504 



CARPENTRY. 



PLATE 

 CXI I. 



Tig. 6. 



Theory of less than fir as a pillar, but more as a tie,) then we 

 Carpentry. mav suppose the greater breadth of the side of the 

 V -""Y"""'' bar to become useful in resisting a strain ; but this is 

 a question of fact and not of theory, and, according to 

 circumstances, the bar may be beet situated either 

 way. In the mean time, it is a question of high im- 

 portance to the engineer ; for until it be ascertained, 

 it is evident that the form of flanges, scantlings of 

 trusses,, &c. must be left entirely to chance, and in 

 making them he may, instead of strengthening his 

 work, be only adding a cumbrous load. 



We may collect from the above enquiry that our 

 triangular section will begin to give way at the point, 

 when the distance of the neutral line from the base is 

 of its distance from the vertex, and therefore -^ of 

 the altitude of the triangle : At that time the verti- 

 cal part being on full stretch, we have its energy act- 

 ing at ^ of -f of the altitude of the triangle from the 

 neutral line MN, and the part CMN will be to the 

 whole bar as the squares of the altitudes CD and CR, 

 viz. 9' : 14>*=81 : 196, that is, the energy of the verti- 

 cal part of the triangle is V\rff ^ tne absolute cohe- 

 sion, acting at the distance of ^ of the depth or al- 

 titude; the other, or trapezoidal part, must of course 

 be exerting the same energy, so that in triangular 

 bars we may suppose the absolute cohesion acting at 

 the distance of .088, or about -,** of the depth of the 

 triangular bar from the neutral line : or the absolute 

 strength is to the resistance to cross strain, as the 

 length of this beam to about -fjr of its depth, so that 

 it is little more than half the strength of a rectangle 

 of equal depth, and the same contents. 



We may learn from this case of how much impor- 

 tance it is that the compression should be attended 

 to. By neglecting to do so, some writers of reputa- 

 tion have fallen into serious mistakes. There is ano- 

 ther case, similar to this, given by Emerson and others : 

 it is said that a hollow tube becomes stronger by 

 bringing the interior opening nearer to the side, which 

 becomes concave by the strain ; but like the above, 

 this depends entirely on the relation between the 

 compressibility and dilatibility of the materials 

 of the tube. In like manner we reject the proposi- 

 tions said to point out the strongest position of a 

 beam having a trapezodial section, and others of the 

 same kind usually given by authors; they are merely 

 deceptions of hypotheses. The truth is, that in most 

 kinds of timber the position of the beam is just the 

 reverse of that usually recommended. In most of 

 them we shall find the compressibility to be much 

 greater than the dilatation. The greater dimension 

 must, therefore, be thrown to that side which is like- 

 ly to be compressed, and a beam broader on one side 

 than the other will be strongest with the broad 

 side uppermost, when it is supported at the two 

 ends. 



In stone, however, the reverse is most likely to 

 be the case ; but is needless now to go farther, the 

 reader must perceive that the whole is matter of ex- 

 periment. 



We may see the propriety of several maxims, which 

 have been long* familiar to the experienced workman. 

 .If a mortice is to be cut out of a beam, the worst 

 place is to take it out of that side which becomes 

 convex by the strain, as (Fig. 7. No. 1.) The concave 



Useful 

 maxims 



PLATE 



cxn. 



'Kg. 7. 

 No. J. 



side (as No. 2.) will be much better, since its place 

 will be, in a great measure, supplied by the tenon ; 

 but towards the middle (No. 3.) will be the best of 

 all, since the parts near the neutral line are suffering 

 little or no strain, and even there it should be placed 

 towards the concave side rather than the convex. 



When afah, or strap, is to be applied to a piece of 

 timber, it must be applied to the convex side, as Nos. 

 4 and 5. 



Though we have supposed, hitherto, that the beam 

 is fixed in a wall, and strained by a load at the outer 

 end, it is not difficult to extend the results we have 

 got to the more usual case of a beam supported at 

 both ends, and loaded in the middle. 



Let AB, Fig. 8, be such a beam, resting on the 

 props E, F, and bearing the weight W on its middle 

 C. It is clear that the props E, F, between them, 

 bear the whole weight of W ; and that the resistance 

 of E, F, produces a strain exactly the same as if two 

 weights c,f, together equal to W, were hung at the 

 ends of the beam a b, balanced at the prop c ; only 

 the forces will have the opposite direction. In this 

 case, we know, that the weights e and / will be to 

 each other inversely, as their distances from the prop 

 c. That is, e if ; : b d : da ; the ends a and b tend 

 to descend, and the beam to part in the section DC. 

 The situation of each half is therefore the same as 

 if the other were built into a wall as far as the part 

 CD ; thence we conclude that a beam, which is sup- 

 ported at both ends, and loaded in the middle, will 

 carry twice as much as a beam which projects half 

 its length from a wall, and loaded at the extremity. 



It is evident that the cross strain produced in CD 

 is the greatest ; and that it diminishes as we pass to- 

 wards the end of the beam. For the same force f 

 acts in gh by a shorter lever bg. If we suppose the 

 weight W shifted off to G, then the pressure on A 

 is as GB, and the lever by which it acts at GH i% 

 AG ; the effort is therefore as AG X GB, or as the 

 product of the two segments of the beam. 



Again, the strain at CD will be as GB X AD, 

 which will also be the strain at GH, when the same 

 weight is applied at CD. We are now therefore 

 sufficiently informed as to the strains produced on 

 one part of the timber, by loads laid on at any other 

 part. 



Suppose, next, the ends of the beam prolonged be- 

 yond the props to the points M and N, and to be 

 there firmly held down, the beam will be able to bear 

 twice the load W that it carried before. For sup- 

 pose the part CD to be sawn through, the weight W 

 is just sufficient to break it at A and B ; and another 

 weight of equal magnitude would have been required 

 to overcome the cohesion at CD, independent of the 

 exterior support. 



It may appear from this, that the joists of floors 

 would be stronger by being built firmly into the walls. 

 But the hold that is thus obtained is much too short 

 to be of great service; and besides, it acts as a power- 

 ful lever, in tending to shatter the wall. But when 

 joists can be carried across partitions, purlins extend?- 

 ed to three or more truss frames, or rafters extended 

 over the purlins, &c. it is of great importance to make 

 them in one piece, as they are then of double the 

 strength. 



3 



Theory of 

 ar P entrv 



cxil. 



Fig. 7. No. 

 2, 3, 4, 5. 



Beams sup- 

 ported at 

 kth ends 

 a ' oa< * e ^ 



Fig. 8. 



