1835.] Account of the Roof of the Kdsipur Foundery. 



115 



When supported at both ends, and loaded in the middle, 

 1 Iw 



~T X ~T3 - s * 



Supported the same, and uniformly loaded, 



1 I w ~ 



T~ X VdP~" ~~ 

 Fixed at both ends, and loaded in the middle, 



1 Iw 



X 



— * ~^-=S. 



6 A H 2 



6. Fixed the same, but uniformly loaded, 



1 

 12 " bd 2 



7. Supported at the ends, and loaded at a point not in the middle, n m being 

 the division of the beam at the point of application, 



I w „ 



n m 



Iw 



bd 2 



S. 



Some authors state the co-efficients for cases 5 and 6 as | and -^ but both 

 theory and practice have shown these numbers to be erroneous. 



By means of these formulae, and the value of S, given in the following table, 

 the strength of any given beam, or the beam requisite to bear a given load, may 

 be computed. T'ais column, however, it must be remembered, gives the ultimate 

 strength, and not more than one-third of this ought to be depended upon for 

 any permanent construction. 



Formula relating to the deflection of beams in cases of Transverse Strains. 



Retaining the same notation, but representing the constant by E, and the 

 deflection in inches by 8, we shall have, 



32 Pw 



Case 1. 1 



12. 



bd38 



Pw 

 b d 3 S 



E. 



= E. 



l 3 w _ 



1 b d 3 d 



E. 



Pw 



Case 4. 



6. 



8 



2 

 ~3~ 



5 



bd 3 d 



Pw 

 b d'S 



= E. 



= E. 



X- 



Pw 



b d 3 d x E. 



Hence again, from the column marked E in the following table, the deflection 

 a given load will produce in any case may be computed ; or, the deflection being 

 fixed, the dimensions of the beam may be found. Some authors, instead of this 



Pw 



measure of elasticity, deduce it immediately from the formula 



^T" — E » 



6b d 2 S 



substituting for to the height in inches of a column of the material, having the 

 section of the beam for its base, which is equal to the weight w, and this is then 

 denominated the modulus of elasticity. It is useful in showing the relation 

 between the weight and elasticity of diiferent materials, and is accordingly intro- 

 duced into most of the printed tables. 



The above formulae embrace all those cases most commonly employed in prac- 

 tice. There are, of course, other strains connected with this inquiry, as in the 

 case of torsion in the axles and shafts of wheels, mills, &c. the tension of bars 

 Q 2 



