The Mohr circle construction

This construction enables the components of a two-dimensional stress (or strain), expressed in terms of a given set of perpendicular axes, to be converted into the components relative to any other set of perpendicular axes. In particular it provides a simple method for determining the principal axes of stress and strain. For fibre symmetry all planes that contain the fibre axis as one of the perpendicular axes are identieal thus the eonstruction is again applicable. 

Let the eomponents of stress be o,a, Oyy, = Oy in terms of perpendieular axes OX, OY. Now eonsider an identical total stress in terms of axes OX, OY, obtained by rotating OX and OY through an angle 6 in the anticlockwise direetion. It ean be shown that the shear stress components o y and o y,i in terms of the new axes are 

For a proof of (Al.l) using elementary mathematies see Hall [1]. The expression may be obtained directly from the general tensor relation a y = Uikajiaja, where a,, ete. represent direction cosines an = cosO, = sin0, fl2i = —sin0, 22 = cos 6. 

A particular angle 9, where 0 0 JtH, ean always be found for whieh in (Al.l) is zero. The condition is 

A Treatise on the Mathematical Theory of Elasticity, 4th Edn, Macmillan, New York, 1944. 

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