Cm factor for non-uniform moment diagram in RC columns

Engineering Structures 31 (2009) 1589–1599

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Cm factor for non-uniform moment diagram in RC columns L. Pallarés ∗ , J.L. Bonet, M.A. Fernandez, P.F. Miguel Departamento de Ingeniería de la Construcción, Universidad Politécnica de Valencia, Spain

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Article history: Received 10 January 2008 Received in revised form 11 September 2008 Accepted 11 February 2009 Available online 17 March 2009 Keywords: Concrete columns Equivalent factor Cm Transverse loads

abstract Generally, reinforced concrete columns are subjected to a non-uniform moment diagram due to unequal eccentricities at the two ends of the columns or to transverse loads along the columns. Most building codes and material specific codes allow the design of columns with the factor Cm , in order to simplify the general analysis of the column to approximate a column with equivalent uniform moment diagram. This paper presents expressions for Cm according to the type of load that is applied along the column. Two Cm expressions are proposed: the first one approaches the solution using a differential equation based on the elastic behavior of the columns; the second one is based on results obtained through a numerical simulation. This study is valid for hinged reinforced high and normal strength concrete columns, subjected to a sustained and short time load with compression and uniaxial bending. The proposed method can be used both to design and analyze RC columns with enough accuracy for engineering practice and involves higher precision and safety factor in comparison with current codes (Eurocode-2 (2004) [European Committee for Standardization. Eurocode 2: Design of concrete structures—Part 1: General rules and rules for buildings. EN 1992-1-1:2004] ACI-318 (08) [ACI Committee 318. Building code requirements for structural concrete (ACI 318–08). Detroit: American Concrete Institute; 2008]). © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, high strength concrete (HSC) has experienced increasing use and acceptance by designers and builders for both reinforced and prestressed concrete. The use of HSC in columns leads to higher load-carrying capacity, less reinforcement, and smaller cross-sections, which in turn increases the available space in a building. Since the behavior of HSC cannot be directly extrapolated from normal strength concrete (NSC), the simplified methods for designing slender columns must be revised. For instance, the method of equivalent factors (Cm ) must be revised. Reinforced concrete columns, whether in buildings or in civil engineering structures are generally subjected to non-uniform moment diagrams. This is either because the eccentricities at its ends are unequal or due to the existence of span loads (Fig. 1). The axial compression load in slender columns induces a second-order additional moment (P–∆) along the column. Solving a differential equation is required to determine the magnitude and location of the maximum moment in the column. In the case of linear elastic material behavior, the solution can be easily determined. However, the case of reinforced concrete columns is considerably more complex.

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Corresponding author. Tel.: +34 963877561; fax: +34 963877569. E-mail address: [email protected] (L. Pallarés).

0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.02.019

To solve this problem through rigorous analysis (for example, finite element method), taking into account the deformed shape and the non-linear behavior of concrete (cracking, creep. . . ) requires high computational time, and its use in design practice requires considerable experience [1]. For this reason, the codes propose different simplified methods to analyze these types of column. Codes ACI-318(08) [2], EC-2 (2004) [3] and CM-90 [4] enable the use of a constant equivalent eccentricity in braced columns subjected to uniaxial bending with unequal eccentricities at the ends (Fig. 2). The product of a factor (Cm ) times the largest eccentricity applied (e2 ) gives the equivalent eccentricity (ee ). The largest moment induced in the actual column (Mtot ) is the same e as that in the equivalent column (Mtot ). Any simplified method for slender column analysis is applicable by means of the ee (magnification moment or complementary moment) in order to obtain the design moment. The method of equivalent factor can be applied to any nonuniform moment diagram. Table 1 shows the expressions for factor Cm proposed by different authors and codes for braced columns with different eccentricities at the ends. As can be observed, there is no homogeneity among the different proposals, neither in the variables considered nor in their influence on the factor Cm . The most commonly used expression is that proposed by Austin [6]. This expression was adopted by most of the codes [2–4] and was deduced from the solution of the differential equation assuming, linear elastic material behavior.

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Fig. 1. Non-uniform moments diagram. (a) Unequal eccentricities at the ends. (b) Span loads.

Fig. 2. Equivalent eccentricities. (a) Moment diagram due to unequal ecccentricities. (b) Equivalent moment diagram. Table 1 Expressions of factor Cm for braced column subjected to uniaxial bending. Proposal

Variables

Factor (Cm )

Campus and Massonet [5]

q

Austin [6] Robinson et al. [7]

β β β

Trahair [8]

β ; N /Ncr

1+β 2

Duan et al. [9] Sarker and Rangan [10] Tikka and Mirza [11] Tikka and Mirzaa [12,13] ACI-318 (08) [2]; CM-90 [4]; EC-2 (2004) [3]

β ; N /Ncr β ; λm β

· (1 − β) 1 + 0.25 · (N /Ncr ) − 0.6 · (N / ) a + (1 − a) · β; a = 0.975 − 0.00375 · λm ; β ≥ −0.50 Proposal 1: 0.6 + 0.4 · β ≥ 0.30 Proposal 2: 0.20 + 0.8 · (0.50 + 0.5 · β)1.1 ≥ 0.30 0.6 + 0.4 · β ≥ 0.40

β

0.3 · 1 + β 2 + 0.4 · β




0.6 + 0.4 · β ≥ 0.40 1.45 − 0.05 · (4 − β)2

+ [0.4 − 0.23 · (N /Ncr )] ·



3

1−β 2 Ncr 1/3

β — Eccentricity ratio at the ends (e1 /e2 ); e1 ; e2 — First order eccentricities at the ends (e2 largest absolute value eccentricity); λm — Mechanical slenderness of the column (lp /i); ‘‘lp ’’ is the buckling length of the column and ‘‘i’’ is the radius of gyration; N — Axial load ; Ncr — Critical buckling load of the column. a

Composite column.

