CARPENTRY. 



Theory ef blc in thil way, since a hollow axle t. 



''"7- which, with the sa"ie legrec of strength, would 

 "*V ' tain much less stulf than a solid one. 'J'tit- Hollow 

 axle is also stiffer than the other, as we have shown 

 ii treating of cross strains, ami it affords a much 

 better fixture for tin- (lances and arms. It is with 

 good reason, then-fore, t.iat this improvement has 

 been of late introduced in the formation of axles and 

 large maats. 



The whole cohesion of the exterior circle of par- 

 ticles being supposed exerted, the force ot any inte- 

 rior circle will be less than ;hat in the proportion of 

 the squares of their diameters or radii. The effect 

 of the whole, therefore, will just be -^dsof the lateral 

 cohesion of the section, or one third the force requi- 

 red to cut the piece across, by means of a square edged 

 tool. The force required for this, as we have already 

 mentioned, appears from experiment to be consider- 

 ably greater than that which would pull the parts 

 of the beam directly asunder. 



To determine the effect of this, in resisting an ex- 

 ternal force which tends to break the axle, Itty re- 

 present the lateral cohesion of any particle, r the ra- 

 dius of the arch, x the di-tance of any circle from 

 the centre ; then x is the fluxionary increment of the 

 radius, or breadth of an indefinitely small concentric 

 ring, we have the force of any particle as its exten- 

 sion, or as the distance x from the centre. That is, 



r : x : :/":, and the cohesion of the part x is " 



T . r ' 



taking ir as the number 3.1416/ we have the cohe- 

 sion of a ring =r-^ 



; its energy is found, by 

 multiplying this by the length of lever or distance x, 



and is therefore = 



the fluent of this is 



ft A 



\ , which, when x=rr, gives for the whole ener- 

 gy of the axle \ sr/> 5 . Now irr 1 is the area, and 

 xfr* is the lateral cohesion of the section; wherefore 

 the energy will be found by supposing the whole 

 lateral cohesion exerted at of the radius from the 

 centre. 



Example. Required the weight which would 

 wrench off an axle of fir 1 foot in diameter, acting 

 on it by a lever or wheel at the distance of 3 feet 

 from the centre. 



Take the absolute cohesion of fir 8000 Ibs. per 

 square inch. 



8000 x 1 44 X. 785*= 904781, the cohesion of the 



nection ; 



36 inc. : 1 : ; 904781 : 37700 Ibs. weight required. 

 Though the length of the axle has not been allu- 

 ded to in these deductions, it is probable that a con- 

 siderable degree of importance should be attached to 

 it. In fibrous matter such as timber, the fibres will 

 be twisted up like a skain, and made to close upon 

 each other. The diameter of the axle will be some- 

 what diminished, and with that the energy of each 

 fibre. The twisting produces also a species of cross 

 strain upon them, while they are violently drawn in 

 length ; so that on the whole, the length of the axle 

 seems to have somewhat of the action of a lever in 

 3 



facilitating the disruption. It i not eaty to ul>jc.i Theory of 



thin to calculation ; but it is obvious that the result for *- ar l" 



a simple twiit will be thereby considerably reduced. '^'V'**' 



The various forms into wh.ch timber may be bent 

 are innumerable, and to disc u completely the strain* 

 to whu h even the most bimple and common may be 

 subjected, would be occupation for volumes ; neve rthe- 

 lesi we cannot entirely omit them. We have already, 

 in examining the effects of bending, when treating of 

 crost. strains, stated some of the most important con- 

 clusions. We shall now, therefore, be very brief. HIAT 

 Let ACB, Fig. 2, be the piece of crooked timber, CXlII. 

 loaded at A to the utmost. Join AB draw the per- * J *' 

 pendicular cCfD. Let ACB represent the neu- 

 tral libre. It is evident that the first change likely to 

 take place, is the slipping of the fibres on each other; 

 for as the beam bends under the load, the concave 

 parts will be somewhat protruded beyond the con- 

 vex : the parts A e B are stretched. And we might 

 consider the whole as a framing, in which Ay, J B, 

 fe are struts, and A e, eB are ties. But this w ;uld 

 necessarily lead us to anticipate other matter; we 

 shall therefore consider the weight A as applied to 

 the virtual lever DC e, of which the fulcrum will be 

 C. The weight of A applied to D will be balanced 

 by the resistance to extension in e C, an equal re- 

 sistance to compression in Cyj added to the weight 

 of A, also compressingy C ; since the force inyC will 

 be equal to the sum of the other two in the opposite 

 direction 



Now the weight of A is constant. The extension 

 in e C, and equal compression in Cy, will therefore 

 increase with, and be proportional to, the distance 

 CD. Its amount would be known as aforesaid, did 

 we know the position of the neutral line. The part 

 C has a tendency to fly out from AB, which is re- 

 sisted by the extension AD; so this tendency of C 

 to fly out is measured by the cosine DC, which show* 

 also the strength of the bndle that should prevent it. 

 Hence we may observe, that cross bridles made to 

 preserve straight posts from yielding, may be vastly 

 smaller than those fir beams which have already taken 

 a set, or were originally crooked. 



If the piece ACB were under distension, the rea- 

 soning and magnitude of the parts would be the 

 same, the mode of action of the concave and convex 

 sides only changing. And in like manner, by at- 

 tending to the virtual leverage, it will not be difficult 

 to form a conception of the mode of action in every 

 oblique strain. When the beam ACB is distended, 

 the parts C e and Cy will tend to part from each 

 other, and hooping the beam there may have a good 

 effect. The same thing is likely to occur in various 

 other instances. 



When the piece is exposed to a cross strain, acting 

 in CD, and the ends A, B have abutment, each part 

 is undergoing a longitudinal strain, the amount of 

 which, in proportion to the force applied, is greater 

 as AD to DC nearly, and may readily be discovered 

 by the resolution of forces. When the parts A and 

 B have no abutment, then the parts B/'A must re- 

 sist by being extended, and the case returns to one 

 in which the parts AB are distended. It is evident 

 that the power of the cross strain will be greater the 

 nearer ACB is to the straight line. In such a case 



