Draw Mohrs Circle for the Stress Stat

Geometric civil engineering calculation technique

Figure i. Mohr'southward circles for a iii-dimensional state of stress

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

Mohr's circumvolve is often used in calculations relating to mechanical engineering science for materials' strength, geotechnical engineering for strength of soils, and structural engineering for strength of congenital structures. It is also used for calculating stresses in many planes past reducing them to vertical and horizontal components. These are called chief planes in which principal stresses are calculated; Mohr'due south circle tin can also be used to find the principal planes and the chief stresses in a graphical representation, and is one of the easiest ways to practise so.^[1]

Subsequently performing a stress assay on a textile body assumed equally a continuum, the components of the Cauchy stress tensor at a particular cloth bespeak 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 system, i.eastward., acting on a differently oriented plane passing through that signal.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {due north} }}$ , ${\displaystyle \tau _{\mathrm {n} }}$ ) of each point on the circle are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate organisation. 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 language engineer Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. His work inspired beau German language engineer Christian Otto Mohr (the circle'south namesake), who extended information technology to both ii- and three-dimensional stresses and developed a failure criterion based on the stress circle.^[2]

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 circle tin be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.

Motivation [edit]

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

Internal forces are produced between the particles of a deformable object, assumed every bit a continuum, as a reaction to applied external forces, i.east., either surface forces or trunk forces. This reaction follows from Euler's laws of motility for a continuum, which are equivalent to Newton's laws of motion for a particle. A measure out of the intensity of these internal forces is called stress. Considering the object is causeless as a continuum, these internal forces are distributed continuously inside the book of the object.

In engineering, east.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a stone mass around a tunnel, airplane wings, or building columns, is adamant through a stress assay. Calculating the stress distribution implies the decision of stresses at every point (cloth particle) in the object. According to Cauchy, the stress at any signal in an object (Figure two), assumed as a continuum, is completely defined by the nine stress components ${\displaystyle \sigma _{ij}}$ of a second club tensor of type (2,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

${\displaystyle {\boldsymbol {\sigma }}=\left[{\brainstorm{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\finish{matrix}}\right]\equiv \left[{\brainstorm{matrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\stop{matrix}}\right]\equiv \left[{\begin{matrix}\sigma _{10}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\cease{matrix}}\correct]}$

Figure 3. Stress transformation at a indicate in a continuum under plane stress atmospheric condition.

After the stress distribution within the object has been determined with respect to a coordinate organization ${\displaystyle (x,y)}$ , it may be necessary to calculate the components of the stress tensor at a particular material betoken ${\displaystyle P}$ 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 point of interest —forming an bending with the coordinate organization ${\displaystyle (x,y)}$ (Figure three). For case, information technology is of interest to find the maximum normal stress and maximum shear stress, also as the orientation of the planes where they act upon. To reach this, information technology is necessary to perform a tensor transformation nether a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation police for the Cauchy stress tensor is the Mohr circle for stress.

Mohr'south circle for ii-dimensional country of stress [edit]

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

In two dimensions, the stress tensor at a given material bespeak ${\displaystyle P}$ with respect to any two perpendicular directions is completely defined by just iii stress components. For the particular coordinate system ${\displaystyle (ten,y)}$ these stress components are: the normal stresses ${\displaystyle \sigma _{ten}}$ 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 can be written as:

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

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

Equation of the Mohr circumvolve [edit]

To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and airplane strain, showtime consider a two-dimensional infinitesimal material chemical element around a material betoken ${\displaystyle P}$ (Figure four), with a unit area in the direction parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ plane, i.e., perpendicular to the page 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:

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

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

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

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

Derivation of Mohr's circle parametric equations - Tensor transformation

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

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

Eliminating the parameter ${\displaystyle 2\theta }$ from these parametric equations will yield the non-parametric equation of the Mohr circumvolve. This tin can exist achieved by rearranging the equations for ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {north} }}$ , first transposing the first term in the first equation and squaring both sides of each of the equations then calculation them. Thus nosotros have