Sarker and Rangan [10] have recently derived an expression for factor Cm that is valid for short-term loads applied to both normal and high strength concrete columns (from 30 MPa up to 101 MPa). The deduced expression depends on the eccentricities at the ends of the column (e1 /e2 ) and on its slenderness (λm ). Sarker and Rangan explain that the use of factor Cm proposed by Austin [6] is unsafe for columns of low and medium slenderness. Finally, Tikka and Mirza [11] deduced an expression for factor Cm for short-term loads from results obtained with a numeric simulation for normal strength concrete columns (from 27.6 MPa up to 55.2 MPa). According to these authors, the axial level (N /Ncr ) and slenderness of the column can be neglected on the computation Cm . However, Tikka and Mirza pointed out the significance of e1 /e2 or [1 + (e1 /e2 )/2]1.1 to compute Cm factor and they also maintain that the expression proposed by code ACI318(08) [2] is safe, which contradicts Sarker and Rangan’s [10] statement. If there is a transverse load applied to the column (Fig. 1b), code ACI-318 (08) [2] (Section 10.10.6.4) proposes taking the factor

Cm equal to 1.0, and EC-2 (2004) [3] does not define the Cm factor explicitly. So when the method based on nominal rigidity is applied, EC-2 (2004) [3] proposes modifying the expression to determine the magnification factor depending on the distribution of moments (constant, triangular, or parabolic). 2. Objective and research significance The objective of this paper is to propose an expression to determine the equivalent factor Cm that is valid for concrete pinended columns subjected to a non-uniform moment diagram. The proposal is valid for the most usual load cases (Table 2), and for uniaxial bending, short and long term behavior and for normal or high strength concrete. The reason for this research is two-fold: on one hand, the purpose is to develop new expressions for the factor Cm in columns subjected to transverse loads, since taking a value of one for this coefficient, as ACI-318(08) [2] proposes, is too conservative. On the

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Table 2 Load cases and proposed theoretical factor Cm . ν = N /Ac fc ; λg = L/b is the geometric slenderness; b is the smallest dimension of the cross-section; L is the buckling length of the column. α = N /Nu ; N is the applied axial load; Nu is the failure axial load of the column. Load case (1)

Theoretical approach (2)

Non-linear analysis (3)

Cm = 1 −

1

Cm = η · α + 1 ≥ Cmin η = 0.6 · (a − 1) ≤ −0.20 Cmin = 0.3 · (a + 2) ≤ 0.80

≥ Cmin η∗ η = 4375 · a + 125 Cmin = 0.30 · (a + 2) ∗

Cm = 1 −

2

Cm = −0.20 · α + 1

Cm = 1 −

3, 4

Cm = η · α + 1 ≥ Cmin η = 0.9 · a − 0.80 ≤ −0.20 Cmin = 0.3 · (a + 2) ≤ 0.80

Cm = 1 −

5

Cm = η · α + 1 ≥ Cmin η = 1.4 · a − 0.95 ≤ −0.35 Cmin = 0.65

Cm = 1 −

6

Cm = η · α + 1 ≥ Cmin η = −0.9 · a + 0.10 Cmin = −0.8 · a + 1.05 ≤ 1

7

Cm = η · α + ξ ≥ Cmin η = −0.4 · (a + 1) ξ = 0.85 Cmin = 0.7 − a ≥ 0.50

8

Cm = η · α + ξ ≥ Cmin η = 0.4 · (β − 1) ξ = 0.15 · β + 0.85 Cmin = 0.5 · (β + 1) ≥ 0.40

other hand, the purpose is to expand Austin’s [6] proposal to high strength concrete columns. 3. Methodology The methodology followed to obtain the proposed expressions for the equivalent factor Cm is as follows: (a) First, a theoretical approach to the problem was carried out assuming linear elastic behavior of the material. In this case, the differential equation that governs the behavior of the column subjected to axial load and to a non-uniform moment diagram was solved.

