 Mohr's circle

Mohr's circle, named after Christian Otto Mohr, is a twodimensional graphical representation of the state of stress at a point. The abscissa, , and ordinate, , of each point on the circle are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector with components . In other words, the circumference of the circle is the locus of points that represent the state of stress on individual planes at all their orientations.
Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. Mohr's contribution extended the use of this representation for both two and threedimensional stresses and developed a failure criterion based on the stress circle.^{[1]}
Other graphical methods for the representation of the stress state at a point include the Lame's stress ellipsoid and Cauchy's stress quadric.
Contents
Mohr's circle for twodimensional stress states
A twodimensional Mohr's circle can be constructed if we know the normal stresses σ_{x}, σ_{y}, and the shear stress τ_{xy}. The following sign conventions are usually used:
 Tensile stresses (positive) are to the right.
 Compressive stresses (negative) are to the left.
 Clockwise shear stresses are plotted upward.
 Counterclockwise shear stresses are plotted downward.
The reason for the above sign convention is that, in engineering mechanics^{[2]}, the normal stresses are positive if they are outward to the plane of action (tension), and shear stresses are positive if they rotate clockwise about the point in consideration. In geomechanics, i.e. soil mechanics and rock mechanics, however, normal stresses are considered positive when they are inward to the plane of action (compression), and shear stresses are positive if they rotate counterclockwise about the point in consideration.^{[1]}^{[3]}^{[4]}^{[5]}
To construct the Mohr circle of stress for a state of plane stress, or plane strain, first we plot two points in the space corresponding to the known stress components on both perpendicular planes, i.e. and (Figure 1 and 2). We then connect points and by a straight line and find the midpoint which corresponds to the intersection of this line with the axis. Finally, we draw a circle with diameter and centre at .
The radius of the circle is , and the coordinates of its centre are .
The principal stresses are then the abscissa of the points of intersection of the circle with the axis (note that the shear stresses are zero for the principal stresses).
Drawing a Mohr's circle
The following procedure is used to draw a Mohr's circle and to find the magnitude and direction of maximum stresses from it.
 First, the x and yaxes of a Cartesian coordinate system are identified as the σ_{n}axis and τ_{n}axis, respectively.
 Next, two points of the Mohr's circle are plotted. These are the points B (σ_{x}, − τ_{xy}) and A (σ_{y}, τ_{xy}). The line connecting these two points is a diameter of the Mohr's circle.
 The center of the Mohr's circle, O, is located where the diameter, AB, intersects the σaxis. This point gives the average normal stress (σ_{avg}). The average normal stress can be read directly from a plot of the Mohr's circle. Alternatively, it can be calculated using
 .
 The Mohr's circle intersects the σ_{n} axis at two points, C and E. The stresses at these two end points of the horizontal diameter are σ_{1} and σ_{2}, the principal stresses. The point σ_{1} represents the maximum normal stress (σ_{max}) and the point σ_{2} is the minimum normal stress (σ_{min}). The equations for finding these values are
 Next we examine the points where the circle intersects the line parallel to τ_{n}axis passing through the center of the circle, O. The vertical diameter of the circle passes through O (σ_{avg}) and goes up to positive τ_{max } and down to negative τ_{min }. The magnitudes of extreme values are equal to the radius of the Mohr's circle, but with different signs. The equation to find these extreme values of the shear stress is^{[6]}
 .
 The next value to determine is the angle that the plane of maximum normal stress makes with the xaxis. Let us create a new Xaxis by drawing a line from the center of the Mohr circle, O, through point A. Let the angle between the Xaxis and the σaxis be ϕ. If θ is the angle between the maximum normal stress and the xaxis, then it can be shown that ϕ = 2θ_{p1}. The angle ϕ is found by:
 .
 To find the angle that the direction that the plane of maximum shear stress makes with the xaxis, we use the relation
 . It is important to pay attention to the use of these two equations as they look similar.
 Often, the final step of the process is to also draw a square stress element indicating the orientations of the maximum normal and shear stresses; the normal stress element at an angle θ and the maximum shear stress element at an angle of θ_{s}.
The previous discussion assumes, implicitly, that there are two orthogonal directions x and y that define a plane in which the stress components . , and are known. It is also implicit that these stresses are known at a point in a continuum body under plane stress or plane strain. The Mohr circle, once drawn, can be used to find the components of the stress tensor for any other choice of orthogonal directions in the plane.
