Draw a Mohrs Circle Diagram

Geometric civil engineering adding technique

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

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

Mohr'south circumvolve is often used in calculations relating to mechanical engineering for materials' strength, geotechnical technology for force of soils, and structural engineering for strength of built structures. Information technology is also used for computing stresses in many planes past reducing them to vertical and horizontal components. These are called primary planes in which primary stresses are calculated; Mohr'south circumvolve can as well be used to find the primary planes and the principal stresses in a graphical representation, and is one of the easiest means to do so.^[i]

After performing a stress analysis on a textile trunk assumed as a continuum, the components of the Cauchy stress tensor at a particular material betoken are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {n} }}$ , ${\displaystyle \tau _{\mathrm {n} }}$ ) of each point on the circle are the magnitudes of the normal stress and shear stress components, respectively, interim on the rotated coordinate arrangement. 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 element.

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

Alternative graphical methods for the representation of the stress country at a signal include the Lamé'south 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 assumed as a continuum.

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

In engineering, e.thousand., structural, mechanical, or geotechnical, the stress distribution within an object, for case stresses in a rock mass around a tunnel, airplane wings, or building columns, is adamant through a stress assay. Calculating the stress distribution implies the determination of stresses at every indicate (material particle) in the object. According to Cauchy, the stress at whatsoever point in an object (Figure 2), causeless as a continuum, is completely defined by the 9 stress components ${\displaystyle \sigma _{ij}}$ of a second gild tensor of type (2,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

Figure 3. Stress transformation at a betoken in a continuum under plane stress weather 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 textile indicate ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (ten',y')}$ , i.e., the stresses acting on a plane with a unlike orientation passing through that point of interest —forming an bending with the coordinate arrangement ${\displaystyle (x,y)}$ (Effigy 3). For instance, it is of interest to discover the maximum normal stress and maximum shear stress, also every bit 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 police force for the Cauchy stress tensor is the Mohr circle for stress.

Mohr's circle for 2-dimensional state of stress [edit]

Figure 4. Stress components at a airplane passing through a betoken in a continuum under airplane stress conditions.

In ii dimensions, the stress tensor at a given textile bespeak ${\displaystyle P}$ with respect to any two perpendicular directions is completely divers by only three stress components. For the detail coordinate system ${\displaystyle (x,y)}$ these stress components are: the normal stresses ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{y}}$ , and the shear stress ${\displaystyle \tau _{xy}}$ . From the rest of athwart momentum, the symmetry of the Cauchy stress tensor tin can be demonstrated. This symmetry implies that ${\displaystyle \tau _{xy}=\tau _{yx}}$ . Thus, the Cauchy stress tensor tin can be written as:

${\displaystyle {\boldsymbol {\sigma }}=\left[{\begin{matrix}\sigma _{x}&\tau _{xy}&0\\\tau _{xy}&\sigma _{y}&0\\0&0&0\\\end{matrix}}\correct]\equiv \left[{\begin{matrix}\sigma _{10}&\tau _{xy}\\\tau _{xy}&\sigma _{y}\\\terminate{matrix}}\right]}$

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

Equation of the Mohr circumvolve [edit]

To derive the equation of the Mohr circumvolve for the two-dimensional cases of plane stress and airplane strain, kickoff consider a 2-dimensional minute material chemical element around a material point ${\displaystyle P}$ (Figure four), with a unit surface area 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:

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

${\displaystyle \tau _{\mathrm {n} }=-{\frac {i}{two}}(\sigma _{10}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$

