 Welcome to lecture series on advanced geotechnical engineering and we are in module 3 and lecture 8 on compressibility and consolidation of soils, module 3 lecture 8 on compressibility and consolidation. So in this lecture we are going to discuss about the compression curves particularly under different conditions like when it is subject sample is subjected to compression and unloading that is recompression and we also discuss about what will happen when the load increment ratios are kept longer or when you have low load increment ratios and what will happen if a particular load is kept for long duration that is the effect of the secondary consolidation. And thereafter we will introduce ourselves to you know the need for the radial consolidation and then in the next lecture we will be discussing about the methods for accelerating consolidation supplements. So let us before going into the details of whatever we have just discussed, now let us see now let us look into the effect of sample disturbance on E log sigma dash curves especially for normally consolidated and war consolidated clays. So first consider normally consolidated clays and in this particular figure where void ratio versus sigma dash on the logarithmic scale is shown here and here if a normally consolidated soil that is the soil which is actually having higher water content and you know where you know this pre consolidation pressure the present effective pressure is only sigma naught dash and that is the maximum pressure whatever the soil has seen then in that case in the field what we actually see is the straight line is the virgin compression curve. Also called as field normally consolidated line or field compression curve but when we take a sample and if it is subjected to you know generally the sampling is done by using an appropriate sampler and which actually gives the least disturbance but if you have a laboratory specimen which is tested which is actually obtained with careful recovery then what we get this E log sigma dash curve is like this is indicated as the curve 2 in the figure but if the same specimen is actually remolded and reconsolidated then actually what will have is the remolded specimen will have you know this type of E log sigma dash curve. So if you look into this all 1, 2, 3 the field compression line and laboratory undistributed sample which is you know which is indicated as curve 2 that is the from the consolidated obtained from the consolidation test on a carefully recovered specimen and curve 3 which is obtained from again from consolidation test on a remolded specimen. So if you look into this at a certain stress when it actually is being subjected then all they will actually meet at a point which is the point for E naught where E naught is the initial void ratio at sigma naught dash that is the current overburden pressure. So if you look into this the difference the distinct difference between field compression curve and these laboratory undistributed sample is because of the you know the degree of disturbance what actually enters into the testing and the curve 3 which is you know if the same soil which is actually tested in 2 is say remolded and reconsolidated then what we get is this so called E log sigma dash curve which is you know in this form and where actually it appears and one more thing is that if you look into this for the remolded specimen very difficult to distinguish you know where is the you know the pre consolidation pressure. If you look into the as you go away from the you know with degree of disturbance the very difficult to distinguish its current as the stress history what the sample actually has been subjected. So in case of a normally consolidated clay what we have is that the field compression curve and you know the E log sigma dash curve for the laboratory test specimen tested with carefully on a carefully recovered specimen and you know the curve 3 which is obtained from the remolded specimen they actually meet at a point which is actually approximated as 0.4 E0 or 0.4 to E0 according to Terzegent pack 1967. Now let us see what will happen for a over consolidated clay. So as you all know that you know the soil actually has seen a pressure which is actually much more than the current present over burden pressure. So you can see that so because of this the initial portion of this curve is actually flatter and then once this stress which is actually is transferred to the soil is actually more than what it has actually seen then it will change into the you know so called normally consolidated mode again. So you can see that you know the two limbs which are actually there this is actually the portion AB and portion BC. So this for this dotted line is the field compression this dotted line BC is the virgin compression curve for the so called you know of the in the we can be obtained for the OC clay in the field and AB portion and BC portion these two are the you know two segments which will be there and the slope of this line AB is recompression index and slope of this line BC is the you know the compression index CC. Now if you are actually testing an undistribute sample the you can see that the degree of in the initial portion is flatter and then you know again it changes into the normally consolidated mode and again it meets at a point C and when the rebound actually happens for a that means there once the unloading actually happens here in the laboratory CF segment is actually is the laboratory you know unloading portion. So you can see here that the according to Schmettman 1953 he concluded that the slope of this line and the slope of this line AB are almost identical. So Schmettman 1953 he concluded that the field compression branch AB as approximately has the same slope as the laboratory unloading branch CF. So in the in case of the over consolidated clay and what we have is that you know the two segments AB and