draw the corresponding 3d mohr's circles to scale

Geometric ceremonious applied science calculation technique

Figure 1. Mohr's circles for a three-dimensional state of stress

Mohr's circumvolve is a two-dimensional graphical representation of the transformation police force for the Cauchy stress tensor.

Mohr'southward circle is frequently used in calculations relating to mechanical technology for materials' strength, geotechnical engineering for strength of soils, and structural applied science 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 primary planes in which main stresses are calculated; Mohr'due south circle tin can also be used to find the master planes and the principal stresses in a graphical representation, and is one of the easiest ways to exercise and so.^[1]

Later performing a stress analysis on a material body assumed every bit 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 interim on a rotated coordinate system, i.e., interim on a differently oriented plane passing through that indicate.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {n} }}$ , ${\displaystyle \tau _{\mathrm {n} }}$ ) of each bespeak on the circle are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the country of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress chemical element.

19th-century High german engineer Karl Culmann was the kickoff to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during angle. His work inspired fellow German engineer Christian Otto Mohr (the circle's namesake), who extended it to both 2- and 3-dimensional stresses and developed a failure criterion based on the stress circle.^[2]

Alternative graphical methods for the representation of the stress country 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 [edit]

Figure 2. Stress in a loaded deformable material body causeless every bit a continuum.

Internal forces are produced between the particles of a deformable object, causeless as a continuum, every bit a reaction to applied external forces, i.due east., either surface forces or body forces. This reaction follows from Euler'south laws of motion for a continuum, which are equivalent to Newton's laws of motility for a particle. A mensurate of the intensity of these internal forces is called stress. Because the object is causeless equally a continuum, these internal forces are distributed continuously within the volume of the object.

In engineering, east.k., structural, mechanical, or geotechnical, the stress distribution within an object, for example stresses in a rock mass around a tunnel, airplane wings, or building columns, is adamant through a stress analysis. Calculating the stress distribution implies the decision of stresses at every point (fabric particle) in the object. According to Cauchy, the stress at whatsoever indicate in an object (Effigy ii), assumed as a continuum, is completely defined by the nine stress components ${\displaystyle \sigma _{ij}}$ of a second order tensor of type (ii,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

${\displaystyle {\boldsymbol {\sigma }}=\left[{\brainstorm{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\terminate{matrix}}\right]\equiv \left[{\brainstorm{matrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{matrix}}\correct]\equiv \left[{\begin{matrix}\sigma _{10}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\terminate{matrix}}\right]}$

Figure 3. Stress transformation at a signal in a continuum under aeroplane stress weather.

Later the stress distribution within the object has been determined with respect to a coordinate organisation ${\displaystyle (x,y)}$ , it may be necessary to calculate the components of the stress tensor at a particular material point ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (x',y')}$ , i.due east., the stresses acting on a airplane with a unlike orientation passing through that point of involvement —forming an bending with the coordinate system ${\displaystyle (x,y)}$ (Effigy iii). For example, it is of involvement to detect the maximum normal stress and maximum shear stress, every bit well equally the orientation of the planes where they act upon. To attain this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.

Mohr'southward circle for two-dimensional state of stress [edit]

Figure 4. Stress components at a plane passing through a point in a continuum under aeroplane stress conditions.

In ii dimensions, the stress tensor at a given material bespeak ${\displaystyle P}$ with respect to whatever two perpendicular directions is completely defined past only three stress components. For the particular coordinate system ${\displaystyle (10,y)}$ these stress components are: the normal stresses ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{y}}$ , and the shear stress ${\displaystyle \tau _{xy}}$ . From the remainder of angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated. This symmetry implies that ${\displaystyle \tau _{xy}=\tau _{yx}}$ . Thus, the Cauchy stress tensor can be written equally:

${\displaystyle {\boldsymbol {\sigma }}=\left[{\begin{matrix}\sigma _{x}&\tau _{xy}&0\\\tau _{xy}&\sigma _{y}&0\\0&0&0\\\finish{matrix}}\correct]\equiv \left[{\brainstorm{matrix}\sigma _{x}&\tau _{xy}\\\tau _{xy}&\sigma _{y}\\\terminate{matrix}}\right]}$

The objective is to utilise the Mohr circle to find the stress components ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on a rotated coordinate system ${\displaystyle (x',y')}$ , i.eastward., on a differently oriented plane passing through ${\displaystyle P}$ and perpendicular to the ${\displaystyle x}$ - ${\displaystyle y}$ plane (Figure 4). The rotated coordinate organisation ${\displaystyle (x',y')}$ makes an angle ${\displaystyle \theta }$ with the original coordinate arrangement ${\displaystyle (x,y)}$ .

