cal flexure Symmetri- which rests simply on a horizontal circle; and is deflected by of flat ring. a load uniformly distributed over its inner edge (or vice versâ, inner for outer). To find the deflection due to this load (which of course is simply added to the deflection due to the weight, determined below). Here G must vanish at each edge. Flexure of flat ring equilibrated by forces symmetrically distri buted over its edges; The radii of the outer and inner edges being a and a', the equations are 1 = 0, ¿C'{(4+ c) log a + ¿(4− c)} + {C''(4 + c) − C''(4 − e) { a = a }C" (A + c) (a2 − a′2) = − } C [(A + c) (a2 log a – a22 log a′) + } (A − c) (a2 − a22)] : and thus, using for C′ its value (12), we find [(2) § 649] Putting the factor of 3 into a more convenient form, and assigning C"" so that the deflection may be reckoned from the level of the inner edge, we have finally Towards showing the distribution of stress through the breadth of the ring, we have from this, by § 649 (6), a2a' 1 a 3A + c log 1 a2 - a'3 8 which, as it ought to do, vanishes when ra', and when ra. which shows that, as is obviously true, the whole amount of the transverse force in any concentric circle of the ring is equal to F. load sym spread over 653. The problem of § 652, extended to admit a load dis- and with tributed in any symmetrical manner over the surface of the metrically ring instead of merely confined to one edge, is solved its area. algebraically in precisely the same manner, when the terms dependent on Z, and exhibited in the several expressions of § 649, are found by integration. One important remark we have to make however: that much needless labour is avoided by treating Z as a discontinuous function in these integrations in cases in which one continuous algebraic or transcendental function does not express the distribution of load over the whole portion of plate considered. Unless this plan were followed, the expression for z, dz/dr, G, and , would have to be worked out separately for each annular portion of plate through which Z is continuous, and their values equated on each side of each separating circle. Hence if there were i annular portions to be thus treated separately there would be 4 arbitrary constants, to be determined by the 4 (i-1) equations so obtained, and the 4 equations expressing that at the outer and inner bounding circular edges G has the prescribed values (whether zero or not) of the applied bending couples, and that z and have each a prescribed value at one or other of these circles. But by the more artful method (due to Fourier and Poisson), the complication of detail required in virtue of the discontinuity of Z is confined to the successive integrations; and the arbitrary constants, of which there are now but four, are determined by the conditions for the two extreme bounding edges. Example.-A circular table (of isotropic material), with a concentric circular aperture, is borne by its outer or inner edge which rests simply on a horizontal circular support, and is loaded by matter uniformly distributed over an annular area of its surface, extending from its inner edge outwards to a concentric circle of given radius, c. It is required to find the flexure. First, supposing the aperture filled up, and the plate uniform from outer edge to centre, let the whole circle of radius c be uniformly loaded at the rate w, a constant, per unit of its area. 0 0 4wc2 4wc2 (2 log(+1) Twe2 (4r2 log" +c2) 1'swc2 (2r2 log" - r2+c2 log" " + {c2 ) I. II. III. IV. V. Circular table of isotropic material, supported symmetrically on its edge, and strained only by its own weight. Reduction problem to case of no Of these results, v. used in (2) gives the general solution; and IV., III, and II. in (6) and (8) give the corresponding expressions for G and . If, first, we suppose the value of G thus found to have any given value for each of two values, r', r', of r, and { to have a given value for one of these values of r, we have three simple algebraic equations to find C, C', C"; and we solve a more general problem than that proposed; to which we descend by making the prescribed values of G and zero. The power of mathematical expression and analysis in dealing with discontinuous functions, is strikingly exemplified in the applicability of the result not only to the contemplated case, in which c is intermediate between r' and r"; but also to cases in which c is less than either (when we fall back on the previous case, of § 652), or c greater than either (when we have a solution more directly obtainable by taking Zw for all values of r). If the plate is in reality continuous to its centre, and uniformly loaded over the whole area of the circle of radius c, we must have C 0 and C"=0 to avoid infinite values of and G at the centre and the equation G=0 for the outer boundary of the disc gives C'at once, completing the determination. If, lastly, we suppose c to be not less than the radius of the disc, we have the solution for a uniform circular disc uniformly supported round its edge, and strained only by its own weight. 654. If now we consider the general problem,—to deterof general mine the flexure of a plate of any form, with an arbitrary distribution of load over it, and with arbitrary boundary appliances, subject of course to the condition that all the applied forces, when the data are entirely of force, must con load over area. Reduction of general stitute an equilibrating system; we may immediately reduce this problem to the simpler one in which there is no load problem to distributed over the area, but arbitrary boundary appliances load over only. We shall merely sketch the mathematical investigation. First it is easily proved, as for a corresponding expression for three independent variables in § 491 (c), that (+1) ƒƒp' log D da'dy' = 2′′p (1), where p' is any function of two independent variables, x', y'; Hence u = Z .(2), case of no area. 4. where D' = √{(x" − x')2+(y′′ − y')'}; and if Z" and Z denote is the general solution of (2). The boundary conditions for ¿ are lar ring the hitherto 655. Mathematicians have not hitherto succeeded in solving Flat circuthis problem with complete generality, for any other form of only case plate than the circular ring (or circular disc with concentric solved. circular aperture). Having given (§§ 640, 653) a detailed lar ring the only case hitherto solved. Flat circu- solution of the problem for this case, subject to the restriction of symmetry, we shall merely indicate the extension of the analysis to include any possible non-symmetrical distribution of strain. The same analysis, under much simpler conditions, will occur to us again and again, and will be on some points more minutely detailed, when we shall be occupied with important practical problems regarding electric influence, fluid motion, and electric and thermal conduction, through cylindrical spaces. Taking the centre of the circular bounding edges as origin for polar co-ordinates, let We shall see, when occupied with the electric and other problems referred to above, that a general solution of this equation, appropriate for our present problem as for all involving the expression of arbitrary functions of 0 for particular values of 9, is ∞ = {(4 ̧ cos i0 + B ̧ sin i0) e' + (A ̧cos i0 + B ̧sin i0)e ̄*}... (12), where A, B, A, B, are constants. That this is a solution, is of course verified in a moment by differentiation. From it we |