Draw Mohrs Circle That Describes the State of Stress

Geometric civil engineering calculation technique

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

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

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

Subsequently performing a stress assay on a material trunk 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 circumvolve is then used to make up one's mind graphically the stress components interim on a rotated coordinate arrangement, 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 bespeak on the circle are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circumvolve is the locus of points that represent the state of stress on individual planes at all their orientations, where the axes correspond the principal axes of the stress chemical element.

19th-century German engineer Karl Culmann was the commencement to excogitate a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. His piece of work inspired boyfriend German language engineer Christian Otto Mohr (the circle's namesake), who extended information technology to both two- and 3-dimensional stresses and adult a failure criterion based on the stress circle.^[two]

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

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

Motivation [edit]

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

Internal forces are produced between the particles of a deformable object, assumed every bit a continuum, as a reaction to applied external forces, i.e., either surface forces or trunk forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of move for a particle. A measure out 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 applied science, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, airplane wings, or building columns, is adamant through a stress analysis. Calculating the stress distribution implies the determination of stresses at every point (textile particle) in the object. According to Cauchy, the stress at any bespeak in an object (Figure 2), assumed every bit a continuum, is completely defined by the 9 stress components ${\displaystyle \sigma _{ij}}$ of a second social club tensor of blazon (2,0) known equally the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

Figure iii. Stress transformation at a point in a continuum under plane stress atmospheric condition.

Subsequently the stress distribution within the object has been determined with respect to a coordinate organization ${\displaystyle (ten,y)}$ , it may exist necessary to calculate the components of the stress tensor at a item material betoken ${\displaystyle P}$ with respect to a rotated coordinate organisation ${\displaystyle (10',y')}$ , i.e., the stresses acting on a plane with a dissimilar orientation passing through that indicate of interest —forming an angle with the coordinate system ${\displaystyle (x,y)}$ (Figure 3). For example, it is of interest to detect the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To accomplish this, it is necessary to perform a tensor transformation under a rotation of the coordinate organisation. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation constabulary. A graphical representation of this transformation constabulary for the Cauchy stress tensor is the Mohr circle for stress.

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

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

In two dimensions, the stress tensor at a given textile point ${\displaystyle P}$ with respect to any ii perpendicular directions is completely defined by only iii stress components. For the particular coordinate arrangement ${\displaystyle (x,y)}$ these stress components are: the normal stresses ${\displaystyle \sigma _{10}}$ and ${\displaystyle \sigma _{y}}$ , and the shear stress ${\displaystyle \tau _{xy}}$ . From the balance of angular 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 be written as:

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

The objective is to use the Mohr circumvolve to find the stress components ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on a rotated coordinate system ${\displaystyle (x',y')}$ , i.e., on a differently oriented plane passing through ${\displaystyle P}$ and perpendicular to the ${\displaystyle x}$ - ${\displaystyle y}$ aeroplane (Effigy 4). The rotated coordinate system ${\displaystyle (10',y')}$ 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 circumvolve for the 2-dimensional cases of plane stress and airplane strain, first consider a two-dimensional infinitesimal material element effectually a fabric point ${\displaystyle P}$ (Figure iv), with a unit area in the direction 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 ${\displaystyle \tau _{\mathrm {north} }}$ are given by:

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

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

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

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

Both equations can too be obtained by applying the tensor transformation law on the known Cauchy stress tensor, which is equivalent to performing the static equilibrium of forces in the direction of ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {due north} }}$ .

Derivation of Mohr's circle parametric equations - Tensor transformation

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

These ii 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 past choosing a coordinate organisation with abscissa ${\displaystyle \sigma _{\mathrm {due north} }}$ and ordinate ${\displaystyle \tau _{\mathrm {due north} }}$ , giving values to the parameter ${\displaystyle \theta }$ will place the points obtained lying on a circle.

