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Advanced Soil Mechanics 5
Tutorial Sheet 1 : Introduction and Elastic Models
1) Calculate values of mean effective stress p′ and deviatoric stress q for a
triaxial test specimen where σ a = 700 kPa , σ r = 400 kPa and u =100 kPa.
2) In a drained triaxial test the initial volume of the sample is 1500 cm3
. Failure
occurs at a nominal (engineering) axial strain of 5%, by which time 34.6 cm3
of water has drained from the sample. Calculate the true values of axial strain
ε a , volumetric strain ε v and deviatoric strain ε d at failure.
3) A saturated soil sample is tested in an undrained triaxial test. The cell pressure
is 200 kPa. The sample fails at a true axial strain of 10%, when the deviator
stress is 120 kPa and the pore pressure is 90 kPa. Calculate the value of mean
effective stress p′ at failure. Calculate also the true values of volumetric
strain ε v and deviatoric strain ε d at failure.
4) A saturated clay behaves as a linear isotropic elastic material. A sample of the
clay is mounted in a triaxial cell and the initial pore pressure is measured as
65kPa. The clay has shear modulus G′ =10 MPa and Poisson’s ratio
ν′ = 0.25. With no drainage allowed from the sample, and the cell pressure
σ r held constant, an axial stress increment σ a = 90 kPa is applied to the
sample. Calculate the axial strain ε a , radial strain ε r , volumetric strain ε v ,
deviatoric strain ε d and pore pressure u . The sample is then allowed to
drain, with the axial and radial total stresses unchanged, until the pore pressure
decays to zero. Calculate the final values of ε a , ε r , ε v and ε d after all
drainage is complete.
5) A second sample of clay, identical to that in Question 4, is sheared under
identical undrained conditions (with an initial pore pressure of 65 kPa) by
applying an axial stress increment of 90 kPa, before being allowed to drain.
However, this time the axial strain ε a is held constant during drainage, rather
than the axial stress. Calculate the final values of ε a , ε r , ε v and ε d after
all drainage is complete.
6) The bulk modulus K′ of a triaxial test sample varies linearly with the mean
effective stress p′ according to K′ = rp′ , where the constant r has a value
of 200. Calculate the volumetric strain ε v if the sample is compressed from
p′ = 200 kPa to p′ = 500 kPa. The sample is then sheared by the application
of a deviator stress q of 100 kPa, while holding p′ constant at 500kPa. This
is followed by a reduction of p′ (at constant q ) to 200 kPa, and then a
reduction of q (at constant p′) to zero. Show that the assumption of a
constant value of Poisson’s ratio ν′ = 0.3 would result in the prediction of a
residual deviatoric strain ε d of -0.11% at the end of the closed stress path,
thus contravening the tenets of elasticity.
7) A natural clay can be considered as a linear cross-anisotropic elastic material.
A sample of the clay is tested in a triaxial apparatus, with the vertical axis of
anisotropy of the soil aligned with the axis of the triaxial apparatus. When an
all-round effective stress increment ′ = r
′ = 20 a δσ δσ kPa is applied, the
measured radial strain increment δε r is 0.245% and the measured axial strain
increment δε a is 0.100%. When an axial effective stress increment
′ = 20 δσ a kPa is subsequently applied (with δσ r
′ = 0 ) the measured strain
increments are δε r = 0.075% and δε a = 0.250% . Confirm that these
measurements are consistent with the assumption of a linear cross-anisotropic
elastic material and calculate the values of Ev
′ , ν vh′ and ( ) Eh ν hh′ ′ / 1 . Suggest
a procedure that could be used to determine the separate values of Eh
′ and
ν hh′ in the triaxial apparatus.
Answers
1) 400 kPa, 300 kPa
2) 5.13%, 2.33% , 4.35%
3) 150 kPa, 0%, 10%
4) 0.30%, -0.15%, 0.30%, 0.00%, 95 kPa, 0.49%, 0.04%, 0.30%, 0.57%
5) 0.30%, 0.09%, 0.14%, 0.48%
6) 0.46%
7) 8 MPa, 0.30, 6.25 MPa.