**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:

s_{n}= __(s _{1}+s_{3})
__+

2 2

s_{s}= __(s _{1}-s_{3})__
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

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
45^{0} 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

not
cause rock fracturing.

On a Mohrs Diagram, the following sense of shear conventions

apply:

Sinistral
(counterclockwise) shear is Positive

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

Coulomb Failure

**MOHR'S CIRCLE
DEPICTION OF:**

**EFFECTIVE
STRESS & FLUID PORE PRESSURE-**

Effective Stress= normal stress minus the pore fluid pressure.

Fluid
Pore Pressure (P_{f})-
hydrostatic pressure exerted by interstitial water.

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

Increase
in P_{f} 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

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