Equation of the Mohr circle [edit]

To derive the equation of the Mohr circumvolve for the two-dimensional cases of plane stress and airplane strain, first consider a ii-dimensional infinitesimal textile element around a material signal ${\displaystyle P}$ (Figure 4), with a unit expanse in the management parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ plane, i.e., perpendicular to the folio or screen.

From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress ${\displaystyle \sigma _{\mathrm {n} }}$ and the shear stress ${\displaystyle \tau _{\mathrm {due north} }}$ are given by:

${\displaystyle \sigma _{\mathrm {n} }={\frac {1}{2}}(\sigma _{x}+\sigma _{y})+{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\cos 2\theta +\tau _{xy}\sin 2\theta }$

${\displaystyle \tau _{\mathrm {due north} }=-{\frac {ane}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$

Derivation of Mohr's circumvolve parametric equations - Equilibrium of forces

From equilibrium of forces in the management of ${\displaystyle \sigma _{\mathrm {n} }}$ ( ${\displaystyle x'}$ -axis) (Figure 4), and knowing that the area of the aeroplane where ${\displaystyle \sigma _{\mathrm {n} }}$ acts is ${\displaystyle dA}$ , we have: ${\displaystyle \ {\begin{aligned}\sum F_{x'}&=\sigma _{\mathrm {n} }dA-\sigma _{ten}dA\cos ^{2}\theta -\sigma _{y}dA\sin ^{ii}\theta -\tau _{xy}dA\cos \theta \sin \theta -\tau _{xy}dA\sin \theta \cos \theta =0\\\sigma _{\mathrm {n} }&=\sigma _{10}\cos ^{2}\theta +\sigma _{y}\sin ^{ii}\theta +two\tau _{xy}\sin \theta \cos \theta \\\end{aligned}}}$ Yet, knowing that ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{two}},\qquad \sin ^{2}\theta ={\frac {i-\cos 2\theta }{2}}\qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ we obtain ${\displaystyle \sigma _{\mathrm {n} }={\frac {1}{2}}(\sigma _{10}+\sigma _{y})+{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\cos two\theta +\tau _{xy}\sin 2\theta }$ At present, from equilibrium of forces in the direction of ${\displaystyle \tau _{\mathrm {northward} }}$ ( ${\displaystyle y'}$ -axis) (Figure 4), and knowing that the expanse of the aeroplane where ${\displaystyle \tau _{\mathrm {n} }}$ acts is ${\displaystyle dA}$ , we have: ${\displaystyle \ {\begin{aligned}\sum F_{y'}&=\tau _{\mathrm {n} }dA+\sigma _{x}dA\cos \theta \sin \theta -\sigma _{y}dA\sin \theta \cos \theta -\tau _{xy}dA\cos ^{2}\theta +\tau _{xy}dA\sin ^{2}\theta =0\\\tau _{\mathrm {north} }&=-(\sigma _{x}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{two}\theta -\sin ^{2}\theta \right)\\\stop{aligned}}}$ Still, knowing that ${\displaystyle \cos ^{2}\theta -\sin ^{2}\theta =\cos two\theta \qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \tau _{\mathrm {north} }=-{\frac {1}{ii}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$

Both equations can likewise 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 ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {due north} }}$ .