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) (Effigy 4), and knowing that the area of the plane where ${\displaystyle \sigma _{\mathrm {due north} }}$ acts is ${\displaystyle dA}$ , we have: ${\displaystyle \ {\begin{aligned}\sum F_{x'}&=\sigma _{\mathrm {n} }dA-\sigma _{x}dA\cos ^{2}\theta -\sigma _{y}dA\sin ^{2}\theta -\tau _{xy}dA\cos \theta \sin \theta -\tau _{xy}dA\sin \theta \cos \theta =0\\\sigma _{\mathrm {north} }&=\sigma _{x}\cos ^{2}\theta +\sigma _{y}\sin ^{2}\theta +two\tau _{xy}\sin \theta \cos \theta \\\end{aligned}}}$ Yet, knowing that ${\displaystyle \cos ^{ii}\theta ={\frac {i+\cos 2\theta }{ii}},\qquad \sin ^{2}\theta ={\frac {ane-\cos two\theta }{2}}\qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ we obtain ${\displaystyle \sigma _{\mathrm {northward} }={\frac {1}{ii}}(\sigma _{x}+\sigma _{y})+{\frac {1}{ii}}(\sigma _{10}-\sigma _{y})\cos 2\theta +\tau _{xy}\sin 2\theta }$ Now, from equilibrium of forces in the management of ${\displaystyle \tau _{\mathrm {northward} }}$ ( ${\displaystyle y'}$ -centrality) (Figure four), and knowing that the surface area of the plane where ${\displaystyle \tau _{\mathrm {northward} }}$ acts is ${\displaystyle dA}$ , we take: ${\displaystyle \ {\brainstorm{aligned}\sum F_{y'}&=\tau _{\mathrm {northward} }dA+\sigma _{x}dA\cos \theta \sin \theta -\sigma _{y}dA\sin \theta \cos \theta -\tau _{xy}dA\cos ^{2}\theta +\tau _{xy}dA\sin ^{two}\theta =0\\\tau _{\mathrm {n} }&=-(\sigma _{ten}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{ii}\theta -\sin ^{two}\theta \correct)\\\end{aligned}}}$ However, knowing that ${\displaystyle \cos ^{ii}\theta -\sin ^{2}\theta =\cos two\theta \qquad {\text{and}}\qquad \sin two\theta =two\sin \theta \cos \theta }$ we obtain ${\displaystyle \tau _{\mathrm {n} }=-{\frac {i}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$

Both equations tin can also be obtained past applying the tensor transformation police force 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 ${\displaystyle \tau _{\mathrm {n} }}$ .

Derivation of Mohr'due south circle parametric equations - Tensor transformation

The stress tensor transformation law can be stated equally ${\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}'&=\mathbf {A} {\boldsymbol {\sigma }}\mathbf {A} ^{T}\\\left[{\begin{matrix}\sigma _{x'}&\tau _{x'y'}\\\tau _{y'x'}&\sigma _{y'}\\\cease{matrix}}\correct]&=\left[{\brainstorm{matrix}a_{x}&a_{xy}\\a_{yx}&a_{y}\\\end{matrix}}\right]\left[{\brainstorm{matrix}\sigma _{10}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\correct]\left[{\begin{matrix}a_{ten}&a_{yx}\\a_{xy}&a_{y}\\\end{matrix}}\correct]\\&=\left[{\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\terminate{matrix}}\right]\left[{\begin{matrix}\sigma _{x}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\right]\left[{\brainstorm{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{matrix}}\right]\end{aligned}}}$ Expanding the right mitt side, and knowing that ${\displaystyle \sigma _{x'}=\sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{ten'y'}=\tau _{\mathrm {n} }}$ , nosotros take: ${\displaystyle \sigma _{\mathrm {n} }=\sigma _{x}\cos ^{2}\theta +\sigma _{y}\sin ^{2}\theta +2\tau _{xy}\sin \theta \cos \theta }$ However, knowing that ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}},\qquad \sin ^{2}\theta ={\frac {one-\cos 2\theta }{2}}\qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \sigma _{\mathrm {north} }={\frac {ane}{ii}}(\sigma _{x}+\sigma _{y})+{\frac {ane}{2}}(\sigma _{x}-\sigma _{y})\cos 2\theta +\tau _{xy}\sin 2\theta }$ ${\displaystyle \tau _{\mathrm {n} }=-(\sigma _{x}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{2}\theta -\sin ^{2}\theta \right)}$ Withal, knowing that ${\displaystyle \cos ^{2}\theta -\sin ^{2}\theta =\cos 2\theta \qquad {\text{and}}\qquad \sin ii\theta =ii\sin \theta \cos \theta }$ we obtain ${\displaystyle \tau _{\mathrm {north} }=-{\frac {1}{two}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$ It is not necessary at this moment to calculate the stress component ${\displaystyle \sigma _{y'}}$ acting on the plane perpendicular to the plane of action of ${\displaystyle \sigma _{x'}}$ as it is not 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 ${\displaystyle \tau _{\mathrm {due north} }}$ are the coordinates. This means that by choosing a coordinate system with abscissa ${\displaystyle \sigma _{\mathrm {due north} }}$ and ordinate ${\displaystyle \tau _{\mathrm {n} }}$ , giving values to the parameter ${\displaystyle \theta }$ will place the points obtained lying on a circle.

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

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

where

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

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

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

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

Sign conventions [edit]

There are 2 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-Circumvolve-space". In improver, within each of the two gear up of sign conventions, the applied science mechanics (structural technology and mechanical engineering) literature follows a dissimilar sign convention from the geomechanics literature. There is no standard sign convention, and the option of a particular sign convention is influenced by convenience for calculation and interpretation for the particular trouble 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 [edit]

From the convention of the Cauchy stress tensor (Figure three 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 management 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 management of the ${\displaystyle y}$ -centrality.

