342b_p.07.pdf

H. Mohy El-Din Afefy, Salah El-Din F. Taher and Salah El-Din E. El-Metwally

October 2009 The Arabian Journal for Science and Engineering, Volume 34, Number 2B 349

A NEW DESIGN PROCEDURE FOR BRACED REINFORCED HIGH STRENGTH CONCRETE COLUMNS

UNDER UNIAXIAL AND BIAXIAL COMPRESSION

Hamdy Mohy El-Din Afefy

Structural Engineering Department Faculty of Engineering, Tanta University

E-mail:[email protected].

Salah El-Din Fahmy Taher*

Faculty of Engineering, Tanta University, Egypt Faculty of Engineering, Tanta University

Salah El-Din E. El-Metwally

Structural Engineering Department Faculty of Engineering, El-Mansoura University, Egypt

E-mail:[email protected]

:خالصـةال

ة تحت تأثير عزوم جانبيا المقيدة ني لتصميم األعمدة الخرسانية المسلحة عالية المقاومة فى المبا أسلوباا البحث هذيقدم ة أحادي اه وثنائي يتكون و .االتجة لتصميم ، ويمثل الجزء االتجاةأحادية المعرضة لعزوم األعمدة طريقة لتصميم األول حيث يمثل الجزء جزأينالبحث من دة الثانى طريق المعرضة األعم

ة األقصى عدم ثبات الحد لنتيجة و. لعزوم ثنائية االتجاه دة لحد النحاف اختالف القصيرة لألعم ة األقصى الكودات وعدم وضوح تعريف الحد ب لحد النحافراح ، الطويلة تم اقتراح طريقة جديدة للتعامل مع تلك الحدود لألعمدة اء واقت ادالت لحساب جساءة االنحن د مع ار تلك ةلألعم وع انهي ى ن اء عل دة بن . األعمزوم إضافة ود المعرض لع ل العم افىء لتحوي ود المك دأ العم ى تطبيق مب ة فقط لكن بطول ، إضافة إل وة محوري ود معرض لق ى عم ة إل وة محوري ى ق إل

ة وبناء على ذلك تم اقتراح . القصيرةلألعمدة لحد النحافة األقصى من الحد التحقيقمختلف، ومن ثم يمكن طريقة لتصميم األعمدة الخرسانية المسلحة عالياه تحت تأثير عزوم ا المقيدة جانبينيالمقاومة فى المبا وة الضغط أحادية االتج اء - باستخدام منحنى ق د . عزم االنحن م استخدام وق سابق األسلوب ت سه ال نف

ع دةللتعامل م اه األعم ة االتج زوم ثنائي سلحة المعرضة لع انية الم ى وذ، الخرس اه عل ك بدراسة آل اتج ده،ل ة خطوط ح تخدام طريق ق باس م التحقي ومن ثة ، الجانبية نتيجة النحافة اإلزاحةثبتت الطريقة المقترحة آفاءتها فى التنبؤ بقيم وقد أ . الكنتور م عرض وفي . وذلك بمقارنتها بالنتائج المعملي ة البحث ت نهاي

ة المفترضة تحت ت ة مثالين تفصيليين لتوضيح الطريق ال وتقوسات مختلف د . أثير أحم ة وق ائج المعملي ة مع النت ا بالمقارن ة المفترضة آفاءته اثبتت الطريق . لتصميم المنشآت الخرسانية المسلحةاألمريكى من الكود المصرى والكود آلوبنود

*Corresponding author: E-mail: [email protected]

Paper Received August 7, 2008; Paper Revised April 6, 2009; Paper Accepted May 27, 2009


The Arabian Journal for Science and Engineering, Volume 34, Number 2B October 2009 350

ABSTRACT

This paper presents a design procedure for braced high-strength reinforced concrete columns under the action of uni-axial and biaxial loading. The paper has two phases; the first phase represents the design procedure of such columns under the action of uniaxial bending, while the second phase shows the implementation of the design procedure on the columns under biaxial bending. Due to the lack of uniformity in the conceptual treatment of the upper slenderness limit for short columns in different codes and the unclear definition of the maximum slenderness limit for slender columns as well, a new approach has been presented. Proposed expressions for the flexural rigidity have been presented based on the mode of failure of such columns. In addition, the equivalent column concept has been implemented to reduce the uni-axially loaded column to an axially loaded one, hence, the upper slenderness limit for the short column condition can be checked properly. As a consequence, a design procedure for the uni-axially loaded column is presented using a strength interaction diagram. Furthermore, the adequacy of the proposed procedure for predicting the second order effect has been verified against the experimental results. The same proposed design procedure in the first phase has been implemented in each direction of the column cross -section then the load contour equation can be used to check the strength of such column under biaxial loading. Finally, two worked examples have been presented covering different curvature modes: single and double. The proposed design procedure shows its competence against the experimental test results and well-designed columns according to the provisions of the current codes as ECCS 203-2001 and ACI 318-05 for braced systems.

Key words: biaxial bending, codes, column, column design, equivalent column, interaction diagram, reinforced concrete



A NEW DESIGN PROCEDURE FOR BRACED REINFORCED HIGH STRENGTH CONCRETE COLUMNS UNDER UNIAXIAL AND BIAXIAL COMPRESSION

1. INTRODUCTION

Over the last few decades, the development in material technology, especially with the availability of superplasticizers, led to the production of higher concrete strength grades [1]. Since then, a series of research studies have been conducted on the behavior of such concrete [2–6]. One application of high-strength concrete has been in the columns of buildings. A large number of studies have demonstrated the economy of using high-strength concrete in columns of high-rise buildings, as well as low to medium-rise buildings [7]. In addition to reducing column size and producing a more durable material, the use of high-strength concrete has been shown to be advantageous with regard to lateral stiffness and axial shortening. Another advantage cited in the use of high-strength concrete columns is the reduction in the cost of formwork.

It has been argued that, when judging the strength of a column, it is not only a matter of ensuring that stresses in the member are kept below a certain specified value, but also of preventing the peculiar state of unstable equilibrium [8]. Buckling has become more of a problem in recent years since the use of high-strength material requires less material for load support-structures and components have become generally more slender and buckle-prone. As a result, the slenderness limits based on normal-strength concrete have to be reassessed to make use of the merits of high-strength concrete grades. This trend has continued throughout technological history. Attempts have been made to modify the theory of analysis of slender columns by introducing effects of inelastic behavior and large deformations.

There are two important limits for the slenderness ratio, which are the upper slenderness limit for the short column and the maximum slenderness limit. The upper slenderness limit is the limit that when exceeded the column is considered a long column On the other hand, the maximum slenderness limit is stipulated to avoid carrying out second order analysis of the column, which is burdensome and more complicated [9].

Most of the slenderness limit expressions provided by codes are derived assuming a certain loss of the column bearing capacity due to the second order effect. Despite this common basis, and even though most relevant factors governing the behavior of slender columns are well identified, a lack of uniformity can be observed in the conceptual treatment of the upper slenderness limit for short columns in different codes [10]. On the other hand, not all the codes use the same parameters in their upper slenderness limit formulae. For instance, The American Concrete Institute Code ACI 318-05 [9] adopts an upper limit for short columns based on the end moment ratio, while the Canadian Code, CSA A23.3-04 [11], also includes the acting load on the column. On the other hand, The Egyptian Code of Practice, ECCS 203 2001 [12], adopts a fixed limit for the upper slenderness limit for the short column regardless of the end moments, the axial load level, or the concrete strength. Not surprisingly, large differences may be obtained when applying the above code provisions. Also, there are different values of the upper slenderness limit for braced and unbraced column conditions.

