Anecdota

Laughter is the Best Medicine

Lecture-29-Calculations of Deflection and Crack Width


Welcome back to prestressed concrete structures. This is the lecture on module six on calculation
of deflection and crack width.In this lecture, we shall first study the calculation of deflection. Under that we shall learn about the deflection
due to gravity loads, deflection due to prestressing force, the calculation of total deflection,
the limits of deflection, determination of moment of inertia and limits of span-to-effective
depth ratio. Next, we shall study about the calculation
of crack width, under which we shall study the method of calculation and the limits of
crack width. First is the calculation of deflection. The deflection of a flexural member is calculated
to satisfy a limit state of serviceability. The deflection is calculated for the service
loads. Since a prestressed concrete member is smaller
in depth than an equivalent reinforced concrete member, the deflection of a prestressed concrete
member tends to be larger. This is one drawback, that if you use very
slender members then we may have deflection problems. Hence the calculation of deflection becomes
necessary, when the members are very slender. We have to make sure that the deflection does
not cause any problem in the functioning of the member. The total deflection in a prestressed concrete
member is a resultant of the upward deflection due to prestressing force and downward deflection
due to the gravity loads. Thus, unlike reinforced concrete member here
we have another component of deflection, which is the deflection due to the prestressing
force. For a simply supported beam, the prestressing
tendon provides an upward thrust to the member, which results in a ….. during the prestressing
of the member. The total deflection is a summation of the
upward deflection due to the prestressing force and the downward deflection due to the
gravity loads. The deflection of a member is calculated at
least for two cases. First is the short term deflection at transfer. This deflection is due to prestressing force,
before long term losses and self-weight without the effect of creep and shrinkage of concrete. Thus, when the prestress is transferred the
member may have a camber depending on the amount of prestressing force and the self
weight of the member. It has to be checked, whether the deflection
is too much during this prestressing operation or not. If there is some finishing over the member
then, it has to be checked whether this finishing will have any negative effect due to the deflection,
whether there will be any cracking on the finishes or not. The second case, for calculating deflection
in the long term deflection at service loads. This deflection is due to prestressing force
after long term losses and the permanent components of the gravity loads including the effect
of creep and shrinkage. Thus, at service loads the deflection has
two components. One is due to the prestressing force after
the long term losses and next is the component due to the gravity loads. For the gravity loads, only the permanent
component of the gravity loads is considered in the long term deflections. The permanent component of the gravity load
includes the dead load and sustained component of the live load.Now, here comes an engineering
judgment; that how much live load should be considered to be sustained that depends upon
the analyst. In our permanent load, we are including only
the sustained part of the live load and not the total live load. If you are interested with the total deflection,
then we may include the deflection due to the additional component of the live load
as well. First, we shall see the calculation of deflection
due to gravity loads. The methods of calculation of deflection are
covered under structural analysis. These methods include the following: double
integration method, moment-area method, conjugate beam method and principle of virtual work. A student should have studied these methods
in a course of structural analysis. In this lecture, we are not going into the
details of the evaluation of deflection based on these methods, but we shall consider the
end results from these methods. Numerical solutions schemes can be implemented
in a computer based on the above methods and in that situation the deflection calculation
can be done in much faster way. For members with prismatic cross-sections
that is the cross-section is constant throughout the span of the member, common support conditions
and subjected to conventional loading, the deflections are available in tables in text
books. The expressions of deflection, which is represented
as delta, for a few cases are provided here. I is the moment of inertia and E is the modulus
of elasticity of concrete. For standard cases, the expressions of deflections
are given in the text books on structural analysis. Here we are giving some examples from them. In this expression, E is the modulus of concrete
and I is the moment of inertia. What is the value of the moment of inertia,
we need to consider; that we shall discuss later. In this figure, we can see two simply supported
beams. For the top one, the simply supported beam
is subjected to a uniformly distributed load and the deflection is given as delta is equal
to 5 by 384 wL four divided by EI. In the figure in the bottom, a simply supported
beam is subjected to a point load at the centre and in that case the deflection delta is equal