BC. So that is the reason why while discussing you know the settlements and you know we have to see the get the pre-consolidation pressure after getting the pre-consolidation pressure then you have to if you are actually having a you know if this is the field compression curve which you are adopting for the design. Suppose if this is the stress which is actually has been subjected then you have to calculate for the settlement for this portion and you have to calculate for the settlement for this portion. So that is what actually we have discussed in the previous lecture. So in this the curve one is the E log sigma dash variation for undistributed over consolidated clay in the field. So this is actually idealizing for the field conditions but if we are actually having the sample recovered from the same side same portion where actually the with a little degree of disturbance then we can see that this how what we get is that laboratory undistributed sample will yield E log sigma curve like this and then you know CF is the unloading branch and the slope of this AB and CF are found to be identical and one more thing is that CR is approximately one-fifth to the one-tenth of you know CC. So in the case of absence of data and it can be assumed that the CR the recompression index is one-fifth to one by ten of CC the CC is nothing but the compression index the slope of the normally consolidated portion. Now let us see what will happen you know if we are actually having deviations from the consolidation test and you know if the standard consolidation tests are conducted with a soil specimen having a thickness of say 25 mm or 25.4 mm in which the load on the specimen is doubled every 24 hours. This is the standard the standard consolidation test or idometer test are conducted with a soil specimen having a thickness of 25.4 mm in which the load on the specimen is doubled every 24 hours in the laboratory. So this means that the delta sigma by sigma dash is equal to one the delta sigma by sigma delta sigma is nothing but the change in load towards the initial load. So delta sigma by sigma dash is equal to one load increment ratio is always maintained as one. So the question is that what will happen if load increment ratio is greater than one what will happen if load increment ratio is less than one. So what will be the effect of any deviation from the standard procedure like if you do not keep for 24 hours if you keep for longer hours what will happen if you keep for less than 24 hours what will happen. So the striking changes in the shape of the compression time curves for one-dimensional consolidation tests are noticed if the magnitude of delta sigma by sigma dash is less than 0.25. So it has been it has been noticed that the several investigators actually have carried out and the striking changes in the shape of the compression time curves were obtained you for the one-dimensional consolidation tests if the magnitude of delta sigma by sigma dash is less than 0.25. So according to Lenard's and Altersheffel 1964 they conducted several tests on mixico-city clay with different load increment ratios and with variation of excess pore water pressure measurement with time. So conducted the several consolidation tests on mixico-city clay with different load increment ratios and variation of excess pore water pressure measurement. So in this particular slide time which is actually shown on the x axis the specimen height is actually shown here you can see that this is delta sigma by sigma dash greater than or equal to one and here delta sigma by sigma dash less than 0.25 that means that as the delta sigma by sigma. So this is equal to or the greater than one or found to be somewhat in this line and when delta sigma by sigma dash is less than one it actually migrates in this direction. So for delta sigma by sigma dash less than 0.25 the position of the end of the primary consolidation is somewhat difficult to resolve and also the C alpha by Cc the C alpha is nothing but the secondary compression index to the primary consolidation the compression index in the primary consolidation range increases with the decrease of the load increment ratio. So what we can see is that for delta sigma by sigma dash less than 0.25 the position of the end of the primary consolidation is somewhat difficult to you know resolve will not able to whether you cannot say whether it is here or here or and also it has been noted down that you can see that the distinct difference of excess pore water pressure dissipations which actually happens excess pore water pressure here is that you know delta sigma by sigma dash is actually happening less than 0.25 it is actually much before and then whereas in case of delta sigma by sigma dash greater than or equal to one it is actually happening at much later that can be understood but here you know the reduction in the early you know this position can be understood because of the low load but here what is actually happening is that here the very difficult to distinguish the so called you know the end of the primary consolidation process and another thing is that the ratio of C alpha by Cc increases with the decrease of the load increment ratio. So as we go in this direction as we go in this direction you know what will happen is that the ratio of C alpha by Cc increases C alpha by Cc that is ratio of secondary compression index to compression index increases. Now when delta sigma by sigma dash is say less than one so here void ratio and log sigma dash are plotted here and this is a case the standard E log peak curve for delta sigma by sigma dash is equal to 1 if it is less than 1 it actually will be above the you know the conventional hydrometer test result if it is greater than 1 it is actually somewhere here. So if you look into this here for a given load and more you know when you have got delta sigma by sigma dash greater than 1 so the void