Chapter 10, Problem 10.2CTP

A soil element beneath a pave ment experiences principal stress rotations when the wheel load, W, passes over it and moves away, as shown in Figure 10.51. In this case, the wheel load has passed over points A and B and is now over point C. The general state of stress at these points is similar to the one shown by a stress block at point D. The phenomenon of principal stress rotation influences the permanent deformation behavior of the pavement layers.

Investigate how the magnitude and the orientations of the principal stresses vary with distance from the point of application of the wheel load. Consider the case shown in Figure 10.51. An unpaved aggregate road with a thickness of 610 mm and unit weight of 19.4 kN/m³ is placed over a soil subgrade. A typical single-axle wheel load, W = 40 kN, is applied uniformly over a circular contact area with a radius of R = 150 mm (tire pressure of 565 kN/m²). The horizontal and shear stresses at each point are calculated from a linear elastic finite element analysis for a two-layer pavement and are presented in the following table.

1. a. Use Eq. (10.28) to calculate the vertical stress increases at soil elements A, B, and C that are located at radial distances 0.457,0.267, and 0 m, respectively, from the center of the load. Determine the total vertical stress (σ_y) due to wheel load, the overburden pressure at each point, and enter these values in the table.
2. b. Use the pole method to determine the maximum and minimum principal stresses (σ₁ and σ₃) for elements A, B, and C. Also determine the orientation (α_s) of the principal stress with respect to the vertical. Enter these values in the table.
3. c. Plot the variations of σ₁ and α_s, with normalized radial distance, r/R, from the center of loading.

To determine

Calculate the total vertical stress $\left({\sigma }_y\right)$ due to wheel load and the over burden pressure at each point.

Explanation of Solution

Given information:

The thickness of unpaved aggregate road $\left(t\right)$ is $610\text{mm}$.

The unit weight of aggregate layer $\left({\gamma }_{\text{agg}}\right)$ is $19.4\text{kN}/{\text{m}}^3$.

The radius of the wheel $\left(R\right)$ is $150\text{mm}$.

A typical single-axle wheel load $\left(W\right)$ is $40\text{kN}$.

The tire pressure $\left(q\right)$ is $565\text{kN}/{\text{m}}^2$.

The radial distance, horizontal stress, and shear stresses at each point are given in the Table.

Calculation:

Consider the unit weight of water $\left({\gamma }_w\right)$ is $9.81\text{kN}/{\text{m}}^3$.

Calculate the increase in vertical stress $\left(\Delta {\sigma }_z\right)$ using the relation as follows.

$\Delta {\sigma }_z=q\left(A^{\prime }+B^{\prime }\right)$ (1)

Here, $A^{\prime }$ and $B^{\prime }$ are the factors and a functions of $\frac{z}{R}$ and $\frac{r}{R}$.

Calculate the depth to radius ratio $\left(\frac{z}{R}\right)$ as shown below.

Substitute $305\text{mm}$ for z and $150\text{mm}$ for R.

$\begin{array}{c}\frac{z}{R}=\frac{305}{150}\\ =2.03\\ \approx 2\end{array}$

For the radial distance $\left(r\right)$ of $0.457\text{m}$:

Calculate the ratio $\left(\frac{r}{R}\right)$ as shown below.

Substitute $0.457\text{m}$ for r and $0.150\text{m}$ for R.

$\begin{array}{c}\frac{r}{R}=\frac{0.457}{0.150}\\ =3.05\\ \approx 3\end{array}$

Similarly calculate the remaining values and tabulate as in Table 1.

Calculate the value of $A^{\prime }$ as shown below.

Refer Table 10.8 “Variation of $A^{\prime }$ with $\frac{z}{R}$ and $\frac{r}{R}$” in the Text Book.

Take the value of $A^{\prime }$ as $0.02221$, for the values $\frac{z}{R}$ of $2$ and $\frac{r}{R}$ of $3$.

Similarly calculate the remaining values and tabulate as in Table 1.

Calculate the value of $B^{\prime }$ as shown below.

Refer Table 10.9 “Variation of $B^{\prime }$ with $\frac{z}{R}$ and $\frac{r}{R}$” in the Text Book.

Take the value of $B^{\prime }$ as $0.00028$, for the values $\frac{z}{R}$ of $2$ and $\frac{r}{R}$ of $3$.

Similarly calculate the remaining values and tabulate as in Table 1.

