 Tugendag waren er alleen twee slijts op de college rama, maar we hebben het al gezegd dat de hele recordering nog steeds op de machine is, zodat ze het op de college rama gaan doen. De recovering rate is 99 procent, dus dat is waarschijnlijk oké. De vorige week hadden we over de zandkuttingen, hoor jij het? Ja, ik hoor het. Oké. Dus we hebben gevoeligd dat je een zandkutting hebt voordat je een zand krijgt, dan krijg je de dilatatie en de perusite is increasing, omdat de perusite in de water in de schierzone gevolgd is, en water wordt alleen gevolgd als je een verschillende pressie hebt. Sinds de buitenste pressie, waarom we het generaal heen, de hydrostatische pressie, zal het niet veranderen. Het enige wat dat kan veranderen is de pressie tussen de partijen, dus de deurpressie. En dat is wat er gebeurt. De deurpressie heeft een limitatie en deze limitatie is als de onderpressie in de deurpressie de watervapere pressie reacht, omdat op dat moment water begint te koeken, en de pressie kan niet meer veranderen. Sinds de, voor normaal temperaties, laten we zeggen 10 graden, deze watervapere pressie is gewoon 0.01 bar, die is 1 procent van een bar, we consideren het 0. Dus in normaal calculaties, we consideren dat de limitatie de absolute 0 pressie is. Niet 0.01, maar absolute 0, omdat de verschil is gewoon 1 procent. In general met soilkutting, of met alles wat het moet doen met soilmechanen, je moet consideren dat de curatie van de soilmechanische parameters niet te hoog is. Dus als je over een angle van internaal friktie gaat, misschien in een laboratie, je kunt het in één graden determineren, maar als je kijkt naar samples van real-life, in c2-samples zoals we zeen, ja, misschien is het 2 of 3 graden. De permeabiliteit, die is een geweldige invloed aan dit proces, ja, het kan tussen 10 minus 3 tot 10 minus 5, die is een factor 100, en dan kun je imaginen, als het makkelijk is om een error met een factor 2 te maken. Nou, als je een error met een factor 2 hebt, de krachten zijn twee keer te hoog of twee keer te hoog. Dus die curaties zijn niet echt te hoog. Dat betekent dat als we de 1% van de waterfaperpressie neglecten, dat is een curatie van 1% en het is echt neglectabel als je met de curatie in de soilmechanische parameters hebt. Dus dat is waarom we zeggen, oké, we neglecten, het maakt alle equaties een beetje meer simpel. Oké, wat hebben we? We wilden de poor pressures calculeren en dit betekent dat in dit gebied je een bepaalde flow van twee plekken, van de bovenste en de bovenste water moet bepaalden in dat gebied, want dat is waar je de perusite increasing hebt. Dit betekent dan weer hoe de water kan bepaalden van vier directies, maar in de realiteit op één punt van de schereplijn de water kan alleen van twee directies komen, niet van vier directies. Later zal ik je laten zien een method waar eigenlijk ik de watervloven van vier directies laat zijn. De method werkt goed, maar is scientifically incorrect. Maar dat gebeurt meer in de signaal dat je een calculatie method hebt die goede resultaten geeft, maar het betekent niet, het is altijd scientifically 100 procent. Maar ik zal je later laten zien. We hebben een mesh nodig, dit was de koortsmes, dit was de fine mesh, de finer de mesh, de meer accurate de calculatie, maar ook de langere calculatie tijd voor de computer. En nu is dat niet echt een probleem, maar ik denk dat je elke keer een uur achter je laptop voordat een calculatie is bepaald en als er een mogelijkheid is om het in vijf minuten te doen, zou je de vijf minuten prefereren, omdat je in één dag meer calculatie doet. In die dagen was het een bedoel van geld en dus als we de mesh koortsmes hadden, was het te makkelijker. Maar nogmaals, je calculaties moeten accurate genoemd zijn. Dit was de resultaten, de potentiële lijnen, waar de grootste onderpressing in dit gebied is, dichtbij de etch van de blad en op de scherplijn. Dit geeft dezelfde foto, maar dan in kleur. En ik heb deze foto gekregen die de flowlijnen geeft. Dus hier zie je dat in dit hele gebied er hardly is niets te verblijven, maar in dit gebied is dit zoals een flowlijn. De division tussen twee kleuren is een flowlijn. Hier heb je flowlijnen en equipotentiële lijnen, samen. Nu hebben we een aantal issues in de equations die we gebruiken. We hebben de hoge forces, de horizontale, de verticale forces, we kunnen ze calculeren, maar er was nog een aantal in die equations. En die een aantal was de scherplijn, de aantal van de scherplijn. Dus, in feite, geïquivalent aan de derivaties die ik heb al gezien, van actieve en passieve scherplijn waar de beta ook de aantal was, en we hadden de beta analitisch opgelaten. In dit geval hebben we ook de beta te vinden, maar de equations zijn zo complicat dat je het niet analytisch kunt doen. Een reden is dat de equations complicat zijn. De tweede reden is, in de equations moet je die poor pressures gebruiken en die poor pressures zijn de resultaten van de Feyenoed Element calculaties, die niet een mooi continiële equations zijn, maar die is een tabel van de nummers die de resultaten van die Feyenoed Element calculaties zijn. Dus, eigenlijk, is het mogelijk om analytisch te zorgen in welke beta de kleinste force zal resultaten. Dus, wat doen we? In ieder geval doen we een aantal calculaties en dragen de forces als een functie van dit scherengel. Dus hier zie je de scherengel van 15 graden tot 45 graden. Dit is gewoon voor het examen. Hier zie je alpha is 60 graden, dus dat is een 60 graden blade. En dan calculate ik de horizontale force, die is de lage lijn en de totale force, om te zien of het een grote verschil zou maken. Maar als je de principale energie gebruikt, moet je alleen de horizontale force nodig zijn, omdat energie, je kan het van de kracht dragen. Nou, kracht is de kracht x velociteit, maar de kracht moet in de richting van de velociteit zijn. Als het perpendicular is, kan het niet aan de kracht contributeren. Sinds we over een horizontale scherengel, dat betekent dat we de horizontale force nodig zijn om de kracht te krijgen. En dan kun je het van tijd dragen en je hebt energie. Een multiplie van tijd en dan heb je energie. Dus eigenlijk moeten we de onderste lijn van de twee die de horizontale force is en kijken waar is die force op de minima, omdat waar die force op de minima is, dat is waarschijnlijk wat natuur zal doen. Het is nooit een garantie, maar vaak is natuur laag en natuur wil niet meer energie spenden dan nodig zijn. Dus based op dat principale, het is een beetje een filosofische principale, we vinden de beta. Dan kun je het voor de 60-degree