Mohr's circle is a two-dimensional graphical representation of the
transformation law for the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
.
Mohr's circle is often used in calculations relating to
mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
for
materials' strength,
geotechnical engineering
Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. It uses the principles of soil mechanics and rock mechanics for the solution of its respective engineering problems. It als ...
for
strength of soils, and
structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
for strength of built structures. It is also used for calculating
stresses in many planes by reducing them to vertical and horizontal components. These are called principal planes in which
principal stress
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
es are calculated; Mohr's circle can also be used to find the principal planes and the principal stresses in a graphical representation, and is one of the easiest ways to do so.
After performing a
stress analysis
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
on a material body assumed as a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
, the components of the Cauchy stress tensor at a particular material point are known with respect to a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. 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 and ordinate
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
(
,
) of each point on the circle are the magnitudes of the normal stress and
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
components, respectively, acting on the rotated coordinate system. In other words, the circle is the
locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of points that represent the state of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress element.
19th-century German engineer
Karl Culmann
Carl Culmann (10 July 1821 – 9 December 1881) was a German structural engineer.
Born in Bad Bergzabern, Rhenish Palatinate, in modern-day Germany, Culmann's father, a pastor, tutored him at home before enrolling him at the military engineerin ...
was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during
bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to ...
. His work inspired fellow German engineer
Christian Otto Mohr
Christian Otto Mohr (8 October 1835 – 2 October 1918) was a German civil engineer. He is renowned for his contributions to the field of structural engineering, such as Mohr's circle, and for his study of stress.
Biography
He was born on 8 Octo ...
(the circle's namesake), who extended it to both two- and three-dimensional stresses and developed a
failure
Failure is the state or condition of not meeting a desirable or intended objective (goal), objective, and may be viewed as the opposite of Success (concept), success. The criteria for failure depends on context, and may be relative to a parti ...
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
Lamé's stress ellipsoid is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing thro ...
and
Cauchy's stress quadric.
The Mohr circle can be applied to any
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
2x2
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
matrix, including the
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
and
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
tensors.
Motivation
Internal forces are produced between the particles of a deformable object, assumed as a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
, as a reaction to applied external forces, i.e., either
surface force
Surface force denoted ''fs'' is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed into two perpendicular components: normal forces and shear forces. A normal force acts normal ...
s or
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s. This reaction follows from
Euler's laws of motion
In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.
O ...
for a continuum, which are equivalent to
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
for a particle. A measure of the intensity of these internal
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s is called
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
. 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
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
,
mechanical
Mechanical may refer to:
Machine
* Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement
* Mechanical calculator, a device used to perform the basic operations of ...
, or
geotechnical
Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. It uses the principles of soil mechanics and rock mechanics for the solution of its respective engineering problems. It als ...
, 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
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
. Calculating the stress distribution implies the determination of stresses at every point (material particle) in the object. According to
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, the ''stress at any point'' in an object (Figure 2), assumed as a continuum, is completely defined by the nine stress components
of a second order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of
type (2,0) known as the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
,
:
:
After the stress distribution within the object has been determined with respect to a coordinate system
, it may be necessary to calculate the components of the stress tensor at a particular material point
with respect to a rotated coordinate system
, i.e., the stresses acting on a plane with a different orientation passing through that point of interest —forming an angle with the coordinate system
(Figure 3). For example, it is of interest to find the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To achieve this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, the Cauchy stress tensor obeys the
tensor transformation law
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.
Mohr's circle for two-dimensional state of stress
In two dimensions, the stress tensor at a given material point
with respect to any two perpendicular directions is completely defined by only three stress components. For the particular coordinate system
these stress components are: the normal stresses
and
, and the shear stress
. From the balance of angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated. This symmetry implies that
. Thus, the Cauchy stress tensor can be written as:
:
The objective is to use the Mohr circle to find the stress components
and
on a rotated coordinate system
, i.e., on a differently oriented plane passing through
and perpendicular to the
-
plane (Figure 4). The rotated coordinate system
makes an angle
with the original coordinate system
.
Equation of the Mohr circle
To derive the equation of the Mohr circle for the two-dimensional cases of
plane stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysi ...
and
plane strain
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
, first consider a two-dimensional infinitesimal material element around a material point
(Figure 4), with a unit area in the direction parallel to the
-
plane, i.e., perpendicular to the page or screen.
From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress
and the shear stress
are given by:
:
:
:
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
and
.
:
These two equations are the
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s of the Mohr circle. In these equations,
is the parameter, and
and
are the coordinates. This means that by choosing a coordinate system with abscissa
and ordinate
, giving values to the parameter
will place the points obtained lying on a circle.
Eliminating the parameter
from these parametric equations will yield the non-parametric equation of the Mohr circle. This can be achieved by rearranging the equations for
and
, first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have
:
where
:
This is the equation of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
(the Mohr circle) of the form
:
with radius
centered at a point with coordinates
in the
coordinate system.
