Draw Critical Element in Mohrs Circle

Geometric civil engineering calculation technique

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

Mohr's circle is a 2-dimensional graphical representation of the transformation constabulary for the Cauchy stress tensor.

Mohr'due south circle is often used in calculations relating to mechanical technology for materials' forcefulness, geotechnical engineering for strength of soils, and structural engineering for strength 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 principal planes in which principal stresses are calculated; Mohr's circumvolve can besides exist used to discover the principal planes and the principal stresses in a graphical representation, and is one of the easiest ways to do and so.^[1]

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

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {n} }}$ , ${\displaystyle \tau _{\mathrm {north} }}$ ) of each betoken on the circle are the magnitudes of the normal stress and shear stress components, respectively, interim on the rotated coordinate organization. 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 represent the main axes of the stress chemical 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 angle. His work inspired fellow German engineer Christian Otto Mohr (the circle's namesake), who extended information technology to both two- and iii-dimensional stresses and adult a failure benchmark based on the stress circumvolve.^[2]

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

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

Motivation [edit]

Figure ii. Stress in a loaded deformable material body causeless as a continuum.

Internal forces are produced between the particles of a deformable object, assumed equally a continuum, as a reaction to applied external forces, i.e., 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 motion for a particle. A measure of the intensity of these internal forces is called stress. Because the object is assumed as a continuum, these internal forces are distributed continuously inside the volume of the object.

In engineering, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, plane wings, or edifice columns, is adamant through a stress analysis. Calculating the stress distribution implies the decision of stresses at every signal (textile particle) in the object. Co-ordinate to Cauchy, the stress at any point in an object (Figure 2), assumed as a continuum, is completely defined past the nine stress components ${\displaystyle \sigma _{ij}}$ of a second gild tensor of type (two,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

Figure 3. Stress transformation at a point in a continuum under airplane stress atmospheric condition.

Later on the stress distribution inside the object has been adamant with respect to a coordinate system ${\displaystyle (x,y)}$ , it may be necessary to summate the components of the stress tensor at a detail cloth point ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (x',y')}$ , i.e., the stresses acting on a plane with a dissimilar orientation passing through that signal of interest —forming an angle with the coordinate system ${\displaystyle (x,y)}$ (Effigy 3). For instance, it is of involvement to find the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To achieve this, information technology 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 police force. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circumvolve for stress.

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

Figure iv. Stress components at a plane passing through a indicate in a continuum under aeroplane stress conditions.

In 2 dimensions, the stress tensor at a given fabric point ${\displaystyle P}$ with respect to any two perpendicular directions is completely defined by but three 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 residue 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 can be written as:

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

The objective is to use the Mohr circle to find the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on a rotated coordinate system ${\displaystyle (ten',y')}$ , i.e., on a differently oriented plane passing through ${\displaystyle P}$ and perpendicular to the ${\displaystyle x}$ - ${\displaystyle y}$ plane (Figure 4). The rotated coordinate system ${\displaystyle (x',y')}$ makes an angle ${\displaystyle \theta }$ with the original coordinate system ${\displaystyle (10,y)}$ .

Equation of the Mohr circumvolve [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 fabric element around a material indicate ${\displaystyle P}$ (Figure 4), with a unit of measurement 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 ${\displaystyle \sigma _{\mathrm {due north} }}$ and the shear stress ${\displaystyle \tau _{\mathrm {n} }}$ are given by:

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

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

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

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

Both equations tin can likewise exist obtained by applying the tensor transformation constabulary 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 ${\displaystyle \tau _{\mathrm {north} }}$ .

Derivation of Mohr's circle parametric equations - Tensor transformation

The stress tensor transformation police force can exist stated as ${\displaystyle {\brainstorm{aligned}{\boldsymbol {\sigma }}'&=\mathbf {A} {\boldsymbol {\sigma }}\mathbf {A} ^{T}\\\left[{\brainstorm{matrix}\sigma _{x'}&\tau _{x'y'}\\\tau _{y'x'}&\sigma _{y'}\\\end{matrix}}\right]&=\left[{\begin{matrix}a_{x}&a_{xy}\\a_{yx}&a_{y}\\\terminate{matrix}}\right]\left[{\begin{matrix}\sigma _{x}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\correct]\left[{\brainstorm{matrix}a_{ten}&a_{yx}\\a_{xy}&a_{y}\\\terminate{matrix}}\right]\\&=\left[{\brainstorm{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\cease{matrix}}\right]\left[{\begin{matrix}\sigma _{x}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\terminate{matrix}}\right]\left[{\brainstorm{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\terminate{matrix}}\right]\end{aligned}}}$ Expanding the right hand side, and knowing that ${\displaystyle \sigma _{ten'}=\sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{10'y'}=\tau _{\mathrm {n} }}$ , nosotros 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 ${\displaystyle \cos ^{ii}\theta ={\frac {one+\cos 2\theta }{2}},\qquad \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}\qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \sigma _{\mathrm {n} }={\frac {ane}{2}}(\sigma _{x}+\sigma _{y})+{\frac {one}{2}}(\sigma _{x}-\sigma _{y})\cos two\theta +\tau _{xy}\sin 2\theta }$ ${\displaystyle \tau _{\mathrm {northward} }=-(\sigma _{10}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{2}\theta -\sin ^{ii}\theta \right)}$ However, knowing that ${\displaystyle \cos ^{2}\theta -\sin ^{2}\theta =\cos two\theta \qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \tau _{\mathrm {n} }=-{\frac {1}{two}}(\sigma _{x}-\sigma _{y})\sin two\theta +\tau _{xy}\cos 2\theta }$ It is non necessary at this moment to calculate the stress component ${\displaystyle \sigma _{y'}}$ interim on the airplane perpendicular to the airplane of action of ${\displaystyle \sigma _{10'}}$ as information technology is non required for deriving the equation for the Mohr circumvolve.

