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# Mechanical Properties of Concrete:Elastic Moduls, Poisson’s Ratio,Fatigue,Impact

Welcome to module 6 lecture 4. And the last
3 lectures of this module, we looked into mechanical properties of concrete namely composite
strength and tensile strength. Also we looked into what are the factors which effect the
strengths particularly compressive strength and including the test factors. Now, compressive
strength is the most important property of concrete being a material, which can with
stand more compressive strength as a brittle material. So it can a lot of compressive forces
compared to you know compressive stress compared to tensile out pull tensile stresses. But
then other important mechanical properties are elastic modules, Poisson s ratio fatigue,
abrasion resistance and impact resistance. So, today we will be looking into this importance
important properties, So, first we will look into elastic modulus
followed by stress strain behavior poissions ratio fatigue impact and abrassion. So, let us look at relevance of elastic I
mean stress strain relationship and elastic modulus and Poisson s ratio, stress strain
relationship. You know sigma I how we denote stress an epsilon. Elastic modulus and Poisson
s ratio are needed to calculate deformation, deflection. And you know it is necessary to
determine also forces some time in indeterminate structures. So, structure analysis would need
this modulus of elasticity particularly and deflection etcetera are dependent on all this.
So, these properties are relevant and the stress strain curves for material is important.
Because in structural design you know they have their relevance. Now, stress strain curves
of materials we can classify into 4 types for all materials in general. Because, we
have left out elastomass but, typically these are the 4. So we will see that concrete is
a of course, non-linear and inelastic or non elastic material. Now, simple linear and elastic material linear
inelastic material strains versus stress it will follow this stress strain curve during
strain remaining this is non-linear and elastic. So, in elastic material will come back to
its original state when load is withdrawn. But, this is not linear it is non-linear. And if, you follow it up further this is off-course
linear but, in elastic or non elastic you know it does not come back. So, there is a
residual permanent strain and this is non-linear and non elastic. So there is permanent strain
it goes like this at some level of stress you go and when you come back it does not
close in. So there is some kind of a hysteresis and it is non-linear non elastic. And concrete
belongs to this so concrete belongs to this material concrete cement paste or similar
material they belong to this. So, it is actually non-linear as well as in elastic. So, when
you release load beyond of course, certain point not in the beginning is at least 1 by
3. If you go on beyond that there will be some permanent strain when you release the
load. Well, since it is non liner we have to define
the modulus of concrete since non-linear. So, it is behavior is something like, this
and then it follows something like this. So what you do we define something called initial
tangent modulus? Initial tangent modulus you know this is what it is. So the tangent if,
you draw here that is we call as initial tangent modulus. But, I can draw tangent anywhere
that is called tangent modulus in that particular load. But, I can join the origin with this
point and I get another modulus the slop of this line and that we call as second modulus.
And while returning I can also calculate out the same second modulus from this point the
slop of this line. So, for concrete modulus can be defined in different way initial tangent
modulus tangent modulus at any stress or strain level. And second modulus again at any stress
or strain level in fact we, can have a cord modulus, joining between 2 points. You can
have a chord modulus. So joining between 1 point and another point so that is how we
define the elastic modulus for concrete. Now if, you look at stress strain behavior
it is non-linear that is what we have shown. And this is attributed to something called
creep that is the deformation under the sustain loading we will come to that sometime its
1 way of look at it. But, there is something more we will look into. We can measure this
under static condition another reason is under dynamic condition. That means static condition
means you apply the load monotonically in steps. And then see how the deformation or
strain changes. So stress at steps and strain at major corresponding stress. And from stress
strain can find out the modulus of elasticity. But, you see from you know concepts of mechanical
wave travelling in solid media. We know the velocity in the media is a function of velocity
in the media is function of E. You know it is a function of E root E over rho actually
you know. So for velocity is proportional to E over rho. So therefore, modulus of velocity
has something to do with this velocity something to do with the mechanical wave or sonic velocities,
pulse velocities. So, when wave is travelling through the solid you know there is a compression
and rare fraction longitudinal wave travelling through it. Or any wave travelling through
it there are deformations. And therefore, the modulus of elasticity plays a role. But,
what kind of modulus of elasticity that, will be bulk modulus. So, you can relate this 2
anyway this can relate this 2 velocity. So if you have actually obtained velocity in
the velocity of wave travelling through this medium you can actually estimate modulus of
elasticity. Now, that is called dynamic modulus so dynamic
modulus is related to dynamic modulus is related to wave propagation within the material itself.
Because velocity of wave propagation or characteristics you know wave propagation is related to elastic
modulus and one can determine that wave. Now, physically off-course, micro cracking
of concrete at usual load results in apparently permanent deformation and that is why it gives
rise to non-linearity. I mean creep is 1 which but, why finally. So, micro cracking of concrete
at usual load results in apparently permanent deformation, giving rise to non-linearity.