ratio you know decrease is high but when you have got delta sigma by sigma dash is less than 1 the percentage decrease in void ratio is low when you relatively when you compare with these two. So when delta sigma by sigma dash less than 1 the ability of individual soil particles to readjust to their position is small. So when delta sigma by sigma dash is less than 1 the ability of soil particles to readjust to their positions is very small and which results in a smaller compression compared to delta sigma by sigma dash greater than or equal to 1. So what we are actually trying to conclude from this slide is that when delta sigma by sigma dash less than 1 the ability of individual soil particles to readjust to their positions is small which results in a smaller compression compared to delta sigma by sigma dash greater than or equal to 1. So that is the reason why we actually have a distinct difference when you actually have variation of the delta sigma by sigma dash. Now what will happen let us see that the load duration suppose if you conventionally we keep 24 each load increment ratio which is actually with delta sigma by sigma dash is equal to 1 is maintained for 24 hours. So in a conventional consolidation test in which a small specimen left under a given load for about a day a certain amount of secondary consolidation takes place before the next load increment is added. So that is possible but so in this particular slide what we have actually seen is that void ratio and the sigma dash that is the log scale that is the you know the log sigma dash curve is log sigma dash axis. Now what we are actually having is that and curve A is the end of the primary consolidation load duration just sufficient for the primary consolidation. So load duration is just sufficient for the primary consolidation when you plot and this what is actually referred here in as curve A and curve B is the load duration with 24 hour load increment duration that means that this test actually conducted with 24 hour load duration. When you have more than 24 hours load duration you can see that the curve actually shifts towards when increasing load duration e log sigma dash curve actually shifts towards left side. So what we have is that you know the sigma C dash the pre-consolation pressure is also changing previously here it should be this much. So there is a shift in the pre-consolation pressure magnitude and when we actually have you know just sufficient for the end of pre-consolation you actually have different pre-consolation. So it can be noted that different values of magnitudes of pre-consolation pressure. So for soils that display secondary compression volume changes occur after the end of primary consolidation because not all soils actually express secondary consolidation. For soils that display secondary compression volume changes occur after the end of primary consolidation if each load increment is in place for a longer period. So most common for the or the normally consolidated and normally likely over consolidated soils up to OCR is equal to 2 and they are referred as the likely over consolidated soils. So if each load increment is in place for a longer period what will happen is that normally consolidated or lightly over consolidated soils they actually subjected to a certain degree of you know secondary compression and heavily over consolidated soils are less prone for the secondary compression. So because heavily over consolidated soils are less prone for the secondary compression and the shape of the curves becomes less distinct with a longer period of duration longer period in duration and it is more difficult to difficult sigma not. So with shapes of the curves they keep on changing and with longer duration it is difficult to predict it difficult to you know difficult to determine sigma not C that is the pre consolidation pressure determination is difficult here. So one if you see this actual portion here and this volume change actually occurs due to the secondary compression this volume change because that same load increment that load duration kept for a longer duration. So what will happen is that the secondary compression component is actually entering into the phenomena into the picture and it actually results in the volume change and it causes this is attributed to the secondary compression. So what we understood is that if you are actually keeping the load duration slightly for longer hours then what we actually have seeing is that there is you know certain amount of for over consolidated soils you know which is has the negligible influence because they are actually not having much of you know the secondary consolidation component will be very very low but when you have the normally consolidated soils and over consolidated soils likely over consolidated soils we have you know we can see the volume changes due to secondary compression significantly and also one more thing we have discussed is that the shape of the curves become less distinct with a longer period and duration and it is more difficult to determine pre consolidation pressure sigma dash C. Now what will happen let us say that sample thickness normally the sample thickness to diameter ratio we actually have said that diameter to thickness ratio is actually maintained as 3 basically to reduce the side wall frictions that is one of the reasons for that and the second reason is that you know to have the you know these epsilon r is equal to 0 and also to represent the one dimensional consolidation only. So for similar load increment ratios proportion of the secondary to primary compression increases with a decrease in the sample thickness. So for if you are having a for a similar load increment ratios the proportion of secondary to primary compression increases with a decrease in sample thickness and also the ratio of secondary to primary compression increases with a decrease in