PLANAR SUBSTRUCTURE MODEL
269
coordinate system of the kthfloor, with aik and bik , where k ¼ 1; 2; . . . ; n and n is the total number of storeys of the frame. The sway stiffness equation of the ith planar frame can be established according to the method described in Chapter 13 and written as ½Kx i fdDx gi ¼ fdFx gi ;
ð15:50Þ
where fdDx gi and fdFx gi are, respectively, the incremental vector of the horizontal displacements of the frame floors along the x-axis and the corresponding incremental force vector and ½Kx i i is the sway stiffness matrix of the ith planar frame in its own plane. Expand Equation (15.50) to the spatial sway stiffness equation of the planar frame in its own coordinates as 2
½Kx i
9 8 9 38 ½0 < > fdDx gi > < fdFx gi > = > = 7 f0g ½0 5 fdDy gi ¼ ; > > > : : ; > ; fd z gi f0g ½0
½0
6 4 ½0
½0
½0
½0
ð15:51Þ
where fdDy gi and fd z gi are, respectively, the incremental vectors of the horizontal floor displacements along they-axis and floor torsions. By the transition relationship between the global coordinates and the local coordinates of the planar frame, one has 8 8 9 9 > > < fdDx gi > < fdDu gi > = = fdDy gi ¼ ½Ti fdDv gi ; > > > > : : ; ; fd z gi fd w gi
ð15:52Þ
8 8 9 9 > > < fdFu gi > < fdFx gi > = = fdFv gi ¼ ½Ti T f0g ; > > > > : : ; ; f0g f0g
ð15:53Þ
where fdDu gi , fdDv gi and fd w gi are, respectively, the incremental vectors of the horizontal floor displacements along the u-axis and v-axis, and floor torsions in the global coordinates, fdFu gi and fdFv gi are the incremental vectors corresponding to fdDu gi and fdDv gi , respectively, and ½Ti is the coordinate transition matrix of the ith planar frame, which can be expressed as 2
½ci
6 ½Ti ¼ 4 ½ si ½0
3
½si
½0
½ci
7 ½0 5;
½0
½I
ð15:54Þ
in which ½ci ¼ cos ji ½I ¼ ci ½I ;
ð15:55aÞ
½si ¼ sin ji ½I ¼ si ½I ;
ð15:55bÞ
where ½I is the unit matrix. Substituting Equations (15.52) and (15.53) into Equation (15.51) leads to 2
c2i ½Kx i
6 4 ci si ½Kx i ½0
ci si ½Kx i s2i ½Kx i ½0
9 9 8 38 fdDu gi > > fdFu gi > ½0 > < = = < 7 ½0 5 fdDv gi ¼ fdFv gi : > > > : : ; ; > fd w gi f0g ½0
ð15:56Þ