Slender Column Design

Slender Column Design Based on ACI 318-08 By Mahmoud E. Kamara, Ph.D., and Lawrence C. Novak, S.E., SECB, LEED AP

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Professional Development Series Slender Column Design Based on ACI 318-08 By Mahmoud E. Kamara, Ph.D., and Lawrence C. Novak, S.E., SECB, LEED AP

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ith advancements in concrete technology and increased use of high-strength concrete, smaller column cross sections are being designed for a given load combination. Also, with new innovations in architectural and structural systems, the demand on columns that span vertically more than one traditional floor height is increasing. New materials, advanced methods of analysis, and innovative structural systems result in materials saving and optimum use of the building space, which leads to more sustainable design. Design of slender columns requires the understanding of the exact behavior of such structural elements. The following article covers the behavior and design of slender columns. The procedure presented here reflects the provisions of the American Concrete Institute’s Building Code Requirements for Structural Concrete (ACI 318-08). All section and equation numbers within this article refer to ACI 318-08.

Background A slender column is defined as a column whose strength is reduced by second-order deformations (secondary moments). Considering a column subjected to a load, P, with a constant eccentricity, e, the lateral deformation of the column axis

causes an increase of ∆ in the original eccentricity of the load at the critical section (see Figure 1). This causes an increase in the bending moment at the critical section known also as P∆ effect. A column deflects due to the effect of buckling and/or the sidesway of the frame containing the column. The terms slender and non-slender columns have been used interchangeably with the terms long and short columns. While a short column may fail because of a combination of moment and axial load that exceeds the strength of the cross section (material failure), a slender column with the same cross section may fail at a lower magnitude of P because of the additional moments resulting from the deflected shape of the column. If a column is very slender, it may reach a deflection due to the effect of the axial load additional moment (P∆) such that deflections will increase indefinitely with the increase in the load. This type of failure is known as a stability failure. Figure 1 shows the three modes of failures for short and slender columns. The theoretical maximum axial load that a column can support when it is at the threshold of buckling is called the critical load. The Swiss mathematician Leonard Euler developed the equation to calculate the critical load for axially loaded columns 200 years ago. The Euler equation states

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Learning Objectives This article discusses slender column design. Upon reading the article and completing the quiz, the reader should be able to understand the behavior and design of reinforced concrete slender columns. The article presents the American Concrete Institute’s Building Code (ACI 318-08) design provisions for slender columns. All referenced items are from ACI 318-08, unless noted otherwise. Also, all notations and definitions in the article are in accordance with Chapter 2 of ACI 318-08.

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Slender Column Design Based on ACI 318-08 that a column will fail by buckling at the critical load of: Pc = π2EI/(,e )2 where EI is the flexural stiffness of the column crosssection; and ,e is the column effective length, which is equal to k ,u , where k is the effective length factor and ,u is the unsupported length of the column. For a “stocky” short column, the value of the buckling load will exceed the direct crushing strength, corresponding to material failure.

by evaluating the stability index, Q (Section 10.10.5.2). A column within a story may be considered nonsway if the stability index for that story is less than or equal to 0.05. The stability index, Q, is calculated as follows: Q=

•

Sway versus nonsway columns Bracing elements in building structures (shear walls or lateral bracing) help reduce excessive sway and minimize the secondary effects on columns. The behavior of a compression member differs depending on whether the member is a part of a sway or nonsway frame. Because of this difference in behavior between sway and nonsway columns, the design is treated differently. As a simplified approach, 10.10.1 permits the compression member to be considered braced against sidesway when the bracing elements have a total stiffness, resisting the lateral movement of a story, of at least 12 times the gross stiffness of the columns within the same story. According to Section 10.10.5.1, a column can be assumed as nonsway if the increase in column end moments due to second-order effects does not exceed 5 percent of the firstorder end moments. Another alternate to evaluate whether a story within a structure is sway or nonsway, for stories where the factored horizontal shear is greater than zero, is

• •

∑Pu ∆o Vus c

≤ 0.05

where: ∑Pu and Vus are the total factored vertical and horizontal story shear in the story being evaluated (from the same load combination) respectively; ∆o = the first order relative lateral deflection between the top and bottom of the story due to Vus ; and ,c = length of the column in a frame measured center to center of the joints in the frame.

Design approach Design of the columns can be based on moment and axial forces from an elastic second-order analysis (10.10.4). To calculate accurate values for the deflections and additional moments, such an analysis must consider the actual stiffnesses of the frame members, the effect of cracking, and the effects of load duration.