That means you know whenever, you are actually for example, if I have a composite specimen
applied load like this. Now if it is sufficient l by d is sufficiently large let us, say l
by d is sufficiently large l by d is equals to 2 or something of that kind. So, whenever
I am applying load remember that we said the cracks will form here there will be cracking
micro cracks will develop. Because of the tension along this direction
you know you recall we discuss there will be tension of this direction. And therefore,
micro cracks will develop. Now, after side inferior of time suppose we draw the load
this you know at this results in deformation along this direction to. So reduction is a
length micro crack will form and there will be reduction in the length. Now, when you
release this load this micro cracks are not going to fill up they remain as permanent
features in the material. And therefore, the it will not go back to its original state.
So, once the micro cracks have formed it will not go to its original state. So, there is
some kind of this will actually you know manifests themselves this cracks manifest in terms of
the deformation permanent deformation. So therefore, permanent deformation this permanent
deformation and also this gives rise to non-linearity. So, permanent deformation non-linearity both
comes to this. Because suddenly there is an non-linear depends at given stress suddenly
there is an increase the modulus of velocity stiffness reduces. Because the cracks has
occurred, so stiffness would reduce. And therefore, the material would tend to show load modulus
of elasticity or lower slop. So material shows a kind of non-linear behavior to start with
it is like this to start with it is like this. You know it goes like this but, then there
is suddenly there are lot of crack formation. So, modulus changes and this if this continues
if this modulus is changing when it will show a kind of non-linear behavior. And if this
crack is on close then it will show permanent deformation. When it is coming back it will
show permanent deformation. So, that is the idea that is the basic you know explanation
to why it shows non-linear behavior. Now, this has been confirmed by test such
as acoustic emission and ultrasonic pulse velocity test. So micro cracking is generally
confirmed by this. Now, let us see let us look at this, so you
see if I look at if I applied load like this. You know if I apply load like this whenever
the specimen is here, if I apply load like this specimen is here if I apply load like
this. Now, this is longitudinal strain that means strain along this direction, the strain
along this direction dimension change. So strain along this direction that is longitudinal
strain. Then this is transverse strain that means if I measure the strain along this direction
that is transverse strain. And volumetric strain because, volume changes original volume
change in volume divided by the original volume. Now, volume actually can be reduction in the
volume change because this is reduction. This side there is an expansion and net effect
initially there is an expansion then there can be a reduction in the volume as such.
Now what is acoustic emission? So, this is Acoustic emission we just mentioned
about acoustic emission emanate. Before we mentioned that you know you can we, you do
acoustic emission acoustic emission and ultrasonic pulse velocity confirms micro cracking. So, what is acoustic emission? We, just quickly
look in to this for our purpose as much as required for our purpose what is the acoustic
emission. You see if you take let us say piece of timber or bamboo simply and try to break
it. You will find that the sound comes out once its stars cracking once it starts breaking.
Now, this sound is you know some of the strain energy is converted into acoustic or sound
energy. You know, some of the strain energy is converted. So even we apply strain energy
part of the strain energy is converted into surface energy for creating crack. But, part
might go as some sort of acoustic energy because you here sound. So, when you break a timber piece you here
sound. So similarly, a bamboo pieces you hear sound that is basically a kind of when it
cracks. So this crack at the time of cracking there s some sound emission. So, what has
been observed is so that is actually acoustic emission in a systematic manner one can actually
trace this find it out. So, emitted sound as I apply the load, as I increase the load,
initially emitted sound is very less and they go on increasing and the sound starts. You
know lot more sound emitted sound becomes more as you know loading progresses. In other
words permanent deformations or non-linearity is related to all these are related to somewhere.
It is lot of cracking and this is confirmed by a ultrasonic pulse velocity. Also you know
sonic pulse velocity. Now, what is pulse velocity in a solid material is more compared to crack
material. Because, velocity of sonic or mechanical waves
in solid concrete various solid concrete could be of the order of around 4 point 5 kilo meter
per second. So, they it could be of the order you know solid material it could be of the
order of around 4 point 5 kilometer per seconds. But, in here we know it is around 340 meter
per second. So if you have lot of cracking occurring cracks have occurred then pulse
will velocity will reduce. So, pulse velocity starts reducing initial reduction is relatively
slow. But, then it stars reducing as soon as lot of cracks starts appearing. So therefore,
the cracking of concrete is confirmed by this kind of testing. And therefore, if I am trying
to plot the stress strain curve you know stress as stress increases longitudinal strain. There
it shows that the non-linearity of the stress strain curve is largely related to micro cracking
of concrete. And so, is that case is permanent deformation
permanent deformation is also related to micro cracking. So, this is the reason why you get
non-linear behavior as well as you know non closure of the stress strain curve theirs
is a hysteresis. Now, I mentioned that I can determine it you know I can have static modulus
of elasticity or dynamic modulus of elasticity. So, we go to do test in 2 different ways,
static test is done in a, you know monotonic loading. But, what do we do is, we actually
cylinder strength. You know we normally test on cylinder we do test on cylinders so, central
1 third is used metering. This is the portion we use for metering, this
portion we use for metering. So, you put gauges actually in good old days we had actually
lamb s extension meter it is called lamb s extension meter. You know it is called lamb
s extension meter that is what we are using. So, we will put the actually the gauge there
gauge you know there so in the central now this will be 300 mm and central100 and 50
mm. So, you push put the gauge here. But, the deformation is relatively small, So how
do you measure this deformation? It has to be amplified in some manner it was actually
amplified optically during those, days earlier days today. Of course, you can put in several
types. You know you can put in transducer of the kind of strain gauges various kind
of sensing devices. You can put in as to measure this you know deformation. So, several actually
deformation measuring devices are there and one can of strain gauges could be there to
measure this. So, earlier this lamb extension meter was
used so no I am not going into that translucent part of this at the moment it is not necessary.