Derivation of Mohr'southward circle parametric equations - Tensor transformation

The stress tensor transformation law can be stated as ${\displaystyle {\brainstorm{aligned}{\boldsymbol {\sigma }}'&=\mathbf {A} {\boldsymbol {\sigma }}\mathbf {A} ^{T}\\\left[{\begin{matrix}\sigma _{x'}&\tau _{x'y'}\\\tau _{y'x'}&\sigma _{y'}\\\finish{matrix}}\correct]&=\left[{\begin{matrix}a_{x}&a_{xy}\\a_{yx}&a_{y}\\\cease{matrix}}\correct]\left[{\begin{matrix}\sigma _{10}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\cease{matrix}}\right]\left[{\begin{matrix}a_{x}&a_{yx}\\a_{xy}&a_{y}\\\end{matrix}}\right]\\&=\left[{\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\stop{matrix}}\right]\left[{\begin{matrix}\sigma _{x}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\right]\left[{\begin{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\finish{matrix}}\correct]\finish{aligned}}}$ Expanding the right hand side, and knowing that ${\displaystyle \sigma _{x'}=\sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{x'y'}=\tau _{\mathrm {n} }}$ , we have: ${\displaystyle \sigma _{\mathrm {northward} }=\sigma _{ten}\cos ^{2}\theta +\sigma _{y}\sin ^{two}\theta +2\tau _{xy}\sin \theta \cos \theta }$ Yet, knowing that ${\displaystyle \cos ^{two}\theta ={\frac {1+\cos two\theta }{2}},\qquad \sin ^{ii}\theta ={\frac {1-\cos 2\theta }{two}}\qquad {\text{and}}\qquad \sin ii\theta =ii\sin \theta \cos \theta }$ we obtain ${\displaystyle \sigma _{\mathrm {northward} }={\frac {1}{2}}(\sigma _{x}+\sigma _{y})+{\frac {1}{ii}}(\sigma _{10}-\sigma _{y})\cos two\theta +\tau _{xy}\sin 2\theta }$ ${\displaystyle \tau _{\mathrm {northward} }=-(\sigma _{x}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{ii}\theta -\sin ^{2}\theta \right)}$ However, knowing that ${\displaystyle \cos ^{2}\theta -\sin ^{2}\theta =\cos ii\theta \qquad {\text{and}}\qquad \sin ii\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \tau _{\mathrm {n} }=-{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$ It is not necessary at this moment to calculate the stress component ${\displaystyle \sigma _{y'}}$ acting on the aeroplane perpendicular to the plane of activity of ${\displaystyle \sigma _{ten'}}$ as it is non required for deriving the equation for the Mohr circle.

These two equations are the parametric equations of the Mohr circle. In these equations, ${\displaystyle 2\theta }$ is the parameter, and ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ are the coordinates. This means that by choosing a coordinate organisation with abscissa ${\displaystyle \sigma _{\mathrm {n} }}$ and ordinate ${\displaystyle \tau _{\mathrm {n} }}$ , giving values to the parameter ${\displaystyle \theta }$ will place the points obtained lying on a circumvolve.

Eliminating the parameter ${\displaystyle 2\theta }$ from these parametric equations volition yield the non-parametric equation of the Mohr circumvolve. This can exist achieved by rearranging the equations for ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , starting time transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus nosotros accept

${\displaystyle {\begin{aligned}\left[\sigma _{\mathrm {northward} }-{\tfrac {i}{2}}(\sigma _{10}+\sigma _{y})\correct]^{2}+\tau _{\mathrm {north} }^{2}&=\left[{\tfrac {1}{ii}}(\sigma _{x}-\sigma _{y})\right]^{2}+\tau _{xy}^{2}\\(\sigma _{\mathrm {n} }-\sigma _{\mathrm {avg} })^{2}+\tau _{\mathrm {due north} }^{2}&=R^{ii}\end{aligned}}}$

where

${\displaystyle R={\sqrt {\left[{\tfrac {1}{two}}(\sigma _{x}-\sigma _{y})\right]^{ii}+\tau _{xy}^{two}}}\quad {\text{and}}\quad \sigma _{\mathrm {avg} }={\tfrac {1}{2}}(\sigma _{x}+\sigma _{y})}$

This is the equation of a circle (the Mohr circle) of the form

${\displaystyle (10-a)^{2}+(y-b)^{two}=r^{2}}$

with radius ${\displaystyle r=R}$ centered at a point with coordinates ${\displaystyle (a,b)=(\sigma _{\mathrm {avg} },0)}$ in the ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {northward} })}$ coordinate organization.

Sign conventions [edit]

There are two separate sets of sign conventions that need to be considered when using the Mohr Circumvolve: One sign convention for stress components in the "physical infinite", and some other for stress components in the "Mohr-Circle-space". In addition, within each of the two set of sign conventions, the applied science 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 item sign convention is influenced by convenience for calculation and interpretation for the particular problem in hand. A more than detailed explanation of these sign conventions is presented below.

The previous derivation for the equation of the Mohr Circle using Figure iv follows the engineering mechanics sign convention. The engineering mechanics sign convention will be used for this article.

Physical-space sign convention [edit]

From the convention of the Cauchy stress tensor (Figure three and Figure four), 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 ${\displaystyle \tau _{xy}}$ is the shear stress interim on the face with normal vector in the positive direction of the ${\displaystyle x}$ -axis, and in the positive direction of the ${\displaystyle y}$ -axis.

In the physical-infinite sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are in to the plane of activeness (compression) (Figure five).

In the physical-space sign convention, positive shear stresses act on positive faces of the cloth element in the positive direction of an axis. Also, positive shear stresses deed on negative faces of the material element in the negative direction of an centrality. A positive confront has its normal vector in the positive management of an centrality, and a negative face has its normal vector in the negative direction of an axis. For instance, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive considering they act on positive faces, and they act likewise in the positive direction of the ${\displaystyle y}$ -axis and the ${\displaystyle x}$ -centrality, respectively (Effigy 3). Similarly, the respective opposite shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ acting in the negative faces have a negative sign because they act in the negative direction of the ${\displaystyle x}$ -centrality and ${\displaystyle y}$ -centrality, respectively.