the load settlement ratio. Now let us see the secondary consolidation or creep effect. So in this particular slide where what we see is the E log sigma dash curve again but we have indicated here 3 curves one is curve A and curve B and there is one curve which is actually for example after 10000 years of consolidation. So in the curve A what we see is that the sedimentation is actually happening and you know this is the current war burden pressure and then you know further it is undergoing normally consolidation and the curve B is you know after 10000 years if the sample is actually subjected to 10000 years of consolidation when it is actually subjected then when it is consolidated and then what we get is this variation. So the continued secondary consolidation of a natural clay deposit has some influence on the pre consolidation pressure. So the continued secondary consolidation of a natural clay has some influence on the pre consolidation pressure sigma dash C. A clay that has recently been deposited and come to equilibrium by its own weight can be called as young and normally consolidated clay. A clay that has recently been deposited and comes to equilibrium by its own weight can be called as young and normally consolidated clay. And curve A is for young normally consolidated clay and wherein sigma C dash is equal to sigma naught dash and at E naught that is the at this particular point that is the initial world ratio and this is the curve here. If the same clay is allowed to remain undisturbed for about 10000 years and for example under the same effective under the same effective war burden pressure sigma naught dash there will be creep of secondary consolidation this will reduce the void ratio from E naught to E1 because of the creep effect the void ratio reduces from E naught to E1. Here you can see that the constant effective stress the sigma is not dash which is actually no change actually happened assume that for the 10000 years then you know the void ratio changes to E1. Now so the clay may now be called as so this clay is actually called as age normally consolidated clay. So previously if there is fresh deposition of clay is happening and the reasonably deposited and comes in equilibrium by its own weight then it is called normally young normally consolidated clay and if this you know subjected to certain amount of secondary creep and then will reduce the void ratio from E naught to E1 as shown in the slide and this clay may be now called as the age normally consolidated clay. Now the clay at E1 that is at this particular point clay at E1 now the initial void ratio for this is that E1 which is actually after 10000 years and the effective war burden pressure is sigma naught dash that is this pressure sigma naught dash and the curve looks like B and the pre-consolidation pressure when determined with the standard procedure will be sigma 1 dash. So when you do the consolidation test on this sample which is actually collected after 10000 years with the sigma 1 dash is the sigma naught dash as the you know initial war burden pressure what you look like is now that pre-consolidation pressure is actually something nothing but sigma 1 dash. So the pre-consolidation pressure when determined by standard procedure it will be sigma 1 dash. Now sigma C dash is equal to sigma 1 dash and which is greater than sigma naught dash so this is sometimes referred to as the quasi pre-consolidation effect more pronounced in the plastic place. So this type of behavior is actually more pronounced in the place of high compressibility and high plasticity that is CH type of place that is the more pronounced in the plastic place. Joram in 1972 a given estimate of the relation between the plasticity index and the ratio of the quasi pre-consolidation pressure to effective war burden pressure that is sigma dash C by sigma naught dash for the late glacial and post-glacial place. So for glacial and post-glacial place so you can see that as the plasticity index values decrease increasing there is an increase in you know ratio of quasi pre-consolidation pressure to the effective war burden pressure sigma naught dash. So with increase in plasticity index then you know this is actually more prominent and it actually increases sigma C dash by sigma naught dash which will be high. So the quasi pre-consolidation effect will be very very pronounced or the significant in plastic place. So you can see from the table which is given here that increase in the plasticity index value where the sigma C dash by sigma naught dash sigma C dash is nothing but quasi pre-consolidation pressure and sigma naught dash is nothing but effective war burden pressure. So sigma C dash and sigma naught dash which is actually they are for late glacial and post-glacial place. Now after having discussed the effect of load duration and effect of you know the load increment ratios and effect of secondary consolidation and effect of sample thickness. Now let us look into the compression curves and different forms of representation and particularly for compression and the swelling that is in the unloaded zone. So if you so if a series of load increments is applied to a specimen which is initially wet and resulting equilibrium states will appear and as shown below. So if you look into this here this specimen this is the E sigma dash that is the compression curve E sigma dash curve where you can see that the sample has been subjected to you know this curve and up to load E naught to E2 then unload it and then reload it and then you can see that you know it joins here and then it goes it get loaded further and then after here unloaded again and reloaded here. So you can see that the cyclic over-consolidated sample recompressing and cyclic this is simply over-consolidated or the swelling is also called this limb is actually called as the swelling portion that is the unloading portion that is the soil