Calculate the increase in vertical stress $\left(\Delta {\sigma }_y\right)$ as shown below.

Substitute $565\text{kN}/{\text{m}}^2$ for q, $0.02221$ for $A^{\prime }$, and $0.00028$ for $B^{\prime }$.

$\begin{array}{c}\Delta {\sigma }_y=565\times \left(0.02221+0.00028\right)\\ =12.7\text{kN}/{\text{m}}^2\end{array}$

Similarly calculate the increase in vertical stress values and tabulate as in Table 1.

Calculate the overburden pressure at the middle of the $\left({\sigma }_o{}^{\prime }\right)$ clay layer as shown below.

${\sigma }_o{}^{\prime }=z{\gamma }_{\text{agg}}$

Substitute $0.305\text{m}$ for z and $19.4\text{kN}/{\text{m}}^3$ for ${\gamma }_{\text{agg}}$.

$\begin{array}{c}{\sigma }_o{}^{\prime }=0.305\times 19.4\\ =5.92\text{kN}/{\text{m}}^2\end{array}$

Calculate the total vertical pressure at each point using the relation.

${\sigma }_y=\Delta {\sigma }_y+{\sigma }_o{}^{\prime }$ (2)

For element A.

Substitute $12.7\text{kN}/{\text{m}}^{\text{2}}$ for $\Delta {\sigma }_y$ and $5.92\text{kN}/{\text{m}}^2$ for ${\sigma }_o{}^{\prime }$ in Equation (2).

$\begin{array}{c}{\sigma }_{y-A}=12.7+5.92\\ =18.62\text{kN}/{\text{m}}^2\end{array}$

Show the increase in vertical stress for each radial distances as in Table 1.

El.	Depth, $z\left(\text{m}\right)$	Radial distance $r\left(\text{m}\right)$	$\frac{z}{R}$	$\frac{r}{R}$	$A^{\prime }$	$B^{\prime }$	$\Delta {\sigma }_y\left(\text{kN}/{\text{m}}^2\right)$	Vertical stress, ${\sigma }_y\left(\text{kN}/{\text{m}}^2\right)$
A	0.305	0.457	2	3	0.02221	0.00028	12.7	18.62
B	0.305	0.267	2	1.78	0.05278	0.04391	54.63	60.55
C	0.305	0	2	0	0.10557	0.17889	160.71	166.63

Table 1

To determine

Calculate the maximum and minimum principal stresses for elements A, B, and C and the orientation of the principal stress with respect to the vertical using pole method.

Explanation of Solution

Given information:

The thickness of road $\left(t\right)$ is $610\text{mm}$.

The unit weight of aggregate layer $\left({\gamma }_{\text{agg}}\right)$ is $19.4\text{kN}/{\text{m}}^3$.

The radius of the wheel $\left(R\right)$ is $150\text{mm}$.

A typical single-axle wheel load $\left(W\right)$ is $40\text{kN}$.

The tire pressure $\left(q\right)$ is $565\text{kN}/{\text{m}}^2$.

The radial distance, horizontal stress, and shear stresses at each point are given in the Table.

Calculation:

Apply the procedure to construct the Mohr’s circle as shown below.

• Find the center of the circle $O_1$ located $\frac{{\sigma }_x+{\sigma }_y}{2}$ from the origin.
• Find the radius $\left(R\right)$ of the Mohr’s circle $\sqrt{{\left[\frac{{\sigma }_y-{\sigma }_x}{2}\right]}^2+{\tau }_{xy}^2}$.
• Sketch the Mohr’s circle once R has been determined.
• Draw a line CP parallel to the plane. Point P is the pole.
• Draw a line PD parallel to the plane AB. The point of intersection with Mohr’s circle is D.
• The coordinates of Q give the stresses on the plane AB.

For element A.

Calculate the center of the Mohr’s circle from the origin $\left(O{O}_1\right)$ as shown below.

$O{O}_1=\frac{{\sigma }_x+{\sigma }_y}{2}$

Substitute $25\text{kN}/{\text{m}}^2$ for ${\sigma }_x$ and $18.62\text{kN}/{\text{m}}^2$ for ${\sigma }_y$.

$\begin{array}{c}O{O}_1=\frac{25+18.62}{2}\\ =\frac{43.62}{2}\\ =21.81\text{kN}/{\text{m}}^2\end{array}$

Similarly calculate the remaining values and tabulate as in Table 2.