Mohr-circle-space sign convention [edit]

Figure v. Engineering mechanics sign convention for cartoon the Mohr circumvolve. This article follows sign-convention # 3, equally shown.

In the Mohr-circle-space sign convention, normal stresses have the same sign equally normal stresses in the concrete-space sign convention: positive normal stresses act outward to the aeroplane of action, and negative normal stresses human action inward to the aeroplane 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-infinite 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 fashion, the shear stress component ${\displaystyle \tau _{xy}}$ is positive in the Mohr-circle space, and the shear stress component ${\displaystyle \tau _{yx}}$ is negative in the Mohr-circle space.

Two options be for drawing the Mohr-circle infinite, which produce a mathematically right Mohr circle:

1. Positive shear stresses are plotted upwards (Figure five, sign convention #1)
2. Positive shear stresses are plotted downwardly, i.e., the ${\displaystyle \tau _{\mathrm {n} }}$ -axis is inverted (Figure 5, sign convention #2).

Plotting positive shear stresses up makes the angle ${\displaystyle 2\theta }$ on the Mohr circle have a positive rotation clockwise, which is opposite to the concrete space convention. That is why some authors^[iii] prefer plotting positive shear stresses downwardly, which makes the angle ${\displaystyle 2\theta }$ on the Mohr circumvolve have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.

To overcome the "consequence" of having the shear stress axis downward in the Mohr-circle space, at that place is an alternative sign convention where positive shear stresses are causeless to rotate the material element in the clockwise direction and negative shear stresses are causeless to rotate the fabric element in the counterclockwise direction (Effigy 5, selection 3). This fashion, positive shear stresses are plotted upward in the Mohr-circle infinite and the bending ${\displaystyle 2\theta }$ has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #ii in Figure 5 because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is too a counterclockwise shear stress, and both are plotted downward. Also, a negative shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is a clockwise shear stress, and both are plotted upwardly.

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'southward circle [edit]

Assuming nosotros know the stress components ${\displaystyle \sigma _{10}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a point ${\displaystyle P}$ in the object nether report, equally shown in Figure 4, the following are the steps to construct the Mohr circle for the state of stresses at ${\displaystyle P}$ :

1. Describe the Cartesian coordinate system ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {northward} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality and a vertical ${\displaystyle \tau _{\mathrm {due north} }}$ -axis.
2. Plot two points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {northward} },\tau _{\mathrm {n} })}$ space corresponding to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Figure 4 and six), post-obit the chosen sign convention.
3. Depict the bore of the circle by joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a straight line ${\displaystyle {\overline {AB}}}$ .
4. Draw the Mohr Circle. The centre ${\displaystyle O}$ of the circumvolve is the midpoint of the diameter line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ axis.

Finding principal normal stresses [edit]

Stress components on a 2D rotating element. Case of how stress components vary on the faces (edges) of a rectangular element as the angle of its orientation is varied. Principal stresses occur when the shear stresses simultaneously disappear from all faces. The orientation at which this occurs gives the primary directions. In this example, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\brainstorm{matrix}\sigma _{xx}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\right]=\left[{\begin{matrix}-10&10\\x&15\terminate{matrix}}\right].}$ The corresponding Mohr'southward circle representation is shown at the bottom.

The magnitude of the master stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle E}$ (Effigy 6) where the circle intersects the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis. The magnitude of the major principal stress ${\displaystyle \sigma _{i}}$ is always the greatest absolute value of the abscissa of any of these two points. Likewise, the magnitude of the minor principal stress ${\displaystyle \sigma _{2}}$ is always the lowest absolute value of the abscissa of these ii points. Every bit expected, the ordinates of these two points are zero, respective to the magnitude of the shear stress components on the master planes. Alternatively, the values of the principal stresses can be found by

${\displaystyle \sigma _{1}=\sigma _{\max }=\sigma _{\text{avg}}+R}$

${\displaystyle \sigma _{2}=\sigma _{\min }=\sigma _{\text{avg}}-R}$

where the magnitude of the boilerplate normal stress ${\displaystyle \sigma _{\text{avg}}}$ is the abscissa of the centre ${\displaystyle O}$ , given by

${\displaystyle \sigma _{\text{avg}}={\tfrac {1}{2}}(\sigma _{x}+\sigma _{y})}$

and the length of the radius ${\displaystyle R}$ of the circle (based on the equation of a circle passing through 2 points), is given by

