Drawing a Mohrs Circle

Geometric civil engineering adding technique

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

Mohr'south circle is a two-dimensional graphical representation of the transformation police force for the Cauchy stress tensor.

Mohr'southward circumvolve is oft used in calculations relating to mechanical applied science for materials' strength, geotechnical engineering science for strength of soils, and structural applied science for strength of congenital structures. Information technology is besides 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'south circle tin can also be used to find the principal planes and the principal stresses in a graphical representation, and is i of the easiest ways to do so.^[1]

Afterward performing a stress assay on a material body causeless as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate organisation. The Mohr circumvolve is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented airplane passing through that point.

The abscissa and ordinate ( ${\displaystyle \sigma _{\mathrm {n} }}$ , ${\displaystyle \tau _{\mathrm {north} }}$ ) of each point on the circumvolve 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 stand for the country of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress chemical element.

19th-century German 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 fellow German engineer Christian Otto Mohr (the circle'southward namesake), who extended it to both two- and three-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 indicate include the Lamé's stress ellipsoid and Cauchy's stress quadric.

The Mohr circle can 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 fabric body assumed as a continuum.

Internal forces are produced between the particles of a deformable object, causeless equally a continuum, as a reaction to applied external forces, i.e., either surface forces or torso forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of motion for a particle. A mensurate 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 engineering science, e.thou., structural, mechanical, or geotechnical, the stress distribution within an object, for case stresses in a stone mass around a tunnel, aeroplane wings, or building columns, is determined through a stress assay. Calculating the stress distribution implies the decision of stresses at every indicate (material particle) in the object. According to Cauchy, the stress at any point in an object (Figure 2), assumed as a continuum, is completely defined past the 9 stress components ${\displaystyle \sigma _{ij}}$ of a 2d order tensor of blazon (2,0) known as the Cauchy stress tensor, ${\displaystyle {\boldsymbol {\sigma }}}$ :

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

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

After the stress distribution within the object has been determined with respect to a coordinate system ${\displaystyle (x,y)}$ , it may be necessary to summate the components of the stress tensor at a particular textile betoken ${\displaystyle P}$ with respect to a rotated coordinate system ${\displaystyle (10',y')}$ , i.e., the stresses acting on a plane with a different orientation passing through that point of involvement —forming an angle with the coordinate system ${\displaystyle (10,y)}$ (Figure 3). For case, it is of interest to observe the maximum normal stress and maximum shear stress, likewise as the orientation of the planes where they act upon. To accomplish this, information technology is necessary to perform a tensor transformation under a rotation of the coordinate arrangement. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation constabulary. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.

Mohr's circumvolve for 2-dimensional state of stress [edit]

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

In 2 dimensions, the stress tensor at a given fabric point ${\displaystyle P}$ with respect to any ii perpendicular directions is completely defined past simply 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 angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated. This symmetry implies that ${\displaystyle \tau _{xy}=\tau _{yx}}$ . Thus, the Cauchy stress tensor tin be written as:

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

The objective is to use the Mohr circumvolve to find the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on a rotated coordinate arrangement ${\displaystyle (ten',y')}$ , i.e., on a differently oriented plane passing through ${\displaystyle P}$ and perpendicular to the ${\displaystyle x}$ - ${\displaystyle y}$ aeroplane (Figure four). 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 aeroplane stress and plane strain, start consider a two-dimensional infinitesimal material element effectually a material point ${\displaystyle P}$ (Effigy four), with a unit expanse in the direction parallel to the ${\displaystyle y}$ - ${\displaystyle z}$ aeroplane, i.e., perpendicular to the page or screen.