blade zien. In fact, de punt waar je de minima voor beide forces hebt, totale force en horizontale force, is bijna bij dezelfde beta en ja, het zou zijn als je er op 20 graden kijkt. Dan kun je naar de 45-degree. Je kunt de verschil zien tussen horizontale en verticale force is een beetje groter, niet veel, maar je kunt ook zien dat de punt waar je de minima vindt gevolgd is. Nou, wat zou het zijn, iets zoals 27, 28 graden. En dan kun je naar de 30-degree blade zien. Je kunt zien dat de verschil tussen de minima is een beetje groter. Nu, ja, waar is deze minima gewoon bijna 30 graden. Dus een simpel rule van de temp based on this is that for a 60-degree blade the shear angle is between 15 and 20 degrees, then if you go to the 45, it's between 25 and 30 degrees, and usually for a 30-degree blade it's around 30 degrees. So if the blade angle decreases, the shear angle increases. And I also made a drawing of the lines for the 15-degree blade, which is a very unpractical blade, but we did tests with such blades and you can see that you are above 35 degrees. So the tendency of an increasing shear angle with a decreasing blade angle is valid. Why is a 15-degree blade not practical? Well, if you have a blade like this, 15 degrees is very flat, and the blade has a point, it means an edge, it means the edge will become very sharp. And if you would use such a blade in reality and you just hit one piece of gravel, the blade is broken because you can't make it strong enough. That's why in general in dredging, if you look at a cutterhead, I would say the blade angles they use are between 50 and 60 degrees. So they are not too small. It has a number of reasons, so one of the reasons is that a very small blade angle makes it very vulnerable. And in dredging, you can say I'm dredging sand or clay, there will always be some gravel somewhere. You will always hit those stones. So it has to be strong enough. But the second reason is that in a cutterhead you are rotating, that means at the bottom of your blade you get wear. And if you have a very flat blade, that also means the wear surface at the bottom will also become very big. In the case you have a lot of wear, your forces increase dramatically and it won't work. If I have a much steeper blade, then also the wear flat is much smaller, so the effect of wear is much smaller. The influence of wear. So how do you determine the influence of wear? In fact this is about the influence of wear on the pore pressures. If theoretically you have a blade with a very sharp edge, that means the water in the back of the blade reaches the edge of the blade. That also means that theoretically the pressure at the point, at the edge of the blade is zero, is hydrostatic. But hydrostatic we call it zero in this respect. You cannot have under pressure at a point where you also have hydrostatic pressure. So that reduces the pressures from the calculations dramatically. We know that in reality you always have some wear, but also that during the cutting process the scent is compacted a little bit. Behind the blade it will move up a little bit and the result is that the water behind the blade can never reach the edge of the blade. To test that we made blades with a wear flat, a horizontal piece behind the edge of the blade and determine the pore pressures. Well, this is the pressure in the shear plane, the average pressure. This is the average pressure on the blade and the W is the length of the wear flat and HI was the thickness of the layer cut. And what do you see that if you increase the size of the wear flat you get, let's say asymptotically to a certain stable situation. And that's what we choose for the size of this wear flat because you like it if it's reproducible and you could say, maybe this under pressure is a little bit too high. Well, the minimum is here so it's not really too high. But it's better if you overestimate the forces a little bit than when you underestimate because if they design a project based on forces that are too low you may damage your equipment. So based on that philosophy we said it's better to be a little bit too high than too low. Then you get the situation of permeability for those calculations. We have the sand before it passes the shear plane which is still the city sand with a low permeability. Then when the sand is sheering the particles roll over each other the porosity is increasing so also the permeability is increasing. But how to deal with that in such a finite element program. Well, the program we had at that time it's still at the university could deal with that you can make two areas with different permeabilities en we looked at the influence of the ratio between those permeabilities on the cutting force. Well, the ratio here you can see the K i is the permeability of the sand in city so before sheering and the K max is the permeability after sheering and you can see that if this ratio is increasing the force is decreasing. Why is that? Well, the K i is fixed because that's the permeability you have from the sand. If the permeability K max is decreasing is getting smaller this axis goes to the right but if that permeability is getting smaller the force is also getting smaller but if you take the reverse of this curve here you see it's a nice straight line and based on that straight line we found that almost exactly you can just take the K i plus the K max and divide it by two and use that permeability in your equations and just take the average of the two permeabilities. Last week I think we stopped with those equations we already looked at this so you had the C1 which was the result of the finite element calculation plus all the signs and cosines that are in the equation including the beta where the cutting forces are at the minimum Rho times G I told you normally I use 10 for that combination but if you put it in a computer program just do it as accurate as possible Vc was the cutting velocity so in meters per second H i the thickness of the layer you are cutting and it was squared because on one hand the pore pressures will be proportional to the layer the thickness of the layer could if it's two times as thick the pore pressures will be doubled but since force is the integral of the pore pressures you get a second time this layer thickness and that's why it's squared B was the width of the blade E was the dilatation so the increase of the volume of the sand with a value similarly I choose something like 0.1, 0.2 in that range but if you know the exact numbers of the porosities you can calculate it but if I don't I have