Thus we see, that the two forces, A e and A which are equivalent to A E. are equivalent also to A k. A i„ A 0, and A 9, 11111 because A f and e E. are eytal and parallel, and E are also parallel, as also e r anti it is evident that if is equal to r E, or to o F, and i A is equal to r e, or to og. Therefore the four forces A y, A o, A A', A i are equal to A and A F. Therefiire A (1 is the compression of the beam A D, or the force pressing it on I), and A F is the force pressing it I in the beam it D. The proportion of these pressures, there fore, is not affected by the form of the joint.
This remark is important ; for many carpenters think the form and direction of the butting joint of great importance ; and even the theorist, by not prosecut lig the general prin ciple through all its consequences, may be led into an error. The form of the joint is of no importance, in as far as it affects the strains in the direction of the beams ; but it is often of great consequence in respect to its own firmness. and the effect it may have in bruising the piece on which it acts, or being crippled by it.
The same compression of A n, and the same thrust on the point n, by the intervention of A D, will obtain, in whatever way the original pressure on the end A is produced. Thus, supposing that a cord is made fast at A, and pulled in the direction A E, and with the same force the beam A D will be equally compressed, and the prop D must re-act with the same force.
But it happens, that the obliquity of the pressure on A n, instead of compressing, stretches it ; and we desire to know what tension it sustains. Of this we have a familiar example in a common roof.
Figure :2!.2.—Let the two rafters A c, A D, press on the tie beam n c. We may suppose the whole weight to press ver tically on the ridge A. as if a weight, B, were hung on there. We may represent this weight by the portion, A b, of. the vertical or plumb line, intercepted between the rike and the beam. Then drawing b d and I. g parallel to A D and A C, A g and A d will represent the pressures on A c and A D. Produce A c till C H he equal to A d. The point c is forced out in this direction, and with a force represented by this line. As this force is not perpendicularly across the beam, it evidently stretches it ; and this extending force must be withstood by an equal iluce pulling it in the opposite direc tion. This must arise front a similar oblique thrust of the
opposite rafter on the other end, D. We concern ourselves only with this extension at present ; but we see that the cohesion of the beam does nothing but supply the balance to the finces. It must still be supported externally, that it may resist. and, by resisting obliquely, be stretched. The points c and D are supported on the walls, which they press in the directions c K and u o, parallel to A b. If we draw n s parallel to n c, and n t parallel to c K (that is, to A 6), Meeting D c produced in t, it ti,Ihms, front the composition of forces, that the point c would be supported by the two forces K C and t c. In like manner, making n N = A g, and com pleting the parallelogram N 0, the point n would be sup ported by the forces o n and 31 D. If we draw g c and d k parallel to o c, it is plain that they are equal to N o and c I; while A c and A k are equal ton o and c K, and A b is equal to the sum of o o and c K (because it is equal to A c A k). The weight of the roof is equal to its vertical pressure on the walls.
Thus we see, that while a pressure on A, in the direction A 6, produces the strains A d and A g, on the pieces A c and A a, it also excites a strain, c 1 or D it, in the piece D c. And this completes the mechanism of a frame; which derives all its efficacy from the triangles of which it is composed, as will appear more clearly as we proceed.
The consideration of the strains on the two pieces A D and A c. by the action of a force at A, only showed them as the means of propagating the same strains in their own direc tion to the points of support. But, by adding the strains exerted inn c, the frame becomes an interinedinm, by which exertions may be made on other bodies, in certain directions and proportions ; so that this frame may become part of a inure complicated one, and, as it were, an element of its con stitution. The similarity of triangles gives the following analogies : no:Diu=A6:6D ct,ornm:eK=ct:Ab Thereforeao:cK=c6:6D.
Or, the pressures on the points c and D, in the direction of the straining force A b, are reciprocally proportional to the portions of D c intercepted by A b.
Also, AL = Do c s, we have AbicK=c64-6D(orcD):6 D, and