APPLIED MECHANICS.] MECHANICS 751 investigation, the resistance of the earth can be treated as one or more upvsard loads applied to the structure. But in other cases the earth is to be treated as one of tlie pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it. 9. Partial Polygons of Resistance. In a structure in which there are pieces supported at more than two joints, let a polygon be con structed of lines connecting the centres of load of any continuous scries of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not com prehended in the partial polygon of resistances, to which the theorem of section 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley s principle of the least resistance, the whole of the relations amongst the loads arid resistances may be found. 10. Line of Pressures Centres and Line of Resistance. The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line tra versing all the centres of resistance of a series of joints, its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be sub divided by the introduction of intermediate joints ad infinitum, and finding the continuous line, curved or straight, in which the in termediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley. 11. Stability of Position, and Stability of Friction. The resist ances at the several joints having been determined by the principles set forth in sections 6, 7, 8, 9, and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative dis placement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either by turning or by sliding. Safety against displacement by turning is called stability of position; safety against displace ment by sliding, stability of friction. 12. Condition of Stability of Position. If the materials of a struc ture were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow r for the finite stiffness and strength of materials, the least dis tance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, some times empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley the modulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice : Retaining walls, as designed by British engineers ...................... ., 1 : 8 Retaining walls, as designed by French engineers ........................ 1 : 5 Rectangular piers of bridges and other buildings, and arch-stones . 1:3 Rectangular foundations, firm ground ........................................ 1 : 3 Rectangular foundations, very soft ground ................................. 1 :2 Rectangular foundations, intermediate kinds of ground ........ , ...... 1 :3 to 1 :2 Thin, hollow towers (such as furnace chimneys exposed to high winds), square .................................................................... 1 :C Thin, hollow towers, circular ..................................................... 1:4 Frames of timber or metal, under their ordinary or average distri bution of load ...................................................................... 1:3 Frames of timber or metal, under the greatest irregularities of load .................................................................................... 1 :3 In the case of the towers, the depth of tJie joint is to be understood to mean the diameter of the tower. 13. Condition of Stability of Friction. If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 3) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is the obliquity of the resistance. From R draw RP parallel and RQ perpendicular to the joint ; then, by the principles of statics, tho component of the resistance normal to the joint is and the component tangential to the joint is CQ = CR . sin^PCR = CP . tan^PCR. If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on the strength of such projections, recesses, or fastenings, and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium, CR, must be a thrust and not npull ; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component ; that is, that CQ}>/.CP, where / denotes the coefficient of friction for the surfaces in question. The angle whose tangent is .the coefficient of friction is called the angle of repose, and is expressed symbolically by <^ = tan- 1 /. Now CQ = CP . tan^PCR ; consequently the condition of stability of friction is fulfilled if that is to say, if the*obliquity of the resistance required at the joint does not exceed the angle of repose ; and this condition ought to be fulfilled under all possible variations of the load. It is chiefly in masonry and earthwork that stability of friction is relied on. 14. Stability of Friction in Earth. The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the por tions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper "On the Stability of Loose Earth," already cited in sect. 2. 15. Parallel Projections of Figures. If any figure be referred to a system of coordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of coordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the coordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose coordinates shall bear respectively to the three corre sponding coordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are called parallel projections of each other. The properties of parallel pro jections of most importance to the subject of the present article are the following : (1) A parallel projection of a straight line is a straight line. (2) A parallel projection of a plane is a plane. (3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio. (4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces ; whence it follows (5) That a parallel projection of a parallelogram is a parallelo gram, and (6) That a parallel projection of a parallelepiped is a parallel epiped. (7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio. Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections : (8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order. For example, all ellipsoids referred to coordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular coordinates. 16. Parallel Projections of Systems of Forces. If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of
forces.