Three-Dimensional Columns
5.1 Modeling Considerations
Torsion has been neglected for several reasons. The global analysis is capable of large displacements, but assumes small rotations. Even a torsionally irregular building subjected to a large near-source pulse excitation may exhibit only small rotations even if its columns experience 20% drift. In the next chapter, some example structures are analyzed for different assumptions to quantify the amount of rotation that the program should be able to accurately model. The resulting twists are small, and the corresponding torsional stresses will also be small. Englekirk writes
. . . torsional stresses, however induced, are seldom calculated, for they have little or no impact on the strength limit state of a member. 1
The three-dimensional rigid body constraints imposed on the frames will cause cor- ner columns to twist a small amount for unsymmetrical loading or an irregular structure. The small rotations will cause low stresses that can be ignored. The pro- gram does not address the lateral-torsional stability of the column in this twisted state. Additionally, for most buildings relying on corner columns, the section is typi- cally a tubular section or a stocky wide flange. These members provide a great deal of torsional resistance and generally do not experience lateral-torsional buckling.
Torsional dofs have not been included, but a member's torsional resistance could be assembled to the rotational dofs of the master joints. This contribution tends to be small for a realistic building and is ignored. It is not believed that this will be a non-conservative assumption because of the aforementioned small rotations.
Orthogonal framing allows the beam and panel zone forces to be accounted for easily, especially considering that torsion has been neglected. Further developments of the three-dimensional column could allow arbitrarily oriented framing, but this has not been developed in this work.
There are several methods for determining the orientation of the members re- sulting from the geometric updating of segment internal nodes. Chen and Atsuta present a method for determining the orientation of a segmented plastic beam- column (Chen and Atsuta 1977, Ch. 11). They keep track of segment orientation for arbitrary displacements and rotations at each end. Their method requires many
1(Englekirk 1994, p. 281)
transformations and is based on a formulation that considers torsion. A method that determines orientation based solely on the location of segment ends is desired since rotational transformations do not commute and the problem is nonlinear. Be- cause the assumed displacements do not have large rotations, there is no concern that the member could accidentally be oriented 90 degrees out of alignment.
A projection method has been developed to maintain the forces and displace- ments associated with the five dofs at each end of a segment. The method conve- niently eliminates torsion consistent with the torsion assumptions. In this manner, the five end forces of an arbitrarily oriented segment are projected onto five in-plane forces associated with the major and minor axes of the segment. Other standard methods would convert the five end forces to six orthogonal forces, introducing a torsion that would then be eliminated.
X' 3
Figure 5.1 Segment local (primed) and global (unprimed) axes.
(a)
u~c ~u~s
u~ ul
IX2
~I
(b) u~
Figure 5.2 (a) Segment local degrees of freedom. (b) Segment global degrees of freedom.
X' 3
Figure 5.3 Projection of global displacements on local X~ and X~ axes.
X'
2~
t::...
rfJ
II
.p
rfJ
0 Q
x4
Figure 5.4 Global directional cosines for X~ axis.
~
t::...
rfJ
as
a. Ix3
cosy3 cosy1
Figure 5.5 Projection of global displacements on local X~ axis.
t::
~
...rfJ
X I
The projection method considers the location of both ends of a segment in global frame coordinates, X1, X 2 and X3. Using the current and previous lengths of the member, the axial deformation is determined. Projecting the current configuration onto the X1-X2 and X2-X3 planes defines the local member axes, Xi and X~,
respectively. The X~ axis is aligned with the current segment axis (fig. 5.1). The displacements in the local dofs can be determined from displacements in the five dofs in the global coordinates. The local and global degrees of freedom are shown in figure 5.2. The contributions of u1 through u3 are shown in figures 5.3 and 5.5. In figure 5.3, the global displacements u2 and u3 are projected onto the local X~ axis and u2 and u1 are projected onto the local Xi axis. Global displacements u1, u2 and u3 can be projected onto the local X~ axis using the global directional cosines 'Yi
(figure 5.4). These projections are instead made using the local angles a:i from figure 5.1. The relation between the directional cosines and local angles is shown in figure 5.5. Note in figure 5.2 that rotational dofs u4 and us are unchanged from global to local axes since the local axes are in the planes perpendicular to the rotational axes. The resulting transformation for all dofs is:
ul sina:1 -COSO:! 0 0 0 U!
u2 tana1 sina2 sina:2 tana3 sina2 0 0 U2
(5.1) u3 0 -cos 0:3 sina:3 0 0 U3 or u=Tu.
u4 0 0 0 1 0 U4
us 0 0 0 0 1 us
The global displacements that result from solving the equations of motion pro- duce the new configuration, which determines the angles used in the transformation above. The transformation is used to obtain the local displacements that in turn are used to determine fiber strains. The fiber strains are used to determine the axial force. Shear forces are obtained from the configuration and end rotations of the segment. Moments are calculated from the shear forces plus the bending due to fiber axial stresses. These end forces are then transferred to global coordinates
using the transpose of the previous transformation (figure 5.6):
(5.2}
A closer look at just the local translational forces shows that the transformation
x2 x3
~~~---~--~--~~
(b)
Figure 5.6 Projection of local segment forces on global axes.
does not preserve orthogonality:
(5.3) pTp
=
p~+
p~+
p~ whereas (5.4) PT p=
PT[T][TT]p(5.5) (5.6) (5.7)
A2 A2 o 2 (1 1 1 ) A2 2 A A
=
p1+
p2 sm a2+
2+
2+
p3+
PIP3 cos a1 cos a3tan a1 tan a3
= P~
+
p~+ p; +
2fi1P3 cos a1 cos a3· 2 ( 1 1 -
since sm a2 1
+
2+
2 )=
1.tan a1 tan a3
The error in the magnitude of the resultant will be small, even for large deformations.
Assume equal forces and displacements in the
Xi
and X~ directions and no force in the X~ direction. This configuration is possible for a building with predominantly torsional response. The assumptions can be represented by(5.8) (5.9) (5.10)
p
=Pl
=P3
equal forcesP2
= 0 no axial loada = a1 = a3 equal displacement.
The error can then be expressed as
(5.11) (5.12)
Error=y'i)T~ pTp
=
J
1+
cos2 a - 1.One common criterion chosen for ending an analysis is when story drift reaches 20%.
If a column manages to reach 20% drift in each direction, cos a1 ~ 0.2 and the error will be less than 2%. Any axial force (2-direction force) will reduce this error since the projection method does not corrupt this component in the transformation from local to global forces.