Draw Mohrs Circle for This State of Stress 102

Geometric civil engineering calculation technique

Figure 1. Mohr's circles for a 3-dimensional land of stress

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

Mohr's circle is ofttimes used in calculations relating to mechanical engineering for materials' strength, geotechnical engineering science for force of soils, and structural technology for strength of built structures. It is also used for calculating stresses in many planes by reducing them to vertical and horizontal components. These are called primary planes in which primary stresses are calculated; Mohr's circle tin also exist used to find the principal planes and the master stresses in a graphical representation, and is 1 of the easiest means to practice and so.^[one]

Later performing a stress assay on a material body assumed as a continuum, the components of the Cauchy stress tensor at a item material point are known with respect to a coordinate system. The Mohr circle is then used to decide graphically the stress components acting on a rotated coordinate arrangement, i.eastward., acting on a differently oriented plane passing through that point.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {due north} }}$ , ${\displaystyle \tau _{\mathrm {northward} }}$ ) of each point on the circumvolve are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the land of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress 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 fellow German engineer Christian Otto Mohr (the circle's namesake), who extended information technology to both two- and three-dimensional stresses and developed a failure criterion based on the stress circumvolve.^[2]

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

The Mohr circumvolve can be 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, equally a reaction to applied external forces, i.east., either surface forces or trunk forces. This reaction follows from Euler's laws of move for a continuum, which are equivalent to Newton's laws of move 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 science, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for example stresses in a rock mass around a tunnel, airplane wings, or building columns, is determined through a stress assay. Calculating the stress distribution implies the determination of stresses at every point (fabric particle) in the object. According to Cauchy, the stress at any point in an object (Figure ii), causeless as a continuum, is completely divers by the nine stress components ${\displaystyle \sigma _{ij}}$ of a second guild tensor of blazon (2,0) known equally the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

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

After the stress distribution within the object has been adamant with respect to a coordinate arrangement ${\displaystyle (x,y)}$ , it may be necessary to summate the components of the stress tensor at a particular material betoken ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (x',y')}$ , i.eastward., the stresses acting on a plane with a different orientation passing through that point of involvement —forming an bending with the coordinate system ${\displaystyle (x,y)}$ (Figure 3). For instance, it is of interest 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, 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 law. 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]

Effigy 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 cloth point ${\displaystyle P}$ with respect to whatever ii perpendicular directions is completely defined past 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 balance of athwart momentum, the symmetry of the Cauchy stress tensor tin can exist demonstrated. This symmetry implies that ${\displaystyle \tau _{xy}=\tau _{yx}}$ . Thus, the Cauchy stress tensor can be written as:

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

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

Equation of the Mohr circle [edit]

To derive the equation of the Mohr circle for the ii-dimensional cases of plane stress and plane strain, offset consider a two-dimensional infinitesimal material chemical element around a material point ${\displaystyle P}$ (Figure four), with a unit area in the management parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ plane, i.eastward., perpendicular to the page or screen.

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

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

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

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

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

Both equations tin can too be 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 ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ .

Derivation of Mohr's circle parametric equations - Tensor transformation

The stress tensor transformation police can be stated as ${\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}'&=\mathbf {A} {\boldsymbol {\sigma }}\mathbf {A} ^{T}\\\left[{\begin{matrix}\sigma _{10'}&\tau _{x'y'}\\\tau _{y'x'}&\sigma _{y'}\\\end{matrix}}\correct]&=\left[{\begin{matrix}a_{x}&a_{xy}\\a_{yx}&a_{y}\\\end{matrix}}\right]\left[{\begin{matrix}\sigma _{10}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\correct]\left[{\brainstorm{matrix}a_{x}&a_{yx}\\a_{xy}&a_{y}\\\terminate{matrix}}\right]\\&=\left[{\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\terminate{matrix}}\right]\left[{\begin{matrix}\sigma _{x}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\right]\left[{\begin{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{matrix}}\correct]\end{aligned}}}$ Expanding the right manus side, and knowing that ${\displaystyle \sigma _{x'}=\sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{x'y'}=\tau _{\mathrm {n} }}$ , we accept: ${\displaystyle \sigma _{\mathrm {due north} }=\sigma _{x}\cos ^{2}\theta +\sigma _{y}\sin ^{ii}\theta +two\tau _{xy}\sin \theta \cos \theta }$ Still, knowing that ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}},\qquad \sin ^{2}\theta ={\frac {1-\cos ii\theta }{ii}}\qquad {\text{and}}\qquad \sin ii\theta =ii\sin \theta \cos \theta }$ we obtain ${\displaystyle \sigma _{\mathrm {n} }={\frac {one}{ii}}(\sigma _{ten}+\sigma _{y})+{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\cos 2\theta +\tau _{xy}\sin ii\theta }$ ${\displaystyle \tau _{\mathrm {n} }=-(\sigma _{x}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{two}\theta -\sin ^{2}\theta \right)}$ However, knowing that ${\displaystyle \cos ^{ii}\theta -\sin ^{ii}\theta =\cos two\theta \qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ we obtain ${\displaystyle \tau _{\mathrm {n} }=-{\frac {1}{two}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos two\theta }$ It is not necessary at this moment to summate the stress component ${\displaystyle \sigma _{y'}}$ acting on the plane perpendicular to the plane of action 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 circumvolve. In these equations, ${\displaystyle 2\theta }$ is the parameter, and ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ are the coordinates. This ways 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 identify 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 by rearranging the equations for ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {due north} }}$ , get-go transposing the first term in the commencement equation and squaring both sides of each of the equations then adding them. Thus we accept

