In the formulas for b, ^ is to be replaced by «.' in calculating b' , b' , 

 b ' , and b' but by i" in calculating b" , b" , b" , and b" . Thus, the 

 quantities s, c, S, and C have different values for the two sections, but 

 q and ? are the same. Substituting 6' = - 6' and 6" = 6", and then sub- 

 stituting in Equations [l9a,b] gives: 



F = (b' + b" ) V - b' 6' + b" e" 

 11 11 12 12 



G = - (b' - b" ) V + b' 6' + b" 6" 

 21 21 22 22 



The difference 6" - 9' represents a jump in slope due to shear 

 warping (a jump being naturally assumed positive from negative toward 

 positive x). Th\is , the beam has no definite slope where F and G are 

 applied (if shear warping is not neglected). If at this point it is 

 desired to treat interaction with another structure, so that a unique 

 value of 6 is needed, perhaps the mean of 6' and 6" may be used, or 



e = i- (e- + e") 



2 

 Then, replacing e" - 6' by -2o? as in Equation [l8] yields: 



6' = 6 - I (e" - e') = 6 + a F 



e" = e + 2 ( e" - 6' ) = e " o ^ 



The last equations for F and G then become: 



[1 + o(b|^ + bj^)] F = (b|^ + b|^^) V - (b|^ - b^^)e [20a] 



- o(b' - b" ) F + G = - (b' - b" )v + (b' + b" )e [20b] 



22 22 21 21 22 22 



The final formulas derived by solving these equations, first for F 

 and G, then, alternatively, for v and 9, are: 



2k 



