analogous) in local coordinates are calculated using the updated geometry as follows:
(5.59) (5.60) (5.61) (5.62) (5.63) (5.64) (5.65)
Ns
p
= 2_
'"'p.S N L..t ts
i=lMn
= Mfi,
segment 1 M12=
M~, segment NsQ1
= L~
(Mn+
M12)SlllO<l
M31
= M£,
segment 1 M32=
M~, segment Ns
1
Q3
= -
L~ (M31+
M32).Slll0<3
These forces are for output purposes only.
The column end forces (equation 5.56) act at the end of the column which is located at the edge of a panel zone. These forces must be transformed to frame nodal forces that are applied at the center of the panel zone. The panel zone depth d~ and widths dJ.i and d3i, and the beam and column rotations
ot,
(}~i' OJ.i and 03i at each end ( i or j) of the columns are used to determine the frame nodal loading from the columns.Incremental displacements
Effect of beam rotation
Figure 5.24 Converting colunm end displacements to frame nodal displacements, X~-X~
plane.
The incremental displacements can be found by imposing small displacements of each of the frame nodal degrees offreedom (figures 5.23 and 5.24). Note that the subscripts for the column ends ( i or j) are not used in these figures which represent the top end of a column. The resulting transformation from nodal increment .6.x to column end increment .6.u00 is
(5.66)
where
(5.67) 8cc
=
1 0 db
2
cos (}li b 0 0 0 0 0 0 0 0 0 0 00 1 d~ TSlll • (Jb li 0 0 2sm d~ • (Jb 3i 0 0 0 0 0 0 0 0 0 0 0 0 1 d~ 2 cos0b 3i 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 - .J. db cos0b 1j 0 0 0 0 0 0 0 0 0 0 0 0 1 -2sm d1 • (Jb lj 0 0 -2sm d1 • (Jb 3j 0 0 0 0 0 0 0 0 0 0 0 0 1 - j d~ cos03j b 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1
The forces can be transformed from the column end forces ( ~ p00 from equation (5.53)) to frame global coordinates ~PF-CC (figures 5.25 and 5.26) by the transpose of the same transformation
(5.68) ~pF-CC
=
[SCC]T ~pCC.Note that the superscript in the figures is F, not F-CC for clarity of the figures.
The column contribution to the frame nodal equations is
(5.69) K~c ~x
=
~PF-ccwhere
(5.70) K~c
=
[scc]r[Kcc][scc].Column forces Frame forces
Figure 5.25 Converting column end forces to frame nodal forces,
Xi
-X~ plane.Column forces Frame forces
Figure 5.26 Converting column end forces to frame nodal forces, X~-X~ plane.
5.3.4 Panel Zone Stiffness
For 3D columns, there are two panel zones per column acting in the planes of the beams framing into the column. Both directions (i
=
1, 3) have a panel zone stiffness which relates the shearing strain 'YfZ of the steel volume acting in the plane of the panel zone and the momentMf z
that resists this strain. As for the 2D case, the shear modulus is determined from a nonlinear stress-strain relationship, and its instantaneous value is denoted GiT· The volume being sheared isif strong axis of wide flange, (5.71) dedb(tw
+
tdp) same as above with doubler plate,if strong axis of box,
if weak axis of wide flange or box
where de is the depth of the column, db is the depth of the deepest beam framing into the column, be is the width of the column, tdp is the thickness of the doubler plate, and t f and tw are the thicknesses of the flange and web, respectively. The shearing strain is related to the beam and column rotations in incremental form by (5.72)
The resulting stiffness equation is
(5.73) K~Tz ~ ax-f'Z ~ = apl!-PZ ~
where Llpf-PZ and LlxfZ are the panel zone incremental moments and rotations associated with the frame dofs in planes i
=
1, 3 and5.3.5 Assembly to Frame Stiffness 5.3.6 Frame Stiffness
The stiffnesses which contribute to the frame have been written in the frame co- ordinate system. The three-dimensional column stiffness, 5.69, and the panel zone stiffness, (5.73), can be assembled to appropriate degrees of freedom and added to the frame stiffness (3.60):
(5.75) KF -T - K F T
+ a
(Kcc T+
KPZ lT+
KPZ) 3T ·All of the stiffness contributions are tangent stiffnesses, so the governing total frame stiffness is also. The stiffness forces at all the degrees of freedom of the frame can also be assembled from the columns, (5.68), and the panel zones, (5.74), as
(5.76)
These contributions to the frame stiffness matrix and stiffness force vector are as- sembled into the global building stiffness matrix ( 4.45) and stiffness force vector
( 4.46), respectively.
5.4 Influence on Three-Dimensional Solution
The addition of the 3D column has several impacts on the solution of the system of equations including extra storage and frame interaction considerations.
The 3D column requires additional storage for fiber properties, member dofs, and frame dofs. Since the program is written in FORTRAN 77 , dynamic allocation is explicitly performed within the program. Because of this, storage is provided for seven dof/node instead of the four dof/node for three-dimensional analysis without 3D columns. Separate variables are used for most of the other 3D column properties to limit excessive storage that is never used.
For most analyses, all of the column horizontal dofs are constrained to the master
joints. For a 3D column, one axis acts in one frame and another axis acts in another frame. Since constraints are applied by frame (see section 4.2), different dofs of a 3D column can be constrained to act with different frames. This is an additional reason for requiring only orthogonal framing to the 3D columns since the current constraint method would not be appropriate for skewed framing.
The foundation elements are modified to be capable of tracking nonlinear re- sponse in two orthogonal directions corresponding to the column.
The stiffness in terms of the column itself and its effects on panel zones, con- straints and foundations have been enumerated. The contribution of mass must be considered separately, though. Recall that the mass associated with a frame node has an out-of-plane contribution to the master joint which results in a non-diagonal mass matrix (section 4.7). Considering the 3D column, out-of-plane motion for one plane will be in-plane motion for the orthogonal plane. Thus, the in-plane mass for the two orthogonal frames to which the column is constrained will fully account for the horizontal translational motion of the mass, which in turn accounts for the rotational motion. No mass or mass cross terms are assembled to the master node for 3D columns.