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Structural DesignSlender Concrete Column Design In Sway Frames

Professor Louie L. Yawc Draft date April 7, 2005

Introduction

Slender column design for sway frames may be accomplished by one of three methods. Thesemethods are specically named in the concrete code, ACI 318-02. The three methods are:

1. Second Order Computer Analysis

2. Direct P- Analysis

3. ACI Sway Moment Magnier Method.

Each of these methods will be described herein. However, several items of note must bestated before we proceed with the explanations for methods 1, 2 and 3.

Load Combinations and Strength Reduction FactorsThe ACI code allows a choice for the set of load combinations and strength reduction factorsthat may be used for design of slender columns in sway frames. The set of load combinationsand stregnth reduction factors must be one of the following sets:

(i) Load combinations and strength reduction factors as given in ACI sections 9.2 and 9.3.

(ii) Load combinations and strength reduction factors as givin in ACI Appendix C.

The d FactorFor load combinations which include lateral loads the factor, d , shall be calculated asfollows:

d = maximum factored sustained shear within a storytotal factored shear in the story. (1)

For load combinations which include gravity loads only the factor, d , shall be calculatedas follows:

d =factored dead load for a storytotal factored load in the story

. (2)

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Structural Design Sway Frame Slender Column Design

The Stability Index, QThe stability index can be used to determine if a particular story in a frame structure should

be called braced or unbraced. The stability index may be calculated as

Q =P u 0

V u u. (3)

Where,P u = the sum of factored vertical loads for the story in question

0 = the 1st order lateral deection of the top of the story relative to the bottom of thestory

u = story heightV u = the total shear acting on the given story.

The story is considered braced if Q 0.05. However, the story is considered unbraced if Q > 0.05.

Calculating EI The formula(s) for calculating EI are specied by ACI. The appropriate formula for cal-culating EI depends on the context in which the value of EI is to be used. Two casesapply.

Case 1 - Calculating EI to determine P c. Under this case EI may be calculated as

EI =0.2E C I g + E s I se

1 + d (4)

,or

EI =0.4E C I g1 + d

. (5)

Case 2 - Calculating EI to determine values, or for use as section property information tobe input into a computer analysis program. For this case the value of EI is calculatedper ACI section 10.11.1.

(EI )beams =0.35E C I g

1 + d(6)

(EI )columns = 0.7E C I g1 + d(7)

Note that for calculating the value of d may be taken as zero. Also, when EI is sodetermined for frames with temporary lateral loads, such as wind or seismic, d will bezero.

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Slender Column AnalysisA slender column analysis is sometimes necessary. The whole point of the analysis is to

determine the magnied moments acting on the column due to the applied factored loadsand slenderness effects. The necessary analysis may be accomplished in one of three waysas previously mentioned above. We now describe, in detail, the procedure for applying eachmethod.

1. Second Order Computer Analysis - Required when k ur > 100

a. Lateral Load Analysis. In this method the analysis of the frame being designed isperformed using a 2nd order computer analysis. A 2nd order computer analysis takesinto accout P effects automatically. To do this analysis each necessary load

combination which includes lateral loads must be used. From the computer analysesthe load combination which causes the worst end moments will be used for design.Since the computer is doing a second order analysis the moments calculated will alreadybe magnied. Hence, the values of P u , M u 1 and M u 2 can be read directly from thecomputer output. This method is the most accurate method of analysis.

b. Special Consideration. None necessary for this method.

c. Gravity Load Stability Check. To insure that gravity loads do not cause an unstablesituation ACI requires a special check for gravity load stability for the frame beingdesigned. Note that under this stability check, since we are using gravity loads, d

will be calculated per equation (2) given above. This special check is carried out asoutlined below.

i. Use gravity load combination 1 .4D + 1 .7L.ii. With the gravity load combination use any reasonable lateral load. A lateral load

of 0.005 times the total story gravity load may be used.

iii. Using loads (i) and (ii) together run a 1st order computer analysis and determinethe 1st order relative lateral deection for the story being designed. For example,using any column in the story, the story deection will be the top column jointlateral deection minus the bottom column joint lateral deection. This is thedeection of just the story under consideration. We call this deection 1 .

iv. Using loads (i) and (ii) together run a 2nd order computer analysis and determinethe 2nd order relative lateral deection for story being designed. We call thisdeection 2 .

v. Calculate the ratio 2 1 . If this ratio exceeds 2.5 the design must be changed. Forexample, the column cross-section sizes used in the story will need to be increasedor the story needs to be braced.

