Page 529 - Aircraft Stuctures for Engineering Student

510 Matrix methods of structural analysis

Fig. 12.6 Forces and moments on a beam element.

The stiffness matrix [Kv] may be built up by considering various deflected states for
the beam and superimposing the results, as we did initially for the spring assemblies
of Figs 12.1 and 12.2 or, alternatively, it may be written down directly from the well-
known beam slope-deflection equations3. We shall adopt the latter procedure. From
slope-deflection theory we have
6EI
4EI
6EI
M.- --v.+-ei+-v.+-ej 2EI (12.39)
I - L2 L L2 L
and
6EI
M ---v.+-e.+-v.+-e. 6EI 4EI (12.40)
2EI
j- ~ 2 1 ~1 L ~ J LJ
Also, considering vertical equilibrium we obtain

Fy,i + Fy, j = 0 (12.41)
and from moment equilibrium about node j we have

Fy,iL + Mi + Mj = 0 (12.42)
Hence the solution of Eqs (12.39), (12.40), (12.41) and (12.42) gives

12EI 6EI 12EI 6EI
-F .=F .=--wi+-ei+-vj+-e. (12.43)
Y?Z YJ L3 L2 L3 L2
Expressing Eqs (12.39), (12.40) and (12.43) in matrix form yields
121~~ -61~~ -121~~ -61~~

6/L2
~
~
6
~
-121~~ 1 ~ 121~~ 11 { ;} (12.44)
6
4/L
2/L
-6/L2
-6/L2 2/L 6/L2 4/L
which is of the form
{PI = [K&51
where [Kv] is the stiffness matrix for the beam.