the axis of ≈, we have by equation (v) of our Art. 55, σy=0 and σxz=P/(wμ). Whence supposing uni-constant isotropy we find : Suppose b' and c' to be the values to be given to b and c that the prism might safely withstand a couple Pa producing flexure only, and b", c" to be the values to be given to b and c that it might safely withstand a shearing force P applied to the undistorted section. Then we easily find P and 1 129 4Sb"c" gives the limiting safe values of b and c for the strain in question. Saint-Venant puts first c' = c" = c and so gets b= 3b' + √(§b')2 +b′′2, whence he deduces and tabulates the values of b/b′ and b/b′′ for various values of b′′/b′ and b'/b" respectively, and also the value of 2c = depth 2a length 2c 2a = or the slide begins to influence sensibly the result, > = or < 10 the flexure begins to influence sensibly the result. Between 2c/2a = and 10 we are compelled to take both into account. Saint- Case (3). This is the treatment of a cylinder on a circular base subjected at the same time to flexure, torsion and extension. Venant neglects the flexural slides and ultimately the extension. obtains an equation similar in character to that of the preceding case and tabulates the values of the radius of safety in terms of the radius of safety in the case of flexure alone for different values of the elastic constant 7. He remarks (p. 503) that it is not necessary to consider values of > for then a stretch would not produce a positive dilatation, ce qui n'est point supposable.' This remark is omitted in the Leçons de Navier where a number of values of 7,>are dealt with. I may add that the problem is far more completely treated in that work (pp. 414-21). Saint-Venant's tables shew that the results obtained are for values of 7, between 1/5 and 1/3 very much the same, or we may adopt generally without fear of error the uni-constant hypothesis 1 = 1/4. This hypothesis Saint-Venant tells us is amply verified by the experiments of M. Gouin (see page 486 of the memoir). 6 I shall have something to say of these experiments when dealing with Morin's Résistance des matériaux, 1853: see our Chap. XI. Case (4). This case gives the calculation of the 'solid of equal resistance' for a bar built-in at one end and acted upon at the other by a non-central load perpendicular to its axis, i.e. combined flexure and torsion. Saint-Venant supposes uni-constant isotropy and neglects the flexural slides. His final equation is Here P is the load acting on an arm k, and r is the sectional radius at distance x from the loaded terminal. (p. 504.) Case (5). An axle terminally supported has weight II and carries two heavy wheels (w and w') upon which act forces, whose moments about the axle are equal and whose directions are perpendicular to the axle. We have thus another case of combined flexure and torsion, which is dealt with as before. [57.] The next case treated by Saint-Venant is of greater complexity; it occupies pp. 507-18 of the memoir. It is the investigation of combined flexural and torsional strain in rectangular prisms (26 × 2c), and possesses considerable theoretical interest. In practice also the non-central loading of beams of rectangular section must be a not infrequent occurrence. Case (6). Saint-Venant in his treatment does not suppose the elasticity round the prismatic axis to be isotropic, but takes the general case of two slide-moduli, supposing, however, that b> c√/μ1. He neglects also the flexural slide-components. Let the torsional slide-components be given by o1 =-,cr and σ=ybτ for z/c=1 and y/b=1 respectively. T must be eliminated by means of the relation M" = Burbe3. If o be the angle the plane of the flexural load makes with the plane through the prismatic axis and the axis of y, and M' the flexural moment at section x, we easily obtain for the stretch s the value Let us substitute these values in equation (ii) of our Art. 53. Taking these expressions alternately for the sides 26 and 2c we obtain : 1- n2 3 M' 27 4bc2 су By means of the Table II. below and Table I. on our p. 39 all the terms of these expressions can be calculated; for y/Y1 and y/y2 are given про for values of and also for values of y/b and z/c respectively. Hence so soon as and the section of danger, i.e. where M' is greatest, are known we can solve the problem by equating to unity the greater of the two maxima written down above and so determine bc for the section. Saint-Venant by using b', c', b", c" with similar meanings to those of our Art. 56, Case (2), throws the equation into a somewhat