${\displaystyle {\begin{aligned}\left[\sigma _{\mathrm {n} }-{\tfrac {ane}{2}}(\sigma _{ten}+\sigma _{y})\right]^{2}+\tau _{\mathrm {n} }^{2}&=\left[{\tfrac {1}{2}}(\sigma _{ten}-\sigma _{y})\right]^{2}+\tau _{xy}^{ii}\\(\sigma _{\mathrm {n} }-\sigma _{\mathrm {avg} })^{2}+\tau _{\mathrm {north} }^{ii}&=R^{2}\cease{aligned}}}$

where

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

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

${\displaystyle (10-a)^{two}+(y-b)^{2}=r^{ii}}$

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

Sign conventions [edit]

At that place are two carve up sets of sign conventions that demand to exist considered when using the Mohr Circle: One sign convention for stress components in the "physical space", and another for stress components in the "Mohr-Circle-space". In addition, within each of the two set up of sign conventions, the engineering mechanics (structural engineering and mechanical engineering) literature follows a dissimilar sign convention from the geomechanics literature. In that location is no standard sign convention, and the pick of a particular sign convention is influenced past convenience for calculation and estimation for the particular problem in hand. A more detailed explanation of these sign conventions is presented below.

The previous derivation for the equation of the Mohr Circle using Figure 4 follows the applied science 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 first subscript in the stress components denotes the confront on which the stress component acts, and the 2d subscript indicates the management of the stress component. Thus ${\displaystyle \tau _{xy}}$ is the shear stress interim on the face with normal vector in the positive management of the ${\displaystyle 10}$ -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 inward to the plane of activity (compression) (Figure 5).

In the physical-space sign convention, positive shear stresses human activity on positive faces of the fabric element in the positive management of an axis. Too, positive shear stresses act on negative faces of the material element in the negative direction 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 example, the shear stresses ${\displaystyle \tau _{xy}}$ and ${\displaystyle \tau _{yx}}$ are positive because they deed on positive faces, and they act as well in the positive management of the ${\displaystyle y}$ -centrality 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 take a negative sign because they act in the negative management of the ${\displaystyle x}$ -axis and ${\displaystyle y}$ -axis, respectively.

Mohr-circle-space sign convention [edit]

Effigy 5. Engineering mechanics sign convention for drawing the Mohr circumvolve. This article follows sign-convention # 3, as shown.

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

Shear stresses, nonetheless, have a unlike convention in the Mohr-circle space compared to the convention in the physical infinite. In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction. This mode, the shear stress component ${\displaystyle \tau _{xy}}$ is positive in the Mohr-circumvolve space, and the shear stress component ${\displaystyle \tau _{yx}}$ is negative in the Mohr-circle space.

Two options exist for drawing the Mohr-circle infinite, which produce a mathematically right Mohr circle:

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 5, sign convention #ii).

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

To overcome the "result" of having the shear stress axis downwardly in the Mohr-circumvolve space, at that place is an culling sign convention where positive shear stresses are causeless to rotate the cloth element in the clockwise management and negative shear stresses are causeless to rotate the fabric element in the counterclockwise management (Effigy 5, pick 3). This style, 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-circumvolve space. This alternative sign convention produces a circle that is identical to the sign convention #ii in Figure five because a positive shear stress ${\displaystyle \tau _{\mathrm {n} }}$ is as well a counterclockwise shear stress, and both are plotted down. Also, a negative shear stress ${\displaystyle \tau _{\mathrm {northward} }}$ is a clockwise shear stress, and both are plotted upward.

This article follows the applied science mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circle space (sign convention #3 in Figure v)

Cartoon Mohr's circle [edit]

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

1. Draw the Cartesian coordinate organisation ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {due north} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {northward} }}$ -axis.
2. Plot two points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ infinite corresponding to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Figure iv and 6), following the chosen sign convention.
3. Draw the diameter of the circle past joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a directly line ${\displaystyle {\overline {AB}}}$ .
4. Draw the Mohr Circle. The eye ${\displaystyle O}$ of the circle is the midpoint of the bore line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ axis.

Finding chief normal stresses [edit]

Stress components on a 2D rotating element. Example of how stress components vary on the faces (edges) of a rectangular element as 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 chief directions. In this instance, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\begin{matrix}\sigma _{twenty}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\right]=\left[{\begin{matrix}-x&10\\10&15\terminate{matrix}}\right].}$ The corresponding Mohr's circle representation is shown at the bottom.