actually bounces back when the load is actually removed. So this is actually plotted on the E lag sigma dash curve when it is actually plotted you can see that this all this whatever may be the loading unloading and loading unloading cycles we have and this line joining this line this line is actually indicates the compression line which is actually obtained and close to the field compression line for normally consolidated soils. So E naught E1, E2, E3 and then unloading and reloading and unloading and reloading. So whatever the cycles we have it actually follows the that slope. So this is further indicated the outermost curve connecting the points when the specimen is in lose possible states is referred as the virgin compression line that is the this line is actually referred as the virgin compression line E naught E1, E2 is referred as virgin compression line and because it is strictly the line connecting the equilibrium states whereas equilibrium states in the sense that for that type of increment load the consolidation is completed and ending and it is strictly the line connecting the equilibrium states whereas the consolidation is a term used only for the time dependent process between any pair of. So it is also you know this is actually called as virgin compression lines because it is strictly the connecting the equilibrium states whereas virgin consolidation is something like is the consolidation is we have a time dependent process between any pair of equilibrium states. So the equation for the virgin compression line is given by E is equal to E naught minus Cz by log to the base 10 by sigma dash by sigma naught dash. So the outermost curve connecting the points when the specimen is in the lose possible states is referred to as the virgin compression line rather than as virgin consolidation line the reason is due to the fact that the line connecting the equilibrium states whereas the consolidation is the term used only for the time dependent process between any pair of equilibrium states. Now this equation of virgin compression line further represented as V in the in terms of sample volume in terms of specific volume that nothing but nu and is equal to nu naught minus lambda natural logarithm of P by P naught which is nothing but sigma by sigma naught and wherein the lambda is a characteristic constant and the nu naught is the initial specific volume. So specific volume is nothing but 1 plus E and then E is the these are actually related as the water content specific gravity and void ratio they are related this one and that. And equation of virgin compression line will be denoted as nu is equal to nu naught minus lambda natural logarithm of P by P naught and equation for swelling and recompression loop will be represented by a single straight line of the form that is nothing but nu is equal to nu naught minus kappa this kappa is the for the you know the recompression loop or the swelling loop. So which is something analogous to swelling index or recompression index natural logarithm of P by P naught where lambda and kappa are the characteristic constants for the soil and virgin compression lambda line represents the irreversible process whereas the swelling and recompression kappa lines represent the reversal process. So lambda line represents the so this is this is when you plot this line is nothing but the lambda line. So the lambda line represents the irreversible process because the soil subjected to irrequerable changes and it is actually called as irreversible process whereas the swelling and recompression kappa lines represent the reversible process. Now this is further indicated as equation for normal compression line OACD is given by E is equal to nu naught minus Cc into log sigma dash z. So here you can see that E versus sigma dash z for the consolidation compression is actually shown and this is for the one dimensional compression and this swelling. So OA and it is actually subject at O it is subjected to an effective overburden pressure of sigma naught O sigma naught sigma naught sigma naught and at point C it actually has been subjected to sigma dash y that is the it is called as the yield stress and the D is again the load was actually terminated here. So you can see that at point C you know the load you know is stopped and then gone unloaded that is called unloading and then that is called also called as the swelling line and this is called recompression line. So BA is the recompression path and then it follows the joints to C and then follows further to D. So this is again in the normal consolidation movement. So the same thing is represented in the void ratio versus log sigma dash z then what will happen is that this is the slope of the Cc that this is called as the normal compression line normal compression line and point from point A and C there at this point and A and C are at this point in the case of normally in case of log sigma dash z you will see at this particular point and B is at this point where that at the end of swelling or a unloading effect and then you know so you can see that sigma naught dash is the overburden pressure here sigma dash y is the yield stress here and so you can see here that E kappa is this particular one at this particular point where you know this recompression or the swelling was complete. So this equation for the swelling and recompression lines are given as E is equal to E kappa minus Cs into log sigma dash z and E is equal to E naught into E is equal to E naught minus Cc log sigma dash z is the normal compression line for OACD the equation for line OACD is given by E is equal to E naught minus Cc log sigma dash z and for swelling and recompression line it is actually given as you know ABC which is nothing but E kappa minus Cs log sigma dash z. Now since delta V that change in volume is equal to change in void ratio and log x to the base tan is equal to 0.43 times natural