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Welcome back to prestressed concrete structures. This is the second lecture of module nine.
In this lecture, we shall study one-way slabs.First, we shall learn about the different types of
slabs and then we will move on to the analysis and design of one-way slabs.
Slabs are an important structural component where prestressing is applied. With increase
in demand for fast track, economical and efficient construction, prestressed slabs are becoming
popular. In the present module, the slabs will be presented in two sections: one-way
and two-way slabs.
Rectangular slabs can be divided into two groups based on the support conditions and
length-to-breadth ratios. In the first type, when a slab is supported
on two opposite edges, then it is a one-way slab spanning in the direction perpendicular
to the edges. Precast planks fall in this group. A ribbed floor, which means slabs with
joists, consisting of double tee sections and a hollow coarse slab are also examples
of one-way slab spanning in the direction.
In the second type, when a rectangular slab is supported on all the four edges and the
length-to-breadth ratio, which we shall denote as L by B, is equal to or greater than two,
the slab is considered to be a one-way slab. The slab spans predominantly in the direction
parallel to the shorter edge.The first type of one-way slab is supported only on the two
opposite edges. In the second type of one-way slab, they are supported on all the four edges.
but the length-to-breadth ratio, where Length is the larger horizontal dimension and breadth
is the smaller horizontal dimension. When this length-to-breadth ratio is greater than
or equal to two, then also the slab is considered to be a one-way slab and the slab spans in
the shorter direction.A slab in a framed building can be a one-way slab depending upon its length-to-breadth
ratio. A one-way slab is designed for the spanning direction only. For a transverse
direction, a minimum amount of reinforcement is provided. Thus, given a building plan,
first you have to determine whether a slab is one-way or not depending upon the length-to-breadth
ratio and if they are supported on all the four sides. If we determine a slab is a one-way
slab based on the criteria that the length-to-breadth ratio is greater than or equal to two then,
we design the slab only for the spanning direction. For the transverse direction, we provide only
minimum amount of reinforcement.
In this figure, on the left side we have a slab which is spanning from left to right
because it is supported only on the two edges; whereas, for the slab on the right side it
is supported on all the four edges and here the length is greater than or equal to the
breadth and the slab is spanning in the shorter direction parallel to B.
Other types of rectangular slabs and non-rectangular slabs are considered to be two-way slabs.
If a rectangular slab is supported on all the four sides and the length-to-breadth ratio
is less than two, then it is a two-way slab. If a slab is supported on three edges or two
adjacent edges, then also it is a two-way slab. A slab in a framed building can be a
two-way slab depending upon its length-to-breadth ratio. A two-way slab is designed for both
the orthogonal directions.The types of slab which do not fall under the category of one-way
slabs come under two-way slabs. For rectangular slabs, if the length-to-breadth ratio is less
than two, when all the sides are supported, then it is a two- way slab and for a two-way
slab, it has to be designed for both the orthogonal directions. Again, given a building plan,
first we have to identify whether a slab is a two-way slab or not and then we have to
proceed as per the two-way slab design. In this module, we shall focus only on the one-way
slab analysis and design.
A slab is prestressed for the following benefits: first, it is the increased span-to-depth ratio.
Typical values of span to depth ratios in slabs are given below. For non- prestressed
slabs, the span to depth ratio is 28: 1; for the prestressed slab, it is 45: 1. Here, you
can see that a prestressed slab can have larger span compared to its depth, and this is a
big benefit when we are looking for large column free space. Thus, one very important
aspect of prestressing a slab is to increase the span to depth ratio.
The second benefit is the reduction in self weight. Once the self weight is reduced, the
design of the supporting beams or columns also leads to smaller sections; the building
mass is reduced. It leads to lower seismic forces and we get a tangible benefit from
the prestressing of the slabs. The third benefit is that the sections remain uncracked under
service loads and this leads to increased durability. Durability is of concern wherever
we have exposure conditions worse than moderate and if the slab is uncracked, then we have
increased durability.
The fourth benefit is a quick release of formwork. Once a slab is prestressed, the formwork supporting
the grill concrete can be removed quicker because, the concrete can sustain its weight
due to the effect of prestressing. Hence, this leads to a fast track construction.
