LATTICE GIRDER-UNIFORM LOAD. 161 same cross section, it is assumed that the resistance to the shearing force is equally distributed amongst them. To fulfil this condition where a pair of diagonals, as in fig. 83, cross each other, with equal and opposite inclinations, the stresses along them must be equal, and of opposite kinds. Then let R' and R' be the stresses along the pair of diagonals, and 2 and — i their inclinations to the horizon, we shall have for the vertical component of the force sustained by them F, R' sin i R' sin (-i) = 2 R' sin i;........(1.) = so that the horizontal components of the stresses along the two diagonals at the plane of section balance each other. Let 2 m be the number of diagonal bars which cross each other at a given vertical section, the amount of the stress along each bar is pull R' = 2 m with which is a {trust} for bars which slope against the shearing force. The pull along the lower bar, and the thrust along the upper bar, at the given vertical section, must constitute a couple which balances the bending couple M, hence their common amount is 165. Lattice Girder-Uniform Load.-If N denote the even number of equal divisions into which the length of a lattice girder is divided by vertical lines traversing all the joints, whether of meeting of diagonal and horizontal bars, or of crossing of diagonal bars, and I the length of one of those divisions, so that N 1, as before, is the span of the girder, then the effect of a load equally distributed amongst all those vertical lines, or amongst the alternate lines, may be found by means of the formulæ for a half-lattice girder, Article 163, as follows: L. When the load is distributed over all the vertical lines, the formulæ for case 1, equations 1, 2, 3, 4, are to be applied to vertical sections, such as CD, traversing the joints of crossing of diagonals; observing only, that the resistance to the shearing force is distributed amongst the diagonals as shown by equation 2 of Article 164. II. When the load is distributed over those vertical lines only which traverse joints of meeting of diagonal and horizontal bars, the formulæ of case 2, equations 5, 6, 7, 8, 9, so far as they relate to sections made at unloaded joints, are to be applied to vertical sections, such as CD, traversing the joints of crossing of diagonals; attending as before to the distribution of the stress amongst the diagonals by equation 2 of this Article. (See p. 639.) 166. Transformation of Frames. The principle explained in Article 66, of the transformation of a set of lines representing one balanced system of forces into another set of lines representing another system of forces which is also balanced, by means of what is called "PARALLEL PROJECTION," being applied to the theory of frames, takes obviously the following form: THEOREM. If a frame whose lines of resistance constitute a given figure, be balanced under a system of external forces represented by a given system of lines, then will a frame whose lines of resistance constitute a figure which is a parallel projection of the original figure, be balanced under a system of forces represented by the corresponding parallel projection of the given system of lines; and the lines representing the stresses along the bars of the new frame, will be the corresponding parallel projections of the lines representing the stresses along the bars of the original frame. This Theorem is called the "Principle of the Transformation of Frames." It enables the conditions of equilibrium of any unsymmetrical frame which happens to be a parallel projection of a symmetrical frame (for example, a sloping lattice girder), to be deduced from the conditions of equilibrium of the symmetrical frame, a process which is often much more easy and simple than that of finding the conditions of equilibrium of the unsymmetrical frame directly. SECTION 2.-Equilibrium of Chains, Cords, Ribs, and Linear Arches. 167. Equilibrium of a Cord.—Let DAC in fig. 84 represent a Fig. 84. B flexible cord supported at the points C and D, and loaded by forces in any direction, constant or varying, distributed over its whole length with constant or varying intensity. Let A and B be any two points in this cord; from those points draw tangents to the cord, A P and B P, meeting in P. The load acting on the cord between the points A and B is balanced by the pulls along the EQUILIBRIUM OF A CORD. 163 cord at those two points respectively; those pulls must respectively act along the tangents A P, BP; hence follows THEOREM I The resultant of the load between two given points in a balanced cord acts through the point of intersection of the tangents to the cord at those points; and that resultant, and the pulls along the cord at the two given points, are proportional to the sides of a triangle which are respectively parallel to their directions. The more the number of loaded points in a funicular polygon (as defined in Article 150) is increased, or, in other words, the more the number of sides in the polygon is multiplied,-the more nearly does it approximate to the condition of a cord continuously