logarithm of x then we get Cc is equal to 2.3 lambda and Cs is equal to 2.3 kappa. So by using this we can actually you know once you substitute this natural logarithm into this logarithmic of base tan by using this we actually get into this particular relation they are one and the same. So for over consolidated soil at a point such as B the yield stress ratio y naught is given by y naught is equal to sigma naught dash y by sigma naught this. So for over consolidation soil for over consolidation soil that is at a point such as B the yield stress ratio y naught is given by y naught is equal to sigma naught dash y by sigma naught dash where sigma naught dash is the current stress and sigma naught dash y is the yield point which lies at the intersection of the swelling through point B and for the normal compression line. So within normal compression line that is this is so we are actually defining here the yield stress the yield stress is nothing but sigma dash y ratio of sigma dash y to sigma naught dash. Sigma naught dash y is actually obtained at this particular point and where it actually has from the normal compression line where it meets this particular first limb where the unloading unloading there which actually meets the normal compression line and that is actually is rendered as sigma dash v and which is sigma dash by v by sigma naught dash where sigma dash is the current, sigma naught dash o is the current stress and sigma dash y is the yield point at which it lies at the intersection of the swelling line through B with normal compression line. Now let us see the effect of one dimensional consolidation and swelling of soil in the ground due to deposition and erosion. So we know that deposition and erosion they are the continuous process but on the soil layer which is actually prone for you know particularly the normally consolidated and now likely over consolidated soils they are actually prone for changes because of these actions like deposition and erosion. So considering this particular slide an element soil element which is actually subjected to vertical stress sigma dash z and horizontal stress sigma dash h and when the erosion actually when the deposition happens what will happen there is a increase in the soil overburden above this particular point that it is subjected to high stress due to the so you can see that z dash B is increase here but when there is erosion takes place the stress on this actually decrease for example you can see this is the you know when the erosion takes place from here to here unloading is actually happening that is sigma dash C. This is actually indicated by as E is equal to WGS so from void ratio and now it is actually indicated with water content here and log sigma dash here you can see that AB is the you know recompression line. So the water content changes from A to B you can see that the water content decreases and the R void ratio decreases so A to B and from B that is sigma dash B when it is unloaded BC is the path which actually takes and stresses this point is sigma dash C. So here what is actually happening is that when you have K0 pressure at rest is defined as sigma horizontal stress by vertical stress when K0 is equal to 1 when it is actually having elastic equilibrium then sigma dash z is equal to sigma dash h so this is that K0 line which actually indicates so this is sigma dash h on the x axis sigma dash z on the y axis so you can see that when the case of deposition is actually happening when the deposition is actually happening you can see that that A to B it actually decreases and B to C you can see that here the K0 values are actually you know can actually go more than 1 particularly because of this stress effect. So the one dimensional consolidation in the swelling of soil ground due to deposition erosion is actually depicted and these actually you know represent the different water contents because of the type of actions which actually have been subjected. So for normally consolidated soil and likely overconsolidated soil sigma dash h is actually less than sigma dash z and K0 sigma dash h is less than sigma 0 h is less than sigma dash z sigma 0 h is less than sigma 0 z and K0 less than 1 that for normally consolidated soil where the deposition is actually happening. So this is this path which is actually shown AB here and while as heavy consolidated soil sigma h greater than sigma z and K0 greater than 1 so that is because of this here it actually can have K0 values more than 1 more than 1 can be obtained for some heavily overconsolidated soils and because of the stress locking which actually can take place because of that you know the exhibit the K0 values very high you know sometimes the K0 value can actually go up to K0 is equal to 3. So here what we are discussing is that for light and light for normally consolidated and likely overconsolidated soils sigma dash h the horizontal stress is less than vertical stress and K0 values less than 1 while for heavily overconsolidated soils sigma dash h is greater than sigma h z and K0 is actually greater than 1. So an approximation is basically often used to estimate K0 and which K0 is actually nothing but K1c where K0 of a normally consolidated clay is which is nothing but 1 – sin phi dash c and phi dash c is the critical state friction angle and root over y0 and the K1c – root over K0nc root over y0 this y0 is nothing but the yield stress ratio which we have defined earlier yield stress ratio is actually nothing but sigma dash y by sigma nought dash where sigma nought dash is the current stress and sigma dash is the stress at the point yield point which is the intersection of swelling line through b and with the normal compression line sigma dash p. Now let us see variation of water content in normally consolidated and overconsolidated soils we actually have been pointing out that for normally consolidated soils the water