The fifth benefit is the reduction in fabrication of reinforcement. In conventional slabs, the
reinforcement is by ordinary steel; it needs more time to place and tight the reinforcing
parts. Whereas in prestressing slabs, the prestressing strands are either placed in
a precast yard or the post-tensioning is done in strands which are placed in ducts and both
these operations are faster than the conventional fabrication of the reinforcement cage. The
sixth benefit is more flexibility in accommodating late design changes. Thus, once a design has
been made for a particular utilization of an area and if the client needs some change
in the utilization of the floor space, the column free space has greater flexibility
to accommodate that change. Although, we have the benefits of this prestressing of slabs,
we have to be careful that the analysis, design and construction has to be done carefully
and the quality of construction has to be strictly adhered too.
Precast planks are usually pre-tensioned. They are manufactured in a yard under controlled
environment, whereas cast-in-situ slabs are usually post tensioned. Post-tensioned slabs
are becoming popular in office, commercial buildings and parking structures, where large
column-free spaces are desirable. The maximum length of a post-tensioned slab is limited
to 30 to 40 meters to minimize the losses due to elastic shortening and friction. Although,
theoretically we can go for even larger spans, but since you have elastic shortening of the
slab and there is loss due to friction, usually in the post-tensioned slabs the spans are
limited to 30 to 40 meters.
Slabs can also be composite for the benefits of reduction of form work, cost and time of
construction; the quality control is also better in a composite slab. A precast plank
can be prestressed and placed in the final location. A topping slab is overlaid on the
precast plank. The grades of concrete in the two portions can be different. This figure
shows a precast plank, but the bottom part has been precast and prestressed in the yard.
These are the prestressing ducts and the top part is a cast-in-situ topping, where there
has been some reinforcement to consider the effects of temperature and shrinkage; the
combination of these two is called a composite slab. That means the bottom part is precast
and prestressed whereas the top part is cast-in-situ.
Next, we move on to the analysis and design of one-way slabs. One-way slabs are analyzed
and designed for the spanning direction similar to rectangular beams. The analysis and design
is carried out for the width of the plank or a unit width, say 1 meter, of the slab.
Ribbed floors are designed as flanged beams. Whatever we have studied for the design of
beams, they are also applicable for the design of one-way slabs. The slabs are designed based
on their width, if it is a precast plank or it will be designed for a unit width, say
1 meter and the analysis and design procedure for one-way beam is applied for the one-way
slabs. If one-way slab is the ribbed floor, then we have to apply the analysis and design
procedure for a flanged section.
For continuous slabs, the moment coefficients of IS: 456-2000, which are given in table
12, can be used. That means, in the conventional analysis, if it is a simply supported slab,
you use the conventional structural analysis formulae, but, if it is a continuous slab,
then we can use moment coefficients that is provided in IS: 456-2000. We need not go for
a rigorous analysis.
The analysis and design procedures for simply supported rectangular beams are covered in
the modules “Analysis of members under Flexure” and “Design of members of Flexures”, respectively.
These materials are briefly reproduced here. Earlier, in module three and module four,
we have studied the analysis and design procedures of beams in detail. Here, we are revising
them briefly
First, we do a preliminary design. In the preliminary design, first the material properties
are selected. That means fck, which is the characteristic compressive strength of concrete
and fpk, which is the characteristic tensile strength of the prestressing steel are selected.
Before starting a project, the weight of concrete and the type of prestressing steel is fixed
during the conceptual process and from that, we know the values of fck and fpk.
Second, determine the total depth of the slab which we shall denote as ‘h’ based on
the span to effective depth ratio given in clause 22.6 of IS 1343-1980. The span to effective
depth ratio will be denoted as L by d and this ratio is limited to avoid deflection
computations. If we fix our total depth, satisfying the span to depth ratio, then we do not have
to check the deflections of the slab if the load is conventional. In order to relate the
effective depth with the total depth, we can use an approximate estimate that d is about
h minus 25 millimeters. Once we select d, based on the l by d ratio, we can select h
and then we have to round off h to a multiple of 10 millimeters. Thus, the total depth of
the slab is determined based on the deflection requirement.