to PL cubed divided by 48EI. For a cantilever beam, under a uniformly distributed
load, delta at the end is given as wL four divided by 8EI. For a cantilever with a point load at the
end, delta is equal to PL cubed divided by 3EI. In most of the prestress applications the
beams are simply supported and hence we can use the standard expressions for the simply
supported beams, in calculating the deflection due to the gravity loads for a prestressed
beam. Next, we are coming to the deflection due
to prestressing force. The deflection due to prestressing force is
calculated by the load-balancing method, which is explained under analysis for flexure. Earlier, we had seen that there are three
methods of analysis of a prestressed member. First is based on the stress concept, the
second is based on the force concept and the third is based on the load-balancing method. Now, the third method is used to calculate
the deflection due to prestressing force. Here, we shall see the expressions of the
deflections due to prestressing force for the standard cases. The deflection due to prestressing force is
represented as deltap. For a parabolic tendon with eccentricity e
at the middle, there is an uniform upward load, which is represented as wup with span
is equal to L. wup is given as 8 times the prestressing force times the eccentricity
divided by L squared. Then, we can calculate the upward deflection,
deltap is equal to 5 by 384 wup times L to the power four divided by EI. Thus, this is the expression of upward deflection
of a prestressed concrete beam, due to the prestressing force. For a singly harped tendon, there is a force
at the location of the harping, which is denoted as Wup. Wup is given as 4 times the prestressing force
times the eccentricity divided by L and from there we can calculate deltap is equal to
Wup times L cube divided by 48EI. This is the expression of the deflection due
to the prestressing force, with a singly harped tendon and the harping point s at the middle
of the beam. Next, we are going on to the expression of
doubly harped tendon. Here, the harping points are symmetric and
each harping point is at a distance a times the span from the support. In this figure, we see that there are two
upward forces, corresponding to the two harping points and each upward force is represented
as Wup. Wup is equal to P times e divided by aL, where
aL is the distance of each harping point from the end. Then, we can calculate the deflection deltap
is equal to a times within bracket 3 minus 4 a square times Wup times L cube divided
by 24EI. Next, we are calculating the total deflection
due to the prestressing force and the gravity loads. As we said before, that the total deflection
is calculated for the two cases: first, the short term deflection at transfer, which we
shall denote as deltast and the second is the long term deflection under service loads,
which is denoted as deltalt. Short term deflection at transfer, deltast
is equal to minus deltapo plus deltasw. Here, deltapo is the magnitude of deflection
due to the prestress at transfer, which is denoted as Po, which is before the long term
losses and deltasw is the deflection due to the self-weight, which is downwards, and the
Po is the prestressing force before the long term losses. Note, that the sign of the two deflections
are opposite. A negative sign has been placed for the deflection
due to the prestressing force, since this is upwards. Thus, in presence of the prestressing force,
the total deflection may become negative, if the value of deltasw is numerically smaller
than deltapo. In that case, the beam will have a hamper
and it will deflect upwards. Now, we are calculating the long term deflection
under service loads. The calculation of long term deflection is
difficult because the prestressing force and creep strain influence each other. The creep of concrete is explained in the
module of material properties. The creep of concrete is defined as the increase
in deformation with time under constant load. Due to the creep of concrete, the prestress
in the tendon is reduced with time. This is an important aspect in the calculation
of long term deflections that the concrete deforms with time due to the permanent load
and due to the deformation of the concrete or due to the shortening of the concrete,
the prestressing force gets reduced; there is a loss in the prestressing force. Thus, creep and the prestressing force influence
each other. Hence, the exact calculation of long term
deflection gets difficult. Now, the creep was discussed in detail in
the module of material properties. Here we are having a quick review of how to
measure the creep strain in concrete. The ultimate creep strain is found to be proportional
to the elastic strain. The ratio of the ultimate creep strain to
the elastic strain is called the creep coefficient theta. Creep coefficient theta, for three values
of age of loading as per the code IS: 1343 – 1980 is given in the table. At seven days of prestressing force the creep
coefficient is 2.2; that means if the prestressing force is transferred when the concrete age
is seven days then the creep strain is 2.2 times the elastic strain. If the prestress is transferred at twenty
eight days, then the creep coefficient is 1.6. Thus, the ultimate creep strain is reduced
to 1.6 times the elastic strain and finally, if the age of loading that means, the age
