## Mohrs Circle for Plane Stress Conditions

41 The Mohr circle will provide a graphical mean to contain the transformed state of stress (axx,ayy,(xy) at an arbitrary plane (inclined by a) in terms of the original one (axx, ayy, axy).

42 Substituting cos 2a = cos2 a - sin2 a sin 2a = 2 sin a cos a into Eq. 2.49 and after some algebraic manipulation we obtain

= 7;(axx + Vyy) + T.((Jxx - (yy)cos2a + aXy sin2a (2.57-a)

43 Points (axx,axy), (axx, 0), (ayy, 0) and [(axx + ayy)/2, 0] are plotted in the stress representation of Fig. 2.6. Then we observe that

xy where

axx ayy

then after substitution and simplifiation, Eq. 2.57-a and 2.57-b would result in

We observe that the form of these equations, indicates that axx and axy are on a circle centered at 2(axx + ayy) and of radius R. Furthermore, since axx, ayy, R and f3 are definite numbers for a given state of stress, the previous equations provide a graphical solution for the evaluation of the rotated stress axx and axy for various angles a.

44 By eliminating the trigonometric terms, the Cartesian equation of the circle is given by

45 Finally, the graphical solution for the state of stresses at an inclined plane is summarized as follows

1. Plot the points (axx, 0), (ayy, 0), C : [1 (axx + ayy), 0], and X : (axx,°xy).

2. Draw the line CX, this will be the reference line corresponding to a plane in the physical body whose normal is the positive x direction.

3. Draw a circle with center C and radius R = CX.

4. To determine the point that represents any plane in the physical body with normal making a counterclockwise angle a with the x direction, lay off angle 2a clockwise from CX. The terminal side CX of this angle intersects the circle in point X whose coordinates are (axx,axy).

5. To determine ayy, consider the plane whose normal makes an angle a + 2n with the positive x axis in the physical plane. The corresponding angle on the circle is 2a + n measured clockwise from the reference line CX. This locates point D which is at the opposite end of the diameter through X. The coordinates of D are (ayy, —axy)

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