Computer Analysis

2

E is the modulus of elasticity of the core 2 I is the moment of inertia of the core A is the area of the perimeter columns L is the height of the building

x is the location of truss measured from the top d is the out-to-out distance of columns

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

Y M L x L x

M= x( −EI)( + ) 2

From our definition, the optimum location of the belt truss is that location for which the deflection YM is a maximum. This is obtained by differentiating Equation (??) with respect to x and equating to zero.

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

1.9.2.3  Computer Analysis

In the preceding discussion, several assumptions were necessary to simplify the problem for hand

56 Structural Analysis and Design of Tall Buildings: Steel and Composite Construction

Belt truss Wind load varies from

20 psf at bottom to 26 psf at top

Outrigger trusses WF columns

typical

WF beams and girders typical Braced

core

(a)

5.18 m

4 at 7.62 m = 30.48 m

6 at 7.62 m=45.72 m (6 at 25 ft=150 ft)

(4 at 25 ft = 100 ft)

5.18 m (17 ft) (17 ft)

Linearly increasing load 26 psf

20 psf (b)

Core (linearly increasing moment of inertia)

46 stories

Columns

(linearly decreasing area up the height)

FIGURE 1.54  Improved model of a single outrigger located at distance x from top: (a) building plan and (b) schematic elevation.

Lateral Load Resisting Systems for Steel Buildings 57

For example:

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

• The cross-sectional areas of both the exterior columns and interior core columns typically reduce up the building height. A linear variation is perhaps more representative of a practi-cal building column, particularly so for a tall building of say 40-plus stories.

• As the areas of core columns decrease up the height, so does its moment of inertia up the height.

Incorporating the aforementioned modifications aligns the analytical model somewhat closer to a practical structure, but renders hand calculations all but impossible. Therefore, a computer-assisted analysis has been performed on a representative 46-story steel building using the modi-fied assumptions previously mentioned. A schematic plan of the building, and an elevation of the improved model, subject to varying lateral loads are shown in Figure 1.54. The lateral deflec-tions at the building top are shown in a graphical format in Figure 1.55 for various outrigger locations.

The deflections shown are in a nondimensional format and are relative to that of the core without the outrigger. Thus, the vertical ordinate with a value of unity at the extreme right of Figure 1.55 is the deflection of the building without the restraining effects of the outrigger. The deflections that include the effect of the outriggers are shown in curve “S.” This curve 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.

It is seen that lowering the outrigger down from the top location decreases the building

58 Structural Analysis and Design of Tall Buildings: Steel and Composite Construction

“optimum  location” only reduces its efficiency. Observe that this level is at distance of 0.435L from the top, very close to the optimum location of x = 0.455L derived earlier in Section 1.9.2.1 for the building with uniform characteristics. Furthermore, it can be seen from Figure 1.55 that the efficiency of the outrigger placed at mid-height, 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 mid-height.

Observe that when the outrigger is at the top, the building drift is reduced to nearly half the deflection of the unrestrained core. A rather impressive reduction indeed, but what is more impor-tant is that the deflection continues to reduce as the outrigger is lowered from level 46 downward.

The deflection parameter reaches a minimum value of 0.25 as shown in Figure 1.55 when the outrig-ger is placed at the optimum location, level 26. Further lowering of the outrigoutrig-ger will not reduce the drift, but increases 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 building.

1.9.2.4  Conclusions

• Given a choice, the best location for a single outrigger is at mid-height of the building.

• An outrigger placed at top, acting as a cap or hat truss is about 50% less efficient than that placed at mid-height. 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 mid-height, the benefits of a cap truss are nevertheless quite impressive, resulting in up to a 50% reduction in building drift.

To better appreciate the benefits of belt and outriggers, let us revisit the graph shown in Figure 1.55.

Instead of the nondimensional deflection parameter, DI, we will use drift index criteria, Δ_t/H, to reveal the efficiency of the system.

The building analyzed is a 600 ft tall, 46-story building with a floor-to-floor height of approxi-mately 13 ft. Let us assume, for purposes explaining the graphs, that the building without the outrigger and belt truss, experiences a deflection 55 in. at top. The drift index Δ_t/H is equal to

55 12 600