Next, we will calculate the self-weight of the slab. Once the depth is known, the weight
per unit width of the slab can be determined. Then, we calculate the total moment, which
we are representing as MT including moment due to self weight, which we are representing
as Msw. Once, we are able to calculate the self-weight, we add that to the other dead
load moment and live load moment to get the total moment MT. After this, we estimate the
lever arm which we denote as z. z is equal to 65% of the total depth, which is 0.65 h,
if the self-weight moment is large; say the self-weight moment is greater than 30% of
the total moment. If the self-weight moment is small, then we can estimate z to be about
50% of the total depth that is z is equal to 0.58. These are the rough estimates of
the lever arm by which the compressive force will move up from the CGS under service conditions.
The sixth step is to estimate the effective prestress, which we shall denote as Pe. Pe
is equal to the total moment, MT divided by z, if the self weight moment is large. Or
else, Pe is equal to the moment due to imposed loads which we are denoting as MIL divided
by z, if Msw is small. These are some rough guidelines to estimate the prestressing force,
given the moment and the estimate of the lever arm. Here, the moment due to imposed loads
is given as MIL is equal to MT minus Msw. That is, it is the moment due to the super
imposed dead load and the live load. How are we calculating it? We are subtracting the
self-weight moment from the total moment and we are saying that is the moment due to imposed
loads. In the seventh step, we are considering that the effective prestress is about 70%
of the characteristic strength of the prestressing steel. From that, we are calculating the area
of the prestressing steel: Ap is equal to Pe divided by fpe.
Thus, first we have estimated the effective prestress. We have estimated the effective
prestressing in tendon. From there, we are having an estimate of the amount of the prestressing
steel for the chosen width of the slab. In the eighth step, check the area of cross section
A, where A is equal to 1000 times h; 1000 is the width of a 1 meter white slab and h
is the total depth. The average stress C divided by A should not be too high as compared to
50% of fcc,allowable; C is equal to the effective prestress, Pe. Thus, we are saying that the
average prestress over the cross-section which is given as C by A is equal to Pe by A, should
not be larger than 50% of the allowable compressive stress for the concrete in the slab. With
these steps, we have found a preliminary design of the unit width of the slab. We move on
to the final design, where we are checking the stresses in the slab under the service
load conditions.
The final design involves the checking of the stresses in concrete as transfer and under
service loads with respect to the allowable stresses. The allowable stresses depend on
the type of slab: Type 1, Type 2 or Type 3. Here, the steps of final design are explained
for Type 1 slabs only. For Type 1 slabs, no tensile stress is allowed at transfer or under
service loads. Thus, the steps of the final design primarily consist of checking the stresses
in the sections and at transfer and under service conditions. The allowable stresses
depend on the type of prestressed member that we are designing for. Earlier, we had known
that the prestressed members are divided into three types: Type 1, where we do not allow
any tensile stress either at transfer or during service; Type 2, where we allow tensile stresses
but we do not allow cracking and Type 3, where we allow cracking, but the crack width is
limited.
In this section, we are studying only the design of Type 1 slabs. The design of Type
2 and Type 3 are very similar to the design of Type 1 slabs, the difference being we have
to incorporate the allowable tensile stress in the expressions. This was covered in detail
in the analysis and design of beams. For small moments due to self-weight, say, when Msw
is less than 30% of the total moment, the steps are as follows:
First, calculate the eccentricity e, to locate the centroid of the prestressing steel. Here,
we are doing an accurate calculation and we are trying to locate the centroid of the prestressing
steel with respect to the centroid of the concrete, the CGC. This distance between the
CGS and the CGC is denoted as the eccentricity e. The lowest permissible location of the
compression due to self-weight is at the bottom kern point to avoid tensile stress at the
top. Earlier, we have studied about these kern locations; the kern points are those
that if the compression is at those points, then there will not be any tension in the
section. Thus when the slab is under minimum load that
is when it is only under its self-weight, the lowest possible position of compression
is the bottom kern point. The distance of the bottom kern point from the CGC is denoted
as kb. If we base our design based on this bottom most position of C, then we get an
economical section.
The design procedure based on the extreme location of C gives an economical section.In
this figure, we are studying the stress conditions; h is the total depth of the slab. First, at
prestressing, the compression is at the same location of the CGS. Then, as the self-weight
acts, the compression moves up from the CGS. It has to be within the kern zone to avoid
any tensile stress in the section. The lowest possible location of C, that means, the minimum
that C has to travel up from the location of CGS is up to the bottom kern point which
is at a distance kb from the CGC. Here, e is the distance between the CGC and the CGS.