In practice, many columns are subjected to bending about both major and minor axes simultaneously, especially the corner columns of buildings [13]. The equations given by strain compatibility and equilibrium can be used to analyze sections subjected to compression force in conjunction with biaxial bending. However, it is so difficult since a trial and adjustment procedure is necessary to find the inclination and depth of the neutral axis satisfying the equilibrium equations. The neutral axis is not usually perpendicular to the resultant eccentricity. In design, a section and reinforcement pattern could be assumed and the reinforcement area successively corrected until the section capacity approaches the required value. Therefore, the direct use of equations in the design of this problem is impracticable without the aid of computer programming.

For high-strength concrete columns, the current codes of practice use the same design procedure based on normal-strength concrete, along with some modification of the equivalent rectangular stress block to take into account the different behavior of high-strength concrete. In the current research, a new methodology is proposed to design high-strength concrete columns in braced systems under the action of both uni-axial and biaxial bending.

2. UNI-AXIAL BENDING For given concrete column dimensions and reinforcement details, a design procedure that comprises the

following criteria is presented:

1. Flexural rigidity, EI

2. Equivalent column length, H*

3. Critical buckling load, Pcr

4. Upper slenderness ratio for short column



5. Design moment, Mdesign

Each criterion is explained in detail in the subsections below.

2.1. Proposed Expression for the Flexural Rigidity

Columns in braced reinforced concrete buildings can be found in two different modes of curvature: single curvature under the effect of either equal end eccentricity or under the effect of unequal end eccentricity, and double curvature as shown in Figure 1. The general curvature-displacement relationship can be defined by the following equation:

( )

''

3 22'1

y

yφ −

=⎡ ⎤+⎢ ⎥⎣ ⎦

(1)

where φ is the curvature and 'y , ''y are the first and the second derivatives of the displacement, respectively. The lateral deformation under the axial load is relatively small, so 'y is small and can be neglected. As a result, the curvature-displacement relationship becomes

''yφ = − (2)

The flexural rigidity, EI, is considered the main quantity in any column analysis, especially in stability analysis and analysis of slender columns. It can be obtained as the slope of the relation curve between the moment and the curvature. In reality, the quantity of EI is constant in the elastic range and, therefore, presents no difficulties. On the other hand, the moment-curvature response is nonlinear in the inelastic range. In turn, the instantaneous bending rigidity should be used as shown in Figure 2.

Single curvature Double curvature

Figure 1. Different curvature modes for braced columns

uP

uP

δ

2e

1e

∗H 0e

1M

2M

∗H uP

2e

oe

∗H

oe

1e

1M

2M

uP



(a) Key points of moment-curvature relationship (b) Typical moment-curvature-thrust relationship Figure 2. Characteristics of moment-curvature relationship

Figure 3 shows the relationship among moment, curvature, and axial load. It can be seen that it is too complicated to get an accurate flexural rigidity because it depends on the axial load level and the mode of failure of such column.

Figure 3. Moment-curvature-thrust model [14] The flexural rigidity is proposed to be calculated from the following equation:

u

u

MEIφ

= (3)

where uM and uφ change with the mode of failure. The curvature, uφ , is calculated from strain distribution over the cross section. Figure 4 shows the key points of the typical interaction diagram. A typical interaction diagram has two clearly differentiated zones, which correspond to brittle failure (compression controlled) as designated by Zone I, and ductile failure (tension controlled) as designated by Zone II. These are separated by the balanced failure as illustrated by point A. Balanced failure is that for which the ultimate concrete strain, cuε , and the yield strain of the reinforcing steel, yε , are simultaneously reached.

For design purposes, the least flexural rigidity is preferably used for conservative design.

For Zone I

The least flexural rigidity occurs at the balanced condition, where the denominator (the curvature) becomes maximum curvature in which the ultimate concrete strain and the yield strain in steel are both simultaneously reached. As a consequence, the flexural rigidity for the compression controlled columns can be calculated as

EI

yM uM

uφyφcrφ

Moment, M

Curvature, φ

bPP =Moment, M

φ Curvature, φ

MbPP

bPP ≺

crM

Load

Moment

Curvature

Point of maximum moment and curvature



ub

b

MEIφ

= (4)

where ubM is the ultimate balanced moment, and bφ is the balanced curvature that can be calculated from the following equation

cu yb d

ε εφ

+= (5)

where cuε is the concrete crushing strain and d is the effective depth of the column cross-section. That approach matches with the Australian Standards AS 3600 [14].

For Zone II

The least flexural rigidity occurs at the pure moment condition where the denominator (the curvature) increases as the steel strain exceeds the yielding strain and the numerator (the moment) decreases. For design purposes, it is better to consider the moment and curvature corresponding to the acting load on the column. Therefore, the flexural rigidity can be calculated from the following expression:

uMEIφ

= (6)

cu s

dε εφ +

= (7)

uM is the ultimate moment corresponding to the acting axial load, and sε is the actual steel strain.

Figure 4. Typical strength interaction diagram

For the equivalent rectangular stress block parameters required for the analysis of the column cross section, the equivalent rectangular stress block model as given in [2] can be implemented where 1α and 1β are the equivalent

rectangular stress block parameters; 1α is the concrete cylinder strength reduction factor, and 1β is a factor relating depth of equivalent rectangular compressive strength block to neutral axis depth. Both parameters can be defined as follows:

1 0.85α = for ' 30cf MPa≤ (8)

( )'1 0.85 0.0014 30 0.72cfα = − − ≥ for ' 30cf MPa (9)

Moment

Load

A bee =

( )0,uoM

Tension failure region

( )ubub PM ,

Compression failure region

( )uoP,0 minee =

Zone I

Zone II



1 0.85β = for ' 30cf MPa≤ (10)

( )'1 0.85 0.0020 30 0.67cfβ = − − ≥ for ' 30cf MPa (11)

2.2. Equivalent Column Length, H*

Von Karman (1910) [8] was the first to recognize the fact that the deflected axis of any column can be represented by a portion of the column deflected shape of an axially loaded pin ended column. For a given beam-column with end moments, an equivalent column exists; that is, any column subjected to axial load and end moment with a given length can be replaced by another column that is subjected to axial load only but has another length which gives the same acting load effect as shown in Figure 5. El-Metwally [15] used the same approach to get the critical buckling length for columns and beam-columns along with their deflected profile for both single and double curvature cases.

The deflected shape of the equivalent pin ended column, H*, can be represented by sinusoidal curve as in the following

*sinoxe e

Hπ

= (12)

where eo is the maximum deflection at the mid-height of the equivalent column that can be calculated as

*2

2o mHe φπ

= (13)

and mφ can be calculated from Equation (5) and Equation (7) for compression-controlled section and tension-controlled section, respectively.

