Fundamentals of Geotechnical Engineering (MindTap Course List)
Fundamentals of Geotechnical Engineering (MindTap Course List)
5th Edition
ISBN: 9781305635180
Author: Braja M. Das, Nagaratnam Sivakugan
Publisher: Cengage Learning
Question
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Chapter 13, Problem 13.22P

(a)

To determine

Find the factor of safety with respect to sliding for case (a).

(a)

Expert Solution
Check Mark

Answer to Problem 13.22P

The factor of safety with respect to sliding for case (a) is 1.5_.

Explanation of Solution

Given information:

The bed slope (n) is 2.

The angle (ϕ) of friction is 10°.

The cohesion (c) is 33.5kN/m2.

The unit weight of the soil (γ) is 17.29kN/m3.

The depth (H) of slope is 15.2 m.

Calculation:

Calculate slope angle (β) using the equation:

β=tan1(1n)

Substitute 2 for n.

β=tan1(12)=26.57°

Assume the developed angle of friction (ϕd) is 5°.

Find the factor of safety (Fϕ) with respect to friction using the equation:

Fϕ=tanϕtanϕd

Substitute 10° for ϕ and 5° for ϕd.

Fϕ=tan10°tan5°=2.01

Find the Taylor’s stability number (m).

Refer Figure 13.17, “Taylor’s stability number” in the textbook.

Take the value of Taylor’s stability number corresponding friction angle (ϕ) of 5° and slope angle (β) of 26.57°

The Taylor’s stability number (m) is 0.098.

Find the developed cohesion (cd) using the equation:

cd=mγH

Substitute 17.29kN/m3 for γ, 0.098 for m, and 15.2 m for H.

cd=0.098×17.29×15.2=25.76kN/m2

Find the factor of safety with respect to cohesion (Fc) using the equation:

Fc=ccd

Substitute 33.5kN/m2 for c and 25.76kN/m2 for cd

Fc=33.525.76=1.30

Similarly find the factor of safety (Fϕ) with respect to friction, the Taylor’s stability number (m), the developed cohesion (cd), and the factor of safety with respect to cohesion (Fc) for the various assumed value of developed angle of friction (ϕd).

Summarize the values of the factor of safety (Fϕ) with respect to friction, the Taylor’s stability number (m), the developed cohesion (cd), and the factor of safety with respect to cohesion (Fc) as in Table 1.

ϕd(deg)Fϕmcd(kN/m2)Fc
52.010.09825.761.30
61.680.09023.651.41
81.250.07820.501.63
1010.06416.821.99

Plot the graph of factor of safety (Fϕ) with respect to friction and factor of safety with respect to cohesion (Fc) as shown in Figure 1.

Fundamentals of Geotechnical Engineering (MindTap Course List), Chapter 13, Problem 13.22P , additional homework tip  1

Refer Figure 1.

Find the factor of safety (FS) with respect to sliding in case (a).

Draw a line from origin of the graph with an angle of 45°. It intersects the curve at the coordinate of (1.5, 1.5). The intersection point is the factor of safety with respect to sliding.

Therefore, the factor of safety with respect to sliding for case (a) is 1.5_.

(b)

To determine

Find the factor of safety with respect to sliding for case (b).

(b)

Expert Solution
Check Mark

Answer to Problem 13.22P

The factor of safety with respect to sliding for case (b) is 1.36_.

Explanation of Solution

Given information:

The bed slope (n) is 1.0.

The angle (ϕ) of friction is 20°.

The cohesion (c) is 19.2kN/m2.

The unit weight of the soil (γ) is 18.08kN/m3.

The depth (H) of slope is 9.15 m.

Calculation:

Calculate slope angle (β) using the equation:

β=tan1(1n)

Substitute 1 for n.

β=tan1(11)=45°

Assume the developed angle of friction (ϕd) is 5°.

Find the factor of safety (Fϕ) with respect to friction using the equation:

Fϕ=tanϕtanϕd

Substitute 20° for ϕ and 5° for ϕd.

Fϕ=tan20°tan5°=4.16

Find the Taylor’s stability number (m).

Refer Figure 13.17, “Taylor’s stability number” in the textbook.

Take the value of Taylor’s stability number corresponding friction angle (ϕ) of 5° and slope angle (β) of 45°.

The Taylor’s stability number is 0.133.

Find the developed cohesion (cd) using the equation:

cd=mγH

Substitute 18.08kN/m3 for γ, 0.133 for m, and 9.15 m for H.

cd=0.133×18.08×9.15=22.00kN/m2

Find the factor of safety with respect to cohesion (Fc) using the equation:

Fc=ccd

Substitute 19.2kN/m2 for c and 22.00kN/m2 for cd

Fc=19.222.00=0.87

Similarly find the factor of safety (Fϕ) with respect to friction, the Taylor’s stability number (m), the developed cohesion (cd), and the factor of safety with respect to cohesion (Fc) for the various assumed value of developed angle of friction (ϕd).

Summarize the values of the factor of safety (Fϕ) with respect to friction, the Taylor’s stability number (m), the developed cohesion (cd), and the factor of safety with respect to cohesion (Fc) as in Table 1.

ϕ(deg)Fϕmcd(kN/m2)Fc
54.160.133220.87
102.060.10517.371.10
151.360.08013.231.45
2010.0589.72

Plot the graph of factor of safety (Fϕ) with respect to friction and factor of safety with respect to cohesion (Fc) as shown in Figure 1.

Fundamentals of Geotechnical Engineering (MindTap Course List), Chapter 13, Problem 13.22P , additional homework tip  2

Refer Figure 2.

Find the factor of safety (FS) with respect to sliding in case (b).

Draw a line from origin of the graph with an angle of 45°. It intersects the curve at the coordinate of (1.36, 1.36). The intersection point is the factor of safety with respect to sliding.

Therefore, the factor of safety with respect to sliding for case (b) is 1.36_.

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Please answer (a)
The figure below shows a slope with an inclination of B = 56°. If AC represents a trial failure plane inclined at an angle 0 = 33° with the horizontal, determine the factor of safety against sliding for the wedge ABC. Given H = 8.4 m; y = 20 kN/m³: ' = 21°; and c' = 36 kN/m?. (Round your final answer to two decimal places.) F, =
Stability of slopes.
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