In the physical-infinite sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are inward to the plane of activity (compression) (Figure 5).

In the physical-infinite sign convention, positive shear stresses act on positive faces of the cloth element in the positive direction of an centrality. Also, positive shear stresses human activity on negative faces of the material chemical element in the negative direction of an axis. A positive face has its normal vector in the positive direction 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 human action on positive faces, and they act likewise in the positive direction 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 accept a negative sign because they act in the negative direction of the ${\displaystyle ten}$ -centrality and ${\displaystyle y}$ -axis, respectively.

Mohr-circle-space sign convention [edit]

Effigy 5. Technology mechanics sign convention for drawing the Mohr circle. This article follows sign-convention # three, as shown.

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 aeroplane of action, and negative normal stresses act inwards to the plane of activity.

Shear stresses, however, accept a different convention in the Mohr-circle space compared to the convention in the physical space. In the Mohr-circumvolve-space sign convention, positive shear stresses rotate the textile chemical element in the counterclockwise direction, and negative shear stresses rotate the textile in the clockwise direction. This way, the shear stress component ${\displaystyle \tau _{xy}}$ is positive in the Mohr-circle infinite, and the shear stress component ${\displaystyle \tau _{yx}}$ is negative in the Mohr-circle space.

2 options exist for drawing the Mohr-circle infinite, which produce a mathematically correct Mohr circle:

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

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

To overcome the "consequence" of having the shear stress axis down 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 management and negative shear stresses are assumed to rotate the material element in the counterclockwise direction (Figure 5, option 3). This mode, positive shear stresses are plotted up in the Mohr-circle infinite and the angle ${\displaystyle ii\theta }$ has a positive rotation counterclockwise in the Mohr-circle infinite. This alternative sign convention produces a circumvolve that is identical to the sign convention #2 in Figure 5 because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is besides a counterclockwise shear stress, and both are plotted downwardly. As well, a negative shear stress ${\displaystyle \tau _{\mathrm {northward} }}$ is a clockwise shear stress, and both are plotted upward.

This article follows the engineering science mechanics sign convention for the physical space and the culling sign convention for the Mohr-circle space (sign convention #3 in Figure 5)

Cartoon Mohr'southward circle [edit]

Assuming nosotros know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a betoken ${\displaystyle P}$ in the object nether study, as shown in Effigy four, the following are the steps to construct the Mohr circle for the state of stresses at ${\displaystyle P}$ :

1. Depict the Cartesian coordinate organisation ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {n} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {n} }}$ -axis.
2. Plot two points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {north} },\tau _{\mathrm {due north} })}$ space corresponding to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Figure 4 and 6), following the called sign convention.
3. Draw the diameter of the circumvolve by joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a directly line ${\displaystyle {\overline {AB}}}$ .
4. Depict 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. Instance of how stress components vary on the faces (edges) of a rectangular chemical element equally the angle of its orientation is varied. Master stresses occur when the shear stresses simultaneously disappear from all faces. The orientation at which this occurs gives the principal directions. In this example, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{twenty}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\terminate{matrix}}\right]=\left[{\brainstorm{matrix}-10&x\\10&15\end{matrix}}\right].}$ The respective Mohr's circle representation is shown at the bottom.

The magnitude of the primary stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle E}$ (Figure 6) where the circle intersects the ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality. The magnitude of the major main 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 chief stress ${\displaystyle \sigma _{2}}$ is e'er the lowest absolute value of the abscissa of these two points. Equally expected, the ordinates of these two points are goose egg, corresponding to the magnitude of the shear stress components on the primary planes. Alternatively, the values of the master stresses tin be plant past

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

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

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

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

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

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

Finding maximum and minimum shear stresses [edit]

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

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

Finding stress components on an arbitrary airplane [edit]

Equally mentioned earlier, 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 material point ${\displaystyle P}$ . These stress components human activity in two perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Figure 5 and half dozen. The Mohr circle is used to notice the stress components ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.due east., coordinates of any point ${\displaystyle D}$ on the circle, acting on whatsoever other plane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the aeroplane ${\displaystyle B}$ . For this, two approaches can be used: the double angle, and the Pole or origin of planes.