Figure 5. Equivalent column concept

H*

H eA

eB

A B

H

R

R

P P MA MB

P*

P*

A

B

P*

P*

Original beam-column under end moments and axial load with length H

Equivalent column with length H* and subjected to axial load P*



This concept can be used to reduce an eccentrically loaded column to a concentrically loaded column with greater length, so the upper slenderness limit for short column can be checked for the equivalent column as follows:

*H

bλ = (14)

where λ is the slenderness ratio and b represents the column side under consideration.

If ,U pper shortλ λ≤ , the second order effect can be neglected.

If ,U pper shortλ λ , the second order has to be taken into consideration.

,U pper shortλ is the upper slenderness limit for short column condition.

First, the axial load on the equivalent column has to be calculated as the resultant of the axial load and the shear load resulting from the end moments. The end eccentricity can then be calculated as the division of the acting moment by the axial load of the equivalent column. Finally, the equivalent column length can be calculated using trial and error for a give column length and end eccentricities.

2.3. Critical Buckling Load, Pcr

The critical buckling load shall be calculated from the following expression

( )( )

2

2*

designcr

EIP

H

π= (15)

where H* is the equivalent column length and ( )designEI shall be calculated from either Equation (4) or Equation (6) according to the mode of failure of the column. It is worth mentioning that the critical buckling load has to be greater than the acting ultimate axial load to avoid any possibility of instability failure.

2.4. Upper Slenderness Limit for Short Column

The most recent upper slenderness limit for short column had been presented by Mari and Hellesland [10]. That limit included the most important parameters governing the behavior of slender concrete columns, such as axial load level, first order end eccentricities, the ratio between permanent and total load, creep coefficient, the amount and distribution of reinforcement, and different loading paths such as constant eccentricity, constant moment, and constant axial load. For 10% loss of capacity, the following expressions had been proposed:

For compression controlled

( )2210

2

0.710.8 1.33 0.4 3.4 1e CBh e h

νλ αν

⎡ ⎤−⎛ ⎞= + − + −⎢ ⎥⎜ ⎟⎝ ⎠ ⎣ ⎦

(16)

where C= -0.3 in the constant eccentricity and constant moment cases, and C=0 in the constant axial load case, B=1, 0, and 1, respectively, in the same three cases (constant eccentricity, constant moment, and constant axial load),

1 2e eα = , e1 and e2 are the minimum and maximum first order end eccentricity, respectively, and ( )'u

c c

Pf bt

νγ

=

where cγ is the strength reduction factor of concrete, and b and t are the short and long cross-sectional dimension of the column, respectively.

For tension controlled

( )2210

2

10.8 3.4 1e CBh e h

λ α⎡ ⎤

= − + −⎢ ⎥⎣ ⎦

(17)

where 0.5 0.3C ν= − ≤ in the constant eccentricity and constant moment cases, and C=0 in the constant axial load case, B=1, 0, and 1, respectively, in the same three cases (constant eccentricity, constant moment and constant axial load).

2.5. Design Moment, Mdesign

In case of ,U pper shortλ λ , the design moment, which takes into account the second order effect, can be calculated as



2*design u oM P e M= ≥ (18)

where uP is the acting ultimate load, oe is the maximum deflection at the mid-height of the equivalent column, and M2 is the bigger end moment. To sum up, Figure 6 shows a flow chart of the design procedure.

3. BIAXIAL BENDING

The strength of columns under biaxial bending can be illustrated by interaction surfaces. By varying the inclination of the neutral axis for the section, it is possible to obtain a series of interaction diagrams at various angles to the major axes of the section. A complete set of diagrams for all angles will describe the interaction surface or the failure surface. The concept of using failure surface had been presented by Bresler, 1960 and Parme, 1963 [16]. The nominal ultimate strength of a section under biaxial bending and compression is a function of three variables, namely Pn, Mnx, and Mny, which may also be expressed in terms of the axial force Pn acting at eccentricities y nx ne M P= and

x ny ne M P= with respect to the x- and y-axes, respectively.

3.1. Load Contour Approach

In this part, the load contour method has been implemented to check the adequacy of high-strength reinforced concrete columns under biaxial loading (refer to Figure 7). The key parameter of this approach is the interaction exponent called nα . The values of that exponent are a function of concrete strength, amount and distribution of reinforcement, cross-section dimensions of the column, elastic properties of both steel and concrete, and angle of eccentricity. There are many approaches for estimating that exponent, such as the Bresler load contour method, the Parme load contour method, and those implemented in international standards as Australian Standard AS3600 and Canadian Standard CSA-A23.3-04.



Figure 6. Flow chart for design procedure of uni-axially loaded column

No

Yes

No

Yes

The following are known: Pu, M1, M2, b, t, H, As and material properties

Calculate the balanced load, Pub

If u ubP P

(Compression controlled) ub

b

MEIφ

=

(Tension controlled) uMEI

φ=

Calculate the critical buckling load, Pcr

Calculate the equivalent column length, H*

Calculate the slenderness ratio, Hb

λ∗

=

If shortλ λ Design for Pu, Mu

Calculate Mdesign,

2design u oM P e M= ≥

Design for Pu, Mdesign



Figure 7. Load contours for constant Pn on failure surface S3 (Bresler, 1960 [19])

More recently, Bajaj and Mendis [17] presented an efficient method to evaluate the biaxial exponent that covers a high-strength concrete range up to 100 MPa. It is also a modified method for the Bresler and Parme approaches. It showed good agreement with the experimental results. That approach can be summarized as described below.

The load contour equation at a constant axial load nP can be represented by the following equation:

1nn

nynx

nox noy

MMM M

αα ⎛ ⎞⎛ ⎞+ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(19)

where nxM and nyM are the acting moments about x- and y- directions, respectively, and noxM and noyM are the uni-axial bending capacities about both directions, respectively.

If 1 2, nynx

nox noy

MMB BM M

= = (20)

An average value has been taken for B1 and B2 and called β as follows:

1 2

2B Bβ +⎛ ⎞= ⎜ ⎟

⎝ ⎠ (21)

log0.5 ,logna n naKα α α

β= = (22)

K is defined as

( )1 for 45 , 0 15 (75 90 )

1.15 for 30 60

o o o o

o o

K and

K or

α α α

α

= = = − = −

= = (23)

where α is the angle of eccentricity of the acting load.

In the current design procedure, the Bajaj and Mendis model has been adopted considering the limiting values for nα as

1 2nα≤ ≤ (24)



3.2. Proposed Design Procedure for Columns Under Biaxial Bending

For given concrete dimensions and reinforcement details of a braced high-strength reinforced concrete column under biaxial bending, the procedure described below can be implemented to check the strength of that reinforced concrete section.