of transfers of prestress is one year then the creep coefficient is 1.1. That is the ultimate creep strain is 1.1 times
the elastic strain. In order to reduce the long term deflections,
we should delay the application of the prestressing force such that the concrete gains adequate
strength and the effect of creep is reduced. In this table the only factor which has been
considered in evaluating the creep strain is the age of loading for the concrete. There are other factors which influence creep. Incase, if more accurate evaluation of creep
is necessary with the time as a variable, then we need to look into specialized literature. The following expression is a simplified form,
where an average prestressing force is considered to generate creep strain. The shrinkage strain is neglected. This expression is an expression of the deflection
due to long term loads, where we have considered an average value of the prestressing force
which causes the creep. That means, you are not considering the creep
and the prestressing force throughout the time. But we are considering it in an average sense,
that as if the average prestressing force is causing the creep in the concrete. With this assumption the expression of the
long term deflection is deltalt is equal to minus deltape, which is the deflection due
to the effective prestressing force at service loads minus deltaPo plus deltape divided by
2. This term is the average deflection due to
the prestressing force times theta, which is the effect of creep. These deflections are calculated based on
elastic calculation and that we are multiplying by theta to get the effect of creep plus deltaDL
plus deltaSL, where deltaDL is the deflection due to dead load and deltaSL is the deflection
due to the sustained load. These deflections are again calculated by
the elastic expression that we had seen earlier, the whole times 1 plus theta; the theta considers
the effect of creep, plus deltaLL, which is the deflection due to additional live load
if you are interested to include that as well. Thus this expression is a simplified form
of calculating the long term deflection by including creep and the change in the prestressing
force and the deflection due to sustain loads in the expression. To summarize the notations in this expression:
deltaPo is the magnitude of deflection due to Po, the prestress before long term losses;
deltape is the magnitude of deflection due to the Pe, where Pe is the effective prestressing
force after long term losses; deltaDL is the deflection due to the dead load including
self-weight; deltaSL is the deflection due to sustained live load; deltaLL is the deflection
due to additional live load. A more rigorous calculation of total deflection
can be done using the incremental time step method. Now, we are learning the summary of a procedure
which is a more rigorous procedure to calculate the deflection which considers the change
of the prestressing force due to creep with time. It is a step-by-step procedure, where the
change in prestressing force due to creep and shrinkage strains is calculated at the
end of each time step. The results at the end of each time step are
used for the next time step. This step-by-step procedure was suggested
by the Precast Prestressed Concrete Institute, the PCI and the reference paper is written
by the PCI committee, the title of the paper is Recommendations for Estimating Prestress
Losses. It was published in the PCI journal, from
the precast prestressed concrete institute; volume 20, number 4, and in the month of July
to august 1975. The pages are from 43 to 75. This method is called the general method of
calculating the prestressing force with time.In this method, a minimum of four time steps
are considered in the service life of a prestressed member. The following table provides the definitions
of the time steps. First, the idea is that the time scale is
discretised into four steps and the discretization is based on the variation of the prestressing
force with time. The method suggests that at least a minimum
four times steps should be considered and those time steps are as follows. From the first time step, for a pre-tension
member, the beginning of the time step is the anchoring of the steel. For post tension the beginning is the end
of curing. The end of the first time step is age of prestressing. Thus within this period there may be some
shrinkage in the concrete and for the pre-tension member there can be some relaxation losses
after the tension has been applied in the steel and then the prestress is transferred
to the member. In the second time step, the beginning of
the time step is the end of the first time step, and the end of the second time step
is thirty days after prestressing or when subjected to superimposed load. Thus, we see let the first one month is important
in the variation of the prestressing force and the creep and shrinkage strains. In the third time step, the beginning is the
end of step 2 and the end is one year of service and the fourth time step begins at the end
of step three and ends at end of service life. Thus, these are the minimum four time steps
that the committee recommends to monitor the prestressing force with time considering the
creep and shrinkage strains in the concrete. The step-by-step procedure can be implemented
in a computer program, where the number of time steps can be increased. Thus, we may not stick to four time steps,