Double angle [edit]

As shown in Effigy 6, to determine the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ interim on a aeroplane ${\displaystyle D}$ at an bending ${\displaystyle \theta }$ counterclockwise to the plane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an angle ${\displaystyle ii\theta }$ in the same counterclockwise management around the circle from the known stress point ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ to bespeak ${\displaystyle D(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ , i.east., an angle ${\displaystyle 2\theta }$ betwixt lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circle.

The double bending arroyo relies on the fact that the angle ${\displaystyle \theta }$ between the normal vectors to whatever two physical planes passing through ${\displaystyle P}$ (Figure 4) is half the bending between 2 lines joining their corresponding stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {due north} })}$ 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 ${\displaystyle ii\theta }$ . It tin also be seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the material element around ${\displaystyle P}$ of Figure v are separated by an angle ${\displaystyle \theta =90^{\circ }}$ , which in the Mohr circle is represented by a ${\displaystyle 180^{\circ }}$ bending (double the angle).

Pole or origin of planes [edit]

Figure 7. Mohr'southward circumvolve for plane stress and plane strain conditions (Pole approach). Any direct line drawn from the pole will intersect the Mohr circle at a betoken that represents the state of stress on a plane inclined at the same orientation (parallel) in space every bit that line.

The second arroyo involves the conclusion of a point on the Mohr circle chosen the pole or the origin of planes. Whatsoever direct line fatigued from the pole volition 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 ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on any detail aeroplane, ane can draw a line parallel to that aeroplane 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. Every bit an instance, let's assume nosotros have a state of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , as shown on Effigy seven. First, nosotros can draw a line from point ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{10}}$ , or, if we cull otherwise, a line from bespeak ${\displaystyle A}$ parallel to the plane of activeness of ${\displaystyle \sigma _{y}}$ . The intersection of any of these ii lines with the Mohr circumvolve is the pole. In one case the pole has been determined, to find the state of stress on a plane 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 tin can draw a line from the pole parallel to that plane (Encounter Figure seven). The normal and shear stresses on that plane are then the coordinates of the bespeak of intersection between the line and the Mohr circumvolve.

Finding the orientation of the principal planes [edit]

The orientation of the planes where the maximum and minimum master stresses human activity, as well 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 bending ∠BOC betwixt ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OC}}}$ is double the bending ${\displaystyle \theta _{p}}$ which the major primary plane makes with plane ${\displaystyle B}$ .

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

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

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

Example [edit]

Assume a cloth element under a state of stress as shown in Effigy 8 and Effigy ix, 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 activity.
• 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 infinite (Figure five), the stress components for the fabric element in this example are:

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

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

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

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

We and so plot two points A(l,40) and B(-10,-xl), representing the state of stress at plane A and B as prove in both Effigy viii and Figure nine. These points follow the engineering 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 cloth element clockwise. This way, the shear stress interim on airplane B is negative and the shear stress interim on plane A is positive. The diameter of the circle is the line joining point A and B. The middle of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {northward} }}$ -axis. Knowing both the location of the centre and length of the diameter, we are able to plot the Mohr circle for this particular country of stress.

The abscissas of both points East and C (Figure 8 and Figure 9) intersecting the ${\displaystyle \sigma _{\mathrm {n} }}$ -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 interim on both the minor and major main planes, respectively, which is zero for main planes.

Even though the idea for using the Mohr circle is to graphically find unlike stress components by actually measuring the coordinates for different points on the circumvolve, it is more convenient to confirm the results analytically. Thus, the radius and the abscissa of the centre of the circumvolve are

${\displaystyle {\begin{aligned}R&={\sqrt {\left[{\tfrac {1}{2}}(\sigma _{x}-\sigma _{y})\right]^{ii}+\tau _{xy}^{two}}}\\&={\sqrt {\left[{\tfrac {1}{2}}(-10-l)\right]^{ii}+40^{2}}}\\&=50{\textrm {MPa}}\\\terminate{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {1}{2}}(\sigma _{ten}+\sigma _{y})\\&={\tfrac {1}{2}}(-10+50)\\&=twenty{\textrm {MPa}}\\\finish{aligned}}}$

and the principal stresses are

${\displaystyle {\brainstorm{aligned}\sigma _{1}&=\sigma _{\mathrm {avg} }+R\\&=seventy{\textrm {MPa}}\\\end{aligned}}}$

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

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 aforementioned planes where the minimum and maximum shear stresses deed, respectively. The magnitudes of the minimum and maximum shear stresses can be found analytically by

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

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

We tin can cull to either use the double angle approach (Effigy 8) or the Pole approach (Figure 9) to find the orientation of the principal normal stresses and chief shear stresses.

