750 MECHANICS APPLIED MECHANICS. [APPLIED MECHANICS. 1. The practical applications of mechanics may be divided into two classes, according as the assemblages of material objects to which they relate are intended to remain fixed or to move rela tively to each other, the former class being comprehended under the term Theory of Structures, " and the latter under the term "Theory of Machines." As the details of the theory of structures are dealt with in other articles, it will be treated of here to such extent only as may be necessary in order to state certain general principles applicable to all these subjects. The greater part of the article will relate to machines. PART I. OUTLINE OF THE THEORY OF STRUCTURES. 2. Siqjport of Structures. Every structure, as a whole, is main tained in equilibrium by the joint action of its own weight, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of HYDROMECHANICS (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure, and of which a summary will _ presently be given, that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures, and which was first demonstrated in a paper "On the Stability of Loose Earth," read to the Royal Society on the 19th of June 1856, and published in the Philosophical Trans actions for that year, viz. : Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, <f> the angle of repose of the earth ; then Q To apply this to liqiiids, <p must be made =0, and then^. =1 ? as Jii is well known. 3. Composition of a Structure, and Connexion of its Pieces. A structure is composed of pieces, such as the stones of a building in masonry, the beams of a timber frame-work, the bars, plates, and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance of fastenings of different kinds, whether made by means of the form of the joint (as dovetails, notches, mortises, and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps, and sockets). 4. Stability, Stiffness, and Strength. A structure may be damaged or destroyed in three ways : first, by displacement of its pieces from their proper positions relatively to each other or to the earth ; secondly, by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected ; thirdly, by breaking of one or more of those pieces. The power of resisting displacement constitutes stability ; the power of each piece to resist disfigurement is its stiffness ; and its power to resist breaking, its strength. 5. Conditions of Stability. The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are_to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position, and direction of the resistances required at each joint in order to produce equilibrium ; and the conditions of stability are, first, that the position, and, secondly, that the direction, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exert ing, conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the resistance, it is limited by con ditions, not of stability, but of strength and stiffness. 6. Principle of Least Resistance. Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, it has been demonstrated by Moseley that the smallest of those alternative systems is that which will actually be exerted, because the resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further. This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate. 7. Relations bctiveen Polygons of Loads and of Resistances. In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two, all the three distances being measured along one direction. Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called the centre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure ; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 1, where p P p P r l> r 2> r 3> r 4 represent the centres of load in a structure of four pieces, and the sides of the polygon of resist ances P 1 P 2 P 3 P 4 represent respec tively the directions and position^ of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, P6 the two resistances by which the piece to which that load is applied is supported ; then will those three lines be respectively the diagonal and sides of a parallelogram ; or, what is the same thing, they will be equal to the three sides of a triangle ; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes. According to a well-known principle of statics, because the loads or external pressures P^, &c., balance each other, they must be proportional to the sides of a closed poly gon drawn respectively parallel to their directions. In fig. 2 construct such a polygon of loads by drawing the lines Lj, &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and paral lelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem [originally due to Rankine] : // from the angles of the polygon of loads there be draivn lines (R v R 2 , &c.), ig. 2. each of which is parallel to the resistance (as P^, &c.) exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads (such as L lt L 2 , &c.) are applied ; then will all those lines meet in one point (0), and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to ivhich they are respectively parallel. When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases" to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point 0. 8. How the Earth s Resistance is to be treated. When the pressure exerted by a structure on the earth (to which the earth s resistance is eq"al and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical
forces, whose amount can be determined at the outset of the