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

where

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

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

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

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]

In that location are two carve up sets of sign conventions that demand to exist considered when using the Mohr Circumvolve: Ane sign convention for stress components in the "physical space", and some other for stress components in the "Mohr-Circumvolve-infinite". In addition, within each of the two fix of sign conventions, the engineering mechanics (structural engineering and mechanical engineering science) literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the pick of a particular sign convention is influenced by convenience for adding and estimation for the item trouble in mitt. A more than detailed explanation of these sign conventions is presented below.

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

Physical-space sign convention [edit]

From the convention of the Cauchy stress tensor (Figure iii and Figure 4), the commencement subscript in the stress components denotes the face on which the stress component acts, and the 2nd subscript indicates the direction 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 ${\displaystyle x}$ -axis, and in the positive management of the ${\displaystyle y}$ -axis.

In the physical-space sign convention, positive normal stresses are outward to the airplane of action (tension), and negative normal stresses are inwards 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 axis. As well, positive shear stresses deed on negative faces of the material element in the negative direction of an axis. A positive face has its normal vector in the positive management of an centrality, and a negative face has its normal vector in the negative direction of an axis. For example, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive because they act on positive faces, and they act equally well in the positive management of the ${\displaystyle y}$ -centrality and the ${\displaystyle 10}$ -centrality, respectively (Effigy 3). Similarly, the corresponding opposite shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ acting in the negative faces have a negative sign considering they deed in the negative direction of the ${\displaystyle x}$ -centrality and ${\displaystyle y}$ -centrality, respectively.

Mohr-circumvolve-space sign convention [edit]

Figure 5. Engineering mechanics sign convention for cartoon the Mohr circle. This article follows sign-convention # 3, as shown.

In the Mohr-circle-space sign convention, normal stresses have the aforementioned 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 deed in to the airplane of activity.

Shear stresses, however, take 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 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.

Two options be for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle:

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

Plotting positive shear stresses upward makes the bending ${\displaystyle two\theta }$ 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 concrete infinite convention for shear stresses.

To overcome the "issue" of having the shear stress axis down in the Mohr-circumvolve infinite, there is an alternative sign convention where positive shear stresses are assumed to rotate the material chemical element in the clockwise direction and negative shear stresses are assumed to rotate the material chemical element in the counterclockwise direction (Figure 5, option 3). This way, positive shear stresses are plotted up in the Mohr-circumvolve infinite and the angle ${\displaystyle ii\theta }$ 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 v because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is also a counterclockwise shear stress, and both are plotted downward. Besides, a negative shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is a clockwise shear stress, and both are plotted up.

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

Drawing Mohr'southward circle [edit]

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

1. Depict the Cartesian coordinate organisation ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality and a vertical ${\displaystyle \tau _{\mathrm {due north} }}$ -centrality.
2. Plot 2 points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {due north} },\tau _{\mathrm {n} })}$ space respective 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 by joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a direct line ${\displaystyle {\overline {AB}}}$ .
4. Depict the Mohr Circle. The centre ${\displaystyle O}$ of the circle is the midpoint of the diameter line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ centrality.

Finding primary normal stresses [edit]

Stress components on a 2D rotating element. Instance of how stress components vary on the faces (edges) of a rectangular element as the angle of its orientation is varied. Principal stresses occur when the shear stresses simultaneously disappear from all faces. The orientation at which this occurs gives the principal directions. In this example, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{xx}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\right]=\left[{\begin{matrix}-10&10\\10&15\finish{matrix}}\right].}$ The corresponding Mohr'southward circle representation is shown at the bottom.