This figure shows the internal force in concrete due to the self-weight of the slab and the
stress condition is when C is located at the bottom kern point, the stress at the top is
zero. The stress at mid-height, which is the average stress, is given as C by A equal to
P0 at transfer divided by A and the stress at the bottom is denoted as fb. This is the
stress diagram for the depth of the slab due to its own weight.
From the stress profiles, the following equations can be derived: e is equal to Msw divided
by P0 plus kb, the derivation of this expression was shown earlier. We are recollecting the
expression of e, which gives us the location of C at the bottom kern point and this is
a function of the self-weight of the moment, the prestressing force at transfer P0 and
the distance of the bottom kern point from CGC. The magnitude of C or T is equal to P0,
the prestress at transfer after initial losses. Thus, by this equation, we are able to determine
the lowest possible location of the centroid and this will lead to an economical section.
The value of P0 can be estimated as follows: P0 is equal to 0.9 Pi. That means 90% of the
initial prestress for pre-tensioned slab and it can be 100% of the initial prestress for
the post tensioned slab. In the pre-tensioned slab, due to elastic shortening P0 is lower
than Pi. Here, Pi is the initial applied prestress and the maximum value of Pi is equal to 80%
of the characteristic strength times the estimated area of the prestressing steel Ap. The permissible
prestress in the tendon is 0.8 fpk and this is used to find out the maximum initial prestress
that we can apply on the estimated amount of prestressing steel. Once, we have determined
the eccentricity, next we are recomputing the effective prestress Pe and the area of
prestressing steel Ap. This is based on the stress diagram under service conditions. Under
service conditions, due to the total moment the maximum distance C can traverse is up
to the top kern point. At this location, the stress at the bottom is zero, the stress at
the top is ft and again the average stress at mid height is C by A is equal to Pe by
A. This is the stress profile under service conditions.
Based on this stress profile, the shift of C due to the total moment gives an expression
of Pe. The derivation of the equation was shown earlier and again here it is recollected.
From that stress profile, we get Pe is equal to the total moment divided by e plus kt,
where e plus kt is the distance by which C has traversed from CGS to the top kern point.
For solid rectangular slabs, kt equal to kp equal to h by 6. Thus, the top and bottom
kern points are located at one sixth of h from the CGC.
Once, we have calculated Pe, next we are recomputing the amount of prestressing steel. Considering
again, the effective prestress fpe is equal to 70% of the characteristic strength fpk¬,
the area of prestressing steel is recomputed as follows: Ap is equal to Pe divided by fpe.
The number of tendons and the spacing is determined based on Ap. Given the value of Pe and fpk,
we are calculating the amount of prestressing steel that we need for the unit width or whatever
width we have selected for the slab. Once we calculate Ap, we can distribute that in
the tendons with suitable spacing. With this, we know the amount of prestressing
force under service conditions and you can update the value of prestressing force at
transfer. Once we have updated P0, we can recompute e again with the updated values
of Ap and P0. Thus, you have to appreciate that the design is sequential process; that
you compute some variables based on the estimated quantities, again you recompute the estimated
variables and converge to a result which is suitable regarding the stresses and the lay
out of the prestressing tendons.
If the variation of e from the previous value is large, another cycle of the computations
of the prestressing variables can be undertaken. For large self-weight moment Msw, if e violates
the cover requirements, e is determined based on the cover. The expression of e that you
have written, was determined based on the force conditions, but, if the self weight
moment is high, then e may come out to be large and we may violate the cover requirements
for the prestressing tendons. In that case, e is selected based on the cover requirements
and then, we check the amount of the prestressing steel and prestressing force.
In the fourth step, we check the compressive stresses in concrete. At transfer, the stress
at the bottom, fp is equal to minus P0 by A, which is the average stress times h divided
by ct. This is equal to minus 2 P0 by A, because h is equal to two times of Ct. Thus we are
able to calculate the stress at the bottom based on the design, prestress at transfer
and the properties of the section. The stress at the bottom fp, should be less than the
allowable compressive stress fcc,allowable. This can be expressed as an equation that
the magnitude of fp should be lower than or equal to fcc,allowable.
