Chapter 10, Problem 10.1P

### Principles of Geotechnical Enginee...

9th Edition
Braja M. Das + 1 other
ISBN: 9781305970939

Chapter
Section

### Principles of Geotechnical Enginee...

9th Edition
Braja M. Das + 1 other
ISBN: 9781305970939
Textbook Problem

# A soil element is shown in Figure 10.34. Determine the following: a. Maximum and minimum principal stresses b. Normal and shear stresses on plane AB Use Eqs. (10.3), (10.4), (10.6), and (10.7).

(a)

To determine

Calculate the maximum and minimum principal stresses.

The maximum principal stress (σ1) is 200.72kN/m2_.

The minimum principal stress (σ3) is 116.28kN/m2_.

Explanation

Given information:

The normal stress along x axis (σx) is 172kN/m2.

The normal stress along y axis (σy) is 145kN/m2.

The shear stress along xy axis (τxy) is +40kN/m2.

Calculation:

Find the horizontal angle as follows:

θ=90°68°=22°

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

σ1=σy+σx2+[σy+σx2]2+τxy2

Substitute 172kN/m2 for σx, 145kN/m2 for σy, and 40kN/m2 for τxy.

σ1=145+1722+[1451722]2+402=158.5+1,782.25=158.5+42.22=200.72kN/m2

Hence, the maximum principal stress (σ1) is 200.72kN/m2_.

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

σ3=σy+σx2[σy+σx2]2+τxy2

Substitute 172kN/m2 for σx, 145kN/m2 for σy, and 40kN/m2 for τxy.

σ3=145+1722[1451722]2+402=158.51,782.25=158.542.22=116.28kN/m2

Hence, the minimum principal stress (σ3) is 116.28kN/m2_.

(b)

To determine

Calculate the normal and shear stresses on plane AB.

The normal stress on plane AB (σn) is 176.6kN/m2_.

The shear stress on plane AB (τn) is 38.15kN/m2_.

Explanation

Given information:

The normal stress along x axis (σx) is 172kN/m2.

The normal stress along y axis (σy) is 145kN/m2.

The shear stress along xy axis (τxy) is +40kN/m2.

The angle (θ) is 22°.

Calculation:

Calculate the normal stress (σn) using the relation.

σn=σy+σx2+σyσx2cos2θ+τxysin2θ

Substitute 172kN/m2 for σx, 145kN/m2 for σy, 40kN/m2 for τxy, and 22° for θ.

σn=145+1722+1451722cos(2×22°)+40sin(2×22°)=158.513.5cos44°+40sin44°=176.6kN/m2

Hence, the normal stress on plane AB (σn) is 176.6kN/m2_.

Calculate the shear stress (τn) using the relation.

τn=σyσx2sin2θτxycos2θ

Substitute 172kN/m2 for σx, 145kN/m2 for σy, 40kN/m2 for τxy, and 22° for θ.

τn=1451722sin(2×22°)40cos(2×22°)=13.5sin44°40cos44°=38.15kN/m2

Therefore, the shear stress on plane AB (τn) is 38.15kN/m2_.

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