The magnitude of the principal stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle E}$ (Figure half-dozen) where the circle intersects the ${\displaystyle \sigma _{\mathrm {northward} }}$ -axis. The magnitude of the major principal stress ${\displaystyle \sigma _{1}}$ is always the greatest absolute value of the abscissa of any of these ii points. Likewise, the magnitude of the minor principal stress ${\displaystyle \sigma _{ii}}$ is always the lowest absolute value of the abscissa of these two points. Equally expected, the ordinates of these 2 points are nada, corresponding to the magnitude of the shear stress components on the main planes. Alternatively, the values of the principal stresses can be found by

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

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

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

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

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

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

Finding maximum and minimum shear stresses [edit]

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

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

Finding stress components on an arbitrary plane [edit]

Equally mentioned before, after the two-dimensional stress analysis has been performed we know the stress components ${\displaystyle \sigma _{10}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a fabric point ${\displaystyle P}$ . These stress components act in two perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Figure five and vi. The Mohr circumvolve is used to notice the stress components ${\displaystyle \sigma _{\mathrm {northward} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.e., coordinates of any point ${\displaystyle D}$ on the circle, interim on whatsoever other aeroplane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the plane ${\displaystyle B}$ . For this, two approaches can exist used: the double angle, and the Pole or origin of planes.

Double angle [edit]

Equally shown in Effigy 6, to make up one's mind the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ acting on a airplane ${\displaystyle D}$ at an bending ${\displaystyle \theta }$ counterclockwise to the aeroplane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an angle ${\displaystyle 2\theta }$ in the same counterclockwise direction around the circumvolve from the known stress bespeak ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ to bespeak ${\displaystyle D(\sigma _{\mathrm {northward} },\tau _{\mathrm {n} })}$ , i.e., an bending ${\displaystyle 2\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circumvolve.

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

This double bending relation comes from the fact that the parametric equations for the Mohr circle are a function of ${\displaystyle ii\theta }$ . It can likewise be seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the fabric element around ${\displaystyle P}$ of Effigy 5 are separated by an bending ${\displaystyle \theta =90^{\circ }}$ , 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 airplane stress and airplane strain conditions (Pole arroyo). Whatever straight line drawn from the pole volition intersect the Mohr circle at a point that represents the country of stress on a plane inclined at the aforementioned orientation (parallel) in space as that line.

The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes. Whatever straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on any detail plane, one can draw a line parallel to that plane through the particular 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 circle. Equally an example, permit's assume we take a state of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , equally shown on Figure 7. First, we tin can draw a line from point ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{10}}$ , or, if we choose otherwise, a line from indicate ${\displaystyle A}$ parallel to the plane of action of ${\displaystyle \sigma _{y}}$ . The intersection of any of these 2 lines with the Mohr circle is the pole. One time the pole has been adamant, to find the country of stress on a plane making an angle ${\displaystyle \theta }$ with the vertical, or in other words a plane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, then we can depict a line from the pole parallel to that plane (Run across Figure 7). The normal and shear stresses on that plane are then the coordinates of the betoken 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 master stresses human activity, besides known every bit principal planes, can be adamant by measuring in the Mohr circle the angles ∠BOC and ∠BOE, respectively, and taking half of each of those angles. Thus, the bending ∠BOC between ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OC}}}$ is double the angle ${\displaystyle \theta _{p}}$ which the major principal aeroplane makes with plane ${\displaystyle B}$ .

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

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

This equation defines ii 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 ${\displaystyle \tau _{\mathrm {northward} }}$ equal to null (the shear stress in the principal planes is e'er zilch).

Example [edit]

Assume a material element nether a state of stress as shown in Figure viii and Figure 9, with the aeroplane of one of its sides oriented 10° with respect to the horizontal airplane. Using the Mohr circle, find:

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

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

Solution: Following the applied science mechanics sign convention for the physical space (Figure 5), the stress components for the fabric element in this example are:

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

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

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

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

We then plot two points A(l,twoscore) and B(-ten,-forty), representing the state of stress at airplane A and B as bear witness in both Figure 8 and Effigy 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 element clockwise. This way, the shear stress interim on aeroplane B is negative and the shear stress acting on airplane A is positive. The diameter of the circle is the line joining point A and B. The centre of the circumvolve is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {north} }}$ -axis. Knowing both the location of the centre and length of the bore, we are able to plot the Mohr circumvolve for this item state of stress.

The abscissas of both points E and C (Effigy eight and Figure 9) intersecting the ${\displaystyle \sigma _{\mathrm {northward} }}$ -centrality are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points East and C are the magnitudes of the shear stresses acting on both the small and major chief planes, respectively, which is zero for principal planes.

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

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

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {1}{2}}(\sigma _{x}+\sigma _{y})\\&={\tfrac {i}{2}}(-ten+l)\\&=20{\textrm {MPa}}\\\terminate{aligned}}}$

and the principal stresses are

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

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

The coordinates for both points H and G (Effigy 8 and Effigy 9) are the magnitudes of the minimum and maximum shear stresses, respectively; the abscissas for both points H and K 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 tin can be found analytically by

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

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

We tin choose to either utilise the double bending approach (Figure viii) or the Pole approach (Figure 9) to find the orientation of the principal normal stresses and principal shear stresses.

