Principles of Geotechnical Engineering (MindTap Course List)
Principles of Geotechnical Engineering (MindTap Course List)
9th Edition
ISBN: 9781305970939
Author: Braja M. Das, Khaled Sobhan
Publisher: Cengage Learning
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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.

Chapter 10, Problem 10.2CTP, A soil element beneath a pave ment experiences principal stress rotations when the wheel load, W, , example  1 Chapter 10, Problem 10.2CTP, A soil element beneath a pave ment experiences principal stress rotations when the wheel load, W, , example  2

  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 (σy) 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 (αs) of the principal stress with respect to the vertical. Enter these values in the table.
  3. c. Plot the variations of σ1 and αs, with normalized radial distance, r/R, from the center of loading.

(a)

Expert Solution
Check Mark
To determine

Calculate the total vertical stress (σy) due to wheel load and the over burden pressure at each point.

Explanation of Solution

Given information:

The thickness of unpaved aggregate road (t) is 610mm.

The unit weight of aggregate layer (γagg) is 19.4kN/m3.

The radius of the wheel (R) is 150mm.

A typical single-axle wheel load (W) is 40kN.

The tire pressure (q) is 565kN/m2.

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

Calculation:

Consider the unit weight of water (γw) is 9.81kN/m3.

Calculate the increase in vertical stress (Δσz) using the relation as follows.

Δσz=q(A+B) (1)

Here, A and B are the factors and a functions of zR and rR.

Calculate the depth to radius ratio (zR) as shown below.

Substitute 305mm for z and 150mm for R.

zR=305150=2.032

For the radial distance (r) of 0.457m:

Calculate the ratio (rR) as shown below.

Substitute 0.457m for r and 0.150m for R.

rR=0.4570.150=3.053

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

Calculate the value of A as shown below.

Refer Table 10.8 “Variation of A with zR and rR” in the Text Book.

Take the value of A as 0.02221, for the values zR of 2 and rR of 3.

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

Calculate the value of B as shown below.

Refer Table 10.9 “Variation of B with zR and rR” in the Text Book.

Take the value of B as 0.00028, for the values zR of 2 and rR of 3.

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

Calculate the increase in vertical stress (Δσy) as shown below.

Substitute 565kN/m2 for q, 0.02221 for A, and 0.00028 for B.

Δσy=565×(0.02221+0.00028)=12.7kN/m2

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

Calculate the overburden pressure at the middle of the (σo) clay layer as shown below.

σo=zγagg

Substitute 0.305m for z and 19.4kN/m3 for γagg.

σo=0.305×19.4=5.92kN/m2

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

σy=Δσy+σo (2)

For element A.

Substitute 12.7kN/m2 for Δσy and 5.92kN/m2 for σo in Equation (2).

σyA=12.7+5.92=18.62kN/m2

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

El.Depth, z(m)

Radial distance

r(m)

zRrR

A

B

Δσy(kN/m2)Vertical stress, σy(kN/m2)
A0.3050.457230.022210.0002812.718.62
B0.3050.26721.780.052780.0439154.6360.55
C0.3050200.105570.17889160.71166.63

Table 1

(b)

Expert Solution
Check Mark
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 (t) is 610mm.

The unit weight of aggregate layer (γagg) is 19.4kN/m3.

The radius of the wheel (R) is 150mm.

A typical single-axle wheel load (W) is 40kN.

The tire pressure (q) is 565kN/m2.

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 O1 located σx+σy2 from the origin.
  • Find the radius (R) of the Mohr’s circle [σyσx2]2+τxy2.
  • 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 (OO1) as shown below.

OO1=σx+σy2

Substitute 25kN/m2 for σx and 18.62kN/m2 for σy.

OO1=25+18.622=43.622=21.81kN/m2

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

Calculate the radius (R) Mohr’s circle as shown below.

R=[σyσx2]2+τxy2

Substitute 25kN/m2 for σx, 18.62kN/m2 for σy, and 17kN/m2 for τxy.

R=[2518.622]2+(17)2=299.1761=17.3kN/m2

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

Calculate the maximum principal stress (σ1) using the relation.

σ1=OO1+R

Substitute 21.81kN/m2 for OO1 and 17.3kN/m2 for R.

σ1=21.81+17.3=39.11kN/m2

Hence, the maximum principal stress (σ1) is 39kN/m2_.

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

Calculate the minimum principal stress (σ3) using the relation.

σ3=OO1R

Substitute 21.81kN/m2 for OO1 and 17.3kN/m2 for R.

σ1=21.8117.3=4.51kN/m2

Hence, the minimum principal stress (σ3) is 4.5kN/m2_.

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 α=90°35°=55°

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, σy(kN/m2)Centre of the circle (OO1)Radius of the circle (R)σ1(kN/m2)σ3(kN/m2)α(deg)
A18.6221.8117.3394.555
B60.5546.2847.2193148
C166.6386.8279.8216770

Table 2

Refer to Table 2.

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

Principles of Geotechnical Engineering (MindTap Course List), Chapter 10, Problem 10.2CTP , additional homework tip  1

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

Principles of Geotechnical Engineering (MindTap Course List), Chapter 10, Problem 10.2CTP , additional homework tip  2

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

(c)

Expert Solution
Check Mark
To determine

Plot the variations of σ1 and     αi with normalized radial distance rR, from the center of loading.

Explanation of Solution

Given information:

The thickness of unpaved aggregate road (t) is 610mm.

The unit weight of aggregate layer (γagg) is 19.4kN/m3.

The radius of the wheel (R) is 150mm.

A typical single-axle wheel load (W) is 40kN.

The tire pressure (q) is 565kN/m2.

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 σ1 and αi with normalized radial distance rR, from the center of loading as shown in Figure 3.

Principles of Geotechnical Engineering (MindTap Course List), Chapter 10, Problem 10.2CTP , additional homework tip  3

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