λ2g · ν

λ2g · ν

η = 4500 Cmin = 0.90 ∗

η∗

≥ Cmin

λ2g · ν

≥ Cmin η∗ η = 4375 · a + 125 Cmin = 0.30 · (a + 2) ∗

λ2g · ν

≥ Cmin η∗ η∗ = 1500 · a + 750 Cmin = 0.30 · a + 0.60

λ2g · ν

≥ Cmin η∗ η∗ = [50 · a · (a − 1) + 12.65] × 104 Cmin = −0.50 · a + 1

λ2g · ν ≥ Cmin Cm = ξ ∗ − η∗ η∗ = −2750 · a + 2500 ξ ∗ = a + 0.75 ≤ 1 Cmin = 0.55

Cm = ξ ∗ −

η∗ =

2500

λ2g · ν η∗

≥ Cmin

1−β ξ ∗ = 0.25 · β + 0.75 Cmin = 0.45 · β + 0.55 ≥ 0.40

i (b) Second, the factor Cm was deduced as the ratio between the first-order moment of model case and the first-order moment of each load case-i. Then, an adjustment by least-squares i approach on the derived theoretical formulae of Cm was carried out for all load cases. (c) A numerical simulation taking into account the non-linear behavior of reinforced concrete using finite element method was carried out. (d) The main variables on which the factor Cm depends have been studied. (e) The accuracy of applying the proposed theoretical expressions compared to results from the numerical simulation was obtained.

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Fig. 3. Theoretical approach. (a) Non-uniform moment diagram. (b) Uniform moment model case.

Fig. 4. Theoretical factor Cm corresponding to uniform transverse load (Load case #1− a = 1).

(f) Finally, new expressions for factor Cm have been proposed using least squares approach on the results obtained in the numerical simulation. 4. Theoretical approach The equilibrium of the column in its deformed shape assuming linear elastic material behavior for the different load-i (Table 2, column 1) cases belonging to a non-uniform moment diagram (Fig. 3a) has to be set out in order to deduce factor Cm . Moreover, the model load case consisting of a column subjected to equal eccentricities at the ends (Fig. 3b) has to be solved. Factor Cm is gathered from equating the maximum total i moment, corresponding to load case-i (Mtot ) to the maximum ,ma´ x e total moment corresponding to the model load case (Mtot ). ,ma´ x e i Mtot ,ma´ x = Mtot,max .

(1)

As an example, Appendix shows the approach used to obtain the analytical expressions of the maximum moment corresponding to the model case and to load case #1 and a = 1 (Table 2, column 1). The same methodology was used to solve all cases. The maximum e i moment (Mtot ,max or Mtot,max ) can be represented as the first order maximum moment (M1i or M e ) multiplied by an amplification factor (δ i or δ e ) that depends on the ratio between the applied axial load and buckling load (α = N /Ncr ). (2)

e e e Mtot ,ma´ x = δ (α) · M . e

i Cm

M1i

Assuming that M = · (Fig. 3) and given that the maximum moment corresponding to load case-i equals that of the i model case (Eq. (1)), the factor Cm belonging to load case-i is derived to be: Me M1i

=

e Mtot ,ma´ x

δ e (α)

Me =

·

δ i (α) i Mtot ,max

=

δ i (α) . δ e (α)

(3)

n X

i Cm · M1i

(4)

i=1 i where Cm is the factor belonging to load case-i and M1i is the first order maximum moment belonging to load case-i. On the other hand, if the maximum moment sections are not the same for each load case and this rule is applied (expression 4), the adopted solution ensures safety. Tables 3a and 3b (column 1) show the accuracy of the proposed formulae (Table 2, column 2) as compared to the results obtained from solving the differential equation. The value of the proposed factor (Cm,prop ) for 18 levels of axial load α (from α = 0.05 to α = 0.95) was compared to the factor obtained from solving the differential equation (Cm,th ). The accuracy was evaluated using the following ratio:

ξ=

i i i Mtot ,max = δ (α) · M1

i Cm =

Once again, it should be highlighted that the factor Cm depends on the ratio α and is non-dependent on the applied axial load. If coefficient Cm is plotted versus α for load case #1 and a = 1 that is the uniform transverse load, an inversely proportional linear trend is observed (Fig. 4). A least-squares approach to linear functions of the type Cm = η ·α +ξ was carried out on the deduced theoretical expressions for the different load-i cases. The proposed formulae can be seen in the column 2 of the Table 2. The same formulation has been proposed for load cases 3 and 4 to ease their application. According to Austin [6] and Galambos [14], if instability occurs in the column due to lateral torsion, factor Cm should not be below 0.4. Mac Gregor et al. [15] pointed out that this behavior is not observed in reinforced concrete columns. Nevertheless, code ACI318(08) [2] applies it due to the uncertainty of the values of e1 /e2 when this ratio is between −0.50 and −1.00. For this reason, the proposal presented in this paper still applies this lower limit when the column is subjected to unequal eccentricities at the ends. If there are rotational restraints at the ends of the column subjected to lateral load, the magnification moment factor δ i (α) is lower than that gathered from a pin-ended column for the same value of α [16]. Therefore, applying factor Cm , as proposed in this article to this type of columns, ensures a conservative result. When a column is subjected to different load cases, if the crosssection with the maximum moment is the same for all the load cases, the addition of load cases is valid [16]:

Cm,prop Cm,th

.

(5)

If the ratio is higher than one, safety is ensured since the factor is overvalued. In the case of columns subjected to span loads, an average ratio of 1.02 is deduced with a coefficient of variation (c.o.v) of 0.04. In the case of columns with unequal eccentricities at the ends, the average ratio is 1.18 and the c.o.v is 0.28. The average ratio and the c.o.v are increased for these types of loads compared with the results obtained for span loads, due to the limiting value on factor Cm (not lower than 0.40).

