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Video 1
In the beginning historical developments, we have seen where the Coulomb in the 18th century. Coulomb in the 18th century proposed the Shear Strength model that is tau equals to σ tan phi. So, the total stress controls the shear strength of the soil that is his approach where he proposed a one-phase model such as he considers soil as nothing, but weak rock mass. So, he does not distinguish between soil and rock only the difference between these two is the strength of the material changes from rock to soil.So, he proposes that the shear strength varies with σ the total stress or the total stress. So,therefore the direct shear test for obtaining a direct shear test was designed and the tau versus σ plots could be obtained by conducting several tests by varying the net normal stress on the soil, normal stress on the soil. So, the shear stress and shear strain profiles are obtained for heavily compacted sand and loosely compacted sands. So, this is a critical shear and this is a critical shear and this is a peak shear. So, this is said one net normal stress this is one net this is at 1 normal stress. So, if this is plotted with normal stress. So, when you get the data if you fit with a straight line fit with a linear line you get the angle of internal friction ϕ` and you had an angle of internal friction ϕ`. So, therefore, the shear strength of the soil can be determined using the shear strength is assumed to be dependent on directly total stress. So, the angle of internal friction of the soil can be determined directly. So, here it should be used phi there is no ϕ` when this was proposed. Then later in Terzaghi in 1920 came up with effective stress principle that is σ` should be equals to σ - u w. So, here mostly the stress is on, mostly the stress is on two-phase system soil solids and water; he considered soil solids and water. And if you have a two-phase system the σ` effective stress = σ - u w σ - u w. So, then the pore water the importance of pore water pressure in the determination of shear strength of the soil came into the picture and the triaxial tests were designed to control the drainage. Then by controlling the drainage valve drained and undrained tests are conducted and in the drain tests essentially you get the same size the Mohr circles. Essentially when the σ 3 is maintained particular value and σ 1 is varied and the soil failsm in this manner. So, these are undrained tests, and if the σ three is increased and the radius of the Mohr circle does not change. And this is another half essentially, you get the shear strength of the soil in an undrained manner the undrained shear strength of the soil is obtained. So, which are used for short term analysis. So, if in the same test if the pore water pressure is measured at the failure. So, then pore water pressure is measured then effective stressenvelopes also can be obtained. And all these things will merge into one single Mohr circle then you get effective stress parameters.So, you can determine effective stress parameters and then you can determine the drained parameters and undrained parameters in the same tests called consolidated undrained test. Similarly, the drain test can be conducted where τf and σ` variation can be obtained. So, for clay soils, there is an intercept that is seen this = tensile stresses. So, and this is an intercept and this is the angle of internal friction. So, using the C` and ϕ` the equation is modified now as C` + σ` tan ϕ`. So, C` and ϕ` are the shear strength parameters. Shear strength parameters are material constants and σ` is effective stress. And Bishop later on in the 1950s came up with two stress state parameters or two stress state variables for understanding the shear strength of unsaturated soils. So, here σ` + σ - u a + χf u a - u w. So, the χf is the effective stress parameter which indicates what is a contribution of u a - u w on the shear stress or shear strength of the soil. So, there were several tests conducted by controlling the suction in the direct shear test and triaxial tests, where the data obtained in this particular manner σ - u a if the tau and σ - u a are plotted for the saturated state. So, this is a failure envelope obtained this is withϕ` and this for u a - u w is 0 and this is for another. Similarly, for other envelopes for different u a - u w say if 50 kilo Pascal and this is for another u a - u w. The angle of internal friction nearly constant hardly varies within 4 to 5 degrees. Similarly, the tau by σ - u a if it is plotted with shear strain. So, this is how the stressstrain curves varied for different matric suction values. So, therefore these can beanalyzed using Bishop’s effective stress principle, extended effective stress principle this data can be very well be analyzed by determining the χf parameter for direct shear. And similarly, for triaxial test we have seen how to determine. Here for different u a - uw values initially the tests are conducted at a completely saturated state and then C` and ϕ` are obtained. Then, when the u a - u w’s are varied that χf parameters are obtained. Then χf relationship with u a - u w are obtained. This relationship is used to define the stress state of the soil very accurately.