But, if you are doing a laboratory test then you will be using any one of these, this is
the old traditional way. And then, you measure the deformation there the deformation there
within that small range. But, before doing that what you do is, you crush the cylinder
to find out to the FCY cylinder failure strength. And then load it up to 1 third of FCY l that
is cylinder strength and then unload it. So, this cycles are repeated to see that at
least first of all the you know there is no in there will be no deformation in the beginning.
Because of plate settling down and things like that number 1 number 2 this is prior
to micro cracking. So, what about a small crack that occurs should occur. And then after
that actually the actual test will be done. So, this will reduced down the effect of creep
some micro cracking would occur and then hysteresis also will be reduced. Because of such micro
cracking and permanent deformation. So besides that there s another important issue why we
do it this way. Actually never the load is never applied 1
time in a structure you know we for example, if you consider the room. You are sitting
and if it a, reinforce concrete slab on which you are sitting the before prior to actually,
codes most of the codes would tell you how to do error so roughly about 15 times I think
then you measure the actual strain versus stress. So, load is then applied monotonically
in steps and corresponding step to corresponding strain are measured and it is plotted. So,
in the testing machine of course, stress cannot be reduced to 0. So, what you do is we find
out chords modulus. Finally after this kind is obtained we find
out chords modulus as given by this formula chords modulus. So we find out a chords formulas
so fb. So, we actually find out chords modulus like this you know because this is difficult
to reduce it to 0. So corresponding 2 point 5 MPa this is epsilon this is sigma point
5 MPa. So, fb so corresponding to minimum stress of point 5 MPa then up to 1 third fc
its within the you know kind of savior micro cracking that to savior micro cracking. So,
fa minus fb epsilon a minus epsilon b. So, you find out this is your epsilon b this is
epsilon a, and sigma or f or whatever you call it. So, you find out a chords modulus
according to this or you know sop the code suggest how much where you have to go to test.
So, fc 1 third fc is the elastic limit and often you might find it there itself. So,
this is how we determine the static modulus of elasticity. Dynamic modulus of elasticity is determined
in this prospective view of the diagram. That you see, you have a clamp support which should
not exceed 120 of the length of the specimen. So, this is length of the specimen this is
length of the specimen this is l the length of the specimen this is l and this should
not exceed l by 20. So, it should be less than l by 20 there is a clamp of the support.
So, a specimen rectangular specimen generally rectangular crosses section. You support it
rectangular cross section actually you know clamp it somewhere in between like this. And
then you have an exciter that actually imparts acoustic imparts actually, acoustic pulses
or acoustic you know wave basically there. So, there is a longitudinal wave will be travelling
along this direction they will be travelling along this direction and there is pickups
which will pick up the, you know pulse transmission. So, signal that is transmitted through it
so this picks up and at as you go on changing the frequency, of this sonic wave. That is
tem travelling along this direction that is exciting this at time will come, at certain
frequency there will be resonance occurring there is resonance occurring. Resonance should
be occurring when actually this dimension of this one matches with the wavelength I
mean some sort of matching with the wavelength. Because the mode would depend upon because,
it is a central support your trying to vibrate it. And it will be related to length of it
is related to the frequency resonance frequency is related to the length of the specimen itself. Now, how it is related, it is related like
this. Longitudinal vibration is propagated you know. So, you will have vibration along
this direction, longitudinal vibration. So it would be actually it would be actually
exciting in this mode. So longitudinal vibration so a compression and rare fraction will be
travelling along this direction. And, from 100 to about 10000 hertz at 100 to 10000 hertz.
This range you actually make it to propagate and you will get fundamental resonance frequency.
When lambda is equals to in a 2L. So, when wavelength is equals to 2 l where L is the
length. So, or Lambda is equals to l is equals to Lambda by 2 at that time resonance would
occur. So, you find out the frequency at which resonance is occurring and you know V is equals
to f Lambda. So the frequency you have determined therefore, lambda is equal to length. You
know Lambda is equals to 2L and V is actually just I mentioned a few minutes before. V is a function of E by rho therefore, you
can actually relate e actually you can relate to V is a function of E by rho. And is also
equals to f by lambda and lambda is actually at resonance frequency is known Lambda is
given by this equation l is equals to. Since you know l So, lambda is 2L, in other words
you can find out E. Because f is known l is known and this is. This is 2L is know this is 2L or you know.