different form. If the section for which M' is greatest be so built-in or symmetrically situated that no distortion is possible the values of the slides must be those of equations (v) of our Art. 55 and not σ1, σ as taken above. TABLE II. Slides at points of the contour of the Cross-Section of a Prism on rectangular base subjected to Torsion. This Table gives Y, Y, in terms of the principal slides 71, 72 at the centre of the corresponding sides 26 and 2c; the values of Y1, 2 are given in Table I. p. 39. (2) c so much less than b that c/b. tano may be neglected as compared with 1, i.e. the case of a 'plate' (pp. 511—2). = = (3) Prism on square base, when tan=0, = 1⁄2, 1, and anything whatever when there is a non-distorted section for section of least safety (pp. 512-4). The fail-points are also determined. = (4) Prism on rectangular base for which b=2c, when tan = 0, = 1, = 1, = 2, = ∞, and anything whatever when there is a nondistorted section for that of least safety (pp. 514-518). The fail-points are also determined. [59.] On pp. 518-22 we have the treatment of a prism on elliptic base subjected at the same time to flexure and torsion. Saint-Venant only works this out numerically for the case of uni-constant isotropy and when tan∞. It is found that after a certain value of the ratio of torsional to flexural couple, the fail-point leaves the end of the major axis (through which the flexural load-plane passes1) and traverses the quadrant of the ellipse till it reaches the end of the minor axis (p. 522). [60.] We now turn to Saint-Venant's final chapter (pp. 522— 558). This consists of three parts: § 135 Résumé général; § 136 Récapitulation des formules et règles pratiques and § 137 Exemples d'applications numériques. In the first article there is little to be noted. A reference is made on p. 528 to the models of M. Bardin shewing the gauchissement of the cross-section to which we have previously referred. Saint-Venant also mentions the visible distortion of the cross-sections obtained by marking them on a prism of caoutchouc and then subjecting it to torsion. 2 In the general recapitulation of formulae we have some results not in the body of the memoir, as on p. 536 (d) where the flexural slides for the prism whose base is the curve = 1 are cited from the memoir on flexure: see our Art. 90. So again on p. 546 for the flexural slides of other cross-sections. The best résumé, however, of formulae as well as numbers for both flexure and torsion is undoubtedly to be found in Saint-Venant's Leçons de Navier to which we shall refer later. The last section § 137 contains some instructive numerical examples of Saint-Venant's treatment of combined strain. 1 Saint-Venant terms this sollicité de champ. When the load-plane is perpendicular to this the prism is sollicité à plat. The memoir concludes with the tables for rectangular prisms. which we have in part reproduced on pp. 39 and 49. [61.] We here bring to a close our review of this great memoir. Since Poisson's fundamental essay of 1828 (see our Art. 434*) no other single memoir has really been so epoch-making in the science of elasticity. It is indeed not a memoir, but a classical treatise on those branches of elasticity which are of first-class technical importance. Written by an engineer who has kept ever before him practical needs, it is none the less replete with investigations and methods of the greatest theoretical interest. Many of its suggestions we shall find have been worked out in fuller detail by Saint-Venant himself, not a few remain to this day unexhausted mines demanding further research. SECTION II. Memoirs of 1854 to 1864. Flexure, Distribution of Elasticity, etc. [62.] Comptes rendus, T. XXXIX. pp. 1027–1031, 1854. Mémoire sur la flexion des prismes élastiques, sur les glissements qui l'accompagnent lorsqu'elle ne s'opère pas uniformément ou en arc de cercle, et sur la forme courbe affectée alors par leurs sections transversales primitivement planes. This is a résumé of the results of the later memoir on flexure (see our Arts. 69 and 93). It cites the general equations for flexure, and the particular results for the case of a rectangular cross-section. [63.] L'Institut, Vol. 22, 1854, pp. 61-63. Solution du problème du choc transversal et de la résistance vive des barres élastiques appuyées aux extrémités. This is an account of SaintVenant's memoir presented to the Société Philomathique. It contains only matter given in the Comptes rendus, and afterwards more completely in the annotated Clebsch: see our Art. 104. |