Stress components on an arbitrary plane
Using the Mohr circle one can find the stress components on any other plane with a different orientation that passes through point . For this, two approaches can be used:
 The first approach relies on the fact that the angle between two planes passing through is half the angle between the lines joining their corresponding stress points on the Mohr circle and the centre of the circle (Figure 1). In other words, the stresses acting on a plane at an angle counterclockwise to the plane on which acts is determined by traveling counterclockwise around the circle from the known stress point a distance subtending an angle at the centre of the circle (Figure 1).
 The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components and on any particular plane, one can draw a line parallel to that plane through the particular coordinates and on the Mohr circle and find the pole as the intersection of such line with the Mohr circle. As an example, let's assume we have a state of stress with stress components , , and , as shown on Figure 2. First, we can draw a line from point parallel to the plane of action of , or, if we choose otherwise, a line from point parallel to the plane of action of . The intersection of any of these two lines with the Mohr circle is the pole. Once the pole has been determined, to find the state of stress on a plane making an angle with the vertical, or in other words a plane having its normal vector forming an angle with the horizontal plane, then we can draw a line from the pole parallel to that plane (See Figure 2). The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.
Mohr's circle for a general threedimensional state of stresses
To construct the Mohr's circle for a general threedimensional case of stresses at a point, the values of the principal stresses and their principal directions must be first evaluated.
Considering the principal axes as the coordinate system, instead of the general , , coordinate system, and assuming that , then the normal and shear components of the stress vector , for a given plane with unit vector , satisfy the following equations
Knowing that , we can solve for , , , using the Gauss elimination method which yields
Since , and is nonnegative, the numerators from the these equations satisfy
 as the denominator and
 as the denominator and
 as the denominator and
These expressions can be rewritten as
which are the equations of the three Mohr's circles for stress , , and , with radii , , and , and their centres with coordinates , , , respectively.
These equations for the Mohr's circles show that all admissible stress points lie on these circles or within the shaded area enclosed by them (see Figure 3). Stress points satisfying the equation for circle lie on, or outside circle . Stress points satisfying the equation for circle lie on, or inside circle . And finally, stress points satisfying the equation for circle lie on, or outside circle .
References
 ^ ^{a} ^{b} Parry
 ^ The sign convention differ in disciplines such as mechanical engineering, structural engineering, and geomechanics. The engineering mechanics sign convention is used in this article.
 ^ Jumikis
 ^ Holtz
 ^ Brady
 ^ Megson, T.H.G., Aircraft Structures for Engineering Students, Fourth Edition, 2007, section 1.8
Bibliography
 Beer, Ferdinand Pierre; Elwood Russell Johnston, John T. DeWolf (1992). Mechanics of Materials. McGrawHill Professional. ISBN 0071129391.
 Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN 0412475502. http://books.google.ca/books?id=s0BaKxL11KsC&lpg=PP1&pg=PA18#v=onepage&q=&f=false.
 Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN 0521498279. http://books.google.ca/books?id=4Z11rZaUn1UC&lpg=PP1&pg=PA16#v=onepage&q=&f=false.
 Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. PrenticeHall civil engineering and engineering mechanics series. PrenticeHall. ISBN 0134843940. http://books.google.ca/books?id=yYkYAQAAIAAJ&dq=inauthor:%22William+D.+Kovacs%22&ei=kFMS5LRKpfCM9vEhIYN&cd=1.
 Jaeger, John Conrad; Cook, N.G.W, & Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). WileyBlackwell. pp. 9–41. ISBN 0632057599. http://books.google.com/books?id=FqADDkunVNAC&lpg=PP1&pg=PA10#v=onepage&q=&f=false.
 Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co.. ISBN 0442041993. http://books.google.ca/books?id=NPZRAAAAMAAJ&source=gbs_navlinks_s.
 Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0415272971. http://books.google.ca/books?id=u_rec9uQnLcC&lpg=PP1&dq=mohr%20circles%2C%20sterss%20paths%20and%20geotechnics&pg=PA1#v=onepage&q=&f=false.
 Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGrawHill International Editions. ISBN 0070858055.
 Timoshenko, Stephen P. (1983). History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN 0486611876.
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