hello everybody welcome to lecture number 16 in this lecture we're going to learn about how to compute primary consolidation settlements in a soil now this lesson follows up from lesson number 15 where we learned a little bit about elastic settlements and also the theory behind consolidation namely obtaining in developing consolidation curves from consolidation testing and also defining the different parts of the consolidation curve well today we're going to put the consolidation curve to use and we do technical engineers use these consolidation curves on a regular basis to predict how much settlement is going to occur in a fine-grained soil when we place an induced stress on top of it so before we dive into that let's let's get into a little bit of fundamental mechanics theory now we're going to try to tie what we'll talk about with the soil to your existing knowledge from engineering mechanics so first things first let's see here let's get my pin the right size so we're talking about having essentially a disc of soil that you know maybe look something like this and there's this little disc or this little puck of soil we place in a consolidometer now inside the consolidometer this guy is completely confined by a steel ring meaning that when we apply a vertical stress to it it's not going to expand whatsoever in the radial direction the only way that it can move is if it goes in the vertical direction so after a consolidation test this little puck here is going to change height a little bit and the change in height of that guy is going to be what we call its consolidation and I'll call that Delta H now the interesting thing that we have to look at here is how do we how do we relate this again back to stress and strains because in your mechanics with the classes that you guys have taken and your engineering education you'll remember that if I have like a steel bar and I apply some sort of stress to it and that bar has some length I'll call it l then we can compute the strain of that bar as simply the stress that we applied times its R divided by its elastic modulus but we can also compute it if that bar experiences a deformation because of that strain then we can also compute it this way as the change in that deformation divided by its original length prior to the deformation so I mean this is basic engineering mechanics so how do we relate this to soil well right off the bat we see that we have a change in height here in our element so that's great but the question remains how do we get this L term in other words the original height of the soil now it's not just the height of the soil that's not that's not what we're talking about let's draw just a very basic three dimensional plot of a soil element here and we're going to assume that this soil element is is comprised of two parts there's the voids which could be water and air and then the solids now if we make the assumption that the solids are equal to 1 the volume of the solids is equal to 1 so in other words this is a unit soil element if that's the case then we know that the volume of the voids is going to be equal to the void ratio and we get that from the fact that the void ratio is equal to the volume of the voids divided by the volume of the solids but at the volume of the solids is equal to one then that just equals the volume of the voids so here's where we are we want to be able to say then that we know that strain is going to be equal to the change in the void ratio of our soil divided by the total the total volume of the soil we if the total volume by the way is going to simply be equal to the volume of the voids plus the volume of the solids which is going to be one plus our void ratio so since if you recall our little circular element I'll redraw it up here is not changing in the radial direction in other words the cross-sectional area of it always remains constant the change in void ratio has to be equal to the change in the height times the cross-sectional area of our soil element where that R is simply equal to the radius of whatever our soil element is and the total volume of that bad boy has to be equal to the height of the element also times its cross-sectional area so in this instance we can see that the cross sectional areas cancel out and we're left then with what we wanted in the first place the change in height over the original height of the element that men can just be written as this relationship here change in void ratio over 1 plus the original void ratio before we started our test that's how we get strain and if I want to get displacement which I'm going to call S sub C as primary consolidation settlement that's simply going to be the strain times the thickness which I'll just call HT of my soil so in other words the change in the void ratio divided by 1 plus its original void ratio times the thickness of that soil layer if I do that then I can estimate how much deformation occurs in that soil layer from that stress that was applied that's going to bring us then to this material right here how do I get my change in void ratio that's the whole reason folks that we have consolidation curves consolidation curves give us a change in void ratio for some predicted change in stress well what's that change in stress come from that folks is from our induced stresses you'll recall from earlier lessons predicting change in stresses or change in vertical stresses from a stress bulb or from isobars using the busan a method those kinds of things so if if we know how much our stress is going to change using then these consolidation curves we can estimate how much a void ratio is going to change then we can plug that right into the these equations here and we can compute our settlement so there's three types of soils that we are going to evaluate here and that and I'm not talking like clays gravel sands I know that's not what I'm talking about I'm it's it's more like case and there's three types of cases that we are going to use when we calculate consolidation settlement the first case we'll call it case number 1 is essentially when our current effective stress of the soil in the out in the field at