So, you can actually find out as 4 n square l square rho 4 n you know 4 n square l square
rho into the to the power 15. Where n is the resonant frequency you know f that I was saying
n is the resonant frequency. So, you determine resonant frequency. So resonant frequency
this is Lambda you know by 2 therefore, velo and then velocity. And then the rho comes
into picture because, velocity is under root E by rho. So, if you want to find out the
E it is proportional to b square so rho b square. So E is actually E will be proportional
to E is proportional to rho V square alright so rho V square. And therefore, you can find
out V is given by n n lambda and from that you can find out. Therefore, knowing the density
of the material so you have a assumed density or measured density of the material. From
that, you can find out the modulus of the elasticity. So, dynamic modulus can be determined in this
manner 2 a first what you do is you actually impart longitudinal vibration or mechanical
wave you know. So, excited through acoustic waves or ultrasonic or sonic waves from 100
to 10000 kilo hertz because the k 10 100 to 10 k alright. So, it is actually sound because
sound is from 20 to 20 kilo hertz. So, it is not ultrasonic this is kilo hertz. So this
is not ultrasonic, this is actually sonic excitation you can provide and then find out
from this modulus of elasticity. So, this is how we find out the modulus dynamic modulus
elasticity. And we said that dynamic modulus of elasticity
is initial tangent modulus. Because no cracks have occurred nothing has occurred in an un
cracked specimen. Is that actually your trying to find out the modulus of the velocity usually
these values are higher. So, now let us look at modulus of elasticity in terms of that,
What are the factors that affect? So, factors affecting modulus of elasticity first one.
You see here, is your cement paste, this is for cement paste. If you see this is for cement
paste, this is stress strain type of cement paste. This is the stress strain carve of
aggregate, generally would show something like that and concrete shows a stress strain
behavior of this kind. So, aggregate stress strain behavior is of course, fixed. You know
that is fixed by I mean you have taken the rock and the wave here will be the fixed. Cement is off-course, is under your control.
So if you improve this cement paste modulus of elasticity concrete modulus of elasticity
also will increase. So, it depends upon both the paste modulus of elasticity and modulus
of elasticity of failure. You know aggregate I mean 1 can think in terms of the range will
of course, be something like 1 by E of concrete should be equals to proportion of the aggregate. P aggregate or proportion of the aggregate,
divided by u of aggregate plus 1 minus P aggregate that is your paste or proportion of paste
divided by E of paste. Or it met might also be written as P aggregate proportion of aggregate
multiplied by E aggregate plus proportion of paste multiplied by E paste. So, this 2
will give you actual and extreme boundary of you knows the concrete modulus of elasticity.
Assuming series and parallel model I am not going to go to the derivation of this. But,
you know this so of course, this he the range this boundary this EC. You have calculated
from this or E C calculated from these they give you 2 different values. So, this gives
you EC 1 let me call it this is EC 2. This will actually give you the wide range of the
modulus of elasticity. But, at the moment it is more important for us to understand
the modulus of elasticity of concrete is a function of modulus of elasticity of the paste.
And modulus of elasticity of the aggregate. So, high modulus aggregate will tend to give
you higher modulus elasticity for concrete and high model. Obviously paste modulus if
you improve by increasing their strength. Then you can actually get higher modulus of
for concrete as well. So, first thing is this other factors which
specimen exhibit higher modulus why because water has got E for E for water for water
is more than E for here both are fluid by the way. So, wet specimen the pores are saturated
with water. For wet specimen pores are saturated with water pores are filled with water they
replace the air. So therefore, modulus of elasticity it is likely to be higher. Because now, it is this pores are filled up
with water instead of air modulus of elasticity of aggregate strongly affects the modulus
of elasticity of concrete. Higher modulus aggregate exhibiting higher modulus of concrete
volume of fraction of aggregate will also play role. That is what we have seen for our
previous formula, I said P aggregate that is proportion of aggregate and P paste. Let us, say paste which will be equals to
1 minus P aggregate minus what so whatever there. So basically proportion will play again
a big role because whichever bounds. You take upper or the lower bound both cases is function
of those 2. You know the series of parallel model which gives you bounds both cases it
will be related to that. Then mostly E can be related to mode FC well compressive strength.
Now, E can be related to compressive strength you know it can be related to compressive
strength. In fact we, have seen that modulus of elasticity governs the strength. Therefore,
we can argue it out that a, strength would be also related to modulus of elasticity.
