506 C A R P E 



Theory of distributed, as for a load laid on directly at the point 

 ?arpeatry. Q . w l, ert .f ore , to make a foist equally strong in all 

 "" Y "" its parts as;ain.<it a load laid on at an,, point taken at 

 random, or asainst a lead uniformly distributed orer 

 it, we mu4 have ih? strength at each point proportion- 

 al to the rectangle under the. two segments of the beam. 

 If therefore the breadth be given, the square of 

 the depth must be everywhere proportional to the 

 rectangle of the segments, that is, the depths must be 

 the ordinates of an ellipse. 



If the depth be given, the breadth must be as 

 AGxGB. 



If the beam be pyramidal, or have its cross sec- 

 tions everywhere similar, the cube of the diameter 

 must be as the rectangle of the segments AG X GB. 

 The joist which is loaded at some particular point 

 will be of equal strength throughout, when the 

 strength of each transverse section between the load 

 and either end is as the distance from that end. It is 

 needless here to repeat the forms, being merely ex- 

 tensions of the Figures 9, 10, 1 1 , to the opposite side, 

 at the point of support, which here becomes the 

 point loaded. 



The strain upon a beam may be merely that ari- 

 sing from its own weight. The determination of the 

 proper form for such a case is not quite so simple as 

 the others, for the very form which is sought must 

 be known ere we can discover the weight to be re- 

 sisted. 



To consider this subject in a familiar way, it is 

 evident that the increase of breadth merely gives no 

 additional strength, since the load to be resisted in- 

 creases along with it. The increase of length dimi- 

 nishes the strength, first, by increasing the load ; se- 

 condly, by increasing also the leverage with which it 

 acts. 



The increase of the depth, though it brings a pro- 

 portionally greater load, yet, since the strength also 

 increases as the square of the depth, the resistance to 

 fracture increases with the depth ; and we may on the 

 whole infer, that the power of a beam to carry its 

 own weight, is directly as its depth, and inversely as the 

 square of the length. We must therefore make these 

 two dimensions proportional ; that is, the depth CD, 

 Fig. 10, the breadth being given, must be propor- 

 tional to the square of the distance BC, or the side 

 BDF must be the curve of a common parabola. 



The reader acquainted with the art of analysis may 

 wish to see this problem resolved in another way. 

 Let the distance BC=x, and CD=y, CDdc an in- 

 definite small increment of the magnitude, it is equal 

 to y x. Its momentum round the fulcrum B is xyx. 

 Suppose now that y is as x m , or as some power of #, 



PLATE 

 CXII. 



Fig. 10- 



then the contents, or weight of BCD is 



also 



=:x m t I x ) of which the fluent or whole momen- 



turn is . The distance of the centre of gravity 



TO + 2 J 



of BCD from the point B, will be found by dividing 

 this by the weight, viz. - , the result is , 



N T R Y. 



ill- r i_ r (x 01 ^ 1 ) Theory tf 

 and the distance ot the same point from c is .r - Caroentrv 



'III -ir- 2 



= . Multiply this by the weight, or , and 



ra + 2 m -j- 1 



we have the strain, or CD= . ; 7^ - This 



strain must be as the square of the depth, tin.- breadth 

 being givi-n, or as ^ z , or as x* m , therefore' m-^-22m t 

 and 772=2, or the depth is a> the square of the distance 

 from the extremity, as before. 



It is evident, that a projecting beam becomes less 

 able to bear its weight as its length is increased. AH 

 enlargement of the depth gives us strength only pro- 

 portioned to that enlargement, while a proportional 

 enlargement of the length gives a strain which is as 

 the square. By enlarging any structure, therefore, 

 we weaken it. By diminishing it proportionally, we 

 strengthen it. This increase of relative Strength al- 

 lows a reduction of weight and stuff, by which means 

 we have a diminution of expence, and an increase of 

 mobility. Hence, a structure may appear very strong 

 in the model, which will not hang together in the 

 great ; and there seems to be a limit set by nature to 

 the increase of structures composed of given mate- 

 rials. The cohesion of the tree is greater than that 

 of the shrub, which again is greater than that of the 

 herb. The sapling gets firmer as it increases to the 

 oak, which, were it to grow 50 times bigger than it 

 is, could not stand, though, with the lightness of fir, 

 it should have the toughness of iron. Were a man 

 twice as long as he is, he would break his bones by 

 falling along. 



We have hitherto considered the strain as acting Obliquely, 

 perpendicularly to the beam : This is the simplest 

 case, and one of most frequent occurrence ; but it 

 often happens that the load acts obliquely to the 

 beam. The slanting rafter, for example, is oblique- 

 ly strained by the weight of the roof, since gravity 

 acts in vertical lines. 



It is not difficult to compute the effect of this mo- 

 dification, as far as to find the equivalent perpendi- 

 cular pressure. Let the load W, Fig. 2. act in PLATE 

 the line CX, and be represented by it. p^'i 



Then, according to the usual theory of the reduc- '^' 

 tion of pressures, its effect will be equivalent to two 

 others, viz. CT directly across the beam, and CV 

 in the direction of the beam ; CX is to CT as ra- 

 dius to the cosine of TC, wherefore the cross strain 

 is diminished in the ratio of the cosine of the angle 

 of obliquity from the perpendicular. This is all that 

 is usually given by authors on this modification of 

 strains ; and they thence infer, that an oblique beam 

 OP will bear as great a load (Fig. 1.) as the hori- P LAT B 

 zontal beam MN, of which the vertical section is the CXIil. 

 same. F 'g- K. 



But this is a very imperfect view of the subject. 

 We might as well infer, that a beam perfectly up- 

 right need have no thickness, which would be ab- 

 surd. There is one part of the strain kept out of 

 view. The beam is loaded lengthwise, by being 

 pressed or drawn longitudinally by the force CV. 



