Draw a Circle C on a Plane
Intro and Derivation
Mohr's circumvolve is a geometric representation of plane (2D) stress transformation and allows united states of america to quickly visualize how the normal (σ) and shear (τ) stress components change as their plane changes orientation.
German civil engineer Otto Mohr adult this method from the expert ol' stress transformation equations. Call up:
If we remove θ by squaring both sides of each equation and so add the ii equations together, nosotros get:
After defining σavg and R, nosotros tin can modify Equation 3 to get Equation 4, which is the equation for a circle with center (σavg,0) and radius R.
Sign Convention
The sign convention for normal stress (σ) is that tension is positive and compression is negative . Shear stress (τ) is illustrated below. Geotechnical engineers may employ the reverse sign conventions, considering they mostly deal with compressive stress.
Mohr's circle is drawn with normal stress (σ) plotted on the abscissa (horizontal axis) and shear stress (τ) plotted on the ordinate (vertical axis). Normal stress (σ) is positive to the right, and shear stress (τ) is positive downward.
Pole Method
Most mechanics of materials textbooks prefer using the Double Angle Method to draw Mohr's circles. The full general thought of this arroyo is that angles betwixt radial lines in the Mohr's circle are twice the bodily angles betwixt the real planes. In other words, a forty° rotation on the plane corresponds to an fourscore° rotation on the circumvolve in the aforementioned direction.
We prefer thePole Method, which is based on a unique point on the Mohr's circle known as thepole. This pole is unique, because any directly line drawn through the pole intersects the Mohr's circumvolve at a bespeak representing the state of stress on a airplane with the same orientation as the line. As shown in the figure to a higher place, a line drawn through the pole and some stress point (σ,𝜏) on the circle isexactly parallel to the airplane with corresponding σ and 𝜏, making this approach very intuitive! Let us illustrate how the pole method works.
To draw a Mohr'southward circle, nosotros demand to know the airplane state of stresses for an element. Consider this:
1. Plot the Stresses
Immediately, nosotros know the two points that form the diameter of the circumvolve, (σx,𝜏xy) = (80,40) and (σy,𝜏yx) = (twenty,-40). (If you are asking why 𝜏yx is -twoscore, remember the sign convention for shear stress!) The middle of the circle is the halfway signal between σten and σy, or σavg.
ii. Locate the Pole
Draw a straight line from one of the known stress points on the circle (σ,τ) in the direction of the plane on which (σ,τ) acts. Starting at point (80,40), a line is drawn parallel to the plane on which (80,40) acts. This is represented past the vertical dotted green line in the effigy beneath. Starting at point (xx,-40), a second line is drawn parallel to the plane on which (xx,-40) acts. This is represented by the horizontal dotted pink line in the same figure beneath. The pole is at the intersection of these dotted lines on the circle. Note that there is simply One pole!
3. Notice Transformed Stresses (e.g. 30° CCW Rotation)
To notice other states of stresses, we beginning drawing the lines from the pole instead of a known stress bespeak (σ,τ). This is the general procedure:
- Draw a line from the pole in the direction of orientation of the transformed plane.
- The point where this line intersects the circle represents the land of stress interim on that plane.
Let u.s. assume a 30° counterclockwise (CCW) rotation of the element. Starting at the pole, to detect the transformed counterpart of (20,-40), nosotros depict a new line that is xxx° CCW from the existing dotted pink line. The transformed stresses on this plane is where the new pink line intersects the circle. Similarly, to detect the transformed counterpart of (80,forty), we depict a new line from the pole that is thirty° CCW from the existing dotted green line. The transformed stresses on this plane is where the new greenish line intersects the circle. Staying consistent with the rules, the new lines are parallel to their corresponding planes. The transformed stresses are:
To go these transformed stress values by hand, we would need a protractor to measure the bending and a ruler to connect the pole to the indicate of intersection on the circumvolve. This can seem like an elaborate procedure, and it goes to show that hand-computing plane stress transformations via Mohr'south circle is simply an approximate method. To get the verbal transformed stress values, we tin use the proficient ol' stress transformation equations.
4. Cheque with Belittling Method
Using the the stress transformation equations (Equations 1 and 2):
Because the center of the circle, or σavg, is the same (the circumvolve does Not move!), we can find σy'.
5. Principal Stresses
Principal stresses act on planes where τ = 0. The larger principal stress is chosen themajor master stress, and the smaller principal stress is chosen theminor chief stress.
Similar to finding transformed stresses, we draw lines from the pole to where τ = 0, or the two "x-intercepts" on the circumvolve. The major and minor primary stresses are 100MPa and 0MPa, respectively. The angle, or the rotation required to reach zero shear stress on the aeroplane, is measured between either pair of the original (dotted) line and the new line connecting the pole to the "10-intercept".
Square vs. Triangle
Some textbooks testify stress elements as squares (like in this commodity), while others show them equally triangles. They are two different representations of the aforementioned element, because the triangles are merely corner cutouts from the foursquare, equally shown beneath by the two a-a and b-b sections.
The advantage of the triangular representation is that we can visually identify the transformed stresses on the hypotenuse. However, since we are dealing with two transformed normal stresses (σ10' and σy'), we still demand 2 triangles to give complete representation of plane stresses. In the effigy beneath, ΔA is the sectioned area (length of hypotenuse × depth of 3D stress block).
TL;DR
- Stresses change every bit the aeroplane on which they act changes orientation. Mohr's circle "maps" these changes.
- Keeping a consistent sign convention is important.
- In Pole Method, things are determined in this order: (σ,τ) → pole → (σ',τ').
- To make up one's mind whatever transformed plane state of stress (σ',τ'), we ALWAYS beginning at the pole and draw a line parallel to the plane. The point of intersection on the circumvolve is (σ',τ').
Source: https://structnotes.com/2018/06/23/mohrs-circle/
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