163

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tion i n °t tl t le le S s °f the said angle, and a force in any direc-tl, e f r he plane of the figure be applied to the angle wherefram e vD'n* 8 unsu PP orte d, and where the bar is not fixed, theforce }> h e by this means rendered immoveable, and theStr a i n ^ '■his disposition will not occasion any transversePentad { he sides of the frame. Suppose the frame to betile e^j na ’ an( t a bar fixed in like manner to two angles, atsides, these two adjoining sides andthe fram' V 1 ^ orm a triangular compartment in the figure ; ifthe thr " e sus P en ded by two of the angles of the triangle,°f th e ba retna ' n ’ n S sides will be moveable at the extremitieshe fi Xe( j , r> an h at the remaining angles ; but if another barend, to ° an y one °f the three angles of the triangle at oneMother + ne 'he angles of the other three sides, to formtvili he '' lan S^ e ! three of the sides of the pentagonal framek.® so lihg w ® immoveable, and the two remaining sides willsides the r Wlse ' I n bke manner, in whatever number ofthe side s rarae eon sists, by first forming a triangle of two oftriangi e t &nd dx ' n g a second bar from any angle of thefinds of t one of the other angles of the figure, at the remote'•Ornoveahi 0 ad j°‘ n ' n o sides of this frame, will form anothern ext ario-i 6 Slde ’ and S' ve another immoveable point at thel he oth e ° 6 , t^ e frame ; if, from this fixed point, or any ofe n< j „ ree points, which are the angles of the triangle,'° oo e 0 p r a third bar be fixed, and the other end of the bartr i;uig| e . ® remaining angles of the frame, so as to form asides p t ' lu secon d bar and one of the adjoining move-files , °t the frame, or a triangle with one of the fixedP r °eeedi no . e . ame and the adjoining moveable side ; and byh^ed, the°f m ^ ds manner successively, until all the sides areatl gles 0 f rame will be made immoveable ; so that if any twofihPiied at US ^ rame he supported, and a force or forces beSure, t ] °[ le > or each of the angles in the plane of the. frames f°^ e ®S ur e will be immoveable.

8ltl gle force & tr >angular form, which have to resist only a®° n strn cte( j’ V" su PPort one weight, are most simply and bestvv ° an»i es 0 three sides ; the frame being suspended fromframe an< ^ t ^ e Porce ° r weight from the other. A trian-an y subdivi’ . Su PP° r ting only one weight, has no occasion for*, 0D ?Pressed S t 1 - 0ns to compart the internal space, provided the• *®'r own Ua her or timbers were inflexible, so as to support

Pable* p^’S^t without bending, and the tensile timbersse ext ension. Though a frame should have tov eral Weights, the external figure may be of any

Ort

l acai ts «pp‘

provided ««. th.is a f s are bung, and the two points f ro ‘“ t ing the

sinn r WUh timber divisions, and tnereoy contiguous

C? joining triangles, of which ^h^w 0 ^^

of t L c 5: rar r>on side ; that is, when two of the an t0

°bservo'^ aCent triangles are coincident. It “ ay . P e y P ible,to t y. - .' ere > that though it may not be a a w yi be

trii de a fra ^®. bo that all the compartment^w^ ^^c °atim eS a yet the succession must not by al y polygonalCes r by the intervention of quadrilateral t0the trus ° r t '' ese eempartments may adjoin w rect iUnear

digob 1 W a so easier constructed, and P other

a 8ure l arge of rain or moi^nre in a roof, than any ^

Wfeigii’ >Jut - in its adoption to large, bulld ' u ° ’ one po int

whi?h « - be su PP° rted ’ and aS l i t becomes neces-sary tn ♦ 1 tbls weight can be suspended, order to

>p on t; ^ other equidistant points in the sides,

8 tafi th ® covering equally : these points may be maae' nt ° a ^.tbe former means of dividing the_> n tw0

''Pper , . aee3s ’ , °n of triangular compartments. u ealia lly

1 8ld ®® of the frame be of equal lengths, and equally

inclined to the horizon, the opposite points may be made tocounteract each other without a concatenation of triangles, byintroducing timbers from point to point, parallel to the hori-zon ; in this the compartments will all be trapezoids, exceptthe upper one, which will be a triangle. These beams maybe supported by vertical bolts passing transversely throughthem from the points where the weights are supported, andthe bolts may be nutted below the beams. This mode ofsecuring the points of support depends entirely upon thedoctrine of equilibrium, and thus a very little difference fromthe equality of forces might easily occasion a change offigure, to which the other method by a series of triangles isnot liable. The securing of the points of support by beamsis not confined to triangular frames, but may be applied toroofs having two or several rafters upon each side, so thattheir lengths and inclinations are equal, and their junctionson the same level. The beauty of every truss is to disposethe timbers in positions as direct to each other as possible :oblique directions require timbers of large scantlings, and exertprodigious thrusts on the abutments, so as to compress thejoggle-pieces, and render the truss in danger of sagging.Trusses are variously constructed, according to the width ofthe building, and contour of the roof, and the circumstancesof walling below.

The general principle of construction is a series oftriangles, of which every two are connected by a commonside.

Let abcdepg (see Plate I.,Centering, Figure 1,) be thecurve of the arch which requires a centre. Let the pointsABC, &c., be connected, so as to form the equilateral polygonabode fg, and join AC, CE, and eg; the timbers thusdisposed will form three triangles, which may be looked uponas so many solids revolvable about the angular points, A, c,e, g ; suppose now these to be in equilibrium, the smallestforce on either side would throw it down ; and therefore,without other connecting timbers, it would be unfit for thepurpose of a centre.

Let a b c de f g (Plate I., Centering, Figure 2,) be thecurve of an arch which requires a centre. First, from theequilateral polygon, abcdifg, with the timbers, ab,cd,GD, &c., and fix the timbers AC, C e, e g, as before, whichwill form three triangles, moveable round aceg; let thetimbers b d and d f be fastened, and thus the whole will beimmutable, so that, if supported at the points A and G, and aforce be applied at any other of the angles, b, c, d, or f, thetimbers will be all in a state of tension, or in a state of com-pression, and the whole may be looked upon as a solid body.For since the sides and angular points of a triangle are fixed,when the triangle is supported at two of the angles, and aforce applied to the other, let us suppose the triangle, abc,to be supported at the points A and b, and the point c, andthe other two sides, b C, c a, will be fixed ; and because BCDis a triangle, and the points b and c are fixed, the point d,and consequently the sides C D and D B, are fixed. In likemanner, since CDE is a triangle, and the points C and Dfixed, the point e will also be fixed, and also the sides d eand e c. The same may be shown in like manner for thepoints f and g. Suppose, then, two equal and oppositeforces applied at the points a and G, in the plane of the figure,the figure can neither be extended out nor compressedtogether. The pieces ah, h b, and G i, if, are of no otheruse than to make the centre stand firmly on its base. Thisdisposition of the timbers will cause them to occupy the leastpossible space.

If the timbers are fixed at the points h, l, m, «, o, p, thesame immutability of figure may be demonstrated ; for sup-pose the points a and h to be fixed, the point k will also

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