Draw a 10 Degree Wedge on Circle in Solid Work

Geometric civil applied science calculation technique

Effigy 1. Mohr'southward circles for a three-dimensional state of stress

Mohr's circumvolve is a ii-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical technology for strength of soils, and structural applied science for forcefulness of built structures. Information technology is as well used for calculating stresses in many planes by reducing them to vertical and horizontal components. These are called primary planes in which principal stresses are calculated; Mohr'due south circle can also be used to find the chief planes and the principal stresses in a graphical representation, and is one of the easiest ways to do and then.^[i]

After performing a stress analysis on a textile body assumed as a continuum, the components of the Cauchy stress tensor at a particular cloth 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., 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, acting 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 stand for the principal axes of the stress element.

19th-century German language engineer Karl Culmann was the first to excogitate a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. His work inspired fellow German language engineer Christian Otto Mohr (the circle's namesake), who extended information technology to both ii- and three-dimensional stresses and developed a failure benchmark based on the stress circle.^[2]

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

The Mohr circumvolve can exist applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.

Motivation [edit]

Figure two. Stress in a loaded deformable material body assumed equally a continuum.

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

In engineering, due east.1000., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a stone mass around a tunnel, airplane wings, or building columns, is determined through a stress analysis. Computing the stress distribution implies the decision of stresses at every point (material particle) in the object. According to Cauchy, the stress at any point in an object (Figure 2), assumed as a continuum, is completely defined by the 9 stress components ${\displaystyle \sigma _{ij}}$ of a second order tensor of blazon (2,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

Figure 3. Stress transformation at a signal in a continuum nether plane stress conditions.

Later the stress distribution within the object has been determined with respect to a coordinate arrangement ${\displaystyle (x,y)}$ , it may be necessary to summate the components of the stress tensor at a item fabric betoken ${\displaystyle P}$ with respect to a rotated coordinate organization ${\displaystyle (x',y')}$ , i.e., the stresses acting on a plane with a different orientation passing through that betoken of interest —forming an angle with the coordinate arrangement ${\displaystyle (x,y)}$ (Effigy iii). For example, it is of involvement to find the maximum normal stress and maximum shear stress, as well equally 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 organization. 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 two-dimensional land of stress [edit]

Effigy 4. Stress components at a airplane passing through a point in a continuum nether plane stress weather condition.

In ii dimensions, the stress tensor at a given cloth bespeak ${\displaystyle P}$ with respect to any two perpendicular directions is completely defined by only iii stress components. For the particular 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 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 tin be written as:

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

The objective is to use the Mohr circle to discover the stress components ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on a rotated coordinate organisation ${\displaystyle (x',y')}$ , i.e., on a differently oriented plane passing through ${\displaystyle P}$ and perpendicular to the ${\displaystyle x}$ - ${\displaystyle y}$ plane (Effigy 4). 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 2-dimensional cases of airplane stress and airplane strain, first consider a two-dimensional minute material element around a material signal ${\displaystyle P}$ (Figure iv), with a unit surface area in the direction parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ airplane, 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 ${\displaystyle \tau _{\mathrm {northward} }}$ are given by:

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

${\displaystyle \tau _{\mathrm {n} }=-{\frac {1}{2}}(\sigma _{10}-\sigma _{y})\sin two\theta +\tau _{xy}\cos two\theta }$

