The following results were obtained from a CU on two samples:
Test no. Confining stress (kPa) Deviatoric stress (kPa) Pore pressure (kPa)
1 100 400 -20
2 300 760 60
a) Determine c and ƒÖ for total stress and effective stress condition.
b) What is the deviatoric stress at failure if the confining stress is 500 kPa?
c) What is pore pressure at failure?
To determine the values of c (cohesion) and φ (angle of internal friction) for total stress and effective stress conditions, we can use the Mohr-Coulomb failure criterion. The failure criterion relates deviatoric stress (σ') to normal stress (σ) using the equation:
σ' = c + (σ - Pore pressure) * tan φ
Where:
σ' - Deviatoric stress
σ - Confining stress (effective stress condition) or Total stress (total stress condition)
Pore pressure - Pore pressure
Given the values in the two samples, we can calculate c and φ for both total stress and effective stress conditions.
a) Determining c and φ for Total Stress Condition:
For total stress condition, we directly use the values of σ, σ', and Pore pressure to calculate c and φ.
Sample 1:
σ = 100 kPa
σ' = 400 kPa
Pore pressure = -20 kPa
Using the formula: σ' = c + (σ - Pore pressure) * tan φ
400 = c + (100 - (-20)) * tan φ
400 = c + 120 * tan φ ---(Equation 1)
Sample 2:
σ = 300 kPa
σ' = 760 kPa
Pore pressure = 60 kPa
Using the formula again:
760 = c + (300 - 60) * tan φ
760 = c + 240 * tan φ ---(Equation 2)
Now, solve the equations simultaneously to find c and φ.
Subtracting Equation 1 from Equation 2:
760 - 400 = c + 240 * tan φ - 120 * tan φ
360 = 120 * tan φ
tan φ = 3
Plugging the value of tan φ back into Equation 1:
400 = c + 120 * 3
c = 400 - 360
c = 40 kPa
So, for total stress condition:
c = 40 kPa
φ = arctan(3)
b) Determining Deviatoric Stress at Failure:
Given confining stress = 500 kPa.
To find deviatoric stress at failure, we can use the relationship between deviatoric and confining stress:
Deviatoric stress at failure = (Confining stress * sin φ) + c
Using the values we calculated above:
c = 40 kPa
φ = arctan(3)
Confining stress = 500 kPa
Deviatoric stress at failure = (500 * sin(arctan(3))) + 40
c) Determining Pore Pressure at Failure:
To determine the pore pressure at failure, we need the deviatoric stress at failure and the confining stress. However, the given information does not provide enough data to calculate this value.