REPORT ON THE STRENGTH OF MATERIALS. 101 



4-. Supported the same, and uniformly loaded, 



1 ^^ _ C 



- X ^2 - b. 



5. Fixed at both ends, and loaded in the middle, 



1 Iw c. 

 "6 ^ 6T^ = ^- 



6. Fixed the same, but uniformly loaded, 



1 Iw _ ^ 



7. Supported at the ends, and loaded at a point not in the 

 middle. Then, n m being the division of the beam at the point 

 of application, 



nm Iw _ jj 



Some authors state the coefficients for cases 5 and 6 as -rr 



1 . 



and jr., bvit both theory and practice have shown these numbers 



to be erroneous. 



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

 follovping Table, the strength of any given beam, or the beam 

 requisite to bear a given load, may be computed. This 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. 



Formules relating to the Deflection of Beams in cases of 

 Transverse Strains. — Retaining the same notation, but repre- 

 senting the constant by E, and the deflection in inches by 8, 

 we shall have. 



Case 1. -r X , ,^^ = E. 



= E. 



3. -^ X ^^ = E. 



Case 4. -rr x , ,o» = E. 



bdn 



2 Pw 



3 ^ bdH 



= E. 



^ 5^ Pw _ -p, 

 12^ bd^^~ 



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

 Ttble, the deflection a given load will produce in any case may be 

 computed ; or, the deflection being fixed, the dimension of the 

 beam may be fovmd. Some authors, instead of this measure of 