Video 2
As we have seen for the Bishops approach the effective stress principle becomes σ ` = σ -m u a + χ times u a - u w. So, this χf is an effective stress parameter this provides the contribution of matrix suction to the shear strength of the soil. So, therefore, here as the χf parameter becomes 1, then this turns out to be Terzaghi’s effective stress principle if χf = 0 and it reference state are at gauge pressure σ ` = σ. So, this is nothing, but the Coulomb’s principle. So, the total stress = effective stress. So, as Bishop has identified 2 stress state variables such as σ - u a and u a - uw, several tests were conducted by controlling the suction interracial and direct shear operators. So, therefore, several data were obtained in direct shear where you get the stress ratios tau divided by σ - u a and gamma. So, the stress ratio gets increased with an increase inthe suction. So, this is for suction equals to 0 that is for the saturated state, and as the suction increases the stress ratio gets increased. So, this is how it increases? Even as the soil is normally consolidated or the soil is loosely compacted, as the suction value increases interestingly at very high suction values it exhibits a peak behavior, such as overconsolidated clays or densely compacted soils. So, this is a very high suction value may be large suction value. So, therefore, if you plot τf versus σ, what we get is the failure envelops with different intercepts, this is for suction s equals to 0 and this is for higher suction and this is a much higher suction. So, as profiles go up if the angle of internal sorry the intercept value keeps on changing increasing. However, the angle of internal friction value nearly the same. Except that the,whatever the small difference we find in the angle of internal friction due to experimental errors, apart from that the angle of friction remains the same. So, therefore, this expression could be the τf equals to σ - u a that is net normal stress + σ ` tanϕc + c` +. So, the effective stress is this. So, this is σ - u a + χf u a - u w times tanϕ`. So, if this expression τf versus σ - u a is plotted. So, this could be written as some c 1` + σ - u a, f tanϕ`. So, this c 1` = c` + χf u a - u w tanϕ`.So, therefore, when τf and σ - u a these 2 are plotted here, the angle of internal friction remains the same ϕ, but c 1` varies as u a - u w changes. So, as u a - u w varies the c 1` becomes c 2`. So, c 1` becomes c 2` and it keeps on increasing as u a - u w increases. So, this is what is observed. So, therefore, this expression could be used to understand or to analyze the test data we obtained from the suction control test. So, then we have utilized this particular expression to determine the χf from measured test data. So, using this expression for direct shear test data for direct shear and triaxial the χf is determined from this expression. So, this is the expression for the Triaxial test. So, in the direct shear test we require c` and ϕ` to be determined as c` andϕ` or the strength parameters of the soil, at saturated state and ϕ` anyways remain the same for even unsaturated soils ϕ` does not change. So, we conduct 2 tests 2 in a saturated state. So, that we can determine c` and ϕ` when we vary u a - u w then we get χf. So, that χf variation with respect to u a - u w can be obtained. If the test is on sands as at saturated state the cohesion intercept is 0 one testcan be conducted at a saturated state. So, that we can obtain the angle of internal friction value, and then we can obtain the functional form of χf. In the triaxial test also when we conduct 2 tests at the saturated state we can obtain c` and ϕ`, then once these 2 are known. The principal stress at failure can be determined by varying u a - u w as it is a suction control test we can control u a - u w and we can obtain the χf variation with respect to ua - u w. Once the χf is known, a functional form is known for us. So, the effective stress is known for that particular soil and the entire stress state of the soil can be determined in unsaturation. So, this is an approach given by bishop and several modifications were suggested by Fredlund Morgenstern and Fredlund Et al.,. Essentially they Fredlund has conducted several null type tests, where he has controlled u a the air pressure, water pressure, and the all-round pressure independently. And, then he increased these valuesindependently such that when the variation in σ - u a and delta σ - u w delta u a - u w. So, these values he started varying independently. So, then he checked the variation in any of these 2 stress variables are kept constant, and then he observed that the volume changes are 0. So; that means, he has kept these 2 parameters 1 and 2, the variation in 1 and 2 are kept constant and other things can be varied then the volume remains the same. Similarly, the 2 and 3 variations in the 2 and 3 parameters are second and third stress state parameters are kept constant. So, then similarly the volume is constant. Similarly, even 1 2 1 3 and different combinations of these stress state variables when they have maintained constant thevariations, so, the volume does not change. That means any of these 2 stress state parameters qualify for independently defining the stress state of the soil. So, therefore, he validates the Bishops approach and confirms that the σ - u a and u a - u w these 2 stressstate variables can be independently used for defining the soil state in unsaturated unsaturation. So, further, he proposes a new shear strength equation new failure criterion, where τf is written as c` + (σ - u a)f tanϕ` + u a - u w times tanϕ b. So, this particular expression isvery advantageous to represent the failure envelope clearly, when it is plotted in 3 dimensions τf in 3 dimensions τf σ - u a and u a - u w, or plotted. So, then the failure envelops can be seen. So, this is how the failure envelope is obtained?So, this angle with respect to u a - u w is ϕb and with respect to σ - u a is ϕ` this is again ϕb and this is the failure envelope. So, the equation defines this particular failureenvelope. Initially, it was thought that the ϕb is constant. So, therefore, we do not require a huge data sets for determining the variation of χf with u a - u w. When ϕb is constant only 1 set of data in saturated state and another set of data in unsaturated states arerequired. So, that in using 1 set of data in the saturated state we can determine σ and c` and ϕ`. And, in an unsaturated state, we can determine ϕb when u a - u w is varied. So, this is what initially thought and then this I was considered to be simplified and this is more advantageous, but later on, it was realized that the ϕb is not constant, ϕb itself is a function of u a - u w. And, as this was plotted τf was plotted with respect to u a - u w. It was observed that initially this nearly equals ϕ` when near the saturation region of SWCC and later on it started decreasing and becomes 0 and even becomes negative. So, here ϕ` is ϕb sorry this is ϕb. And, this ϕb is 0 and ϕb even becomes negative for some soils this was observed experimentally. So, therefore, as ϕb itself is not constant and which varies with u a - u w or the amount of water that is present in the soils. So, we do not draw any additional advantage. This is just similar to the Bishops approach except that a new variable is given here. So, this is similar to when you compare this expression and the Bishop’s expression. So, here the tanϕ b = χf tanϕ χf tanϕ` as χf is not varied, not constant. So, therefore, ϕb is not constant. So, therefore, ϕb anyways whatever we are obtaining ϕb from either considering several set of data, at different matrix suction values to determine ϕb this approach will be similar to obtaining χf using a set of several sets of several set of data. From suction control direct shear and suction control triaxial test.
