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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.

    RockMechs

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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 ):

    http://www.rockmechs.com/stress-strain/stress/principal-stresses-and-invariants/http://www.rockmechs.com/stress-strain/stress/mohr-circle-3d/attachment/mohr_circle_3d/

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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.

    http://www.rockmechs.com/stress-strain/stress/normal-and-shear-stress-on-plane/

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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,

    http://www.rockmechs.com/stress-strain/stress/normal-and-shear-stress-on-plane/http://www.rockmechs.com/stress-strain/stress/mohr-circle-3d/attachment/mohr_circle_3d_example/

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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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