mohr circle 2
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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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