Draw Mohrs Circle That Describes the State of Stress in

Geometric civil engineering 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 law for the Cauchy stress tensor.

Mohr's circle is ofttimes used in calculations relating to mechanical engineering for materials' strength, geotechnical engineering for strength of soils, and structural engineering for forcefulness of built structures. Information technology is also used for calculating stresses in many planes by reducing them to vertical and horizontal components. These are called primary planes in which chief stresses are calculated; Mohr'southward circle can likewise exist used to find the chief planes and the principal stresses in a graphical representation, and is one of the easiest ways to do so.^[1]

After performing a stress analysis on a material torso causeless every bit a continuum, the components of the Cauchy stress tensor at a particular cloth point are known with respect to a coordinate system. The Mohr circumvolve is then used to decide graphically the stress components interim on a rotated coordinate system, i.e., acting on a differently oriented aeroplane passing through that betoken.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {n} }}$ , ${\displaystyle \tau _{\mathrm {n} }}$ ) of each point on the circumvolve 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 state of stress on individual planes at all their orientations, where the axes correspond the principal axes of the stress chemical element.

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

Alternative graphical methods for the representation of the stress state at a indicate include the Lamé'due south stress ellipsoid and Cauchy's stress quadric.

The Mohr circle tin be practical to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.

Motivation [edit]

Figure 2. Stress in a loaded deformable material trunk assumed as a continuum.

Internal forces are produced between the particles of a deformable object, assumed every bit a continuum, as a reaction to applied external forces, i.e., either surface forces or body forces. This reaction follows from Euler'southward laws of move for a continuum, which are equivalent to Newton'south laws of motion for a particle. A measure of the intensity of these internal forces is called stress. Because the object is assumed equally a continuum, these internal forces are distributed continuously within the book of the object.

In engineering, e.chiliad., structural, mechanical, or geotechnical, the stress distribution inside an object, for case stresses in a rock mass effectually a tunnel, airplane wings, or building columns, is determined through a stress assay. Computing the stress distribution implies the determination of stresses at every point (material particle) in the object. According to Cauchy, the stress at whatever bespeak in an object (Effigy 2), assumed as a continuum, is completely defined by the nine stress components ${\displaystyle \sigma _{ij}}$ of a second social club tensor of type (2,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

Figure three. Stress transformation at a point in a continuum nether plane stress atmospheric condition.

After the stress distribution within the object has been adamant with respect to a coordinate system ${\displaystyle (x,y)}$ , it may be necessary to calculate the components of the stress tensor at a particular fabric indicate ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (10',y')}$ , i.due east., the stresses interim on a plane with a different orientation passing through that point of interest —forming an bending with the coordinate arrangement ${\displaystyle (x,y)}$ (Figure 3). For example, it is of interest to find the maximum normal stress and maximum shear stress, as well every bit the orientation of the planes where they act upon. To achieve this, information technology is necessary to perform a tensor transformation nether 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 police for the Cauchy stress tensor is the Mohr circle for stress.

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

Effigy four. Stress components at a aeroplane passing through a point in a continuum under airplane stress atmospheric condition.

In two dimensions, the stress tensor at a given material point ${\displaystyle P}$ with respect to whatever ii perpendicular directions is completely defined past just iii stress components. For the particular coordinate organization ${\displaystyle (x,y)}$ these stress components are: the normal stresses ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{y}}$ , and the shear stress ${\displaystyle \tau _{xy}}$ . From the balance 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 as:

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

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

Equation of the Mohr circle [edit]

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 effectually a material point ${\displaystyle P}$ (Figure 4), with a unit area in the direction parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ airplane, i.east., perpendicular to the page or screen.

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

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

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

Derivation of Mohr'southward circle parametric equations - Equilibrium of forces

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

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 ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ .

