![]() | ||
Mohr's circle, named after Christian Otto Mohr, is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Contents
- Motivation for the Mohr Circle
- Mohrs circle for two dimensional state of stress
- Equation of the Mohr circle
- Sign conventions
- Physical space sign convention
- Mohr circle space sign convention
- Drawing Mohrs circle
- Finding principal normal stresses
- Finding maximum and minimum shear stresses
- Finding stress components on an arbitrary plane
- Double angle
- Pole or origin of planes
- Finding the orientation of the principal planes
- Example
- Mohrs circle for a general three dimensional state of stresses
- References
After performing a stress analysis on a material body assumed as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point.
The abscissa,
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 three-dimensional stresses and developed a failure criterion based on the stress circle.
Alternative graphical methods for the representation of the stress state at a point include the Lamé's stress ellipsoid and Cauchy's stress quadric.
The Mohr circle can be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.
Motivation for the Mohr Circle
Internal forces are produced between the particles of a deformable object, assumed as a continuum, as a reaction to applied external forces, i.e., either surface forces or body forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of motion for a particle. A measure of the intensity of these internal forces is called stress. Because the object is assumed as a continuum, these internal forces are distributed continuously within the volume of the object.
In engineering, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, airplane wings, or building columns, is determined through a stress analysis. Calculating the stress distribution implies the determination of stresses at every point (material particle) in the object. According to Cauchy, the stress at any point in an object (Figure 2), assumed as a continuum, is completely defined by the nine stress components
After the stress distribution within the object has been determined with respect to a coordinate system
Mohr's circle for two-dimensional state of stress
In two dimensions, the stress tensor at a given material point
The objective is to use the Mohr circle to find the stress components
Equation of the Mohr circle
To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point
From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress
Both equations can also be obtained by applying the tensor transformation law on the known Cauchy stress tensor, which is equivalent to performing the static equilibrium of forces in the direction of
These two equations are the parametric equations of the Mohr circle. In these equations,
Eliminating the parameter
where
This is the equation of a circle (the Mohr circle) of the form
with radius
Sign conventions
There are two separate sets of sign conventions that need to be considered when using the Mohr Circle: One sign convention for stress components in the "physical space", and another for stress components in the "Mohr-Circle-space". In addition, within each of the two set of sign conventions, the engineering mechanics (structural engineering and mechanical engineering) literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the choice of a particular sign convention is influenced by convenience for calculation and interpretation for the particular problem in hand. A more detailed explanation of these sign conventions is presented below.
The previous derivation for the equation of the Mohr Circle using Figure 4 follows the engineering mechanics sign convention. The engineering mechanics sign convention will be used for this article.
Physical-space sign convention
From the convention of the Cauchy stress tensor (Figure 3 and Figure 4), the first subscript in the stress components denotes the face on which the stress component acts, and the second subscript indicates the direction of the stress component. Thus
In the physical-space sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are inward to the plane of action (compression) (Figure 5).
In the physical-space sign convention, positive shear stresses act on positive faces of the material element in the positive direction of an axis. Also, positive shear stresses act on negative faces of the material element in the negative direction of an axis. A positive face has its normal vector in the positive direction of an axis, and a negative face has its normal vector in the negative direction of an axis. For example, the shear stresses
Mohr-circle-space sign convention
In the Mohr-circle-space sign convention, normal stresses have the same sign as normal stresses in the physical-space sign convention: positive normal stresses act outward to the plane of action, and negative normal stresses act inward to the plane of action.
