Principles Of Geotechnical Engineering, Si Edition
Principles Of Geotechnical Engineering, Si Edition
9th Edition
ISBN: 9781305970953
Author: Braja M. Das, Khaled Sobhan
Publisher: Cengage Learning
Question
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Chapter 15, Problem 15.27P
To determine

Find the minimum factor of safety Fs using the Bishop and Morgenstern’s method.

Expert Solution & Answer
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Answer to Problem 15.27P

The the minimum factor of safety Fs using the Bishop and Morgenstern’s method is 1.19_.

Explanation of Solution

Given information:

The height (H) of the slope is 30 m.

The inclination β of a slope (3:1slope) is 18.43°.

The unit weight γ is 18kN/m3.

The angle of friction ϕ is 23°.

The cohesion c is 27kN/m2.

The value non-dimensional quantity ru is 0.5.

Calculation:

Determine the minimum factor of safety Fs from Table (15.5), using the procedure as follows:

  • Step 1. Obtain the values of angle of friction, inclination of a slope, and the value of cγH
  • Step 2. Obtain the value of nondimensional quantity.
  • Step 3. Refer Table (15.5) “Values of m and n for cγH=0” in the text book. Obtain the values of m and n for D=1,1.25,and1.5.
  • Step 4. calculate the factor of safety value, using the values of m and n for each value of D.
  • Finally, the required value of factor of safety is the smallest one obtained in step 4.

Trial 1:

Determine the value of cγH.

Substitute 27kN/m2 for c, 18kN/m3 for γ, and 30 m for H.

cγH=2718(30)=0.05

Refer Table (15.5d) “Stability coefficients m and n for cγH=0.05 and D=1.00” in the text book.

Take the stability coefficient m as 2.014 when the angle of friction ϕ is 22.5°.

Take the stability coefficient m as 2.193 when the angle of friction ϕ is 25°.

Determine the stability coefficient m when the angle of friction ϕ is 23° using the interpolation.

m=(2322.5)(2.1932.014)(2522.5)+2.014=2.049

Take the stability coefficient n as 1.568 when the angle of friction ϕ is 22.5°.

Take the stability coefficient n as 1.757 when the angle of friction ϕ is 25°.

Calculate the stability coefficient n when the angle of friction ϕ is 23° using the interpolation.

n=(2322.5)(1.7571.568)(2522.5)+1.568=1.606

Calculate the factor of safety using the formula.

Fs=mnru

Substitute 2.049 for m, 1.606 for n, and 0.5 for ru.

Fs=2.0491.606(0.5)=1.25

Trial 2:

Refer Table (15.5e) “Stability coefficients m and n for cγH=0.05 and D=1.25” in the text book.

Take the stability coefficient m as 2.024 when the angle of friction ϕ is 22.5°.

Take the stability coefficient m as 2.222 when the angle of friction ϕ is 25°.

Determine the stability coefficient m when the angle of friction ϕ is 23° using the interpolation.

m=(2322.5)(2.2222.024)(2522.5)+2.024=2.064

Take the stability coefficient n as 1.690 when the angle of friction ϕ is 22.5°.

Take the stability coefficient n as 1.897 when the angle of friction ϕ is 25°.

Determine the stability coefficient n when the angle of friction ϕ is 23° using the interpolation.

n=(2322.5)(1.8971.690)(2522.5)+1.690=1.731

Determine the factor of safety using the formula.

Fs=mnru

Substitute 2.064 for m, 1.731 for n, and 0.5 for ru.

Fs=2.0641.731(0.5)=1.19

Trial 3:

Refer Table (15.5f) “Stability coefficients m and n for cγH=0.05 and D=1.50” in the text book.

Take the stability coefficient m as 2.234 when the angle of friction ϕ is 22.5°.

Take the stability coefficient m as 2.467 when the angle of friction ϕ is 25°.

Calculate the stability coefficient m when the angle of friction ϕ is 23° using the interpolation.

m=(2322.5)(2.4672.234)(2522.5)+2.234=2.281

Take the stability coefficient n as 1.937 when the angle of friction ϕ is 22.5°.

Take the stability coefficient n as 2.179 when the angle of friction ϕ is 25°.

Calculate the stability coefficient n when the angle of friction ϕ is 23° using the interpolation.

n=(2322.5)(2.1791.937)(2522.5)+1.937=1.985

Calculate the factor of safety using the formula.

Fs=mnru

Substitute 2.281 for m, 1.985 for n, and 0.5 for ru.

Fs=2.2811.985(0.5)=1.29

The required value of factor of safety is the smallest one obtained from trial 2.

Thus, the minimum factor of safety Fs using the Bishop and Morgenstern’s method is 1.19_.

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