Dynamic modulus is somewhat close to initial tangent modulus. Therefore, it is more than e static. And general
relationship empirical relationship people have found out for modulus of elasticity,
is E is a function of k into come cube compressive strength to the power n. And you know IS 4562
1000 for design, uses this fc as if ck characteristic strength and n as point 5. So, E is written
can be written as 5 thousand under root fck. It is meant for design this actually conservative
value in fact it will be something like this. You know if I just plot it here, before this
I will just plot it here. It will be something like this, you know if
I plot f c concrete q strength and E then or under root fc. Then you get all sorts of
scattered results and you can plot a curve. So, this is what you is trying to do linear
curve, what some people put you the some lc to the power n may be point 6. Or whatever,
values are and 1 can because this is you know scatters are there. Now, code would use a
value of modulus of elasticity the code you know the actual values will be there. And
the relationship that code tend to use is somewhere there conservative value of E it
takes lower value of E. Because, affects of creep long term behavior or several other
things comes into picture. So codes takes this sort of stand actually, actual values
could be different. Now, since this is you know under root of
c I said it can be fc to the power something of, so different codes may use different one,
but I s 456 off course, uses E is equals to 5000 under root fck. So, you know the line
is somewhere the slop is 5000 and under root fc that is what is used somewhere down there.
Because, it takes long time behavior and other things into account and therefore, try to
use a conservative value. So that is related to what code uses and it can be related to
composite strength. Now, next look into stress strain curve if you look at stress strain
curve at a constant strain rate. Now last class last lecture if you remember we talked
that means you apply load at a particular phase 1 for 14 m p a per second. So, you apply load at a constant rate and
we have seen rate of loading actually changes the behavior you know like behavior changes.
So, supposing I apply now I can apply in terms of strain at certain micro strain per second.
Some strain rate so if I apply load at fixed strain rate, and I told you also that is difficult
to apply. This you need a machine which is capable of applying the server control you
know it is should capable of applying the maintaining the strain rate, at a fixed rate
whenever required and that would require a control or in a server control system and
so on so forth. So, if you are applying a load at a constant
rate you will find that they are depending upon normal wait con you know normal weight
concrete behaves in this manner, while light weight concrete shows this kind of behavior.
Because, you are able to now since strain rate is there so this proportion of the current
curve becomes available and this behavior is similar in compression as well as in flexure
some sort of behavior like this. So, this is the e max or rather f max or sigma max,
maximum stress and corresponding to this we call as a epsilon 0. That is the strain at
maximum stress strain at maximum for light weight this would be the maximum stress sigma
max and epsilon 0 would be this much. So epsilon 0 and sigma max this is the 2 ones and you
see this kind of behavior. So people have tried to fit it an approximate equations to
this so empirical equations to this. And these empirical equations that have you
know representing this is the simplest there are maybe many more complex relationships.
But one of the simplest relationship is of this form, Where E is the initial tangent
modulus and you know, it is which is twice sigma max divided by epsilon 0 and sigma is
a stress at any point, epsilon is a stress at any point, epsilon 0 is the strain at maximum
stress. So, epsilon 0 is strain at the maximum stress. So this sort of formula is available
through which you can define the Stress strain curve and there are complex formula also available.
So this was stress strain curve of concrete. Now, we actually if you I just before I go
to this in flexural design. Therefore, one should be using this sort of
curve and compression you know. So, this is if you remember flexural design flexural stress
strain curve given in the code would be something like this. You know stress concrete stress
strain curve of concrete if you recollect this is strain inflection, you know if, it
is a beam section of a beam if I am looking at a section of a beam this is a reinforcement
this is the concrete, anyway. So, stress strain curve will be like this
stress versus strain curve if you see in idealized stress strain curve it will be 0.002 here,
And 0.0035 stress strain curve. This is actually you know fc or whatever you call it the stress
maximum pillar stress. Now, this is an idealized stress strain curve idealized stress strain
curve of concrete in fact stress strain curve of concrete should look like something of
this kind you know, some stress strain curve of concrete will look like let me draw it
in red color. Stress strain of concrete will look like this
maximum stress and here. Now, this is idealized stress strain curve beyond that you assume
is as if it is straight line. So, actual stress strain curve of concrete will like this and
this is important in flexural design. So, this measured stress strain curve that we
have seen. You know whether stress strain curve actual
stress strain curve observed as we have seen this is idealized into this sort of a straight
line curve. And off course certain factors are applied to bring down the load as well
that I am not discussing at the moment. So, this importance of the stress strain curve
is here because in fractural design you got to use this. So idealized curve is something
of this kind this is arrived from this real behavior this is arrived from this real behavior.
