MATERIALS, STRKNGTH OP. 



MATERIALS, STRENGTH OF. 



ratio of / to / U different in different material ; and if we take 

 f--,f (which is the cue in some kinds of wood), the said sum will 



be = |/W nearly. 



But when the beam ia strained by a weight w" applied at P, so th.it it 

 takes the inclined position mm, it we join o and , and let fall the 

 perpendicular p B on A B, we shall hare w . ;> n , or (if I be the length of 

 the beam) w . I cos oj> B, for the momentum of the weight. Then 



w.i cosOj)B= g /W becomes the equation of equilibrium, w repre- 



senting the weight which will just break the beam; and when 

 /O/<B, or the deflection, is small, its cosine may be considered as 

 equal to unity. It follows that the strength by which beams of the 



vd 



like material resist this kind of strain will vary as -j" 



If a perfectly elastic beam or bar were attached horizontally at one 

 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 erfd 

 of the beam (that is, the distance to which this end would be drawn in 



w.i 

 a vertical direction from the original position of the beam) A = J^Jj 



(Poisson, ' Mecanique,' torn, i., No. 310), where A = that deflection ; 

 I = the length of the beam ; a = the area of the transverse section ; 

 d = the depth ; and 8 = the element of deflection. Therefore, if A 

 be found from experiment on a beam or bar in which w, /, a, d, 

 are given, we may from this equation obtain 8 ; 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 ita length is expressed by 

 a 8 D (74., No. 808), where D is the element of dilatation. Now, if w 

 be the weight which would produce the deflection S and dilatation D, 



D 1 

 we should havetc = a8D; whence = r; and the first member of 



to a 



this equation being substituted for its equivalent in the above 



D . W. i> 



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. I, if we put for this elongation when w = w, we 



shall have A = E . Whence the elongation of an elastic rod by a 



<f* 



weight or power acting in the direction of ; its 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 s in the preceding figure, is fixed at one end in a wall, and when 

 a beam p' H P in the annexed diagram (/</. 2), of equal dimensions with 



Fig. 3. 



respect to breadth and depth, but twice as long, is supported on a prop 

 at iU middle point (the weight at each extremity of the Utter being 

 equal to that at the extremity of the former), are the same. Also the 

 angle TO p of deflection (o v being in the direction of p" o produced), 

 when a beam is supported on a prop at Q, is equal to L o p B,'.or tuop, 

 in the preceding figure (o n being drawn perpendicular to the wall, 

 or parallel to the horizon, and the beam UN being equal in every 

 respect to one of the half-beams on the prop). For the angles M o m 

 are equal in both cases ; since the weight at if produces only the same 

 effect as the reaction of the wall : and hence it follows that the angle 

 H o r of deflection, with respect to the horizontal line n' n, will bo 

 equal to only half the angle u op. The same relation subsists between 

 the deflections when the beam P'K p is supported on the props at the 



' V- ...... 



It will follow, from what was at first stated, that a beam attached at 

 one end to a wall in a horizontal position will bear suspended from the 

 other extremity only half the weight which the same beam will bear on 

 its middle point when made to rest loosely on the two props. If the 

 ends of the beam were prevented from rising on the props, 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 result* obtained from ex- 

 periments nude by Messrs. Banks, Barlow, and Tredgold, on wood and 

 iron, when supported loosely on props and subject to a transverse 

 train at the tnidilli- 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 weigh U in pounds: and the 

 last, the deflections at the middle point* in inches. 



Length. Area. Weight. Deflection, 



Young oak (I 

 Sbtp limber 



New England flr 

 Ulna flr . 

 Teak . . 



rant-Iron tun 

 Ditto . 



481 



304 

 OS 7 

 430 

 313 

 810 

 748 

 869 



Since the strengths of beams attached at one end or supported on 

 props, the other dimensions being the same, vary as the squares of the 

 vertical depths, it follows that the most advantageous position, when the 

 areas of the transverse sections are equal, U that in which the broadest 

 surface is in a vertical position. In this manner girders and joists in 

 edifices are invariably placed. 



When a beam or bar is attached at one end to a wall, or when jt 

 turns upon its middle point like the great lever of a steam-engine, if it 

 be required that the beam should be equally strong in its whole length, 

 it should be mode to taper towards its extremities. When the depth 

 of the beam is constant, the breadth at any point should he ] 

 tionol to its distance from the extremity. When the breadth is to be 

 constant, the vertical face of the beam should have the form of the 

 common parabola; and when both breadth and depth vary, a longitu- 

 dinal section of the beam should have the figure of that which is called 

 a cubical parabola. 



If a weight be applied at any point in the length of a beam which 

 is supported on two props, the strain produced by it will be the 

 greatest when it is placed in the middle ; and the strain varies as the 

 product of the distance of the weight from the points of support. For 

 let A B (jig. 3) represent a beam supported at A and B, and let o be any 



Fig. 3. 



B 

 H 



point in it. Imagine a weight w to be applied at any point p ; then, 

 by the nature of the lever, A B : p B : : w : the pressure exerted bj 



the point A, namely^' 1 ; and this term expresses the re-action of 



A B 



the prop at A in consequence of the weight at P. Then also ' A o 



A B 



is equal to the strain at c produced by this re-action. Again, imagine 

 a weight w' to be applied at r'; then we shall have, as before, 



, and this last term expresses the reaction of the 

 prop at B in consequence of the weight at if : also '. B o is equal * 



AB : P'A : : w* : 



w'. r' A 



AB 



AB 



to the strain at c produced by this reaction. The sum of these strains 

 is equal to the whole strain at produced by the two weights. But 

 when p and p" coincide with c, we have p' A=AC, TB = BC; and the 



sum of the two strains is. 



jo. AC w*. AO. BC. 



- T * 



or, putting w" for w 



All AB 



+ w', we have the strain at c, in consequence of the weight w" placed 

 there, equal to w''. AC ' BO ; aud if w" be a constant weight applied 



AB 



at any point in A B, the strain will vary as A c, B c. This rectangle, 

 and consequently the strain, is a maximum when c is in the middle of 

 the line. 



If a weight be diffused over a beam which is fixed at one end to a 

 wall, it may be considered as acting at its centre of gravity, which, if 

 the diffusion be uniform, will be in the middle of the length of tho 

 beam. The momentum of the strain will consequently be cu 

 half of that which would result from an equal weight attached to the 

 opposite end. 



When a body is compressed in a direction perpendicular to the 

 length of the fibres, the points of support being very near and on 

 opposite sides of the place at which the force is applied, the strain to 

 which the body ia then subject has been colled by Dr. Toung tho 

 force of dctrusion. This species of strain sometimes occurs in the 

 construction of machinery; but few experiments h.< , 1,1:1110 



to determine the strength by which materials resist it. I 

 however it appears that the strength is proportional to the area of 

 the transverse section, and that it varies from four-thirds to 

 the strength by which the name material would resist a strain in tin- 

 direction of its length. 



Such machines as capstans ami windlasses, and axles which revolve 

 with their wheels, ore, when in action, subject to be twisted, so that 

 tli.'ir fibres tend to become curved in ob]i<ini- ilirections ; and the 

 strain thus produced is called that of torsion. The most natural way 

 of investigating the strength of materials to resist this kind of strain in 

 probably that which was adopted by Dr. Hobison : this mechanician 

 imagined the cylindrical Ixxly to be composed of an infinite num' 

 concentric hollow i -cried in each other ; and, supposing the 



whole to be cut by a plane perpendicular to the axis, lie- i-..n.-civd that 

 two particles in the circumference of any one of the concentric circles 

 would resist tho effort to separate them, by a force proportional to 