to make an estimate Nkm is that average permeability so it's just the average of the permeability before shearing and after shearing for the Fv you get the same reasoning the only difference is you have a C2 now if you don't know anything about the sand you should make a what I call an educated guess and that would be that C1 is 0.5 and C2 is 0 and that means you don't have a vertical force then in the cavitating equations the water starts cooking in the pores you have a D1 the educated guess for D1 is 5 so 10 times C1 but it's 5 but because the equation is completely different you cannot compare C1 and D1 again the density of water times the G which we estimate 10 between brackets Z plus 10 Z is the water depth and the 10 is because of atmospheric pressure because rho times G times 10 is exactly 100 kPa which is 1 bar so the 10 is just the atmospheric pressure and then for each 10 meters of water depth you can add a 10 again the HI, the layer thickness why is it not squared here well the pressure is a constant once you are cavitating the pressure is a constant and the pressure doesn't depend on the layer thickness anymore so the only reason I have an H is because I have to integrate that pressure over the layer thickness and that's when I get one H in this equation and then you see the B, the width of the blade which makes sense the blade has a width which is 2 times probably the force will be 2 times it's not 100% because you have some side effects but in the 2 day, 2D situation if you make those 2 equations equal you get an equation like this and if this equation is 1 you are at the transition between the noncavitating and the cavitating process if it's 1, it's just making the 2 equations equal we talked about cavitation cavitation, if it occurs it will start in the shear plane and in that shear plane this is the area where you saw on previous pictures that we had the biggest under pressures where you have the biggest under pressures cavitation will start but cavitation means the increase of porosity has been filled with vapor bubbles not with water so during the process it will not just disappear you need water to fill up that volume so the sand is moving like this over the blade on this side water is flowing to that cavitation area filling up the pores with water and it takes a while actually before the cavitation has disappeared the cavitation can also occur here before the shear plane just because that area is catching water from the surroundings it could already start a little bit sooner if you measure here I show you the points where we had pressure transducers in the blade so we call it differential pressure transducers we can measure the pore pressure in the sand not the pressure of the sand of the particles just the pore pressure and as long as you have this cavitation spot in front of your blade measure a high under pressure and then where the cavitation spot has disappeared the under pressure will drop so you get something like this anyway that's what we would expect this shows again where did we put the pressure transducers because we already expected that near the edge of the blade there will be high gradients of the pressure while if you go to the back of the blade the change of the pressure is much less so we like to put the transducers as close as we could to the edge but you can see in this area because there you have the edge of the blade you cannot really put those transducers just no space why does it look the way it looks well in fact the actual pressure transducer was put behind the blade when the sand is moving on the blade you get a lot of wear so in fact what I'm telling you is a little bit how to do such an experiment so if I would put this pressure transducer in the blade after one test no more transducer because the sand would wear it off and it would be so damaged you can't use it again ok so what to do so we decided we make holes but if you just make a hole in the blade the hole will be filled up with sand and then what you like to keep it clean so we decided we make small pills like aspirins to put in front but those pills have to be on one hand they have to be very hard because they have to resist the wear of the sand on the other hand they have to be completely permeable because you want to measure all the changes in the pressure in the water en if those pills are not permeable then the pressure transducer will not measure anything now the problem is if you make those pills so we found the right material to make those pills you make aspirins in fact they are full of air and if you just put them in and then put water on top the air is still in those small pills because why would it go out that means air is very compressible so air in such a small pill would behave like what we call a low pass filter and filter out all the high frequency changes in the pressure you wouldn't measure a thing only over longer periods of if there are pressure difference you could measure something so what we did we made a lot of those pills we borrowed a vacuum tank in one of the other departments put them in a glass of water and put absolute vacuum in that vacuum tank so almost zero bar absolute pressure in the vacuum tank and then you can see all the air go out and we left it there for a week and then after a week you could assume all the air was out of those pills but you have to keep them underwater you cannot take them out and put them in the test facility because there will be air in again so we had to take the glass of water with those pills step with a wetsuit step in the laboratory in the tank with water pictures of that and put the glass underwater taking care those pills would never be in contact with air otherwise it won't work well that was a lot of work but it worked now first if you look at the under pressures based on those finite element calculations here you see the blade with the locations of the transducers and A B is the shear plane so A was the edge of the blade and B was where the shear plane reaches the surface well it's logic that at the point where the shear plane reaches the surface the pressure is zero then it's increasing and near the edge of the blade you could have a maximum although it doesn't have to be a full maximum as you can see here and in fact at the blade we used that wear surface to take wear and other effects into account then on the blade you can see that in fact just behind the edge of the blade you find the maximum but it completely depends on the blade angle and the layer thickness and so on and then over the blade you see