1 CHAPTER 2C- Mohr’s Circle Dr. Zuhailawati Hussain EBB 334 Mechanical Metallurgy

2 Mohr’s circle is an ingenious graphical representation of the plane stress transformation equations, Eqs.8.5. It permits an easy visualization of the normal stress and shear stress on arbitrary planes, and it greatly facilitates the solution of plane-stress problem like Example Problem 8.1 through 8.3. MOHR’S CIRCLE FOR PLANE STRESS

3 DERIVATION OF MOHR’S CIRCLE Let us begin our derivation of Mohr’s circle for plane stress by rewriting Eqs.8.5 in the form (8.21)

4 Squaring both sides of each of these equations, and adding the resulting squares, we get Or (8.22)

5 This equation of a circle in (σ,  ) coordinates, with center at (σ avg,0 ) and radius R. The plane-stress transformation equation 8.5 are just parametric equations of a circle, with the parameter being θ, and with the coordinates of point N on the circle representing the normal stress σ n and shear stress  nt on the n plane at orientation θ  θ xn. To determine more of the properties of Mohr’s circle, consider now the circle shown in Fig After discussing many properties of Mohr’s circle, using Fig.8.16, we will suggest a procedure for you to use in constructing a Mohr’s circle from given stress data, and we will illustrate how to solve problems like Example Problem 8.1 through 8.3 using Mohr’s circle.

6 Mohr’s circle is drawn on a set of rectilinear axes with the horizontal axis (axis of abscissas) representing the normal stress σ, and with the vertical axis (axis of ordinates) representing the shear stress, . Note that the positive  axis is downward. The sign convention for θ is the same one that was introduced in section 8.3. The angle θ  θ xn is, as previously, measured counterclockwise from the x axis to the n axis on the body undergoing plane stress. Correspondingly, an angle of 2θ  2θ xn is measured counterclockwise on Mohr’s circle.

7 Every point on Mohr’s Circle corresponds to the stresses σ and  on a particular face; for the generic point N, the stresses are (σ n,  nt ). To emphasize this, we will label points on the circle with the same label as the face represented by that point, except that a capital letter will designate the point X on the circle; the n face is represented by point N on the circle, and so on.

8 To reinforce the sign convention for plotting shear stresses on Mohr’s Circle, the small shear stress icons in Fig.8.16 indicate that the shear stress on a face plots as positive shear (i.e.,plots downward) if the shear stress on the face would tend to rotate the stress element counterclockwise. Conversely, the shear stress on a face plots as negative shear (i.e., plots upward) if the shear stress on that face would tend to rotate a stress element clockwise. Note that this sign convention for plotting  causes the y face to be represented by the point Y at (σ y,-  xy ), just as was explained in Fig.8.11b.

9 Before we look at example of how Mohr’s circle is used, let us establish the fact that Mohr’s circle does, indeed, provide a graphical representation of the stress transformation equations,Eqs.8.5. Equations 8.23 and 8.24 come directly from the trigonometry of the circle in Fig.8.16., (8.23) and (8.24)

10 The angle  on Fig.8.16 is introduced just to facilitate the derivation that follows. The trigonometric identities for the cosine and sine of the sum of two angles are (8.25) Figure 8.16: Properties of Mohr’s circle for plane stress.

11 Letting  =2θ, we can convert Eqs.8.24 to the form (8.26a) (8.26b) If we multiply Eq.8.26a by cos 2θ and Eq.8.26b by sin 2θ and add the resulting equation, we get (8.27) But, combining this with Eq.8.23a leads to (8.5a) repeated

12 Which, of course, is just Eq.8.5a. Similarly, multiplying Eq.8.26a by sin 2θ and Eq.8.26b by cos 2θ and subtracting the latter from the former we get (8.28) But, Eq8.23b reveals that this can be written (8.5b) (repeated) Which is just Eq.8.5b. Again, we have shown that Eq,8.5 are just the parametric equations of a circle in (σ,  ) coordinates.

13 Having established that Mohr’s circle of stress is a graphical representation of the transformation equations for plane stress, let us examine other properties that can be easily deduced from Mohr’s circle.

14 PROPERTIES OF MOHR’S CIRCLE Referring to Fig.8.16, we can conclude the following: The center of Mohr’s circle lies on the σ axis at (σavg,0). Points on the circle that lie above the σ axis (i.e.,  negative) correspond to faces that have a clockwise- acting shear; points that lie below the σ axis (i.e.,  positive) correspond to faces that have a counterclockwise-acting shear, as illustrated by Fig The radius of the circle is determined by applying the Pythagorean theorem to the triangle with sides  xy and giving (8.11) repeated

15 Figure 8.17 : The Mohr’s circle shear-stress sign convention.

16 Figure 8.18: Consistent angles. Two planes that are 90˚ apart on the physical body are represented by the two points at the extremities of a diameter, such as points X and Y or P1 and P2 in Fig If we rotate counterclockwise by an angle θ ab to go from face a to face b of the physical body, we must rotate in the same direction through the angle 2θ ab to get from point A on Mohr’s circle to point B. Figure 8.18 illustrates this property of the Mohr’s circle sign convention. In equations, a positive angle is always counterclockwise. The principle planes are represented by points P1 and P2 at the intersections of Mohr’s circle with the σ axis (Fig.8.16). The corresponding principle stresses are σ 1 =σ avg +R and σ 1 =σ avg -R.

