CARPENTRY. 



Theory of Let ui now proceed to the strain which arise* fr-.m 

 n load distributed uniformly over the beam; of which 

 kiml the weight of the beam itself ia one of the 



I. u:lu. r i:i. 



most important. 



\Vo must suppose the whole of the load thus dif- 

 fused, to be united in its centre of gravity, which, in 

 the case of uniform distribution, will be the middle 

 of the piece. 



Thus, to find the strain at GH arising from the 



ht of the beam AB, we may suppose the weight 



of the beam accumulated in the point D, and, the 



strain, which the weight of a part AD produces at 



GH, will be as the weight AD X DA x GB. 



Therefore the strain on the middle of a beam uni- 

 formly loaded, is half the weight of the load acting 

 by the lever DB; and the strain at G is half the 

 load acting by the lever GB, and is therefore greatest 

 of all at the point D, or in the middle of the beam. 



Since the strain in the various sections of the beam 

 also varies considerably, it becomes of importance to 

 determine the various proportions of the beam, so 

 that it may be equally strong in every part. The 

 strength must be everywhere proportioned to the 

 strain ; should it be anywhere greater, this addition- 

 al power will be useless, for it cannot be brought in- 

 to action. 



Suppose the beam fixed in the wall, and strained 

 by a load at the extremity. 



If the upper and lower sides are parallel, i. e. if 

 the depth be given, the horizontal section must be a 

 triangle ; or the two sides are vertical planes, meet- 

 ing in an edge at the extremity. For the strength, 

 as we have shown above, is as the breadth, multiplied 

 into the square of the depth, and divided by the 

 length. That the strength may be everywhere the 

 same, the breadth must therefore be proportional to 

 sig. 9. the length, for the depth is given. See Fig. 9. where 

 BC=CE. 



If the breadth be given, we must have the length 

 proportional to the square of the depth, which is 

 done by making the depths the ordinates of a corn- 

 Tig. 10. mon semiparabola, (Fig. 10.) where BC*==CD*. 



If the beam be pyramidal or conical, having its base 

 in the upright wall, i. e. if its vertical sections be si- 

 milar figures, as circles, squares, &c. ; then since d l l 

 is as d 3 , we must form the diameter or side so as to 

 have its cube proportional to the length, or so as to 

 Pig. 11. k e the ordinate of the cubic parabola, (Fig. 11.) BC 



The same forms are proper for the arms of a lever, 

 observing that the greater of the three pressures is 

 to be applied at the junction of the bases, or thickest 

 places. 



It is worthy of remark, that although the different 

 forms of Figs. 9, 10, 11, are all equally strong, they 

 are not, however, equally stiff; Fig. 9. having the 

 upright end, will bend least on the whole; Fig. 10. 

 will bend most, a property which may have its use in 

 forming springs, &c. 



It is not necessary, theoretically speaking, that the 

 upper or under side of Fig. 10. should be straight. 

 The proper depth being preserved, we may place the 

 beam either way, divide the curvature between the 

 upper and lower sides, or give the beam any proposed 

 camber. In straight-fibred timber, however, we will 



VOL. V. PART JI. 



naturally put the straight face to that *ide which it Tfu 

 stretched by the strain ; as well to prevent their parts 

 from tearing, as to procure greater stiffness. The * ""V 

 same remarks may be applied to the other figure*. 

 They will occur to the intelligent engineer. It i 

 evident that the same remarks and construction will 

 apply to a joist which i* loaded at a particular point, 

 and supported at both ends. It is in the name state 

 as the lever above-mentioned. 



When the weight to be supported is uniformly dis- 

 tributed over the beam, the forms are considerably 

 different from these. 



The strain on any section arises from the weight 

 distributed over the part beyond it. This weight, 

 being distributed uniformly, will be as the length be- 

 yond the section, and it may be supposed accumula- 

 ted in the centre of gravity, which will be in the mid- 

 dle of that length ; so that it acta with a leverage al- 

 so proportional to the length ; and the strain produ- 

 ced is therefore proportional to the square of the 

 length. Now the strength of the section is as the 

 breadth multiplied into the square of the depth. 



If the breadth, therefore, be given, that is, if the 

 beam must have upright parallel sides, we must make 

 the square of the depth proportional to the square of 

 the length beyond the section, that is, the depth as 

 the length ; and the beam becomes a triangular prism, 

 as Fig. 9. only having the extreme edge at B placed 

 not vertically, but horizontally. 



If the depth be given, or if the upper and under 

 sides be parallel planes, then the breadth must be 

 formed proportional to the square of the length be- 

 yond the section ; and the beam becomes the same as 

 Fig. 10. only placed on the flat side, or the horizontal 

 section is formed by arches of the common parabola. 



If the beam must be of a conical or pyramidal 

 shape, or rather if the vertical sections must be every- 

 where similar figures, then the cube of the diameter 

 must be as the square of the distance from the end, 

 and the sides of the beam are arches of the semicubi- 

 cal parabola. 



It is evident that the same observations as formerly p. . 



l l i it* LATZ 



made respecting camber may be again repeated here. CXII. 



It will not now be difficult to apply these deduc Fig. 8. 

 tions to the form of a joist uniformly loaded through 

 its whole length. 



We have already shown that the strain on any point 

 C, Fig. 8, from a load laid on another point G, is 

 proportional to the rectangle GB X AC, being the 

 distances ot the points G and C from the ends tiext 

 each. The strains, therefore, which are produced at 

 two points on the same side, will be to each other as 

 their distances from the end ; and the strain under the 

 load is as the rectangle of the segments on each side, 

 and is therefore greatest in the middle. 



For the strain produced at any point G by a load " 

 uniformly distributed, we may suppose the load on 

 the part GB accumulated in its centre of gravity, 

 which will therefore be distant ^GB from B ; its ef- 

 fort at G will be proportional to ^GB X GA ; and 

 the load in the part AG will, for the same reason, 

 produce a strain as iGAxGB; the sum of these 

 will therefore be likewise proportional to the rect- 

 angle of the segments AG and GB ; and we have 

 the same proportion of strain for the load uniformly 

 3 s 



