TRANSVERSE STRAIN.] 



APPLIED MECHANICS. 



805 



Strength of No. 1 is as 

 Strength of No. 2 is as 



3 ins. X 4x4 



10 



6 ins. X 6 X 6 

 15 



= 4-3 



= 14-4. 



And by the simple proportion, 

 Strength No. 1. Strength No. 2. Tons. Tons. 



4-8 : 14-4 : : 2:6, load carried by 



beam No. 2. 



In the absence, then, of sufficient knowledge of the 

 effect of material at different parts of the section in con- 

 I tributiug to strength, for every different form of section 

 we are under the necessity of making such experiments 

 as shall fix data for calculation, to be applied to cases 

 where the section is similar, but the dimensions different. 

 A number of such experiments have been tried from 

 time to time, and attempts have been made to settle, by 

 theoretical reasoning as well as by practical results, the 

 best form of section to be used ; that is to say, the form 

 which gives the greatest strength with the least material. 

 We are not aware, however, that for any material, the 

 subject has been so far investigated as to warrant us in 

 laying down any absolute proportions ; and the form of 

 section must therefore be determined, in a great measure, 

 by balancing a number of circumstances, and adopting 

 such arrangements as shall combine the results most 

 a Ivantageously. 



In a recent engineering work, a triumph of skill and 

 perseverance over difficulties at first sight apparently in- 

 surmountable we mean the Britannia Tubular Bridge 

 the Menai Straits an enormous amount of pre- 

 liminary investigation was conducted before the form 

 and dimensions of the structure were finally determined. 

 Models were made of all convenient forms, and tested 

 against each other : one was found too weak in one place, 

 another too weak in another place. Fresh models were 

 made, with the weak parts strengthened by additional 

 materials, and the mass of material in strong parts re- 

 moved : the most suitable and convenient form was thus 

 decided. A much larger model was made and tested as 

 to deflection and fracture ; and after a large comparison 

 of results, the actual dimensions and details of construc- 

 tion of the full-sized structure were determined, and 

 carried out with merited success. 

 Fig. 66. 



Hitherto we have referred only to the circumstance of 

 a beam projecting and loaded at one end. This, how- 

 Fig. 67. 



I 



ever, is by no means the most ordinary condition under 

 which materials are exposed to transverse strain. Beams 

 are usually supported at both ends, and carry their load in 



the middle. We have, therefore, now to ascertain What 

 relation exists between the strength of such abeam and that 

 of a beam projecting and loaded at one end. If we suppose 

 a beam built into a wall and projecting equally on both 

 sides (Fig. 66), each end being loaded with an equal weight, 

 it is clear that the wall supports double the weight sus- 

 pended at each end. If we conceive the forces acting on 

 such a beam inverted or turned upside down, we estab- 

 lish the conditions of a beam supported at both ends, 

 and loaded in the middle (Fig. 07) with double the 

 weight wliich each of the end supports has to bear. 

 Now, if, in the first case, where the beam projected both 

 ways from the wall, each end were loaded with the 

 weight capable of breaking the beam, that is, up to its 

 transverse strength ; in the second case, the beam, which 

 is twice the length of each projecting arm of the first, 

 may be loaded with double the weight which hung from 

 each end of the first, and this weight will measure its 

 strength. We, therefore, come to the conclusion that 

 the transverse strength of a beam, supported at both 

 ends and loaded in the middle, is double that of a beam 

 half the length at one end and loaded at the other. We 

 have already shown that if we double the length of a 

 projecting beam, we can load it with only half the weight; 

 therefore, if each projecting arm of the beam were made 

 equal in length to the beam supported at both ends, the 

 former would bear only one-fourth of the weight which 

 the latter can bear. For equal lengths, therefore, the 

 strength of the beam supported at both ends and loaded 

 in the middle, is just four times that of the beam fixed 

 at one end and loaded at the other ; or, conversely, the 

 strength of the beam projecting is only one-fourth of 

 that of the beam supported at both ends. Experiments 

 upon transverse strain have been generally made upon 

 beams supported on both ends ; and tables of practical 

 data founded on these experiments have been formed. 

 Such tables generally contain the transverse strength of 

 beams, having square and also circular sections one inch 

 in diameter, and having a length of one foot between 

 the supports, their load being supposed to be placed in 

 the middle of their length. When the load is placed 

 out of the middle it may be increased, because its lever- 

 age to break the beam, is diminished the farther it is 

 from the middle point. 



The mode of reckoning this diminution of strain may 

 be best illustrated by an example. Suppose wo found 

 that a beam, 10 feet long, bore 42 cwt. at its middle 

 point, and desired to ascertain how much it would bear 

 suspended 2 feet from the middle that is, 7 feet from 

 Fig. 68. one end and 3 feet from 



the other (Fig. 08) we 

 should proceed as follows : 

 Square half the length 

 of the beam, or multiply 

 6 by itself, giving 25, and 

 this by 42 cwt., the load 

 sustained at the middle, product 1,050 ; now multiply 7 

 by 3 (the two portions into which the beam is divided), 

 product 21, and divide the former product 1,050 by 

 this, that gives a quotient 50 cwt. , the load which the 

 beam would carry 2 feet from the centre. 



The simple principle of this computation is, that the 

 load hung from any point of a beam, is to the load 

 wliich may be hung from any other, as the product of 

 the two lengths into which the second point divides 

 the beam, is to the product of the two lengths into 

 which the first point divides it. Thus, in the case 

 we have given above, knowing that the beam 

 bears 42 cwt. at the middle, or when it is divided 

 into two lengths, each 5 feet, we say, 



f X 3 = 21 : 5 X 6 = 25 : : 42 : 50 

 It frequently happens, that beams have to bear a 

 load not hung at any one point, but distributed uni- 

 _, formly over their length ; as in the case of roofs, 

 floors, and girder-bridges. Here, it may be readily 



7\ 



seen, that the strain is only half that which it would 

 be if the whole load were collected at the middle. If we 

 suppose a beam projecting 6 feet from a wall, loaded at 



