For the Plane of Stress Below Draw Mohr Circle
Intro and Derivation
Mohr's circle is a geometric representation of airplane (2d) stress transformation and allows usa to quickly visualize how the normal (σ) and shear (τ) stress components modify equally their plane changes orientation.
German civil engineer Otto Mohr developed this method from the practiced ol' stress transformation equations. Recall:
If we remove θ by squaring both sides of each equation and so add the two equations together, we get:
Later on defining σavg and R, we can modify Equation three 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 pinch is negative . Shear stress (τ) is illustrated below. Geotechnical engineers may use the reverse sign conventions, because they more often than not 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 centrality). 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'south circles. The general idea of this arroyo is that angles between radial lines in the Mohr's circle are twice the actual angles between the existent planes. In other words, a 40° rotation on the plane corresponds to an 80° rotation on the circle in the same management.
We adopt thePole Method, which is based on a unique point on the Mohr's circle known equally thepole. This pole is unique, because whatsoever straight line fatigued through the pole intersects the Mohr's circumvolve at a bespeak representing the country of stress on a plane with the same orientation every bit the line. Every bit shown in the figure above, a line drawn through the pole and some stress point (σ,𝜏) on the circle isexactly parallel to the aeroplane with corresponding σ and 𝜏, making this arroyo very intuitive! Permit u.s. illustrate how the pole method works.
To draw a Mohr's circumvolve, we need to know the aeroplane land of stresses for an chemical element. Consider this:
one. Plot the Stresses
Immediately, we know the two points that form the diameter of the circumvolve, (σx,𝜏xy) = (80,40) and (σy,𝜏yx) = (xx,-40). (If you are request why 𝜏yx is -40, remember the sign convention for shear stress!) The heart of the circle is the halfway point between σten and σy, or σavg.
2. Locate the Pole
Draw a straight line from ane of the known stress points on the circumvolve (σ,τ) in the direction of the airplane on which (σ,τ) acts. Starting at point (eighty,40), a line is drawn parallel to the aeroplane on which (80,40) acts. This is represented by the vertical dotted green line in the figure below. Starting at point (20,-40), a second line is drawn parallel to the aeroplane on which (xx,-40) acts. This is represented by the horizontal dotted pink line in the same figure below. The pole is at the intersection of these dotted lines on the circle. Notation that in that location is only Ane pole!
3. Find Transformed Stresses (e.g. 30° CCW Rotation)
To find other states of stresses, we showtime drawing the lines from the pole instead of a known stress point (σ,τ). This is the general procedure:
- Draw a line from the pole in the direction of orientation of the transformed airplane.
- The point where this line intersects the circumvolve represents the state of stress interim on that plane.
Let us presume a xxx° counterclockwise (CCW) rotation of the element. Starting at the pole, to find the transformed counterpart of (20,-40), we draw a new line that is 30° CCW from the existing dotted pinkish line. The transformed stresses on this plane is where the new pink line intersects the circle. Similarly, to find the transformed counterpart of (fourscore,40), we draw a new line from the pole that is 30° CCW from the existing dotted dark-green line. The transformed stresses on this aeroplane is where the new dark-green line intersects the circumvolve. Staying consistent with the rules, the new lines are parallel to their corresponding planes. The transformed stresses are:
To get these transformed stress values by hand, we would need a protractor to mensurate the angle and a ruler to connect the pole to the signal of intersection on the circumvolve. This can seem like an elaborate process, and information technology goes to testify that hand-computing plane stress transformations via Mohr'southward circle is only an guess method. To become the exact transformed stress values, nosotros can apply the good ol' stress transformation equations.
iv. Check with Belittling Method
Using the the stress transformation equations (Equations 1 and 2):
Considering the centre of the circle, or σavg, is the same (the circle does NOT move!), we tin can find σy'.
v. Principal Stresses
Principal stresses act on planes where τ = 0. The larger master stress is called themajor principal stress, and the smaller principal stress is called thepocket-sized principal stress.
Similar to finding transformed stresses, we draw lines from the pole to where τ = 0, or the two "10-intercepts" on the circle. The major and pocket-size primary stresses are 100MPa and 0MPa, respectively. The angle, or the rotation required to reach cipher shear stress on the plane, is measured between either pair of the original (dotted) line and the new line connecting the pole to the "x-intercept".
Square vs. Triangle
Some textbooks show stress elements every bit squares (like in this article), while others show them as triangles. They are two different representations of the same element, because the triangles are just corner cutouts from the square, equally shown below by the two a-a and b-b sections.
The advantage of the triangular representation is that we tin visually identify the transformed stresses on the hypotenuse. However, since we are dealing with two transformed normal stresses (σx' and σy'), we still need 2 triangles to requite 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 as the plane on which they deed changes orientation. Mohr'southward 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 determine whatever transformed airplane state of stress (σ',τ'), we ALWAYS get-go at the pole and draw a line parallel to the airplane. The point of intersection on the circle is (σ',τ').
Source: https://structnotes.com/2018/06/23/mohrs-circle/
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