Theory of 

 rarpeiitry. 



The cross 

 btrain in- 

 vestigated 



CXII. 

 Fig. 2. 



And Ma- 

 riotte. 



500 CARP 



IV. Of the Resistance to Cross Strains. 



The usual form of this strain is when the points of 

 bearing are at some distance from each other, and the 

 load is'somewhere applied between them. This case 

 has attracted the attention of many writers, and has 

 been made the subject of numerous experiments. The 

 benefit derived from these experiments, however, has 

 not been so great as might have been expected ; and 

 we are yet very far from being able, with precision, 

 to determine the strength of timber against a cross 

 strain. 



The first who attempted to give any theory for the 

 transverse strength of timber was the celebrated Ga- 

 lileo. He supposes the prismatic body, ABCD, is 

 fixed at one end into a wall, from which it projects 

 horizontally, and is acted on by the weight W, hang- 

 ing from the farther end. Let the body be supposed 

 to break across in any line EF. In the instant of 

 fracture, the surface EF is supposed to exert every- 

 where an equal cohesion. As the section across is, 

 therefore, everywhere the same, it is evident that the 

 energy of the weight W will increase with the length 

 of the lever CF, and of course the greatest strain will 

 be just at the wall, and in the line DA. The action 

 at the point C tends to make the body ABCD turn 

 round the point D as round a centre ; and therefore 

 the parts at AH must separate from each other in 

 the horizontal direction. If we suppose all the par- 

 ticles in the section AD to be exerting an equal force, 

 and that the disruption takes place at once, then the 

 total force may be supposed accumulated in the cen- 

 tre of magnitude or gravity G ; and the energy of 

 the weight W, at the instant of fracture, will be to 

 the absolute.strength of the beam in resisting a direct 

 pull, as the distance of the centre G from D, to the 

 length DC. 



Now if the beam be rectangular, the centre of mag- 

 nitude G will be in the middle of its height or depth 

 AD ; wherefore we should have for the strength of 

 the beam as the length of the bearer to half its 

 depth, so the absolute cohesion in length to the 

 transverse strength. 



It will be improper for us to pursue this view of 

 the subject any farther. It has led to very erroneous 

 conclusions : and, from being all that is to be found 

 in our common treatises, has become, in them, the 

 foundation of very false maxims of practice. 



Succeeding writers, such as Mariotte, Leibnitz, 

 &c. have perceived that this supposition of equal co- 

 hesion, exerted by all the particles in the instant of 

 fracture, was not conformable to the proceeding of 

 nature. We know that there is no body, however 

 hard, but is somewhat extended before breaking. 

 When a force is applied across the beam at C, the 

 beam is bent downwards, becoming convex on the 

 upper side. That side is therefore on the stretch, 

 and the particles at AH being farther removed from 

 each other, are exerting greater cohesive forces. We 

 know, that tliese greater forces are, while the body 

 is. not crippled, proportional to the extensions. 

 Suppose, then, the beam to be so much bended, 

 that the particles at AH are just giving way; it 

 is plain that a total fracture must immediately en- 

 sue, for the force which exceeds the whole cohe.. 



ENTRY, 



sion of the particle at A, and a certain portion of Theory of 

 the cohesion of the rest, will still more exceed the Carp e11 

 cohesion of the particle next within A, and a smaller 

 portion of the cohesion of the rest. Now, since the 

 force of any fibre is as its distance from the ful- 

 crum D, in order to find the amount of the whole, 

 take DMzzx, DA a, and the greatest force of the 

 exterior fibre AH=/J then the force of any fibre 



MN, being (DA : DM : : a : * : :/:) X , and from 

 the usual fluxionary notation, the force of a fibre x 



will be' , this resists the strain by the leverage 



a ' . 



fx 1 x 

 MD=#. Its energy therefore is- , and that of 



2 f> 



the whole section, or f is X T S > which when 



** a a, 



x=zaiayfa*. Nowy is the force of absolute co- 

 hesion, and therefore the force resisting the trans- 

 verse strain is the same as if the whole absolute co- 

 hesion were accumulated in a point G at one-third 

 of AD from D. 



This hypothesis assigns a smaller relative strength 

 to the beam than the hypothesis of Galileo. In that, 

 the section is supposed to have an energy equal to 

 half the absolute strength of the rectangular beam. 

 In this it is found to be only one-third. 



But ere we proceed to draw any conclusions from 

 what we have now stated, it will be proper to ob- 

 serve, that even this supposition does not fully ex- 

 plain the mechanism of the transverse strain. The 

 force bending the body ABCD, not only stretches 

 the fibres on the convex side of the beam, but com- 

 presses the lower or concave side. There must be 

 some material support as a fulcrum for the lever when 

 it stretches the fibres or tears them asunder, and this 

 can be found in no other way than from those par- 

 ticles which are thus compressed. Let CDG be 

 considered as a bent lever, having the fulcrum DE, 

 by means of which the weight W, and cohesion of 

 the surface AD, are in equilibrio. We know that 

 . the pressure on D will be the same as if the other 

 pressures were both applied to that point, each in its 

 proper direction. The pressure at G is to that at 

 C, as DC to DG. Completing, therefore, the pa- 

 rallelogram GC, and drawing the diagonal KD, it 

 will express the direction and magnitude of the 

 equivalent pressure on the fulcrum D. This pres- 

 sure is opposed, by the resistance to compression of 

 the particles towards D, which may be expressed by 

 DC ; and, again, by the cohesion of the particles in 

 the line of section DA, which of course is expressed 

 by DG. The particle D is not only pushed -back- 

 ward, but drawn downward. This last observation 

 is easily verified. Take a stick of soft wax and bend 

 it gently, the lower or concave side will be observed 

 to swell out, and the convex side to get flat before it 

 cracks and gives way. Take a parcel of the leaves 

 of this book, and holding them firmly together at 

 two parts of the edge so as to give them adhesion, 

 attend to the effects of bending them, the leaves in 

 the hollow side will bulge out and separate, those on 

 the opposite side become flat and clap closely toge- 

 ther. Attend now to the cracks and fissures, that is, 



Their theo- 

 ries insuffi- 

 cient. 



Effects of 

 compressi- 

 bility. 



PLATE 

 CXII. 



Fig. 2. 