Calculate the radius $\left(R\right)$ Mohr’s circle as shown below.

$R=\sqrt{{\left[\frac{{\sigma }_y-{\sigma }_x}{2}\right]}^2+{\tau }_{xy}^2}$

Substitute $25\text{kN}/{\text{m}}^2$ for ${\sigma }_x$, $18.62\text{kN}/{\text{m}}^2$ for ${\sigma }_y$, and $17\text{kN}/{\text{m}}^2$ for ${\tau }_{xy}$.

$\begin{array}{c}R=\sqrt{{\left[\frac{25-18.62}{2}\right]}^2+{\left(17\right)}^2}\\ =\sqrt{299.1761}\\ =17.3\text{kN}/{\text{m}}^2\end{array}$

Similarly calculate the remaining values and tabulate as in Table 2.

Calculate the maximum principal stress $\left({\sigma }_1\right)$ using the relation.

${\sigma }_1=O{O}_1+R$

Substitute $21.81\text{kN}/{\text{m}}^2$ for $O{O}_1$ and $17.3\text{kN}/{\text{m}}^2$ for R.

$\begin{array}{c}{\sigma }_1=21.81+17.3\\ =39.11\text{kN}/{\text{m}}^2\end{array}$

Hence, the maximum principal stress $\left({\sigma }_1\right)$ is $\underset{\_}{39\text{kN}/{\text{m}}^2}$.

Similarly calculate the remaining values and tabulate as in Table 2.

Calculate the minimum principal stress $\left({\sigma }_3\right)$ using the relation.

${\sigma }_3=O{O}_1-R$

Substitute $21.81\text{kN}/{\text{m}}^2$ for $O{O}_1$ and $17.3\text{kN}/{\text{m}}^2$ for R.

$\begin{array}{c}{\sigma }_1=21.81-17.3\\ =4.51\text{kN}/{\text{m}}^2\end{array}$

Hence, the minimum principal stress $\left({\sigma }_3\right)$ is $\underset{\_}{4.5\text{kN}/{\text{m}}^2}$.

Similarly calculate the remaining values and tabulate as in Table 2.

The maximum principal stress acts on the plane inclined at an angle of $35°$ with the horizontal.

The angle $\begin{array}{c}\alpha =90°-35°\\ =55°\end{array}$

Similarly calculate the remaining values and tabulate as in Table 2.

Show the calculated the maximum and minimum principal stresses of each element as shown in Table 2.

El.	Vertical stress, ${\sigma }_y\left(\text{kN}/{\text{m}}^2\right)$	Centre of the circle $\left(O{O}_1\right)$	Radius of the circle $\left(R\right)$	${\sigma }_1\left(\text{kN}/{\text{m}}^2\right)$	${\sigma }_3\left(\text{kN}/{\text{m}}^2\right)$	$\alpha \left(\mathrm{deg}\right)$
A	18.62	21.81	17.3	39	4.5	55
B	60.55	46.28	47.21	93	1	48
C	166.63	86.82	79.82	167	7	0

Table 2

Refer to Table 2.

Sketch the Mohr’s circle for the soil element A as shown in Figure 1.

Sketch the Mohr’s circle for the soil element B as shown in Figure 2.

For element B, there is no shear stress. Hence, the horizontal and vertical stresses are principal stresses.

To determine

Plot the variations of ${\sigma }_1$ and     ${\alpha }_i$ with normalized radial distance $\frac{r}{R}$, from the center of loading.

Explanation of Solution

Given information:

The thickness of unpaved aggregate road $\left(t\right)$ is $610\text{mm}$.

The unit weight of aggregate layer $\left({\gamma }_{\text{agg}}\right)$ is $19.4\text{kN}/{\text{m}}^3$.

The radius of the wheel $\left(R\right)$ is $150\text{mm}$.

A typical single-axle wheel load $\left(W\right)$ is $40\text{kN}$.

The tire pressure $\left(q\right)$ is $565\text{kN}/{\text{m}}^2$.

The radial distance, horizontal stress, and shear stresses at each point are given in the Table.

Calculation:

Refer to Table 2.

Sketch the variations of ${\sigma }_1$ and ${\alpha }_i$ with normalized radial distance $\frac{r}{R}$, from the center of loading as shown in Figure 3.