we can have even larger number of time steps, which can be implemented in a computer program. In this method, we need more accurate expressions
of the creep and shrinkage strains, which are functions of time. Next, we are calculating the limits of deflection. Clause 19.3.1 of IS: 1343 – 1980 specifies
limits of deflection such that the efficiency of the structural element and the appearance
of the finishes or partitions, are not adversely affected. The limits of deflection are summarized next. The total deflection due to all loads, including
the effects of temperature, creep and shrinkage, should not exceed span by 250. This is the first requirement that the total
deflection at any point during the service life of the structure should not exceed span
divided by 250. The next requirement is that the deflection
after erection of partitions or application of finishes, including the effects of temperature,
creep and shrinkage, should not exceed span by 350 or 20 millemeters, whichever is less. Thus, we see that if there are partitions
or finishes we may need to calculate deflections before the finishes or partitions are applied,
because we are calculating the additional deflection after the partitions are finishes
are placed. This additional deflection should be limited
to span by 350 or 20 millemeter, whichever is less. The third limit is that if finishes are applied
at the top of a beam, then the total upward deflection due to the prestressing force should
not exceed span by 300. Thus, if the finishing is applied before the
prestressing force has been transferred, then the upward deflection should not exceed span
by 300. These are the limits that the code specifies. For special structures additional limits may
be considered depending upon the situation. Next, we are moving on to the determination
of moment of inertia. For Type 1 and Type 2 members, since they
are designed to be uncracked under service loads, the gross moment of inertia which is
represented as Ig can be used to calculate deflections. That means, the moment of inertia can be calculated
from the total section and it can be substituted in the expressions of deflection for Type
1 and Type 2 members. For Type 3 members, these members are expected
to be cracked under service loads. Strictly, the gross moment of inertia cannot
be used in the calculations. IS: 1343 – 1980, clause 22.6.2, recommends
the following: When the permanent load is less than or equal
to 25% of the live load, the gross moment of inertia can be used. If the permanent component of the live load
is very small, then most of the time the section will remain uncracked. Hence we can use the gross moment of inertia. If the permanent component of the live load
exceeds 25%, then the curve recommends that the span-to-effective depth ratio, which is
denoted as L by d should be limited to bypass the calculation of deflection.Thus, if you
limit the span-to-depth ratio to a certain value, which we shall learn next, then we
can bypass the calculation of deflection, because it is considered that the deflection
will not be of any problem. If the span by depth ratio exceeds the limits,
then the gross moment of inertia can be used, when the tensile stress under service loads
is within the allowable value. The calculation of gross moment of inertia
is simpler as compared to an effective moment of inertia. In reinforced concrete, we use an effective
moment of inertia to consider the variation of moment of inertia along the span. For prestressed concrete, even for Type 3
members if you limit the tensile stresses to the allowable values, then we may use only
the gross moment of inertia, because the calculation of effective moment of inertia is more involved. Hence the gross moment of inertia can be used
to calculate the deflections. Next, we are learning about the limits of
span-to-effective depth ratio. The calculation of deflection can be bypassed
if the span-to-effective depth, which is represented as L by d ratio, is within the specified limit. The limits of L by d ratio, as per clause
22.6.2 IS: 1343 – 1980 are as follows. For span less than 10 meters, for cantilever
beams the span-to-effective depth ratio should be less than seven; for simply supported beams
the ratio should be less than 20; for continuous beams the ratio should be less than 26. If the span exceeds 10 meters, then we have
to modify those limits and the modifications are given as follows. For simply supported beams the L by d ratio
should be less than 20 times 10 divided by L and for continuous beams L by d should be
less than 26 times 10 divided by L. Here, L is in meters. Deflection calculations are necessary for
cantilevers with L greater than 10 meters. Thus, these are the limits of span-to-depth
ratio. If they are satisfied in a member, then the
deflection calculation can be bypassed. But if they are not satisfied, then we need
to do the deflection calculations and make sure that the deflections are within the specified
limits as per the code. Next, we are moving on to the second serviceability
check, which is the calculation of crack width. The crack width of a flexural member is calculated
to satisfy a limit state of serviceability. Among prestressed concrete members, there
is cracking under service loads only for Type 3 members. Hence the cracking and the calculation of
crack width is relevant only for type 3 members. Hence, the calculation of crack width is relevant