${\displaystyle R={\sqrt {\left[{\tfrac {one}{2}}(\sigma _{ten}-\sigma _{y})\right]^{2}+\tau _{xy}^{two}}}}$

Finding maximum and minimum shear stresses [edit]

The maximum and minimum shear stresses correspond to the ordinates of the highest and lowest points on the circumvolve, respectively. These points are located at the intersection of the circle with the vertical line passing through the center of the circumvolve, ${\displaystyle O}$ . Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circle'south radius ${\displaystyle R}$

${\displaystyle \tau _{\max ,\min }=\pm R}$

Finding stress components on an arbitrary airplane [edit]

Equally mentioned before, after the two-dimensional stress assay has been performed we know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a fabric point ${\displaystyle P}$ . These stress components act in 2 perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Effigy 5 and 6. The Mohr circle is used to detect the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.e., coordinates of whatever indicate ${\displaystyle D}$ on the circumvolve, acting on any other airplane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the plane ${\displaystyle B}$ . For this, 2 approaches tin can exist used: the double angle, and the Pole or origin of planes.

Double angle [edit]

Equally shown in Figure 6, to determine the stress components ${\displaystyle (\sigma _{\mathrm {northward} },\tau _{\mathrm {n} })}$ interim on a plane ${\displaystyle D}$ at an angle ${\displaystyle \theta }$ counterclockwise to the plane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an angle ${\displaystyle ii\theta }$ in the aforementioned counterclockwise direction effectually the circle from the known stress point ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ to point ${\displaystyle D(\sigma _{\mathrm {due north} },\tau _{\mathrm {n} })}$ , i.e., an angle ${\displaystyle 2\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circle.

The double angle approach relies on the fact that the bending ${\displaystyle \theta }$ betwixt the normal vectors to any ii physical planes passing through ${\displaystyle P}$ (Figure four) is half the bending between 2 lines joining their corresponding stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {due north} })}$ on the Mohr circumvolve and the centre of the circumvolve.

This double bending relation comes from the fact that the parametric equations for the Mohr circumvolve are a function of ${\displaystyle 2\theta }$ . It tin can also exist seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the fabric element around ${\displaystyle P}$ of Effigy 5 are separated by an bending ${\displaystyle \theta =90^{\circ }}$ , which in the Mohr circle is represented past a ${\displaystyle 180^{\circ }}$ angle (double the angle).

Pole or origin of planes [edit]

Figure 7. Mohr's circle for plane stress and plane strain weather (Pole approach). Any straight line fatigued from the pole will intersect the Mohr circumvolve at a point that represents the state of stress on a airplane inclined at the same orientation (parallel) in infinite equally that line.

The second approach involves the decision of a bespeak 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 country of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on whatever particular plane, one tin can draw a line parallel to that plane through the particular coordinates ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {north} }}$ on the Mohr circle and find the pole as the intersection of such line with the Mohr circle. As an case, let's assume nosotros have a country of stress with stress components ${\displaystyle \sigma _{10},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , as shown on Figure 7. First, nosotros can describe a line from point ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{10}}$ , or, if nosotros cull otherwise, a line from indicate ${\displaystyle A}$ parallel to the airplane of action of ${\displaystyle \sigma _{y}}$ . The intersection of any of these two lines with the Mohr circle is the pole. Once the pole has been determined, to notice the country of stress on a plane making an bending ${\displaystyle \theta }$ with the vertical, or in other words a plane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, then nosotros can draw a line from the pole parallel to that plane (Meet Figure vii). The normal and shear stresses on that plane are and then the coordinates of the point of intersection between the line and the Mohr circle.

Finding the orientation of the primary planes [edit]

The orientation of the planes where the maximum and minimum principal stresses deed, also known as primary planes, can be adamant past measuring in the Mohr circle the angles ∠BOC and ∠BOE, respectively, and taking half of each of those angles. Thus, the bending ∠BOC between ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OC}}}$ is double the angle ${\displaystyle \theta _{p}}$ which the major principal plane makes with plane ${\displaystyle B}$ .

Angles ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ tin can also be found from the post-obit equation

${\displaystyle \tan ii\theta _{\mathrm {p} }={\frac {2\tau _{xy}}{\sigma _{y}-\sigma _{ten}}}}$

This equation defines two values for ${\displaystyle \theta _{\mathrm {p} }}$ which are ${\displaystyle xc^{\circ }}$ apart (Effigy). This equation tin exist derived directly from the geometry of the circle, or by making the parametric equation of the circle for ${\displaystyle \tau _{\mathrm {north} }}$ equal to zero (the shear stress in the chief planes is always cipher).