Next, we are checking the compressive stress in concrete at service conditions. The expression
at the stress in the top is similar that ft is equal to minus Pe by A, which is the average
prestress times h by Cb. This relationship is again coming from the triangular stress
distribution and this is equal to minus 2 Pe by A, because h is equal to two times of
Cb. The stress at the top ft should be less than fcc,allowable, where fcc,allowable is
the allowable compressive stress in concrete at service. Here also, we are expressing this
relationship as the magnitude of fp should be less than fcc,allowable. Note that the
allowable compressive stress at transfer and the allowable compressive stress at service
are different.
For Type 2 and Type 3 members, the tensile stress should be restricted to the allowable
values. So when we show this design for Type 1 members, the allowable tensile stress was
zero. If we do the design for the Type 2 and Type 3 members, then the allowable tensile
stress is as per the values given in the curve. For a continuous slab, a suitable profile
of the tendons is selected; this is covered in the module of cantilever and continuous
beams. So, the value of eccentricity that we have found is at the critical location.
For a simply supported beam, the critical location is at the middle, we can have a parabolic
profile or if the depth is small, then we can have a straight profile with constant
e. For a continuous slab, the prestressing tendon
goes above CGC at the support locations and then we need to determine several values of
the eccentricity, one at the span and another at the support. This procedure was shown in
the module of the continuous beams. For continuous beams or slabs then, important design element
is selecting an appropriate profile of the tendons.
When the value of e is fixed, say in either pre-tension or post-tension operations, the
design steps are simpler. If the tendons are placed at the CGC, say for which e is equal
to zero, then the uniform compressive stress due to prestress counteracts the tensile stress
due to service loads. Sometimes, the eccentricity is fixed based on the construction requirement.
If the eccentricity is zero, that means CGS lies at the CGC, in that case there is a uniform
compression in the slab and this uniform compression counteracts the tensile stress that is generated
due to the bending. To have zero stress at the bottom under service conditions, the value
of Pe can be directly calculated from the following equation:
Pe by A, which is the uniform compressive stress is equal to MT divided by Zb, which
is the tensile stress generated due to bending under service conditions. From this, we directly
get the value of the prestressing force, Pe is equal to A times MT divided by Zb, and
this expression is a simple expression which we can use if the eccentricity is zero. This
is based on the concept that the uniform compressive stress balances the tensile stress under service
conditions due to flexure.
Zp is the section modulus. The above expression is same as Pe equal to MT divided by kt which
is equation 9b-3 with e equal to 0. Earlier we have seen one expression, where
Pe is equal to MT divided by e plus kt. In that equation if you substitute e equal to
0, then we get Pe is equal to M divided by kt and this expression is same as Pe equal
to A times MT divided by zb. Thus both these expressions are same, but the one given here
is a convenient expression when the eccentricity is zero. The stresses at transfer can be checked
with an estimate of P0 from Pe. That means, once we have calculated Pe, we can estimate
P0 from Pe and based on P0 we can calculate the stresses at transfer.
The fifth step is to check the shear capacity. The shear is analogous to that generates in
a beam due to flexure. The calculations can be per unit weight of the slab. The critical
section for checking the shear capacity is at a distance effective depth d, from the
face of the column or the supporting beam across the entire width of the slab. The critical
section is transverse to the spanning direction. Thus, for one-way slabs the checking for shear
is similar to beams, where the critical section is considered to be at a distance effective
depth from the face of the support and the critical section is perpendicular to the spanning
direction. The shear demand Vu in the critical section generates from the gravity loads in
the tributary area. That means once we know the tributary area for the slab, you can calculate
the shear demand that comes in the critical section.
For adequate shear capacity the resistance VuR should be greater than Vu, where VuR is
equal to Vc, the shear capacity of uncracked concrete of unit width of the slab. We do
not usually provide shear reinforcement in slabs. Thus, the shear capacity is equal to
the shear capacity of the concrete for the uncracked section. The expression of Vc is
given in the module of analysis and design for shear and torsion. If this is not satisfied,
it is preferred to increase the depth of the slab to avoid shear reinforcement. Finally,
we provide transverse reinforcement based on temperature and shrinkage.
