MECHANICS. 



arms, of which W acts on the long arm en 1 , c 

 and cd being two short arms at right angles to en' 

 The fibres of the upper part of the section (which 

 we suppose not yet ruptured, but only extended" 



U 



"", f* 

 ']' 



i 



Fig- 31- 



are acting all along the arm ce, as at ik, ae, draw- 

 ing it towards the wall ; while those in the lower 

 part are resisting compression, and pushing the 

 arm cd away from the wall. Both these sets of 

 forces resist the tendency of W to turn the lever 

 about the axis c, and thus together constitute the 

 strength of the beam. The effects or moments of 

 the two sets must evidently be equal to each 

 other ; for the neutral axis so adjusts itself that 

 the amount of resistance on the one side balances 

 that on the other. If the tensive strength of the 

 fibres is equal to their compressive strength, the 

 neutral axis will be in the middle of a rectangular 

 beam ; if the tensive strength is greater, the 

 neutral axis will be nearer the upper side ; and 

 vice versa. 



But the most important point to observe in this 

 investigation is that the fibres are not all equally 

 well situated for offering resistance. A fibre at a, 

 for instance, acts with twice the leverage of one at 

 /, supposing i in the middle of ca. If we call the 

 tensive strength of a fibre of certain area I, its 

 moment or efficacy at a will be expressed by ca 

 X I, or simply ca ; while the moment of a similar 

 fibre at / will be ci X i, or ci. The sum of the 

 moments of a row of fibres from c to a would thus 

 be expressed by the number of such fibres multi- 

 plied by their mean distance from c, which is \ ca. 

 Since the number of fibres along the line ca 

 increases with the length of the line, we may take 

 that length to represent the number ; and there- 

 fore, the sum of the moments will be ca X \ ca, or 

 \ ca 2 . But the resistance to compression made 

 on the arm cd, is equal to the resistance on ca ; 

 therefore, the whole resistance to fracture made 

 by a line of fibres from a to f, is expressed by (2 

 times \ co* or) ca\ And if we put b for the breadth 

 of the beam, the whole strength of the beam is 

 expressed by b X ca\ 



From a variety of causes, the formula above 

 given does not enable us to calculate at once the 

 transverse strength of a beam from merely know- 

 ing the tensive strength of a rod of its fibres. 

 One of these causes is that we have no means of 

 determining where the neutral axis is situated. The 

 formula, however, serves this valuable purpose, 



that when the actual transverse strength of one 

 beam of certain dimensions has once been deter- 

 mined by experiment, it enables us, from this 

 data, to calculate the strength of another beam of 

 the same general form, but of different dimensions. 

 For, whatever fraction ac is of a/, in a beam of 

 any material, it is the same fraction in all beams 

 of that material, or the length of ac varies with 

 the length of af; and, therefore, to express the 

 relative strengths of beams, we may put af in the 

 formula instead of ac. If, then, we put d for the 

 depth, or af, the formula becomes b X d 2 , which 

 expresses the leading proposition in the strength 

 of materials, that THE TRANSVERSE STRENGTH OF 



A BEAM INCREASES AS THE BREADTH MULTI- 

 PLIED BY THE SQUARE OF THE DEPTH. 



The truth of this important proposition is fully 

 confirmed by experience, and is constantly acted 

 upon in all kinds of constructions. Rafters, joists, 

 &c. are never made square in the section ; but as' 

 deep as possible, and no thicker than is necessary 

 to prevent them from bending laterally, or buckling 

 as it is called. Strength is 

 thus gained, and material 

 saved. Ex. Let A (fig. 32) 

 be the section of a square 

 wooden beam of 10 inches 

 in the side, and B that of 

 another beam of the same 

 wood, 5 inches broad by 20 

 deep. The areas of the two 

 sections are equal, being 100 

 square inches each ; but the 

 transverse strength of A is as 10 x io z = rooo, 

 and that of B is as 5 X 20* = 2000. B has thus 

 double the strength with the same amount of 

 material. When the breadth of a beam is doubled, 

 retaining the same depth, the strength is merely 

 doubled ; when the depth is doubled, retaining the 

 same breadth, the strength is quadrupled. 



In order to make this expression available for 

 finding the actual weight that any beam will bear, 

 small beams or prisms are taken, of the most usual 

 materials, having a section of one square inch, 

 and a length of 12 inches, and loaded with weights 

 till they break. The breaking weights of these 

 small bars are then used as constants or co- 

 efficients, for calculating the breaking weights of 

 other beams of the same materials. The following 

 are a few of the results of experiments made on 

 mall bars fixed and loaded as in fig. 31 : 



Ash 2030 



Im 1030 



ir, Scotch 1140 



iied Pine 1340 



0,k, English 



Cast-iron 8100 



Wrought -iron 9000 



To find the weight that will just break a beam of 

 _iven dimensions, we have only to multiply the 

 constant for the material in the table by the 

 breadth and square of the depth, and divide by 



he length the dimensions being all reduced to 

 nches ; or, employing the formula, W = Constant 

 X b X d* -r- /. Ex. What weight will break a 

 >eam of Scotch fir, fixed at one end, and loaded 



t the other, its breadth being 3 inches, depth 1 1 

 nches, and length 9 feet? Ans. W = I 140 X 3 X 



i 2 -f- 108 = 3832 Ibs. nearly. 

 When the beam is cylindrical, find the strength 

 _f a square beam, whose side is equal to the 



liameter of the cylinder, and take two-thirds of 

 he result. Ex. What is the strength of a round 



sir 