This design procedure can be summarized as the following:

1. Calculate the flexural rigidities in both directions as explained in Part I, EIx, EIy.

2. Calculate the curvature at both directions, xφ , yφ .

3. Calculate the equivalent column lengths in both directions, * *,x yH H .

4. Check the upper slenderness limits for short column conditions.

5. Calculate the design moments in both directions, ,nx nyM M .

6. For a given ultimate load Pu, calculate the corresponding uni-axial moments in both directions, ,ox oyM M .

7. Calculate the biaxial interaction exponent, nα , from Equation (22)

8. Check the load contour equation , Equation (19)

4. VERIFICATION OF THE PROPOSED FLEXURAL RIGIDITY EXPRESSION

There are two approaches in the International Standards for the calculations of the flexural rigidity. Those are the empirical expressions based on the concrete and steel moduli of elasticity and expressions based on the slope of the moment curvature relationship at a balanced condition. To check the accuracy of both approaches, a comparative study has been done to compare the expressions adopted in ACI 318-05, CSA A23.3-04, and the proposed expressions, taking into account the relevant material characteristics in each one as follows:

'4734c cE f= (ACI 318-05) (25)

The modulus of elasticity for concrete is defined as the slope of a line drawn from a stress of zero to a compressive stress of 0.45 '

cf ,

'4500c cE f= (CSA A23.3-04) (26)

for concrete range 'cf =20-40 MPa. The modulus of elasticity for concrete is taken as the average secant modulus for

a stress of 0.4 'cf .

200sE = GPa and yf = 400 MPa for reinforcing steel.

For the proposed expressions, two values had been obtained: the upper limit that corresponds to the compression controlled column and the lower limit that corresponds to the tension controlled column at the pure moment conditions.

4.1. Case Studies

Table 1 shows the studied cross-sections accompanied by their characteristics where the main parameters were concrete cross-section, reinforcement ratio of longitudinal steel, and the concrete cylinder strength. 13 concrete cross-sections had been used. Each one had two reinforcement ratios between 1% and 3%. The concrete cylinder strength was changed from 30 MPa to 50 MPa. A total of 78 study cases had been considered.

In this part, the effect of long term loading is neglected, i.e., βd is considered as zero. In addition, ACI 318-05 I and ACI 318-05 II represent the flexural rigidities calculated from Equation (27) and Equation (28), respectively. CSA-A23.3-04 I and CSA-A23.3-04 II represent the flexural rigidities calculated from Equation (27) and Equation (28), respectively, considering the relevant modulus of elasticity as calculated from Equation (25) and Equation (26).

0.21c g s se

d

E I E IEI

β+

=+

(27)

0.41

c g

d

E IEI

β=

+ (28)



where Ec and Es are the concrete and steel modulus of elasticity, respectively, and Ig and Isc are the second moment of inertia for the whole concrete section and reinforcing steel about the centroidal axis of the column cross-section, respectively.

To compare the foregoing expressions for the flexural rigidities, all flexural rigidities had been normalized to gEI where E, had been calculated from Equation (25).

Table 1. Characteristics of the Study Cases

Section No. b, mm

t, mm As sρ , % Ig, mm

4(x10 6) Is,mm4 (x10 6)

1 300 300 6φ 14 1.03 675 9.6 2 200 300 6φ 12 1.13 450 7.1 3 300 400 6φ 16 1 1600 24.5 4 1000 1000 16φ 28 0.98 83333 1387 5 400 400 8φ 16 1 2133.3 36.8 6 300 300 6φ 16 1.33 675 12.5 7 500 500 8φ 20 1 5208.3 95.4 8 400 300 6φ 16 1 900 18.8 9 300 200 6φ 12 1.13 200 4.3

10 200 200 4φ 12 1.13 133.3 2.9 11 200 300 6φ 14 1.54 450 9.6 12 300 400 6φ 20 1.57 1600 38.5 13 150 200 4φ 12 1.5 100 2.9 14 500 500 8φ 25 1.57 5208.3 149 15 200 200 4φ 14 1.54 133.3 3.9 16 300 200 6φ 14 1.54 200 5.91 17 1000 1000 16φ 38 1.81 83333 2558 18 400 300 6φ 20 1.57 900 29.4 19 400 400 8φ 22 1.9 2133.3 69.8 20 150 150 4φ 12 2 42.2 1.6 21 150 200 4φ 14 2 100 3.94 22 150 100 4φ 11.3 2.67 12.5 0.5 23 150 100 4φ 12 3 12.5 0.55 24 100 150 4φ 11.3 2.67 28.13 1.4 25 150 150 4φ 14 2.74 42.2 2.22 26 100 150 4φ 12 3 28.13 1.63

Figure 8 shows comparisons among all studied expressions for the flexural rigidities at concrete cylinder strength of 30MPa,

40MPa, and 50MPa. It can be seen that the proposed expressions almost bounded both the American and Canadian values, which took the type of failure into account. The proposed expressions showed their rationality in application through introducing a range for the flexural rigidity according to the mode of failure.



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Rel

ativ

e fle

xura

l rig

idity

with

resp

ect t

o EI

g

ACI 318-05 IACI 318-05 IICSA-A23.3-04 I CSA-A23.3-04 II EI, Proposed for compression controlled (Upper limit)EI, Proposed for tension controlled (Lower limit)

00.10.20.30.40.50.60.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Rel

ativ

e fle

xura

l rig

idity

with

re

spec

t to

EIg

MPafc 40' =

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Section

Rel

ativ

e fle

xura

l rig

idity

with

re

spec

t to

EIg

MPafc 50' =

MPafc 30' =

Figure 8. Comparisons among different approaches for the flexural rigidity expressions

5. VERIFICATION OF THE PREDICTED SECOND ORDER EFFECT AGAINST THE EXPERIMENTAL RESULTS

To check the adequacy of the proposed procedure for estimating the second order effect, 20 constant curvature tested columns have been chosen out of the experimental work program carried out by Chuang and Kong (1997)[3]. The tested columns had concrete strength varied from 31–96MPa and the slenderness ratio varied from 15–31.7. The characters A, B, and C appearing in the specimen titles are the column groups, the numbers next to those characters represent the slenderness ratio, and the last numbers represent the end eccentricity ratio. Table 2 shows the characteristics of such specimens. The measured lateral deflections had been compared to the obtained maximum deflections calculated based on the equivalent column concept. In addition, those results had been compared to the estimated values by the Egyptian Code of Practice, ECCS 203-2001, calculated from the following equation:

2.2000

b bλδ = (29)

where δ is the additional lateral deflection due to the second order effect, b is the column side under consideration, and bλ is the slenderness ratio.



Table 2. Characteristics of the Experimentally Tested Specimens by Chuang and Kong [3]

Specimen cuf , MPa H, m b, m H/b e/b

A-15-0.25 31.1 3 0.20 15 0.25 A-17-0.25 38.2 3.4 0.20 17 0.25

A-18-0.25 32.8 3.6 0.20 18 0.25 A-19-0.25 32.3 3.8 0.20 19 0.25 A-15-0.50 33.0 3 0.20 15 0.50 A-17-0.50 40.3 3.4 0.20 17 0.50

A-18-0.50 32.7 3.6 0.20 18 0.50 A-19-0.50 30.3 3.8 0.20 19 0.50 B-17-0.25 37.2 3.4 0.20 17 0.25 B-17-0.50 38.6 3.4 0.20 17 0.50

B-18-0.50 42.5 3.6 0.20 18 0.50 B-19-0.50 45.0 3.8 0.20 19 0.50

C-27.5-0.50 42.6 3.3 0.12 27.5 0.50 C-30-0.50 41.5 3.6 0.12 30 0.50

C-31.7-0.50 43.7 3.8 0.12 31.7 0.50 HB-17-0.25 96.2 3.4 0.20 17 0.25 HB-18-0.25 94.8 3.6 0.20 18 0.25 HB-17-0.50 94.1 3.4 0.20 17 0.50 HB-18-0.50 95.9 3.6 0.20 18 0.50 HB-19-0.50 96.1 3.8 0.20 19 0.50

Figure 9 shows comparisons among the three values: proposed method, experimental results, and estimated values according to the Egyptian code provisions.