ACI approach Slenderness Limits — Lower-bound slenderness limits are given, below which secondary moments may be disregarded and only axial load and moment from first-order analysis need be considered to design a column cross-

Figure 1: Strength of short and slender columns

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ADVERTISEMENT section and reinforcement. For columns in a nonsway frame, effects of slenderness may be neglected when k ,u /r is less than or equal to 34 – 12(M1 /M2 ), where M2 is the larger end moment and M1 is the smaller end moment. The ratio M1/M2 is considered positive if the column is bent in single curvature, and considered negative if bent in double curvature. M1 and M2 are factored end moments obtained by an elastic frame analysis, and the term [34 – 12(M1 /M2 )] must not be taken greater than 40. For nonsway columns bent in double curvature with equal end moments (M1 = M2 ), slenderness effects may be neglected if ,u /h ≤ 12 for rectangular columns (assuming k = 1), where h is the smaller dimension of the column cross section. This constitutes a large percentage of columns in buildings. For compression members in a sway frame, effects of slenderness may be neglected when k ,u /r is less than 22. The total moment, including the second-order effects in a column, must not exceed 1.4 times the moment due to first-order effects (Section 10.10.2.1). If 10.10.2.1 is not satisfied, the structural design must be revised. Methods of Analysis — The Code recognizes the following three methods to account for slenderness effects: 1) Nonlinear second-order analysis (Section 10.10.3).

The demand on columns that span vertically more than one traditional floor height is increasing.

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Slender Column Design Based on ACI 318-08 In this analysis, consideration must be made for material nonlinearity, member curvature and lateral drift, duration of loads, shrinkage and creep, and interaction with the supporting foundation. For this type of complex analysis, tools are typically currently not readily available to the design engineer. 2) Elastic second-order analysis (Section 10.10.4). In this analysis, consideration must be made for the influence of axial loads, the presence of cracked regions along the length of the member, and the effects of load duration. 3) Moment magnification procedure (Section 10.10.5). An approximate analysis of slenderness effects based on a moment magnifier (see 10.10.6 and 10.10.7) is permitted. In this method, moments computed from first-order analysis are multiplied by a moment magnifier to account for the second-order effects. The moment magnifier is a function of the factored axial load, Pu , and the critical buckling load, Pc , for the column. This method is discussed in detail in sections 10.10.6 and 10.10.7 for nonsway and sway columns, respectively. Figure 2 provides a flowchart for the code analysis for slender columns. The following sections of this article present the ACI 318-08 moment magnification procedure. While the ACI approximate moment magnification procedure focuses primarily on magnifying the end moments, the maximum moment in a column may occur between its ends. Section 10.10.2.2 requires that second-order effects be considered along the length of compression members. For both sway and non-sway columns, this requirement can be accounted for by the moment magnification procedure outlined in 10.10.6.

Unsupported and effective length of column The unsupported length, ,u , of a column is the clear distance between lateral supports as shown in Figure 3. It is important to note that the length ,u may be different for buckling about each of the principal axes of the column cross-section. The basic equations for the design of slender columns were derived for columns with hinged ends, and

thus, must be modified to account for the effects of end restraint. Effective column length, k ,u , is the length used to estimate slender column strength, and considers end restraints as well as nonsway and sway conditions. For columns in a nonsway frame, the effective length k ,u falls between ,u /2 and ,u , where ,u is the actual unsupported length of the column. For columns in a sway frame, the effective length k ,u is always greater than the actual unsupported length of the column, ,u . Use of the alignment charts given in the ACI 318-08 Figure R10.10.1.1 allows graphical determination of the effective length factors for compression members in nonsway and sway frames

Radius of gyration In general, the radius of gyration, r, is: Ig Ag where Ig and Ag are the moment of inertia and the area for the gross section, respectively. As a simplification, r may be taken as 0.30 times the dimension in the direction of analysis for a rectangular section and 0.25 times the diameter of a circular section.