17 The planes of maximum shear stress are represented by points S 1 and S 2 that lie directly below and above the center of the Mohr’s circle (Fig.8.16). The corresponding stresses are (σ avg,R) on the face S 1, (σ avg,-R) on face S 2. Since the stresses on the orthogonal planes n and t are represented by the points at each end of a diameter of Mohr’s circle., (8.7) repeated

18 To solve plane-stress problems, such as determining the stresses on a particular face (e.g.,Example Problem 8.1 and 8.2) or determining principle stresses and maximum in-plane shear stresses (Example prob 8.3), the following procedure is suggested: Draw Mohr’s Circle 1. Establish a set of (σ,  ) axes, with the same scale on both axes. Remember, the +  axis points downward. It is good idea to use paper that has a grid, like graph paper or “engineering paper”. Use a scale that will result in a circle of reasonable size. 2. Assuming that σ x,σ y,  xy are given (or can be determine from a given stress element), locate point X at (σ x,  xy ) and point Y (σ y,-  xy ). 3. Connect points X and Y with straight line, and locate the center of the circle where this line crosses the σ axis at (σ avg,0). 4. Draw a circle with center at (σ avg,0) and passing through points X and Y. It is best to use a compass to draw the circle. PROCEDURE FOR CONSTRUCTING AND USING MOHR’S CIRCLE OF STRESS

19 Compute the Required Information 5. Form the triangle with sides  xy and, and compute 6. If the stresses on a parrticular face, call it face n, are required, locate point Non the circle by turning an angle 2θ counterclockwise (or clockwise) on the circle, corresponding to rotating an angle θ counterclockwise (clockwise) from some reference face on the stress element. Using trigonometry, calculate σn and  nt. 7. If the principle stresses and the orientation of the principle planes are required, use σ 1 =σ avg +R, σ 2 =σ avg -R to calculate the principle stresses, and use trigonometry to determine some angle, such as 2θ xp1 that can be used to locate a principle plane, say P 1, with respect to some known face say the x face. 8. Use a procedure similar to step 7 if the maximum in-plane shear stress and the planes of maximum shear stress are required.

20 For the plane stress state in Example Problem 8.1 and 8.2 (Fig.1), do the following: (a) Draw Mohr’s circle. (b) Determine the stresses on all faces of an element that is rotated 30  counterclockwise from the orientation of the stress element in Fig.1. (c) Determine the orientation of the principle planes; determine the principle stresses. (d) Determine the orientation of the planes of maximum shear stress; determine the value of the maximum shear stress. Example 8.4:

21 Solution: (a) Mohr’s circle: on grid paper (Fig.2), plot point X at (20MPa,-10MPa) and plot point Y, at (- 10MPa,10MPa). The center of the circle is obtain by connecting X and Y. the diameter crosses the σ axis at C: (σavg,0) where (1) The circle is drawn with center at C and passing through points X and Y. The radius R is calculated from the shaded triangle XCB in Fig.2. (2)

22 (b) Stresses on x’ and y’ Faces: Locate the points on Mohr’s circle that correspond to rotating the stress element by 30 . This means rotating 60  counterclockwise from the XY diameter on Mohr’s circle. We label these two points X’ and Y’.

23 To determine the stresses at points X’ and Y’, we need to establish the geometry and trigonometry of the triangle X’ CA. To determine X’ CA we need to first determine the (clockwise) angle, in Fig.2. Using the triangle XCB, we get (3a) (3b) Therefore, (4) From the triangle X’CA, (5a) Or (5b) Therefore, Ans. (b) (6)

24 Also, from triangle X’CA we get Or Ans.(b) (7) Equation (6) and (7) are the same answers that we obtained in Example problem 8.1 by using formulas directly.

25 c) Principle Planes and Principle Stresses: We have already calculated θxp1 in Eq.(3). From Fig.2, (8a) (8b) Also, from Fig2, Ans.(c) (9)

26 (d) Maximum In-Plane Shear Stress: The planes of maximum in-plane shear stress are represented by the points S1 and S2 on Mohr’s circle. From Fig. 2, so Also, by referring to Fig.2, we see that The maximum in-plane shear stress occurs on plane s1 and on plane s2 and is given by On the planes of maximum shear stress, the normal stress is Ans.(d)