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  • 1. MIT, Unified (16.001/16.002) Materials and Structures Fall, 2008 UNIFIED HANDOUT MATERIALS AND STRUCTURES - #M-6 Fall, 2008 Mohr’s Circle Mohr’s circle is a geometric representation of the 2-D transformation of stresses and is very useful to perform quick and efficient estimations, checks of more extensive work, and other such uses. (Note: a similar formulation can be used for tensorial strain) CONSTRUCTION: Given the following state of stress: y2 y1 with the definition (by Mohr) of positive and negative shear: “Positive shear would cause a clockwise rotation of the infinitesimal element about the element center.” Thus, from the illustration above, σ12 is plotted negative on Mohr's circle, and σ21 is plotted positive on Mohr's circle. Paul A. Lagace  2007 Handout M-6 p. 1
  • 2. MIT, Unified (16.001/16.002) Materials and Structures Fall, 2008 • Begin the construction by doing the following: 1. Plot σ11, - σ12 as point A 2. Plot σ22, σ21 as point B 3. Connect A and B Step 2 Step 3 Step 1 • Then complete the circle by doing Step 4: 4. Draw a circle of diameter of the line AB about the point where the line AB crosses the horizontal axis (denote this as point C) Step 4 Extensional Paul A. Lagace  2007 Handout M-6 p. 2
  • 3. MIT, Unified (16.001/16.002) Materials and Structures Fall, 2008 USE OF THE CONSTRUCTION: • To read off stresses for a rotated system: 1. Note that the vertical axis is the shear stress axis and the horizontal axis is the extensional stress axis. 2. Positive rotations are measured counterclockwise as referenced to the original system and thus to the line AB. y2 ~ y2 ~ y1 REAL SPACE θ y1 3. Rotate line AB about point C by the angle 2θ where θ is the angle between the unrotated and rotated systems. 4. The points D and E where the rotated line intersects the circle are used to read ~ off the stresses in the rotated system. The vertical location of D is - σ12; the ~ ~ horizontal location of D is σ11. The vertical location of E is σ21 , the horizontal ~ location of E is σ22 (Recall Mohr definition with regard to negative/positive sense of shear stress on Mohr's circle). “MOHR SPACE” ~ –σ ~ σ Extensional ~ σ ~ σ Paul A. Lagace  2007 Handout M-6 p. 3
  • 4. MIT, Unified (16.001/16.002) Materials and Structures Fall, 2008 • We can immediately see the following: 5. The principal stresses, σI and σII , are defined by the points F and G ~ (along the horizontal axis where σ12 = 0). The rotation angle to the principal axis is θp which is 1/2 the angle from the line AB to the horizontal line FG. 6. The maximum shear stress is defined by the points H and H’ which are the endpoints of the vertical line. The line is orthogonal to the principal stress line and thus the maximum shear stress acts along a plane 45° (= 90°/2) from the principal stress system. Mohr's Circle (Principal Stresses) Shear H σ12max B G C 2θp σII Extensional Ext F σI A H' Paul A. Lagace  2007 Handout M-6 p. 4
  • 5. MIT, Unified (16.001/16.002) Materials and Structures Fall, 2008 Full two-dimensional stress transformation equations (θ as on p.3 figure): ~ σ 11 = cos 2 θ σ 11 + sin 2 θ σ 22 + 2 sin θ cos θ σ 12 ~ σ 22 = sin 2 θ σ 11 + cos 2 θ σ 22 − 2 sinθ cos θ σ 12 ~ σ 12 = - sin θ cos θ σ 11 + sin θ cos θ σ 22 + (cos 2θ − sin 2 θ ) σ 12 Note: θ is not the direction cosine angle in the tensor transformation relations, e.g.: Rather, recall: σ αβ = lαθ lβτ σ θτ ˜ ˜ ˜ l~ mn = cosine of angle from ~ m to yn y by convention, angle is measured positive counterclockwise (+ CCW). For the situation developed here for Mohr’s circle, the direction cosines are: l 11 = cos( 360 o − θ) = cos(θ) ˜ l 21 = cos(270 o − θ) = −sin(θ) ˜ l 12 = cos(90 o − θ) = sin(θ) ˜ l 22 = cos( 360 o − θ) = cos(θ) ˜ but this does yield the same set of operating equations! € Paul A. Lagace  2007 Handout M-6 p. 5