M EC

155

ME C

and

o r

tlipf" r ^ lere f° r e a g is the compression of the beam ad,Oq t jj e ° rce pressing it on d, and A f is the force pressing itfor e • t>eam H d. The proportion of these pressures, there-T[j. S ri0 *' affected by the form of the joint,forjjj 18 Ee ®ark is important; for many carpenters think theand ei; direction of the butting joint of great importance ;cipi e ,,® n theorist, by not prosecuting the general prin-The f 0 rou S' 1 a d its consequences, may be led into an error,effects ^ °f the joint is of no importance, in as far as it

but it is

ect s tv» •

ne strains in the direction of the beams :

oft en q j 1

the efF ® r . eat consequence in respect to its own firmness, and°r fiQ,- Ct lt; ma y have in bruising the piece on which it acts,crippled by it.

point jj Sa / ne com pression of A b, and the same thrust on theVi>ay t i ’ X d'e intervention of A d, will obtain, in whateverSup posi or, g*nal pressure on the end A is produced. Thus,<lire ct - t ^ at a c °rd is made fast at A, and pulled in theeqo a il' on and with the same force, the beam ad will beforc e ^ cotn P l 'essed, and the prop d must re-act with the same

4 °dcn happens, that the obliquity of the pressure on

know , eat °f compressing, stretches it ; and we desire toexanini ' at tens > on it sustains. Of this we have a familiar^igu m a coramon roo f-

^ ea »t —f jet the two rafters, A c, A d, press on the tie-

ticalj q C ’ ma y suppose the whole weight to press ver-j^ 011 "re ridge a, as if a weight, b, were hung on there.'■'ertj c ,J 1 ^ re present this weight by the portion, A b, of the

^um.

t9 a ^d . r**'? 4 VAiavviii o ^ 14 uiiu 17 y F aittAACA tuuuu A^ r °duc , w dl represent the pressures on a c and ad,cut ^ 6 A c dll c h be equal to A d. The point C is forcedI’Ue, . 118 direction, and with a force represented by this11 evij^dus force is not perpendicularly across the beam,"■ithatoojj h Stretches d > an d this extending force must beduti. Th' ^ an e< l ua ^ f° rce pulling it in the opposite direc-18 must arise from a similar oblique thrust of the

or plumb line, intercepted between the ridge and1'hen drawing b d and b g parallel to A d and a c,

“HOSit^ n *” ai *oO llUIll ct S1LLU1

0tl iy ^ter on the other end, d. We concern ourselves

fusion e , extens i°n at present; but we see that thethe o 1 °* * ’

'® extend ' Jeam does nothing but supply the balance toJjut j t In g forces. It must still be supported externally,b e Point^ res ' st ’ an d> dy resisting obliquely, be stretched,d Ss in tti C an .^ D are supported on the walls, which theyu j. e directions c k and d o, parallel to A b. If wef 6et 'ng J> P ara ilel to d c, and h i parallel to C k (that is, to a b),that* 3 P r °duced in i, it follows, from the composition ofbi C a,1 d x c A Point C would be supported by the two forcesh the* manner, making D n = A g, and com-

h° rfe d by ,P ai ’ a delogram dmno, the point d would be sup-^/ullel to r, 16 ^ orces 0 D an d m d. If we draw g c and dkto a 4 c and*’ is plain that they are equal to N o and c i;

he n d A k are equal to do and c k, and A b is equalV ® ? 0 an d C k (because it is equal to a c + a k).

’Ik ° ° *^ le r °°^ * S e< f ua ^ * 0 ats vertical pressure on the

4 Ud Produo S6e ’ t * lat while a pressure on A, in the directionAha it t ' le 8tra ‘ ns A d and A g, on the pieces ac

Oil tlli s eV' 80 ex cites a strain, c i or d m, in the piece d c.\vilj ta efk C;ic P'etes the mechanism of a frame ; which derives

'r, a Ppear ^ 10m the triangles of which it is composed, as4ho 1 naor» _

c °nsid

4c > by tv, eratl0n t ' le atnains on the two pieces adti Qn ^ ana e action of a force at A, only showed them as

s | a °re clearly as we proceed.

’on the two

e ni

; lOh,

the P(°P a gating the same strains in their own direc-in j^Pmnts of support. But, by adding the strainss ’^ e frame becomes an intermedium, by whichy be made on other bodies, in certain directions

and proportions ; so that this frame may become part of amore complicated one, and, as it were, an element of its con-stitution. The similarity of triangles gives the followinganalogies :

do:dm = aJ:1dC i, or d m : C k = c b : a bTherefore DO:CK = ci:Jo.

Or, the pressures on the points C and d, in the directionof the straining force A b, are reciprocally proportional to theportions of d c intercepted by a b.

Also, a b = d o -f- c k, we have

a5 : ck = cJ + 6d (or c d) : b D, and

aJ:do = cd:Sc.

In general, any two of the three parallel forces, A b, D O, C K,are to each other in the reciprocal proportion of the partsof c d, intercepted between their directions and the directionof the third.

And this explains a still more important office of theframe adc. If one of the points, such as d, be supported,an external power acting at a, in the direction a b, and withan intensity which may be measured by a b, may be set inequilibrio with another acting at c, in the direction C L,opposite to C K, or A b, and with an intensity representedby C K : for since the pressure C H is partly withstood by theforce i C, or the firmness of the beam d c supported at e, theforce k c will complete the balance. When we do not attendto the support at D, we conceive the force a b to be balancedby k c, or k c to be balanced by a b. And, in like manner,we may neglect the support or force acting at a, and con-sider the force d o as balanced by C k.

Thus the frame becomes a lever, and we are able totrace the interior mechanical procedure which gives it itsefficacy.

The strains or pressures aJ, do, and c k, not being inthe directions of the beams, may be called transverse. Wesee that by their means a frame of carpentry may be con-sidered as a solid body: but the example is too limited forexplaining the efficacy which may be given to such construc-tions. We shall therefore give a general proposition, whichwill more distinctly explain the procedure of nature, andenable us to trace the strains as they are prapagated throughall the parts of the most complicated framing, finally producingthe exertion of its most distant points.

Figure 23.—Let the strong lines abcd represent a frameof carpentry. Suppose it is to be pulled at the point a by aforce acting in the direction A e, but that it rests on a fixedpoint, C, and that the other extreme point, b, is held back bya power which resists in the direction bf: it is required todetermine the proportion of the strains excited in its differentparts, the proportion of the external pressures at A and B,and the pressure which is produced on the obstacle orfulcrum c.

It is evident that each of the external forces at A and btend one way, or to one side of the frame ; that each wouldcause it to turn round c, if the other did not prevent it; andthat, notwithstanding their action, if it turned neither way,the forces in actual exertion would be in equilibrio by theintervention of the frame. It is no less evident that theseforces concur in pressing the frame on the prop C. There-fore, if the piece CD were away, and if the joints C and d beperfectly flexible, the pieces c a, c b, would be turned roundthe prop c, and the pieces ad, d b, would also turn withthem, and the whole frame change its form. The powerat A presses its end against the prop j and in doing this, itputs the bar A d on the stretch, and also the bar D b. Theirplaces might therefore be supplied by cords or metal wires.Hence it is evident that d c is compressed, as is also a c • and,

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