that particular depth is approximately equal to that soils pre-consolidation stress so in other words the soil is already normally consolidated the soil is already feeling as much stress as it has ever felt in its previous existence so in other words if this curve right here is my consolidation curve that means that my stress that the soil is at in the ground is sitting right here on the line right at that inflection point and any induced stress that I add to it is going to cause a major change in the void ratio that soil is going to compress a lot so all I need then is to be able to compute what that is but folks this is just essentially the slope of a line right we have rise over run and you have to remember that this is a log scale it's a log scale so if I want to compute rise the change in void ratio I need the slope of my line which is C sub C that's this guy right there that's the slope of my normally consolidated portion of my curve and then I also need to get the the run for computing that that rise so I get the run with this equation right here which is just my original stress plus my change in stress my induced stress divided by my original which happens to be the pre-consolidation stress approximately so making that linear calculation I get my change in void ratio and then I'm using the theory we talked about before I just plug this down into this equation right here and this then becomes my computed settlement for that layer where H that H right there is the thickness of the soil layer so that's case one normally consolidated soil now what if the stress of my soil in the field isn't anywhere near the pre-consolidation stress and instead is way up here on the overconsolidated portion of the curve so so my soil is really over consolidated and after I apply my induced stress my induced stress moves me closer to pre consolidation but I'm still as you can see I'm still not there yet I'm still on the overconsolidated portion of that curve if that's the case then my soil started over consolidated in other words my initial stress was less than my pre consolidation pressure and my soil ended over consolidated so the initial stress plus that induced stress that I computed and I am adding to the soil of that layer is still less than the pre-consolidation stress if that's the case then we call this case two case two again and is the soil started over consolidated and remained over consolidated after I applied my undue stress so if that's the case it's the same as before but this time we're using a different slope we're on the overconsolidated portion of the curve and this is C sub R now C so R so you'll see that this looks very similar to this equation from case one but we're using C sub R instead of C sub C which corresponds to the over or I'm sorry the normally consolidated portion of Arkansaw our consolidation curve so I plug those into my stress-strain relationship and I get a settlement for that layer for case two okay so we have case one which is normally consolidated soils in case two which was over consolidated soils that remain over consolidated after stress was applied but but what would happen if I apply my induced stress and the soil started pre consolidated but that new stress was enough to kick it past the pre-consolidation stress and to get me down onto this normally consolidated portion what would I do in that case for that case where the soil starts over consolidated but then becomes normally consolidated because of our loading we call that case three and here's a diagram of it so in other words I start somewhere on the pre or on the over consolidated portion of that curve and then my induced stress pushes me over to the normally consolidated portion of my consolidation curve so I have a I have two void ratios that I have to account for I have the change in void ratio from my over consolidated portion on the curve and then I have my change in void ratio once I got on to the normally consolidated portion of the curve for this instance then I have to have those two change in void ratio is calculated and I'm going to calculate them with these two equations right here and then I plug both of those in to my stress-strain relationship and this is how I compute settlement for a soil layer that we are classifying as case three so again remember case one is where let's write these out case one is where the effective stress is essentially equal to the pre-consolidation stress that means the soil is normally consolidated case two you can recognize a case two if the initial effective stress in the ground is less than the pre-consolidation stress of the soil that you get from your consolidation curve but then the initial stress plus whatever your induced stress is is still less than your pre-consolidation stress and so that is an over consolidated soil case three a little trickier your initial effective stress is less than your pre-consolidation stress but your initial effective stress plus your induced stress at that depth is greater than or equal to the pre-consolidation stress in that instance it starts normally consolidated but it ends I'm sorry I got that backwards it starts over consolidated and it ends normally consolidated so hopefully that makes sense to everybody those are our three cases so from all of my years of practice as a geotechnical engineer and and most of my practice with settlements occurred in the areas of Salt Lake City Utah and in the Pacific Northwest where they have lots of pockets of really really soft soil so this is something that I have a lot of experience in here are some things that I learned from my practice that were valuable to me that I'd like to share with you so first of all because the pre-consolidation stress and conversely the over consolidation ratio you'll you'll recall that the over consult although my pin keeps doing that the over consolidation ratio is simply equal to the the ratio of the pre-consolidation stress divided by the effective stress of the soil so if the pre-consolidation stress is little greater than the effective stress the soil is overconsolidated if the pre-consolidation stress is equal to the current