CLASS 4: MOHRS CIRCLE

Mohrs Circle-    developed by Otto Mohr (1835-1918).

a convenient graphical means to depict states of stress;

A force applied to an area (stress) may be resolved into a

normal force (Fn) perpendicular to a plane and a

shear force (Fs) , parallel to a plane in questions.

1    Sigma 1- Maximum Compressive Stress

2     Sigma 2- Intermediae Compressive Stress

3    Sigma 3- Minimum Compressive Stress

Stress is a vector quantity that can be considered as:

n    Normal Stress-

oriented perpendicular to a plane

s     Shear Stress-

oriented parallel to a plane in question

Theta- angle formed by an inclined plane with the maximum and minimum compressive stress directions, and measured from the minimum stress position.

Important Normal Stress and Shear Stress Equations:

sn= (s1+s3) + (s1-s3) cos 2q

2                2

ss= (s1-s3) sin 2q

2

Importance of Mohrs Diagram:

1.For any value of maximum compressive stress value and minimum compressive stress value, one can determine the normal and shear stress for any planes that lie at an angle theta

2.Depicts the attitude of planes along which shear stress is the greatest for a given stress state

3. The most important aspect of Mohrs diagram is that it facilitates a quick, graphical determination of stresses on planes of any orientation.

4.Mohr diagrams are excellent for visualizing the state of stress but difficult for calculating stress. Stress tensors are used to calculate stress.

PLOTTING MOHR'S CIRCLE:

Mohr's circle is  plotted on two perpendicular axes: The vertical axis

(ordinate) depicts shear  stress and the horizontal axis (abscissa) depicts

normal stress.

By convention compressive  stress is positive (+) and

tensile stress is negative (-).

Principle Stresses sigma 1 (maximum compressive stress) and sigma 3 (minimum compressive stress) plot as two points on the horizontal axis. These two points define the diameter of a circle. The Circle is plotted on the abscissa

These points establish a radius (R) whereby:

The center (C) is then plotted:

We can determine the normal and shear stress on any plane

oriented at an angle theta from the abscissa , as measured

counterclockwise from the minimum compressive stress

direction. Because of the properties of a circle, the angle

between Point P, the center of the circle and the maximum

compressive stress direction = 2 theta, as measured

counterclockwise from the center of the circle.

Mohrs Circle can graphically depict stress on any plane inclined

relative to the principal plane.

Normal and shear stresses can be determined graphically using

the circle or by using equations.

Maximum shear stress occurs on planes oriented 450 to the maximum and minimum compressive stress directions; thus, these points plot at the top and bottom of Mohr's Circle

Differential stress, that is the difference between the maximum

and minimum compressive stress, is the most important factor in

rock fracturing. The intermediate principal stress generally does

not cause rock fracturing.

On a Mohrs Diagram, the following sense of shear conventions

apply:

Sinistral (counterclockwise) shear is Positive (+) and

Dextral (clockwise) shear is Negative (-).

Angles 2 theta associated with planes experiencing

sinistral shear plot in the upper hemisphere.

Angles 2 theta associated with planes experiencing

dextral shear plot in the lower hemisphere

Note that the axes of Mohrs diagram do not have a

geographic orientation.

However, prior to constructing a Mohrs diagram it is useful to

sketch a block diagram of the orientations of the principal

stress axes and the plane in question to ascertain the

relative sense of shear and orientation of principal stress axes

Mohrs Envelop of Failure:

Represented by a straight line with a slope equal to Coulombs coefficient

A number of Mohrs circles are plotted and a line tangential to the circles is drawn.

Constructed using a series of experiments in which the principal stresses change.

Failure occurs when the Mohrs circle intersects the envelope of failure.

COULOMB'S COEFFICIENT

= tan

(mu) Coulomb�s Coefficient  (coefficient of internal friction)

slope of the line (envelop of failure)

(phi)  angle of internal friction

Experiments are usually constructed with an axial load

(maximum compressivestress) is applied to a rock cylinder

under a confining pressure.

Coulomb Failure

MOHR'S CIRCLE DEPICTION OF:

EFFECTIVE STRESS &  FLUID PORE PRESSURE-

Effective Stress=  normal stress minus the pore fluid pressure.

Fluid Pore Pressure (Pf)- hydrostatic pressure exerted by interstitial water.

Mohr circle remains same size but is translated to the left along the horizontal axis.

Increase in Pf results in:

a reduction in the strength of the rock

facilitates hydraulic fracturing.

Check out the following Webpage for Stress visualization: http://www.geology.sdsu.edu/visualstructure/vss/htm_hlp/index.htm