Example [edit]

Assume a fabric element nether a state of stress as shown in Effigy 8 and Effigy 9, with the airplane of one of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circle, notice:

• 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 police.

Solution: Post-obit the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the cloth element in this example are:

${\displaystyle \sigma _{x'}=-10{\textrm {MPa}}}$

${\displaystyle \sigma _{y'}=50{\textrm {MPa}}}$

${\displaystyle \tau _{10'y'}=40{\textrm {MPa}}}$ .

Following the steps for cartoon the Mohr circle for this particular country of stress, we first draw a Cartesian coordinate system ${\displaystyle (\sigma _{\mathrm {north} },\tau _{\mathrm {north} })}$ with the ${\displaystyle \tau _{\mathrm {n} }}$ -axis upward.

We then plot 2 points A(50,40) and B(-10,-40), representing the country of stress at plane A and B equally testify in both Figure 8 and Effigy nine. These points follow the engineering science mechanics sign convention for the Mohr-circle space (Effigy five), which assumes positive normals stresses outward from the fabric element, and positive shear stresses on each airplane rotating the material element clockwise. This way, the shear stress interim on aeroplane B is negative and the shear stress acting on airplane A is positive. The diameter of the circumvolve is the line joining betoken A and B. The center of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {north} }}$ -axis. Knowing both the location of the centre and length of the diameter, nosotros are able to plot the Mohr circle for this item land of stress.

The abscissas of both points E and C (Figure 8 and Figure 9) intersecting the ${\displaystyle \sigma _{\mathrm {northward} }}$ -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 detect 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

${\displaystyle {\begin{aligned}R&={\sqrt {\left[{\tfrac {i}{two}}(\sigma _{x}-\sigma _{y})\right]^{ii}+\tau _{xy}^{ii}}}\\&={\sqrt {\left[{\tfrac {ane}{2}}(-ten-fifty)\right]^{2}+xl^{2}}}\\&=fifty{\textrm {MPa}}\\\end{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {1}{two}}(\sigma _{x}+\sigma _{y})\\&={\tfrac {1}{2}}(-x+50)\\&=20{\textrm {MPa}}\\\finish{aligned}}}$

and the principal stresses are

${\displaystyle {\brainstorm{aligned}\sigma _{ane}&=\sigma _{\mathrm {avg} }+R\\&=lxx{\textrm {MPa}}\\\cease{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{2}&=\sigma _{\mathrm {avg} }-R\\&=-30{\textrm {MPa}}\\\end{aligned}}}$

The coordinates for both points H and G (Figure 8 and Effigy 9) are the magnitudes of the minimum and maximum shear stresses, respectively; the abscissas for both points H and One thousand are the magnitudes for the normal stresses interim on the same planes where the minimum and maximum shear stresses human action, respectively. The magnitudes of the minimum and maximum shear stresses tin be found analytically by

${\displaystyle \tau _{\max ,\min }=\pm R=\pm 50{\textrm {MPa}}}$

and the normal stresses acting on the same planes where the minimum and maximum shear stresses act are equal to ${\displaystyle \sigma _{\mathrm {avg} }}$

We tin can choose to either use the double angle approach (Effigy 8) or the Pole approach (Effigy nine) to discover the orientation of the primary normal stresses and principal shear stresses.

Using the double angle approach we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Figure viii) to notice double the angle the major principal stress and the small-scale principal stress brand with plane B in the physical space. To obtain a more than authentic value for these angles, instead of manually measuring the angles, we can utilize the analytical expression

${\displaystyle {\brainstorm{aligned}2\theta _{\mathrm {p} }=\arctan {\frac {2\tau _{xy}}{\sigma _{10}-\sigma _{y}}}=\arctan {\frac {ii*twoscore}{(-10-50)}}=-\arctan {\frac {iv}{3}}\end{aligned}}}$

One solution is: ${\displaystyle 2\theta _{p}=-53.13^{\circ }}$ . From inspection of Effigy 8, this value corresponds to the bending ∠BOE. Thus, the minor principal angle is

${\displaystyle \theta _{p2}=-26.565^{\circ }}$

And so, the major principal angle is

${\displaystyle {\begin{aligned}2\theta _{p1}&=180-53.xiii^{\circ }=126.87^{\circ }\\\theta _{p1}&=63.435^{\circ }\\\end{aligned}}}$

Remember that in this particular instance ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ are angles with respect to the airplane of activeness of ${\displaystyle \sigma _{x'}}$ (oriented in the ${\displaystyle 10'}$ -axis)and not angles with respect to the plane of action of ${\displaystyle \sigma _{x}}$ (oriented in the ${\displaystyle 10}$ -axis).