Derivation of Mohr's circle parametric equations - Tensor transformation

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

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

Eliminating the parameter ${\displaystyle two\theta }$ from these parametric equations volition yield the non-parametric equation of the Mohr circumvolve. This tin be achieved by rearranging the equations for ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , first transposing the first term in the beginning equation and squaring both sides of each of the equations then adding them. Thus nosotros have

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

where

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

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

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

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

Sign conventions [edit]

There are two separate sets of sign conventions that need to be considered when using the Mohr Circle: I sign convention for stress components in the "physical space", and some other 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. At that place is no standard sign convention, and the choice of a particular sign convention is influenced by convenience for calculation and estimation 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 Effigy iv follows the engineering mechanics sign convention. The engineering mechanics sign convention will be used for this commodity.

Physical-infinite sign convention [edit]

From the convention of the Cauchy stress tensor (Figure three and Figure four), the commencement subscript in the stress components denotes the confront 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-space sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are in to the plane of action (compression) (Figure v).

In the physical-infinite sign convention, positive shear stresses deed on positive faces of the textile element in the positive direction of an axis. Also, positive shear stresses human activity on negative faces of the material element in the negative direction of an axis. A positive face up has its normal vector in the positive direction of an axis, and a negative confront has its normal vector in the negative management of an axis. For case, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive because they human activity on positive faces, and they act as well in the positive management of the ${\displaystyle y}$ -axis and the ${\displaystyle x}$ -axis, respectively (Figure 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 ${\displaystyle x}$ -axis and ${\displaystyle y}$ -axis, respectively.

Mohr-circumvolve-space sign convention [edit]

Figure 5. Technology mechanics sign convention for cartoon the Mohr circle. This article follows sign-convention # three, every bit shown.

In the Mohr-circumvolve-infinite sign convention, normal stresses have the same sign as normal stresses in the physical-space sign convention: positive normal stresses act outward to the airplane of action, and negative normal stresses act inward to the plane of action.

Shear stresses, all the same, take a different convention in the Mohr-circle infinite compared to the convention in the concrete space. In the Mohr-circumvolve-space sign convention, positive shear stresses rotate the fabric chemical element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction. This way, 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.

Ii options be for drawing the Mohr-circle space, which produce a mathematically right Mohr circumvolve:

1. Positive shear stresses are plotted upward (Figure 5, sign convention #1)
2. Positive shear stresses are plotted downwardly, i.e., the ${\displaystyle \tau _{\mathrm {north} }}$ -centrality is inverted (Figure 5, sign convention #two).

Plotting positive shear stresses upwards makes the angle ${\displaystyle 2\theta }$ on the Mohr circle have a positive rotation clockwise, which is opposite to the physical space convention. That is why some authors^[3] adopt plotting positive shear stresses downward, which makes the bending ${\displaystyle 2\theta }$ on the Mohr circle have a positive rotation counterclockwise, similar to the physical infinite convention for shear stresses.

To overcome the "consequence" of having the shear stress axis downward in the Mohr-circle infinite, there is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise management and negative shear stresses are assumed to rotate the cloth element in the counterclockwise direction (Figure 5, choice iii). This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle ${\displaystyle 2\theta }$ has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #2 in Figure 5 because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is likewise 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 commodity follows the engineering mechanics sign convention for the physical infinite and the culling sign convention for the Mohr-circle infinite (sign convention #3 in Figure 5)

Cartoon Mohr's circle [edit]

Assuming we know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a signal ${\displaystyle P}$ in the object nether written report, as shown in Figure 4, the post-obit are the steps to construct the Mohr circumvolve for the state of stresses at ${\displaystyle P}$ :

1. Draw the Cartesian coordinate system ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {due north} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {northward} }}$ -axis.
2. Plot ii points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ space respective to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Effigy four and 6), following the chosen sign convention.
3. Draw the diameter of the circumvolve by joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a straight line ${\displaystyle {\overline {AB}}}$ .
4. Describe the Mohr Circle. The centre ${\displaystyle O}$ of the circle is the midpoint of the diameter line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ centrality.

Finding principal normal stresses [edit]

Stress components on a 2nd rotating chemical 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 principal directions. In this case, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{xx}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\correct]=\left[{\begin{matrix}-ten&x\\ten&xv\terminate{matrix}}\right].}$ The respective Mohr'southward circle representation is shown at the bottom.