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

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

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

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

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

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

Derivation of Mohr's circle parametric equations - Tensor transformation

The stress tensor transformation police force can be stated as ${\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}'&=\mathbf {A} {\boldsymbol {\sigma }}\mathbf {A} ^{T}\\\left[{\brainstorm{matrix}\sigma _{x'}&\tau _{x'y'}\\\tau _{y'10'}&\sigma _{y'}\\\end{matrix}}\right]&=\left[{\begin{matrix}a_{ten}&a_{xy}\\a_{yx}&a_{y}\\\end{matrix}}\correct]\left[{\begin{matrix}\sigma _{10}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\end{matrix}}\right]\left[{\begin{matrix}a_{x}&a_{yx}\\a_{xy}&a_{y}\\\end{matrix}}\correct]\\&=\left[{\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{matrix}}\correct]\left[{\begin{matrix}\sigma _{ten}&\tau _{xy}\\\tau _{yx}&\sigma _{y}\\\stop{matrix}}\right]\left[{\begin{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\terminate{matrix}}\right]\end{aligned}}}$ Expanding the right hand side, and knowing that ${\displaystyle \sigma _{x'}=\sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{x'y'}=\tau _{\mathrm {n} }}$ , we have: ${\displaystyle \sigma _{\mathrm {n} }=\sigma _{x}\cos ^{2}\theta +\sigma _{y}\sin ^{2}\theta +2\tau _{xy}\sin \theta \cos \theta }$ However, knowing that ${\displaystyle \cos ^{ii}\theta ={\frac {1+\cos 2\theta }{2}},\qquad \sin ^{two}\theta ={\frac {1-\cos 2\theta }{2}}\qquad {\text{and}}\qquad \sin ii\theta =2\sin \theta \cos \theta }$ nosotros obtain ${\displaystyle \sigma _{\mathrm {due north} }={\frac {i}{2}}(\sigma _{x}+\sigma _{y})+{\frac {1}{2}}(\sigma _{ten}-\sigma _{y})\cos ii\theta +\tau _{xy}\sin 2\theta }$ ${\displaystyle \tau _{\mathrm {n} }=-(\sigma _{x}-\sigma _{y})\sin \theta \cos \theta +\tau _{xy}\left(\cos ^{2}\theta -\sin ^{2}\theta \right)}$ However, knowing that ${\displaystyle \cos ^{two}\theta -\sin ^{2}\theta =\cos 2\theta \qquad {\text{and}}\qquad \sin 2\theta =2\sin \theta \cos \theta }$ we obtain ${\displaystyle \tau _{\mathrm {n} }=-{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$ Information technology is non necessary at this moment to calculate the stress component ${\displaystyle \sigma _{y'}}$ acting on the plane perpendicular to the plane of action of ${\displaystyle \sigma _{x'}}$ every bit it 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 ${\displaystyle \tau _{\mathrm {n} }}$ are the coordinates. This ways that past choosing a coordinate organization with abscissa ${\displaystyle \sigma _{\mathrm {due north} }}$ and ordinate ${\displaystyle \tau _{\mathrm {n} }}$ , giving values to the parameter ${\displaystyle \theta }$ volition place the points obtained lying on a circle.

Eliminating the parameter ${\displaystyle 2\theta }$ from these parametric equations will yield the not-parametric equation of the Mohr circle. This tin be achieved past rearranging the equations for ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ , first transposing the commencement term in the first equation and squaring both sides of each of the equations and then calculation them. Thus nosotros take

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

where

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

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 ${\displaystyle (\sigma _{\mathrm {north} },\tau _{\mathrm {n} })}$ coordinate system.

Sign conventions [edit]

There are 2 separate sets of sign conventions that need to be considered when using the Mohr Circle: One sign convention for stress components in the "physical infinite", and another for stress components in the "Mohr-Circumvolve-infinite". In addition, within each of the ii set 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 option of a particular sign convention is influenced by convenience for calculation and interpretation for the item problem in mitt. A more detailed explanation of these sign conventions is presented below.

The previous derivation for the equation of the Mohr Circumvolve using Figure iv follows the applied science mechanics sign convention. The engineering mechanics sign convention volition be used for this article.

Physical-infinite sign convention [edit]

From the convention of the Cauchy stress tensor (Effigy 3 and Figure four), the starting time subscript in the stress components denotes the face on which the stress component acts, and the second subscript indicates the direction of the stress component. Thus ${\displaystyle \tau _{xy}}$ is the shear stress acting on the face up with normal vector in the positive direction of the ${\displaystyle x}$ -axis, and in the positive direction of the ${\displaystyle y}$ -axis.

In the physical-infinite sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are inward to the plane of action (pinch) (Effigy 5).

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

Mohr-circumvolve-space sign convention [edit]

Effigy 5. Engineering mechanics sign convention for cartoon the Mohr circumvolve. This commodity follows sign-convention # 3, every bit 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 human action outward to the airplane of action, and negative normal stresses human activity inwards to the airplane of activeness.