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

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

where

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

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

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

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

Sign conventions [edit]

At that place are two split up sets of sign conventions that demand to be considered when using the Mohr Circumvolve: I sign convention for stress components in the "concrete space", and some other for stress components in the "Mohr-Circle-infinite". In addition, within each of the two fix of sign conventions, the engineering mechanics (structural engineering and mechanical engineering) literature follows a unlike sign convention from the geomechanics literature. There is no standard sign convention, and the choice of a particular sign convention is influenced past convenience for calculation and interpretation for the detail problem in hand. 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 engineering science mechanics sign convention will be used for this article.

Physical-space sign convention [edit]

From the convention of the Cauchy stress tensor (Effigy 3 and Figure iv), the first subscript in the stress components denotes the face on which the stress component acts, and the 2nd subscript indicates the management of the stress component. Thus ${\displaystyle \tau _{xy}}$ is the shear stress acting on the face up with normal vector in the positive management of the ${\displaystyle 10}$ -axis, and in the positive management of the ${\displaystyle y}$ -centrality.

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

In the physical-infinite sign convention, positive shear stresses human activity on positive faces of the fabric element in the positive direction of an centrality. Besides, positive shear stresses act on negative faces of the material element in the negative direction of an centrality. A positive confront 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 case, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive because they deed on positive faces, and they act too in the positive management of the ${\displaystyle y}$ -axis and the ${\displaystyle x}$ -axis, respectively (Figure 3). Similarly, the respective opposite shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ acting in the negative faces have a negative sign because they human activity in the negative direction of the ${\displaystyle x}$ -axis and ${\displaystyle y}$ -centrality, respectively.

Mohr-circumvolve-space sign convention [edit]

Figure five. Engineering mechanics sign convention for drawing the Mohr circle. This article follows sign-convention # 3, as shown.

In the Mohr-circle-space sign convention, normal stresses accept the same sign as normal stresses in the physical-infinite sign convention: positive normal stresses act outward to the plane of activeness, and negative normal stresses human action inwards to the airplane of action.

Shear stresses, however, accept a different convention in the Mohr-circle space compared to the convention in the physical space. In the Mohr-circle-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-circle space.

Two options exist for cartoon the Mohr-circle infinite, which produce a mathematically correct Mohr circle:

1. Positive shear stresses are plotted up (Figure 5, sign convention #1)
2. Positive shear stresses are plotted downwards, i.eastward., the ${\displaystyle \tau _{\mathrm {n} }}$ -axis is inverted (Figure v, sign convention #2).

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

To overcome the "consequence" of having the shear stress axis downward in the Mohr-circle space, in that location is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise direction and negative shear stresses are assumed to rotate the material element in the counterclockwise direction (Effigy 5, choice 3). This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle ${\displaystyle 2\theta }$ has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #2 in Figure 5 because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is also a counterclockwise shear stress, and both are plotted downward. Likewise, a negative shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is a clockwise shear stress, and both are plotted upward.

This article follows the technology mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circle infinite (sign convention #3 in Figure 5)

Drawing Mohr's circle [edit]

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

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

Finding principal normal stresses [edit]

Stress components on a second rotating 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 principal directions. In this example, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{xx}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\cease{matrix}}\correct]=\left[{\begin{matrix}-10&10\\10&15\cease{matrix}}\right].}$ The corresponding Mohr's circle representation is shown at the bottom.

The magnitude of the chief stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle Due east}$ (Figure 6) where the circle intersects the ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality. The magnitude of the major primary stress ${\displaystyle \sigma _{1}}$ is ever the greatest absolute value of the abscissa of whatever of these ii points. Likewise, the magnitude of the pocket-sized principal stress ${\displaystyle \sigma _{2}}$ is always the lowest accented value of the abscissa of these two points. Equally expected, the ordinates of these two points are zip, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the principal stresses can be institute by

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

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

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

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

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

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

Finding maximum and minimum shear stresses [edit]

The maximum and minimum shear stresses 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 eye of the circle, ${\displaystyle O}$ . Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circumvolve's radius ${\displaystyle R}$

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

Finding stress components on an capricious plane [edit]

As mentioned before, afterward the two-dimensional stress analysis 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 deed in 2 perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Effigy 5 and half-dozen. The Mohr circle is used to notice the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {northward} }}$ , i.e., coordinates of whatsoever point ${\displaystyle D}$ on the circumvolve, interim on any other airplane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the airplane ${\displaystyle B}$ . For this, two approaches can be used: the double angle, and the Pole or origin of planes.