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Table 3a Accuracy in columns subjected to transverse load. Case

Variable

Value

Theoretical approach (1) N◦

1 1 1 1 1 2 2 2 3 3 3 3, 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7

a a a a a q1 /q2 q1 /q2 q1 /q2 a a a a a a a a a a a a a a a a a a

1 0.2 0.4 0.6 0.8 1.5 3 20 0.2 0.5 0.7 1 0.2 0.5 0.7 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.2 0.4

19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 12 6

ξm 1.00 1.08 1.02 1.01 1.01 1.00 0.99 1.01 1.04 1.01 1.00 1.01 1.06 1.04 1.04 1.09 1.03 1.00 1.01 1.02 1.01 0.99 0.99 0.99 0.95 1.10

c.o.v

0.00 0.03 0.01 0.01 0.00 0.00 0.00 0.00 0.03 0.01 0.02 0.00 0.03 0.02 0.02 0.06 0.04 0.02 0.01 0.01 0.00 0.00 0.01 0.01 0.02 0.02

ξmin 0.99 1.01 1.00 1.00 1.00 0.99 0.99 1.00 1.00 1.00 0.98 1.00 1.01 1.01 1.01 1.01 0.99 0.97 1.00 1.00 1.00 0.98 0.97 0.98 0.91 1.07

Non-linear analysis p1%

0.99 1.01 1.00 1.00 1.00 0.99 0.99 1.00 1.00 1.00 0.98 1.00 1.01 1.01 1.01 1.01 0.99 0.97 1.00 1.00 1.00 0.98 0.97 0.98 0.91 1.07

p5%

0.99 1.02 1.01 1.00 1.00 0.99 0.99 1.01 1.00 1.00 0.98 1.00 1.02 1.01 1.01 1.01 0.99 0.98 1.00 1.01 1.00 0.99 0.97 0.98 0.92 1.08

ξmax 1.00 1.12 1.05 1.02 1.01 1.00 1.00 1.02 1.10 1.03 1.05 1.02 1.12 1.06 1.08 1.19 1.11 1.05 1.02 1.03 1.02 1.00 1.00 1.00 0.98 1.12

Theoretical-Cm (2)

Proposed-Cm (3)

N◦

ξm

c.o.v

ξmin

p1%

p5%

ξmax

N◦

ξm

c.o.v

ξmin

p1%

p5%

ξmax

349 213 222 210 212 209 219 181 215 196 188 241 215 201 191 213 22 220 30 251 212 21 208 221 223 248

1.04 1.05 1.01 1.01 1.01 1.02 1.03 1.02 1.01 1.00 1.03 1.03 1.02 1.02 1.05 1.04 0.95 1.00 0.98 1.04 1.01 0.98 0.97 0.99 0.90 0.92

0.06 0.11 0.08 0.05 0.05 0.04 0.04 0.05 0.12 0.07 0.05 0.05 0.12 0.07 0.05 0.13 0.14 0.11 0.11 0.09 0.00 0.02 0.04 0.05 0.14 0.20

0.90 0.73 0.78 0.85 0.87 0.91 0.91 0.90 0.70 0.79 0.88 0.91 0.70 0.80 0.88 0.72 0.73 0.71 0.74 0.77 1.00 0.93 0.87 0.83 0.57 0.46

0.91 0.75 0.81 0.86 0.88 0.93 0.93 0.92 0.71 0.81 0.89 0.92 0.72 0.83 0.90 0.73 0.73 0.72 0.74 0.80 1.00 0.93 0.89 0.85 0.58 0.48

0.94 0.80 0.85 0.90 0.91 0.94 0.94 0.93 0.74 0.87 0.92 0.94 0.75 0.87 0.93 0.75 0.73 0.76 0.78 0.86 1.00 0.95 0.91 0.88 0.64 0.56

1.20 1.26 1.18 1.15 1.14 1.13 1.14 1.13 1.24 1.13 1.16 1.15 1.28 1.23 1.18 1.30 1.16 1.22 1.17 1.27 1.02 1.01 1.11 1.11 1.12 1.55

349 213 222 210 212 209 219 181 214 196 188 241 215 201 191 213 22 220 30 251 212 21 208 221 223 248

1.07 1.15 1.12 1.08 1.07 1.07 1.07 1.07 1.15 1.11 1.08 1.08 1.17 1.14 1.10 1.22 1.06 1.14 1.04 1.13 1.00 1.01 1.05 1.10 1.25 1.35

0.03 0.11 0.07 0.05 0.04 0.03 0.04 0.04 0.11 0.06 0.05 0.04 0.11 0.07 0.05 0.12 0.04 0.09 0.04 0.08 0.00 0.01 0.03 0.05 0.13 0.23

0.98 0.98 0.99 0.98 0.98 1.00 0.99 0.98 0.97 0.99 0.98 0.99 0.98 0.98 0.99 1.00 1.00 0.98 0.97 0.98 0.99 0.99 0.99 0.98 0.99 0.95