Video 3
Later on, Lu and Likos in 2006 have come up with a suction stress characteristic curve called SSCC. So, in the suction stress characteristic curve if defined suction stress as, stress which takes care of all the chemical potential that has decreased due to the presence of electrostatic potentials, capillary suction force, capillary forces due to surface tension, and van der wall attraction forces between the particles. And all are cementation all any other forces that can come into soils in unsaturation and even at saturated state. So, therefore, they introduced suction stress, in the suction stress approach they have in the expression that is σ ` is written as σ - u a - σ s. So, this is suction stress. So, suction stress should absorb all different components of all different types of physicochemical capillary force effects in this particular parameter, but later on, this equation got diluted by introducing in Lu Et al., 2010. This expression in Lu and Lu and et al., 2010 later on this expression got reduce to simply - Se *( u a - u w), which was similar to the Bishop’s approach. So, if you write the substituted σs as - Se times u a - u w. So, this is simply Se times u a - u w. So, this Se is a degree of saturation, but this is a normalized degree of saturation itself got normalized S e = S degree of the situation at any given point - residual degree ofsaturation at residual point divided by (1 – Sr).If Sr = S = Sr this becomes 0 if S becomes 1. So, then this whole thing becomes 1 so, Se varies from 0 to 1. So, then they have substituted they expressed Se or u a - u w in terms of other they expressed S e terms of u a - u w, or they expressed u a - u w in terms of Se using interrelationship proposed by Van Genuchten equation. So, as Van Genuchten in1980; he has given expression for volumetric water content normalized volumetric water content that is 1 by 1 + α times he essentially says h over n whole power m. And, later on, he related m and n, and m is related to n by this particular formula. For simplification so, that you have only 2 parameters to estimate. So, a similar expression is used by Lu Et al., in 2010, they have used this particular form and they have used α and u a - u w power n power 1 - 1 by n. So, this expression they used then the effective stress equation is written as in terms of S e, this can be written as Se powern by 1 - n - 1 whole power 1 by n or in terms of u a - u w this becomes 1 + whole power n - 1 by n. So, these 2 expressions were given and again the estimation of this is a continuous form. So, if you have the data of SWCC then you can fit this particular expression Van Genuchten expression and you can obtain α and n parameters. So, if you know the α and n parameters you can obtain the suction stress characteristic curve by plotting σ s versus u a - u w if you do not have this data again using the same expression similar to bishops. So, the σ S can be written as - τf - c` - (σ - ua) f tanϕ` `, this is for direct shear and for triaxial. So, this becomes σ - u a (σ 1 - ua ) f – (σ 3 - ua )f tan square 45 + ϕ` by 2 - 2 c` tan 45 + ϕ` by 2 by 2 tan 45 + ϕ` by 2 tanϕ`. So, here similar to Bishop the c` and ϕ` can be determined, if the tests are conducted at saturated state, and then beyond that after that when the direct shear test is conducted atparticular u a - u w. Then σs can be determined at different u a - u w S, and σ S and u a - u w can be plotted to obtain the Suction Stress Characteristic Curve SSCC. Similarly, this is with the triaxial test, triaxial shear stress data is analyzed in this particular manner. So, essentially all these approaches are similar to the Bishops approach, were either to determine χf if χ is has known the tanϕ b. The ϕb parameter in Fredlund and Morgenstern can be obtained and the suction stress characteristic can also be derived if χf is known.So, therefore, essentially all these approaches are the same, and still so, to determine the effective stress of unsaturated soil we require χf or ϕb are σ S. So, therefore, we can determine effective stress and we can understand the failure envelopes in unsaturated soils.