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2. Direct P Analysis

a. Lateral Load Analysis. To determine the magnied moments the following proceduremay be used. During this process d is generally taken as zero.

i. For the worst load combination that includes lateral loads a 1st order frame anal-ysis must be done in such a way to obtain nonsway and sway moments. Suchmoments need to be taken from the computer analysis at the top and bottom joints of the column being designed. The moments will be factored and we callthem M 1 ns and M 1 s for the bottom joint end moments and M 2 ns and M 2 s for thetop joint end moments of the column being designed. Be careful to maintain theappropriate signs on these moments when you extract them from the computeranalysis.

ii. The stability index, Q, must be calculated per equation (3) and is based on theworst load combination determined during the process of step (i).

iii. Calculate s by using the following formula:

s =1

1 Q 1.5. (8)

If s is greater than 1 .5 then analysis method 1 or 3 must be used. Alternatively,the column being designed would need to be redesigned with a bigger cross-section.Note that this limitation on s is exceeded when Q becomes greater than 1 / 3.

Also, remember that s is never to be used less than 1 .0.iv. Calculate the magnied moments as

M 1 = M 1 ns + s M 1 s (9)

M 2 = M 2 ns + s M 2 s (10)

The larger absolute moment, M 1 or M 2 , shall be used for design of the columnunder consideration. This larger moment is usually called M 2 and it is a factoredmoment.

b. Special Consideration. The maximum moment may occur between the ends of thecolumn being designed. This is ordinarily not the case for columns in sway frames. Forsway frames the maximum column moment usually occurs at one end of the column.However, under certain conditions this may not be the case. Hence ACI requires thatwe check for such a condition. Such a condition occurs when

u

r>

35

P uf c A g. (11)

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If equation (11) is true then the column must be designed as a nonsway column basedon

M c = ns M 2 (12)ns =

C m1 P u0 .75 P c

1 (13)

C m = 0 .6 + 0.4M 1M 2

0.4 (14)

with M 1 and M 2 calculated by using equations (9) and (10) and s calculated perequation (8). The factor d (proabably zero) is dened per the load combinationunder consideration and k is dened for a column in a non-sway frame and will likelyneed to be determined by using calculated values. Once M c is calculated the columnis designed for P u and the factored moment M c.

c. Gravity Load Stability Check. MacGregor and Wight (4th edition) indicate on page583 that this check is unnecessary when s M s has been computed by using the directP method of analysis. Hence, since our present method of analysis is the directP method, we do not have to do a gravity load stability check here.

3. ACI Sway Moment Magnier Method

a. Lateral Load Analysis. To determine the magnied moments the following proceduremay be used. During this process d is generally taken as zero.

i. For the worst load combination that includes lateral loads a 1st order frame anal-ysis must be done in such a way to obtain nonsway and sway moments. Suchmoments need to be taken from the computer analysis at the top and bottom joints of the column being designed. The moments will be factored and we callthem M 1 ns and M 1 s for the bottom joint end moments and M 2 ns and M 2 s for thetop joint end moments of the column being designed. Be careful to maintain theappropriate signs on these moments when you extract them from the computeranalysis.

ii. Calculate P u for the story of the column being designed.

iii. Calculate P c for each column in the story of the column being designed. Thencalculate P c for the given story.iv. Calculate s .

s =1

1 P u0 .75 P c(15)

v. Calculate the magnied moments per equations (9) and (10), but in this caseuse s as calculated by equation (15). The larger absolute moment, M 1 or M 2 ,shall be used for design of the column under consideration. This larger momentis usually called M 2 and it is a factored moment.

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b. Special Consideration. Follow the same procedure as given in this handout in section2(b) except for the following:

Use s as determined in this section. Use M 1 and M 2 as determined in this section.

c. Gravity Load Stability Check. To insure that gravity loads do not cause an unstablesituation ACI requires a special check for gravity load stability for the frame beingdesigned. Note that under this stability check, since we are using gravity loads, dwill be calculated per equation (2) given above. This special check is carried out asoutlined below.

i. Use gravity load combination 1 .4D + 1 .7L to calculate P u for the story of the

column being designed. Also, calculate P c for the columns in the story of thecolumn being designed. The value of d used shall correspond to the gravity loadcombination being used.

ii. From P u and P c obtained in (i) above, calculate s as determined in equation(15). The value of s so determined shall not exceed 2.5. If it does, the story of the frame under consideration is unstable and needs to be redesigned. This likelywill require that bigger column cross-sections be used.

Conclusion. After having completed one of the three methods of analysis listed above theconcrete column cross-section is typically designed using an interaction diagram as is donefor short columns. Of course in this case we do the design using the governing values of factored axial load and factored moment as determined from one of our three methods of slender column analysis.References

American Concrete Institute, ACI 318-02, Building Code Requirements for Structural Concrete and Commentary , 2002.

MacGregor and Wight, Reinforced Concrete - Mechanics and Design , 4th edition, Prentice-Hall, 2005.

Nilson, Darwin, and Dolan, Design of Concrete Structures , 13th edition, McGraw-Hill,2004.

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