0

10

20

30

40

50

60

70

80

90

100

A-15-0.25

A-17-0.25

A-18-0.25

A-19-0.25

A-15-0.50

A-17-0.50

A-18-0.50

A-19-0.50

B-17-0.25

B-18-0.25

B-19-0.25

B-17-0.50

B-18-0.50

B-19-0.50

C-27.5-0.25

C-0.30-0.25

C-31.7-0.25

C-27.5-0.50

C-0.30-0.50

C-31.7-0.50

Specimen

Late

ral d

efle

ctio

n, m

m

Calculated lateral deflection (Proposed procedure)

Measured lateral deflection (Chuang and Kong, 1997)

Additional lateral deflection (ECCS 203-2001)

Figure 9. Comparison among the experimentally recorded lateral deflections and the predicted lateral deflections using the

proposed procedure and the additional lateral deflections according to the Egyptian Code of Practice, ECCS 203-2001



It can be seen that the estimated values are closer to the experimental results than those of the calculated values based on the Egyptian code provisions. In addition, the proposed procedure shows its rationality where with increasing the height of the columns the second order lateral deflections increase while, in some experimental results, the lateral deflections decrease, which are attributed to experimental errors. Table 3 shows the obtained equivalent column lengths and the comparison among the abovementioned approaches. It can be seen that the proposed procedure gives more accurate results compared to the estimated values by the Egyptian code, ECCS 203-2001. In addition, the obtained lateral deflections seem to be more conservative for design purposes compared with the values of the Egyptian code.

Table 3. Comparison Among the Recorded Lateral Deflections and Estimated Lateral Deflections According to the Proposed Procedure and the ECCS 203-2001 Provisions

Specimen

H, m

Equivalent column H*,m

experimentale , mm

proposede ,mm

ECCSe ,mm

exp

proposed

erimental

ee

exp

ECCS

erimental

ee

A-15-0.25 3.0 5.24 29 30.69 22.5 1.058 0.776 A-17-0.25 3.4 5.5 41 38.9 28.9 0.949 0.705 A-18-0.25 3.6 5.63 39 43.15 32.4 1.106 0.831 A-19-0.25 3.8 5.77 43 47.84 36.1 1.113 0.84 A-15-0.50 3 6.68 31 31.13 22.5 1.004 0.726 A-17-0.50 3.4 6.9 55 39.9 28.9 0.725 0.525 A-18-0.50 3.6 7.02 58 44.82 32.4 0.773 0.559 A-19-0.50 3.8 7.14 45 49.81 36.1 1.107 0.802 B-17-0.25 3.4 5.34 23 42.7 28.9 1.857 1.257 B-17-0.50 3.4 6.65 38 43.77 28.9 1.152 0.761 B-18-0.50 3.6 6.76 37 48.56 32.4 1.312 0.876 B-19-0.50 3.8 6.9 37 54.78 36.1 1.481 0.976

C-27.5-0.50 3.3 4.73 72 71.04 27.225 0.987 0.378 C-30-0.50 3.6 4.96 60 84 32.4 1.4 0.54

C-31.7-0.50 3.8 5.11 94 92.94 36.1 0.989 0.384 HB-17-0.25 3.4 5.32 35 42.79 28.9 1.223 0.826 HB-18-0.25 3.6 5.46 30 47.74 32.4 1.591 1.08 HB-17-0.50 3.4 6.64 34 44.55 28.9 1.31 0.85 HB-18-0.50 3.6 6.76 40 49.82 32.4 1.246 0.81 HB-19-0.50 3.8 6.88 39 55.2 36.1 1.415 0.926

Average 1.19 0.771

Coefficient of variation 0.23 0.282

6. VERIFICATION OF THE DESIGN PROCEDURE To verify the above-mentioned procedure, two examples had been worked out in detail covering single and

double curvature modes and loading conditions. More details about such examples are presented in Appendices I and II.

Example one is high-strength reinforced concrete fixed hinged ended braced column and is chosen from the experimental work program of the first author that had been conducted for that purpose. The column is CHIIUN specimen where the cross-section was 100 mm x 150 mm, and the longitudinal reinforcement composed of four bars of 11.3 mm each in diameter and yield strength of 400 MPa. The column height was 1900 mm. In addition, the column bent about the minor axis at 40 mm end eccentricity at the top hinged end so that the eccentricity at the lower fixed end was 20 mm. Figure 10 shows the test specimen under the test rig. More details about the test specimens and the experimental work program can be found elsewhere [18,19]. The failure characteristics of such column are as follows: ' 70.85cf MPa= , Pu=292 kN, maximum lateral deflection at the failure zone = 12.7 mm. The comparison for that column is carried out using strength interaction diagram. The strength interaction diagram for the column section has been constructed along with the moment-load curve for the column at the failure zone making use of Figure 11. Example two represents a well-designed column section to support the given straining actions.



LVD

T St

and

LVDT

LVDT

LVDT

Floor Slab, (1m thickness)

Load CellSteel Assembly to prevent

Reaction Beam

Mai

n Fr

ame's

Col

umn

Lateral Displacement

Spec

imen

Bungee Cable

Hei

ght o

f Col

umn

Figure 10. Test set up

Figure 11. Load eccentricity and the second order lateral deflection at the column failure zone 7. AN OVERVIEW ON THE DESIGN PROCEDURE

To sum up, a parametric study has been presented to study the main criteria of the proposed design procedure. Different columns have been studied having square cross sections of 200 mm sides and reinforced with 4 bars of high tensile steel of 16 mm diameter with steel yield strength of 400 MPa. The main studied parameters are the concrete cylinder strength, the height of the columns, and the curvature mode. Three different concrete strengths have been considered: 30, 60, and 90 MPa, along with three different heights having three different curvature modes as shown in Figure 12.

P

Eccentricities corresponding to external loading

Eccentricity, e

Eccentricity, e/2

Measured Lateral Deflection

H

Failure location(Based on test

efailure δfailure

( )* failure failureM P e δ= +



Figure 12. Studied curvature modes

7.1. An Overview of the Flexural Rigidity

Two modes of failure of column have been considered: compression-controlled and tension-controlled. For the compression controlled, the flexural rigidity at the balanced condition is considered, while the moment and curvature corresponding to one half of the balanced load is considered for calculating the flexural rigidities of the tension-controlled. Table 4 shows the obtained flexural rigidities at different concrete strengths.