Shear stresses, however, have a different convention in the Mohr-circle space compared to the convention in the physical infinite. In the Mohr-circle-infinite 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 fashion, 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 drawing the Mohr-circle space, 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 downward, i.e., the ${\displaystyle \tau _{\mathrm {n} }}$ -axis is inverted (Effigy 5, sign convention #ii).

Plotting positive shear stresses upward makes the angle ${\displaystyle two\theta }$ on the Mohr circle accept a positive rotation clockwise, which is reverse to the concrete space convention. That is why some authors^[iii] adopt plotting positive shear stresses downward, which makes the bending ${\displaystyle two\theta }$ on the Mohr circle have a positive rotation counterclockwise, similar to the physical infinite convention for shear stresses.

To overcome the "issue" of having the shear stress axis downward in the Mohr-circle space, there 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 causeless to rotate the material element in the counterclockwise direction (Figure v, option 3). This manner, positive shear stresses are plotted up in the Mohr-circle space and the bending ${\displaystyle ii\theta }$ has a positive rotation counterclockwise in the Mohr-circle infinite. This alternative sign convention produces a circle that is identical to the sign convention #2 in Figure v because a positive shear stress ${\displaystyle \tau _{\mathrm {northward} }}$ is besides 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 upward.

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

Cartoon Mohr's circle [edit]

Assuming we know the stress components ${\displaystyle \sigma _{x}}$ , ${\displaystyle \sigma _{y}}$ , and ${\displaystyle \tau _{xy}}$ at a signal ${\displaystyle P}$ in the object under 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 organisation ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ with a horizontal ${\displaystyle \sigma _{\mathrm {northward} }}$ -axis and a vertical ${\displaystyle \tau _{\mathrm {n} }}$ -axis.
2. Plot 2 points ${\displaystyle A(\sigma _{y},\tau _{xy})}$ and ${\displaystyle B(\sigma _{x},-\tau _{xy})}$ in the ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {northward} })}$ space respective to the known stress components on both perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ , respectively (Figure 4 and six), post-obit the called sign convention.
3. Describe the bore of the circumvolve past joining points ${\displaystyle A}$ and ${\displaystyle B}$ with a directly line ${\displaystyle {\overline {AB}}}$ .
4. Describe the Mohr Circle. The centre ${\displaystyle O}$ of the circle is the midpoint of the diameter line ${\displaystyle {\overline {AB}}}$ , which corresponds to the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ axis.

Finding principal normal stresses [edit]

Stress components on a second rotating element. Example of how stress components vary on the faces (edges) of a rectangular chemical 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 instance, when the rectangle is horizontal, the stresses are given by ${\displaystyle \left[{\brainstorm{matrix}\sigma _{20}&\tau _{xy}\\\tau _{yx}&\sigma _{yy}\end{matrix}}\right]=\left[{\brainstorm{matrix}-10&ten\\10&15\end{matrix}}\right].}$ The corresponding Mohr'due south 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 6) where the circle intersects the ${\displaystyle \sigma _{\mathrm {north} }}$ -axis. The magnitude of the major main stress ${\displaystyle \sigma _{1}}$ is ever the greatest accented value of the abscissa of whatever of these two points. Besides, the magnitude of the minor primary stress ${\displaystyle \sigma _{ii}}$ is always the everyman absolute value of the abscissa of these ii points. As expected, the ordinates of these two points are nil, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the principal stresses can be plant by

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

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

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

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

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

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

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 circumvolve with the vertical line passing through the heart of the circle, ${\displaystyle O}$ . Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circle's radius ${\displaystyle R}$

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

Finding stress components on an capricious aeroplane [edit]

As mentioned before, later 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 two perpendicular planes ${\displaystyle A}$ and ${\displaystyle B}$ passing through ${\displaystyle P}$ as shown in Figure 5 and 6. The Mohr circle is used to notice the stress components ${\displaystyle \sigma _{\mathrm {n} }}$ and ${\displaystyle \tau _{\mathrm {northward} }}$ , i.east., coordinates of any signal ${\displaystyle D}$ on the circle, acting on any other plane ${\displaystyle D}$ passing through ${\displaystyle P}$ making an angle ${\displaystyle \theta }$ with the plane ${\displaystyle B}$ . For this, 2 approaches tin can exist used: the double angle, and the Pole or origin of planes.