Using the double angle approach we measure the angles ∠BOC and ∠BOE in the Mohr Circumvolve (Figure viii) to discover double the bending the major principal stress and the modest principal stress make with plane B in the physical space. To obtain a more than accurate value for these angles, instead of manually measuring the angles, we tin use the belittling expression

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

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

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

And then, the major main angle is

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

Recollect that in this detail example ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ are angles with respect to the airplane of activity of ${\displaystyle \sigma _{ten'}}$ (oriented in the ${\displaystyle x'}$ -axis)and not angles with respect to the airplane of action of ${\displaystyle \sigma _{x}}$ (oriented in the ${\displaystyle 10}$ -axis).

Using the Pole arroyo, we first localize the Pole or origin of planes. For this, we draw through point A on the Mohr circumvolve a line inclined 10° 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 (Figure 9). To confirm the location of the Pole, we could describe a line through point B on the Mohr circle parallel to the aeroplane B where ${\displaystyle \sigma _{x'}}$ acts. This line would as well intersect the Mohr circle at the Pole (Figure 9).

From the Pole, we draw lines to different points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a plane in the physical space having the same inclination as the line. For instance, the line from the Pole to bespeak C in the circle has the same inclination as the plane in the concrete infinite where ${\displaystyle \sigma _{1}}$ acts. This plane makes an bending of 63.435° with plane B, both in the Mohr-circle space and in the concrete space. In the same way, lines are traced from the Pole to points Due east, D, F, G and H to observe the stress components on planes with the aforementioned orientation.

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

Figure ten. Mohr's circle for a three-dimensional state of stress

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

Considering the principal axes as the coordinate system, instead of the general ${\displaystyle x_{one}}$ , ${\displaystyle x_{two}}$ , ${\displaystyle x_{3}}$ coordinate system, and assuming that ${\displaystyle \sigma _{1}>\sigma _{2}>\sigma _{3}}$ , then the normal and shear components of the stress vector ${\displaystyle \mathbf {T} ^{(\mathbf {north} )}}$ , for a given airplane with unit vector ${\displaystyle \mathbf {due north} }$ , satisfy the following equations

${\displaystyle {\brainstorm{aligned}\left(T^{(n)}\right)^{two}&=\sigma _{ij}\sigma _{ik}n_{j}n_{1000}\\\sigma _{\mathrm {n} }^{2}+\tau _{\mathrm {n} }^{2}&=\sigma _{1}^{two}n_{1}^{two}+\sigma _{two}^{two}n_{two}^{two}+\sigma _{three}^{2}n_{3}^{2}\end{aligned}}}$

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

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

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

Since ${\displaystyle \sigma _{1}>\sigma _{two}>\sigma _{three}}$ , and ${\displaystyle (n_{i})^{two}}$ is not-negative, the numerators from these equations satisfy

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

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

${\displaystyle \tau _{\mathrm {n} }^{two}+(\sigma _{\mathrm {northward} }-\sigma _{one})(\sigma _{\mathrm {n} }-\sigma _{2})\geq 0}$ every bit the denominator ${\displaystyle \sigma _{3}-\sigma _{1}<0}$ and ${\displaystyle \sigma _{three}-\sigma _{2}<0.}$

These expressions tin can exist rewritten as

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

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

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

See also [edit]

• Disquisitional plane analysis

References [edit]

1. ^ "Chief stress and principal airplane". world wide web.engineeringapps.net . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. ane–xxx. ISBN0-415-27297-1.
3. ^ Gere, James Yard. (2013). Mechanics of Materials. Goodno, Barry J. (8th ed.). Stamford, CT: Cengage Learning. ISBN9781111577735.

Bibliography [edit]

• Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials . McGraw-Hill Professional. ISBN0-07-112939-one.
• Brady, B.H.G.; East.T. Brown (1993). Stone Mechanics For Underground Mining (3rd ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. sixteen–26. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering science. Prentice-Hall civil engineering and technology mechanics series. Prentice-Hall. ISBN0-xiii-484394-0.
• Jaeger, John Conrad; Cook, North.G.W.; Zimmerman, R.Due west. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. 9–41. ISBN978-0-632-05759-seven.
• Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with applied 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 (ii ed.). Taylor & Francis. pp. i–30. ISBN0-415-27297-i.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third 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-half-dozen.

External links [edit]

• Mohr's Circle and more circles by Rebecca Brannon
• DoITPoMS Educational activity and Learning Package- "Stress Assay and Mohr's Circle"

mankemorne1937.blogspot.com

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

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