a drop to zero because at the end of the blade it's in contact with water with hydrostatic pressure so the pressure is zero so this is what we calculated and then you like to know does it really happen but before I show you some data from the laboratory the method which is not scientifically correct but which works very well in fact some students did whole master degree assignments with this method to solve certain problems and they actually solved it well this method we call the parallel resistor method I don't think in soil mechanics you will ever get it but it works very well you can say if I have a point on the shear plane then the water could flow from four directions theoretically so here this is direction one direction two, direction three direction four of each direction I can determine the length for example I can say this is a circle segment so if I know the location on the shear plane I can calculate the length of that circle segment here you have a circle segment plus a straight line here you also have two circle segments plus here a piece of straight line so it's a little bit engineering intuition how to implement the method but this is the way I did it now the resistance of one line so the water flow over one line is the length of that line divided by the permeability so if you take the length of a flow line divided by the permeability it gives you a resistance from electronics or electrical engineering so if you have parallel resistors there is a rule how to determine the total resistance of a number of parallel resistors in fact the one divided by the total resistance equals one divided by the first resistance plus one divided by the second resistance et cetera et cetera so this is the way how to determine the total resistance and in fact for each line you know the S divided by K because you know in which area you are before shearing or after shearing so you know the K value and you have to make an estimate for the length I use circle segments but if you say I want to make it more fancy the difference is not very much if you choose a little bit different approach the values will almost be the same then if I know this resistance the under pressure at that point on the shear plane you can calculate with this equation so in fact the density of water G cutting velocity E and the sign of beta so basically this is a constant and then what I do I divide the shear plane in let's say 100 steps, 100 small parts and on each part I determine this under pressure that way I get a nice curve of the under pressure over the whole shear plane I can integrate it and that gives me the average under pressure on the shear plane another advantage of this is that for each point I can check whether I exceed the water vapor pressure and if I do I can limit the outcome to the water vapor pressure so the transition between no cavitation and cavitation can be determined very smoothly using this method I just want to show you the method why did some students use it well we had students who wanted to investigate if you have the drag head of a hopper dredge and you have those teeth IHC not so long ago a couple of years ago they put water jets inside the teeth in order to compensate for the under pressure in the sand they said if we can blow water into the sand before it starts dilatating we don't have those under pressures and they call it actually the dragon head and the student wanted to calculate how does it work well with this method you are more flexible in determining under pressures because everywhere you have the over pressure of the water jet you don't have under pressure so you can define that they actually graduated and what they calculated matched reality in fact I tuned this method on the finite element program the finite element method and the difference is just a few percent so and this I can put in any computer program and I do not have to wait until the finite element program has finished it's like 20 lines of code in a computer program and you can use it for everything and then if the accuracy is a few percent that's good enough this picture you already saw when I discussed soil mechanics I think what is it well in order to predict cutting forces very often what you have is SPT values and things like that not although here in Holland we have good laboratories here you have somewhere here you have the old soil mechanics laboratory and you have the deltares and so on and they can do sophisticated tests but that's not everywhere in the world so one reason is it's not always available so for example you have a project on Borneo do you think somewhere in Borneo they have a very sophisticated soil mechanics laboratory probably not still you have to do a project there and you like to know something about the soil so you can do SPT you can do CPT but usually it's tests that you can do locally if you want to know more about the material you have to send it by plane to a good laboratory and wait for the results well what does this picture show you here you have SPT value here you have relative density relative density was if it's zero it means you have the highest porosity of your sand if it's 100 percent you have the lowest porosity of your sand so you can see if the porosity is of the sand is decreasing the relative density is increasing and then the SPT value is also increasing it just gives you a feeling how does it change from very loose to very dense material then if we know the SPT value you also have the angle of internal friction as a function of the SPT value this gives you that function so you can also see here loose up to very dense so in fact that's the relative density again and so if I would have 40 SPT which we consider a hard sand okay 40 that would mean I'm here I would have an angle of internal friction of 38 39 degrees this method is not very accurate in fact the red lines give the upper and lower limit which is plus and minus 3 degrees but if you don't know anything you have a method to determine some values of the sand well if this way I estimate my angle of internal friction and I take two-thirds of that value I have my angle of external friction so I can make some estimates well last thing before the break then we get to specific energy what is specific energy specific energy and it counts for all types of soil not just for sand all types of soil if you divide the cutting power by the production q is the volume flow of material so that's the production cubic meters per second if I divide those two I get energy per cubic meter well the power is force times velocity