Using the double angle approach we measure the angles ∠BOC and ∠BOE in the Mohr Circumvolve (Effigy 8) to find double the angle the major main stress and the minor principal stress brand with aeroplane B in the physical space. To obtain a more than accurate value for these angles, instead of manually measuring the angles, nosotros can use the analytical expression

${\displaystyle {\begin{aligned}two\theta _{\mathrm {p} }=\arctan {\frac {2\tau _{xy}}{\sigma _{x}-\sigma _{y}}}=\arctan {\frac {2*40}{(-ten-50)}}=-\arctan {\frac {4}{3}}\terminate{aligned}}}$

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

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

Then, the major main angle is

${\displaystyle {\begin{aligned}two\theta _{p1}&=180-53.thirteen^{\circ }=126.87^{\circ }\\\theta _{p1}&=63.435^{\circ }\\\cease{aligned}}}$

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

Using the Pole approach, we outset localize the Pole or origin of planes. For this, we depict through signal A on the Mohr circumvolve a line inclined x° 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 circumvolve (Figure 9). To ostend the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the plane B where ${\displaystyle \sigma _{10'}}$ acts. This line would also intersect the Mohr circle at the Pole (Figure 9).

From the Pole, we depict lines to dissimilar 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 concrete space having the same inclination equally the line. For instance, the line from the Pole to point C in the circle has the same inclination every bit the plane in the physical infinite where ${\displaystyle \sigma _{one}}$ acts. This plane makes an angle of 63.435° with airplane B, both in the Mohr-circle space and in the physical space. In the aforementioned mode, lines are traced from the Pole to points Eastward, D, F, G and H to find the stress components on planes with the same orientation.

Mohr's circumvolve for a general three-dimensional country of stresses [edit]

Figure 10. Mohr's circle for a 3-dimensional land of stress

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

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

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

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

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

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

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

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

${\displaystyle \tau _{\mathrm {northward} }^{2}+(\sigma _{\mathrm {northward} }-\sigma _{3})(\sigma _{\mathrm {northward} }-\sigma _{1})\leq 0}$ as the denominator ${\displaystyle \sigma _{2}-\sigma _{three}>0}$ and ${\displaystyle \sigma _{ii}-\sigma _{i}<0}$

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

These expressions tin be rewritten as

${\displaystyle {\brainstorm{aligned}\tau _{\mathrm {n} }^{2}+\left[\sigma _{\mathrm {n} }-{\tfrac {1}{two}}(\sigma _{2}+\sigma _{3})\right]^{2}\geq \left({\tfrac {one}{2}}(\sigma _{2}-\sigma _{three})\right)^{two}\\\tau _{\mathrm {n} }^{2}+\left[\sigma _{\mathrm {n} }-{\tfrac {one}{two}}(\sigma _{one}+\sigma _{3})\right]^{2}\leq \left({\tfrac {1}{2}}(\sigma _{1}-\sigma _{iii})\right)^{2}\\\tau _{\mathrm {n} }^{2}+\left[\sigma _{\mathrm {n} }-{\tfrac {one}{2}}(\sigma _{1}+\sigma _{2})\right]^{2}\geq \left({\tfrac {ane}{two}}(\sigma _{i}-\sigma _{2})\right)^{2}\\\stop{aligned}}}$

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

These equations for the Mohr circles show that all admissible stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ lie on these circles or inside the shaded expanse enclosed by them (see Figure ten). Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{ane}}$ lie on, or exterior circle ${\displaystyle C_{one}}$ . Stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{2}}$ lie on, or inside circle ${\displaystyle C_{2}}$ . And finally, stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{three}}$ lie on, or outside circumvolve ${\displaystyle C_{3}}$ .

Meet also [edit]

• Critical airplane assay

References [edit]

1. ^ "Principal stress and master aeroplane". www.engineeringapps.internet . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. ane–30. ISBN0-415-27297-1.
3. ^ Gere, James Yard. (2013). Mechanics of Materials. Goodno, Barry J. (8th ed.). Stamford, CT: Cengage Learning. ISBN9781111577735.

Bibliography [edit]

• Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials . McGraw-Colina Professional. ISBN0-07-112939-1.
• Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Cloak-and-dagger Mining (3rd 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. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, North.Yard.Due west.; Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. 9–41. ISBN978-0-632-05759-vii.
• 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 (two ed.). Taylor & Francis. pp. 1–30. ISBN0-415-27297-1.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (3rd ed.). McGraw-Hill 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 circles by Rebecca Brannon
• DoITPoMS Teaching and Learning Parcel- "Stress Analysis and Mohr'due south Circle"

stapletonlonot1975.blogspot.com

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

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