Double angle [edit]

As shown in Effigy 6, to decide the stress components ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ acting on a plane ${\displaystyle D}$ at an angle ${\displaystyle \theta }$ counterclockwise to the airplane ${\displaystyle B}$ on which ${\displaystyle \sigma _{x}}$ acts, we travel an bending ${\displaystyle 2\theta }$ in the same counterclockwise direction around the circumvolve from the known stress point ${\displaystyle B(\sigma _{10},-\tau _{xy})}$ to indicate ${\displaystyle D(\sigma _{\mathrm {northward} },\tau _{\mathrm {n} })}$ , i.e., an angle ${\displaystyle ii\theta }$ between lines ${\displaystyle {\overline {OB}}}$ and ${\displaystyle {\overline {OD}}}$ in the Mohr circle.

The double angle approach relies on the fact that the angle ${\displaystyle \theta }$ between the normal vectors to any two physical planes passing through ${\displaystyle P}$ (Figure 4) is half the bending betwixt ii lines joining their corresponding stress points ${\displaystyle (\sigma _{\mathrm {n} },\tau _{\mathrm {n} })}$ on the Mohr circle and the centre of the circle.

This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of ${\displaystyle 2\theta }$ . It tin can also be seen that the planes ${\displaystyle A}$ and ${\displaystyle B}$ in the cloth chemical element around ${\displaystyle P}$ of Figure 5 are separated by an bending ${\displaystyle \theta =xc^{\circ }}$ , which in the Mohr circle is represented by a ${\displaystyle 180^{\circ }}$ angle (double the angle).

Pole or origin of planes [edit]

Figure vii. Mohr's circle for plane stress and plane strain conditions (Pole arroyo). Any direct line fatigued from the pole volition intersect the Mohr circle at a bespeak that represents the state of stress on a airplane inclined at the same orientation (parallel) in space equally that line.

The 2nd approach involves the determination of a indicate on the Mohr circle called the pole or the origin of planes. Whatever straight line fatigued from the pole volition intersect the Mohr circle at a signal that represents the land of stress on a plane inclined at the aforementioned orientation (parallel) in infinite as that line. Therefore, knowing the stress components ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ on any item aeroplane, one tin can describe a line parallel to that aeroplane through the particular coordinates ${\displaystyle \sigma _{\mathrm {northward} }}$ and ${\displaystyle \tau _{\mathrm {n} }}$ on the Mohr circle and find the pole every bit the intersection of such line with the Mohr circle. As an example, let's assume nosotros accept a country of stress with stress components ${\displaystyle \sigma _{x},\!}$ , ${\displaystyle \sigma _{y},\!}$ , and ${\displaystyle \tau _{xy},\!}$ , as shown on Effigy 7. First, we can draw a line from bespeak ${\displaystyle B}$ parallel to the plane of action of ${\displaystyle \sigma _{x}}$ , or, if nosotros choose otherwise, a line from signal ${\displaystyle A}$ parallel to the plane of action of ${\displaystyle \sigma _{y}}$ . The intersection of any of these two lines with the Mohr circle is the pole. Once the pole has been determined, to detect 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 tin can draw a line from the pole parallel to that aeroplane (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 circumvolve.

Finding the orientation of the master planes [edit]

The orientation of the planes where the maximum and minimum principal stresses act, besides known as 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 betwixt ${\displaystyle {\overline {OB}}}$ 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 also be constitute from the post-obit equation

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

This equation defines two values for ${\displaystyle \theta _{\mathrm {p} }}$ which are ${\displaystyle xc^{\circ }}$ apart (Effigy). This equation can be derived direct from the geometry of the circumvolve, or by making the parametric equation of the circle for ${\displaystyle \tau _{\mathrm {n} }}$ equal to zero (the shear stress in the main planes is always nil).

Example [edit]

Assume a material element nether a land of stress as shown in Figure eight and Figure 9, with the aeroplane of one of its sides oriented 10° with respect to the horizontal airplane. Using the Mohr circumvolve, observe:

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

Bank 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 material element in this instance are:

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

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

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

Post-obit the steps for drawing the Mohr circle for this particular state of stress, we beginning draw a Cartesian coordinate arrangement ${\displaystyle (\sigma _{\mathrm {due north} },\tau _{\mathrm {n} })}$ with the ${\displaystyle \tau _{\mathrm {n} }}$ -axis upwardly.