0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 0.97 0.99 0.98 1.00 0.99 1.00 0.99 1.01 1.00 0.99 0.97 0.99 1.00 0.99 1.00 1.00 1.00 1.00

1.01 1.00 1.01 1.00 1.01 1.01 1.02 1.01 0.99 1.01 1.01 1.01 1.01 1.02 1.01 1.04 1.01 1.01 0.97 1.02 1.00 1.00 1.01 1.02 1.02 1.05

1.19 1.52 1.35 1.23 1.19 1.17 1.19 1.19 1.52 1.28 1.22 1.21 1.55 1.33 1.25 1.65 1.14 1.46 1.10 1.37 1.02 1.04 1.18 1.24 1.67 2.79

ξm : average ratio; C.V.: coefficient of variation; ξmin : minimum ratio; p1% : one percentile value; P5 : five percentile value; ξmax : maximum ratio. Table 3b Accuracy in columns subjected to unequal eccentricities at the ends. e1 /e2 N◦

ξm

0.75

c.o.v.

ξmin p1% p5%

ξmax N◦

ξm

0.50

c.o.v.

ξmin p1% p5%

ξmax N◦

ξm

0.25

c.o.v.

ξmin p1% p5%

ξmax N◦

ξm

c.o.v.

ξmin p1% p5%

ξmax

0

Theoretical approach (1)

Cm -theoretical (2) Cm -proposed (3)

Non-linear analysis Austin [6] (4)

18 1.03 0.02 1.00 1.00 1.00 1.06 17 1.06 0.04 1.00 1.00 1.00 1.11 16 1.09 0.05 1.00 1.00 1.00 1.14 14 1.10 0.06 1.00 1.00 1.00 1.16

173 1.03 0.02 0.97 0.98 1.01 1.08 160 1.07 0.04 0.92 0.94 1.01 1.18 138 1.10 0.06 0.88 0.95 1.00 1.26 119 1.13 0.08 0.89 0.91 0.99 1.34

173 1.02 0.01 0.97 0.98 1.00 1.03 160 1.04 0.03 0.92 0.93 0.96 1.08 138 1.06 0.05 0.90 0.91 0.94 1.15 119 1.08 0.08 0.89 0.89 0.93 1.31

173 1.04 0.01 1.00 1.01 1.02 1.06 160 1.09 0.03 0.99 1.00 1.03 1.16 138 1.14 0.05 1.02 1.03 1.05 1.25 119 1.21 0.07 1.02 1.06 1.08 1.43

e1 /e2

Theoretical approach (1)

Non-linear analysis

−0.25

13 1.11 0.04 1.04 1.05 1.05 1.17 11 1.28 0.16 1.02 1.03 1.04 1.59 8 2.02 0.35 1.15 1.17 1.21 3.08

109 1.13 0.10 0.80 0.81 0.90 1.34 76 1.28 0.11 0.94 0.95 1.04 1.57 61 1.61 0.29 1.00 1.01 1.07 3.00

−0.50

−0.75

Cm -theoretical (2) Cm -proposed (3) Austin [6] (4) 109 1.25 0.12 0.96 0.97 1.03 1.61 76 1.44 0.15 1.01 1.02 1.13 2.02 61 1.72 0.27 0.96 0.98 1.07 3.00

109 1.08 0.14 0.74 0.77 0.84 1.32 76 1.21 0.16 0.82 0.86 0.92 1.57 61 1.60 0.30 0.94 0.98 1.01 3.00

ξm : average ratio; C.V.: coefficient of variation; ξmin : minimum ratio; p1% : one percentile value; p5% : five percentile value; ξmax : maximum ratio.

5. Numerical simulation

• One-dimensional finite element with thirteen degrees of freedom [17].

The factor Cm for the different load cases in Table 2 (column 3) that belongs to a reinforced concrete column was deduced by finite element analysis. The numerical model included the following basic features:

• Non-linear behavior of concrete (CM-90 [4], CEB-FIP (1995) [18]).

• Non-linear behavior of steel: bilinear diagram (CM-90 [4]). • Geometric nonlinearity: large displacements and large deformations.

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Table 4 Range of the parameters. Parameters

Considered values

Type of load Geometric slenderness of the column (λg = lp /h)

• See Table 2 (column 1) • λg = 10, 20, 30

Longitudinal reinforcement

• Pin-ended column 10 levels of load (α ∗ = N /Nu ) where N is the

Restraints Axial load

Strength concrete (fc ) Reinforcement area (ω = As · fy/Ac · fc) Creep coefficient (ϕ )

applied axial load and Nu the failure load of the column • fc = 30 MPa and 80 MPa • ω = 0.06, 0.25, 0.50, 0.75

• ϕ = 0, 1, 2, 3

and the experimental axial load, Nexp . The ratio varies from 0.64 to 1.19, with an average value of 0.98 and a c.o.v of 0.12. The accuracy of this numerical model was also verified by comparing it with 58 experimental tests available in the literature [7,32,10] belonging to pin-ended columns subjected to unequal eccentricities at the ends. In the analyzed columns (e1 /e2 = −1–0.54; e2 /h = 0.09–1.5; l/i = 33–138; ρrs = 0.5%–4%; fy = 280–505 MPa; fc = 24.3–101 MPa) the ratio varies from 0.71 to 1.22 with an average value of 0.96 (on the safety side) and a c.o.v of 0.12. Furthermore, the results of the numerical model in comparison with 56 tests carried out by Pallarés et al. [33] are shown in References [33,34]. 6. Estimate of the equivalent factor Cm through numerical simulation Factor Cm is numerically obtained for each axial load level-j (Nj ) applied to the column (Fig. 5) as the ratio between the maximum first-order moment (M e )NS belonging to the model case and the maximum first-order moment belonging to load case-i (M1i )NS : Cm =

Fig. 5. Equivalent factor method.