Double angle [edit]

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

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

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

Pole or origin of planes [edit]

Effigy 7. Mohr's circle for plane stress and aeroplane strain conditions (Pole arroyo). Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a aeroplane inclined at the aforementioned orientation (parallel) in space every bit that line.

The second approach involves the determination of a indicate on the Mohr circumvolve chosen the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the country of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on whatever item plane, 1 can draw a line parallel to that plane through the particular coordinates ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {northward} }}$ on the Mohr circle and notice the pole as the intersection of such line with the Mohr circle. As an instance, let's assume we accept a state of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , equally shown on Figure 7. First, nosotros can depict a line from signal ${\displaystyle B}$ parallel to the airplane of action of ${\displaystyle \sigma _{x}}$ , or, if we choose otherwise, a line from signal ${\displaystyle A}$ parallel to the plane of action of ${\displaystyle \sigma _{y}}$ . The intersection of whatsoever of these two lines with the Mohr circumvolve is the pole. One time the pole has been determined, to find the state of stress on a plane making an bending ${\displaystyle \theta }$ with the vertical, or in other words a plane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, then we can describe a line from the pole parallel to that airplane (Encounter Figure seven). The normal and shear stresses on that aeroplane are then the coordinates of the signal 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 primary stresses act, also known every bit principal planes, can be determined 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 primary airplane makes with plane ${\displaystyle B}$ .

Angles ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ tin likewise exist found from the post-obit 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 can be derived directly from the geometry of the circle, or by making the parametric equation of the circumvolve for ${\displaystyle \tau _{\mathrm {north} }}$ equal to zero (the shear stress in the principal planes is always nix).

Case [edit]

Assume a textile element nether a state of stress equally shown in Effigy 8 and Figure ix, with the airplane of one of its sides oriented 10° with respect to the horizontal aeroplane. Using the Mohr circumvolve, find:

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

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

Solution: Post-obit the engineering science mechanics sign convention for the concrete infinite (Effigy 5), the stress components for the material element in this example are:

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

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

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

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

We so plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as show in both Effigy eight and Figure nine. These points follow the engineering mechanics sign convention for the Mohr-circumvolve space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise. This way, the shear stress acting on plane B is negative and the shear stress acting on aeroplane A is positive. The bore of the circle is the line joining indicate A and B. The eye 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 detail state of stress.

The abscissas of both points Due east and C (Effigy 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 pocket-size and major principal planes, respectively, which is zilch for principal planes.

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

${\displaystyle {\brainstorm{aligned}R&={\sqrt {\left[{\tfrac {one}{2}}(\sigma _{ten}-\sigma _{y})\right]^{two}+\tau _{xy}^{2}}}\\&={\sqrt {\left[{\tfrac {1}{two}}(-10-50)\right]^{2}+twoscore^{2}}}\\&=50{\textrm {MPa}}\\\finish{aligned}}}$

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {1}{ii}}(\sigma _{10}+\sigma _{y})\\&={\tfrac {one}{2}}(-10+fifty)\\&=20{\textrm {MPa}}\\\end{aligned}}}$

and the primary stresses are

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

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

The coordinates for both points H and Chiliad (Figure viii 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 exist found analytically by

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

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

Nosotros can choose to either use the double bending approach (Figure eight) or the Pole arroyo (Figure ix) to find the orientation of the primary normal stresses and principal shear stresses.