only for type 3 members. We have learnt earlier that Type 1 member
is designed such that, there is no tensile stress in the member under service loads. Type 2 member is designed such that, there
can be tensile stress in the member but the tensile stress is less than the cracking stress
at service loads. For type 3 member cracking is allowed, but
it is limited by limiting the crack width. Thus, the calculation of crack width is relevant
only for Type 3 members and the crack widths are calculated for the service loads. The Type 3 members have regular reinforcing
bars, which are non-prestressed in the tension zone close to the surface, in addition to
the prestressed tendons. This is to limit the crack width. The Type 3 members are provided with non-prestressed
steel, so that the cracking is distributed and the crack width is limited. The crack width is calculated for the flexural
cracks. The flexural cracks start from the tension
face and propagate perpendicular to the axis of the member. This type of cracks is mentioned in the module
of analysis for shear. If these cracks are wide, it leads to corrosion
of the reinforcing bars and prestressed tendons. We had learnt about the flexural cracks when
we studied shear and this flexure cracks, if they tend to become wide then it leads
to the problem of corrosion and the durability. Also the appearance is bad. The crack width calculation is related to
the crack width of the flexural cracks. The surface crack width of a flexural crack
depends on the following quantities: amount of prestress, tensile strength in the longitudinal
bars, thickness of the concrete cover, diameter and spacing of longitudinal bars, depth of
member and location of neutral axis, bond strength and tensile strength of concrete. The crack width calculation is an involved
process. There is a fracture mechanics approach to
calculate the crack width, but the recommendations in the reinforced concrete design are simpler
for our day-to-day use in design checks. But we have to appreciate that the crack width
depends on several factors and the expressions that has been developed is an estimate of
the crack width. When we are experimentally calculating the
crack width, we may find variations in the observed crack width. We are moving on to the method of calculation
of crack width. The IS: 456 – 2000, Annex F, gives a simplified
procedure to determine crack width. The design surface crack width, which will
be represented as Wcr at a selected location in the section with maximum moment is given
as follows. First, we are calculating the crack width
at the location of maximum moment and then for that particular section, we can select
any location along the periphery of the beam and it is selected in the tensile region of
the beam and the crack width expression is given by this equation. Wcr is equal to 3 times distance measured
which is acr times epsilonm, which is a measure of the strain at the location of the calculation
of crack width, divided by 1 plus 2 times acr minus Cmin, where Cmin is the minimum
cover to the nearest longitudinal bar, divided by h minus x where, h is the total depth and
x is the depth of the neutral axis. Thus, h minus x is the depth of the zone of
concrete under tension. To summarize the notations acr is the shortest
distance from the selected location on the surface to a longitudinal bar, Cminimum is
the minimum clear cover to the longitudinal bar, h is the total depth of the member, x
is the depth of the neutral axis. Epsilonm is an average strain at the selected
location. We shall discuss later what is meant by an
average? The values of Cminimum and h are obtained
from the section of the member. Thus, among the variables we can directly
get h and Cminimum from the sectional properties. Next, we need to calculate acr and then we
need to calculate x and epsilonm. Evaluation of acr – the location of the crack
width calculation can be at the soffit or the sides of the beam. The value of acr depends on the selected location. The following sketch shows the values of acr
at a bottom corner A, at a point in the soffit of the beam, B and at a point at the side,
C. The crack width can be calculated anywhere along the periphery of the beam which is under
tension. The zone below the neutral axis is the zone
under tension. We can calculate the crack width at a point
in the corner, which is represented as A or a point in the soffit, which is represented
as B or a point in the side, which is represented as C. For this three points, the distance to the
nearest longitudinal bar has been represented by acr. Thus, the acr can be found out from the design
section, based on the location of calculation of crack width. Usually, the crack width is calculated at
a point in the soffit, which is equidistant from two longitudinal bars. This point is the location of maximum estimated
crack width. Usually it is found that a point at the soffit,
which is in between two longitudinal bars, that point has the maximum crack width. Hence, first we calculate the crack width
at the soffit, which is in between to the longitudinal bars. In this sketch of the cross section of the
beam, Cminimum is the clear cover, s is the spacing of the longitudinal bars, db is the
diameter of the longitudinal bars, acr is the distance from the point of investigation
to the nearest longitudinal bar and dc is the effective cover to the reinforcing bars. Thus, the value of acr is obtained from the
