Chapter Nine

Stresses in a Soil Mass

9.1 Introduction Construction of a foundation causes changes in the stress, usually a net increase. The net stress increase in the soil depends on: • the load per unit area to which the foundation is subjected • the depth below the foundation at which the stress estimation is desired. It is necessary to estimate the net increase of vertical stress in soil that occurs as a result of the construction of a foundation so that settlement can be calculated.

9.2 The 2:1 Method

A simple method to determine the approximate increase in vertical stress beneath a footing , it assumes that the stress dissipates with depth in the form of trapezoid that has 2:1 ( verical: horizontal) inclined sides.

The main disadvantage is that with 2:1 method, the stress increase under the corner of uniformly loaded area equals the stress increase under the center, whereas in the actual situation, the stress increase under the center is greater than that under the corner; thus this method is estimating the average settlement of the loaded area

In 1883, Boussinesq published equations and charts based on the theory of elasticity in any point in homogeneous, elastic and isotropic medium, the elastic solutions use a specific type of applied load such as: • a point load • uniform load • triangular distribution.

9.3 Boussinesq

9.3.1 Stresses Caused by a Point Load

I = influence value

Vertical Stress Caused by a Rectangular Loaded Area

Vertical Stress at Any Point Below a Uniformly Loaded Circular Area

9.3.5 Vertical Stress Due to Embankment Loading

INFLUENCE CHART FOR VERTICAL PRESSURE • It is usually necessary to compute vertical stresses due to an irregularly shaped

loaded area at various points inside and/or outside an area. • For this, NEWMARK developed an influence chart from which the vertical stress may

be computed. • The procedure for obtaining vertical pressure at any point below a loaded area is as follows: 1. Determine the depth z below the uniformly loaded area at which the stress

increase is required. 2. Plot the plan of the loaded area with a scale of z equal to the unit length of the

chart 3. Place the plan (plotted in step 2) on the influence chart in such a way that the

point below which the stress is to be determined is located at the center of the chart.

4. Count the number of blocks (N) of the chart enclosed by the plan of the loaded area.

5. The increase in the pressure at the point under consideration is given by:

Normal and Shear Stresses on a Plane

• Most common type of failure in soils is due to shear • At failure, shear stress along the failure surface (τ) reaches the shear strength (τf). • Because the shear strength (τf) of the soil depends on the normal stress (σ), we should determine both the normal and shear stresses in soils

• The 2D state of stress will be considered for simplification

SIGN CONVENTION Compression is positive, while tension is

negative Counter clockwise shear is positive, while

clockwise shear is negative

• Knowing the stresses on any two perpendicular planes passing through a point is useful in finding the stresses on all other planes passing through the same point

• Summing the components of forces that act on the element in the direction of N and T, we have

Or

and

Principal stresses • Those are the planes passing through the same point where the shear stress is equal to

zero.

• this means that there are two planes at right angles to each other on which the shear stress is zero.

• The normal stresses that act on the principal planes are referred to as principal stresses: major principal stress σ 1 the largest normal stress acting at any plane

minor principal stress σ 3 the smallest stress acting on any plane.

• The values of principal stresses can be found by substituting τn = 0, we get

The Mohr Circle The normal stress and shear stress that act on any plane can also be determined by plotting a Mohr’s circle

• The points R and M in Figure represent the stress conditions on planes AD and AB, respectively.

• O is the point of intersection of the normal stress axis with the line RM.

• The circle MNQRS drawn with O as the center and OR as the radius is the Mohr’s circle for the stress conditions considered.

• The stress on plane EF can be determined by moving an angle 2θ in a counter clockwise direction from point M along the circumference of the Mohr’s circle to reach point Q.

• The abscissa and ordinate of point Q, respectively, give the normal stress σn

and the shear stress τn on plane EF

R =

• The radius of the Mohr’s circle is equal to

Principal stresses

The Pole Method of Finding Stresses Along a Plane • Another important technique of finding stresses along a plane from a Mohr’s circle is the pole method. • According to the pole method:

draw a line from a known point on the Mohr’s circle parallel to the plane on which the state of stress acts. The point of intersection of this line with the Mohr’s circle is called the pole.

Example1

A soil element is shown in Figure .The magnitudes of stresses are σx = 2000 lb/ft τ = 800 lb/ft and σy = 2500 lb/ft , θ = 20o. Determine: a. Magnitudes of the principal stresses b. Normal and shear stresses on plane AB

Example2 For the stressed soil element shown in Figure, determine a. Major principal stress b. Minor principal stress c. Normal and shear stresses on the plane AE Use the pole method.