Three-Dimensional Modeling
4.3 Shear Spring Elements
necessary.
F
Figure 4.3 Load-deflection relation for shear springs.
Figure 4.4 Shear spring stiffnesses.
The stiffness formulation of the shear springs is now derived. Each shear spring i of each story j can have orthogonal tangent nonlinear story stiffnesses k~,j and kt,j (figure 4.4). The displacements at the shear spring location first need to be written in terms of the master node displacements using the three-dimensional small rotation transformation. For a general location c on a particular floor, this geometric
transformation is
.6..xc 1 0 _ycMN .6..xMN
(4.19) _6.yC 0 1 xcMN _6.yMN or .6.xc
=
T .6.xM N_6.()C 0 0 1 _6.()MN
where xcMN = x c - xMN and ycMN = y c - yMN (figure 4.5). Note that the
Figure 4.5 Transformation of displacements from master node to location c.
rotation of location c is also included in the transformation. This is an unnecessary variable to keep track of, but the forces in a shear spring at location c will contribute to the torsion at the master node, so the master node rotation ()M N must remain in the formulation. This transformation written for shear spring i connecting floors j and j
+
1 becomes( 4.20)
Considering only story j bounded by floor j below and floor j
+
1 above, the forcedisplacement relationship for a shear spring is
k~,j 0 0 -kL 0 0 Llx~+I Llp~,j+l
0 kt,j 0 0 -kt,j 0 Lly;+l Llpt,j+l
0 0 0 0 0 0
Lle;+l
Llp~,j+l(4.21)
-k~,j 0 0 k~,j 0 0 .£lxi.
J Llp~,j
0 -kt,j 0 0 kt,j 0 Lly; Llpt,j
0 0 0 0 0 0 .£lei
J Llp~. ,J
A notation for the 3 x 3 sub-matrices is introduced as k~,j 0 0
( 4.22) ki_,SS
=
J 0 kt,j 0 ' 0 0 0 allowing equation 4.22 to be written as
(4.23) [ k;•ss -ki_,SS J
-k]ss] {
ki_,SS J ax~ a xi. +I J}={
ap~+I ap~}
The transformation T~ from eqs. 4.19 and 4.20 gives the exact relationship between the forces of a shear spring and their contribution to the master node at level j:
(4.24)
which can also be expressed in incremental form. Substituting (4.20) and (4.24) into (4.23), the contribution of a single shear spring ito the master node dofs becomes
( 4.25)
ax¥N
J
which can be written as
( 4.26)
for story j. Note that a shear spring on story j contributes to the master node dofs on floor j
+
1 above and floor j below the story. The overall story stiffness is the sum of all the shear spring stiffnesses associated with that story (a total ofnj
8shear springs in story j):
(4.27) where
The shear building stiffnesses KfH are tangent stiffnesses, and the total shear build- ing tangent stiffness involves all of the master node dofs by the set of equations
(4.28) where
ns
KSH T
= a
[K~H] .j=l J
In the assembly above, n8 is the number of stories. This stiffness only contributes to the master node degrees of freedom. The forces at the master nodes can be assembled similarly. Using equation 4.24, the forces can be summed for all shear springs i associated with floor j, and these master node forces can be assembled into the p8 H vector of all building forces. This summation and assembly can be expressed as
(4.29)
As mentioned above, the shear springs can be placed at arbitrary locations within each story of the building. In the absence of specific information, one strategy is to place a shear spring at each column location, including columns that are present in the model (frame columns) and those that are not (gravity columns). The x and y stiffness and strength of each shear spring would be made the same, equal to the desired lateral stiffness and strength in a story divided by the number of shear
springs in that story. This method will automatically establish specific values for the torsional stiffness and strength in the story due to the shear springs. In the deformed state, the alignment of frame columns and shear springs positioned in the same location will not be perfect since the shear springs are moving as a rigid body connected to the master nodes, whereas the average of a frame's column movements is constrained to rigid body motion with the master nodes.