effective stress then that ratio is 1 and the over consolidation ratio indicates the soil is normally consolidated so because the pre-consolidation stress and conversely the over consolidation ratio vary substantially with depth it's not wise to just we can't just take one value and assign it to our entire big thick soil layer because you have to remember that these pre consolidation stresses are changing with depth so in other words let me diagram this out for you let's say I have a soil profile and let's say I've got a big thick clay layer now if I go down and I just do one sample right here and I test what that pre-consolidation stress is so there's Sigma prime P and and this other axis the vertical axis is depth so if I'm here at this depth were a sample I'm a measure a pre-consolidation stress now if I make the mistake of just assuming that that is uniform through my whole layer I'm probably going to be way off on my settlement calculations and the reason for that is if I come down to a shallower depth and I sample there I may find that my pre-consolidation stress is a little different and if I sample at an even shallower depth I may find my pre-consolidation stress is is even higher and if I sample right near the ground surface I could get a really high pre-consolidation stress if I go down deeper eventually what I may find is that my points may start doing something like this where we start to develop a curve that looks something like this okay now if let's change the color here if I compute the effective stress with depth and so other words I just calculate effective stress it might look something like this so maybe this line right here represents just my effective stress so what this would tell me is for this particular sketch right here from this point upwards I am over consolidated because my pre-consolidation stress is larger than my normally consolidated stress and from this point downward the two are approximately equal and so in that particular instance I would probably want to know that's not a good color let's use I hate being colorblind as stinks let's try this one I would probably want to subdivide this upper layer into zones corresponding to where I did all my tests so each one of these would be a thickness in my soil profile and down here where the OCR or the over consolidation ratio isn't changing maybe I make that a thicker layer but the the point is where we tend to see lots of changes in over consolidation ratio and a lot of changes in the pre-consolidation stress that is usually an indicator that we want to add more sub layers to maintain the resolution and the accuracy in our calculations that that were desired but but that's not the only thing - because we also have to remember that there's a stress bulb there and so similarly if I have a footing here that's applying my induced stress or an embankment or whatever we have to remember that there's a bulb of influence or stress influence with depth where remember all of these correspond to my change in stress so we have to account for this - in our sub layering generally where we see a lot of increase in stress happening in other words really really tight induced stress contours on our stress bulb that's also another indication that we might want to add more sub layers so look there's no hard or fast rule to this this is an art and just the in general the more layers you have the more accurate your answer will be but you don't want to have more layers than your available data justifies if that makes sense so that's the first point that I want to share is is creating sub layers in your profiles now the second thing that I'd like to share is not everybody in the world has access to good consolidation lab testing equipment and so obtaining a good measurement of C sub C and C sub R the slopes on your consolidation curves they can be hard to do but even if you don't have access to good solid Asian lab equipment it's you're not entirely hopeless we've learned through research that the outer Berg limits do correlate pretty well to these consolidation parameters so using atterberg limits you and some different researchers have developed a couple of different relationships that can approximate these consolidation parameters from the atterberg limit data so here's a couple of them two different ones for C sub C and then in general the reconsolidation index or C sub R is generally 10 to 20 percent of C sub C not always but that's just based on my observation and and those of others for typical soils but look if possible you definitely want to rely on lab measurements they're way more reliable and not as not as scary when you're trying to guess using and correlated data the last thing I want to identify and highlight to you is the importance of that pre-consolidation stress right that sigma prime p governs everything when it comes to accurate consolidation predictions if you're off on your pre-consolidation stress you're going to be off on everything so it's very important to get an accurate estimate of your pre consolidation of stress and if you'll call from the previous lesson the the challenge with getting an accurate pre-consolidation stress is not disturbing the soil sample when you pull that soil out of the ground every bump every jiggle every wiggle every time you twist it turn it drop it place it somewhere every little vibration damages the soil fabric a little bit more and makes it so that the pre-consolidation stress is more and more difficult to accurately interpret through your testing so it's always the best idea possible to get as undisturbed samples as possible I even in my own practice I mean I've had instances where we were bringing samples back to our lab from sites from another state and we purchased we literally purchased the seat next to me on the airplane and I was strapping those bad boys in because I wanted to keep them as undisturbed as possible so again good quality testing to get that pre-consolidation stress is key if you don't have access to consolidation testing and you can't measure that pre-consolidation stress you can rely on empirical correlations there's a lot that's