MECHANICS 745 Now, from the value of TJ above, we have 1 c Hence we have H B +CcosD, = A -B -CsinD, Eliminating and D, we have C = Ag 7 + ?)- + (A + B)cosi7 + (A -B)sin?7,
Eliminate the ratio A/15, which is all that these equations furnish, and we have From this equation the values of i must be determined. It is clear that the multiplier of cosz7 is always greater than 2, except in the special case of i = 0, which we obviously need not consider, as it gives J/ = 0, and therefore belongs to the statical problem already considered. Hence as, to make cosi7 negative, il must be greater than 4 ir ; and, as g^ 77 + g " rr = 5 nearly, it is clear that the excess of the first value of il over ir is somewhere about 3. The next value falls short of ir by a quantity of the order y^, the next exceeds fir by a quantity of the order f^Vin &c. The required values arrange themselves in two groups, one of either group being taken alternately. The first group involves arcs a little greater than but rapidly approaching to the values of (4;/i+l)^ir; the second consists of arcs a little less than but rapidly approaching to those of (4m + 3) |ir. Effectsof 278. Some of the simplest, but at the same time pressure most practically useful, of questions connected with elas- on tubes ti c ity of solids relate to the changes of form or volume ""herical experienced by circular cylindrical tubes or spherical shells -hells, exposed to hydrostatic pressure. A steam-boiler, the cylinders and tubes of an hydraulic press, a fowling-piece or cannon and (on a much smaller scale) Orsted s piezo meter, deep-sea thermometers, &c., afford common in stances. All that is necessary for attacking such questions is given in 45, 46 of the article ELASTICITY. For it is there shown that, if a homogeneous isotropic elastic solid be subjected to a simple longitudinal stress P, uniform and in a definite direction throughout its whole substance, the l_ 1 . in result will be linear extension = P ^ } in the direc- 9k Equal pres- within and without meter. tion of P, and linear contraction = P f - 6 --g ) in all direc-
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tions perpendicular to P. The quantities n and , as explained in the article referred to, are respectively the " rigidity " and the reciprocal of the " compressibility " of the solid operated on. 279. The case of the piezometer, in which the vessel holding the liquid whose compression is to be measured is exposed both inside and outside to the same hydrostatic pressure, is seen to correspond to three equal stresses in directions at right angles to one another. These direc tions mny be any whatever, and in each of them the linear extension is obviously Cylinder under internal e> two. In a transverse section of the cylinder, the second of these is radial and the third is tangential to a coaxal cylinder passing through the element considered. We suppose the cylinder closed at both ends, and we make the further assumption (quite exact enough for practical applications, and most important from the point of view of simplicity of calculation) that all transverse sections of the cylinder remain, after distortion, transverse sections. This is equivalent to assuming P x to be constant through out the walls of the cylinder. Hence, if there be interior pressure only, the value of this stress must be _ pressure on end of interior of cylinder Tie* 1 area of transverse section of walls of cylinder a - where II is the interior hydrostatic pressure, and a w a^ are the internal and external radii. This stress represents a longitudinal tension of the walls of the cylinder. Let us consider an element of the cylindrical wall, defined as follows in the unstrained state : Draw two transverse sections at distances x and a;+8x from one end, two planes through the axis making an (infinitesimal) angle with one another, and two cylinders of radii r and r + Sr about the common axis. In the strained state is unchanged, but x becomes x + , and r becomes r + p. The distance between the transverse U- J ~Gn 9k J } 8k P is negative, as the stress is a pressure. Hence the strain consists in a simple alteration of volume measured by P/&. E very part of the walls of the vessel, as well as its external bulk, and its interior content, is altered to the same extent. 280. In the case of a cylinder, when the internal and external pressures are different, it is clear from symmetry that the stresses may be resolved at any point of the walls into three at right angles to one another, the first (PJ parallel to, the second (P 2 ) at right angles to, the axis, and the third (P 3 ) perpendicular to each of the other sections was So;; it becomes v -- - C --8x, so that the linear exten- dx sion parallel to the axis is ~ . The distance between the cylinders was Sr ; it becomes 8r+ -r-Sr, so that the radial extension is -3-. dr " r . The breadth of the base of the wedge-shaped element was r6 ; it becomes (r + p)0, so that the linear extension perpendicular alike to the radial line and to the axis is - . r If we now write, for simplicity, + - f= 3?i 9k 6n 9k the three requisite equations between stresses and strains are at once obvious in the form ePi-/P 2 -/P. dr We have, however, four unknown quantities , p, Po, and P 3 , so that another equation is required. This must be supplied by one of the statical conditions of equilibrium of the element above defined, when in its strained state. There is obviously equilibrium in the axial direction, and also parallel to the base of the element ; but radially we have a case resembling that of an element of a cord as in 264. Neglecting small quantities of a higher order than those retained, this consideration gives d d ,p dr dr Solving the four equations, and taking account of the boundary conditions P.j= -Hwhen r = a , and P. 2 = when r-a 1} we obtain the following values : or, re-introducing the values of c and/, r/| Wi* 1 t/.r jj -- a I 3/v Hft- /i dp Un ~ 281. Thus the nature of the distortion produced in the walls of a cylindrical tube by internal pressure may be described as made up
XV. --94