Equations [6] indicate that this choice of y and "z has uncoupled tensile force V from the 

 bending moments M and M^. Point e, so located, is called the elastic axis, and is located 

 at the center of effective tension-carrying area. 

 We make these further definitions: 



iy .-IK. 



■' e 1 1 



= total area of cross section. 



= area moment of inertia of cross section about 

 axis through e and parallel to the z-axis. 



lyz = Sy^.Zg A. = area product of inertia of cross section relative 

 to axes through e. 



I^^ = Sz .^A. = area moment of inertia of cross section about 

 axis through e and parallel to the y-axis. 



Then Equations [6] may be written: 



V^ =(B + Cy + Dz)A 



¥ = I C + 1 D 



y y z zz 



M = -I C -I D 



z y y y z 



We can solve Equations [7] for B, C, D in terms of V , M , M , obtaining 



-1 Lj 11 x'v'z' f^ 



V. 



(zl -yl )M -(yl -zl )M 



yy "^ yz' y ^•^ zz yz'^ 2 



I I -I 2 



yy zz yz 



-I M - I M 



y z y zz z 



I I -I 2 



yy zz yz 



I m" + 1 ¥ 



yy y yz z 



I I 



yy zs 



H7] 



^ [8] 



Substituting into Equation [1], the initial expression for a at node i, gives 

 (a ). = B + Cy. + Dz. = (B + Cy + Dz) + Cy . + Dz . 



Vxx'i •'i 1^ ■' ' ■'ei ei 



V (z I -y I )M - (y I - z . I ) M 



X ^ ei yy ^ei yz' y ^'^ e i zz ei yz' z 



(a ). = 



I I -I 2 



y y zz y z 



[9] 



The elastic parameters for bending and tension may be obtained from the associated strains 

 as follows (see page 232 of Reference 7): 



63 