We then plot two points A(50,40) and B(-10,-xl), representing the country of stress at airplane A and B equally show in both Figure 8 and Effigy 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure five), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise. This mode, the shear stress acting on plane B is negative and the shear stress acting on plane A is positive. The diameter of the circle is the line joining signal A and B. The center of the circle is the intersection of this line with the ${\displaystyle \sigma _{\mathrm {n} }}$ -axis. Knowing both the location of the centre and length of the diameter, nosotros are able to plot the Mohr circle for this detail state of stress.

The abscissas of both points E and C (Effigy 8 and Figure 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 interim on both the modest and major master planes, respectively, which is zero for master planes.

Even though the idea for using the Mohr circumvolve is to graphically find different stress components past really measuring the coordinates for dissimilar points on the circle, information technology is more convenient to confirm the results analytically. Thus, the radius and the abscissa of the eye of the circle are

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

${\displaystyle {\begin{aligned}\sigma _{\mathrm {avg} }&={\tfrac {one}{2}}(\sigma _{x}+\sigma _{y})\\&={\tfrac {1}{ii}}(-x+50)\\&=20{\textrm {MPa}}\\\end{aligned}}}$

and the principal stresses are

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

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

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

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

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

We can choose to either use the double angle approach (Figure viii) or the Pole arroyo (Figure ix) to find the orientation of the primary normal stresses and master shear stresses.

Using the double bending arroyo we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Figure 8) to notice double the angle the major main stress and the minor 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 can use the belittling expression

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

One solution is: ${\displaystyle ii\theta _{p}=-53.xiii^{\circ }}$ . From inspection of Effigy viii, this value corresponds to the angle ∠BOE. Thus, the pocket-sized principal angle is

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

Then, the major master angle is

${\displaystyle {\begin{aligned}2\theta _{p1}&=180-53.13^{\circ }=126.87^{\circ }\\\theta _{p1}&=63.435^{\circ }\\\terminate{aligned}}}$

Remember that in this particular example ${\displaystyle \theta _{p1}}$ and ${\displaystyle \theta _{p2}}$ are angles with respect to the plane of action of ${\displaystyle \sigma _{10'}}$ (oriented in the ${\displaystyle x'}$ -centrality)and non angles with respect to the plane of action of ${\displaystyle \sigma _{x}}$ (oriented in the ${\displaystyle x}$ -axis).

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

From the Pole, we draw lines to dissimilar points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a 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 aforementioned inclination as the aeroplane in the physical infinite where ${\displaystyle \sigma _{1}}$ acts. This aeroplane makes an angle of 63.435° with plane B, both in the Mohr-circle space and in the concrete space. In the same mode, lines are traced from the Pole to points E, D, F, G and H to detect the stress components on planes with the same orientation.

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

Effigy 10. Mohr's circle for a 3-dimensional country of stress

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

Considering the principal axes as the coordinate system, instead of the full general ${\displaystyle x_{one}}$ , ${\displaystyle x_{ii}}$ , ${\displaystyle x_{3}}$ coordinate system, and bold that ${\displaystyle \sigma _{1}>\sigma _{ii}>\sigma _{3}}$ , then 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

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

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

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

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

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

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

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

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

These expressions can exist rewritten equally

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

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

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

See as well [edit]

• Critical airplane assay

References [edit]

1. ^ "Master stress and primary aeroplane". 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. 1–xxx. ISBN0-415-27297-1.
3. ^ Gere, James Chiliad. (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.G.; East.T. Brownish (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN0-412-47550-2.
• Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge Academy Press. pp. 16–26. ISBN0-521-49827-9.
• Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil applied science and engineering mechanics series. Prentice-Hall. ISBN0-13-484394-0.
• Jaeger, John Conrad; Cook, Due north.K.Due west.; Zimmerman, R.W. (2007). Fundamentals of stone mechanics (Fourth ed.). Wiley-Blackwell. pp. 9–41. ISBN978-0-632-05759-7.
• Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co. ISBN0-442-04199-3.
• Parry, Richard Hawley Gray (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. one–30. ISBN0-415-27297-1.
• 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 cursory account of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN0-486-61187-6.

External links [edit]

• Mohr'southward Circle and more than circles by Rebecca Brannon
• DoITPoMS Teaching and Learning Package- "Stress Analysis and Mohr'due south Circle"

gylescrist2001.blogspot.com

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

Share :