As per IS: 456-2000, clause 26.5.2.1, the minimum amount of transverse reinforcement
Ast,minimum in millimeter square per unit width of slab is given as follows: Ast,minimum
equal to 0.15% of the total section, which is 1000 times h for Fe 250 grade of steel
and Ast,minimum equal to 0.12% of 1000 h for Fe 415 grade of steel. Usually, the transverse
reinforcement is provided by non-prestressed reinforcement.
A minimum reinforcement is sufficient for the transverse moment due to Poisson’s effect
and small point loads. For heavy point load, transverse reinforcement needs to be computed
explicitly. Thus, whatever transverse reinforcement we provide is adequate for any transverse
bending due to the Poisson’s effect or due to small point loads. If there is a large
point load then we have to explicitly calculate the transverse reinforcement.
Next, we are learning this principle from a design example. Design a simply supported
precast prestressed, Type 2, composite slab for the following data: width of the slab
is 0.3 meters; clear span is equal to 2.9 meters; effective span l is equal to 3.1 meter;
thickness of the precast slab is 50 millimeters; thickness of the cast-in slab is 50 millimeters;
the grade of concrete in the precast slab is M60 and the grade of concrete in the topping
slab is M15. Note that the two grades of concrete are different. The topping slab is a linear
concrete as compared to the precast portion. The tendons are located at the mid depth of
the precast slab. The live load is 2 Kilonewton per meter square and the load due to floor
finish is 1.5 Kilonewton per meter square.
This problem has been taken from a guide book titled Partially Precast Composite PSC Slab,
authored by Professor A. R. Santhakumar and it has been published by the building technology
center of Anna University, Chennai.
We are calculating the moments. The load per unit area: first, we are calculating the weight
of the precast slab. Next, we are calculating the weight of the topping slab. Then, we are
calculating the weight of the floor finish and finally we are adding the live load. This
gives us a total load of 6 Kilonewton per meter square of the slab.
The total moment MT along the width of the slab is given as the weight times the width
times the span square divided by 8. This is a simple expression for a simply supported
slab and substituting the values, we get MT equal to 2.16 Kilonewton meters.
The individual moments are calculated based on the proportionality of the loads. We can
calculate Msw, which is the moment due to self-weight of the precast slab, which comes
out to be 0.45 Kilonewton meters. Mtop, the moment due to the weight of topping slab comes
out to be again to 0.45 Kilonewton meters. Mfinish, which is the moment due to the weight
of the floor finish is equal to 0.54 Kilonewton meters and MLL, moment due to live load is
coming out to be 0.72 Kilonewton meters. Once, we have calculated MT, we can quickly calculate
the individual moments by taking the load proportionality.
Next, we are calculating the geometric properties. For the precast section, the cross-section
area is 300 millimeters times the depth 50 millimeters which is 15,000 millimeters square.
The moment of inertia, I1 is equal to 1 by 12 times B times depth cube which comes out
to be 3125000 millimeter to the power 4.
We are calculating the distance to the extreme fibers, cb and ct which are both 25 millimeters.
The section modulus can be calculated as I1 divided by cb or ct and both of them are equal
to 125,000 millimeter cube.
For the composite section, since the grade of concrete are different for the precast-
prestressed and cast-in-situ (CIS) portions, an equivalent or transformed area, is calculated.
The CIS portion is assigned a reduced width based on the equivalent area factor which
is also called the modular ratio. The equivalent area factor is determined based on the grades
of concrete and we get a value of 0.5.
Thus, the composite section is being transformed to an equivalent section with a reduced width
of the topping slab which is 50% of its original value. For the composite transformed section,
we are finding out the centroid, calculating the area of the top, the area of the bottom
and the total area, from which we are locating the centroid which is at a distance of 41.7
millimeters from the bottom. This is the CGC of the equivalent transformed section.
We are calculating the moment of inertia of this equivalent section. First, we are calculating
the Itop of the top part for the axis passing through the CGC. We are calculating Ibottom
of the bottom part and then we are adding these two values to get the total I for the
composite section.
Distance of the extreme fibers is given as 41.7 and 58.3 and the section modulus for
the composite transformed section can be calculated by I divided by the respective y values which
gives the section moduli. We are calculating the prestress. Since the
tendons are located at the mid depth of the precast slab, e is 0 for the precast slab.