The magnitude of the master stresses are the abscissas of the points ${\displaystyle C}$ and ${\displaystyle East}$ (Figure 6) where the circle intersects the ${\displaystyle \sigma _{\mathrm {northward} }}$ -centrality. The magnitude of the major primary stress ${\displaystyle \sigma _{i}}$ is always the greatest absolute value of the abscissa of any of these two points. Likewise, the magnitude of the minor principal stress ${\displaystyle \sigma _{2}}$ is always the lowest accented value of the abscissa of these two points. As expected, the ordinates of these two points are aught, respective to the magnitude of the shear stress components on the chief planes. Alternatively, the values of the principal stresses can be 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 average normal stress ${\displaystyle \sigma _{\text{avg}}}$ is the abscissa of the center ${\displaystyle O}$ , given by

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

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

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

Finding maximum and minimum shear stresses [edit]

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

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

Finding stress components on an capricious plane [edit]

As mentioned before, subsequently 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 cloth bespeak ${\displaystyle P}$ . These stress components act in two perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Effigy 5 and 6. The Mohr circle is used to notice the stress components ${\displaystyle \sigma _{\mathrm {north} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , i.e., coordinates of any point ${\displaystyle D}$ on the circle, acting on whatever other plane ${\displaystyle D}$ 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]

As shown in Figure six, to make up one's mind the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ acting on a plane ${\displaystyle D}$ at an bending ${\displaystyle \theta }$ counterclockwise to the plane ${\displaystyle B}$ on which ${\displaystyle \sigma _{ten}}$ acts, we travel an angle ${\displaystyle 2\theta }$ in the aforementioned counterclockwise direction around the circle from the known stress point ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ to signal ${\displaystyle D(\sigma _{\mathrm {n} },\tau _{\mathrm {northward} })}$ , i.east., an angle ${\displaystyle 2\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circle.

The double angle arroyo relies on the fact that the angle ${\displaystyle \theta }$ between the normal vectors to whatsoever 2 physical planes passing through ${\displaystyle P}$ (Effigy four) is half the angle between ii lines joining their corresponding stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {north} })}$ on the Mohr circumvolve 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 can also be seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the cloth element effectually ${\displaystyle P}$ of Figure 5 are separated past an angle ${\displaystyle \theta =ninety^{\circ }}$ , which in the Mohr circle is represented by a ${\displaystyle 180^{\circ }}$ bending (double the bending).

Pole or origin of planes [edit]

Effigy vii. Mohr's circle for aeroplane stress and plane strain weather condition (Pole approach). Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the country of stress on a airplane inclined at the same orientation (parallel) in infinite as that line.

The second approach involves the conclusion of a signal on the Mohr circle called 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 land of stress on a airplane inclined at the same orientation (parallel) in space equally that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on whatsoever particular plane, 1 can draw a line parallel to that plane through the particular coordinates ${\displaystyle \sigma _{\mathrm {due north} }}$ and ${\displaystyle \tau _{\mathrm {north} }}$ on the Mohr circumvolve and find the pole as the intersection of such line with the Mohr circle. As an example, let's presume we have a state of stress with stress components ${\displaystyle \sigma _{10},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , as shown on Figure 7. Start, we can depict a line from point ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{x}}$ , or, if we choose otherwise, a line from point ${\displaystyle A}$ parallel to the aeroplane of action of ${\displaystyle \sigma _{y}}$ . The intersection of whatever of these 2 lines with the Mohr circle is the pole. One time the pole has been determined, to discover the state of stress on a plane making an angle ${\displaystyle \theta }$ with the vertical, or in other words a plane having its normal vector forming an angle ${\displaystyle \theta }$ with the horizontal plane, then we can draw a line from the pole parallel to that airplane (See Figure seven). The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.

Finding the orientation of the principal planes [edit]

The orientation of the planes where the maximum and minimum principal stresses human activity, also known every bit main planes, can be determined past measuring in the Mohr circumvolve the angles ∠BOC and ∠BOE, respectively, and taking one-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 airplane makes with airplane ${\displaystyle B}$ .

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

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

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 circle for ${\displaystyle \tau _{\mathrm {due north} }}$ equal to zero (the shear stress in the principal planes is e'er zero).