that's general for power the production is the layer thickness times the width of the blade times the cutting velocity and then you can see the cutting velocity is in both so apparently I can divide the VC out of the equation if I do that for the cavitating cutting process and I do that because in dredging with cutter heads normally the velocities are so high that it's always cavitating so that's what we are interested in you get this simple equation for the specific energy I already told you the d if you don't know anything take a value of 5 okay so if I take a 5 rho times g is 10 and suppose the water depth is also 10 you get 5 times 10 times 20 is 1000 kpa which means my specific energy for such ascent would be 1000 kpa here you see the equation again now you can also reverse this and say okay so if I know the installed power of my cutter head I just reverse the equation and I can determine the production so the production is the installed power divided by the specific energy so suppose I have a cutter head with 1 megawatt installed power 1 megawatt is 1000 kilowatt if I divide 1000 kilowatt by 1000 kpa I get 1 dat would mean such a cutter head has a production of 1 cubic meter per second this is based on just energy considerations so I'm not looking at the cutter dredge and check if the cutter dredge is capable of having such a swing speed that it can do 1 cubic meter per second I didn't look at that all I look at is the power of the cutter head enough to cut 1 cubic meter per second well in this case in this specific example it is because specific energy is 1000 installed power is 1000 if you divide it you get 1 cubic meter per second well this is the way to determine your production there are other limitations which I will discuss later but at the exam you could get questions if I have such and such a sand determine the specific energy I have a cutter head with so much installed power what would be the production but I can also reverse it and say if this is the production what should be the installed power or whatever you can do it in many directions based on the previous slides where I had a relation between the SPT value en de angle of internal friction I derived those equations those 3 equations for 30, 45 en 60 degree blades where if you know the phi the angle of internal friction you can determine the d1 and if you do that for something like 35, 40 degrees for the 60 degree blade you should get close to the 5 but in fact the d1 could range from let's say 2 to 10 that's roughly the range ok we have a break well we saw the specific energy so here you see the equations again and here I made a graph with horizontally the SPT value vertically the specific energy of the sand and a set of water depth so we start at 0 meter water depth and we end at 30 with steps of 5 meter and you can see where you are well normally hard sand would be in the range of 40 to 50 SPT and if you look at that 40 to 50 SPT you move up well the first line would be 0 meter water depth and you would arrive at something like 600 kPa specific energy but in the example I gave before the break we said if we have 10 meters we would arrive at about 1000 well look at the 10 meters which is the straight line here 40 to 50 straight line we are close to the 1000 kPa so that works but if the water depth is bigger for every 10 meters of water depth you can multiply the specific energy with a factor so 0 is 1 10 meters is 2 20 meters is 3 et cetera et cetera so every time you have to add up the number 1 so if 0 meters water depth would be 500 kPa then 10 meters is 1000 20 meters is 1500 30 meters is 2000 and that means if I go from 10 meters to 30 meters the specific energy doubles and that means the production goes to 50% so with the same cutter dredge the amount of sand I can dredge but this is the maximum it doesn't always mean I'm dredging the maximum but if I would dredge the maximum then at 30 meters the maximum is 50% of the 10 meters this chart gives you the productions per 100 kW installed power on the cutter head and you can see how it is decreasing with increasing SPT value but here both axes are logarithmic by the way yesterday I already put all those slides on blackboards so you can check it and I'm almost ready with the lecture notes so I will also put them but I still have to write some text the lecture note you will get is not yet 100% finished because it's too much if I would finish it in one year I would work full time on just writing lecture notes but there is enough for all the lectures to understand and to see all the equations that you need experiments well, we used to have a laboratory and I just saw Mr. Nief from China his father is the dredging professor in Changzhou in China and he just told me they are building a new laboratory, some of the biggest dredging laboratory in the world, somewhere in China but at the location of his father in Changzhou, close to Shanghai they also have a laboratory like this we used to have one but because of Delft University policy another group needed that space so we lost that laboratory we still have a laboratory but not as big anymore we used to have a laboratory with a big concrete tank that tank was 2,5 meters in width about 36 meters long and on top of the tank there was a carriage, a vehicle that could drive over the tank and under that vehicle you could mount blades or cutter heads or whatever you want it was driven by a winch an infinite winch and we had a cabin with all the measurement equipment to get all the data from all the measurements this is a cross section of that tank so here you see the concrete tank here you see a side tank why did we have a side tank because if you do cutting tests you get a lot of turbidity of the sand moving up and that would mean with an underwater camera you cannot see anything because of all the turbidity so for example when we do tests with a cutter head the sand goes into the suction mount which is this pipe is transported through a pump dumped in the side tank and that way we could keep the water clean at the bottom, this is the sand we had a drainage system so we could pump in or out water why did we do that well if you pump water in and in fact we needed the fire department connections to have enough flow we could make all the sand fluid and then if you do that so you pump water up all the garbage that is in the water will float to the top like clay particles or real garbage and you could scrape it away one time and that way keep the sand clean because if you get too many clay particles and fines in the sand it