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

Eliminating the parameter ${\displaystyle 2\theta }$ from these parametric equations will yield the non-parametric equation of the Mohr circle. This tin exist achieved past rearranging the equations for ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {northward} }}$ , first transposing the first term in the first equation and squaring both sides of each of the equations and so adding them. Thus we have

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

where

${\displaystyle R={\sqrt {\left[{\tfrac {1}{2}}(\sigma _{10}-\sigma _{y})\correct]^{two}+\tau _{xy}^{ii}}}\quad {\text{and}}\quad \sigma _{\mathrm {avg} }={\tfrac {1}{2}}(\sigma _{10}+\sigma _{y})}$

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

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

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

Sign conventions [edit]

In that location are ii divide sets of sign conventions that need to be considered when using the Mohr Circumvolve: I sign convention for stress components in the "concrete space", and another for stress components in the "Mohr-Circumvolve-space". In add-on, within each of the two set of sign conventions, the applied science mechanics (structural technology and mechanical applied science) literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the choice of a particular sign convention is influenced by convenience for adding and interpretation for the particular trouble in paw. A more detailed caption of these sign conventions is presented below.

The previous derivation for the equation of the Mohr Circle using Effigy 4 follows the engineering mechanics sign convention. The technology mechanics sign convention will be used for this article.

Physical-space sign convention [edit]

From the convention of the Cauchy stress tensor (Figure 3 and Figure 4), the first subscript in the stress components denotes the face on which the stress component acts, and the second subscript indicates the direction of the stress component. Thus ${\displaystyle \tau _{xy}}$ is the shear stress interim on the confront with normal vector in the positive direction of the ${\displaystyle x}$ -centrality, 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 inwards to the plane of action (compression) (Figure v).

In the physical-space sign convention, positive shear stresses act on positive faces of the textile chemical element in the positive direction of an axis. Also, positive shear stresses act on negative faces of the material element in the negative management of an centrality. A positive face has its normal vector in the positive direction of an axis, and a negative face has its normal vector in the negative direction of an axis. For case, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive considering they act on positive faces, and they human action as well in the positive direction of the ${\displaystyle y}$ -axis and the ${\displaystyle 10}$ -centrality, respectively (Figure 3). Similarly, the respective reverse shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ acting in the negative faces have a negative sign because they human action in the negative management of the ${\displaystyle x}$ -axis and ${\displaystyle y}$ -centrality, respectively.

Mohr-circumvolve-space sign convention [edit]

Figure five. Applied science mechanics sign convention for cartoon the Mohr circle. This article follows sign-convention # 3, as shown.

In the Mohr-circumvolve-infinite sign convention, normal stresses take the aforementioned sign as normal stresses in the physical-space sign convention: positive normal stresses act outward to the plane of action, and negative normal stresses act in to the aeroplane of action.

Shear stresses, nevertheless, accept a different convention in the Mohr-circle space compared to the convention in the physical infinite. In the Mohr-circumvolve-space sign convention, positive shear stresses rotate the material chemical element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise management. 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-circumvolve infinite.

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

1. Positive shear stresses are plotted upward (Figure 5, sign convention #1)
2. Positive shear stresses are plotted downward, i.e., the ${\displaystyle \tau _{\mathrm {n} }}$ -axis is inverted (Figure five, 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^[three] adopt plotting positive shear stresses downward, which makes the angle ${\displaystyle 2\theta }$ on the Mohr circle have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.

