1 1.351 HOMEWORK SET 6 Due: Friday April 3 1. The shear strength of a soil is most frequently characterized by a frictional failure criterion. Two commonly used failure criteria are: Mohr-Coulomb: f = σ 1 - σ 3 2 - σ 1 + σ 3 2 sin φ = 0 (1a) Extended von Mises: f = J 2 - h 2 σ 2 = 0 (1b) where: σ 1 , σ 3 are the major and minor principal stresses, φ is the friction angle. J 2 = 1 2 s ij s ij is the second invariant of deviatoric stresses, σ = 1 3 σ kk is the mean (octahedral) stress and h is a constant. In conventional practice, the friction angle (φ = φ TC ) is reported from measurements of the shear strength in triaxial compression type tests (in which σ xx =σ zz < σ yy ; Fig. 1): a) Derive an expression for the frictional parameter, h (eqn. 1b) in terms of this friction angle. b) Assuming that failure of the soil is best described by the extended von Mises criterion, what is the frictional angle mobilized in a triaxial extension test (i.e., φ TE for the case when σ xx =σ zz > σ yy ). c) Plot values of φ TE as a function of the measured friction angle, φ TC for the extended von Mises failure criterion. y x z Figure 1.
1. The shear strength of a soil is most frequently characterized
by a frictional failure criterion. Two commonly used failure
criteria are:
Mohr-Coulomb: f = 1 - 32
- 1 + 32
sin = 0 (1a)
Extended von Mises: f = J2 - h22 = 0 (1b) where: 1, 3 are the
major and minor principal stresses, is the friction angle. J2= 12
si jsi j is the
second invariant of deviatoric stresses, = 13 kk is the mean
(octahedral) stress and h is a
constant. In conventional practice, the friction angle ( = TC)
is reported from measurements of the shear strength in triaxial
compression type tests (in which xx=zz < yy; Fig. 1): a) Derive
an expression for the frictional parameter, h (eqn. 1b) in terms of
this friction angle. b) Assuming that failure of the soil is best
described by the extended von Mises criterion, what is
the frictional angle mobilized in a triaxial extension test
(i.e., TE for the case when xx=zz > yy).
c) Plot values of TE as a function of the measured friction
angle, TC for the extended von Mises failure criterion.
y
x
z
Figure 1.
2
2. Figure 2 shows a slope with inclination, i, in a dry soil
with c = 0 kPa, and ' = 30o, calculate the ultimate surcharge load,
qult, which can be applied vertically to the top of the slope. Do
not consider the self weight of the soil, and assume a normal
resisting load, p, to act normal to the slope (Fig. 2). There are
closed-form solutions for this problem. However, I would like you
to solve this problem using the graphical construction method
proposed by Scott (1963) - and presented in Appendix C available on
stelllar web site.
Report your results in the form of a dimensionless plot of
qult/p versus slope inclination angle, i. State clearly any
additional assumptions you make.
Dry soil' = 300, c = 0
surcharge, qult (force/unit area)
H
i
resisti
ng loa
d, p
Figure 2
3
3. A bulldozer blade of height, H=5m, applies a horizontal (H)
and vertical (shear) force (V) as it pushes vertically against a
layer of clay with unit weight, =20kN/m3 and (undrained) shear
strength, c=20kPa, as shown in Figure 3. Assuming that the
bulldozer blade remains vertical and that the maximum shear
resistance at the blade-soil interface is given by ci=0.7c:
a) Using the mechanism in figure 3a, compute an upper bound
estimate of the forces acting on the bulldozer blade.
b) By considering the stress states at elements A and B located
at a depth z in figure 3b, find a lower bound solution for the same
problem.
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