influences the permeability and you don't want that you want to have a very controllable permeability so that's when we pump water in when we pump water out we create a huge under pressure at the bottom of the tank en dat way you can compact the sand much better later I will show you how we really compact the sand well this was the carriage so you see the big tank and the side tank here we have a so called dustpan so after a test we can suck out the sand and pump it back to the main tank this is a cutter hat that we used 40 centimeter in diameter this one was for cutting rock and here this is that same cutter hat this is a cutter hat with smooth blades and we use this one for cutting clay I already told you if you have clay with a high adhesion it will stick between anything it can stick so if I would cut clay with this cutter hat within 5 minutes it's completely filled with clay and blocked and I can't dredge anymore but if I would use this one it's okay here you see the carriage with a blade at the bottom in fact I just did this in coral drawer because the real blade had too much rust on it so I just painted it this is the real blade and you see the sand here this is how we mounted the blade under the carriage so what you can see is a center blade and two side blades and each blade is mounted in force transducers and moment transducers torque transducers and why is it well the theory is two dimensional so if you want to check your theory you need to have 2D tests so on the center blade we measured the 2D process and on the side blades you also have the side effects which are 3D so we could distinguish between the center blade and the side blade for later research we put a window here with a camera behind it so we could actually see we were moving over the blade now when you do a cutting test the distance between those blades has to be very small because you don't want sand to go in between but on the other hand they shouldn't touch each other because you get a force influence if they actually touch each other but what happened at some tests when building the laboratory some people were welding in the neighborhood of the tank and after finishing the welding they just throw those small pieces of welding material into the sand and when we did tests so that's like I would say 3mm, 4mm thick bars from the welding device actually they went through this gap the gap was only 0.1mm but the forces were so high that the pressure pressed them in between and behind the blade we had those pressure transducers with pipes and so on and those welding things they went straight through it and everything was broken so the forces and the stresses can become so high that something you don't expect will still happen this is the whole mounting system I just show it for your information so the center blade and the two side blades here you see what we call dynamometers to measure forces and torques so we could actually measure everything those are the blades so you can see how they were constructed this is the actual blade but this is the holder and it has this shape because you don't want the holder to influence the cutting process it shouldn't touch the sand here you can see a blade with a wear flat so we welded a triangle of steel at the tip of the blade to simulate wear of the blade and we also did tests with that this is 45 degrees, 60 degrees, 30 degrees and we also had a 15 degree blade but then you can already imagine it's too flat it becomes too flat the tank had windows and we actually mounted cameras on the outside connected to the vehicle so if the whole thing is moving the camera is also moving and then in front of the blade you can record what is going on but only if the water is clean but we also used underwater cameras in fact we mounted a camera in the back of the blade so because the first thing you do when you do research you want to observe the physics don't just start with equations or whatever first observe the physics do some orientation tests see what is actually going on so you get a feeling this is what is happening and describe it in words not in equations, describe it in words because if you are not capable of describing in words what is going on you can never explain to other people what you are doing so the first step is just in words explain the physics in words here you see that window again and this is when we prepare the sand because you can tell there are sand in the tank we just do a test but you want the tests to be reproducible every time the sand should be the same so this is an eagleizer here to make the sand flat have a flat bed and here you see two concrete vibration needles en we use those to compact the sand because you don't want blue sand blue sand doesn't give a problem we like very dense sand how to make the sand dense well, the car, the vehicle will move with very low speed over the sand those vibration needles vibrate the sand that way they compact it and at the same time we found that if we put the drainage system on and suck out the water creating under pressure in the sand we get a better reproducibility of the sand so then each time the sand is almost the same 100% is not possible but almost what did we get out of such a test this is just a sheet as an example, the raw data we have six signals horizontal force, vertical force and then four pore pressure transducers where P1 is at the end of the blade and P4 is at the tip of the blade and you can see that both the forces and the pressures are not nice constant signals in time but they go up and down we had already told you that the failing process of sand is the shear type where you have a build up of pressure when the pressure is high enough a shear plane shoots to the surface that releases the pressure on the blade it moves forward, builds up the pressure again you get a shear plane again etc so this up and down, what you can see here every peak is a shear plane shooting to the surface and based on that if you count the number of peaks in such a signal you know the number of shear planes that you had and you divide it by the time and you can find the distance, the thickness of each shear plane so that's the reason why it's not a nice constant force but it goes up and down what is that? oh, ja, now it works ok, this is a picture of measurements of those poor pressure transducers in fact over a whole test we determined the average of all those signals and the average we put in graphs well what you see here, this is the blade and you see the location of the pressure transducers this is two series with