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ock Mechanics for Engineers

RockMechs > Stress & Strain > Stress > Mohr’s circle in 3 dimensions

Mohr’s circle in 3 dimensions

Mohr’s diagram is a useful graphical representation of the stress

state at a point. In this graphical representation the state of stress

at a point is represented by the Mohr circle diagram, in which the

abscissa and give the normal and shear stress acting on a

particular cut plane with a fixed normal direction. In the general 3

dimensional case, for a given state of stress at a point, the Mohr

circle diagr am has three circles as shown in Fig. 1. Mohr’s circle

diagram is used frequently in conjunction with failure criteria like

the Mohr-Coulomb failure criterion.

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Figure 1: Mohr’s circle in three dimensional case (σ1 ≥ σ2 ≥ σ3)

Assume that the stress state at a point is given by the stress

tensor:

(1)

The center of each circle in Mohr’s diagram lies on axis and is

given by:

(2)

while the radii of the circles are calculated by:

(3)

for centers , and , respectively. If the principal stresses are

known (may be calculated by the stress tensor as shown in

Principal stresses and stress invariants) then the above equations

(2) and (3) take the form (for the case ):

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(4)

and

(5)

Consider an arbitrary cut plane that passes through the

considered point. All the admissible values of and for this

plane lie inside or on the boundaries of the region bounded by the

circles , and (see Fig.1). The proof, however, will not be

given in this article but it can be found in many related books.

In order to calculate the normal and shear stresses acting on any

plane, through Mohr’s circle diagram, it is necessary to know the

direction cosines of the normal unit vector of the plane with

respect to the principal directions. Assume that , and are

the direction cosines of the plane with respect to the principal

directions of , and , respectively. For a given value of the

point lies on the arc as shown in Fig. 1. To construct

this arc we draw line that passes through and is parallel to

axis. Then we measure angle from that line. This line

intersects the circle at points and . By using center as

center (the only center that does not depend on ) we draw the

arc . Similarly, for direction cosine the point lies on

the arc . We draw line and measure angle . The

intersection points are and . Using center we draw the arc

. Finally, we can do the same for direction cosine . We

measure angle from and using center we draw the

arc . Since, only two values of , and are independent, it

is adequate to use only two direction cosines in order to determine

the values . The normal and shear stress is given by the

coordinates of intersection point . All arcs pass through that

point, hence, one can use for example and to calculate point

and use to verify the procedure.

The Mohr’s circle diagram may be used to calculate graphically

the normal and shear stresses on a plane. Otherwise, the method

described in Calculation of normal and shear stress on a plane

may be used.

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Example

Consider the following stress state acting on a point:

(6)

Calculate the normal and shear stress on the plane with normal

vector:

(7)

Solution

From equations (4) and (5) we calculate the centers , and

and the radii , and :

(8)

Next we draw Mohr’s circle diagram as shown in Fig. 2.


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Figure 2: Mohr’s circle diagram example (3d).

From the direction cosines we calculate the angles , and :

(9)

Using the above angles (we need only two, for example and )

we draw the arcs and we find the normal and shear stress on the

plane:

(10)

We can also confirm the solution by using the methodology

described in the article: Calculation of normal and shear stress ona plane.

Suggested Bibliography

L.E. Malvern. Introduction to the Mechanics of a Continuous

Medium. Prentice Hall, Englewood Cliffs, New Jersey, 1969.

J.C. Jaeger, N.G.W. Cook and R.W. Zimmerman. Fundamentals

of Rock Mechanics. Blackwell Publishing, Malden MA, 4th edition,

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2007.

W.F. Chen and D.J. Han. Plasticity for Structural Engineers.

Springer-Verlag, New York, 1988.

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