NOTES ON MOHR’S CIRCLES AND SHEAR STRENGTH

Plot of σn vs τn for various values of θ.

When the soil sample has confining pressure, σ3 , as well as axial pressure, σ1 ; the normal and shear stresses are:

The normal and shear stress curves move apart

Plot of σn vs τn

The equations for normal and shear stress can be found from the geometry of the circle

Shear strength comes from two sources1. Cohesion- Cohesive shear strength, c2. Friction - Angle of internal friction, φ

or Angle of shearing resistance

Strength of Clay

In fact, the cohesion in clay (all soils, actually) is due to the negative porewater pressure in that soil.

Strength of Sand

In the absence of pore water, the strength of sand is due almost entirely to friction.

SAND STRENGTH

Strength Tests for Sand

1. Direct Shear (Poor test)2. Triaxial Test (Much better test)

-Uniform stress: fields in the center of the sample-Uniform strain: fields in the center of the sample

Most soils are a mixture of fine and coarse grained sizes so they usually have both c and φ properties. How much c depends on whether the sample is loaded slowly enough to allow the pore water pressure to dissipate. Tests are classified by the rate with which -Confining pressure-Deviator stressare applied.

Symbol for Test

Rate of Application

Type of TestConfining Pressure

(consolidation prior to test)

Deviator Stress(drainage permitted

during test)

Q (UU) Rapidly Rapidly Unconsolidated – Undrained c, φ

R (CU) Slowly RapidlyConsolidated- Undrained (may

make measurements of pore water pressures)

S (CD) Slowly Slowly Consolidated- Undrained c', φ'

Total Stress properties are measured in the UU testEffective pressure properties are measured in the CD testEffective pressure properties may be determined in the CU test if pore water pressures are measured during the undrained portion of the test.

There are 3 pairs of quantities: σ1, σ3 ; σf, τf (failure plane); c,φ

The radiusof the circle is r = (σ1-σ3)/2The projection of the failure line is x=c/tanφ

Normally, if you are given any three of the quantities you can find the other three.

1. Suppose you know σ1, σ3 and cFind expressions for σf, τf and φ or tanφ

2. Suppose you know σ1, σf and τf

Find expressions for σ3, c and φ

3. Suppose you know σ1, σf and cFind expressions for σ3, τf and φ

Problem 01

Problem 02

Problem 03

Problem 01Draw the sketch of the Mohrs Circle, marking all that is known and all that is to be found.

Problem 02

Problem 03