CARPENTRY. 



.307 





PL.AT* 



ITB 



rxnr. 

 Fig. i. 



r.flect of 

 bending. 



provide for this furcc in addition to the 

 strength required agaiiibt the crous strain. And, in- 

 t of the btn ngth winch this load requires, 

 .nice to the cross strain itself is greatly mo- 

 dified by it. The quantity of the beam in Fig 4-, 

 which is under compression, wi'l be increased ; when 

 \CV is an acute angle, the neutral point D will 

 fore be nearer the upper side, and the quantity 

 of fibres under dilatation will be diminished. The 

 reverse will be the case when the oblique force acts 

 in any direction beyond the perpendicular CW, so 

 that VCX becomes an obtuso angle. The dilated 

 part will then increase, and the part under compres- 

 sion will diminish. We know too little of the cor- 

 puscular forces, to say exactly what may be the ef- 

 fect of tins ; but it is clear, that it may tend mate- 

 rially to weaken the beam. The compression or ex- 

 tension may be so very great, that a very small cross 

 strain may suffice to complete the rupture ; yet this 

 is a case which is sometimes unavoidable. The most 

 enormous strains to which framings of carpentry are 

 subjected are of this kind ; and the skill of the engi- 

 neer is particularly shown in reducing every load to 

 a direct pressure or pull in the length of the beam, 

 and to avoid everything which may tend in the small- 

 est degree to disturb this arrangement. By the set- 

 tling and sagging of the work, pieces come often to 

 exert very various and yet very considerable strains, 

 which were originally supposed and intended to be 

 inactive. 



When the oblique load upon a beam is such, that 

 it increases with the increase of length, as the roof 

 over a rafter or the like, we have another reason for 

 giving the beam a greater scantling. If MN and OP, 

 Fig. 1, be loaded equally per foot in length, OP 

 carries more in the ratio of PD to DN. To be 

 equally strained, therefore, DP must be made broad- 

 er, or placed nearer its neighbours, in the ratio of 

 DP to DN ; or its depth increased, in the ratio of 

 the square root of DN to the square root of DP. 

 In the case of a rafter, it is to be observed, that the 

 weight increases as we pass downwards ; and hence 

 they must be thickened below. In a kingpost, on 

 the other hand, the weight of the post itself is a load 

 which increases towards the top ; and we place the 

 thick part of that post (as it is improperly called) to 

 the upper end. 



We may now return again to the case of a beam 

 compressed lengthwise, which is liable to be so much 

 modified by the resistance to bending. 



It is needless to inquire how a pressure acting on 

 a straight beam or post in the direction of its length 

 can bend it. It is evident, that the case arises from 

 the unequal compressibility of the sides. One side 

 of the beam must be supposed to act as a fulcrum, 

 to enable the incumbent weight to compress the other 

 ide. But this state of the case soon ceases. As 

 the compression goes on, the fulcrum or centre of 

 support shifts gradually towards the concave side ; 

 and when the post is so far bent, that the line of di- 

 rection of the load passes without it, the concave 

 parts must be in a state of compression, ard if the 

 post supports the load, the convex parts must be in 

 a state of extension. This situation of things bears 

 an intimate resemblance to the action of a cross strain 



' 



on one end of a beam, of which the other end u built Theory cf 

 into a wall ; we may say, it is exactly the same v 

 the condition of such a beam, when pulled in the ' "~'~ 

 same direction as that in which the load acts. 



Let ACB, in Fig. 2, represent the position of 

 the neutral fibre ; then drawing the perpendicular 

 DC, we may say, that the strain on the section ef 

 is the same as if the weight A were applied at the 

 end D of the lever DC e t of which e is the fulcrum, 

 and balanced by the resistance to compression in the 

 part C/, and to dilatation in the part Ce; or as these 

 two resistances must be equal, we may conceive it 

 balanced by twice the resistance to dilatation in the 

 part C e, in addition to its own weight. 



If therefore we knew exactly where the neutraf 

 point C was situated, we would thereby be enabled 

 at once to determine the strain on a post or strut, 

 which was bent to a given degree. But, unfortu- 

 nately, theory seems here to fail us. The proportion 

 between the extension and compression produced by 

 equal forces can only be learned by experiment. 



We may, however, draw some useful general infer- 

 ences. 



I. It is evident, that the greatest strain will be at 

 the place where the post is most bent from the straight 

 line. For the arm of the lever DC will then be the 

 greatest, and it is not likely that the arm C e will be 

 any where less. 



II. The strain appears to be nearly proportional 

 to the distance of the section from the line AB. 

 But we cannot say so with certainty, as there may 

 possibly be the same change in the arm beyond the 

 neutral line, as in the part CD, by a shift in the vir- 

 tual fulcrum. Euler, therefore, the only author of 

 reputation who has treated of this subject, and after 

 him Emerson upon Euler's principles, have fallen 

 into considerable errors, when they say the strain is 

 proportional to the ordinate CD, or rather yD, as 

 they have it. In the first and second hypotheses 

 above mentioned for the cross strain, this would be 

 indeed true ; but we have already shown the falsity 

 of these hypotheses, and the necessity of attend- 

 ing to the compression. If that be necessary in 

 treating of the strain directly across, it is still more 

 indispensible here, where the very essence of the 

 strain consists in its compressing the timber. 



In the great and most dangerous loads, as the 

 trussed centres for heavy vaultings, or the like, the 

 compressed part occupies the greater part of the sec- 

 tion, which, by removing the neutral point very near 

 to the convex side, not only increases the strain, by 

 extending the arm DC, but diminishes the power of 

 resistance, by leaving fewer fibres for distension ; and 

 these few acting by a shorter arm of the lever than 

 before. Upon this principle, we may again account 

 for the facility with which a very trifling cross strain 

 is enabled to break a beam which is under very great 

 compression. 



III. At the point B, where the crooked beam is 

 intersected by the line of direction of the load, there 

 is, properly speaking, no cross strain ; it does not 

 follow, however, that there is no strain at all, as Eu- 

 ler has hastily supposed. The truth is, the strain is of 

 the simple lateral kind, and the upper part AB tends 

 to part from the lower, by sliding along the lection 