a wear flat and without a wear flat the blue line is the result of the finite element programming so the numerical solution you can see the red with the wear is a little bit above the theory the green is a little bit below the theory but in general it matches pretty well and if I would change the size of the wear flat in the finite element calculations I could reduce or increase this peak that you have near the edge that one was for a very long blade you can imagine, well if I show it again if you have a very long blade most of the under pressures will occur near the tip of the blade but then it reduces rapidly but if I have a very small blade and small is in relation with the layer thickness then I get a shape like this and again you see the measurements with wear and without wear so with wear is always a little bit higher because the water cannot flow there so the under pressures will be a bit higher but the difference is not so much that I really need two separate finite element calculations for it here I have measurements the previous two graphs were dimensionless so I divided by many things to make it dimensionless here you have measurements in absolute pressures but in meters water column not in Pascal, but meters water column well the maximum you have is 10 meters water column because the amount of water above the center is so little we can neglect it that means if I get an under pressure of 10 meters water column I have full cavitation well what we got you can see what happens this is the cutting velocity and you can see the lowest one the blue line here well I'm far from having cavitation but if I go to the highest speeds up here then near the edge of the blade I have 9 meters under pressure or something which is already close to cavitation one thing you have to keep in mind when you do such experiments and that's again air if you would just have like less than 1% air in the water and air is compressible also expandable then I made calculations for that that with just a little bit air in the water I would never reach the 10 meter water column under pressure I would be stuck at about 9 meters just because of less than 1% air in the water and if you see how the water is pumped into such a tank with a lot of turbidity you can imagine er altijd be some air dissolved in the water and in fact if you have very tiny air bubbles in the sand it's almost impossible to get it out because you cannot put the whole tank in a vacuum tank it's too big so once you have some air in the sand it will be there and then with thermodynamics you can calculate how the air will expand if you get those under pressure so that's the reason why here we don't actually reach the 10 meters it's just because there was some air in the water this is the same but for a sand with a lower permeability because the lower the permeability the higher the under pressures and here you can see for small velocities we don't have cavitation but for the highest velocities you can see that shape that I showed you theoretically so everywhere you have that cavitation area close to the blade you will have an under pressure which is close to the cavitation under pressure then some forces horizontal forces this is from three sets of measurements and the blue line is the theoretical line and you can see that the measured points are a little bit higher than the theory but not that much if in reality you can make an estimate within 10% every company will be happy so the difference that you see here is not much and in fact you could say why not adapt the theory so it matches exactly you could do that in a set of tests and then it's below and then you have to adapt it again so the theory with the constants that I have in the theory are already based on many experiments and in fact I always feel theory doesn't have to match 100% with experiments as long as you get close enough this is the vertical force if you look at the horizontal force you can see a scale to 5 if you look at the vertical force you see a scale up to 1.5 and that's because the vertical forces are always much smaller than the horizontal forces a positive force means it pulls the blade into the soil a negative force means it's pushing it out and in this case everything is positive here you can see some tests in a sand at different layer thicknesses with speeds up to almost 1.5 meter per second and red is 25 millimeters green 50 millimeters and blue is 100 millimeters and especially with the blue line well here you can see it better it follows the curve this part is no cavitation and the horizontal part here is with cavitation so the fact that in the beginning it's increasing almost proportional to the velocity but above a certain point it's almost independent of the velocity you can see here so it matches pretty well horizontal force this is the specific energy and you can see that the specific energy in this case is bigger for a thinner layer snowplow effect we already talked about the snowplow effect that in general blades are not perpendicular to the cutting velocity but they are under an angle if you have a 3D curved cutter head and you divide the cutter head in small parts such a segment will never be perpendicular to the absolute velocity so it has a certain deviation angle with the velocity well here we have tests with zero degrees deviation so the blade is perpendicular to the velocity horizontal force, vertical force and specific energy and this would be the side force because of the snowplow effect but it's zero if it's perpendicular but in this case you have a 45 degree blade so it deviated 45 degrees so it's a snowplow under 45 degrees and suddenly you can see that you also get a large side force and the curves is what we calculated and the points is what we measured so we actually did this kind of research and you see that it gets close enough horizontal force and side force get close enough for the vertical force you could say there is some deviation but then on the other hand the forces, the vertical forces in general are so small compared to the other one here you have ten well most of the forces are like 1.5 and in the negative direction almost nothing that means if I have a force where the vertical force is almost zero then the factor is horizontal if something happens and this factor is moving a little bit downwards it has a very large influence on the vertical component but hardly anything on the horizontal component and that's what's happening here so the two horizontal forces this one is straight forward and this one is to the side they have quite