# Elements or Carpentry

## piece, ac, pieces, ad, fig, angle, strain, force, sine and direction

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The same consecreeees result • from an improper -change of Abe position of .AC. If it is placed as in fig.•. the stains on both are vastly increased. In -short, the ride: is., general ; that the more open we make the angle against which the push is. exerted, . the greater are th•straiae which are brought on the struts or ties which form Ihe sides of the angle.

The reader . may not readily conceive the piece • AC of fig. 8. as sustaining a compression ; for the weight . B appears to hang from AC as much as from AD. But his 'doubts will be removed by con sidering. whether he, could employ a rope in place .of He cannot:, but AD may be exchanged for a rope. AC is therefore a strut, and not a tie.

In fig. g. , Plate . XLIX., AD is again a strut, . on the block•D, and AC is. a tie : and the • batten AC may be; repleemiby .a rope. While AD is oompresaed by the .force.,A,G, . AC is • stretched ,.by the force AF.

If. we • give AG the position represented by. the dotted lines, the compression of AD is now AG', ..and. the force stretching ACi is now AF' • •both much . rester • than they were before. • Thi; . disposition Is analogous to fig. 0. •and. to ,the dotted lines in fig. 8. Nor will the young •artist•have any doubts of • AC' being on the stretch, if be consider whether AD • can be replaced by a rope. It, cannot, -.but • AC' may ; and it is • therefore ..net compressed, .but :stretched.

In fig. 10. all the three pieces, AC, ADeand.A.B• are ties, on the stretch. This, is .the complete version of fig. 8. 4• and the dotted position of AC duces.the same changes in the force, AF, AG•, as. to 8. Thus hive we gone .over • all. the varieties which can .happen in the bearings of three pieces on one point. All calculations about the strength of pentry . are reduced to this case for when more Ales or braces meet • in a point (a thing that rarely happens), we reduce them to three, by substituting for any two•be force which results front their bination, and then combiniug,this with another ; . and so on.

The young artist meat be, particularlyeareful not -to mistake the kind of stain that is exerted on any of the and ,suppose aepiette .1c be a ce which is really *Ale. -It is tasy to avoid all mistakes :in this matter by .tire. following rule, .404oh. has no exception, (See Note AA.) Take notice of the direction in which. the. piece - acts-from which • the strain proceeds. Draw a line in that direction from the ,point on which the strain is exerted.;•and let its length (measured on .some scale ofequal parts) the magnitude of . this action in pounds, hundreds, or tons. From its re mote extremity 'draw lines • parallel, to the pieces on .which the strain is exerted. The line parallel to one piece will necessarily cut the other, or- its direction produced: if it cut the piece itself, that piece is .compressed by the strain,, and it is performing the office of a . strut or .brace.; if it cut its direction • produced, the • piece• is stretched, and it is a tie. Jo short, the tanks on the pieces AC, AD, are to be estimated ia.the direction of the points F and G

from! the stained point A. Thus, in fig.. 8. the upright piece BA, loaded with the weight B, presses the point A in the direction AE; so does the rope.AB in. thti other figures, or ..tbe batten AB in fig. 5.

• In general, if the piece is within the angle formed by the pieces which are strained, the • strains Which they sustain are- of the opposite kind to that which it exerts. If it be pushing, they are , drawing i but if it be within the angle fdrmed by their • directions produced, the strains which they sustain the same kind. All the three are either draw ing or pressing. If.the straining piece lie within the angle formed by one piece and the produced direc • tion of the other, iteown•strein, whether compression -.or extension, is of the same. kind with that of the . most remote of the other two, and opposite to.that of •the• nearest. NbUesi in, 61.•9. where AB, is draw • !PM the roma piece. AC Kelso drawing, while AD -is pushieg or resisting compression.

In all that . has -been said • on this subject, we have not spoken. of any joints. In the calculations with which we are occupied at present, the resist • ance of joints•,haa nnebarei and 'we must not .sup _ pose that they exerteny -force .which. tends to pre . vent the. angles from changing. ,The joints are sup posed perfectly tienibie, or to be like compass • joints ; the pin of .which only keeps the pieces to ) gether when one of . the pieces .draws or pulls. The carpenter must always suppose them all compass joints when . be calculates the . thrusts and draughts. of the different.pieces of his frames. The strains Au: joints, and their power to prodlice or ba lance them, are . a -digerent..kind, and require a very different examination.

Seeing , that the angles which the pieces, mate with each, other sue: of. such. importance to the Inag nitude and the proportion of the excited strains, it is proper to find out some way of readily and compen diously conceiving and expressing this analogy.

In general, the strain on any piece is propor tional to the straining force. This is evident.

Secondly, the strain on any piece AC is propor tional to the sine of the angle which the straining force makes with the other piece directly, and to the sine of the angle which the pieces make with each other inversely.

For it is plain, that the three pressures AE, AF, and AG, which are exerted at the point A, are in the proportion of the lines AE, AF, and FE (be cause FE is equal to AG). But because the sides of a triangle are proportional to the sines of the op posite angles, the strains are proportional to the sines of the angles AFE, AEF, and FAE. But the sine of AFE is the same with the sine of the angle CAD, which the two pieces AC and AD make with each other ; and the sine of AEF is the same with the sine of EAD, which the straining piece BA makes with the piece AC. Therefore we have this analogy, Sin. CAD : Sin. EAD =

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