Derivation of Mohr'due south circle parametric equations - Equilibrium of forces

From equilibrium of forces in the direction of ${\displaystyle \sigma _{\mathrm {northward} }}$ ( ${\displaystyle x'}$ -axis) (Figure 4), and knowing that the area of the airplane where ${\displaystyle \sigma _{\mathrm {n} }}$ acts is ${\displaystyle dA}$ , we have: ${\displaystyle \ {\begin{aligned}\sum F_{x'}&=\sigma _{\mathrm {n} }dA-\sigma _{x}dA\cos ^{ii}\theta -\sigma _{y}dA\sin ^{two}\theta -\tau _{xy}dA\cos \theta \sin \theta -\tau _{xy}dA\sin \theta \cos \theta =0\\\sigma _{\mathrm {n} }&=\sigma _{10}\cos ^{2}\theta +\sigma _{y}\sin ^{ii}\theta +two\tau _{xy}\sin \theta \cos \theta \\\terminate{aligned}}}$ However, knowing that ${\displaystyle \cos ^{2}\theta ={\frac {one+\cos 2\theta }{two}},\qquad \sin ^{2}\theta ={\frac {1-\cos two\theta }{ii}}\qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \sigma _{\mathrm {northward} }={\frac {ane}{ii}}(\sigma _{10}+\sigma _{y})+{\frac {1}{ii}}(\sigma _{x}-\sigma _{y})\cos two\theta +\tau _{xy}\sin 2\theta }$ Now, from equilibrium of forces in the management of ${\displaystyle \tau _{\mathrm {n} }}$ ( ${\displaystyle y'}$ -axis) (Effigy iv), and knowing that the area of the plane where ${\displaystyle \tau _{\mathrm {n} }}$ acts is ${\displaystyle dA}$ , we accept: ${\displaystyle \ {\begin{aligned}\sum F_{y'}&=\tau _{\mathrm {north} }dA+\sigma _{x}dA\cos \theta \sin \theta -\sigma _{y}dA\sin \theta \cos \theta -\tau _{xy}dA\cos ^{two}\theta +\tau _{xy}dA\sin ^{2}\theta =0\\\tau _{\mathrm {northward} }&=-(\sigma _{10}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{ii}\theta -\sin ^{2}\theta \correct)\\\end{aligned}}}$ However, knowing that ${\displaystyle \cos ^{two}\theta -\sin ^{two}\theta =\cos 2\theta \qquad {\text{and}}\qquad \sin two\theta =two\sin \theta \cos \theta }$ we obtain ${\displaystyle \tau _{\mathrm {north} }=-{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\sin ii\theta +\tau _{xy}\cos 2\theta }$

Both equations tin 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 ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ .

Derivation of Mohr's circle parametric equations - Tensor transformation

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

These 2 equations are the parametric equations of the Mohr circumvolve. 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 system 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 circle.

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

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

where

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

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

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

with radius ${\displaystyle r=R}$ centered at a point with coordinates ${\displaystyle (a,b)=(\sigma _{\mathrm {avg} },0)}$ in the ${\displaystyle (\sigma _{\mathrm {due north} },\tau _{\mathrm {n} })}$ 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 addition, within each of the two set of sign conventions, the applied science mechanics (structural engineering and mechanical applied science) literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the selection of a particular sign convention is influenced by convenience for calculation and interpretation for the detail problem in hand. A more detailed explanation of these sign conventions is presented below.

The previous derivation for the equation of the Mohr Circle using Figure 4 follows the engineering mechanics sign convention. The engineering science mechanics sign convention volition exist used for this article.

Concrete-space sign convention [edit]

From the convention of the Cauchy stress tensor (Effigy 3 and Effigy four), the outset 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 acting on the face with normal vector in the positive direction of the ${\displaystyle ten}$ -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 inward to the plane of action (compression) (Figure v).

In the concrete-infinite sign convention, positive shear stresses act on positive faces of the material element in the positive direction of an axis. Also, positive shear stresses act on negative faces of the material element in the negative direction of an axis. A positive face has its normal vector in the positive direction of an axis, and a negative face up has its normal vector in the negative direction of an centrality. For example, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive because they act on positive faces, and they act as well in the positive management of the ${\displaystyle y}$ -axis and the ${\displaystyle x}$ -centrality, respectively (Figure 3). Similarly, the respective opposite shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ interim in the negative faces have a negative sign considering they act in the negative management of the ${\displaystyle x}$ -axis and ${\displaystyle y}$ -centrality, respectively.

Mohr-circle-space sign convention [edit]

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

In the Mohr-circle-infinite sign convention, normal stresses have the same sign every bit normal stresses in the physical-space sign convention: positive normal stresses act outward to the plane of action, and negative normal stresses human activity inward to the plane of activity.

Shear stresses, all the same, have a different convention in the Mohr-circle space compared to the convention in the physical infinite. In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise management, 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-circumvolve infinite, and the shear stress component ${\displaystyle \tau _{yx}}$ is negative in the Mohr-circle space.

Ii options exist for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle:

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

Plotting positive shear stresses upward 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 angle ${\displaystyle two\theta }$ on the Mohr circle take a positive rotation counterclockwise, like to the concrete space convention for shear stresses.

To overcome the "issue" 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 cloth element in the counterclockwise management (Effigy v, selection 3). This way, positive shear stresses are plotted upwards in the Mohr-circle infinite 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 ${\displaystyle \tau _{\mathrm {due north} }}$ is also a counterclockwise shear stress, and both are plotted downward. Besides, a negative shear stress ${\displaystyle \tau _{\mathrm {north} }}$ is a clockwise shear stress, and both are plotted upward.

