## Ei

The compatibility relation at x is given by W M M

6EI EI Kx where

W = intensity of the wind load per unit height of the structure

Mx = the restoring moment due to outrigger restraint

Kx = spring stiffness at x equal to AE/(L - x) x (d2/2)

Figure 3.23b. Conceptual model for a single outrigger and belt truss model. Restraining "spring" occurs at a distance x from top.

E = modulus of elasticity of the core I = moment of inertia of the core A = area of the perimeter columns L = height of the building x = location of truss measured from the top d = distance out-to-out of columns

Next, obtain the deflection at the top due to Mx:

M 2EI

From our definition, the optimum location of the outrigger is that location for which the deflection YM is a maximum. This is obtained by substituting for Mx from Eq. (3.27) into Eq. (3.28) and differentiating it with respect to x and equating to zero. Thus, dy/dx of

Simplifying this equation, we get a cubic equation in x.

This cubic equation has a single positive root, x = 0.455L.

Therefore, to minimize drift, a single outrigger must be located at a distance x = 0.455 L from the top or, say, approximately at midheight of the building.

In the discussion, several assumptions were made to simplify the problem for hand calculations. However, in a practical building, many of these assumptions are rarely satisfied. For example:

• The lateral load does not remain constant up the building height. It varies in a trapezoidal or a triangular manner, the former representative of wind loads and the latter, seismic loads.

• The cross-sectional areas of both the exterior and interior columns typically reduce up the building height. A linear variation is perhaps more representative of a practical building column.

• As the areas of core columns decrease up the height, so does the moment of inertia of the core. Therefore, a linear variation of the moment of inertia of the core, up the height is more appropriate.

Incorporating the aforementioned modifications aligns the analytical model closer to a practical structure, but renders the hand calculations all but impossible. Therefore, a computer-assisted analysis has been performed on a representative 46-story building using the modified assumptions previously mentioned. A schematic plan of the building, and an elevation of the idealized structural system and lateral loading are shown in Figs. 3.24 and 3.25. The lateral deflections at the building top are shown in a graphical format in Fig. 3.26 for various outrigger locations.

The deflections shown in a nondimensional format in Fig. 3.26 are relative to that of the core without the outrigger. Thus, the vertical ordinate with a value of unity at the extreme right of Fig. 3.26 is the deflection of the building without the restraining effect of the outrigger. The deflections including the effect of the outriggers are shown in curve 'S.' It is obtained by successively varying the outrigger location starting at the very top and progressively lowering its location in single-story increments, down through the building height.

Figure 3.26. Deflection index versus level of outrigger location.

Deflection at top w/o outrigger

Figure 3.26. Deflection index versus level of outrigger location.

Deflection at top w/o outrigger

Note: Deflection index =

### Deflection at top with outrigger

It is seen that lowering the outrigger down from its top location decreases the building drift progressively until the outrigger reaches level 26. Moving it either above or below this "optimum location" only reduces its efficiency. Observe that this level is at distance (46 - 26/46)L = 0.435L from the top, very close to the optimum location of x = 0.455L for the building with uniform characteristics. Furthermore, it can be seen from Fig. 3.26 that the efficiency of the outrigger placed at midheight; that is, at level 23, is very close to that when it is at the optimum location. Therefore, as a rule of thumb, the optimum location for a single outrigger may be considered at midheight.

Observe that when the outrigger is at the top, the building drift is reduced to nearly half the deflection of the unrestrained core. Thus, for example, if the drift of the unrestrained core is, say, 20'' at the top, the corresponding deflection with an outrigger at level 46 is reduced to 0.48 x 20 = 9.6 in. A rather impressive reduction indeed, but what is more important is that the deflection continues to reduce as the outrigger is lowered from level 46 downward. It reaches a minimum value of 0.25 x 20 = 5 in. as shown in Fig. 3.26 when the outrigger is placed at the optimum location, level 26. Further lowering of the outrigger will not reduce the drift, but increase it. Its beneficial effect vanishes to nearly nothing when placed very close to the bottom of the building, say, at level 2 of the example problem.

Using the results of the example problem, the following conclusions can be drawn:

• Given a choice, the best location for a single outrigger is about midheight of the building.

• An outrigger placed at the top, acting as a cap or hat truss, is about 50% less efficient than that placed at midheight. However, in many practical situations, it may be more permissible to locate the outrigger at the building top. Therefore, although not as efficient as when at midheight, the benefits of a cap truss are nevertheless quite impressive, resulting in up to a 50% reduction in building drift.

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