Table 4. Relative Flexural Rigidities to Those Calculated According to ACI 318-05 Provisions

Tension controlled ( 0.5u ubP P= )

Compression controlled ( u ubP P= )

EI proposed,

N.mm2 05318−ACI

proposed

EIEI

EI proposed, N.mm2 05318−ACI

proposed

EIEI

' 30cf MPa=

0.790*1012

0.57

1.836*1012

1.33

' 60cf MPa=

1.062*1012

0.54

2.628*1012

1.34

' 90cf MPa=

1.260*1012

0.53

3.312*1012

1.38

The above values have been compared to those calculated by the ACI 318-05 equation, Equation (28). The

results of comparison are showed in Table 4. It can be seen that the tension-controlled sections gave lower values than that obtained from the ACI equation, while for the compression-controlled phase, the proposed expression gave higher values. It can be concluded that for both phases the effect of concrete grade on the ratio between the obtained flexural rigidities from the proposed expressions and those obtained from the ACI equation are small to some extent.



7.2. An Overview of the Equivalent Column Length

Table 5 shows the relative values of the obtained equivalent column lengths compared to the actual lengths for different concrete strengths, heights, and curvature modes.

It can be seen that the equivalent column length increases for compression-controlled columns compared to tension-controlled columns for the same column length and having the same curvature mode. In addition, with increasing column length, the equivalent column decreases for the same phase. Furthermore, the constant curvature mode gives the highest equivalent columns, while the double curvature mode gives the lowest equivalent columns for the same column height for the studied eccentricities. It is worth mentioning that the concrete strength has almost no effect on the equivalent column lengths.

Table 5. Relative Values of the Equivalent Column Lengths to the Actual Lengths



H = 2m H = 4m H = 6m H = 2m H = 4m H = 6m

Constant curvature 2.2 1.42 1.22 2.89 1.71 1.38

Single curvature 2.02 1.34 1.17 2.68 1.58 1.3

' 30cf MPa=

Double curvature 2.01 1.15 1.06 2.55 1.35 1.13


Single curvature 2.01 1.34 1.17 2.68 1.58 1.3

' 60cf MPa=



Single curvature 2.01 1.33 1.16 2.68 1.58 1.3 ' 90cf MPa=


7.3. An Overview of the Second Order Effect

Table 6 shows comparisons between the design moment as multipliers of the maximum first order moment, M2, taking into account the second order effect and those obtained from both the Egyptian Code of practice, ECCS 203-2001, and the American Code, ACI 318-05.

It can be noted that the proposed design method gives more conservative design moments than that obtained from both ECCS and ACI for all cases of the tension-controlled columns, while it gives less conservative design moments in some cases of the compression-controlled columns.



Table 6. Comparison Between the Design Moments Obtained from the Proposed Method Against Those Obtained from the Egyptian Code of Practice, ECCS 203-2001, and the American Code, ACI 318-05



H = 2m H = 4m H = 6m H = 2m H = 4m H = 6m

Mproposed /MECCS 1.18 1.48 1.71 1.04 1.10 1.13

Con

stan

t cu

rvat

ure

Mproposed /MACI 1.22 1.46 0.82 0.97 0.52 --

Mproposed /MECCS 1.11 1.51 1.74 1.01 1.08 1.11

Sing

le

curv

atur

e

Mproposed /MACI 1.11 1.61 0.95 1.01 0.55 --

Mproposed /MECCS 1.13 1.45 1.82 1.77 1.03 1.05

30MPa

Dou

ble

curv

atur

e

Mproposed /MACI 1.13 1.45 1.58 1.77 0.80 --

Mproposed /MECCS 1.18 1.49 1.71 1.04 1.10 1.13

Con

stan

t cu

rvat

ure

Mproposed /MACI 1.18 1.21 -- 0.90 -- --

Mproposed /MECCS 1.12 1.52 1.75 1.01 1.08 1.11

Sing

le

curv

atur

e

Mproposed /MACI 1.12 1.34 -- 0.97 -- --

Mproposed /MECCS 1.12 1.46 1.84 1.77 1.03 1.05

60MPa

Dou

ble

curv

atur

e

Mproposed /MACI 1.12 1.46 -- 1.77 -- --

Mproposed /MECCS 1.18 1.49 1.72 1.04 1.10 1.13

Con

stan

t cu

rvat

ure

Mproposed /MACI 1.17 1.14 -- 0.89 -- --

Mproposed /MECCS 1.12 1.52 1.75 1.01 1.08 1.11

Sing

le

curv

atur

e

Mproposed /MACI 1.12 1.25 -- 0.95 -- --

Mproposed /MECCS 1.12 1.46 1.84 1.77 1.03 1.05

90MPa

Dou

ble

curv

atur

e

Mproposed /MACI 1.12 1.46 -- 1.77 -- --

-- The actual compression force is greater that the critical buckling load for the ACI predictions

It is worth mentioning that in both types of column failure, the proposed design method gives more conservative design moments compared to those of the Egyptian Code of Practice, ECCS 203-2001.

8. CONCLUSIONS

The proposed design procedure, which is limited to braced buildings, showed its application in the case of the availability of ready-made strength interaction diagrams for high-strength concretes. In addition, it shows its efficiency in verifying the column design under different loading conditions. Furthermore, it can be used to design most complicated columns having different boundary conditions with different curvature modes. Finally, the obtained design moments are more conservative than those obtained from the current Egyptian Code of practice. The generalization of the proposed methodology to include unbraced systems may be a subject of future study.

ACKNOWLEDGMENTS

The research presented in this paper is an extension of the Ph. D. thesis of the first author, Structural Engineering Department, Faculty of Engineering, Tanta University, Egypt, 2007.



REFERENCES

[1] J. Foster and M. M. Attard, “Experimental Tests on Eccentrically Loaded High Strength Concrete Columns”, ACI Structural Journal, 94(3)(1997), pp. 295–303.

[2] E. Canbay, G. Ozcebe, and U. Ersoy, “High Strength Concrete Columns Under Eccentric Load”, Journal of Structural Engineering, ASCE, 132(7)(2006), pp. 1052–1060.

[3] H. Chuang and F. K. Kong, “ Large Scale Tests on Slender Reinforced Concrete Columns ”, Journal of the Institution of the Structural Engineers, Nanyang Technological University, Singapore, 75(23-24)(1997), pp. 410–416.

[4] C. Claeson and K. Gylltoft, “Slender High-Strength Concrete Columns Subjected to Eccentric Loading”, Journal of Structural Engineering, ASCE, 124(3)(1998), pp. 233–240.

[5] H. P. Hong, “Strength of Slender Reinforced Concrete Columns Under Biaxial Bending”, Journal of Structural Engineering, ASCE, 127(7)(2001), pp. 758–762.

[6] S. Lee and J. Kim, “The Behavior of Reinforced Concrete Columns Subjected to Axial Force and Biaxial Bending”, Engineering Structures, Elsevier Science Ltd, 23(2000), pp. 1518–1528.

[7] ACI-ASCE Committee 441, “High-Strength Concrete Columns: State of the Art”, ACI Structural Journal, 94(3)(1997), pp. 323–335.

[8] W. F. Chen and E. M. Lui, Structural Stability: Theory and Implementation, New York: Elsevier Science Publishing Co., Inc., 1987, 483 pp.

[9] ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05)”, American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp.

[10] R. Mari and J. Hellesland, “Lower Slenderness Limits for Rectangular Concrete Columns”, Journal of Structural Engineering, ASCE, 131(1)(2005), pp. 85–95.