following equation. That acr is the radial distance from the point
of investigation to the center of the nearest bar and this radial distance is from the Pythagorean
theorem is given by s by 2 whole square which is half the distance between the bars, plus
dc, which is the vertical distance from the soffit to the central line of the bars, whole
square minus the radius of the bar, which is db by 2. This expression gives as the value of acr
from a point which is at the soffit and in between two longitudinal bars. In this expression db is the diameter of a
longitudinal bar. dc is the effective cover, which is equal
to Cminimum plus db by 2, s is the center-to-centre spacing of longitudinal bars. The values of db, dc and s are obtained from
the section of the member. Thus, once the member has been designed for
flexure, these variables are available and we can calculate the distance acr, to which
is the distance from the point of investigation to the nearest longitudinal bar. Next, we are evaluating the depth of the neutral
axis x and the average strain at the level of the investigation of crack width. The values of x and epsilonm are calculated
based on a sectional analysis under service loads. The sectional analysis should consider the
tension carried by the uncracked concrete in between two cracks. The stiffening of a member due to the tension
carried by the concrete is called the tension stiffening effect. The value of epsilonm is considered to be
an average value over the span. This is a new concept which we are observing. For flexure, usually we do a sectional analysis
at the critical section, which is cracked section. But when we are trying to find out the crack
width, if you do a cracked section analysis, then the crack width are over estimated. The reason behind this is that, if we just
do a cracked section analysis then we are neglecting the effect of concrete, which is
in between the cracks. The concrete in between the cracks has some
tensile strength and that tensile strength reduces the crack width if you just calculate
based on a cracked section. The effect of tension in the concrete in between
two cracks is called the tension stiffening effect. It reduces the crack width and it reduces
the deflection from the values calculated based only on cracked section. In this figure you can see that the cracked
section is at the location of a crack, whereas the uncracked concrete is in between the two
cracks, which helps the beam to reduce deflection. The contribution of uncracked concrete is
called the tension stiffening effect. When we are calculating epsilonm at the soffit
of the beam, since epsilonm various along the length of the span, we are calculating
an average value, which should include the tension stiffening effect. IS: 456 – 2000 recommends two procedures for
the sectional analysis, considering the tension stiffening effect. The first one is a rigorous procedure with
explicit calculation of tension carried by the concrete. The second one is a simplified procedure based
on the conventional analysis of a cracked section, neglecting the tension carried by
concrete. An approximate estimate of the tension carried
by the concrete is subsequently introduced. Thus, IS: 456 gives us two procedures to do
the sectional analysis to calculate x and epsilonm. The first one is a rigorous procedure, where
we consider a section with tension in the concrete below the neutral axis. The second procedure is the conventional cracked
section analysis, where we neglect any tension in the concrete below the neutral axis. But then the strain is modified to take account
of the tension in the concrete. In this lecture, we shall explain the simpler
procedure which is based on a conventional cracked section analysis. In this type of analysis for a rectangular
zone under tension, the expression of epsilonm is equal to epsilon1 minus in the numerator
b times h minus x times a minus x divided by 3 times Es times As times d minus x. For a prestressed member EpAp plus EsAs is
substituted in place of EsAs. We shall understand the notation of each of
these terms in the next slide. a is the distance from the compression face
to the locations at which crack width is calculated, which is same as h when the crack width is
calculated at the soffit. b is the width of the rectangular zone. In most of our applications we will have a
rectangular zone at the bottom and hence this formula will be applicable, where b is the
width of the rectangular zone. d is the effective depth of the longitudinal
reinforcement. That means it is the effective depth of the
non-prestressed steel, As is the area of non-prestressed reinforcement, Ap is the area of prestressing
steel. Es is the modulus of elasticity of non-prestressed
steel, Ep is the modulus of elasticity of prestressed steel. All these variables are available from the
section and the material properties of the beam. The two variables epsilon1 and epsilons, which
is used in epsilon1 needs to be calculated. Epsilon1 is the strain at the selected location,
which is if it is the soffit, then it is a strain at the soffit and which by similarity
of triangles is given as epsilons, which is the strain in the longitudinal reinforcement
based on the cracked section analysis times a minus x, which is the distance of the soffit
from the neutral axis divided by d minus x, which is the distance of the centroid of the
