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Unformatted text preview: MOHR’S CIRCLE FOR PLANE STRESS 1. Equations of Mohr’s Circle
Fundamental Transformation Equations: 0 :ax+o~y +O'x—O'y
x' 2 2 cos 26 + rxy sin 26 (1) O'x—O'y _
Tm] =— 2 Sln 26+er cos26 (2) Solving for sin26 and 00526 from (1) and (2) (simultaneous equations). Substituting sin 26 and cos 26 into the following equation
sin2 26 + cos2 26 =1 (3) Then, the parameter, 26 , can be eliminated and we obtain the following equation: 2 2
0' +0' 0' —0'
0X — x y +73, = x y +12
1 2 I): 2 xy (4)
Recall
2
R: [ax—0y] +7 2
2 Ky
(5)
0x + Q
OaVL’ : 2 Substituting (5) and (6) into (4), we obtain
2
(0):, _ Gave) + lily, = R2 Mohr’s Circle for Plane Stress 1 10/02/2008 2. Construction of Mohr’s Circle > Sign convention KN Clockwise shear stresses
T
N \ (a) (b) \j Counterclockwise shear stresses
(C)
> Stress element ) Two ends of the diameter > Center of the circle 3. Information identiﬁed from Mohr’s Circle
> Principal stress
> Maximum shear stress > Stress on an inclined element (Note that an angle 2 6corresponds to
an angle 6 on a stress element.)
4. Mohr’s Circles of Special Cases > Biaxial Stress > Pure Shear > Uniaxial Stress Mohr’s Circle for Plane Stress 2 10/02/2008 560 CHAPTER 7 Analysis of Stress and Strain FIG. 716 Construction of Mohr’s circle
for plane stress Construction of Mohr’s Circle Mohr’s circle can be constructed in a variety of ways, depending upon
which stresses are known and which are to be found. For our immediate
purpose, which is to show the basic properties of the circle, let us
assume that we know the stresses 0,, cry, and 73Cy acting on the x and y
planes of an element in plane stress (Fig. 7l6a). As we will see, this
information is sufﬁcient.to construct the circle. Then, with the Circle
drawn, we can determine the stresses 0,”, cry}, and Txm acting on an
inclined element (Fig. 7v16b). We can also obtain the principal stresses
and maximum shear stresses from the circle. ’ With 0,, 0y, and 7,7,. known, the procedure for constructing
Mohr’s circle is as follows (see Fig. 7‘16c): 1. Draw a set of coordinate axes with 0“ as abscissa (positive to the
right) and Txm as ordinate (positive downward). 2. Locate the center C of the circle at the point having coordinates
0,1 2 Cave, and 71m : 0 (see Eqs. 7—313 and 732). 3. Locate point A, representing the stress conditions on the x face
of the element shown in Fig. 746a, by plotting its coordinates
all : (IX and Tle : 7”. Note that point A on the circle corre—
‘sponds to 6:0. Also, note that the x face of the element
(Fig. 7463) is labeled “A” to show its correspondence with point
A on the circle. 4. Locate point B, representing the stress conditions on the y face
of the element shown in Fig. 7—163, by plotting its coordinates 4 . — W? 0—1 7 .. 7.... FIG. 7’16 (Repeated) SECTION 7.4 Mohr’s Circle tor Plane Stress 561 Waxl : (IV and Tam : #7”. Note that point B on the circle gonespdﬁds {6 6)? 790°] In additiOn; the yrace Of the Blane/ﬁr '7 " " (Fig. 746a) is labeled “8” to show its correspondence with point
B on the circle. 5. Draw a line from point A to point B. This line is a diameter of the
circle and passes through the center C. Points A and B, representing
the stresses on planes at 900 to each other (Fig. 7—1621), are at Oppt}
site ends of the diameter (and therefore are 180° apart on the circle). 6. Using point C as the center, draw Mohr’s circle through points A and B.
The circle drawn in this manner has radius R (Eq. 7—3lb), as shown in
the next paragraph. Now that we have drawn the circle, we can verify by geometry that
lines CA and CB are radii and have lengths equal to R. We note that the
abscissas of points C and A are (UK + (ry)/2 and (IX, respectively. The
difference in these abseissas is (0X — (ry)/2, as dimensioned in the fig—
ure. Also, the ordinate to point A is Try. Therefore, line CA is the
hypotenuse of a right triangle having one side of length (0X e (ry)/2 and
the other side of length Txy. Taking the square root of the sum of the
squares of these two sides gives the radius R: ...
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This note was uploaded on 11/04/2008 for the course CE 207 taught by Professor Elghandour during the Spring '06 term at Cal Poly.
 Spring '06
 Elghandour

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