Mohr’s Circle

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Mohr’s CircleAcademic Resource Center

Introduction The transformation equations for plane stress canbe represented in graphical form by a plot knownas Mohr’s Circle. This graphical representation is extremely usefulbecause it enables you to visualize therelationships between the normal and shearstresses acting on various inclined planes at apoint in a stressed body. Using Mohr’s Circle you can also calculate principalstresses, maximum shear stresses and stresses oninclined planes.

Stress Transformation Equationssx 1s x s y s x -s ycos2q t xy sin 2q22s x s yt x1y1 sin 2q t xy cos2q212

Derivation of Mohr’s Circle If we vary θ from 0 to 360 , we will get allpossible values of σx1 and τx1y1 for a givenstress state. Eliminate θ by squaring both sides of 1 and 2equation and adding the two equationstogether.

Derivation of Mohr’s Circle (cont’d)

Mohr’s Circle Equation The circle with that equation is called a Mohr’sCircle, named after the German Civil EngineerOtto Mohr. He also developed the graphicaltechnique for drawing the circle in 1882. The graphical method is a simple & clear approachto an otherwise complicated analysis.

Sign Convention for Mohr’s Circle Shear Stress is plotted as positive downward θ on the stress element 2θ in Mohr’s circle

Constructing Mohr’s Circle:Procedure1. Draw a set of coordinate axes with σx1 as positiveto the right and τx1y1 as positive downward.2. Locate point A, representing the stress conditionson the x face of the element by plotting itscoordinates σx1 σx and τx1y1 τxy. Note thatpoint A on the circle corresponds to θ 0 .3. Locate point B, representing the stress conditionson the y face of the element by plotting itscoordinates σx1 σy and τx1y1 -τxy. Note thatpoint B on the circle corresponds to θ 90 .

Procedure (cont’d)4.Draw a line from point A to point B, a diameter of the circlepassing through point c (center of circle). Points A and B are atopposite ends of the diameter (and therefore 180 apart on thecircle).5. Using point c as the center, draw Mohr’s circlethrough points A and B. This circle has radius R.The center of the circle c at the point havingcoordinates σx1 σavg and τx1y1 0.

Stress Transformation: GraphicalIllustration

Explanation On Mohr’s circle, point A corresponds to θ 0. Thusit’s the reference point from which angles aremeasured. The angle 2θ locates the point D on the circle, whichhas coordinates σx1 and τx1y1. D represents thestresses on the x1 face of the inclined element. Point E, which is diametrically opposite point D islocated 180 from cD. Thus point E gives the stresson the y1 face of the inclined element. Thus, as we rotate the x1y1 axes counterclockwise byan angle θ, the point on Mohr’s circle correspondingto the x1 face moves ccw by an angle of 2θ.

Explanation Principle stresses are stresses that act on aprinciple surface. This surface has no shear forcecomponents (that means τx1y1 0) This can be easily done by rotating A and B to theσx1 axis. σ1 stress on x1 surface, σ2 stress on y1 surface. The object in reality has to be rotated at an angleθp to experience no shear stress.

Explanation The same method to calculate principle stresses is used to findmaximum shear stress. Points A and B are rotated to the point of maximum τx1y1value. This isthe maximum shear stress value τmax. Uniform planar stress (σs) and shear stress (τmax) will be experiencedby both x1 and y1 surfaces. The object in reality has to be rotated at an angle θs to experiencemaximum shear stress.

Example 1Draw the Mohr’s Circle of the stress element shownbelow. Determine the principle stresses and themaximum shear stresses.What we know:σx -80 MPaσy 50 MPaτxy 25 MPaCoordinates of PointsA: (-80,25)B: (50,-25)

Example 1 (cont’d)

Example 1 (cont’d)Principle Stress:

Example 1 (cont’d)Maximum Shear Stress:

Example 2Given the same stress element (shown below), findthe stress components when it is inclined at 30 clockwise. Draw the corresponding stress elements.What we know:σx -80 MPaσy 50 MPaτxy 25 MPaCoordinates of PointsA: (-80,25)B: (50,-25)

Example 2 (cont’d)Using stress transformation equation (θ 30 ):sx -s x s y1 s x -s ycos2q t xy sin 2q22s x s yt x1y1 sin 2q t xy cos2q2σx -25.8 MPaσy -4.15 MPaτxy 68.8 MPa

Example 2 (cont’d)Graphical approach using Mohr’s Circle (and trigonometry)

principle surface. This surface has no shear force components (that means τx 1 y 1 0) This can be easily done by rotating A and B to the σx 1 axis. σ 1 stress on x 1 surface, σ 2 stress on y 1 surface. The object in reality has to be rotated at an angle θ p to experience no shear stress.

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Those who already know Mohr’s circle may recall that an angle gets dou-bled when portrayed in Mohr’s circle , which can be very confusing. Section 2 introduces a little known enhancement to Mohr’s circle (namely, the Pole Po int) that rectifies this prob - lem. Some engineering applications of the 2D Mohr

Mohr's Circle is still widely used by engineers all over the world. Derivation of Mohr's Circle To establish Mohr's Circle, we first recall the stress transformation formulas for plane stress at a given location, Using a basic trigonometric relation (cos22θ sin22θ 1) to combine the two above equations we have,

The radius of this circle (default: 1 .0). The number of circle objects created. Constructs a default circle object. Constructs a circle object with the specified radius. Returns the radius of this circle. Sets a new radius for this circle. Returns the number of circle objects created. Returns the area of this circle. T he - sign indicates

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screening, ad design, and layout will be charged to the Advertiser. Closing Dates:Check CARNEGIE magazine Editorial Calendar or contact Mohr & More Media Sales at 412.366.2080. Send all artwork, contracts, and copy to: Terry Mohr Mohr & More Media Sales 103 Ennerdale Lane Pittsburgh, PA

screening, ad design, and layout will be charged to the Advertiser. Closing Dates:Check CARNEGIE magazine Editorial Calendar or contact Mohr & More Media Sales at 412.366.2080. Send all artwork, contracts, and copy to: Terry Mohr Mohr & More Media Sales 103 Ennerdale Lane Pittsburgh, PA

Exegetische Studien zum lukanischen Verständnis von ßaodsia xoD Oeoö von Alexander Prieur J.C.B. Mohr (Paul Siebeck) Tübingen . Die Deutsche Bibliothek - CIP-Einheitsaufnahme Prieur, Alexander: Die Verkündigung der Gottesherrschaft: exegetische Studien zum lukanischen Verständnis von basileia tu theu / von Alexander Prieur. - Tübingen: Mohr, 1996 (Wissenschaftliche Untersuchungen zum .

6) An element in plane stress on the surface of a part is subjected to the following stresses: σ x 35ksi tension, σ y 15ksi compression, τ xy 10psi. Create Mohr’s circle, and from it, determine the principal stresses and the maximum shear stress. Label the principal stresses and maximum shear stress on the Mohr’s circle.