Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/836

812 812 ELASTICITY (2.) Contour lines for St Venant s &quot; etoile ct quatre points arrondis&quot; This diagram (fig. 8) shows the contour lines, in all respects as in case (1), for the case of a prism having for section the figure indicated. The portions of curve outside the continuous closed curve are merely indications of mathematical extensions irrelevant to the physical problem. (3.) Contour lines of normal section of triangular prism, as ivarped by torsion, shown as in case (1; (fig. 9. (4.) Contour lines of normal sections of square prisms 03 warped by torsion (fig. 10). ^ Fig. 10. (5.) Diagram of St Venant s curvilineal squares for which torsion problem is algebraically solvable. This diagram (fig. 11) shows the series of lines represented by the equation z 2 + j/ 2 - a(x* - Gx^y z + y 4 ) = 1 - a, with the indicated values for a. It is remarkable that the values a = 5 and a = - 1( /2 - 1) give similar but not equal curvilineal squares (hollow sides and acute angles), one of them turned through half a right angle relatively to the other. 70. Torsional Rigidity less in proportion to sum of principal Fiexural Rigidities than according to false extension (section 66) of Coulomb s Law. Inasmuch as the moment of inertia of a plane area about an axis through its centre of inertia perpendicular to its plane is obviously equal to the sum of its moments of inertia round any two axes through the same point at right angles to one another in its plane, the fallacious extension of Coulomb s law, referred to in section 66, would make the torsional rigidity of a bar of any section equal to the product of the ratio of the modulus of rigidity to the Young s modulus into the sum of its fiexural rigidities (section 61) in any two planes at right angles to one another through its length. The true theory, as we have seen (section 67), always gives a torsional rigidity less than this. How great the deficiency | may be expected to be in cases in which the figure of the section presents projecting angles, or considerable pro minences (which may be imagined from the hydrokinetic analogy given in section 67), has been pointed out by M. de St Venant, with the important practical application, that strengthening ribs, or projections (see, for instance, the second of the annexed diagrams), such as are intro duced in engineering to give stiffness to beams, have the reverse of a good effect when torsional rigidity or strength is an object, although they are truly of great value in increasing the fiexural rigidity, and giving strength to bear ordinary strains, which are always more or less fiexural. &quot;With remarkable ingenuity and mathematical skill he has drawn beautiful illustrations of this important practical principle from his algebraic and transcendental solutions. 84S46. 88326 8186. 86C6. Fig. 12. Diagrams showing torsional rigidities. Thus, for an equilateral triangle, and for the rectilineal and three curvilineal squares shown in the diagrams (fig. 12), he finds for the torsional rigidities the values stated. The number immediately below the diagram indicates in each case the fraction which the true torsional rigidity is of the old fallacious estimate (section 66), the latter being the product of the rigidity of the substance into the moment of inertia of the cross section round an axis perpendicular to its plane through its centre of inertia. The second number indicates in each case the fraction which the torsional rigidity is of that of a solid circular cylinder of the same sectional area. 71. Places of greatest Distortion in Tivisted Prisms. M. de St Venant also calls attention to a conclusion from his solutions which to many may be startling, that in hia simpler cases the places of greatest distortion are those points of the boundary which are nearest to the axis of the twisted prism in each case, and the places of least distortion those farthest from it. Thus in the elliptic cylinder the