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
Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
(
structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
and
mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
) literature follows a different sign convention from the
geomechanics
Geomechanics (from the Greek prefix ''geo-'' meaning "earth"; and "mechanics") is the study of the mechanical state of the earth's crust and the processes occurring in it under the influence of natural physical factors. It involves the study of th ...
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
is the shear stress acting on the face with normal vector in the positive direction of the
-axis, and in the positive direction of the
-axis.
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
and
are positive because they act on positive faces, and they act as well in the positive direction of the
-axis and the
-axis, respectively (Figure 3). Similarly, the respective opposite shear stresses
and
acting in the negative faces have a negative sign because they act in the negative direction of the
-axis and
-axis, respectively.
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
is positive in the Mohr-circle space, and the shear stress component
is negative in the Mohr-circle space.
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
-axis is inverted (Figure 5, sign convention #2).
Plotting positive shear stresses upward makes the angle
on the Mohr circle have a positive rotation clockwise, which is opposite to the physical space convention. That is why some authors
prefer plotting positive shear stresses downward, which makes the angle
on the Mohr circle have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.
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
has a positive rotation counterclockwise in the Mohr-circle space. This ''alternative'' sign convention produces a circle that is identical to the sign convention #2 in Figure 5 because a positive shear stress
is also a counterclockwise shear stress, and both are plotted downward. Also, a negative shear stress
is a clockwise shear stress, and both are plotted upward.
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
,
, and
at a point
in the object under study, as shown in Figure 4, the following are the steps to construct the Mohr circle for the state of stresses at
:
# Draw the Cartesian coordinate system
with a horizontal
-axis and a vertical
-axis.
# Plot two points
and
in the
space corresponding to the known stress components on both perpendicular planes
and
, respectively (Figure 4 and 6), following the chosen sign convention.
# Draw the diameter of the circle by joining points
and
with a straight line
.
# Draw the Mohr Circle. The centre
of the circle is the midpoint of the diameter line
, which corresponds to the intersection of this line with the
axis.
Finding principal normal stresses
The magnitude of the
principal stresses
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
are the abscissas of the points
and
(Figure 6) where the circle intersects the
-axis. The magnitude of the major principal stress
is always the greatest absolute value of the abscissa of any of these two points. Likewise, the magnitude of the minor principal stress
is always the lowest absolute value of the abscissa of these two points. As expected, the ordinates of these two points are zero, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the principal stresses can be found by
:
:
where the magnitude of the average normal stress
is the abscissa of the centre
, given by
:
and the length of the radius
of the circle (based on the equation of a circle passing through two points), is given by
:
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,
. Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circle's radius
:
Finding stress components on an arbitrary plane
As mentioned before, after the two-dimensional stress analysis has been performed we know the stress components
,
, and
at a material point
. These stress components act in two perpendicular planes
and
passing through
as shown in Figure 5 and 6. The Mohr circle is used to find the stress components
and
, i.e., coordinates of any point
on the circle, acting on any other plane
passing through
making an angle
with the plane
. For this, two approaches can be used: the double angle, and the Pole or origin of planes.
Double angle
As shown in Figure 6, to determine the stress components
acting on a plane
at an angle
counterclockwise to the plane
on which
acts, we travel an angle
in the same counterclockwise direction around the circle from the known stress point
to point
, i.e., an angle
between lines
and
in the Mohr circle.
The double angle approach relies on the fact that the angle
between the normal vectors to any two physical planes passing through
(Figure 4) is half the angle between two lines joining their corresponding stress points
on the Mohr circle and the centre of the circle.
This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of
. It can also be seen that the planes
and
in the material element around
of Figure 5 are separated by an angle
, which in the Mohr circle is represented by a
angle (double the angle).
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
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 7. 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 7). The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.
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
and
is double the angle
which the major principal plane makes with plane
.
Angles
and
can also be found from the following equation
:
This equation defines two values for
which are
apart (Figure). This equation can be derived directly from the geometry of the circle, or by making the parametric equation of the circle for
equal to zero (the shear stress in the principal planes is always zero).
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:
* The orientation of their planes of action.
* The maximum shear stresses and orientation of their planes of action.
* The stress components on a horizontal plane.
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
with the
-axis upward.
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
-axis. Knowing both the location of the centre and length of the diameter, we are able to plot the Mohr circle for this particular state of stress.
The abscissas of both points E and C (Figure 8 and Figure 9) intersecting the
-axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points E and C are the magnitudes of the shear stresses acting on both the minor and major principal planes, respectively, which is zero for principal planes.
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
:
One solution is:
.
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
and
are angles with respect to the plane of action of
(oriented in the
-axis)and not angles with respect to the plane of action of
(oriented in the
-axis).
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
acts. The Pole is where this line intersects the Mohr circle (Figure 9). To confirm the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the plane B where
acts. This line would also intersect the Mohr circle at the Pole (Figure 9).
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
acts. This plane makes an angle of 63.435° with plane B, both in the Mohr-circle space and in the physical space. In the same way, lines are traced from the Pole to points E, D, F, G and H to find the stress components on planes with the same orientation.
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
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
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
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix (mathematics), matrix of coefficients. This me ...
which yields
:
Since
, and
is non-negative, the numerators from 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