been published in the literature but honestly they're all over the place and they're not very reliable that the best is to use lab based methods so here's the deal this is a head start on your homework that's going to be due in a couple days for special problem number seven so this is a problem I created what you have is you have a two layer soil system you have a lean clay underlaying by a high plasticity fat clay and then you have bedrock down below that 15 feet of each and what we're doing is we're coming and replacing three feet of a new fill you know the type of fill is really irrelevant the only thing we care about is its unit weight because you can represent this fill is just a whole bunch of arrows they think of this as an infinite load that we're just applying in an infinite extent across the site and so that's change in stress is simply going to equal our unit weight times our thickness which in this case is a hundred and twenty two pounds per cubic feet three feet of thickness so that's going to equal our our induced stress okay the other thing that's given to us in this problem is this chart right here which gives us of this this little teeny line here this thin dashed line this is our estimated effective stress with depth and then this thicker dashed line is our interpolated or our art we'll use is called our measured pre-consolidation stress with depth so what this is telling us is from this depth here this this upper leaned clay it's obvious that the pre-consolidation stress is greater than the effective stress so that means that up here the soil is overconsolidated and then down here the soil is normally consolidated because the pre-consolidation stress is effectively equal to the effective stress so what we're going to do in this problem is we're going to compute this settlement anticipated that the primary consolidation settlement that's anticipated from the placement of this three feet of new fill and and here's how we're going to do it obviously I don't want to treat this upper layer as one cohesive unit because you can see my pre-consolidation stress and my over consolidation ratio are changing with depth through that whole layer now I could try to be a little smarter and maybe do an average value for the whole layer but why not just subdivide it into smaller layers and then compute the or use the pre-consolidation stress at the midpoint of each one of those layers so for this problem and you can divide it into as many layers as you want my recommendation is to divide it in this upper layer and to maybe maybe three sub layers each one five feet thick this bottom layer because it's normally consolidated it wouldn't kill you if you left the whole thing at 15 feet thick but it's generally good practice to at least have you know a couple sub layers so you could divide us into two sub layers of seven and a half feet thick but honestly you know it's it's your problem you decide how you want to do it for each of these soils by the way you'll notice that all of the consolidation parameters are provided I'll erase it so that you can see those clearly there we go so we have C sub our C sub C's and initial void ratios for all of these soil layers and we also have the saturated unit weight so that you guys can calculate total stresses and you can calculate the effective stresses because you know the water table is at the top of that soil layer so this is a fun little problem a nice introduction to calculating primary consolidation settlement I'm going to give you a hint on how to set up this problem in general it's a really good idea to just build a table that looks just like this where we have the depth to the midpoint of my sub layer then we have the thickness of that sub layer we have the void ratio of that sub layer and C sub C and C sub R so obviously these are given for each soil and then all these other things we have to compute right so we have total stress that we have to compute at the midpoint of our sub layer or the depth that we indicated we have the static pore pressure from hydrostatic pressure we have to compute and then we also have the effective stress at that depth of the midpoint of our each sub layer then we also have the induced stress which for this problem we already said was going to equal a hundred and twenty two pounds per cubic feet times three feet and because that's an infinitely wide distribution or we're applying that new soil layer across the whole site we're going to assume that that change in stress is uniform with depth all the way down to infinity as deep as we're going to go so that's going to be equal to 366 pounds per square foot and then the pre-consolidation stress you can get from the chart that's given to you just interpolate it try to you know do your best to read it from the chart you don't have to be perfect but you know you want to be pretty close and then of this value this is your effective stress from the effective stress column plus your induced stress so you're just adding those two columns together to get this value and then you're comparing your effective stresses with your pre consolidation stresses and then your post induced stress stresses to estimate your case and remember you're going to have case one case two or case three and depending on which case you have then you're going to use the equation I gave you to calculate the consolidation settlement so at this point I'm inviting you to pause this video and just knock this problem out it'll probably take you quite a bit of time maybe an hour or so if it's your first time but just fill in all of these columns for all of your soil sub layers you know again for for us we're dividing this into at least the way I recommended you set it up I think we we're going to have five sub layers we're the first three sub layers these three sub layers right here we're from the lean clay and then the bottom two sub layers were from the fat clay the lower clay so you know I'll go ahead and just fill out this chart as best as you can and then compute all of your settlements induced in each one of these sub layers that's going to come from this final column your total