The magnitude of the principal stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle E}$ (Figure half-dozen) where the circle intersects the ${\displaystyle \sigma _{\mathrm {north} }}$ -axis. The magnitude of the major main stress ${\displaystyle \sigma _{1}}$ is always the greatest absolute value of the abscissa of whatsoever of these ii points. Likewise, the magnitude of the small main stress ${\displaystyle \sigma _{ii}}$ is always the lowest accented value of the abscissa of these ii points. Equally expected, the ordinates of these two points are zippo, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the principal stresses can exist 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 average normal stress ${\displaystyle \sigma _{\text{avg}}}$ is the abscissa of the centre ${\displaystyle O}$ , given by

${\displaystyle \sigma _{\text{avg}}={\tfrac {1}{ii}}(\sigma _{10}+\sigma _{y})}$

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

${\displaystyle R={\sqrt {\left[{\tfrac {1}{two}}(\sigma _{x}-\sigma _{y})\right]^{two}+\tau _{xy}^{two}}}}$

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 circle, 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 circumvolve's radius ${\displaystyle R}$

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

Finding stress components on an arbitrary plane [edit]

As mentioned before, after the two-dimensional stress analysis has been performed nosotros know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a cloth signal ${\displaystyle P}$ . These stress components act in 2 perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Figure 5 and 6. The Mohr circle is used to find the stress components ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.due east., coordinates of whatever point ${\displaystyle D}$ on the circle, acting on any other plane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an bending ${\displaystyle \theta }$ with the plane ${\displaystyle B}$ . For this, two approaches tin be used: the double bending, and the Pole or origin of planes.

Double angle [edit]

As shown in Figure half dozen, to determine the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ acting on a aeroplane ${\displaystyle D}$ at an angle ${\displaystyle \theta }$ counterclockwise to the airplane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an angle ${\displaystyle 2\theta }$ in the aforementioned counterclockwise management around the circumvolve from the known stress betoken ${\displaystyle B(\sigma _{10},-\tau _{xy})}$ to point ${\displaystyle D(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ , i.e., an angle ${\displaystyle ii\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circle.

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

This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of ${\displaystyle 2\theta }$ . It can as well be seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the textile chemical element around ${\displaystyle P}$ of Figure 5 are separated by an angle ${\displaystyle \theta =90^{\circ }}$ , which in the Mohr circle is represented by a ${\displaystyle 180^{\circ }}$ angle (double the angle).

Pole or origin of planes [edit]

Figure 7. Mohr's circle for plane stress and aeroplane strain weather (Pole arroyo). Whatever direct line drawn from the pole will intersect the Mohr circle at a bespeak that represents the state of stress on a plane inclined at the same orientation (parallel) in space equally that line.

The second approach involves the decision of a point on the Mohr circumvolve chosen the pole or the origin of planes. Whatsoever straight line fatigued from the pole will intersect the Mohr circle at a point that represents the state of stress on a aeroplane inclined at the aforementioned orientation (parallel) in space as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on whatever particular airplane, 1 tin draw a line parallel to that plane through the particular coordinates ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {north} }}$ on the Mohr circumvolve and find the pole every bit the intersection of such line with the Mohr circle. Every bit an case, let'south presume we accept a state of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , every bit shown on Figure seven. First, we can describe a line from point ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{x}}$ , or, if nosotros choose otherwise, a line from point ${\displaystyle A}$ parallel to the plane of action of ${\displaystyle \sigma _{y}}$ . The intersection of any of these two lines with the Mohr circle is the pole. One time the pole has been determined, to notice the state of stress on a airplane making an angle ${\displaystyle \theta }$ with the vertical, or in other words a plane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, then we can draw a line from the pole parallel to that airplane (Come across Figure 7). The normal and shear stresses on that plane are then the coordinates of the indicate 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 primary stresses act, also known equally chief planes, tin exist 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 ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OC}}}$ is double the angle ${\displaystyle \theta _{p}}$ which the major principal plane makes with aeroplane ${\displaystyle B}$ .

