.502 



CARPENTRY. 



Estimated 



T-heory of sider it so, by which the repulsions oppose the effort 

 Carpentry. o f W, the centre of compressions acting as a fulcrum. 

 If we suppose the extension in the instant of frac- 

 ture to be equal to the compression, then the dis- 

 tances of the centres of effort from the neutral point 

 D are equal, and the sum of them, or length of that 

 arm of the lever by which the cohesion acts, is equal 

 to one-third of the whole depth. 



The point D, therefore, becomes the virtual ful- 

 crum of the lever, by which the forces of attraction 

 and repulsion in the section A R resist fracture : the 

 action of the part above D is precisely the same as 

 in the last hypothesis ; and in a rectangular beam, 

 upon the supposition of the force increasing as the 

 extension, the centre of effort E will be at one-third 

 the distance DA from D. But in addition to this, 

 the resistance to compression in the part DR acts 

 precisely in the same way, and its effect may be sup- 

 posed condensed in the centre F at one-third of DR 

 from D ; the whole resisting energy will therefore be 

 the absolute- cohesion of DA drawn into one-third 

 DA, and added to the force of repulsion in DR 

 drawn into one-third D R. If we suppose the forces 

 of adhesion and repulsion to be equal, DR and DA 

 will be equal, and taking, as before, a the whole 

 depth, f the force of a fibre, whether against com- 

 pression or dilatation, the energy of the section 

 will be ! a X -5- /+| X \ af= ^ 2 f, that is, the 

 whole absolute strength may be conceived to act at 

 the distance of one sixth of the depth of the beam 

 from the fulcrum ; or the absolute strength is to the 

 resistance to cross strain as the length of the beam 

 to one sixth of its depth. 



We have found, then, that the principle of com- 

 pression makes an important change in our views of 

 the strength of timber. It shows the transverse 

 strength to be only one half of what the supposition 

 of Mariotte, and only one third of what the hypo- 

 thesis of Galileo had made it! The difference might 

 be still greater, were the body much more contrac- 

 tible than expansible. For in that case, the quantity 

 of fibres in a state of expansion being smaller, would 

 not only have less force in themselves, but would act 

 with a shorter arm of the lever. Thus, in the same 

 figure as before, suppose DArrr-j-AR, and we have 

 DE=i AR, the resistance of the upper or dilated 

 part would be ^ afx % a, and the resistance of the 

 lower part being the same, the whole is -^ a^f, or 

 nearly |- of the strength which Mariotte's supposition 

 would give. 



Observe, that although a body may, by the faci- 

 lity of extension, be readily bent or broken, though 

 not very compressible, yet in this respect the other 

 two theories would not be found so very erroneous, 

 as it is the resistance to extension alone which they 

 have considered. 



It would be of vast service in calculating the 

 strength of timber, if we could distinguish, even 

 with tolerable correctness, the amount of the fibres 

 under compression from those which were in a state 

 of dilatation ; or, what would be much the same, could 

 we learn what proporti"n there is bftween the com- 

 pressibility and extensibility of the timber. But this 

 is matter of fact, and not to be looked for as the re- 

 sult of any theory. The strain now under consider- 



with diffi- 

 culty. 



square 



depth. 



ation seems the best calculated for prosecuting this Theory ofl 

 research. Thus, if we find that a piece of fir timber r 

 an inch square, requires 9000 Ibs. to pull it directly 

 asunder, and that 150 Ibs. will break it transversely, 

 by acting ten inches from the section of fracture, we 

 may conclude that the neutral point is in the middle 

 of the depth, and that the attractive and repulsive 

 forces are equal. 



We should imagine that the sensible compressions 

 in fibrous bodies, as timber, are likely to be much 

 greater than the real corpuscular extensions. An 

 undulated fibre can only be drawn strait, and then 

 corpuscular extension begins ; but it may be bent 

 up and coiled to an indefinite degree, without alter- 

 ing much the situation of its particles, and of course 

 without affecting sensibly the corpuscular compres- 

 sion. 



We might pursue this enquiry to a great extent, 

 but it is rather foreign to our present object ; we 

 shall find some other opportunity of returning to it ; 

 and, in the mean time, hasten to draw practical in- 

 ferences from what we have stated. 



It is worthy of remark, that in all the hypotheses The 

 for expressing the resistance to fracture, the strength strength it 

 of the beam is as the breadth and square of the depth 

 directly, and length inversely; the only difference 

 among them being in the value of the constant nu- 

 merical coefficient. 



The strength, therefore, of a beam depends chief- 

 ly on its depth, or rather on that dimension which is 

 in the direction of the strain, and this is confirmed by 

 constant experience. The strongest joist which can 

 be cut out of a round log, is not that which contains 

 the greatest quantity of timber, but that which has 

 the breadth into the square of its depth the greatest 

 possible. Take AC the diameter of the log = a, PI.ATE 

 Fig. h., the breadth of the joist DC =r .r, its depth 

 DA will be n-v/AC 1 *DC*=*Sa*x\ and the pro- 

 duct AD Z X DC, that is x (a 1 x*) must be a max- 

 imum, and therefore its fluxion z x- 3.r 2 ;r=0,or 



a z =3a; i anda:=./ = a ^J =. .577 > and ma- 

 *' 3 ^ 3 



king cE = T Ac, the perpendicular ED cutting the 

 circumference in D, determines the breadth and depth 

 of the joist. The triangles ADC, AFC having the 

 same base, they are as the altitudes FH, DE; that 

 which has the greatest altitude is AFC, half of the 

 inscribed square. The joist ABCD is not only 

 stronger than A G C F, but is also lighter and cheaper. 

 The strength of the two are as 10 to 9-186, their 

 weight and bulks are as 10 to 10.607 ; the joist 

 ABCD is, therefore, nearly 15 per cent, better than 

 AGCF. 



The strongest joist which has a given girt, is that 

 of which the depth is twice the breadth ; for let 

 AD-f DC the half girt = a, and AD the depth zr x, 

 then the breadth, ur DC= x, the strength is as 

 ax 1 3 , and when that is a maximum 9,axx 3x*x 

 =r or 2azz3.r. that is, the depth must be one third 

 of the girc, or twice the breadth. In joists of the 

 same content of timber, the thinnest placed on edge 

 is the strongest. 



Upon the same principles hollow tubes are stiffer 

 than solid rods, containing the same quantity of mat- 