To overcome the "issue" of having the shear stress axis down in the Mohr-circumvolve space, at that place is an alternative sign convention where positive shear stresses are causeless to rotate the fabric element in the clockwise management and negative shear stresses are assumed to rotate the material chemical element in the counterclockwise management (Figure 5, option 3). This way, positive shear stresses are plotted up in the Mohr-circle space and the bending ${\displaystyle two\theta }$ has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #ii in Figure 5 because a positive shear stress ${\displaystyle \tau _{\mathrm {north} }}$ is also a counterclockwise shear stress, and both are plotted downward. Also, a negative shear stress ${\displaystyle \tau _{\mathrm {northward} }}$ is a clockwise shear stress, and both are plotted upward.

This commodity follows the engineering mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circle infinite (sign convention #3 in Figure five)

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 under study, as shown in Figure 4, the following are the steps to construct the Mohr circle for the state of stresses at ${\displaystyle P}$ :

1. Draw the Cartesian coordinate system ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {n} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {north} }}$ -axis.
2. Plot 2 points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{10},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {northward} },\tau _{\mathrm {n} })}$ infinite corresponding to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Figure 4 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 ${\displaystyle \sigma _{\mathrm {n} }}$ axis.

Finding principal normal stresses [edit]

Stress components on a 2nd rotating element. Example of how stress components vary on the faces (edges) of a rectangular chemical element equally the angle of its orientation is varied. Primary 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 past ${\displaystyle \left[{\brainstorm{matrix}\sigma _{twenty}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\right]=\left[{\brainstorm{matrix}-x&ten\\10&15\end{matrix}}\correct].}$ The corresponding Mohr's circle representation is shown at the bottom.

The magnitude of the principal stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle Due east}$ (Figure 6) where the circumvolve intersects the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis. The magnitude of the major primary stress ${\displaystyle \sigma _{1}}$ is ever the greatest absolute value of the abscissa of any of these two points. Likewise, the magnitude of the modest master stress ${\displaystyle \sigma _{two}}$ is always the everyman absolute value of the abscissa of these ii points. As expected, the ordinates of these two points are goose egg, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the primary stresses tin exist found by

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

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

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

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

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

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

Finding maximum and minimum shear stresses [edit]

The maximum and minimum shear stresses correspond to the ordinates of the highest and everyman points on the circle, respectively. These points are located at the intersection of the circle with the vertical line passing through the center of the circle, ${\displaystyle O}$ . 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]

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

Double angle [edit]

As shown in Effigy 6, to decide the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ acting on a plane ${\displaystyle D}$ at an angle ${\displaystyle \theta }$ counterclockwise to the plane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an angle ${\displaystyle 2\theta }$ in the same counterclockwise direction around the circle from the known stress point ${\displaystyle B(\sigma _{ten},-\tau _{xy})}$ to point ${\displaystyle D(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ , i.eastward., an angle ${\displaystyle ii\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circumvolve.

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

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

Pole or origin of planes [edit]

Figure vii. Mohr'south circle for plane stress and plane strain atmospheric condition (Pole approach). Whatsoever straight line drawn from the pole will intersect the Mohr circumvolve at a point that represents the state of stress on a aeroplane inclined at the same orientation (parallel) in space as that line.

The second arroyo involves the determination of a point on the Mohr circle called the pole or the origin of planes. Whatsoever straight line drawn from the pole will intersect the Mohr circumvolve at a indicate that represents the land 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 particular plane, 1 tin can draw a line parallel to that airplane through the item coordinates ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on the Mohr circle and observe the pole as the intersection of such line with the Mohr circumvolve. As an instance, let's assume we have a state of stress with stress components ${\displaystyle \sigma _{10},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , every bit shown on Effigy 7. First, we can draw a line from bespeak ${\displaystyle B}$ parallel to the aeroplane of action of ${\displaystyle \sigma _{x}}$ , or, if we 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. Once the pole has been determined, to observe 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 aeroplane, then nosotros can draw a line from the pole parallel to that plane (Encounter Figure vii). 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 master planes [edit]

The orientation of the planes where the maximum and minimum principal stresses human activity, too known as main planes, can exist adamant past 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 plane ${\displaystyle B}$ .

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

${\displaystyle \tan two\theta _{\mathrm {p} }={\frac {2\tau _{xy}}{\sigma _{y}-\sigma _{10}}}}$

This equation defines 2 values for ${\displaystyle \theta _{\mathrm {p} }}$ which are ${\displaystyle 90^{\circ }}$ apart (Figure). This equation can be derived straight from the geometry of the circle, or by making the parametric equation of the circumvolve for ${\displaystyle \tau _{\mathrm {n} }}$ equal to zilch (the shear stress in the principal planes is always zero).

Example [edit]

Assume a material element under a state of stress equally shown in Figure 8 and Figure 9, with the plane of i of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circle, find:

• The orientation of their planes of action.
• The maximum shear stresses and orientation of their planes of activity.
• The stress components on a horizontal plane.

