woodcomputer
Computer
- Jan 13, 2025
- 2
Disclaimer: I am not a geotechnical engineer of any kind, nor a student with that kind of training or anything adjacent to it. I'm just a guy who would like to better understand some of the peripheral concepts that are tangential to my actual field of work. I'm going to draw some embarrassingly false conclusions as I lay out what I think I understand, so please bear with me.
I'm trying to better understand the relationship between the angle of repose and angle of internal friction of soils. I have surfed around for articles that try to clarify this relationship, but all I've found either hardly say anything enlightening at all, or are too thick in the weeds for me.
I am at least aware that the two are not the same thing. But I do know they are related to one another. In fact, I think it's not uncontroversial to say that in many cases, one can approximate the other, especially given how many results I've found online that seem to conclude that they are roughly equivalent, even if they aren't technically equivalent. I've even attempted to read Rankine's original 1857 paper on the stability of loose earth, and see that he inserts the angle of repose of soil where we would today use internal friction angle in those equations. I don't feel he'd have done such a thing if the two values weren't surreptitiously close at least some of the time. So, I make the following assumption:
Assumption 1: Angle of repose being approximately equal to internal friction angle (θ ≈ φ) is not a given, but is not uncommon.
Now, I turn my attention to Mohr-Coulomb diagrams for triaxial shear tests.
If I understand it correctly, the plot of a Mohr's circle from a triaxial test encodes measurements of compressive stress and shear stress in the soil normal to some imaginary plane, for all possible rotational angles of that plane. A horizontal plane corresponds to the rightmost point on the circle, a vertical (rotated 90°) plane corresponds to the leftmost point on the circle, and all rotational angles of the plane A in-between correspond to an angle 2A of rotation about the circle counterclockwise.
If we draw the failure envelope line on the diagram and observe the point where the line is tangent to Mohr's circle, we can obtain an angle of rotation about the circle to that point. I can trivially show with some geometry that this angle must be 90° + φ, where φ is the angle of internal friction measured on the same diagram. Through the double-angle relation of the diagram, that should correspond to an imaginary plane cutting through the soil sample at an angle of 45° + (φ / 2) from the horizontal. It seems intuitive to me that this must be the plane along which the real world soil sample actually shears. So,
Assumption 2: When a soil sample fails in shear due to vertical load, it shears along a plane at angle 45° + (φ/2) from the horizontal.
This assumption gets some extra points in my eyes, because that expression looks familiar to me from Rankine theory (even though it came from Mohr-Coulomb theory).
However, it also seems intuitive to me that when a soil fails in shear due to a vertical load, it will fail along a plane angled at its angle of repose. That is, as I understand it, what the angle of repose is. So,
Assumption 3: When a soil sample fails in shear due to vertical load, it shears along a plane at angle θ from the horizontal.
Combining the three assumptions, however, I get a contradiction:
φ ≈ θ = 45° + (φ/2)
φ ≈ 45° + (φ/2)
This is obviously nonsense. So, clearly, I am not understanding things and I have made false assumptions.
Could anyone help me identify in which ways I am misinformed?
I'm trying to better understand the relationship between the angle of repose and angle of internal friction of soils. I have surfed around for articles that try to clarify this relationship, but all I've found either hardly say anything enlightening at all, or are too thick in the weeds for me.
I am at least aware that the two are not the same thing. But I do know they are related to one another. In fact, I think it's not uncontroversial to say that in many cases, one can approximate the other, especially given how many results I've found online that seem to conclude that they are roughly equivalent, even if they aren't technically equivalent. I've even attempted to read Rankine's original 1857 paper on the stability of loose earth, and see that he inserts the angle of repose of soil where we would today use internal friction angle in those equations. I don't feel he'd have done such a thing if the two values weren't surreptitiously close at least some of the time. So, I make the following assumption:
Assumption 1: Angle of repose being approximately equal to internal friction angle (θ ≈ φ) is not a given, but is not uncommon.
Now, I turn my attention to Mohr-Coulomb diagrams for triaxial shear tests.
If I understand it correctly, the plot of a Mohr's circle from a triaxial test encodes measurements of compressive stress and shear stress in the soil normal to some imaginary plane, for all possible rotational angles of that plane. A horizontal plane corresponds to the rightmost point on the circle, a vertical (rotated 90°) plane corresponds to the leftmost point on the circle, and all rotational angles of the plane A in-between correspond to an angle 2A of rotation about the circle counterclockwise.
If we draw the failure envelope line on the diagram and observe the point where the line is tangent to Mohr's circle, we can obtain an angle of rotation about the circle to that point. I can trivially show with some geometry that this angle must be 90° + φ, where φ is the angle of internal friction measured on the same diagram. Through the double-angle relation of the diagram, that should correspond to an imaginary plane cutting through the soil sample at an angle of 45° + (φ / 2) from the horizontal. It seems intuitive to me that this must be the plane along which the real world soil sample actually shears. So,
Assumption 2: When a soil sample fails in shear due to vertical load, it shears along a plane at angle 45° + (φ/2) from the horizontal.
This assumption gets some extra points in my eyes, because that expression looks familiar to me from Rankine theory (even though it came from Mohr-Coulomb theory).
However, it also seems intuitive to me that when a soil fails in shear due to a vertical load, it will fail along a plane angled at its angle of repose. That is, as I understand it, what the angle of repose is. So,
Assumption 3: When a soil sample fails in shear due to vertical load, it shears along a plane at angle θ from the horizontal.
Combining the three assumptions, however, I get a contradiction:
φ ≈ θ = 45° + (φ/2)
φ ≈ 45° + (φ/2)
This is obviously nonsense. So, clearly, I am not understanding things and I have made false assumptions.
Could anyone help me identify in which ways I am misinformed?
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