sulting equation in respect to x, we shall obtain for the distance of the point m from c the expression 370. THE LENGTH OF THE NEUTRAL LINE, THE BEAM BEING LOADED IN THE CENTRE. Let the directions of the resistances upon the extremities of the beam be supposed nearly perpendicular to its surface; then if x and y be the co-ordinates of the neutral line from the point a, we have (equation 501), representing the horizontal distance AB by 2a, and observing that in this case 1 d'y and that the resistance at A or B=1P, == R dx2' Integrating between the limits x and a, and observing that Now if s represent the length of the curve ac, a dy2 a dy += ƒ (1+dx)'dr=ƒ' {1+¿ (d')"}dz nearly; since Eliminating P between this equation and equation (511), and representing the deflexion by D, 371. A BEAM, one portion oF WHICH IS FIRMLY INSERTED IN MASONRY, AND WHICH SUSTAINS A LOAD UNIFORMLY DISTRIBUTED OVER ITS REMAINING PORTION. Let the co-ordinates of the neutral line be measured from the point B where the beam is inserted in the masonry, and let the length of the portion AD which sustains. the load be represented by a, and the load upon each unit of its length by μ; then, representing by a and y the co-ordinates of any point P of the neutral line, *The following experiments were made by Mr. Hatcher, superintendant of the work-shop at King's College, to verify this result, which is identical with that obtained by M. Navier (Resumé de Leçons, Art. 86.). Wrought iron rollers 7 inch in diameter were placed loosely on wrought iron bars, the surfaces of contact being smoothed with the file and well oiled. The bar to be tested had a square section, whose side was 7 inch, and was supported on the two rollers, which were adjusted to 10 feet apart (centre to centre) when the deflecting weight had been put on the bar. On removing the weights carefully, the distance to which the rollers receded as the bar recovered its horizontal position was noted. and observing that the pressures applied to AP, and in equilibrium, are the load μ(a-x) and the elastic forces developed upon the transverse section at P, we have by the principle of the equality of moments, taking P as the point from which the moments are measured, and observing that since the load (a-x) is uniformly distributed over AP it produces the same effect as though it were collected over the centre of that line, or at distance (a-x) from P; observing, moreover, that the sum of the moments of the elastic forces upon the section at P, about that point, is represented (Art. 358.) by Integrating twice between the limits 0 and x, and observing =0 and y=0, since the portion BC of the beam is rigid, we obtain El121⁄4=—¿μ(a−x)3+ {μa3 EI dy dx (523), Ely=2(ax)1 + {μа3x—‚1μà1. . . . (524), which is the equation to the neutral line. Let, now, a be substituted for x in the above equation; and let it be observed that the corresponding value of y represents the deflexion D at the extremity A of the beam; we shall thus obtain by reduction Representing by ẞ the inclination to the horizon of the tangent to the neutral line at A, substituting a for x in equation dy (523), and observing that when x=a, tan. B, we obtain dx 372. A BEAM SUPPORTED AT ITS EXTREMITIES, AND SUSTAINING A LOAD UNIFORMLY DISTRIBUTED OVER ITS LENGTH. Let the length of the beam be represented by 2a, the load upon each unit of length by ; take randy as the co-ordinates of any point P of the neutral line, from the origin A; and let it be observed that the forces applied to AP, and in equilibrium, are the load u upon that portion of the beam, which may be supposed collected over its middle point, the resistance upon the point A, which is represented by ua, and the elastic forces developed upon the section at P; then by Art. 360., Integrating this equation between the limits x and a, and observing that at the latter limit dy =0, since y evidently attains its maximum value at the middle C of the beam, dy El = {μ(x3—a3) —¦μa(x2—a2) . .... (528). dx Integrating a second time between the limits 0 and x, and observing that when x=0, y=0, Ely={μ({x1—a3x)—§μa(}x3—a2x). .... (529), which is the equation to the neutral line. Substituting a for x in this equation, and observing that the corresponding value of y represents the deflexion D in the centre of the beam, we have by reduction Representing by 6 the inclination to the horizon of the tan gent to the neutral line at A or B, and observing that when Let it be observed that the length of the beam, which in equation (511) is represented by a, is here represented by 2a, and that equation (530) may be placed under the form (2aμ) (2a)3 48EI D={. ; whence it is apparent that the deflexion of a beam, when uniformly loaded throughout, is the same as though ths of that load (2aμ) were suspended from its middle point. 373. A BEAM IS SUPPORTED BY TWO STRUTS PLACED SYMMETRICALLY, AND IT IS LOADED UNIFORMLY THROUGHOUT ITS WHOLE LENGTH; TO DETERMINE ITS DEFlexion. Let CD=2a, CA a1, load upon each foot of the length of the beam =; then load on each point of support =μa. D B Take C as the origin of the co-ordinates; then, observing that the forces impressed upon any portion CP of the beam, terminating between C and A, are the elastic forces upon the transverse section of the beam at P, and the weight of the load upon CP; and observing that the weight CP of the load upon CP, produces the same effect as though it were collected over the centre of that portion of the beam, so that its moment about the point P is represented by μ. CP. CP, or by CP2; we obtain for the equation to the neutral line in respect to the part CA of the beam (Art. 360.) |