## Optimum Location of Two Outriggers

In the preceding analyses, only one compatibility equation was necessary because the one-outrigger structure is once redundant. On the other hand, a two-outrigger structure is twice redundant, requiring a solution of two compatibility equations.

As before, the sectional areas of the exterior columns and the moment of inertia of the core are assumed to decrease up the height. A trapezoidal distribution is assumed as before, for the lateral load. Schematic elevation and a conceptual analytical model used in the analysis are shown in Figs. 3.27 and 3.28.

The method of analysis for calculating the deflections at the top is similar to that used for the previous example. The moments at the outrigger locations are chosen as the unknown arbitrary constants M1 and M2, and the structure is rendered statically determinate by removing the rotational restraints at the outrigger locations. Next, the compatibility equations for the rotations at the truss locations are set up and solved simultaneously to obtain the values to M1 and M2. The final deflection at the top is obtained by a superposition of the deflection due to the external load and a counteracting deflection due to the moments M1 and M2.

The magnitude of the deflection at the top is given for three types of buildings by assuming that the lateral loads are resisted by: 1) the core alone; 2) the core acting together with a single outrigger; or 3) the core acting in conjunction with two outriggers. (See Fig. 3.29.)

As before, the vertical ordinate shown with a value of unity is the deflection index at the top derived by neglecting the restraining effect of outriggers. The resistance is provided by the cantilever action of the braced core alone. Curve S represents the top

Figure 3.29. Deflection index versus level of outrigger locations.

Deflection at top w/o outriggers

Figure 3.29. Deflection index versus level of outrigger locations.

Deflection at top w/o outriggers

Note: Deflection index =

Deflection at top with outriggers deflection of the core restrained by a single outrigger located anywhere up the height of the structure.

The curves designated as 4, 8,..., 46 represent the deflections at the top for two outriggers located anywhere up the height of the structure. To plot each curve, the location of the upper outrigger was considered fixed in relation to the building height, while the location of the lower outrigger was moved in single-story increments, starting from the floor immediately below the top outrigger.

The number designations of the curves represent the floor number at which the upper outrigger is located. The second outrigger location is shown by story levels on the vertical axis. The horizontal distance between the curves and the vertical axis is the relative building drift for the particular combination of truss locations given by the curve designation and the story level. For example, let us assume that the relative deflection at the top is desired for a combination (20, 15), the numbers 20 and 15 representing the floors at which the upper and lower outriggers are located. To find the deflection index for this particular combination, the procedure is to select the curve with the designation 20, go down the vertical axis to level 15, and draw a horizontal line from this level to curve 20. The required relative top deflection is the horizontal distance between level 15 and curve 20 (distance HH1 in Fig. 3.29). Similarly, the length KK1 gives the relative deflection at the top for the combination (28, 4). It is seen from Fig. 3.29 that the relative location of the trusses has a significant effect on controlling the drift. Furthermore, it is evident that a deflection very nearly equal to the minimum can be achieved by placing the trusses at levels other than at their optimum locations. For the example building, a relative deflection of 0.15, which differs negligibly from the optimum value of 0.13, is achieved by placing the outriggers at (40, 23), (32, 33), etc.

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