This article follows the engineering mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circumvolve space (sign convention #3 in Effigy 5)

Drawing Mohr's circle [edit]

Assuming we know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a point ${\displaystyle P}$ in the object under study, as shown in Figure four, the post-obit are the steps to construct the Mohr circle for the land of stresses at ${\displaystyle P}$ :

1. Draw the Cartesian coordinate organization ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {n} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {n} }}$ -centrality.
2. Plot 2 points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {north} },\tau _{\mathrm {n} })}$ 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. Describe the diameter of the circle by joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a straight line ${\displaystyle {\overline {AB}}}$ .
4. Draw the Mohr Circumvolve. The eye ${\displaystyle O}$ of the circle is the midpoint of the diameter line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ axis.

Finding principal normal stresses [edit]

Stress components on a 2D rotating chemical element. Example of how stress components vary on the faces (edges) of a rectangular chemical element as the bending 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 main directions. In this example, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{xx}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\correct]=\left[{\begin{matrix}-10&x\\ten&15\end{matrix}}\correct].}$ The corresponding 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 half dozen) where the circle intersects the ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality. The magnitude of the major master stress ${\displaystyle \sigma _{1}}$ is ever the greatest accented value of the abscissa of any of these two points. Likewise, the magnitude of the pocket-sized principal stress ${\displaystyle \sigma _{2}}$ is always the lowest absolute value of the abscissa of these two points. As 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 primary stresses tin can be plant by

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

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

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

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

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

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

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 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 capricious aeroplane [edit]

Equally mentioned earlier, after the ii-dimensional stress assay has been performed we 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 Effigy 5 and vi. The Mohr circle is used to find the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.east., coordinates of any signal ${\displaystyle D}$ on the circle, acting on any other plane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the aeroplane ${\displaystyle B}$ . For this, two approaches tin can be used: the double bending, and the Pole or origin of planes.

Double angle [edit]

Every bit shown in Figure half-dozen, to decide the stress components ${\displaystyle (\sigma _{\mathrm {north} },\tau _{\mathrm {n} })}$ acting on a airplane ${\displaystyle D}$ 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 effectually the circle from the known stress point ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ to betoken ${\displaystyle D(\sigma _{\mathrm {north} },\tau _{\mathrm {n} })}$ , i.eastward., an angle ${\displaystyle 2\theta }$ betwixt lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circumvolve.

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

This double bending relation comes from the fact that the parametric equations for the Mohr circle are a function of ${\displaystyle 2\theta }$ . It can likewise exist seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the material element around ${\displaystyle P}$ of Figure 5 are separated past 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 plane strain conditions (Pole approach). Whatever straight line fatigued from the pole will intersect the Mohr circumvolve at a point that represents the land of stress on a plane inclined at the same orientation (parallel) in space equally that line.

The second approach involves the determination of a signal on the Mohr circumvolve called the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a signal that represents the state of stress on a plane inclined at the aforementioned orientation (parallel) in space as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on any detail airplane, one can draw a line parallel to that airplane through the particular coordinates ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on the Mohr circle and find the pole as the intersection of such line with the Mohr circle. As an case, allow'due south assume we take a state of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , as shown on Figure 7. First, we can draw a line from point ${\displaystyle B}$ parallel to the airplane of activity of ${\displaystyle \sigma _{x}}$ , or, if we choose otherwise, a line from point ${\displaystyle A}$ parallel to the aeroplane of activity of ${\displaystyle \sigma _{y}}$ . The intersection of whatever of these two lines with the Mohr circumvolve is the pole. One time the pole has been adamant, to find the state of stress on a plane making an angle ${\displaystyle \theta }$ with the vertical, or in other words a aeroplane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, and so we tin can describe a line from the pole parallel to that plane (See Figure 7). The normal and shear stresses on that airplane are so the coordinates of the bespeak of intersection between the line and the Mohr circle.

Finding the orientation of the principal planes [edit]

The orientation of the planes where the maximum and minimum principal stresses act, also known every bit main planes, can 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 bending ${\displaystyle \theta _{p}}$ which the major principal 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 ${\displaystyle 90^{\circ }}$ apart (Figure). This equation tin can be derived directly from the geometry of the circle, or by making the parametric equation of the circle for ${\displaystyle \tau _{\mathrm {n} }}$ equal to zero (the shear stress in the chief planes is always goose egg).

Example [edit]

Assume a fabric chemical element under a state of stress as shown in Figure 8 and Figure 9, with the plane of 1 of its sides oriented ten° with respect to the horizontal plane. Using the Mohr circle, observe:

• The orientation of their planes of activeness.
• The maximum shear stresses and orientation of their planes of activeness.
• The stress components on a horizontal airplane.

Check the answers using the stress transformation formulas or the stress transformation law.

Solution: Following the applied science mechanics sign convention for the concrete space (Figure 5), the stress components for the material chemical element in this instance are:

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

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

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

Following the steps for drawing the Mohr circle for this particular state of stress, nosotros first depict a Cartesian coordinate arrangement ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ with the ${\displaystyle \tau _{\mathrm {n} }}$ -centrality upwards.

