonded tendon to reach the yield point within the deformation range
anticipated in a moderately severe earthquake. A significant practical benefit of this is that
permanent structural damage to the floor system will be greatly reduced in comparison to
other structural systems.
Anchorage systems used for un-bonded tendons are required to meet static and dynamic test
requirements much more severe than the loadings that would be anticipated in an earthquake
of high intensity. These specifications also require the anchorage system to withstand,
without failure, 500,000 cycles from 60 to 66 percent of the minimum specified ultimate
strength of the tendon material.
In addition to the satisfactory performance of post-tensioned specimens subjected to
earthquake loadings in laboratory research projects, post-tensioned structures have responded
well in numerous earthquakes in the United States and in other countries. Many post-
tensioned structures have been built in areas of 'high seismicity and many have been
subjected to earthquakes of significant magnitude. A post-tensioned ductile moment resisting
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 12
Chapter 5 post-tensioned flat slab
space frame with un-bonded tendons in combination with bonded non-prestressed
reinforcement, as well as other post-tensioned buildings, performed very well in the 1971
San Fernando earthquake in California. In summary, both test results and performance in
earthquakes indicate advantages for the use of un-bonded tendons for structures in areas of
high seismicity, particularly when used in conjunction with shear walls or other lateral force
resisting elements.
5.7 SELECTION OF NON-PRESTRESSED REINFORCING
5.7.1 BAR SIZE
To take full advantage of the maximum lever arm for reinforcement in both directions, the
top bar diameters should match those of the adjacent tendons. Thus, it is reasonable to use 16
mm bars over the supports, a sheathed 12.7 mm diameter strand is slightly larger than a 16
mm bar. For bottom bars, it is better to use smaller bars, such as 12 mm bars, for the
distributed tendon direction because these are distributed uniformly among the tendons, and
larger bars for the banded tendons because bars for the banded direction are normally
grouped together and placed within the band width.
5.8 ANALYSIS TOOL
For design and analysis of Post-tensioned flat slab, ADAPT-6.15 software is used. Three
span continuous two-way flat slab geometry is used for the analysis. Following points and
values of various parameters were used for the analysis and design with ADAPT software.
Clear span (from face of support) is used to reduce moments.
Equivalent frame modeling is used for analysis and design.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 13
Chapter 5 post-tensioned flat slab
Including self weight of slab, 2 kN/m2 floor finish load and 5 kN/ m2 live load is
taken.
5.8.1 MATERIAL
28 days compressive strength at is 30 N/mm2 :- M30
Ultimate creep co-efficient :- 2
Steel reinforcement grade :- Fe415
For post-tensioning tendon, 12.7 mm diameter – 7wires strand is used.
Ultimate strength of tendon :- 1860 N/mm2
Effective stress of tendon :- 1200 N/mm2
Minimum strand cover from top fiber 25mm
Minimum strand cover from bottom fiber 25mm
5.8.2 Allowable stresses
Initial tensile stress of top fiber at time of jacking (transfer) :- 0.25√ f ci'
Final tensile stress of top fiber at time of service load :- 0.5√ f c'
Initial tensile stress of bottom fiber at time of jacking (transfer) :- 0.25√ f ci'.
Final tensile stress of bottom fiber at time of service load :- 0.5√ f c'
Initial compression stress of top fiber at time of jacking (transfer) :- 0.6f ci'.
Final tensile stress of top fiber at time of service load :- 0.45f c'.
5.8.3 VALUES FOR POST-TENSIONING
Minimum average pre compression :- 0.86 N/mm2
Maximum average pre compression :- 2.07 N/mm2
Minimum percentage of dead load to balance :- 25%
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 14
Chapter 5 post-tensioned flat slab
Maximum percentage of dead load to balance :- 100%
5.8.4 VALUES FOR CALCULATION OF FRICTION STRESS LOSSES
Ratio of jacking stress to ultimate stress :- 0.8
Strands modulus of elasticity :- 2 x 10^5 N/mm2
Angular coefficient of friction Mu :- 0.07 /rad.
