A corresponding expression for the strength arising from the resist ance of the fibres below the neutral axis to the force of compression I lb 7 would be the integral of f dx (between x=0, x= d), that 2 is, nearly rg &P f' (f' being the force by which a fibre at Q N would resist compression), and the sum of the two integrals will be the whole strength of the beam to resist a transverse strain. Now the ratio of to f is different in different materials; and if we take f= f (which is the case in some kinds of wood), the said sum will I be = f 1x1= nearly.
But when the beam is strained by a weight at P, so that it takes the inclined position so N if we join o and p, and let fall the perpendicular p u ou A a, wo shall have w .p is, or (if I be the length of the beam) w.1 cos op n, for the momentum of the weight. Then w.1 cos op a= f bd' becomes the equation of equilibrium, w repre senting the weight which willjust break the beam ; and when L Op a, or the deflection, is small, its cosine may be considered as equal to unity. It follows that the strength by which beams of the like material resist this kind of strain will vary as T .
If a perfectly elastic beam or bar were attached horizontally at ono end to a wall, and were strained by a weight w at the other end, the mathematical theory would give for the deflection of the opposite erid of the beam (that is, the distance to which this end would be drawn in w / a vertical direction from the original position of the beam) A= (Poisson, Mdcanique,' torn. i., No. 310), where A = that deflection ; = the length of the beam ; a = the area of the transverse section ; d = the depth; and 5 = the element of deflection. Therefore, if A be found from experiment on a beam or bar in which w, 1, a, d, are given, we may from this equation obtain 5; and subsequently the value of A may be found for any beam, the materials being of the same kind. Again, the straining power by which a beam fixed at one end to a wall is dilated in the direction of its length is expressed by a a D (lb., No. 303), where D is the element of dilatation. Now, if to be the weight which would produce the deflection 5 and dilatation D, D 1 5 we should have w= a 5D ; whence — = — • and the first member of to a this equation being substituted for its equivalent in the above expression for A, the latter becomes A= ; or since the elongation of the whole beam is proportional to the length, and may be represented by D.1, if we put E for this elongation when w=w, wo
shall have A=E — 12 . Whence the elongation of an elastic rod by a weight or power acting in the direction of length is to the deflec tion of the same rod by a weight or power acting perpendicularly to its length, as the square of the depth or thickness is to the square of the length.
The relations between the strength and strain when a beam or bar, as m re in the preceding figure, is fixed at one end in a wall, and when a beam ell I' in the annexed diagram (fig. 2), of equal dimensions with Fig. 2.
respect to breadth and depth, but twice as long, is supported on a prop at its middle point (the weight at each extremity of the latter being equal to that at the extremity of the former), are the same. Also the angle v 0 r of deflection (0 v being in the direction of lo 0 produced), when a beam is supported on a prop at e, is equal to L o p ret,or L B op, in the preceding figure (o n being drawn perpendicular to the wall, or parallel to the horizon, and the beam stis being equal in every respect to one of the half-beams on the prop). For the angles re o trt are equal in both eases; since the weight at r' produces only the same effect as the reaction of the wall : and hence it follows that the angle It o r of deflection, with respect to the horizontal line te rt, will be equal to only half the angle u op. The same relation subsists between the deflections when the beam P'111. is supported on the props at the extremities. • It will follow, from what was at first stated, that a beam attached at one end to a wall in a horizontal position will boar suspended from the other extremity only half the weight which the same beam will hear on middle point when made to rest loosely on the two props. If the ends of the beam were prevented from rising on the prop, the strength would, on account of the additional weight necessary to pro duce deflection or fracture at each end, be increased In the ratio of 3 to 2 nearly.
The following table contains a few of the results obtained from ex periments made by Messrs. Banks, Barlow, and Tredgold, on wood and iron, when supported loosely on props and subject to a transverse strain at the middle point- The first column contains the length of the beam or bar In feet; the second, the areas of the transverse sections in square inches ; the third, the breaking weights in pounds; and the last, the deflections at the middle points in inches.