By P. R. Lancaster, D. Mitchell
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Extra resources for Advanced Solid Mechanics: Theory, worked examples and problems
CP2 - p3), ~CP3- Pl) and hence calculate the normal stresses on these shear planes. 6 the resultant shear stress and its direction cosines in relation to the given axes. What are the principal stresses and the direction cosines of the greatest of these? 6 A resistance strain gauge rosette has arms a, b and c. The following readings are obtained -3e a Show that the maximum shear stress at the point is given by ~[28)~ 1 + \} 3 assuming that the measurements are taken on the surface of a thin plate subjected to a two-dimensional stress ~stem.
X r be derived. lSa the stresses at a point are shown in both rectangular and polar coordinates. lSb. 16). 11 (a) Fig. 15 T~--------------~a•L-----------~--~ Fig. 16 28 To obtain the value of ar note that 6 is the angle measured anticlockwise from the x-axis to the normal to the plane ac. On the Mohr circle this corresponds to measuring 26 anticlockwise from the datum ab. Note that points in the Mohr circle correspond to planes in the stress diagram. Thus ar + a6 --:2,--- + R cos (26 - a) where R is the radius of ar + a6 ----2--- cry and + ~1ohr circle.
In what follows, the polar equivalents of direct and shear stresses will be derived along with the stress compatibility equation. An example will follow in which the stress/strain equations are used with their displacements derived in chapter 1. ) y Fig. 4 X No . the two independent relationships which can be obtained from figut·e 2. 1 ar r ae = and the operator a;ax is axa = cos 8 ~ ar _ sin e a - r - ar (2. 12) and similarly a ay = sin 8 ~ ar + cos e ~ r ar Hence by differentiation ::: = aax(~ = (cos 8 aar _ si~ e aar]( cos 8 ~!
Advanced Solid Mechanics: Theory, worked examples and problems by P. R. Lancaster, D. Mitchell