content will be very high and overconsolidated soils the normal content is less and so consider the variation of water content with depth for a deposit which is lightly eroded you can see that eroded that is the depth of erosion which is small or heavily eroded that is depth of erosion is very large. So we have two states this is the stress point a and b and stresses are points a and b. So this particular soil status the elements a and b actually are referring to the way the depth of erosion is relatively small and in this case the stress element c and d the depth of erosion is relatively large that means that this the element c and d is subjected to very high past pressures a and b subjected to low pressures low degree of pressures. Now what will happen is that for the likely eroded soil the difference between the water content at point a and b is relatively large while for heavily eroded soil the difference between water content and c and d is much smaller. So here what we have is that water content versus sigma dash at z is shown here. So you can see that this is the compression line loading and unloading so this first limb actually is refers to sigma dash a and this is unloading so we have done loading portion and loading portion it is simplified here and this is point b the limb point b and you can see here there is a distinct change in the water content here, here there is marked there is no negligible or marginal change in the water content with for the in this particular portion. So for the likely eroded soil the difference between water content a and b is relatively large while for heavily eroded soil the difference between the water contents at c and d is smaller. So that is what actually has been indicated here. So here if you look into this is for the normally consolidated soils are heavily likely over consolidated soils and these are the heavily over consolidated soils so you can see that the c and d so this is water content versus the depth and element a sigma dash a and sigma dash b low water contents at c and d are attributed to the very large past stresses. So the soil actually has been subjected to past stresses so the low water contents here recorded and you can see that you know this is also negligible variation but here you can see that with the depth of course there is a decrease in water content but thing is that here there is a large variation but here there is a negligible or marginal variation. So the reason is actually basically attributed to that the soil elements at c and d were actually has been subjected to very large past stresses. Now let us after having discussed these you know the effects of the load increment ratios and the duration let us look into some example problems and in this example an idometer test was performed in a clay sample of the TMM thick and drained at both sides that means that we have put kept to porous stones at top and bottom and taken from the mid-statum at the mid-depth of the clay and 70% of consolidation was attained in 6.67 minutes and what we need to find out is that time required to obtain 70% consolidation of the clay stratum and the magnitude of the settlement in that time so magnitude of the settlement in that time. So an idometer test was actually performed in a clay sample of the TMM thick and drained on both sides and taken from mid-statum depth shown and 70% of consolidation was attained in 6.67 minutes so find the time required to attain 70% consolidation of the clay stratum and magnitude of the settlement in that time. So this is you know a particular cross section a levee cross section where the sand embankment or a levee embankment is actually place and this is the clay and we have got you know 2 meter and 7 meter is the thickness of the clay and this is the element A which is E 0 is equal to 1, C C is equal to 0.2 and gamma sat is equal to 20 kilo per meter cube and here is gamma 18 kilo per meter cube water table location is somewhere here. Now the solution works out like this since the soil is same is the same clay in the laboratory and in the field and both are actually 70% consolidation. So first what we can do is that you know we can by the question of consolidation will be same because what has been done is told is that soil is same in clay in the clay in the field since the soil is same soil is the same clay in the laboratory and in the field and both are actually having 70% consolidation. So time factor T v is equal to T cv by h square so we can write it as C v is equal to T suffix v h dr square by small t and which is can be related as because C v in laboratory is equal to C v in prototype C v in field they are same so T v h dr square by T laboratory is equal to T capital v suffix v h dr square by T field they are same. So with that what we get is that T field by T laboratory is equal to h field square by h laboratory. So T field can be obtained like this T lab into hf square by hl square so we actually have measured T lab as 6.67 minutes into hf is actually given as 7 meter thick clay so we actually have said that this clay is actually having a thickness of 7 meters. So 7 divided by the thickness of the sample is 30 mm but is actually is doubly drainage so what we are doing is that 30 by 2 that is nothing but 1.5 centimeter so to equalize the units we have taken in 3 centimeter units and with this what we will actually get is that the time which actually takes place in the for 70% consolidation is about 2.76 years that is one thing which we have to note down the time in the field is you know about 2.76 years so this if this type of situations are there in the field there is a possibility that the settlement actually can affect the structure and if the settlements are actually occurring say beyond say 30 years or 40 years or so and if the design life of the structure is actually small then we need not really worry about that but if this is the situation