2 



IT 



9_N V+N V+N M 



=N,,V +N,,V +N„M^ 



-j 31y 23z 33x 



where the N's are constants for a given section and Nj, = N21 , etc. 



Next, redefine quantities with respect to axes through a point s at y = y , z = f by 

 these equations: 



Vy = V, 



M =M -zV +yV 



XX y •> z, 



Uy = u^ - z (?, e,- e^ 



u. = u, + y 0, 



z z 



Substituting in the above equations for deformations gives 



(9U 



dx 



dx 



1 -z 



1 +y 



1 



N N N 



'^ 1 1 ^^12 ^13 



^12 22 ^23 



1 



1 



-z + y 1 



V, r 



M 



Nil -2zNi3 + f2N^^ Ni2+yNi3 -iN23-flN33 N,3--zN37 



^12 +yNl3 -^^23 -1'^N33 N22 +2^N23 +PN33 N23+fN33 



N23 + yN33 N33 



^ N23 _ N\3 



Next choose y = and z = + Now the above equations may be written: 



N N H J 

 ^^33 33 



M. 



- " ■ « 



Then iu! = [n] Iv! , !u1 = [Qj] [n] [Q2] IvS 



1 







-^ 







1 



y 











1 



U, > and J V_ 



1 















1 







-z 



y 



1 



50 