Wobble co-efficient of friction k :- 0.0046 rad/m
Anchor set :- 6 mm.
5.8.5 CALCULATION OF LONG TERM STRESS LOSS
Age of concrete at stressing :- 5 days.
Concrete modulus of elasticity :- 20000 N/mm2
5.8.6 LOAD COMBINATIONS
Service combination factors for DL, LL and prestress ( balanced loading) :-
1 DL + 1 LL + 1 prestress.
Strength combination factors for DL, LL and hyperstatic (secondary) actions:-
1.5 DL + 1.5 LL + 1 hyp.
Analysis and design is done by ADAPT-6.0 software with the use of design code ACI 318.
Here explanation of software shows, how input data are to be entered to the software. At
starting of software window, it asks general settings of title. Two way slab structural system
is used for the analysis. equivalent frame modeling is used for design. After giving structural
system and design settings the next window will be of span geometry (fig. 5.7), this screen is
used to enter the cross-sectional geometry of the slab at mid span. All dimensions are defined
in the legend at the top of the screen and illustrated in the appropriate section figure. The
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 15
Chapter 5 post-tensioned flat slab
section span can be changed by clicking on the section column. We can use the “Typical”
input row to enter similar dimensions. Type the values in to the appropriate cell in the top
row and then press enter. The typical value will be copied to all the spans.
Fig. 5.7 Window showing input data of span geometry in software
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 16
Chapter 5 post-tensioned flat slab
Fig. 5.8 Window showing input data of support geometry in software
As shown in fig. 5.8. This screen is used to input Column/wall height, widths and depths.
After entering support condition the next window will be of boundary condition. Design of
post-tensioned flat slab is done with no end support fixity. Boundary condition for the
column is assumed as upper and lower columns are fixed. For the input of loading
information, software gives the various types of loading pattern which we can directly entre
the value of loading by simply dragging the loading type indicated with graphics icon
(fig.5.9).
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 17
Chapter 5 post-tensioned flat slab
Fig. 5.9 Window showing input data of loading in software
It is required to provide the material property for concrete as 28 days compressive strength of
concrete for slab as well as for column and also and ultimate creep coefficient. Then it is also
necessary to input the material property of reinforcement, post-tensioning system whether
bonded or un-boned, area of tendon, allowable stresses and post-tensioning values to the
software as listed above. Software will ask to specify the tendon profile among three types,
reversed parabola, partial parabola and harped parabola (Fig. 5.10). X1/L and X3/L are the
inflection points.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 18
Chapter 5 post-tensioned flat slab
Fig. 5.10 Window showing criteria of tendon profile in software
Then specify minimum covers for post-tensioning tendons and mild steel reinforcement, also
specify minimum bar length and bar extension of mild steel reinforcement and load
combinations. Finally we can see the 3D model view of strip of slab of input data as shown
in Fig. 5.11, Fig. 5.12 and Fig. 5.13.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 19
Chapter 5 post-tensioned flat slab
Fig. 5.11 3D view of interior panel of Post-Tensioned flat slab
Fig. 5.12 3D view of exterior panel of Post-Tensioned flat slab
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 20
Chapter 5 post-tensioned flat slab
Fig. 5.13 3D view of interior panel of Post-Tensioned flat slab with drop cap
Fig. 5.14 Graphs showing tendon profile, tension and compression stresses and required and
provided Post-Tensioning forces
Fig. 5.14 displays the top diagram, the tendon height diagram shows the elevation of tendon
profile. The second diagram, stress diagrams, plots the maximum compressive and tensile
stresses at the top and bottom face of the member. The third diagram, Post-Tensioning
diagrams shows the required and provided post-tensioning force at 1/20 th points along each
span. The vertical line represents the required post-tensioning and the horizontal line
represents the provided post-tensioning at that section. The program also gives the required
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 21
Chapter 5 post-tensioned flat slab
non-prestressed reinforcement and stress ratio for punching shear which shall be below 1
otherwise shear reinforcement is require near column.
Design of Reinforced Concrete & Post-Tensioned flat slab using software. 22