So, there is the importance of the stress strain curve now next to get next property,
which is Poisson s ratio. This is important to find out lateral deformation when you apply
a longitudinal stress initial situation and so on. So therefore, if you apply load when
strains are here and this is compressive stress. So, what happens is this is a longitudinal
strain, this is the volumetric stress that is what we talked about earlier you know,
when we talked of after some pulse velocity and acoustic emission and this is the lateral
strain which is in the opposite direction it is in the tensile direction. So, mu is
defined as lateral by longitudinal strain. So, this is lateral by longitudinal strain
mu is defined by this actually so from this if you try to calculate out the mean you know
ratio for example, this divided by this ratio, you know this divided this ratio. Or let us
say it 400 strain of longitudinal strain corresponding to this may be whatever the value is and similarly,
you know so you can actually find out the mu values. Now, mu values if we calculate out in this
manner and then plot, we will see that which is roughly around point 0.2 it is roughly
around point 0.2 up to about 70 percent of the load for 2 different concrete good old
days people conducted experiments and found out. But, beyond that point actually it starts
increasing it starts increasing can go close to about point 0.3 or so on so forth. With the you know like if your stress to strength
ratios, so when before failure, so before failure Poisson s ratio increases and we understand
this. Because, lot of micro cracking would have occurred and that would have caused lateral
strain to expand you know increase it at a faster rate than the longitudinal strain. And this is manifested here also this is manifested
in this diagram also. Suddenly there is lateral strains increasing not you know here it was
increasing in slow manner. But, it is increase started increasing in the first manner. While,
this is increases although there is a faster increase here I mean you know this is not
really it is almost close to linear sort of situations like variations are there. But,
this increases at a very fast rate this is increases still at a fast rate and this we
can understand from the physical scenario. Because, micro cracking in the wood would
result in lot of transverse deformation longitudinal deformation also would be there but, lateral
deformation will increase. Significantly, because of the micro cracking
so beyond 70 percent of the load actually your Poisson s ratio increases significantly.
And you know up to its constant almost constant up to 70 percent of the strength stress 70
percent of the strength. Generally, it varies in range of 0.15to 0.2
that is what say determined from static modulus step test but, you can determine dynamically
Poisson s ratio also, from ultrasonic pulse velocity test, from fundamental. Or you know
sonic velocity test from fundamental resonant frequency of longitudinal vibration of concrete
beam. Similar test that we did from modulus of elasticity. Now, how do you determine this,
how do you determine this actually, because the bulk modulus you know it is related to
bulk modulus. So, this is the relationship between you know this is involved in the velocity
equation. The young s modulus is related to the bulk modulus in this manner when we use
the Poisson s ratio. So, velocity frequency length and is related to modulus of you know
Poisson s ratio is related to that. So this is the velocity, pulse velocity, resonant
frequency, beam length and Poisson s ratio from that one can find out. So from dynamic
testing also you can find out Poisson s ratio. And generally dynamic modulus again gives
you higher value of Poisson s ratio. Because, you know higher value of Poisson s ratio usually
0.2 to 0.4. And higher the strength of concrete Poisson s ratio is lower so that is related
to Poisson s ratio. Now, let us look at another way mechanical
property is called Fatigue. You know fatigue is related to reversal of stresses. And you
can understand the importance of this one in many structures. The load that is applied
in most of the structure the imposed load there is actually quancy static but, that
we do not call it as fatigue. You know like because even in a building where you are sitting
let us now, if it is on the first floor or you know in a higher floor rather ground floor. The slab that you are actually sitting on
or your chairs are the load you are coming. And then going out so there is a kind of stresses
its quancy static actually what happens is, by enlarge we assume that to be static. And
because, major load will come from the permanent features, such as furniture and things like
that the impose load, human contribution is would be relatively less. So, we are assuming
to be practically you know static so fatigue is not there but supposing it is a bridge
or assume a structure in marine environment. Where waves come and hits here, you know waves
come and hits. So, it is actually under kind of a reversal of stresses will be occurring.
on to vehicular load comes on to the bridge deck. And then it moves bridge deck and then it
moves therefore, this is there is a reversal of stresses there reversal of stresses you
know on to the deck. So therefore, this reversal of stresses is related to what is called Fatigue
stress reversal of stresses is related to fatigue strength. So, let us see and in fact
the load want the any structure or any structural element can carry under reversal of stresses
is much lower than the load it can carry in static conditions. So, Fatigue is related to repeat loading under
lower with number of repetition and it is a fatigue failure. A simple example is given,
supposing I have a wire small wire, and I try to pull it. It will I can never break
it you know binding wire which are used for bonding the steel in concrete while casting
concrete. But, if I try to bent it in this manner that
means bent it in this manner you know reverse is the stress. There is a reversal of stresses
so you bent first you bent it in this manner then bent in the reverse direction. So, first
you bent it in this manner then reverse direction you go on doing this after sometime it will
break. So repeated loading you can you know it fails at much lower load and that is basically
much lower load and that is basically a fatigue failure therefore. Fatigue failure occurs at much lower load
so the mechanism you know that is what is the you know that is what we define as Fatigue.
And let us see how what happens in case of concrete. Now steel has an interesting behavior
we define something called endurance limit. What happens is as you go on number of repetition
you know if you apply a load say load to which is 50 percent of the strength and then apply
reverse the stresses may be bring it to 0. Some reversal bring it to 0 take it to near
100 percent number of I mean one thing the amplitude. You know or the maximum stress positive and
the negative or maximum reversal range of the stress range in which it is reversing.