Nosotros and then plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as testify in both Figure 8 and Figure 9. These points follow the technology mechanics sign convention for the Mohr-circle space (Effigy 5), which assumes positive normals stresses outward from the textile chemical element, and positive shear stresses on each airplane rotating the fabric chemical element clockwise. This way, the shear stress acting on plane B is negative and the shear stress acting on aeroplane A is positive. The diameter of the circle is the line joining indicate A and B. The center of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {north} }}$ -centrality. Knowing both the location of the centre and length of the bore, nosotros are able to plot the Mohr circle for this particular land of stress.

The abscissas of both points East and C (Figure 8 and Effigy 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 acting on both the minor and major principal planes, respectively, which is zippo for principal planes.

Fifty-fifty though the idea for using the Mohr circle is to graphically find different stress components by really 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 middle of the circle are

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

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

and the main stresses are

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

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

The coordinates for both points H and G (Figure 8 and Figure 9) are the magnitudes of the minimum and maximum shear stresses, respectively; the abscissas for both points H and G are the magnitudes for the normal stresses acting on the same planes where the minimum and maximum shear stresses deed, respectively. The magnitudes of the minimum and maximum shear stresses can be found analytically past

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

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

Nosotros can choose to either use the double angle approach (Figure 8) or the Pole approach (Figure nine) to find the orientation of the principal normal stresses and principal shear stresses.

Using the double angle arroyo we mensurate the angles ∠BOC and ∠BOE in the Mohr Circle (Figure 8) to find double the bending the major principal stress and the pocket-size main stress make 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

${\displaystyle {\begin{aligned}2\theta _{\mathrm {p} }=\arctan {\frac {ii\tau _{xy}}{\sigma _{x}-\sigma _{y}}}=\arctan {\frac {2*40}{(-10-fifty)}}=-\arctan {\frac {iv}{iii}}\end{aligned}}}$

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

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

Then, the major principal angle is

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

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

Using the Pole approach, we first localize the Pole or origin of planes. For this, nosotros draw through point 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 circle (Effigy 9). To confirm the location of the Pole, we could draw a line through betoken B on the Mohr circle 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 draw lines to different points on the Mohr circumvolve. The coordinates of the points where these lines intersect the Mohr circle signal the stress components interim on a aeroplane in the physical space having the same inclination every bit the line. For case, the line from the Pole to indicate C in the circle has the same inclination as the plane in the physical space where ${\displaystyle \sigma _{i}}$ acts. This plane makes an angle of 63.435° with plane B, both in the Mohr-circle space and in the physical space. In the same way, lines are traced from the Pole to points E, D, F, G and H to find the stress components on planes with the same orientation.

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

Figure 10. Mohr'south circle for a three-dimensional state 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 ${\displaystyle \left(n_{ane},n_{2},n_{iii}\right)}$ must be first evaluated.

Considering the principal axes as the coordinate system, instead of the general ${\displaystyle x_{i}}$ , ${\displaystyle x_{2}}$ , ${\displaystyle x_{3}}$ coordinate system, and bold that ${\displaystyle \sigma _{i}>\sigma _{ii}>\sigma _{iii}}$ , so the normal and shear components of the stress vector ${\displaystyle \mathbf {T} ^{(\mathbf {n} )}}$ , for a given airplane with unit vector ${\displaystyle \mathbf {n} }$ , satisfy the post-obit equations

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

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

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

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

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

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

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

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

These expressions can be rewritten as

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

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

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

See besides [edit]

• Critical plane analysis

References [edit]

1. ^ "Master stress and principal plane". www.engineeringapps.cyberspace . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Greyness (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. one–30. ISBN0-415-27297-1.
3. ^ Gere, James One thousand. (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-Loma Professional. ISBN0-07-112939-1.
• Brady, B.H.Thousand.; Due east.T. Dark-brown (1993). Rock Mechanics For Underground Mining (Tertiary ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Printing. pp. 16–26. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical technology. Prentice-Hall civil engineering science and engineering mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, North.Chiliad.W.; Zimmerman, R.W. (2007). Fundamentals of stone mechanics (Fourth ed.). Wiley-Blackwell. pp. ix–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 Gray (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. i–xxx. ISBN0-415-27297-1.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (3rd ed.). McGraw-Colina 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'southward Circle and more circles by Rebecca Brannon
• DoITPoMS Educational activity and Learning Bundle- "Stress Analysis and Mohr'due south Circle"

kovacsthiss1950.blogspot.com

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

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