Using the Pole approach, nosotros outset localize the Pole or origin of planes. For this, nosotros draw through point A on the Mohr circumvolve a line inclined ten° with the horizontal, or, in other words, a line parallel to plane A where ${\displaystyle \sigma _{y'}}$ 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 signal B on the Mohr circumvolve parallel to the plane B where ${\displaystyle \sigma _{ten'}}$ acts. This line would also intersect the Mohr circle at the Pole (Figure 9).

From the Pole, we describe lines to different points on the Mohr circumvolve. The coordinates of the points where these lines intersect the Mohr circumvolve betoken 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 circumvolve has the same inclination as the aeroplane in the concrete space where ${\displaystyle \sigma _{1}}$ acts. This airplane 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 Due east, D, F, Thou and H to detect the stress components on planes with the same orientation.

Mohr's circle for a general iii-dimensional state of stresses [edit]

Figure 10. Mohr'south circumvolve for a three-dimensional land of stress

To construct the Mohr circle for a general three-dimensional case of stresses at a point, the values of the principal stresses ${\displaystyle \left(\sigma _{1},\sigma _{2},\sigma _{3}\right)}$ and their principal directions ${\displaystyle \left(n_{1},n_{two},n_{3}\right)}$ must exist outset evaluated.

Because the principal axes equally the coordinate system, instead of the general ${\displaystyle x_{1}}$ , ${\displaystyle x_{ii}}$ , ${\displaystyle x_{3}}$ coordinate system, and assuming that ${\displaystyle \sigma _{1}>\sigma _{2}>\sigma _{3}}$ , and so the normal and shear components of the stress vector ${\displaystyle \mathbf {T} ^{(\mathbf {n} )}}$ , for a given plane with unit vector ${\displaystyle \mathbf {n} }$ , satisfy the following equations

${\displaystyle {\begin{aligned}\left(T^{(due north)}\right)^{2}&=\sigma _{ij}\sigma _{ik}n_{j}n_{k}\\\sigma _{\mathrm {n} }^{2}+\tau _{\mathrm {n} }^{two}&=\sigma _{1}^{two}n_{1}^{two}+\sigma _{2}^{2}n_{two}^{ii}+\sigma _{3}^{2}n_{three}^{2}\terminate{aligned}}}$

${\displaystyle \sigma _{\mathrm {n} }=\sigma _{1}n_{1}^{ii}+\sigma _{2}n_{2}^{2}+\sigma _{3}n_{3}^{ii}.}$

Knowing that ${\displaystyle n_{i}n_{i}=n_{1}^{2}+n_{2}^{ii}+n_{three}^{2}=1}$ , we can solve for ${\displaystyle n_{1}^{2}}$ , ${\displaystyle n_{2}^{two}}$ , ${\displaystyle n_{three}^{ii}}$ , using the Gauss elimination method which yields

${\displaystyle {\begin{aligned}n_{1}^{2}&={\frac {\tau _{\mathrm {n} }^{2}+(\sigma _{\mathrm {north} }-\sigma _{ii})(\sigma _{\mathrm {northward} }-\sigma _{3})}{(\sigma _{one}-\sigma _{2})(\sigma _{1}-\sigma _{3})}}\geq 0\\n_{2}^{ii}&={\frac {\tau _{\mathrm {n} }^{2}+(\sigma _{\mathrm {n} }-\sigma _{iii})(\sigma _{\mathrm {n} }-\sigma _{1})}{(\sigma _{two}-\sigma _{3})(\sigma _{two}-\sigma _{1})}}\geq 0\\n_{3}^{2}&={\frac {\tau _{\mathrm {n} }^{2}+(\sigma _{\mathrm {n} }-\sigma _{i})(\sigma _{\mathrm {n} }-\sigma _{2})}{(\sigma _{3}-\sigma _{one})(\sigma _{3}-\sigma _{2})}}\geq 0.\end{aligned}}}$

Since ${\displaystyle \sigma _{i}>\sigma _{two}>\sigma _{3}}$ , and ${\displaystyle (n_{i})^{2}}$ is not-negative, the numerators from these equations satisfy

${\displaystyle \tau _{\mathrm {n} }^{2}+(\sigma _{\mathrm {north} }-\sigma _{2})(\sigma _{\mathrm {n} }-\sigma _{iii})\geq 0}$ as the denominator ${\displaystyle \sigma _{one}-\sigma _{ii}>0}$ and ${\displaystyle \sigma _{1}-\sigma _{3}>0}$