Let us say we keep it fixed. Now, after certain period of time you will find that it fails.
Now if you lower the load, if you lower the range in which you are operating then, it
will take longer number more number of cycles. So, the number of cycles is the function of
level of the stress, you know reversal where you are reversal of the stress is occurring.
Now, endurance limit is defined in this manner it is much better to define this and we come
back to this. If you look at this diagram on this axis number
of cycles drawn in log scale. Because, 10 to the power 2 etcetera this is log scale
and ratio of the fatigue strength to short term static strength. So, actually if you
are operating at you know fatigue strength you will find that in case of steel, this
is mild steel in tension you do reversal of stresses. But, beyond 10 to the power 7 cycle or some
cycles slightly above 10 to the power 6 cycles and 7 cycles you know. The endurance the fatigue
the load at which reversal this is occurring at 0.4 or so close to 0.4. So, around you
know like 250 mp is the yield strength of mild steel if endurance limit is around 100
close to 100 mpa. So, that means if you apply less then you know 100 mpa and do stress reversal
actually it is not going to fail. Now, endurance limit is defined therefore,
endurance limit is defined as fixed endurance limit that is repetition beyond which the
fatigue strength remains constant. So, What is fatigue strength? Fatigue strength is the
number of repetition or the strength which, you know it can with stand at certain number
of repetition. So, it is related to both repetition and the level of this one. Now, endurance
limit is that level of the strength beyond which actually, lower than that strength it
can with stand large number of repetition. You know because, the fatigue strength because
almost constant. Now, concrete does not have a fixed endurance limit concrete does not
have any fixed endurance limit concrete does not have any fixed endurance limit. In tension concrete in tension it shows continuously
if you know the fatigue strength will go on reducing as you increase the number of cycles.
So, here also concrete in compression same thing. So therefore, how do we define the
strength there is no endurance limit how do define this enduring limit. So, this you know
steel is advantageous in that sense from usefulness point of view I mean what I say design calculation
point of view or understanding point of view because you have got a fixed endurance limit.
So therefore, if you know it is likely to come under cyclic load you assume, that it
would it can withstand around that endurance limit is you know is which will be the, which
you can use in design. In case of concrete you cannot do that, because
it will go on reducing. So, what we will do is we take the fatigue strength corresponding
to 10 to the power 6 cycles as the fatigue compression strength. Similarly, in this case
10 to the power 6 tensile fatigue con tensile strength. So, fatigue compressive strength
and tensile strengths are defined with respect to 10 to the power 6 cycles because it will
go on reducing. For steel of course, you can use the endurance limit, so, it will go on
reducing it with number of cycles increased number of cycles. So, this is how we define. Now in concrete
2 types of fatigue failure is distinguished the one if a Sustained load near the static
strength, under increasing load causes failure. So, its static fatigue or creep ruptures. Now, concrete shows creep at ordinary temperature
that means if you put a load we will discuss this in subsequent module. That means if you
keep the load constant and over the time period over the period of time we will find the deformation
is increasing. And it can have a failure also depending upon the situation. So, under sustained
load concrete exhibits kind of deformation and that is related to phenomenon of creep.
So, if you sustain load near the static strength, now fatigue is also nearly sustained because,
you are reversing the stresses and near the static strain so you increase the load and
it cause the failure. And this is related to static fatigue or creep
rupture you know. So, if you are reversing this stresses close to the static the static
strain. Then, there can failure can be by creep rupture because sustain load is sustain
very static load is sub strain plus minus something is always occurring you know plus
minus something is occurring. So, this sustain load can result in creep rupture. Because,
case where you are operating at much lower level. But, then it because repeatedly you
have done you know reversal of stresses have been done over a large number of cycles so
these are the 2 situations. And we can see the behavior in this manner.
