Page 527 - Aircraft Stuctures for Engineering Student

508 Matrix methods of structural analysis

Fig. 12.5 Local and global coordinate systems for a member in a pin-jointed space frame.

In Fig. 12.5 the member ij is of length L, cross-sectional area A and modulus
of elasticity E. Global and local coordinate systems are designated as for the two-
dimensional case. Further, we suppose that
e,, = angle between x and 2

8, = angle between x and jj

e,, = angle between z and jj

Therefore, nodal forces referred to the two systems of axes are related as follows
-
F, = F, COS exj + F,, COS e,, + F, COS e,?
-
F~ = F~ cos e,, + cos e,, + F, COS e,,: (12.34)
-
F~ = F, cos ezj + F,, cos e,, + F, COS ezi
Writing
A, = cos e,,, A, = COS e,,, A, = COS e,
pj = cos e,,,, p- - COS e - pF = COS e,, (12.35)
Y -
YY’
uj = cos eaf, v, = COS e,,, uz = COS e=?
we may express Eq. (12.34) for nodes i andj in matrix form as
‘A, pf u, 0 0 0-
A, p, u, 0 0 0
A, p2 u, 0 0 0 (12.36)
0 0 0 A, pj vj
0 0 0 A, p, v,
0 0 0 A, pz u,-