CARPENTRY. 



503 



. ( ,f whirh prii.i ij.lc we have some beautiful in- 



. tin- quills of birds, th Ij-.mc I ui .mtmals, 



ka of reeds and grasses, &c. ; and our best en- 



i nutate this procedure of nature in form- 



: In .r mill axles, uprights, and the like, of cast iron 



Hut another maxim which we find given by wri- 

 <>f reputation is not to be relied on. A triangu- 

 lar bar projecting from a wall with a weight at its 

 extremity, is said to be jthrice as strong when one side 

 is uppermost as it will be when one angle is upper- 

 most. This is only true upon the first and second 

 hypothesis, win-re no account is taken of the com- 

 pression of the concave side. Having already shown 

 the falsity of this principle, we would observe, 

 that in the supposition of an equal resistance to 

 condensation and dilatation, it is a matter of indif- 

 ference which side of such a bar is uppermost; for 

 if the broad side be favourable for resisting dila- 

 tation, the opposite angle is equally unfavourable for 

 resisting compression. Should the resistance to com- 

 pression be very great, the proposition would be but 

 partly true, for, in truth, it supposes that resistance 

 infinite. Now we know that, in many kinds of tim- 

 ber, the fibres undergoing compression are more nu- 

 merous than those which are in dilatation. In such 

 cases the reverse of this proposition is the truth, and 

 the bar will be stronger with the broadside to resist 

 the compression. The exact mode of action is by no 

 means so readily discovered in this particular case ; 

 but as this theorem has given rise to several erroneous 

 maxims of practice, we shall endeavour to give the 

 reader a correct idea of it, by subjecting the simpler 

 case to mathematical investigation. 



Let A BC be the cross section of fracture of the tri- 

 angular bar ; MN the neutral line, or separation be- 

 >[) the parts compressed and those dilated; oprq 

 an indefinitely thin slip of the fibres. The length of 

 the slip op will be proportional to its distance from 

 the angle ; the stretch or compression which it un- 

 dergoes, will be proportional to its distance from the 

 neutral line ; and upon the supposition formerly made, 

 of the resisting force being proportional to the ten- 

 sion, being the law of elastic bodies, the resistance it 

 makes will also be proportional to that distance. Last- 

 ly, the lever by which it acts in resisting a bend or 

 fracture, is the same distance, and therefore its ener- 

 gy is as the square of its distance from MN. Take 

 this distance DE=x, and making DC=r,MN:=6, we 

 have for the length of the slip t op, CD : MN : : CE : op, 



b x . , , . . 



or a : 6 : : a x : b -- , its breadth is x, its magm- 







tude opqr is bx -- ; and making, as before,ythe 

 force of each fibre exerted at the greatest distance fl, 



at the 



the strength of this will be ^ ^* 



a a* 



distance x ; its energy therefore by means of the le- 



. bfx*x bfx*x _,, _ - .. , 



x, is - ** - . 1 he fluent of which, 



verage 



bfx* 



. .//-., I 



is = a, be = , j bfa z=. 



That is, as '. a /(/represents the absolute cohesion of 

 the i MN ; the actual force it as if the ab- 



solute cohesion of MNc acted at the distance of i of 

 its perpendicular from the neutral line MN, which is 

 the solution usually given for the transverse strength 

 of the triangle. 



Before proceeding farther we may observe, that this 

 shews us the ratio of the strength of a square, or 

 rhomboidal joist, lying on edge, compared with the 

 same joist lying on one side. Upon the supposition 

 of the equality of compression and dilatation, the 

 triangle below will just exert the same strength as 

 the one above, and the whole will be as if twice the 

 absolute cohesion of the half beam, that is, as if the 

 absolute cohesion of the whole beam were exerted at 

 the distance of one sixth of the half diameter from the 

 centre. In the other case it will be as if the abso- 

 lute cohesion were exerted at the distance of one- 

 third of the side from the centre, and therefore it is 

 stronger in the proportion of 166 to 118. But it 

 must be observed, that the supposition of equal com- 

 pression and dilatation is the most unfavourable for 

 the diagonal position, the strength of the broadest 

 part of the beam being lost by being in the neutral 

 line. 



Let us next inquire into the action of the broader 

 part of the triangles, viz. MABN. Here, as before, 

 the length of the slip s t r n increases with the dis- 

 tance from C, and calling DF = x, we find, 



, . bx 

 a:a-\-x::b:sr=.b-\ ; 



and, by reasoning as before, the energy of the whole 



MABN will be 



+ 



r 



It is evident that we cannot, in this case, take x=<i 

 upon the supposition of equality of compression and 

 dilatation. Let us, therefore, assume the whole 

 energy of the side AN, to be equal to that of the 



side MCN, then making 

 we find 



3a 



total enrgy, is -r 7 r which, when 



and taking the root of this equation, we get x = p=rri 



or nearly a, in which case it must be observed that 

 the fibres of the broad side are strained little more 

 than one half of those at the opposite edge. It is 

 perhaps possible, therefore, that a triangular bar may 

 fail at the edge, and yet not be broken by the strain, 

 it would be very desirable that experiments were made 

 on this point. 



Upon the principle of equality of compression and 

 dilatation, which is certainly the most probable at the 

 beginning of the bending, it does not appear how the 

 superabundant strength of the broad side of the beam 

 can be brought into action ; its superiority, therefore, 

 in one position over another, is a mere deception of 

 hypothesis. But if we suppose the'compressibility 

 and dilatibility different, which without doubt is ul- 

 timately the case in most bodies, (thus oak carries 



"" 



< 