Example [edit]

Presume a material chemical element under a country of stress as shown in Figure 8 and Effigy 9, with the plane of i of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circumvolve, find:

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

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

Solution: Post-obit the engineering mechanics sign convention for the physical infinite (Figure 5), the stress components for the material element in this instance are:

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

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

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

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

We then plot two points A(l,40) and B(-x,-twoscore), representing the state of stress at plane A and B as testify in both Figure 8 and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure v), which assumes positive normals stresses outward from the cloth element, and positive shear stresses on each plane rotating the material chemical element clockwise. This way, the shear stress interim on airplane B is negative and the shear stress interim on plane A is positive. The diameter of the circle is the line joining signal A and B. The heart of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ -centrality. Knowing both the location of the centre and length of the diameter, nosotros are able to plot the Mohr circle for this particular state of stress.

The abscissas of both points E and C (Figure 8 and Effigy 9) intersecting the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points E and C are the magnitudes of the shear stresses acting on both the minor and major primary planes, respectively, which is zero for main planes.

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

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

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {ane}{two}}(\sigma _{x}+\sigma _{y})\\&={\tfrac {i}{2}}(-10+50)\\&=xx{\textrm {MPa}}\\\cease{aligned}}}$

and the principal stresses are

${\displaystyle {\begin{aligned}\sigma _{1}&=\sigma _{\mathrm {avg} }+R\\&=70{\textrm {MPa}}\\\end{aligned}}}$

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

The coordinates for both points H and G (Figure viii and Figure ix) 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 human activity, respectively. The magnitudes of the minimum and maximum shear stresses can be found analytically by

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

and the normal stresses 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 angle approach (Figure viii) or the Pole approach (Figure nine) to find the orientation of the principal normal stresses and primary shear stresses.

Using the double angle arroyo we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Effigy 8) to detect double the angle the major principal 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 utilise the analytical expression

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

1 solution is: ${\displaystyle 2\theta _{p}=-53.13^{\circ }}$ . From inspection of Effigy eight, this value corresponds to the bending ∠BOE. Thus, the small-scale main bending is

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

Then, the major principal angle is

${\displaystyle {\begin{aligned}two\theta _{p1}&=180-53.13^{\circ }=126.87^{\circ }\\\theta _{p1}&=63.435^{\circ }\\\end{aligned}}}$

Think that in this particular case ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ are angles with respect to the plane of action of ${\displaystyle \sigma _{10'}}$ (oriented in the ${\displaystyle x'}$ -axis)and not angles with respect to the aeroplane of action of ${\displaystyle \sigma _{x}}$ (oriented in the ${\displaystyle x}$ -centrality).

Using the Pole approach, we kickoff localize the Pole or origin of planes. For this, nosotros draw through signal 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 ix). 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 ${\displaystyle \sigma _{10'}}$ acts. This line would also intersect the Mohr circle at the Pole (Effigy 9).

From the Pole, we depict lines to different points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle signal 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 point C in the circle has the same inclination every bit the plane in the physical space where ${\displaystyle \sigma _{1}}$ acts. This airplane makes an angle of 63.435° with airplane B, both in the Mohr-circle infinite and in the physical space. In the same mode, lines are traced from the Pole to points Eastward, D, F, M and H to find the stress components on planes with the same orientation.

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

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

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

Because the main axes as the coordinate system, instead of the general ${\displaystyle x_{ane}}$ , ${\displaystyle x_{2}}$ , ${\displaystyle x_{3}}$ coordinate organisation, and assuming that ${\displaystyle \sigma _{1}>\sigma _{2}>\sigma _{3}}$ , so 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

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

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

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

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

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

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

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

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

These expressions can be rewritten as

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

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

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

See also [edit]

• Critical plane analysis

References [edit]

1. ^ "Chief stress and chief plane". www.engineeringapps.net . Retrieved 2019-12-25 .
2. ^ Parry, Richard Hawley Grayness (2004). Mohr circles, stress paths and geotechnics (two ed.). Taylor & Francis. pp. one–30. ISBN0-415-27297-ane.
3. ^ Gere, James M. (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. ISBN0-07-112939-1.
• Brady, B.H.Thousand.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. Due south. (1996). Elasticity and geomechanics. Cambridge University Printing. pp. 16–26. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil engineering and technology mechanics serial. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, N.G.W.; Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. nine–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 Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN0-415-27297-one.
• Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGraw-Hill International Editions. ISBN0-07-085805-v.
• Timoshenko, Stephen P. (1983). History of force 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 Circumvolve and more circles past Rebecca Brannon
• DoITPoMS Teaching 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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