[11] Canadian Standards Association, A23.3-04: Design of Concrete Structures, Fifth Edition, Canadian Standard Association, 178 Rexdale Boulevard, Toronto, Ontario, Canada, 2004.

[12] Housing and Building Research Center, The Egyptian Code for Design and Construction of Reinforced Concrete Structures, ECCS 203-2001, 2001,

[13] S. I. Abdel-Sayed and N. J. Gardner, “Design of Symmetric Square Slender Reinforced Concrete Columns Under Biaxially Eccentric Loads”, American Concrete Institute, Detroit, SP 50-6, (1975), pp. 149–164.

[14] Center for Construction Technology Research, University of Western Sydney, “Cross-Section Strength of Columns, Part 1: AS 3600 Design ”, Reinforced Concrete Building Series, Design Booklet RCB-3.1 (1), One Steel Reinforcing Pty Ltd and University of Western Sydney, (2000).

[15] S. E. El-Metwally, “Method of Segment Length for Instability Analysis of Reinforced Concrete Beam-Columns”, ACI Structural Journal, 91(6)(1994), pp. 666–677.

[16] Wang and C. G. Salmon, “ Reinforced Concrete Design”, Harper & Row, Publishers, Fourth Edition, New York, 1985.

[17] S. Bajaj and P. Mendis, “New Method to Evaluate the Biaxial Interaction Exponent for RC Columns”, Journal of Structural Engineering, ASCE, 131(12)(2005), pp. 1926–1930.

[18] H. M. Afefy, E. E. Etman, S. F. Taher, S. E. El-Metwally, and K. M. Sennah, , “Behavior of Fixed-Hinged Ended Braced Reinforced Concrete Columns Under Biaxial Loading”, in Proceedings of the 6th Alexandria International Conference on Structural and Geotechnical Engineering, AICSGE6, Structural Engineering Department, Faculty of Engineering, Alexandria University, Vol. II, April, (2007), pp. RC215-RC233.

[19] H. M. Afefy, “Experimental and Numerical Instability Analysis of High Strength Reinforced Concrete Systems,” PhD thesis, Faculty of Engineering, Tanta University, Tanta, Egypt, 2007, 244 pp.

[20] K. Kong and R. H. Evand, Reinforced and Prestressed Concrete, 2nd edition, Thomas Nelson Ltd, 1980.



APPENDIX I

Verification of the Design Procedure Against the Experimental Results Under Uniaxial Loading

Example 1( double curvature) CHIIUN specimen [19]

Flexural rigidity

( )'1 0.85 0.0014 30 0.72cfα = − − ≥

( )1 0.85 0.0014 70.85 30 0.793α = − − =

( )'1 0.85 0.0020 30 0.67cfβ = − − ≥

( )1 0.85 0.0020 70.85 30 0.7683β = − − =

1600 600 85 51 * 0.7683*51 39.18

600 600 400b b by

C d mm a C mmf

β⎛ ⎞ ⎛ ⎞= = = ⇒ = = =⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠

' '150.003 0.0021 (0.002)s y s yc f f

cε ε−⎛ ⎞= = ⇒ =⎜ ⎟

⎝ ⎠

'1 * * * 0.793*70.85*39.18*150 330.19c cC f a b kNα= = =

' ' ' 200* 400 80000 80s s sC A f N kN= = = =

200* 400 80000 80s yT A f N kN= = = =

' 330.19ub c sP C C T kN= + − =

( )292u ubP P ⇒≺ Tension controlled

u

b

MEIφ

=

where Mu is the acting moment at Pu=292 kN

' '15 150.003 600s sc cf

c cε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

'1 * * * 0.793*70.85*0.768 *150 6472.4c cC f a b c cα= = =

200* 400 80000 80s yT A f N kN= = = =

' 292u c sP C C T kN= + − =

15292*1000 6472.4 200*600 80000ccc−⎛ ⎞= + −⎜ ⎟

⎝ ⎠

2 39.09 278.1 0 45.24 , 34.74c c c mm a− − = ⇒ = =

' '150.003 0.00201 (0.002)s y s yc f f

cε ε−⎛ ⎞= = ⇒ =⎜ ⎟

⎝ ⎠

6472.4 292.8cC c kN= =

292.8*0.03264 2*80*0.035 15.16 .uM kN m= + = ( )mmd

ycub /10000588.0

85002.0003.0

=+

=+

=εε

φ

a=39.18 mm Plastic centriod

'sC =80 kN

cC =330.2 kN T=80 kN

182.7mm

'sε

s yε ε=

.N A



6

11 215.16*10 2.58*10 .0.0000588

EI N mm= =

Equivalent column length

M2=11.68 kN.m, M1=5.84 kN.m (double curvature case)

Reaction due to end moments, 1 2 11.68 5.84 9.221.9

M MV kNH+ +

= = =

Axial load on the equivalent column, * 2 2 292.14uP P V kN= + =

22 * 40Me mm

P= ≅ , 1

1 * 20Me mmP

= ≅

( )2* *2*2

2 2

* 0.0000588 0.000005957o

H He Hφ

π π= = =

*sinoxe e

Hπ

∗

⎛ ⎞= ⎜ ⎟⎝ ⎠

*2 11 *

0.000005957 sin 20 (1)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

*2 22 *

0.000005957 sin 40 (2)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

1 2 1900 (3)x x+ = →

⇒

**2 1

* *2

180 335700840 0.000005957 sin 1900 sin180HH

H H−

⎛ ⎞⎡ ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠

by trial and error * 2.616H m= , 1 0.427x m= , 2 1.473x m=

*20.000005957 40.77oe H mm= =

The critical buckling load

( ) ( )

2 2 11

2 2*

*2.58*10 372087 3722616cr u

average

EIP N kN PH

π π= = = =

Check the upper slenderness limit for short column

Consider the case of constant eccentricity and using Equation (17) for tension controlled, 20.5 0.088, 1, 0.4, 0.5, 0.412C B e hν α ν= − = = = = − = ⇒ , 19.83upper shortλ =

* / 2.616/ 0.1 26.16 19.83H bλ = = = ⇒ long column, consider the second order effect

e

mm 1900 2x 1x

2e

1e

oe

oe

*H

*H



Design moment

2

2292.14*40.77 /1000 11.91 .design u o

design

M P e MM kN m M

= ≥

= =

Pu =292.14 kN, Mdesign = 11.91 kN.m

For the verification, the strength interaction diagram had been constructed for the cross section as shown in Figure 13. The diagram also includes the load moment curves for tested specimen calculated at the actual failure zone along the column height as explained in Figure 11. Since the column has already failed under the given straining actions and the design strength does not reach the failure strength interaction diagram for the column section, the column failure can be attributed to the instability failure.

0

200

400

600

800

1000

0 2 4 6 8 10 12 14 16

M , (kN.m)

P,

(kN

)

CHIIUN specimen

ECCS 203-2001

CHIIUN

Figure 13: Comparison between experimental results and those obtained from the design procedure



APPENDIX II

Verification of the Design Procedure for Column Under Biaxial Loading

Example 2 (single curvature)

This example is a safe designed column taken from reference [20] Example 7.5-3. It is required to apply the proposed design procedure to check that column.

