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Building Design and Construction Handbook - Merritt - Ventech!

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CONCRETE CONSTRUCTION 9.127<br />

TABLE 9.25 Maximum Spacing of Column<br />

Ties*<br />

Vertical<br />

bar<br />

size,<br />

number<br />

5 10<br />

6 12<br />

7 14<br />

Size <strong>and</strong> spacing of<br />

ties, in<br />

No. 3 No. 4 No. 5<br />

8 16 16<br />

9 18 18<br />

10 18 20<br />

11 † 22 22<br />

14 † 24 27<br />

18 † 24 30<br />

* Maximum spacing not to exceed least column<br />

dimensions.<br />

† Not allowed.<br />

For most unbraced frames, an investigation will be required to determine the<br />

magnification factor to allow for the effects of sidesway <strong>and</strong> end rotation. The<br />

procedure for determination of the required increase in primary moments, after the<br />

determination that slenderness effects cannot be neglected, is complex. For direct<br />

solution, the requirements of Sec. 10.10, ACI 318-99 can be met by a P-� analysis.<br />

(See, for example, J. G. MacGregor <strong>and</strong> S. E. Hage, ‘‘Stability Analysis <strong>and</strong> <strong>Design</strong><br />

of Concrete,’’ Journal of the Structural Division, ASCE, Vol. 103, No. ST10,<br />

October 1977.)<br />

The direct P-� method of MacGregor <strong>and</strong> Hage is based upon an equation for<br />

a geometric series that was derived for the final second-order deflection as a function<br />

of the first-order elastic deflection. This direct P-� analysis provides a very<br />

simple method for computing the moment magnifier � when the stability index Q<br />

is greater than 0.04 but equal to or less than 0.22.<br />

� � 1/(1 � Q) 0.04 � Q � 0.22 (9.102)<br />

where Q � Pu�u/(Huhs) Pu � sum of the factored loads in a given story<br />

�u � elastically computed first-order lateral deflection due to Hu (neglecting<br />

P-� effects) at the top of the story, relative to the bottom<br />

Hu � total factored lateral force (shear) within the story<br />

hs � height of story, center-to-center of floors or roof<br />

The approximate method of ACI 318-99 may also be used to determine the<br />

moment magnifier. This approximate method is a column-by-column correction<br />

based upon the stiffness of the column <strong>and</strong> beams, applied primary design column<br />

end moments, <strong>and</strong> consideration of whether the entire structure is laterally braced<br />

against sidesway by definition. (See ACI 318-99, Sec. 10.11).<br />

The ACI 318 <strong>Building</strong> Code permits slenderness effects to be neglected only<br />

for very short, braced columns, with the following limitations for columns with<br />

square or rectangular cross-sections:

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