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

loading. And when you unload it comes back just back to the original with no residual

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its original state when load is withdrawn.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

initially apply a several cycles of loading and unloading at 1 third of the estimated

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,

even loading. There will be several loading unloading that would occur actually during

construction phase or even later. On several loading and unloading that occurs.

So therefore, determining the stress strain curve after some loading and unloading that

what. So, what do you do we actually do about 15 times loading and unloading. You know the

codes most of the codes would tell you how to do error so roughly about 15 times I think

most of the codes would say fifteen cycling 15 times. You know loading and unloading and

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

of constant rate of loading. Now, you can do 2 tests by constant loading rate of loading,

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

you apply load you know, you can see this is compressive loading this is tensile the

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.

Bridges this vehicular load comes in vehicular load comes on to you know vehicular load comes

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

repeated loading materials fail at load lower than static strength. The failure load being

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,

load is sustain very close to the static strain. Repeated sighting loading is off course other

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

sustaining the load for 50 days. 3 minutes stressed and rapid loading stress strain curve

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

failure at some higher load. So, it fails somewhere there but, if you are loading under

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.

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