Section properties for frame analysis The frame analysis must consider section properties determined taking into account the influence of axial loads, the presence of cracked regions along the length of the member, and the effects of load duration. To account for the presence of cracked regions, the member stiffness is multiplied by a stiffness reduction factor. Table 1 summarizes the reduced stiffness for different structural members as given in Section 10.10.4. It is important to note that for service load analysis of the structure, it is satisfactory to multiply the moments of inertia given in Table 1 by 1/0.70 = 1.43 (R10.10.4.1). As an alternate to the stiffness values in Table 1, the Code provides more refined values for EI (Equations 10-8 and 10-9) to account

Table 1: Reduced stiffness for different structural members for ultimate load analysis Structural Member

Moment of inertia, I

Beams

0.35 Ig

Columns

0.70 Ig

Walls — uncracked

0.70 Ig

Walls — cracked

0.35 Ig

Flat plates and flat slabs

0.25 Ig

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Area

1.0 Ag


for axial load, eccentricity, reinforcement ratio, and concrete compressive strength. The stiffnesses calculated from these two equations are applicable for all levels of loading. For compression members:

(

I = 0.80 + 25

Ast Ag

)(

1–

Mu Pu h

– 0.5

Pu Po

)

Ig ≤ 0.875Ig (Equation 10-8)

For flexural members:

(

I = (0.10 + 25 ρ) 1.2 – 0.2

bw d

)

Ig ≤ 0.5Ig

For compression and flexural members, I need not be taken less than 0.35Ig . For continuous members, I may be taken as the average for the values calculated from Equation 10-9 for the positive and negative moment cross sections. To account for the presence of sustained lateral loads, the moment of inertia, I, for compression members should be divided by (1+ βds ), where βds is the ratio of maximum factored sustained shear within a story to the maximum factored shear in that story associated with the same load combination. βds must not be taken greater than 1.0 (see Section 10.10.4.2 and R10.10.4.2)

(Equation 10-9)

Figure 2: Code provision for slender column


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Slender Column Design Based on ACI 318-08 ACI 318-08 moment magnification procedure Columns in nonsway frames — For each load combination, the factored moments at the top and bottom of the column are calculated using first-order frame analysis. For each column, the smaller and larger factored end moments are designated as M1 and M2 , respectively. The magnified moment, Mc (for each load combination), is calculated by multiplying the larger factored end moment, M2 , by a magnification factor for nonsway frame δns . Following is a summary for the code moment magnification equations for nonsway frame: Mc = δns M2

(Equation 10-11)

EI =

(0.2Ec Ig + Es Ise) 1 + βdns

(Equation 10-14); or

EI =

0.4Ec Ig 1 + βdns

(Equation 10-15)

Alternatively, EI may be calculated using the value of I from Equation 10-8 divided by (1+ βdns ). βdns is a factor to account for the reduction in the column stiffness due to the effect of sustained axial load:

βdns =

Maximum factored axial sustained load Maximum factored load associated with the same load combination

≤1

where: Cm δns = ≥ 1.0 Pu 1– 0.75Pc Pc =

π 2E I (ku)2

(Equation 10-12); and

(Equation 10-13)

Pc is the critical buckling load for the column. When calculating Pc , the effect of cracking, creep, and the nonlinear behavior of concrete on the stiffness, EI, can be accounted for using the following equations:

For heavily reinforced columns, Equation 10-15 underestimates the effect of the reinforcement, therefore Equation 10-14 is more accurate for those columns. It is important to note that of the above-mentioned three options to calculate EI, values obtained from Equation 10-8 divided by (1+ βdns ) are the most accurate. The term Cm is a factor relating the actual moment diagram to an equivalent uniform moment diagram and is calculated as follows: Cm = 0.6 + 0.4

M1 M2

Figure 3: Unsupported length, ,u

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(Equation 10-16)

where M1 /M2 is positive if the column is bent in single curvature and negative if bent in double curvature. For columns with transverse loads between support, Cm must be taken as 1.0. Figure 4 shows some values of Cm for different end moment cases. The factored end moment, M2 , must not be taken less than M2,min = Pu (0.6 + 0.03h), where h is the column cross section dimension in inches at the direction of analysis.

Columns in sway frames For each load combination, the factored nonsway moments, Mns , and the factored sway moments, Ms , are calculated at the top and bottom of the column using firstorder elastic frame analysis. The magnified sway moments are added to the unmagnified nonsway moments, Mns , at each end of the column. The magnified moments at each end of the column (M1 and M2 ) are calculated as follows: M1 = M1ns + δs M1s

(Equation 10-18)

M2 = M2ns + δs M2s

(Equation 10-19)

where δs is the moment magnification factor for frames not braced against sidesway. δs accounts for the effects of lateral drift resulting from lateral and gravity loads. The Code gives two alternate methods to calculate δs .