If, you look at this diagram usually you know in this case what you are doing 50 you are
is like this is excess is strain this is stress rapid loading stress strain curve is like
this. And here it might fail. If, you finish it within 3 minutes it will show you failure
like this. I do not have a here I am only controlling the rate of loading. In terms
of load part you know pie unit, time not in times of strain control situations. So, what
will happen very rapid load loading no you the micro cracks will be formed. But, at certainly there will be a sudden brittle
let us say 3 minute failure up to failure. Which could be the case in case of static
monotonic loading so it will fail somewhere there not much deformation or such thing it
will not show it will show this if you have a strain curve control situation. So, a kind
of what you call strain softening would occur you know as the strain increases strength
it can load it can carry is lower. So, if you know if go on increasing this is
mono normal stress strain diagram but, if is 3 days this is you know this about 3 days
you take if you 3 days if you take then you find that, if the load for 3 days. So, it
is 80 percent of the static load, it will actually 80 percent of the static load. It
will fail keep the load for 3 days. It might fail somewhere 80 percent I mean there is
a notional thing one cannot take it for all the time granted the values might differ from
test to test. And if you keep it for 50 days it is actually fail at somewhere there. So,
depending upon this is 20 minutes 3 minutes 2 days 3 days and 50 days. So increasing the
test duration then which means that my rate of loading I am decreasing and at lower rate
of loading we have seen it fails earlier. So, 50 days same load you are applying in
50 days you find that the load carrying capacity is much greater than less. So, I can actually
have a kind of a envelope. So you can call it static fatigue failure envelope. Because,
this is related to micro cracking or creep behavior and so on. So, very close to the
load quickly, if you load it fails at higher load you take a longer period of time it will
actually, fail after certain period of time. Because, this is sustain for fifty days so
it will actually fail. So this so you know this is related to static fatigue failure
so, close to this one if you take you know if you do reversal for 50 days it will fail. So, if you do reversal somewhere here if you
do lot of reversal there time frame it will be related to this actually failure would
occur even though you would not gone too much of a cycle. For example, for 0 cycles this
situation is for 0 cycles. So, even if I have got too much of cycle it will still fail so
this is actually related to static failure, static fatigue failures you know so or creep
rupture as we are calling it a creep rupture. And this is again strain versus stress graph.
So, limit of strain after approximately 30 years. So if I look at it static rupture curve
is here maximum strain after 30 years this envelope gives you 50 days. We looked into
so, 0.6 of the strength 60 percent of the strength if you load it to after 30 years
deformation will be somewhere here. You know after 30 years deformation will be somewhere
here off course, it will not fail but, if you put it at 80 percent after 30 years it
will actually fail. So this is it and this is the usual short term strength which is
about 3 minutes or so, 0.3 is less than 30 percent of the strain it will actually after
30 years deformation will increase but, it will still not fail. So, it fails up to something
like 80 percent of the static strength or 30 after 30 years and so but, if you have
something more than that it will fail. So this is how you know this is related to this
fatigue. And if you do reversal of stresses one cycle
the behavior is like this. See, 1000 and first so that this hysteresis area reduces significantly.
And then this area would increase. So, what will do is you know the fatigue we have introduction
to fatigue I have given to you. Now, and you can see that after long period of time because
of long cycling number of number of you know cycling causes more micro cracks to appear.
So therefore, this increases in the beginning there is a reduction but, under long cycling
actually no micro cracks formed under fatigue loading. Because, there will be frictional
and heat generation as it happens or whatever mechanism are there, so fatigue causes failure
in this one where something called modified good mans diagram. Here, this diagram helps us in finding out
what is the strain level or reversal of stress concrete can with stand. For example, this
curve you know this is a minimum compressive stress line this is the minimum tensile stress
line. Now, Uni axial compression if you are operating at 40 percent of that strength level.
Or let us say 20 percent 10 percent of the stress level you can do reversal up to this
and come back in compression mode only. So, failure would occur at 10 to the power 6 cycles.
And if you are applying let us say tensile stresses you can go up to 20 percent of the
tensile strength 30 percent of the tensile strength. And reversal can occur from 10 percent to
the compression to 30 percent of the tensile. If, this in compression Uni axial tension
and compression this is in flexure. And if you are doing purely tensile so that 10 percent
of the tensile load you can go up to this and come back. So, Modified Goodman diagram
can be used to find out what level of stress reversal you can do, what level of stress
reversal you can do at what level of static strain static stress level at 10 percent of
the stress level you can reverse the stresses given form this diagram. Say if it is compressive
stress. If, it is tensile stress one can determine from this diagram so, Modified Goodman diagram
can be used to find out what level of ranges of stress you can use. Impact Strength is again can be related to
compressive strength and it depends strongly upon aggregate type. Lower for lower water
stored concrete and lower m.s.a improves impact strength and that is shown by these diagram
different types of aggregate. This is the number of actually impacts actually it can
take the weight is measured depending upon compressive strength. Both impact and both impact strength as well
as you knows Abrasion their function of compressive strength. And impact is loading at a very
high load you know high rate very fast rate. So, rate of loading is very high impact is
related to that. Abrasion is again related to compressive strength,
you know as you were rubbing how much now how many what kind of depends upon different
types of test. Actually you can rub it and find out percentage of the material will be
generated. And this is also related to compressive strength. So Bond reinforcement again for plane buds
and reformed buds related to compressive strength. So, bond strength you know so bond by pull
out test one can find out. So what we find is all these tests are all these strengths
are related to compressive strength. Bond fatigue not bond impact and abrasion resistance.
These are all related to compressive strength. Essentially, higher the compressive strength
of concrete all other properties can be related to that and they improve. So, higher strength
means better impact resistance, better abrasion and better you know better abrasion, better
impact resistance or impact strength. So, all properties actually higher modulus of
elasticity. So I think we have discussed in this one Elastic
modulus, Poisson s ratio and fatigue and impact with this module will complete. And next 7th
modulus we will look into creep and fatigue.thanks.