Strength Analysis - Finite Element Analysis for Satellite Structures - page 86

h i u fg¼ F fg K ms

1 1 ½ K mm K ms ½ K ss ½ ½ K sm ½ K ss ½ F fg ð 4 : 6 Þ

Or alternatively:

^ ^

f ^ g¼f F ½ K g ð 4 : 7 Þ

where:

^

1 ½ K ¼ K mm ½ K ms ½ K ss ½ ½ 4 : 8 Þ

K sm

^

1 f F g¼ F fg K ms ½ K ss ½ ð 4 : 9 Þ

F fg

f ^ g¼ u fg ð 4 : 10 Þ

^ ^

½ K and f F g are the superelement reduced stiffness matrix and load vector,

respectively.
In the preceding development, the load vector for the superelement has been
treated as a total load vector. The same derivation may be applied to any number
of independent load vectors, which in turn may be individually scaled in the
superelement use pass. For example, the analyst may wish to apply thermal,
pressure, gravity, and other loading conditions in varying proportions. Expanding
the right-hand sides of Eqs. ( 4.3 ) and ( 4.4 ):

F fg¼ X

N

F m fg

ð 4 : 11 Þ

i ¼ 1

F fg¼ X

N

F s fg ð 4 : 12 Þ

i ¼ 1

where:
N = number of independent load vectors.
Substitution into Eq. ( 4.9 )

g¼ X

N

1 X

N

^

f F F m fg K ms ½ K ss ½ F s fg ð 4 : 13 Þ

i ¼ 1 i ¼ 1

To have independently scaled load vectors in the use pass, expand the left-hand
side of Eq. ( 4.13 ) as:

g¼ X

N

^ ^

f F f F i g ð 4 : 14 Þ

i ¼ 1

Substituting Eq. ( 4.14 ) into Eq. ( 4.13 ):