TORSION OF AXIALLY GROOVED SHAFTS : WITH THE EFFECT OF KEYWAYS

Abstract

The paper consists of two parts, The first part is the torsion problems of long solid cylinders, whose sectional forms are represented by β=0 curves of the equation x+iy=e^<i(α+iβ)>+Ae^<in(α+iβ)>+Be^<i(2n-1)(α+iβ)> where A and B are the arbitrary constants to define the forms of the section. Such shafts under consideration are long enough to apply St. Venant's theorv of torsion. In this part, it is also analvtically proved that the shearing stress-intensity is very great under torsion at the bottom of radial crack from the surface and how the radial depth of such a crack influences on the stress distribution of the section. The results are shown in Fig. 2 and 3. Furthermore the author calculates the relation of the maximum shearing stress to the depth of the groove when the radius of curvature at the bottom of groove is constant and that relation of the stress to the radius of curvature at the constant depth (Fig. 6). In the second part, he applies the above theory to the shaft with a keyway. His assumption is that the shearing stress at the corner A or B(Fig. 7) is nearly equal to that stress at the bottom of the groove which is represented by the dotted line, if the radii of curvatures at the both cases are equal. Then, the shearing stress at the corner of the keyway becomes very great when the radius of curvature is less than r/20,and that stress is nearly constant when the radius of curvature greater than r/20,r being the radius of the shaft. And the stress for the case r/20 is as large as the twice of the maximum shearing stress of the perfect circular cylinder. On the other hand, the torsional moment decreases with the keyway for twisting the same angle. Therefore, the resisting moment of the shaft with keyway (w=D/4 and h=D/6,where w and h are breadth and thickness of key) is 1/2 〓32 of that of perfect circular cylinder of the same diameter. Finally the theoretical values have been proved by experiments to be nearly true with mild steel shafts.