Mohrs Circle in Two Dimensions

combine the two equations after squaring them, to get  

EquatioD (1.45) is further modified for the purpose of the graphical presentation as follows  

This circle is in a Cartesian coordinate system with center coordinates a, b and radius r. This circle equation applies to any point on the circle where the radius is the hypotenuse of a right-angled triangle whose other sides are of length (x — a) and (y — b). If the circle is centered at the origin (0, 0), then the equation simplifies to  

Equation (1.47) of the circle can be redrawn by replacing coordinates a, b, and r with Cave, and R, as dehned in Eq. (1.47a). In this circle, the points along the abscissa (at zero shear x y ) are the principal stresses, as indicated in Eq. (1.46), in which is the principal stress designated as Ti. Any point on the circle can be obtained by the Pythagorean theorem. Such constructions serve as the basis for Mohr s circle, yielding the particular stress at each point. Bear in mind that the normal and shear stress components in the z direction are zero or negligible. 

A short exercise can illustrate how to use a Mohr s circle to get the principal stresses. A priori the angles are not needed for this. Rgure 1.22 indicates the method when no rotation of the coordinate system has occurred. The magnitudes of ffx ty and Tyx (= Txy) are indicated in Fig. 1.22. 

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