Matrix Structural Stiffness - Numerical Structural Analysis - page 193

Members 6 and 7 have the same orientation. Selecting the top-left end as
the i i-end, the rotation is 36.87° or −323.13°. Instead of using angles, in
this case it is easier to use trigonometry directly. The cosine is 0.8 and the
sine is 0.6. The axial stiffness is 666.7 k/in.

426 7 . − 320 0 . − 426 7 . 320 0 .

− 320 0 . 240 0 . 320 0 . − 240 0 . K K

KK

[ ] = [ ] =

2 2 23

KK
23

=

42

− 4 26 7 . 320 0 . 426 7 . − 320 0 .

32 33

320 0 . − 240 0 . − 320 0 . 240 0 .

KK
KK

=

44 42

24 22

The global joint stiffness matrix can be assembled using each of the mem-
ber's contributions (step 2).

KK K
KKKKK
KK K
KK KK

0 0

11 12 14

21 22 23 24 25

=

K 0 0

g 32 33 35

0

41 42 44 45

0 KKKKK

52 53 54 55

427 320 − 427 − 320 0 0 0 0 0 0

320 796 − 320 − 240 0 0 0 − 566 0 0

− 427 − 320 1707 0 − 427 320 − 427 320 − 427 − 320

− 320 − 240 0 960 320 − 240 320 − 240 − 320 − 240

0 0 − 427 320 427 − 320 0 0 0 0

=

K g

0 0 320 − 240 − 320 796 0 0 0 − 566

0 0 − 427 320 0 0 843 − 320 − 417 0

0 − 566 320 − 240 0 0 − 320 796 0 0

0 0 − 427 − 320 0 0 − 417 0 843 3 20

0 0 − 320 − 240 0 − 566 0 0 320 795

The global joint loading can be determined from Equation 4.35 directly
since all of the applied loads are at the joints and in the global system
(step 3). Note that there are no loads applied directly to the members, so
there are no fixed-end forces and moments. The load matrix is in units of
kips (k).