estimated settlement will be the summation of that column and that will give you your total predicted primary consolidation settlement now if you get stuck have no fear that's okay you just go to the next slide and I've computed all the values for you for this problem using those assumed sub layers so you can see what I put in there and you can see how I came to my answer of almost half a foot of predicted consolidation settlement or just under six inches and so you know this will this will help you get going on that special problem homework so go ahead pause the video before we move on just go ahead and do that and you'll feel a whole lot better about yourself okay hopefully you're back and you had a good experience in doing that little consolidation calculation there again it did challenging and beefy if it's the first time you've done it but you'll get the hang of it and pretty soon those things those types of calculations can come pretty quickly the last thing I want to talk about in a lesson today is the last and the third type of settlement that we wanted to talk about in this class and that's secondary consolidation settlement so now remember secondary consolidation settlement occurs as the long term degradation and decomposition of the soil particles themselves under and due stress and continued exposure to chemical reactions in in the groundwater and and just in the soil itself so the soil is literally breaking down and decomposing and so that just causes settlement to occur and it's given that nature it's a lot more common in organic soils and normally consolidated fine-grained soils those two types of soils can have some pretty massive amounts of secondary consolidation how do we deal with it how do we predict it well it's pretty easy from your consolidation test you're going to remember that there were a couple of load cycles where you measured the dial reading with time in other words you didn't just load it up and walk away and come back a day later you came back at various components during that load cycle and you recorded what that dial reading was with time that type of measurement is necessary in order to compute secondary consolidation settlement because what we end up getting is a curve that looks like this where you have this initial steep portion of this dial reading and guess what folks that that initial portion right there that initial portion is our primary consolidation okay but that's not what we're interested in what we're interested in is this secondary portion right here where the dial continues to go you can see that the the slope of our dial reading has changed but the so it's not compressing or consolidating near as fast that's what we call secondary consolidation and that's what we want to be able to predict and measure so with and these displacement curves from our consolidation testing what we need to do is first we draw a tangent line to the two bellies of the curve like that and where those lines intersect that is what we call the time for primary consolidation to end so everything is to the left of that is primary consolidation everything as to the right of that is secondary consolidation so if I know some time out there that I'm interested in I have a starting time and an ending time I can estimate from that slope what the change in height or in other words the settlement is going to be so with this dial reading curve I'm just going to use this equation right here super super simple folks it's just my change in height divided by the log of my time - divided by time one so I can get that that's my run to compute my rise my rise is my Delta H all times the thickness of my soil layer really really simple that's the equation you want to use okay and we can get an idea if you want to use an approximation method just a really really loosey-goosey idea of what that's going to be if we get the this point zero four five times are normally consolidation index all right that's called our consolidation index C sub C divided by one plus e-naught what this is is this gives us then the slope of that line okay this really is what we're computing here is the slope of the line this slope right there that's C alpha prime okay once I get that slope I can compute secondary consolidation using this equation right here the thing to remember is that time to this is the tricky part time two is the time for primary consolidation plus whatever time frame of interest I'm looking at so here's an example to show you let's say we have a 10 feet thick clay layer and it took five years for that layer to finish primary consolidation and let's say that it has a modified recompression or secondary compression index of 0.01 let's say I want to compute the secondary consolidation five years after a primary consolidation is ended okay so here's my equation I'm just going to plug and chug C alpha prime is equal to 0.01 its given the thickness that's ten feet it's given okay all I need to do is get my time's right so we know we know that the time for primary consolidation is five years okay totally get that but time to is got to be the time for primary consolidation plus a time of interest so that's going to be five years plus an additional in this case five years after that so this is gonna be a total of ten years just like that so then I just plug and chug I compute the answer and I get point zero three feet as my computed of secondary settlement in that layer or 0.35 inches so that would be five years after primary consolidation has ended so the cool thing about secondary consolidation settlement is you don't need to subdivide your layers necessarily unless unless this C sub alpha changes significantly with depth if it doesn't change significantly with depth then you can just have a big thick layer and you have one calculation so that's an advantage of this method here all right folks that's all I have for you in this lesson I appreciate your attention and I will be gone to the United Kingdom again until November 3rd of this year so if you have any questions just reach out to me through email and I will get back to you as soon as I can until then have a great day