Angles ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ can likewise be found from the following equation

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

This equation defines two values for ${\displaystyle \theta _{\mathrm {p} }}$ which are ${\displaystyle 90^{\circ }}$ apart (Figure). This equation can be derived direct 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 ever aught).

Case [edit]

Assume a cloth chemical element under a state of stress as shown in Figure 8 and Figure 9, with the plane of i of its sides oriented ten° with respect to the horizontal aeroplane. Using the Mohr circumvolve, discover:

• The orientation of their planes of action.
• The maximum shear stresses and orientation of their planes of activity.
• 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 infinite (Figure 5), the stress components for the material element in this instance are:

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

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

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

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

We then plot two points A(50,40) and B(-10,-forty), representing the state of stress at plane A and B as prove in both Figure viii and Figure nine. These points follow the applied science mechanics sign convention for the Mohr-circumvolve space (Figure v), which assumes positive normals stresses outward from the material chemical element, and positive shear stresses on each airplane rotating the material element clockwise. This manner, the shear stress acting on plane B is negative and the shear stress acting on aeroplane A is positive. The diameter of the circumvolve is the line joining indicate A and B. The middle of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis. Knowing both the location of the middle and length of the diameter, we are able to plot the Mohr circle for this particular country of stress.

The abscissas of both points Due east and C (Figure viii and Figure 9) intersecting the ${\displaystyle \sigma _{\mathrm {north} }}$ -axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points Due east 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.

Fifty-fifty though the idea for using the Mohr circle is to graphically notice different stress components by really measuring the coordinates for different points on the circle, it is more user-friendly to confirm the results analytically. Thus, the radius and the abscissa of the centre of the circle are

${\displaystyle {\begin{aligned}R&={\sqrt {\left[{\tfrac {one}{ii}}(\sigma _{x}-\sigma _{y})\right]^{2}+\tau _{xy}^{2}}}\\&={\sqrt {\left[{\tfrac {one}{2}}(-ten-50)\right]^{2}+twoscore^{2}}}\\&=50{\textrm {MPa}}\\\finish{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {i}{2}}(\sigma _{x}+\sigma _{y})\\&={\tfrac {ane}{2}}(-ten+l)\\&=xx{\textrm {MPa}}\\\terminate{aligned}}}$

and the principal stresses are

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

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

The coordinates for both points H and G (Figure eight and Effigy 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 human activity, respectively. The magnitudes of the minimum and maximum shear stresses tin exist constitute analytically past

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

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

We can cull to either use the double angle approach (Figure eight) or the Pole approach (Figure ix) to notice 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 Circumvolve (Figure 8) to find double the angle the major main stress and the small principal stress make with plane B in the physical infinite. To obtain a more accurate value for these angles, instead of manually measuring the angles, we can use the belittling expression

${\displaystyle {\begin{aligned}two\theta _{\mathrm {p} }=\arctan {\frac {2\tau _{xy}}{\sigma _{ten}-\sigma _{y}}}=\arctan {\frac {2*xl}{(-10-50)}}=-\arctan {\frac {four}{3}}\end{aligned}}}$

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

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

And so, the major principal angle is

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

Retrieve that in this item case ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ are angles with respect to the plane of action of ${\displaystyle \sigma _{x'}}$ (oriented in the ${\displaystyle ten'}$ -axis)and not angles with respect to the plane of activeness of ${\displaystyle \sigma _{x}}$ (oriented in the ${\displaystyle x}$ -axis).

Using the Pole approach, we kickoff localize the Pole or origin of planes. For this, we describe through signal A on the Mohr circle a line inclined x° with the horizontal, or, in other words, a line parallel to airplane A where ${\displaystyle \sigma _{y'}}$ acts. The Pole is where this line intersects the Mohr circumvolve (Effigy 9). To ostend the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the airplane B where ${\displaystyle \sigma _{x'}}$ acts. This line would also intersect the Mohr circle at the Pole (Figure 9).