• Short and long term behavior: shrinkage and creep (CEB (1978, 1983) [18,19]). Long term effects of the axial load have been taken into consideration in this work applying the entire axial load as permanent. More detailed information on this numerical model can be found in Bonet (2001) [20]. That paper presented the accuracy of this model as compared to the results of 173 experimental tests found in the bibliography [21–31]. These tests belong to pin-ended columns subjected to equal eccentricities at the ends (e1 /e2 = 1; relative eccentricity (e2 /h) between 0.10 and 5.14; slenderness (l/i) between 10 and 207; reinforcement ratio (ρrs ) between 0.9% and 9.8%; yield point of steel (fy ) between 387 and 684 MPa; concrete strength (fc ) between 18.58 and 97 MPa; creep coefficient (φ ) between 0 and 3.29; skew angle from 0 to 90◦ ). The accuracy is assessed through the ratio between the numerical axial load NNS

(M e )NS
. i M1

(6)

NS

The numerical model described in the previous section was used in this paper to analyze the main variables that affect the equivalent factor Cm . Table 4 shows the analyzed parameters and the range of variation. The combination of the different values adopted for those parameters produced approximately 7000 numerical simulations. The size and shape of the cross-section (square 300 × 300 mm), the yield point of the steel (fy = 500 MPa) and the cover of the longitudinal reinforcement (10% of the depth) were all considered as fixed values. Previous studies [20] pointed out that these variables are not significant to assess the equivalent factor Cm . When columns are loaded laterally or with unequal eccentricities at their ends (Fig. 6), the ultimate bending strength of the column is always larger than that of the model case (Cm smaller than one) for the same level of axial load. When columns are subjected to a transverse load (Fig. 6a) the maximum moment section (including P–∆) is located between the ends of the column, therefore, the first-order maximum moment is lower than the ultimate strength of the section. Nevertheless, when applying unequal moments at the ends (Fig. 6b), the maximum total moment can be placed at the ends of the column, which is equal to the ultimate strength of the cross-section. As a result, these load cases have been excluded from the analysis. In short, 5968 numeric tests have been considered in the analysis.

Fig. 6. Cross-section (λg = 0) and column (λg 6= 0) interaction diagram. (a) Uniform transverse load. (b) Unequal eccentricities at the ends.

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7. Discussion of the numerical results

Fig. 7. Equivalent factor Cm versus the ratio axial load applied and failure axial load (α ∗ = N /Nu ) (Load case #1 − a = 1).

This section analyzes the variation of equivalent factor Cm according to the established parameters. For example, the results obtained in load case #1 and a = 1 (Table 2, column 1) and distribution of longitudinal reinforcement type 1 (Table 2) has been chosen. Fig. 7 shows the equivalent factor versus the ratio between the axial load applied in the column and the failure load of the column (α ∗ ) for different slenderness. The failure axial load (Nu ) coincides with the critical axial load if the column becomes unstable or conversely with the ultimate axial load of the cross-section (Nuc ). It can be observed that as α ∗ increases, generally factor Cm decreases. This result was already observed in the theoretical approach. Also, as the slenderness of the column increases, factor Cm reduces. Fig. 8 shows factor Cm versus the load ratio ν = N /[Ac · fc ], where N is equal to the applied axial load, Ac is the cross-section area and fc is the compressive strength concrete. It can be observed

Fig. 8. Influence in the equivalent factor Cm of: (a) Strength and slenderness of tests; (b) Reinforcement areas (c) Creep coefficient; (d) Location of reinforcement; (e) The type of load; (f) The type of case.