Shear stresses, however, have a different convention in the Mohr-circle space compared to the convention in the physical space. In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction. This way, the shear stress component
Two options exist for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle:
- Positive shear stresses are plotted upward (Figure 5, sign convention #1)
- Positive shear stresses are plotted downward, i.e., the
τ n
Plotting positive shear stresses upward makes the angle
To overcome the "issue" of having the shear stress axis downward in the Mohr-circle space, there is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise direction and negative shear stresses are assumed to rotate the material element in the counterclockwise direction (Figure 5, option 3). This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle
This article follows the engineering mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circle space (sign convention #3 in Figure 5)
Drawing Mohr's circle
Assuming we know the stress components
- Draw the Cartesian coordinate system
( σ n , τ n ) with a horizontalσ n τ n - Plot two points
A ( σ y , τ x y ) andB ( σ x , − τ x y ) in the( σ n , τ n ) space corresponding to the known stress components on both perpendicular planesA andB , respectively (Figure 4 and 6), following the chosen sign convention. - Draw the diameter of the circle by joining points
A andB with a straight lineA B ¯ - Draw the Mohr Circle. The centre
O of the circle is the midpoint of the diameter lineA B ¯ σ n
Finding principal normal stresses
The magnitude of the principal stresses are the abscissas of the points
where the magnitude of the average normal stress
and the length of the radius
Finding maximum and minimum shear stresses
The maximum and minimum shear stresses correspond to the ordinates of the highest and lowest points on the circle, respectively. These points are located at the intersection of the circle with the vertical line passing through the center of the circle,
Finding stress components on an arbitrary plane
As mentioned before, after the two-dimensional stress analysis has been performed we know the stress components
Double angle
As shown in Figure 6, to determine the stress components
The double angle approach relies on the fact that the angle
This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of
Pole or origin of planes
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
Finding the orientation of the principal planes
The orientation of the planes where the maximum and minimum principal stresses act, also known as principal planes, can be determined by measuring in the Mohr circle the angles ∠BOC and ∠BOE, respectively, and taking half of each of those angles. Thus, the angle ∠BOC between
Angles
This equation defines two values for
Example
Assume a material element under a state of stress as shown in Figure 8 and Figure 9, with the plane of one of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circle, find:
Check the answers using the stress transformation formulas or the stress transformation law.
Solution: Following the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the material element in this example are:
Following the steps for drawing the Mohr circle for this particular state of stress, we first draw a Cartesian coordinate system
We then plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as show in both Figure 8 and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise. This way, the shear stress acting on plane B is negative and the shear stress acting on plane A is positive. The diameter of the circle is the line joining point A and B. The centre of the circle is the intersection of this line with the
The abscissas of both points E and C (Figure 8 and Figure 9) intersecting the
Even though the idea for using the Mohr circle is to graphically find different stress components by actually measuring the coordinates for different points on the circle, it is more convenient to confirm the results analytically. Thus, the radius and the abscissa of the centre of the circle are
and the principal stresses are
The coordinates for both points H and G (Figure 8 and Figure 9) are the magnitudes of the minimum and maximum shear stresses, respectively; the abscissas for both points H and G are the magnitudes for the normal stresses acting on the same planes where the minimum and maximum shear stresses act, respectively. The magnitudes of the minimum and maximum shear stresses can be found analytically by
and the normal stresses acting on the same planes where the minimum and maximum shear stresses act are equal to
We can choose to either use the double angle approach (Figure 8) or the Pole approach (Figure 9) to find the orientation of the principal normal stresses and principal shear stresses.
Using the double angle approach we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Figure 8) to find double the angle the major principal stress and the minor principal stress make with plane B in the physical space. To obtain a more accurate value for these angles, instead of manually measuring the angles, we can use the analytical expression
From inspection of Figure 8, this value corresponds to the angle ∠BOE. Thus, the minor principal angle is
Then, the major principal angle is
Remember that in this particular example
Using the Pole approach, we first localize the Pole or origin of planes. For this, we draw through point A on the Mohr circle a line inclined 10° with the horizontal, or, in other words, a line parallel to plane A where
From the Pole, we draw lines to different points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a plane in the physical space having the same inclination as the line. For instance, the line from the Pole to point C in the circle has the same inclination as the plane in the physical space where
Mohr's circle for a general three-dimensional state of stresses
To construct the Mohr circle for a general three-dimensional case of stresses at a point, the values of the principal stresses
Considering the principal axes as the coordinate system, instead of the general
Knowing that
Since
These expressions can be rewritten as
which are the equations of the three Mohr's circles for stress
These equations for the Mohr circles show that all admissible stress points