The value of Pe is calculated directly from the following stress profiles. This Pe generates
a uniform compressive stress. Then, the self weight and the topping load generates a stress
profile like this and finally, we have the finished loads and the live load moment which
is acting over the full composite section. From these stress profiles, you can say, to
avoid tensile stress at the bottom under service conditions, the resultant stress is equated
to zero.
Once, we write the expression of the resultant stress and we equate it to zero, we get an
expression of the effective prestress that we need to apply at the precast plant. From
this expression we can find out the value of Pe.
In the above expression, the first term inside the bracket corresponds to the precast section.
The moments due to self-weight and topping slab are resisted by the precast section alone.
The second term inside the bracket corresponds to the equivalent section. The moments due
to the weight of the floor finish and live load are resisted by the equivalent section.
Thus, we are using the principles of a composite section to analyze this composite slab.Substituting
the values, we get the value of the prestress as 153816 Newton
Assuming 20% loss, the prestress at transfer P0 is almost equal to 1.2 times Pe, which
is equal to 184579 Newton. Selecting wires of diameter 7 millimeters and ultimate strength
fpk equal to 1500 Megapascal, we are calculating the area of the prestressing steel. Area of
one wire is 38.48 millimeter square and the maximum allowable tension in one wire is equal
to 0.8 fpk times Ap, which is 46176 Newton.Thus, the number of wires required is 184579, which
is the prestressing force at transfer divided by the maximum prestressing force applied
in one wire which gives us a value of 4 wires needed within 300 millimeters. Required pull
in each wire is given as the total prestressing force P0 divided by 4 and thus the total prestressing
force is equal to P0 equal to 4 times 46145 equal to 184580 Newton. These values turn
out to be close because we have got a value 4 which is very close to the calculated 3.99.
The effective prestressing force is 80% of the prestressing force at transfer which comes
out to be 147664 Newton.
We are checking the stresses. At transfer, the compressive strength is 70% of fck which
is 42 Megapascal. The allowable compressive stress is 0.44 fci which comes out to be 18.5
Megapascal. For Type 2 members, the allowable tensile stress is 3 Megapascal.
At transfer, we see that for the precast portion, the average stress is minus 12.3 Megapascal
which is less than the allowable value of 18 Megapascal. Hence, it is satisfactory for
the transfer load case. At mid-span of the precast section, we are including the moment
due to self-weight. There also we find that the stress at the top which is of higher value
15.9 is less than the allowable value and hence this stress profile is acceptable at
transfer.
After casting of the topping slab at 28 days, the allowable compressive stress is now 0.44
of 60 which is 26.4 Megapascal and allowable tensile stress still remains as 3 Megapascal.
After the casting of the topping slab, we are calculating the stress conditions. At
the mid span, we have the force due to the prestressing force and due to the self weight
and topping slab. These values are coming out to be minus 19.5 which is the higher one
and this is also less than the allowable compressive value.
At service conditions, for the prestressing portion the allowable compressive stress is
21. The allowable tensile stress is 3.The expression of the stress at the mid-span for
the composite section is given by: minus Pe by A, the prestressing force, then the effect
of the self-weight and the topping moment, and the effect of the finishing moment and
the live load moment.We are calculating the stresses at the junction. We find out it is
minus 17.6 and the stress at the bottom is 0.4 Megapascal; these values are for the precast
portion.
For the cast-in-situ portion, the stress at the top is minus 4.3 Megapascal and the stress
at the junction is minus 0.6 Megapascal.Thus, we get the stress profile for the composite
section and we see that, most of the stress is concentrated in the precast portion. Here
also, this value minus 17.6 is less than the allowable compressive stress and hence the
stress profile under service condition is acceptable.We are checking the shear capacity.
We are first finding out the shear capacity to be 42 Kilonewton and the shear demand to
be 4.2 Kilonewton.
Since the shear strength is greater than the demand, the shear capacity is adequate. We
are providing some transverse reinforcement as per the minimum requirement that is 8 millimeter
diameter at 300 millimeter on center.
Thus, this is the final design of the precast plank. We have laid the prestressing steel,
the topping steel and we have cast the concrete. Finally, we get the precast plank for the
design forces. Thus, in this module we covered one-way slabs and then we studied the analysis
and design of one-way slabs. Thank you.