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that when both the axial load and the slenderness of the column increase, factor Cm is reduced. Fig. 8a analyzes the influence of concrete strength. It may be seen that concrete strength does not have a significant impact on the value of the equivalent factor. As expected, as the reinforcement area increases for an equal level of load, P–∆ effects are smaller and therefore the equivalent factor is bigger (Fig. 8b). As is well known, P–∆ effects increase with creep. As can be observed in the Fig. 8c, increasing the creep coefficient reduces the equivalent factor Cm . This effect is less pronounced when the slenderness of the column is greater. In this way, for the slenderness of 30, the influence of the creep coefficient on factor Cm is not important. This effect is due to the fact that the behavior of slender concrete columns is more affected by the geometric nonlinearity (instability of the column) than by the material behavior (ultimate load of the cross-section). Since for the same total maximum moment (including second order effects) δ e (α) is always larger than δ i (α) for each load case-i, the creep factor (ϕ ) impact resulted more relevant on the δ e (α) coefficient than on the δ i (α), and then Cmi became smaller. Fig. 8d shows the influence of the reinforcement location on the equivalent factor Cm . It can be observed that the equivalent factor depends on the reinforcement location, so that the larger the radius of gyration of the reinforcement (is /h, where is represents the radius of gyration of the reinforcement and h is the depth) smaller are P–∆ effects and the larger is the equivalent factor Cm . Finally, as expected from the theoretical results, equivalent factor Cm significantly depends on the type of load (Fig. 8e). 8. Proposed method for determining Cm First, the application of the expressions obtained from the theoretical approach (Table 2, column 2) was verified with the results obtained through numerical simulation. In these expressions, the relation α is replaced by α ∗ which is equal to the applied axial load (N) divided by the failure load (Nu ) of the column. The failure load of the column is the minimum of the ultimate strength of the cross-section Nuc and the buckling load (Ncr ). Nuc was computed as fc · Ac + As · fy where fc and fy are the concrete and steel strength respectively; Ac and As are the concrete and steel area respectively. Ncr was computed numerically through finite element analyses. Tables 3a and 3b (column 2) show the degree of accuracy that is assessed from the following ratio:

ξ=

Cm,prop Cm,SN

(7)

where Cm,prop is the value of the proposed equivalent factor and Cm,NS is the equivalent factor obtained through numerical simulation. If the ratio is higher than one then a conservative result is ensured. Generally, obtained average ratios are higher than 1.0 and coefficient of variation (c.o.v) are small for all load cases. In the case of columns subjected to transverse loads, the average ratio is 1.0 and the c.o.v is 0.08. The case of columns subjected to unequal eccentricities at the ends, the average ratio is 1.18 with a c.o.v of 0.08. Despite the accuracy of the results obtained by applying the proposed expressions (Table 2, column 2), from the point of view of applying these formulae, it is necessary to know the location and the reinforcement area in order to estimate the ultimate strength of the cross-section (Nuc ) or the stiffness EI to determine axial compression buckling load (Ncr ). For this reason, new formulae for factor Cm that are independent of the location and reinforcement area have been proposed.

Fig. 9. Coefficient of equivalence Cm according to the product (ν · λ2g ) (Load case #1 − a = 1).

A comparative study among the candidate functions for factor Cm has been carried out with the results obtained in the numerical simulation, concluding that the expression with the best degree of accuracy is:

λ2g · ν Cm = ξ ∗ − η∗

(8)

where ξ ∗ and η∗ depend on the type of load. This result coincides with the conclusions obtained in the theoretical approach (Section 4) and in the analysis of the results (Section 7), where it has been shown that the factor Cm depends linearly on N /Ncr since this relation is proportional to the product [ν · λ2g ]. The proposal is defined as a superior envelope of the results obtained through numerical simulation. As an example, the Fig. 9 shows a support subjected to a uniform span load (load case #1 − a = 1), with the results obtained using a numerical simulation compared to the proposed factor Cm . It should be noted that although factor Cm depends on creep coefficient (φ ), when Cm is represented as function of [ν · λ2g ], the results obtained for short term load (φ = 0) and for long term load (φ = 2) are slightly overlapping. Table 2 (column 3) shows the proposed formulae for the studied load cases. To ease the application of the formulae, a single expression has been proposed for uniform, trapezoidal, and triangular transverse distribution load (load case #1–4, Table 2). This proposal keeps the criteria for Cm higher than 0.40 when unequal eccentricities act at the ends of the column. 9. Validation of the proposed factor Cm 9.1. Column subjected to transverse load Table 3a (column 3) shows the accuracy obtained with the proposed formulae (Table 2, column 3). The proposal is conservative for all load cases. The average ratio is 1.12 and the c.o.v is 0.07. As an example, the Fig. 10 shows factor Cm , obtained through numeric simulation, compared to the proposal for four load cases. It can be observed how the proposal is safe. It can also be observed that the ACI-318(08) [2] proposal (Cm = 1) for these load cases, is very conservative. 9.2. Column subjected to unequal eccentricities at the ends Table 3b (column 3) shows the accuracy of the proposed formulae (Table 2, column 3) and that obtained from the Austin’s

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Fig. 10. Validation of the proposal factor Cm . Column subjected to transverse load.

proposal (1961) [6], which has been adopted by many current standards (Table 3b, column 4). As an example, Fig. 11 shows the results obtained through numerical simulation, the proposal for Cm (Table 2, column 3) and Austin’s proposal (1961) [6]. Regarding the proposal, an average ratio of 1.20 with a c.o.v of 0.08 was obtained and results are conservative. Austin’s (1961) [6] proposal obtains an average ratio of 1.10 with a c.o.v of 0.08. This proposal result is unsafe for low and medium values of [ν · λ2g ]. This fact occurs with a slenderness of 10, 20, and 30. Sarker and Rangan (2003) [10] already pointed that in the case of medium and low slenderness columns there were some unsafe cases. Finally, the fact that the medium ratios and variation coefficients obtained in this paper for Austin’s (1961) [6] proposal for each relation e1 /e2 are very similar to those obtained by Tikka and Mirza (2004) [11] should be highlighted. 10. Conclusions This paper presents a proposal to determine the equivalent factor Cm in columns subjected to a non-uniform moment diagram. The most usual load cases were considered. Some expressions were proposed as an approximation to the solution of the differential equation (Table 2, column 2) to determine Cm with an elastic and linear behavior. A good degree of accuracy was achieved regarding the theoretical solution, the average ratio being 1.02 (conservative) when transverse loads act and 1.18 for unequal eccentricities at the ends. These expressions can also be applied successfully to reinforced concrete columns, but they require a determination of the buckling load (Ncr ) or the ultimate strength of the cross-section (Nuc ). The average ratio for columns with transverse loads is 1 and for

