Beams fixed at the Ends. 



247 



if the beam was divided into n parts. In fact, c is the 

 average value of m for the whole span. The application of 

 (3) to beams of uniform section, fixed at the ends with one 

 load in the middle or with a uniformly distributed load, is 

 very simple. 



IV. The condition (1) is applicable when I at every cross 

 section of the beam is previously settled in any arbitrary 

 manner. Let us give to I such a value that the beam shall 

 be of uniform strength everywhere, that is that 

 M 



y =fc or fv 



(4) 



where y is the greatest distance of any point in the section 

 from the neutral line on the compression- or tension-side, 

 and f c and f f are the constant maximum stresses in com- 

 pression or tension to which the material is subjected in every 

 section. 



f e may generally be taken as equal to f t ; and if d 

 the depth of the beam, y may generally be taken as equa 

 to \d. Hence (4) becomes 



M 



jd=±2f, (5) 



the + sign being taken when M is positive, and the — sign 

 when M is negative. 



Hence (1) becomes ^ 1 _ , 



^Td = ' (6) 



the negative sign being taken from the ends of the beam to 

 the points of inflexion, and the positive sign being taken 

 between the two points of inflexion. 



Now to satisfy (6) we have merely to solve the following 







y 









r/ 





^\S 



3 



E. 









\ G 





|f 





a 





A 



,-<-"" T 







i 



: l C 



problem : — In the figure E A C GrE is a diagram whose ordi- 

 nates represent the value of -~, or the reciprocal of the depth 



