negative; so that the neutral line passes from one side to the other of the line joining the centres of gravity of the cross sections of the lamina, at the point where =0, or at the point where the normal to the neutral line is parallel to the direction of P. 356. Case of a rectangular beam. If the form of the beam be such that it may be divided into lamina parallel to ABCD of similar forms and equal dimensions, and if the pressure P, applied to each lamina may be conceived to be the same; or if its section be a rectangle, and the pressures applied to it be applied (as they usually are) uniformly across its width, then will the distance of the neutral line of each lamina from the centre of gravity of any cross section of that lamina, such as PT, be the same, in respect to corresponding points of all the laminæ, whatever may be the deflection of the beam; so that in this case the neutral surface is always a cylindrical surface. 357. Case in which the deflecting pressure P, is nearly per pendicular to the length of the beam. In this case, and therefore sin. 4, is exceeding small, so long as the deflexion is small at every point R of the neutral line; so that his exceedingly small, and the neutral line of the lamina passes very nearly, or accurately, through the centre of gravity of its section PT. 358. THE RADIUS OF CURVATURE OF THE NEUTRAL SURFACE OF A BEAM. Since the pressures applied to the portion APTD of the lamina ABCD are in equilibrium, the principle of the equality of moments must obtain in respect to them; taking, there fore, the point R, where the neutral axis of the lamina intersects PT, as the point from which the moments are measured, and observing that the elastic pressures developed by the extension of the material in RP and its compression in RT both tend to turn the mass APTD in the same direction about the point R, and that each such pressure upon an element Ak of the section PT is represented (equation 498) by -pk, and therefore the moment of that pressure about the E R E point R by pak, it follows that the sum of the moments R about the point R of all these elastic pressures upon PT is represented by Epak, or by if I be taken to represent E El the moment of inertia of PT bout R. Observing, moreover, that if p represent the length of the perpendicular let fall from Rupon the direction of any pressure P applied to the portion APTD of the beam, Pp will represent its moment, and Pp will represent the sum of the moments of all the similar pressures applied to that portion of the beam; we have by the principle of the equality of moments, EI R = =ΣPp; 359. The neutral surface of the beam is a cylindrical surface, whatever may be its deflection or the direction of its deflecting pressure, provided that its section is a rectangle (Art. 356.); or whatever may be its section, provided that its deflection be small, the direction of the deflecting pressure nearly perpendicular to its length, and its form before deflexion symmetrical in respect to a plane perpendicular to the plane of deflexion. In every such case, therefore, the neutral lines of all the lamina similar to ABCD, into which the beam may be divided, will have equal radii of curvature at points similar to R lying in the same right line perpendicular to the plane of deflection; taking, therefore, equations similar to the above in respect to all the laminæ, multiplying both sides of each by I, adding them together, and observ ΣΙ ΣΡΟ ing that R and E are the same in all, we have = R E In this case, therefore, I may be taken in equation (700) to represent the moment of inertia of the whole section of the beam, and P the pressure applied across its whole width. 360. The radius of curvature of a beam whose deflexion is small, and the direction of the deflecting pressures nearly perpendicular to the length of the beam. In this case the neutral line is very nearly a straight line, perpendicular to the directions of the deflecting pressures; so that, representing its length by x, we have, in this case, px; and equation (500) becomes which relation obtains, whatever may be the form of the transverse section of the beam, I representing its moment of inertia in respect to an axis passing through its centre of gravity and perpendicular to the plane of deflexion. 361. The moment of inertia I of the transverse section of a beam about the centre of gravity of the section. In treating of the moments of inertia of bodies of different geometrical forms in a preceding part of this work (Art. 82, &c.), we have considered them as solids; whereas the mo ment of inertia I of the section of a beam which enters into equation (500) and determines the curvature of the beam when deflected, is that of the geometrical area of the section. Knowing, however, the moment of inertia of a solid about any axis, whose section perpendicular to that axis is of a given geometrical form, we can evidently determine the moment of the area of that section about the same axis, by supposing the solid in the first place to become an exceedingly thin lamina (2. e. by making that dimension of the solid which is parallel to the axis exceedingly small in the expression for the moment of inertia), and then dividing the resulting expression by the exceedingly small thickness of this lamina. We shall thus obtain the following values of I: 362. For a beam with a rectangular section, whose breadth is represented by b and its depthi I= ;'z'c' by c (equation 61), 363. For a beam with a triangular section, whose base is b and its height cI(equation 63), 364. For a beam or column with a circular section, whose radius is e (equation 66), B 365. To determine the moment of inertia I in respect to a beam whose transverse section is of the form represented in the accompanying figure, about an axis ab passing through its centre of gravity; let the breadth of the rectangle AB be represented by b, and its depth by d,, and let b, and d, be similarly taken in respect to the rectangle EF, and b, and d, in respect to CD; also let I, represent the moment of inertia of the section about the axis ed passing through the centre of CD, A,, A, A,, the areas of the rectangles respectively, and A the area of the whole section. Now the moments of inertia of the several rectangles, about axes passing through their centres of gravity, are represented by bd,, bd,b,d,', and the distances of these axes from the axis cd are respectively (d,+d,), (d,+d,), 0. Therefore (equation 58), 3 12 I,=,'1⁄2b,d,' +1(d,+d,)2 A ̧+¿1⁄2b„d,”+1(d,+d,)2 A ̧+11⁄2b,d;'; but A ̧=b,d,, A‚=b,d, A ̧=b,d,; .. I‚—‚¿‚1⁄2‚(A‚d‚' + A‚d,2+A ̧d ̧2)+‡(d,+d ̧)2 A‚ +‡(d,+d ̧)2 A‚. Also if h represent the distance between the axes ab and cd, then (Art. 18) A=4(d,+d,)A,−1(d,+d,) A ̧, and (equation 58) I=I,—'A. 2 ..I=‚1⁄2(A‚d,2+A„d,”+A‚d ̧1) +‡ {(d, +d3)3A, +(d,+d,)aA,} {(d›+d,)A,−(d,+d ̧)A,}' Ꭺ (502). If d, and d, be exceedingly small as compared with d neglecting their values in the two last terms of the equation and reducing, we obtain If the areas AB and EF be equal in every respect, I= {d,'+3(d,+d,)'} A,+1,A‚d,' . .. (504). .... 366. THE WORK EXPENDED UPON THE DEFLEXION OF A BEAM TO WHICH GIVEN PRESSURES ARE APPLIED. pended upon the extension or compression of pq is therefore represented by E. Ax 12 RRT (p2Ak). R2 And the same being true of the work expended on the compression or extension of every other fibre composing the elementary solid VTPQ, it follows that the whole work expended upon the deflexion of that element of the beam EAX ΕΙ 1 is represented by Epak, or by ; for Σp3ak repre sents the moment of inertia I of the section PT, about an axis perpendicular to the plane of ABCD, and passing through the point R. If, therefore, a, be taken to represent the length of that portion of the beam which lies between D |