Chapter 13, Problem 13.1P

(a)

To determine

Whether the statement “A lower safety factor implies that a greater fraction of the shear strength is mobilized along the failure surface” is true or false.

Answer to Problem 13.1P

The given statement is $\underset{\_}{\text{True}}$.

Explanation of Solution

Write the equation of factor of safety $\left(F{S}_s\right)$ with respect to strength.

$\begin{array}{l}F{S}_s=\frac{{\tau }_f}{{\tau }_d}\\ {\tau }_d=\frac{{\tau }_f}{F{S}_s}\end{array}$ (1)

Here, ${\tau }_f$ is the average shear strength of the soil along the potential failure surface and ${\tau }_d$ is the average shear stress mobilized along the potential failure surface.

Equation (1) clearly shows that the average shear stress mobilized $\left({\tau }_d\right)$ along the potential failure surface is inversely proportional to the factor of safety $\left(F{S}_s\right)$ with respect to strength.

Hence, the lower value of safety factor implies that the greater fraction of the shear strength.

Therefore, the given statement is $\underset{\_}{\text{True}}$.

(b)

To determine

Whether the statement “An infinite slope of cohesive soil cannot remain stable when $\beta \text{>}{\varphi }^{\prime }$” is true or false.

Answer to Problem 13.1P

The given statement is $\underset{\_}{\text{False}}$.

Explanation of Solution

Condition for stability:

The factor of safety is greater than 1.

Write the equation of factor of safety $\left(F{S}_s\right)$ with respect to strength.

$F{S}_s=\frac{c^{\prime }}{\gamma H{\cos }^2\beta \tan \beta }+\frac{\tan {\varphi }^{\prime }}{\tan \beta }$

Consider cohesionless soil, the effective cohesion $\left(c^{\prime }\right)$ is zero.

$\begin{array}{c}F{S}_s=\frac{0}{\gamma H{\cos }^2\beta \tan \beta }+\frac{\tan {\varphi }^{\prime }}{\tan \beta }\\ =\frac{\tan {\varphi }^{\prime }}{\tan \beta }\end{array}$ (2)

From Equation (2), when $\beta >{\varphi }^{\prime }$, the term $\tan {\varphi }^{\prime }/\tan \beta$ is less than 1. Therefore, cohesionless is not stable when $\beta >{\varphi }^{\prime }$.

But in case of cohesive soil, effective cohesion $\left(c^{\prime }\right)$ is greater than zero. Therefore, the term $\left[c^{\prime }/\left(\gamma H{\cos }^2\beta \tan \beta \right)\right]$ compensated the term $\tan {\varphi }^{\prime }/\tan \beta$ and it makes that the factor of safety is more than 1. Hence, the infinite slope of cohesive soil remains stable when $\beta \text{>}{\varphi }^{\prime }$.

Therefore, the given statement is $\underset{\_}{\text{False}}$.

(c)

To determine

Whether the statement “In an undrained slope of 1.0 (horizontal): 1.5 (vertical), the critical circle is always a toe circle.” is true or false.

Answer to Problem 13.1P

The given statement is $\underset{\_}{\text{True}}$.

Explanation of Solution

For slope angle $\left(\beta \right)$ greater than $\text{53}\text{.8}°$, the critical circle is always a toe circle.

Find the slope angle $\left(\beta \right)$ using the equation:

$\tan \beta =\frac{\text{Vertical}\text{rise}}{\text{Horizontal}\text{rise}}$

Substitute 1.5 for vertical rise and 1.0 for horizontal rise.

$\begin{array}{l}\tan \beta =\frac{1.5}{1.0}\\ \beta ={\tan }^{-1}\left(1.5\right)\\ \beta =56.3°\end{array}$

Here, the slope angle $\left(56.3°\right)$ is greater than $53°$. Hence, the critical failure circle is a toe circle.

Therefore, the given statement is $\underset{\_}{\text{True}}$.

(d)

To determine

Whether the statement “The mid-point circle passes through the toe” is true or false.

Answer to Problem 13.1P

The given statement is $\underset{\_}{\text{False}}$.

Explanation of Solution

• In case of base failure, the failure circle is called a mid-point circle.
• The surface of sliding that passes at some distance below the toe of the slope at the time of failure is called base failure. Hence, the mid-point circle does not pass through the toe.

Therefore, the given statement is $\underset{\_}{\text{False}}$.

(e)

To determine

Whether the statement “Taylor’s stability charts can be used only for homogeneous soils” is true or false.

Answer to Problem 13.1P

The given statement is $\underset{\_}{\text{True}}$.

Explanation of Solution

• Taylors stability chart is used to take the value of stability number (m) with corresponding friction angle of soil $\left(\varphi \right)$ and the slope angle $\left(\beta \right)$.
• Taylor’s stability chart is developed for slopes in homogeneous soil with friction angle $\left(\varphi \right)$.greater than zero. Hence, the Taylor’s stability charts can be used only for homogeneous soils.

Therefore, the given statement is $\underset{\_}{\text{True}}$.