loaded; while at the same time, the number of lines radiating from the point O in the diagram of forces (exemplified in fig. 75) increases with the number of sides of the funicular polygon, and the polygon of external forces of fig. 75* approximates to a continuous line, curved or straight. A diagram of forces for a continuously loaded cord may be constructed in the following manner (fig. 84*). Let radiating lines be drawn from the point O parallel to the tangents of the cord at any points which may be under consideration:-for example, let O C, OD, be parallel to the tangents at the points of support, and O A, O B, parallel to the tangents at the points A and B of fig. 84 respectively Let the lengths of those radiating lines represent the pulls along the cord at the points to whose tangents they are parallel; and let a line D A B C, curved or straight, as the case may be, be drawn so as to pass through the extremities of all the radiating lines which represent the pulls along the cord at different points. Then from Theorem I. it appears, that a straight line drawn from B to A in fig. 84*, will represent in magnitude and direction the resultant of the load on the cord between A and B (fig. 84). Now, suppose the point marked A in fig. 84 to be taken gradually nearer and nearer to B; then will OA in fig. 84* approach gradually nearer and nearer to OB; and while the direction of the straight line drawn from B to A gradually approaches nearer and nearer to the direction of the tangent at the point B to the line CBAD in fig. 84*, the resultant load between B and A represented by that straight line gradually approaches nearer and nearer in direction to the direction of the load at the point B in fig. 84; therefore, the direction of the load at any point B of the cord (fig. 84), is represented by the direction of a tangent at B (fig. 84*), to the line CBAD. Hence follows D Fig. 84*. THEOREM II. If a line (called a line of loads) be drawn, such that while its radius-vector from a given point is parallel to a tangent to a loaded cord at a given point, its own tangent is parallel to the direction of the load at the point in the cord, then will the length of a radius-vector of the line of loads represent the pull at the corresponding point of the cord; and a straight line drawn between any two points in the line of loads will represent in magnitude and direction the resultant load between the two corresponding points in the cord. The supporting forces required at the points C and D (fig. 84), are obviously represented in magnitude and direction by the extreme radiating lines, OC, OD. A loaded cord, hanging freely, is obviously stable, but capable of oscillation. C 168. Cord under Parallel Loads.-If the direction of the load be everywhere parallel and vertical, the line of loads becomes a vertical straight line, as CBAD (fig. 84**). To express this case algebraically, let A in fig. 84 be the lowest point of the cord, so that the tangent AP is horizontal. Then in fig. 84**, O A will be horizontal, and perpendicular to CD. Let Fig 84** H: = horizontal tension along the cord at A; ROB = pull along the cord at B; P = AB load on the cord between A and B; To deduce from these formulæ an equation by which the form of the curve assumed by the cord can be determined when the distribution of the load is known, let that curve be referred to rectangular horizontal and vertical co-ordinates, measured from the lowest point A, the co-ordinates of B being, A X = x, XB = y; then a differential equation which enables the form assumed by the cord to be determined when the distribution of the load is known. 169. Cord under Uniform Vertical Load.—By an uniform vertical load is here meant a vertical load uniformly distributed along a CORD UNDER UNIFORM VERTICAL LOAD. 165. horizontal straight line; so that if A (fig. 85), be the lowest point of the rope or cord, the load suspended between A and B shall be proportional to AX = x, the horizontal distance between those points, and capable of being expressed by the equation Р = px;....... = (1.) = y. where p is a constant quantity, denoting the intensity of the load in units of weight per unit of horizontal length: in pounds per lineal foot, for example. It is required to find the form of the curve DA B C, and the relations amongst the load P, the horizontal pull at A (H), the pull at B (R), and the co-ordinates AX X, BX First Solution.-Because the load between A and B is uniformly distributed, its resultant bisects A X; therefore, the tangent BP bisects AX: this is a property characteristic of a PARABOLA whose vertex is at A, therefore, the curve assumed by the cord is such a parabola. Also, the proportions of the load, and the horizontal and oblique tensions are as follows: Second Solution.-In the present case equation 2 of Article 168 becomes which being integrated with due regard to the condition that when x= 0, y = 0, gives P x2 (4.) the equation of a parabola whose focal distance (or modulus, to use the term adopted in Dr. Booth's paper on the "Trigonometry of the Parabola," Reports of the British Association, 1856), is, |