columns subjected to unequal eccentricities at the ends is 1.18 (conservative). Other alternative expressions have been proposed to determine Cm (Table 2, column 3) with the results obtained through numerical simulation for reinforced concrete columns. The proposed formulae are valid for uniaxial bending forces, short and long term load and high and normal strength concrete. The accuracy of the proposed formulae has been assessed: the average ratio is 1.12 and 1.20 in the case of transverse loads and unequal eccentricities respectively. The following conclusions can be drawn from the analysis carried out in this paper: 1. In elastic and linear behavior, the equivalent factor Cm depends on the type of load and the axial load (α ) obtained as the ratio of the applied axial and buckling load. The equivalent factor is independent of the value of the transverse load. 2. The variables that impact in the equivalent factor Cm in reinforced concrete columns are: the axial load, the slenderness of the column, the creep coefficient, the location and the reinforcement area and the type of load. 3. The proposal of ACI-318(08) [2] about taking the factor Cm equals 1.0 for columns subjected to transverse loads is excessively conservative. The application of the proposal (Table 2, column 3) implies an improvement of accuracy and provides the optimization of the reinforcement. 4. In the case of columns subjected to unequal eccentricities at the ends, Austin’s (1961) [6] proposal, adopted by the most of the standards, is unsafe for low and medium values of [ν · λ2g ]. The proposal (Table 2, column 3) improves the level of safety. In short, the proposal showed in this paper (Table 2, column 3) is easy to apply, depends on known parameters at the design stage of the columns and improves the level of safety and accuracy obtained by other proposals.

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Fig. 11. Validation of the proposal factor Cm . Column subjected to unequal eccentricities at the ends.

Acknowledgements

The maximum total bending moment is located in this case in the mid-span section (x = L/2):

The content of this article falls within the research framework carried out by the Instituto de Ciencia y Tecnología del Hormigón (ICITECH) of the Universidad Politécnica de Valencia. This paper was supported by the Ministry of Education and Science (Reference BIA2006-06429). The authors thank the institutions mentioned above and Mr. M. Eatherton for their contribution to this work.

A.1. Determination of the total maximum moment belonging to the model case The equilibrium of the deformed shaped of any section that is located at x from one of the end of the column (Fig. 3b) is considered. So the following formula is obtained: M e + N · y(x) = EI · φ(x)

(A.1)

e

where M and N are the bending moments and axial compression load applied at the ends, y(x) is the lateral deflection, E is the modulus of elasticity of the material. I is the moment of inertia and φ(x) is the curvature. Assuming small deformations, the curvature is the second derivative of lateral displacement φ(x) = −d2 y/dx2 = −y00 (x). Taking into account the restrained conditions of the column [y(x = 0) = 0; y(x = L) = 0], the differential equation is solved (A.1) and the total moment is expressed by (A.2):

√ π · x  sin(√α π·x )(1 − cos(π √α))  L = M · cos α + √ L sin(π α) e

2 1 + cos(π

e e √ = M · δ (α) α)

(A.3)

where δ e is the amplification factor of the model case, that is a function of α . A.2. Determination of the total maximum moment belonging to load case #1 (a = 1)

Appendix

e Mtot

√ e e Mtot ,max = M · p



(A.2) where α = N /Ncr in which Ncr , the buckling axial load of the column, is equal to π 2 · EI /L and L the length of the column.

The equilibrium of the deformed shaped of any section that is located at ‘‘x’’ from one of the end of the column (Fig. 3a) is considered. So the formula (A.4) is obtained: q·l

q · x2

+ N · y = −EI · y00 (A.4) 2 2 where q is the uniform of distributed load. Taking into account the restrained conditions of the column, the total bending moment at any section of the column is obtained by integrating the differential equation:    √  √ q · L2 8 π αx i Mtot = · 2 sin(π α) −1 + cos √ 8 L π α · sin(π α)  √     √ π αx 1 − cos(π α) . + sin (A.5) x−

L

The maximum bending moment is located at the mid-span section (x = L/2): 1 Mtot ,max

=

q · L2 8

·

8

√ π 2 α · sin(π α)

= M11 · δ 1 (α)

√

"

2

p

1 + cos(π

# √ −1 α) (A.6)

L. Pallarés et al. / Engineering Structures 31 (2009) 1589–1599

where δ 1 is the amplification factor belonging to load case #1 (a = 1) that is a function of α and, M11 is the maximum first order moment of the column equal to [q · L2 /8]. References

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