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

Solution: Following the technology mechanics sign convention for the physical space (Figure five), the stress components for the fabric chemical element in this example are:

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

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

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

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

We and so plot two points A(50,forty) and B(-x,-40), representing the state of stress at airplane A and B every bit show in both Effigy eight and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circumvolve space (Figure 5), which assumes positive normals stresses outward from the textile element, and positive shear stresses on each plane rotating the material chemical element clockwise. This fashion, the shear stress interim on plane B is negative and the shear stress acting on airplane A is positive. The diameter of the circle is the line joining indicate A and B. The centre 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, we are able to plot the Mohr circle for this particular country of stress.

The abscissas of both points E and C (Figure 8 and Effigy ix) intersecting the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points Eastward and C are the magnitudes of the shear stresses interim on both the minor and major main planes, respectively, which is zippo for principal planes.

Fifty-fifty though the idea for using the Mohr circumvolve is to graphically find different stress components by actually measuring the coordinates for dissimilar 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

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

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {1}{ii}}(\sigma _{10}+\sigma _{y})\\&={\tfrac {ane}{2}}(-x+l)\\&=20{\textrm {MPa}}\\\stop{aligned}}}$

and the principal stresses are

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

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

The coordinates for both points H and Yard (Figure 8 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 aforementioned planes where the minimum and maximum shear stresses act, respectively. The magnitudes of the minimum and maximum shear stresses can exist constitute analytically past

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

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

We can choose to either apply the double angle approach (Figure 8) or the Pole approach (Effigy 9) to notice the orientation of the principal normal stresses and master shear stresses.

Using the double angle arroyo we mensurate the angles ∠BOC and ∠BOE in the Mohr Circle (Figure viii) to observe double the angle the major main stress and the minor 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, nosotros tin use the analytical expression

${\displaystyle {\begin{aligned}ii\theta _{\mathrm {p} }=\arctan {\frac {2\tau _{xy}}{\sigma _{x}-\sigma _{y}}}=\arctan {\frac {2*40}{(-10-50)}}=-\arctan {\frac {4}{3}}\cease{aligned}}}$

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

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

Then, the major primary angle is

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

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

Using the Pole approach, we starting time localize the Pole or origin of planes. For this, nosotros describe through point A on the Mohr circle a line inclined 10° with the horizontal, or, in other words, a line parallel to plane A where ${\displaystyle \sigma _{y'}}$ acts. The Pole is where this line intersects the Mohr circle (Figure 9). To confirm the location of the Pole, we could describe a line through point B on the Mohr circle parallel to the plane B where ${\displaystyle \sigma _{x'}}$ acts. This line would also intersect the Mohr circle at the Pole (Figure 9).

From the Pole, we describe 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 airplane in the physical space having the aforementioned inclination every bit the line. For instance, the line from the Pole to indicate C in the circumvolve has the same inclination as the plane in the physical infinite where ${\displaystyle \sigma _{1}}$ acts. This plane makes an bending of 63.435° with aeroplane B, both in the Mohr-circle space and in the physical infinite. In the same way, lines are traced from the Pole to points Due east, D, F, Thou and H to find the stress components on planes with the same orientation.

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

Effigy 10. Mohr's circle for a three-dimensional state of stress

To construct the Mohr circumvolve for a general three-dimensional case of stresses at a betoken, the values of the principal stresses ${\displaystyle \left(\sigma _{1},\sigma _{two},\sigma _{three}\right)}$ and their principal directions ${\displaystyle \left(n_{1},n_{ii},n_{iii}\correct)}$ must exist get-go evaluated.

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

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

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

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

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

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

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

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

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

These expressions can be rewritten as

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

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

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

See also [edit]

• Critical airplane analysis

References [edit]

1. ^ "Main stress and principal aeroplane". www.engineeringapps.net . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (two ed.). Taylor & Francis. pp. 1–30. ISBN0-415-27297-one.
3. ^ Gere, James Thou. (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-Hill Professional person. ISBN0-07-112939-one.
• Brady, B.H.Yard.; E.T. Brown (1993). Stone Mechanics For Underground Mining (Third ed.). Kluwer Bookish Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. Southward. (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 applied science. Prentice-Hall civil engineering and applied science mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Melt, N.1000.West.; Zimmerman, R.Due west. (2007). Fundamentals of rock mechanics (Quaternary ed.). Wiley-Blackwell. pp. 9–41. 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. 1–30. ISBN0-415-27297-1.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Tertiary ed.). McGraw-Hill International Editions. ISBN0-07-085805-5.
• Timoshenko, Stephen P. (1983). History of strength of materials: with a brief business relationship of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN0-486-61187-vi.

External links [edit]

• Mohr'south Circle and more than circles by Rebecca Brannon
• DoITPoMS Education and Learning Package- "Stress Analysis and Mohr's Circle"

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

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