SPACE BAR MODEL
261
where 2
1 ½Bg ¼ 4 0 0
0 cos jg 0
0 sin jg 0
3 0 0 5: sin jg
0 0 cos jg
ð15:4Þ
Combining the displacement vectors at ends i and j of the element leads to the transformation equation as f gg g ¼
f ggi g f ggj g
¼
½Bg 0
0 ½Bg
fDggi g fDggj g
¼ ½Tg fDgg g:
ð15:5Þ
By a similar derivation, the transformation equation for the force vectors from the local to the global coordinates is fFgg g ¼ ½Tg T f fgg g;
ð15:6Þ
where f fgg g and fFgg g are the force vectors corresponding to f gg g and fDgg g, respectively. Assume that the incremental stiffness equation of the beam element with joint panels in the local coordinates is fdfgg g ¼ ½kgg fd gg g:
ð15:7Þ
Substituting Equations (15.5) and (15.6) into Equation (15.7) yields the incremental stiffness equation of the beam element with joint panels in the global coordinates as fdFgg g ¼ ½Kgg fdDgg g;
ð15:8Þ
where ½Kgg is the stiffness matrix of the beam element with joint panels in the global coordinates, given by ½Kgg ¼ ½Tg T ½kgg ½Tg :
ð15:9Þ
15.1.1.2 Coordinate transformation of column element An arbitrary column element in the global coordinates is shown in Figure 15.4, where jc is the angle between the first principal axis of the column section, ox, and the global axis ou. In the local coordinates, the displacement vectors at ends i and j of the column element with joint panels are f cgs g ¼ ½ zs ; xs ; ys ; gys ; ys ; xs ; gxs ; zs T ;
v
y
s ¼ i; j:
x
jc o
O′
u
Figure 15.4 Column element and global coordinates
ð15:10Þ