longitudinal reinforcement from the neutral axis. This expression of epsilon1 is substituted
in the expression of epsilonm. The cracked section analysis of a Type 3 member
should be based on strain compatibility of concrete and prestressing steel. The depth of neutral axis x can be calculated
by a trial and error procedure till the equilibrium equations are satisfied. The equilibrium and compatibility equations
are provided here.We shall discuss the cracked section analysis of a Type 3 member based
on the compatibility of strain between the concrete at the level of the prestressing
steel and the prestressing steel. From this cracked section analysis by a trial
and error procedure, we can find out the depth of the neutral axis which is represented as
x. The following sketch shows the beam cross
section, strain profile, stress diagram and force couples under service loads. The contribution of non-prestressed reinforcement
is also included.In this figure, we see that for a rectangular section b is the breadth,
dp is the depth of the prestressing steel, d is the depth of the non-prestressed reinforcement. In the strain diagram epsilonc is the strain
in the concrete at the top, epsilons is the strain in the non-prestressed steel and epsilonp
is the strain in the level of the concrete at the non-prestressed steel. Epsilon dec is the strain at decompression
and this is added to epsilonp to get the total strain in the prestressing steel. Thus, epsilonp is the strain in the concrete
at the level of the prestressing steel plus epsilondec. The strain diagram considers the strain compatibility
of the concrete and the prestressing steel at the level of the prestressing steel. The stress diagram is available from the elastic
analysis, but here the stress in the concrete is linear with the stress of fc and the stress
in the prestressing steel and the non-prestressed steel are fp and fs respectively. The compression occurs at one-third the depth
of the neutral axis and the tension are represented as Tp for the prestressing steel and Ts for
the non-prestressed steel. From the stress diagram and the force couples,
we can write the expression of the forces C is equal to half times Ec times epsilonc
times x times b, which is the area of the stress triangle. Tp is equal to area of the prestressing steel
times Ep times epsilonp and Ts is equal to As times Es times epsilons. Thus, we are using linear elastic constitutive
relationships to write the expression of the forces. The first equilibrium equation is sigmaF is
equal to 0, which is equilibrium of the axial forces, thus Tp plus Ts should be equal to
C and then we write the expression of TP and Ts and C. We get the first equilibrium equation; x should
be such that this equation is satisfied. The second equation is the moment equation,
sigmaM is equal to 0 and taking the moment about the prestressing steel, we have M is
equal to Ts times the distance between the prestressing and the non-prestressed steel
plus C times the distance between the prestressing steel and the location of C. When we substitute
the expressions of Ts and C, we get an expression of the moment and the value of the moment
should be equal to the moment due to service loads. Thus, x should be such that we need to satisfy
both this equilibrium equations. The compatibility equations are the first
equation relates the compatibility between the prestressing steel and the concrete. x by dp, from the similarity of triangles
is epsilonc divided by epsilonc plus epsilonp minus epsilondec and the second compatibility
equation is the non-prestressed steel, which is d minus x divided by x is equal to epsilons
divided by epsilonc. This equation we are familiar during the analysis
of partially prestressed sections. The constitutive relationships have been considered
in the expressions of C, Ts and Tp and we had used the elastic relationships to calculate
C, Ts and Tp from the respective strains. Thus, this is the procedure of doing a cracked
section analysis and to evaluate x which should be substituted in the expression of epsilonm
and Wcr, to calculate the crack width for a Type 3 member. The limits of crack width are as follows. Clause 19.3.2 of the code specifies the limits
such that the appearance and durability of the structure elements are not affected. The limits of crack width are: it should be
less than 0.2 millimeters for moderate and mild environments and 0.1 millimeters for
severe environments. The type of environments are explained in
table 9, an appendix A of IS: 1343 – 1980. Once we calculate the crack width, we should
make sure that the crack width is within the limits depending on the environment the structure
is in.Thus, in today’s lecture, we first went to the calculation of deflection. We knew the deflection due to gravity loads,
we learned about the deflection due to prestressing force. Then, how to calculate the total deflection,
then we studied the limits of deflection the structure needs to satisfy, we learned about
the determination of moment of inertia. The limits of span-to-effective depth ratio
if we satisfy, then we can bypass the deflection calculations. Next, we studied the calculation of crack
width. First we found out the method of calculation
and next, we found out the limits of crack width. The deflection calculation and the crack width
calculation help us to satisfy the limits state of serviceability. Thank you.

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