Two assumptions concerning the size of angles are made in the three-dimensional stiffness formulations. First, the total angle of twist of a building in the plane of a floor is assumed to be small. This will be called the small angle assumption. It affects the constraint equations and the P-.6. effect of out-of-plane displacement of frame columns. Second, the incremental rotation of a master node is assumed to be small. This will be called the small rotation assumption. It affects the shear spring stiffness, the P-.6. effect of gravity loads not carried by frames and the story-damping formulation.
4.5.1 Small Angle Assumption
The frame module is a two-dimensional entity which provides resistance only in the original plane of the frame. Nodal updating in this plane accounts for moment
amplification (member buckling) and in-plane P-~. For three-dimensional analysis, the constraints connecting frames together use the original orientation of the frame, (3, ana the fixed perpendicular distance to the master node, R. A frame will assemble in its local in-plane direction regardless of whether one story has rotated relative to another. Since these geometric quantities are not updated, the constraint equations are linear. Implicit here is the assumption that the total rotation of the building is small. This assumption is not detrimental to the solution since rotations will not exceed a few degrees for the variety of buildings investigated in this work.
The P-~ contribution of frame columns in their out-of-plane direction uses the same small angle assumption of the constraint equations: the out-of-plane direction is assumed to be (3
+
1r /2, without updating the current angle. The current orien- tation is used to determine locations of frame columns, but the assembly assumes the original orientation to determine the direction of the load, so there will be some error due to cumulative rotations. However, the out-of-plane P-~ effect is secondary in nature. There will be a small error in the orientation of force couples that are due to the out-of-plane displacement of frame columns only. In most cases, this will be a small error (in angle) of a small fraction (out-of-plane frame columns vs. in-plane frame columns plus gravity columns) of a secondary effect (P-~ forces). Further attention is not warranted based on the level of accuracy of other assumptions being used in the program.4.5.2 Small Rotation Assumption
The stiffness formulations in this work are for nonlinear analysis with a time-stepping solution. Small increments in displacements occur in each step, so displacement transformations take this into account. Specifically, the displacement transformation used for the shear springs (eq. 4.19) assumes the rotational increment is small.
The same transformation provides exact results for forces (eq. 4.24). The P-~
contribution of loads carried by gravity columns is assembled to the master nodes and uses the same small rotational increment assumption used for the shear spring contributions. The damping formulation to be developed later in the chapter also
uses the small rotation assumption.
The small rotation assumption has been used commonly to linearize the equa- tions of motion (Weaver and Nelson 1966; Wilson and Allahabadi 1975; Wakabayashi 1986; Chopra 1995) by removing trigonometric functions of the rotations from trans- formation equation. This assumption is valid for the nonlinear inelastic problems solved here. The program solves iteratively for small time steps, so the incremental displacements are small and the approximation will not lose any accuracy. Some buildings are eccentric and exhibit torsional response from the lateral excitations, but even in the inelastic range for severe records the cumulative rotations are not large enough to affect accuracy with the small rotation assumption being used for the increments.
If small rotations are not assumed, the transformation of displacements from the master node to any location c will be of the form
(4.44)
Llxc
=
LlxMN+
XcMN cos(LlOMN)- ycMN sin(LlOMN)-XcMN Llyc=
LlyMN+
XcMN sin(LlOMN)+
ycMN cos(LlOMN) _ ycMN Llec=
LleMNwhere xcMN = Xc - XMN and ycMN = yc- yMN (figure 4.5). If the small rotation assumption is used, the transformation is equation (4.19), repeated here in equation form for comparison.
Llxc = LlxMN- ycMN sin(LleMN) Llyc
=
LlyMN+
xcMN sin(LlOMN) Llec=
LleMNA pure torsion example can quantify the error associated with the linearized equations for small rotations. Assume a point is being rotated about the master node with the starting point at (Xo, Yo)= (1,0) and the master node located at (XMN, yMN) = (0, 0) (figure 4.9). If equal incremental twists occur each step until a total rotation of II/ 4 radians is achieved, then there will be a cumulative error
X,Y
~
xMNyMN ,
Figure 4.9 Pure torsion example.
!)..() Cumulative Cumulative Average Average
(xii) Error in X Error in Y Error in X Error in Y
0.050 7.0% 5.6% 1.4% 1.1%
0.025 3.3% 3.0% 0.33% 0.30%
0.005 0.63% 0.61% 0.013% 0.012%
Table 4.1 Small rotation error for large twist.
based on the increment size. The results for several increment sizes are shown in table 4.1.
The total twist is 45°, which is much larger than most buildings will achieve before collapse. The incremental rotations 0.05II, 0.025II and 0.005II radians cor- respond to go, 4.5° and 0.9°, respectively. Considering the small time steps used for time history analysis in this work, 1° would still exceed the maximum rotation observed in a time step. If the incremental rotations were this size and occurred monotonically as in the above example to always add to the cumulative error, a total twist of 45° will still have less than 1% total error.
The small rotation assumption is concerned with increments, not with total rotations. If the increments were small enough, this assumption would be valid for large total rotations of building floors. The small angle assumption is concerned with the total angle, so even if the small rotation assumption does not introduce large cumulative errors, the constraint equations and out-of-plane P-D.. forces can have large errors for large total rotations.