OF TII• COMPOUND SPRAINS.
The variety of compound strains is inexhaustible; We can only be expected here to treat of the simplest and most important.
IV. Of the Resistance to Cross Strains.
The usual form of this strain is when the points of bearing arc 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 bene fit derived from these experiments, however, has not been so great as might have been expected; and we arc yet very far from being able, with precision, to deter mine the strength of timber against a cross strain.
'File first who attempted to give any theory for the transverse strength of timber was the celebrated Gali leo. lie supposes the prismatic body, ABCD, (Plate CXII. Fig 2.) is fixed at one end into a wall. from which it projects horizontally., and is acted on by the weight \V, hanging from the farther end. Let the body be sup posed to break across in any line EV. In time instant of fracture, the surface EF is supposed to exert every where an equal cohesion. As the section across is, therefore, every where the same, it is evident that the energy of the weight \V 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 I) as round a centre; and there fore the pants at All nmst separate from each other in the hot izontal direction. If me suppose all the particles in the section 'AD to .be exerting an equal force, and that the disrup tiOn takes place at once, then the total force may be sup posed accumulated in the centre of magnitude or gra vity G ; and the energy of the weight \V, at the instant or fracture•, will be to the absolute strength of the beam in resisting a direct pull, as the distance of the centre (; from I), 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 Al); wherefore we should have fin' the strength of the beam as the of the bearer to half its depth, so the absolute cohesion in length to the transverse -trengt.i.
It will be improper for us to pursue this %kw ul the subject any frther. It has led to very erroneous con clusions: and, front being ill that is to bt found in our commor matises, has beconr , in them, the foundation of very false maxims of prt,ctice.
Succeeding writers, suzil as Mariotte, Leibnitz, have perceiv ed that this supposition of equal cohesion, exerted by all the particles in 'lie instant of fracture, was not conformable to the proceeding of nature. We know that there is no body, however hard, but is sum, what extended before breaking. Wheal a force is applied across the beam at C. the beam is bent downward, be coming convex on the upper side. That side is there fore on the stretch, and the particles at AH being farther removed from each other, are exerting greater cohesive forces. We know that these greater forces are, while the body is not crippled, proportional to the extensions. Suppose, then, the beam to be so much beaded, that the particles at AH are just giving way ; it is plain that a total fracture must immediately ensue, for the force which exceeds the whole cohesion of the particle at A, and a certain portion of the cohesion of the rest, will still more exceed the 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 fulcrum D, in order to find the amount of the whole, take DM=x, DA=a, and the greatest force of the exterior fibre AH=f; then the force of any fibre MN, being