Given:

• Column total height, H= 6.5 m

• Ultimate load, Pu = 2500 kN

• 2 250 .xM kN m= , 1 200 .xM kN m=

• 2 120 .yM kN m= , 1 100 .yM kN m=

• 40cuf MPa=

• longitudinal steel is eight bars of 38 mm diameter,

410yf MPa=

y-direction ( M2=250 kN.m, M1=200 kN.m)

Flexural rigidity

0.003 * 450 267.34100.003200000

cub

cu y

c d mmεε ε

⎛ ⎞⎛ ⎞ ⎜ ⎟

= = =⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎜ ⎟⎝ ⎠ +⎜ ⎟⎝ ⎠

0.8 0.8* 267.3 213.84b ba c mm= = =

217.30.003* 0.000194267.3sε = =

3217.30.003* 0.0024 0.00205267.3s yε ε= = ⇒

0.67 * * * 0.67 *40* 213.84*400 2292.4c cuC f a b kN= = =

1 3402* 410/1000 1394.82sT kN= =

2 0.000194* 200000* 2268/1000 88sC kN= =

3 3402*410 /1000 1394.82sC kN= =

2 3 1 2380.4ub c s s s uP C C C T kN P= + + − = ⇒≺ Compression controlled

2292.4*0.143 1394.82*0.200*2 885.92 .ubM kN m= + = ,

( )0.003 0.00205 0.00001122 1/450

s yb mm

dε ε

φ+ +

= = =

613 2885.92*10 7.896*10 .

0.00001122ub

b

MEI N mmφ

= = =

x500 mm

y

400 mm

267.3bc mm=

182.7mm

3sε

2sε

1sε

182.7mm

.N A

Strain distribution

23402sA mm=

23402sA mm=

22268sA mm=

cC213.84a mm=

3sC

2sC

1sT

.N A

Plastic centroid




*

sinoxe e

Hπ⎛ ⎞= ⎜ ⎟

⎝ ⎠

*2*2

2 0.000001136o bHe Hφπ

= = ,

22 100Me mm

P= = , 1

1 80Me mmP

= =

*2 11 *

0.000001136 sin 80 (1)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

*2 22 *

0.000001136 sin 100 (2)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

2 1 6500 (3)x x= + →

Using mathematical manipulation results in the following equation

**2 1

* *2

180 70422535100 0.000001136 sin sin 6500180HH

H H−

⎛ ⎞⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠

by trial and error * 11.34H m= , 1 2.092x m=

*20.000001136 146.08oe H mm= =


( ) ( )

2 2 13

2 2*

*7.896*10 6060114 6060 .11340cr u

EIP N kN P o kH

π π= = = = ⇒


Consider the case of constant eccentricity and for compression controlled, 20.3, 1, 0.4, 0.5, 0.56C B e h α ν= − = = = − = ⇒ , 7.14upper shortλ =


2

2500*146.08/1000 365.2 .design u o

design

M P e M

M kN m

= ≥

= =

e

*H

6500mm

2x1x

*H1 80e mm= oe

x



Mox at Pu=2500 kN

1 1450 4500.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2250 2500.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3 350 500.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0.67 * 40*0.8* * 400 8576cC c c= =

2500P kN=∑

2 32500000 c s sC C C T∴ = + + −

250 50 4502500000 8576 2268*600 3402*600* 3402*600c c ccc c c

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2 343.2 158675.4 0 262, 209.9c c c a mm+ − = ⇒ = =

1 10.0024 410s y sf MPaε ε= ⇒ =

2 20.000137 27.5s sf MPaε = ⇒ =

3 30.00215 410s sf MPaε = ⇒ =

62246.9*0.145 410*3402* 200* 2*10 883.7 .u oxM M KN m−= = + = ⇒

x-direction ( M2=120 kN.m, M1=100 kN.m)

Consider the same sequence as in y-direction (b=500 mm, t=400 mm)

207.0bc mm= , 166.3ba mm=

2 0.000114sε = , 3 0.00227 0.00205s yε ε= ⇒

0.67 * * * 2228.4c cuC f a b kN= =

1 3402* 410/1000 1394.82sT kN= =

2 0.000114* 200000* 2268/1000 51.71sC kN= =

3 3402*410 /1000 1394.82sC kN= =

2 3 1 2280ub c s s s uP C C C T kN P= + + − = ⇒≺ Compression controlled

2228.4*0.117 1394.82*0.15*2 678.8 .ubM kN m= + = ,

( )0.003 0.00205 0.0000144 1/350

s yb mm

dε ε

φ+ +

= = =

613 2678.8*10 4.704*10 .

0.0000144ub

b

MEI N mmφ

= = =

182.7mm

3sε

2sε

1sε

.N A

cC213.84a mm=

3sC

2sC

1sT

.N A

Plastic centroid




*

sinoxe e

Hπ⎛ ⎞= ⎜ ⎟

⎝ ⎠

*2*2

2 0.000001459o bHe Hφπ

= = ,

22 48Me mm

P= = , 1

1 40Me mmP

= =

*2 11 *

0.000001459 sin 40 (1)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

*2 22 *

0.000001459 sin 48 (2)xe HH

π⎛ ⎞= = →⎜ ⎟⎝ ⎠

2 1 6500 (3)x x= + →

**2 1

* *2

180 27416038.448 0.000001459 sin sin 6500180HH

H H−

⎛ ⎞⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠

by trial and error * 8.753H m= , 1 1.0196x m=

*20.000001459 111.8oe H mm= =


( ) ( )

2 2 13

2 2*

* 4.704*10 6059729 6060 .8753cr u

EIP N kN P o kH

π π= = = = ⇒


Consider the case of constant eccentricity and for compression controlled, 20.3, 1, 0.4, 0.5, 0.56C B e h α ν= − = = = − = ⇒ , 7.14upper shortλ =


2

2500*111.8 /1000 279.5 .design u o

design

M P e M

M kN m

= ≥

= =

e

*H

mm6500

2x 1x

mme 401 = oe xmme 482 =



Moy at Pu=2500 kN

1 1350 3500.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2200 2000.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3 350 500.003 600s s

c cfc c

ε − −⎛ ⎞ ⎛ ⎞= ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0.67 * 40*0.8* *500 10720cC c c= =

2500P kN=∑

2 32500000 c s sC C C T∴ = + + −

2 507.76 149154.9 0 208.3, 166.64c c c a mm+ − = ⇒ = =

1 10.00204 410s y sf MPaε ε= ≅ ⇒ =

2 20.0001195 23.9s sf MPaε = ⇒ =

3 30.0023 410s sf MPaε = ⇒ =

679 .u oyM M KN m= = ⇒

Check of column strength

1 2365.2 279.50.413, 0.412883.7 679

nynx

nox noy

MMB BM M

= = = = = =

1 2 0.413 0.412 0.41252 2

B Bβ + +⎛ ⎞= = =⎜ ⎟⎝ ⎠

log0.5 0.87 1 1logna nα α

β= = ⇒ =≺

( ) ( )0.413 0.412 0.825 1nn

nynx

nox noy

MM safeM M

αα ⎛ ⎞⎛ ⎞+ = + = ⇒⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

≺

003.0=cε

3sε

2sε

1sε

AN.

342b_p.07.pdf

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