4.4 P-.6. Effects
Planar frames can act as a three-dimensional structure using the constraint equa- tions, but additional P-~ effects must be accounted for and assembled to the master node degrees of freedom. Vertical loading carried by gravity columns not included in the model will create a P-~ effect discussed in the next subsection. The P-~
effects on frame columns in the plane of the frames is automatically accounted for through geometric updating, but the P-~ effect in the out-of-plane direction must also be included, and is discussed in the following subsection.
4.4.1 Gravity Columns
Vertical loading not carried by the frame columns will create a P-~ effect. Since the frames and shear springs provide all of the lateral resistance, when the top of a gravity column moves relative to the bottom, the resulting P-~ horizontal couple will be carried by the frames and shear springs, not the relatively flexible gravity column. Gravity columns are typically not included in the model, and if they are, they are represented as single column frames. To include this P-~ effect, cumulative gravity loads are placed in plan at gravity column locations. In this way, the P-~
effect can be determined in each direction for gravity columns and tied to the frames through the master node constraints.
Many programs account for P-~ effects by assigning a force couple at floor levels to induce the incurred moment and by reducing the stiffness of the story between the floors based on the force and displacement levels. These programs can give accurate
results, but there are drawbacks to these techniques. For example, ETABS assumes the input mass at each floor can be used as a constant load on the columns of the structure on a floor-by-floor basis. This allows the geometric stiffness reduction that is dependent on axial load to be assembled to the building stiffness prior to analysis. In this way, ETABS can approximate the P-~ effect without any iteration (Wilson 1997). The drawback is that if a column has uplift from overturning, it will still receive a P-~ stiffness reduction as if a constant downward load were being applied. This approach assumes small lateral displacements relative to story heights and ignores nonlinear beam-column stiffness effects altogether in its global approach (Wilson and Habibullah 1987). While this method is simplistic, it will be used for the gravity column P-~ effects.
The cumulative gravity load
Pj
associated with gravity column i of story j produces a negative geometric stiffness written in the same form as the 3 x 3 shear spring stiffness ( eq. 4. 22):_ _}_ pi 0 0 Lj
( 4.30) ki.'P-.6.
=
0 _ _}_ pi 0J Lj
0 0 0
This can be assembled the same way the shear spring stiffness was by replacing k~,ss with k~,P-.6. in equation ( 4.25) and summed for all gravity loads for story j as
(4.31)
The resulting geometric stiffness equation is the same as developed in Wilson and Habibullah (1987). The geometric stiffness associated with the gravity column loads for the entire structure is
(4.32)
where n8 is the number of stories.
As in the case of shear springs, the transformation
T;
from equation 4.19 pro- vides the exact relation between the forces at gravity column location i and the master node at level j:( 4.33)
The forces can be summed for the
nf-!1
locations at each floor j where gravity loads have been placed and assembled into the total force vector as(4.34)
4.4.2 Frame Columns Out-of-Plane
Out-of-plane frame displacement will create a geometric effect that must be carried by frames and shear springs providing transverse resistance. The out-of-plane dis- placements and column forces are tracked, and their contribution as force couples is summed at the master nodes. The original orientation of a frame (f3) is used to determine the out-of-plane direction ({3
+
1r /2) in which to apply the couples.The out-of-plane geometric stiffness reduction is not considered in the updating of the global tangent stiffness matrix. While the tangent stiffness is a function of the current deformation state, no other stiffness contribution is directly a function of the current forces. Neglecting the contribution prevented the load vector from being passed to the stiffness assembly routine. Convergence did not appear to be greatly affected.
The out-of-plane geometric stiffness effects are included when updating the resid- ual force vector, summing the effects at the master nodes. For the gravity loads of the previous subsection, the force couples are applied to the master nodes based on relative displacements of gravity columns at adjacent floors. These displacements are determined by transforming the master node displacements to the gravity col- umn locations based on rigid body motion (equation 4.19). For frames, the force couples are determined from components of displacement perpendicular to the orig-
inal orientation of the frame (figure 4.6). This is consistent with other assumptions, including the constraint stiffness that also uses the original frame orientation. A