Alternate 1: δs =

1 ≥1 1–Q

(Equation 10-20)

If δs calculated by Equation 10-20 exceeds 1.5, the magnified moments must be calculated using second-order elastic analysis or Alternate 2. Alternate 2: 1

δs = 1–

∑Pu 0.75∑Pc

≥1

(Equation 10-21)

where ∑Pu is the summation of all factored load in the story and ∑Pc is the summation of the critical buckling loads for all sway-resisting columns in the story. Pc is calculated using Equation 10-13.

Conclusion With increasing use of high-strength concrete and new architectural innovation, demand on incorporating slender columns in design is increasing. The ACI 318 Code provides

Figure 4: Values of Cm for different end moment cases


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Slender Column Design Based on ACI 318-08 three alternates for slender column design: Nonlinear second-order analysis (10.10.3), elastic second-order analysis (10.10.4), and moment magnification procedure (10.10.5). The moment magnification procedure is an approximate simplified procedure and is introduced in this article. The ACI 318 methods introduced in Sections 10.10.3 and 10.10.4 are more suited for computer design and analysis. Although, the moment magnification procedures introduced in Section 10.10.5 are intended for hand calculations, software programs that implement these procedures from different sources are available. Commercially available analysis and design programs, especially when linked with a multidisciplinary building information model

(BIM) software, assist the structural engineer to work more efficiently, particularly considering today’s architecturally complex building geometries and the iterative nature of design of slender columns, including a high number of loading combinations.

REFERENCES • ACI Committee 318, 2008, Building Code Requirements for

Structural Concrete (ACI 318-08) and Commentary, American Concrete Institute, Farmington Hills, Mich. • Kamara, M.E., Novak, L. C., and Rabbat, B. G., editors,

2008, Notes on ACI 318-08 Building Code Requirements for Structural Concrete with Design Application, Portland Cement association, Skokie, Ill., EB708.

Mahmoud E. Kamara, Ph.D., is the senior structural engineer for the Portland Cement Association. He can be reached at [email protected]. Lawrence C. Novak, S.E., SECB, LEED AP, is the director of Engineered Buildings for the Portland Cement Association. He can be reached at [email protected].

• Alsamsam, I. M., and Kamara, M. E., 2004, Simplified Design

of Reinforced Concrete Buildings of Moderate Size and Heights, Portland Cement Association, Skokie, Ill., EB104, Third Edition.

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Structural Engineering & Design Professional Development Series Reporting Form Article Title: Slender Column Design Based on ACI 318-08 Publication Date: January 2010 Sponsors: Bentley Systems, Inc./Portland Cement Association

Valid for credit until: January 2012

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Quiz instructions On the Professional Development Series Reporting Form on page 10, circle the correct answer for each of the following questions.

1. The equation for the theoretical maximum axial load a column can support when it is at the threshold of buckling was developed by: a) Isaac Newton b) Timoshenko c) Leonard Euler d) Saint Venant 2. ACI 318 Code permits the column to be considered braced against sidesway if the ratio of the total stiffness of the bracing elements to the total stiffness of the columns within a story is equal to or greater than: a) 6

a) 10 b) 12 c) 22 d) 34 7. The ratio of the total moment including the second-order effect to the moment due to firstorder effects must not exceed: a) 1.1 b) 1.4 c) 1.5

b) 8

d) 2

c) 10 d) 12 3. ACI Code permits the column to be considered as nonsway if the stability index, Q, is less than or equal to: a) 0.05

8. For a column in sway frame with unsupported length ,u = 15 feet, which of the following could not represent the effective length k ,u ? a) 10 feet b) 16 feet c) 20 feet

b) 0.025

d) 1,000 feet

c) 0.02 d) 0.01 4. For a nonsway column with M1 = M2 (bent in single curvature), slenderness may be neglected if k ,u /r is less than or equal to: a) 46

9. For a column in nonsway frame with unsupported length ,u = 20 feet, which of the following could not represent the effective length k ,u ? a) 10 feet b) 16 feet

b) 40

c) 20 feet

c) 34

d) 1,000 feet

d) 22 5. For a nonsway column with M1 = -M2 (bent in double curvature), slenderness may be neglected if k ,u /r is less than or equal to:

6. For a sway column, slenderness may be neglected, if k ,u /r is less than or equal to:

10. Which of the following is not an ACI Coderecognized method to account for slenderness effect?

a) 46

a) Moment magnification procedure

b) 40

b) Moment redistribution procedure

c) 34

c) Elastic second-order analysis

d) 22

d) Nonlinear second-order analysis Professional Development Advertising Section — Bentley Systems, Inc.

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Slender Column Design

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