From the Pole, nosotros 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 signal C in the circumvolve has the same inclination as the airplane in the physical infinite where ${\displaystyle \sigma _{1}}$ acts. This plane makes an angle of 63.435° with aeroplane B, both in the Mohr-circumvolve space and in the physical space. In the same way, lines are traced from the Pole to points Due east, D, F, G and H to detect the stress components on planes with the same orientation.

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

Figure 10. Mohr'due south circle for a three-dimensional state of stress

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

Considering the master axes as the coordinate arrangement, instead of the general ${\displaystyle x_{1}}$ , ${\displaystyle x_{2}}$ , ${\displaystyle x_{3}}$ coordinate arrangement, and assuming that ${\displaystyle \sigma _{ane}>\sigma _{two}>\sigma _{3}}$ , then the normal and shear components of the stress vector ${\displaystyle \mathbf {T} ^{(\mathbf {n} )}}$ , for a given plane with unit vector ${\displaystyle \mathbf {northward} }$ , satisfy the following equations

${\displaystyle {\brainstorm{aligned}\left(T^{(n)}\right)^{2}&=\sigma _{ij}\sigma _{ik}n_{j}n_{yard}\\\sigma _{\mathrm {n} }^{2}+\tau _{\mathrm {due north} }^{2}&=\sigma _{1}^{two}n_{1}^{2}+\sigma _{2}^{2}n_{ii}^{two}+\sigma _{iii}^{2}n_{three}^{2}\end{aligned}}}$

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

Knowing that ${\displaystyle n_{i}n_{i}=n_{i}^{2}+n_{2}^{2}+n_{3}^{2}=1}$ , we can solve for ${\displaystyle n_{one}^{2}}$ , ${\displaystyle n_{two}^{2}}$ , ${\displaystyle n_{3}^{two}}$ , using the Gauss elimination method which yields

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

Since ${\displaystyle \sigma _{1}>\sigma _{2}>\sigma _{3}}$ , and ${\displaystyle (n_{i})^{2}}$ is non-negative, the numerators from these equations satisfy

${\displaystyle \tau _{\mathrm {northward} }^{ii}+(\sigma _{\mathrm {due north} }-\sigma _{two})(\sigma _{\mathrm {north} }-\sigma _{3})\geq 0}$ as the denominator ${\displaystyle \sigma _{one}-\sigma _{ii}>0}$ and ${\displaystyle \sigma _{1}-\sigma _{three}>0}$

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

${\displaystyle \tau _{\mathrm {n} }^{ii}+(\sigma _{\mathrm {due north} }-\sigma _{i})(\sigma _{\mathrm {n} }-\sigma _{two})\geq 0}$ every bit the denominator ${\displaystyle \sigma _{3}-\sigma _{i}<0}$ and ${\displaystyle \sigma _{iii}-\sigma _{2}<0.}$

These expressions can be rewritten every bit

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

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

These equations for the Mohr circles show that all open-door stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {due north} })}$ lie on these circles or inside the shaded area enclosed by them (run across Figure x). Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{i}}$ lie on, or outside circle ${\displaystyle C_{1}}$ . Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{2}}$ lie on, or inside circumvolve ${\displaystyle C_{2}}$ . And finally, stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {due north} })}$ satisfying the equation for circle ${\displaystyle C_{3}}$ lie on, or outside circle ${\displaystyle C_{iii}}$ .

See also [edit]

• Critical aeroplane analysis

References [edit]

1. ^ "Main stress and chief plane". www.engineeringapps.net . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–xxx. ISBN0-415-27297-1.
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. ISBN0-07-112939-ane.
• Brady, B.H.K.; E.T. Brownish (1993). Rock Mechanics For Cloak-and-dagger Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN0-521-49827-nine.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical applied science. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, N.G.W.; Zimmerman, R.Due west. (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-three.
• Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–thirty. ISBN0-415-27297-i.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGraw-Colina International Editions. ISBN0-07-085805-5.
• Timoshenko, Stephen P. (1983). History of strength of materials: with a cursory 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 than circles by Rebecca Brannon
• DoITPoMS Education and Learning Package- "Stress Analysis and Mohr's Circle"

huntcombou.blogspot.com

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

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