there is actually need for understanding and eliminating these settlements otherwise the structure is actually going to be subjected to distress because of the consolidation settlements. So the amount of the settlement of the takes place for 70% consolidation can be estimated like this which is we know the degree of consolidation is nothing but 70% consolidation at let us say settlement at 70% consolidation to final consolidation settlement. So we know the final consolidation settlement expression so we use this so delta h 70% is equal to 0.7 Cc by h, h full thickness 7 meter we have to take now 1 plus E0 log P0 that is sigma 0 dash plus delta sigma dash by sigma 0 dash so the in-situ stress at mid-clear stratum before the surcharge applied we can calculate that is the effective stress we can actually calculate and delta P dash or delta sigma dash is nothing but 72 kilo Pascal's and this can be calculated as this unit weight of this which can be calculated that is the unit weight of the sand surcharge and with that is the increase in the load. So with that what we will get is that the 70% of the settles amount of settlement that takes place at 70% consolidation can be obtained as 70% of delta h which is nothing but 0.7 times Cc, Cc is nothing but 0.2 which is actually given into h which is 7 meters divided by 1 plus E0 and E0 is equal to 1 so it becomes 1 plus 1 and log to the base 10 sigma 0 dash that is the stress at this particular point is effective overburden pressure and that is nothing but 71.7 which is actually computed here plus 72 is due to this levy which is actually there. So because of that what is actually estimates is that so at the center the settlement is actually of the order of 148 mm and which actually is you know above the tolerable limit of say 50 mm so in that case you know the structure is actually going to suffer distress for example this is a then this cause you know the differential settlements for a levy and which actually can hamper with the integrity of the system. So the amount of the settlement that takes place at 70% consolidation here what we need to understand is that the degree of consolidation is determined by settlement at any time to final consolidation settlement. So settlement at 70% consolidation divided by the final consolidation settlement so because of that what we have done is that degree of consolidation is given as 0.7 for that what we have taken is that the settlement at 70% consolidation is actually obtained as 0.7 into the expression for the final consolidation settlement we have used here. So with that what we have got is that at 70% degree of consolidation what is the settlement of the particular levy at the mid stratum depth. So mid stratum that the settlement has been computed and this is actually worked out to be around 14.8 centimeters. So here also we can look into that here that if you are having if you take if you divide the clay into number of small incremental thickness let us say 1 meter, 1 meter, 1 meter then each depth each thickness if the settlement is delta 1, delta 2, delta 3 like the delta 7 then you know what we need to do is that we have to calculate what is the increase in load due to the so called the trapezium load which is there that is if you do the method of superposition I can use for this triangular load this trapezium loading and this trapezium loading then we can actually get the settlement at this particular point with that you know we may actually get the more accurate estimation of the settlement. Let us take another example where in oidometer test is actually performed on 100 meter thick specimen drained and top at bottom and it was observed that 45% consolidation the time factor was 0.15 and which was attained at 78 hours in the laboratory and we need to determine that time required to attain 70% consolidation is TV is 0.45 in a job site where the clay stratum is shown in the figure. So we have got sand layer and sand layer at the top and bottom and clay layer which is at the center here that is you know 7.5 meters thickness and let us assume that one dimensional consolidation is actually valid here and there is a building load which actually imposes certain amount of delta sigma on the soil. So this is the water table location here so what we need to find out is that we need to find out you know what is the time which actually takes in the field. So if you calculate if you see that the coefficient of consolidation is actually same for the lab and field samples again by using the same discussion whatever we had which actually says that CV is equal to TV into HDR square by T with that we can actually compute time in field and which actually works out to be the time in field is actually something like 134 years to attain 70% consolidation. So sometimes actually this appears that for a given type of clay you know that the settlements actually continues for about 134 years of time. So if you are actually constructing an important structure on that and if the clay is actually undergoing the consolidation like this and this is actually is a proof that for the you know the leading tower of PISA even after 400 to 500 years actually undergoing consolidation this is a classical example. So if you are actually having this type of situations where the settlements are actually coming predominant settlements are coming within the design life of a building then it is actually better to accelerate the consolidation settlements. So in such situations if the significant amount of consolidation settlements can occur in the design life of a structure there is a need for the acceleration accelerating the consolidation of a soil. So for that there are you know couple of methods which actually we will discuss in the next lecture. So with that what will happen is that we will be covering in this particular module the methods for accelerating the consolidation settlements.