a magnitude the vertical force is always small compared to those two we also like to determine the friction based on tests you can do it in a laboratory but you can also from your cutting test you can also try to derive it in fact what do you measure a horizontal force and a vertical force then if you use poor pressure measurements or finite element calculations you can determine a normal force a friction force and the two poor pressure forces oh it looks like the battery is empty so factorial you can make such a graph and that way determine what would be the delta value that we actually had during the test so we also did that and that's how we found that using two-thirds of the internal friction angle is a good way ja heb jij nog een nee verkeerde batterij gebruiken we die gewoon ok, so that was the normal cutting process and in some cases I already explained in the beginning in the cutting equations we divide by the sign of the sum of four angles and if the sum of those four angles gets close to 180 degrees it means you will divide by almost zero and then the cutting forces would become infinite well, nature will not do that so nature will find another way to deal with that problem so nature will create a wedge and this wedge can be static and it can be dynamic if we call it static it means the sand inside the wedge is not moving anymore it doesn't have velocity if it's dynamic it means it's still moving but maybe not with a constant speed or anything maybe you have a velocity gradient somewhere but it could still move so if we say ok, this is what's going to happen if the sum of those four angles gets too close to 180 degrees then how to determine the shape of that wedge because you can see the wedge has an angle theta, we call it the wedge angle if I know this wedge angle I could calculate everything because I can put it in a finite element program to determine my under pressures I can solve the equilibrium of forces moments, all those things but I have to know this theta well, first of all and you already saw those pictures but not for sand first of all we have to look at all the forces that we have and in this picture I only put the forces that we have in sand in the picture you saw last week I think I also, I really put all the forces so also adhesion, cohesion and so on here you only see the forces that you have in saturated sand so what do we have on the shear plane normal force and shear force and the pore pressure force and on the blade also normal force, shear force and pore pressure force all the other forces we are going to neglect they are either not there of they are so small that they do not influence the process then on the wedge it looks more complicated but in fact it's not really you have the same set of forces on each plane a normal force, shear force and pore pressure force and the thing is how to get them how to determine them but in this case having two equations for vertical and horizontal equilibrium is not enough we also need the equilibrium of moments and this is the forces on the blade you already saw such pictures before and here you have the forces that determine the moments so you have the normal forces and the pore pressure forces you have to know both in order to determine the equilibrium of moments we take the equilibrium of moments around the edge, the tip of the blade but you could do it in any point if there is an equilibrium of moments it should exist around each point the thing is, if I use the tip of the edge many forces go through the tip of the edge so I do not have to take them into account for the equilibrium of moments like the shear force on this plane goes through the tip of the blade so it doesn't result in a moment and then the equations become more simple so the choice of where around which point do you take the moments just depends on what makes the equations as simple as possible we have the same problem with the pore pressure the water flows to the shear plane but actually so this plane is sheering you have dilatation here here you do not have dilatation since the sand can only have dilatation once and if it is a static wedge in front of the blade that sand can only have dilatation once it expands it will not expand more so in the wedge in the wedge and on this plane I do not have dilatation only in the shear plane same picture as before for determining where does the flow come from with the boundary conditions of the finite element program again a coarse mesh although this one already looks pretty fine and here you have a fine mesh and this is the result and what you see from those calculations you get the highest under pressures in the shear plane but in the wedge not much is happening what I found is for the pore pressure calculations you can almost say I just assume the wedge is also made of steel and then the results of pore pressure calculations do not really differ much from what we did with the finite element program so in our calculations the water could flow through the wedge although we didn't have dilatation there but the water could flow through but it didn't really influence the pore pressures in the shear plane a lot so you could almost use the same values well here you see the flow lines and you can see the yellow which is high flow but in the wedge you can see there is not much flow based on the colors and this graph shows how the pore pressures develop along the different lines so from A to B or B to AB is where the shear plane reaches the surface so there your pressures are zero because you have hydrostatic pressure then you see the curve going to A A is the tip of the wedge not of the blade but of the wedge then from A to D you have the bottom of the wedge so you go from the tip of the edge to the bottom to the tip of the blade you can see how that develops then DC is the blade itself and CA is the top of the blade to the tip of the wedge in that direction and you can see how it develops the top one is for a case where you have a blade and such a wedge with a shape like this so a 90 degree blade with quite a wedge this one is the case where you have like such a blade and then a small wedge in front of it and that's why the bottom of the wedge is very small and this distance is the bottom of the wedge and then you can see what that looks like en in fact such a calculation should almost give exactly the same result as the case where you don't have any wedge so that's a way to check if the calculations are correct one more slide I think no we will stop here and then tomorrow continue with a good laser pointer