${\displaystyle \tau _{\mathrm {north} }^{2}+(\sigma _{\mathrm {n} }-\sigma _{3})(\sigma _{\mathrm {n} }-\sigma _{1})\leq 0}$ as the denominator ${\displaystyle \sigma _{two}-\sigma _{3}>0}$ and ${\displaystyle \sigma _{2}-\sigma _{1}<0}$

${\displaystyle \tau _{\mathrm {n} }^{2}+(\sigma _{\mathrm {n} }-\sigma _{i})(\sigma _{\mathrm {n} }-\sigma _{2})\geq 0}$ as the denominator ${\displaystyle \sigma _{three}-\sigma _{one}<0}$ and ${\displaystyle \sigma _{3}-\sigma _{two}<0.}$

These expressions can exist rewritten as

${\displaystyle {\begin{aligned}\tau _{\mathrm {n} }^{2}+\left[\sigma _{\mathrm {north} }-{\tfrac {ane}{2}}(\sigma _{2}+\sigma _{iii})\correct]^{2}\geq \left({\tfrac {ane}{2}}(\sigma _{2}-\sigma _{3})\right)^{ii}\\\tau _{\mathrm {n} }^{2}+\left[\sigma _{\mathrm {n} }-{\tfrac {one}{2}}(\sigma _{1}+\sigma _{iii})\right]^{two}\leq \left({\tfrac {one}{two}}(\sigma _{i}-\sigma _{3})\right)^{2}\\\tau _{\mathrm {n} }^{two}+\left[\sigma _{\mathrm {north} }-{\tfrac {i}{two}}(\sigma _{1}+\sigma _{2})\right]^{two}\geq \left({\tfrac {1}{2}}(\sigma _{1}-\sigma _{2})\correct)^{ii}\\\terminate{aligned}}}$

which are the equations of the three Mohr's circles for stress ${\displaystyle C_{ane}}$ , ${\displaystyle C_{ii}}$ , and ${\displaystyle C_{3}}$ , with radii ${\displaystyle R_{1}={\tfrac {ane}{2}}(\sigma _{2}-\sigma _{3})}$ , ${\displaystyle R_{two}={\tfrac {1}{2}}(\sigma _{i}-\sigma _{3})}$ , and ${\displaystyle R_{3}={\tfrac {1}{2}}(\sigma _{1}-\sigma _{ii})}$ , and their centres with coordinates ${\displaystyle \left[{\tfrac {1}{two}}(\sigma _{two}+\sigma _{iii}),0\correct]}$ , ${\displaystyle \left[{\tfrac {1}{2}}(\sigma _{1}+\sigma _{3}),0\correct]}$ , ${\displaystyle \left[{\tfrac {1}{ii}}(\sigma _{1}+\sigma _{two}),0\right]}$ , respectively.

These equations for the Mohr circles bear witness that all admissible stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ prevarication on these circles or inside the shaded surface area enclosed by them (meet Figure 10). Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ satisfying the equation for circle ${\displaystyle C_{one}}$ prevarication on, or exterior circumvolve ${\displaystyle C_{1}}$ . Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{2}}$ prevarication on, or within circle ${\displaystyle C_{ii}}$ . And finally, stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {northward} })}$ satisfying the equation for circle ${\displaystyle C_{iii}}$ lie on, or outside circumvolve ${\displaystyle C_{3}}$ .

See also [edit]

• Critical plane assay

References [edit]

1. ^ "Principal stress and primary plane". world wide web.engineeringapps.cyberspace . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN0-415-27297-one.
3. ^ Gere, James M. (2013). Mechanics of Materials. Goodno, Barry J. (eighth ed.). Stamford, CT: Cengage Learning. ISBN9781111577735.

Bibliography [edit]

• Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials . McGraw-Colina Professional person. ISBN0-07-112939-one.
• Brady, B.H.G.; Eastward.T. Brownish (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. South. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall ceremonious engineering science and engineering mechanics serial. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, N.G.W.; Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. nine–41. ISBN978-0-632-05759-7.
• Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co. ISBN0-442-04199-3.
• Parry, Richard Hawley Grayness (2004). Mohr circles, stress paths and geotechnics (ii ed.). Taylor & Francis. pp. 1–thirty. ISBN0-415-27297-1.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGraw-Hill International Editions. ISBN0-07-085805-5.
• 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. ISBN0-486-61187-6.

External links [edit]

• Mohr's Circle and more circles by Rebecca Brannon
• DoITPoMS Teaching and Learning Package- "Stress Assay and Mohr'due south Circle"

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Source: https://en.wikipedia.org/wiki/Mohr%27s_circle

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