Using the double bending approach we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Figure 8) to find double the angle the major chief stress and the minor principal stress make with airplane B in the physical infinite. To obtain a more than accurate value for these angles, instead of manually measuring the angles, we can use the analytical expression

${\displaystyle {\brainstorm{aligned}2\theta _{\mathrm {p} }=\arctan {\frac {2\tau _{xy}}{\sigma _{x}-\sigma _{y}}}=\arctan {\frac {2*forty}{(-10-50)}}=-\arctan {\frac {4}{three}}\end{aligned}}}$

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

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

So, the major principal angle is

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

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

Using the Pole approach, nosotros beginning localize the Pole or origin of planes. For this, nosotros describe through indicate A on the Mohr circle a line inclined ten° with the horizontal, or, in other words, a line parallel to plane A where ${\displaystyle \sigma _{y'}}$ acts. The Pole is where this line intersects the Mohr circle (Figure nine). To confirm the location of the Pole, we could depict a line through signal 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, nosotros draw lines to different points on the Mohr circumvolve. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a plane in the physical space having the same inclination as the line. For case, the line from the Pole to point C in the circumvolve has the aforementioned inclination as the plane in the physical space where ${\displaystyle \sigma _{one}}$ acts. This plane makes an angle of 63.435° with plane 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, Yard and H to discover the stress components on planes with the same orientation.

Mohr'southward circumvolve for a general three-dimensional land of stresses [edit]

Figure 10. Mohr's circle for a three-dimensional country of stress

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

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

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

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

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

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

Since ${\displaystyle \sigma _{1}>\sigma _{2}>\sigma _{three}}$ , and ${\displaystyle (n_{i})^{2}}$ is non-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 ${\displaystyle \sigma _{ane}-\sigma _{3}>0}$

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

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

These expressions can be rewritten as

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

which are the equations of the three Mohr's circles for stress ${\displaystyle C_{1}}$ , ${\displaystyle C_{2}}$ , and ${\displaystyle C_{3}}$ , with radii ${\displaystyle R_{1}={\tfrac {1}{ii}}(\sigma _{2}-\sigma _{3})}$ , ${\displaystyle R_{2}={\tfrac {i}{2}}(\sigma _{1}-\sigma _{3})}$ , and ${\displaystyle R_{iii}={\tfrac {1}{two}}(\sigma _{1}-\sigma _{2})}$ , and their centres with coordinates ${\displaystyle \left[{\tfrac {1}{2}}(\sigma _{ii}+\sigma _{3}),0\right]}$ , ${\displaystyle \left[{\tfrac {1}{2}}(\sigma _{ane}+\sigma _{iii}),0\right]}$ , ${\displaystyle \left[{\tfrac {one}{2}}(\sigma _{1}+\sigma _{2}),0\correct]}$ , 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 within the shaded surface area enclosed by them (see Figure 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 ${\displaystyle (\sigma _{\mathrm {due north} },\tau _{\mathrm {due north} })}$ satisfying the equation for circle ${\displaystyle C_{ii}}$ lie on, or inside circle ${\displaystyle C_{2}}$ . And finally, stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ satisfying the equation for circle ${\displaystyle C_{3}}$ lie on, or outside circumvolve ${\displaystyle C_{3}}$ .

See also [edit]

• Critical airplane assay

References [edit]

1. ^ "Main stress and principal plane". world wide web.engineeringapps.internet . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grayness (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–xxx. ISBN0-415-27297-1.
3. ^ Gere, James G. (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. ISBN0-07-112939-one.
• Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Tertiary ed.). Kluwer Bookish Publisher. pp. 17–29. ISBN0-412-47550-two.
• 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 technology mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, N.G.W.; Zimmerman, R.Westward. (2007). Fundamentals of rock mechanics (4th 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-iii.
• Parry, Richard Hawley Grayness (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–xxx. 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 forcefulness 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-half dozen.

External links [edit]

• Mohr's Circle and more circles by Rebecca Brannon
• DoITPoMS Educational activity and Learning Parcel- "Stress Analysis and Mohr'southward Circle"

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

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