Integration of Equation [l6a] through the slice gives: 



The first integral is negligible because of the thinness of the slice, 



whereas : 



■^ )dx = P" - P' ; / F dx = F 

 Sx y j 1 



* 



Hence, the total external force is 



F = P' - P" [iTa] 



Similar treatment of Equation [l6b] gives, since /Pdx is negligible: 



G = M' - M" [1Tb] 



Also, differentiation of Equation [2c], elimination of y by means of 

 Equation [2d], and integration with respect to x give: 



-— dx = =r- dx + 2o 3— dx 



3x j EI I 9x 



Here the M integral is negligible, whereas / (3e/3x)dx = e" -e' , where 



e' and e" denote the slope of the beam or Sv/sx at the left and right sides 



of the slice, respectively. Hence, using Equation [iTa]: 



e" - e' = 2a(P" - P') = - 2oF [18] 



The equations for a beam forced at one end csji now be applied to the 

 two sections into which the beam has been divided. Let the lengths of the 

 sections be £' and l" , so that £'+£," = £., the total length. Let all 

 quantities referring to the left or £,' section be distinguished by one 

 prime and those referring to the l" section, which extends toward positive 

 x, by two primes . 



* Compare with Equation [18] of Reference 12, 



22 



