Investigation of Biaxial Bending of Reinforced Concrete ... · PDF fileInvestigation of Biaxial Bending of Reinforced Concrete Columns Through Fiber ... column subjected to biaxial

Investigation of Biaxial Bending ofInvestigation of Biaxial Bending ofInvestigation of Biaxial Bending ofInvestigation of Biaxial Bending ofInvestigation of Biaxial Bending ofReinforced Concrete Columns ThroughReinforced Concrete Columns ThroughReinforced Concrete Columns ThroughReinforced Concrete Columns ThroughReinforced Concrete Columns ThroughFiber Method ModelingFiber Method ModelingFiber Method ModelingFiber Method ModelingFiber Method ModelingBernardo A. LejanoBernardo A. LejanoBernardo A. LejanoBernardo A. LejanoBernardo A. LejanoCivil Engineering DepartmentDe La Salle [email protected]

The analysis and design of reinforced concrete columns with biaxial bending are difficultbecause a trial and adjustment procedure is necessary to find the inclination and the depth ofthe neutral axis that satisfies equilibrium conditions. This study addresses the problem ofaccurately predicting the behavior of a reinforced concrete column with biaxial bending throughfiber method modeling in order to be able to establish its capacity at the ultimate stage. Thefiber method has been found to be an effective method in predicting the flexural response ofstructural members, especially when bending moments and axial loads dominate the behavior.In implementing the fiber method, Bazant’s Endochronic theory was used as a constitutivemodel for concrete while the Ciampi model was used for steel. The effects of different structuralparameters were considered in establishing the interaction surfaces. Numerical analyses ofsquare reinforced concrete columns with symmetrical reinforcement were conducted. Thestrength of concrete considered varied from 21 MPa (3,000 psi) to 62 MPa (9,000 psi). Theresult of the fiber method modeling agreed well with some available experimental data. Thedevelopment of interaction diagrams for the biaxial bending of column sections provides structuraldesigners with an alternative way to analyze and design such column sections.

INTRODUCTION

Generally, the column is the most critical part ofa building, bridge, or any structural skeletal framesystem. A failure of one of these columns couldlead to disastrous damage. These columns areusually loaded by biaxial bending, aside from axialcompression or tension. However, most of thedesigns for reinforced concrete columns is basedon unidirectional bending. This is understandablebecause most of the available design charts are forthe unidirectional bending of reinforced concrete

columns. These design charts can be extended tobiaxial bending through the use of the “load contourmethod” and “reciprocal load method” developedby Bresler (1960). It is upon this premise that thisresearch aims to investigate the behavior ofreinforced concrete columns subjected to biaxialbending and to develop design charts for this kindof loading.

Most of the design charts available today areonly for the uniaxial bending of columns. Thedevelopment of design charts for biaxial bendingof column sections will provide structural designers

Journal of Research in Science, Computing, and Engineering 4:3 (2007), pp. 61-73

© 2007 De La Salle University-Manila, Philippines

62 VOL. 4 NO. 3JOURNAL OF RESEARCH IN SCIENCE, COMPUTING, AND ENGINEERING

with an alternative way to analyze and design suchcolumn sections. This will not only make the designwork easier but will also increase accuracy and, inturn, provide greater safety to the structure. In thecourse of developing the design charts, a betterunderstanding of the behavior of biaxially-loadedcolumns will be achieved. The computer programthat will be developed may be used as instructionalmaterial to help both students and practicingprofessionals understand and visualize whathappens to a column subjected to biaxial bending.

Review of Related Literature

Over the past years, different approximationmethods for the analysis and design of biaxialbending of reinforced concrete columns have beenused by structural designers. The most popularamong these are Bresler’s (1960) Load ContourMethod and the Reciprocal Load Method. Shownbelow is Bresler’s equation:

(Mz / Moz)α + (My / Moy)α = 1 (Eq. 1)

Where Mz = actual moment about z-axisMy = actual moment about y-axisMoz = moment capacity about the x-axis

under unidirectional bendingMoy = moment capacity about the x-axis

under unidirectional bending

However, Nilson (1997), in his book entitled“Design of Concrete Structures”, states thefollowing: “Although the load contour method andreciprocal load method are widely used in practice,each has serious shortcomings. With the loadcontour method, selection of the appropriate valueof the exponent α is made difficult by a number offactors relating to the column shapes and bardistribution.”

According to Park and Paulay (1975), the studyof biaxial bending over the past years may beclassified into the following: (a) methods ofsuperposition; (b) methods of equivalent uniaxialeccentricity; and (c) methods based onapproximation of shape interaction surface. Moran

(1972) discussed simplified methods ofsuperposition to reduce the inclined bending tobending about the major axes, thus allowing theuniaxial bending approach for the case of symmetricalreinforcement. This method has been used in thecode of Valenzuela. In the 1968 Spanish Code, anapproximate analytical expression was adopted inorder to be able to determine the equivalent uniaxialeccentricity of a section. Aside from Bresler’spopular methods, Pannell (1963), Furlong (1961),and Meek (1963) also made suggestions for theshape of the interaction surface. In the case of Weber(1966), he produced a series of design charts forsquare columns by linear interpolation betweenbending about the major axis and bending about adiagonal. A major drawback of these previousstudies is the problem of finding an accurate modelfor the stress-strain relationship of concrete.

A new and simpler way, as compared to theFinite Element Method, is to study the behavior ofreinforced columns which may have been pioneeredby Kaba and Mahin (1984). They presented theconcept of the fiber method in their refinedmodeling of reinforced concrete columns forseismic analysis. The fiber method was found tobe an effective method in predicting the flexuralhysteretic response of reinforced concretemembers, especially when bending and axial loaddominate the behavior. Their work though waslimited to the uniaxial bending of columns. As anextension, this research will utilize the fiber methodand extend its formulation to biaxial bending, andultimately produce design charts.

Statement of the Problem

The analysis and design of column sections withbiaxial bending is difficult because a trial andadjustment procedure is necessary to find theinclination and depth of the neutral axis that satisfiesequilibrium conditions. This study addresses theproblem of accurately predicting the behavior ofcolumn section with biaxial bending in order to ableto establish its capacity at the ultimate stage. Inparticular, the specific problems that were tackledin this study are the following:

LEJANO, B. A. 63INVESTIGATION OF BIAXIAL BENDING

1. How to extend the fiber method formulationin order for it to be applicable to the biaxialbending of reinforced concrete columns.

2. Encode the formulation into an efficientcomputer program.

3. What will be the effect of structuralparameters in the ultimate capacity andperformance of a column subjected tobiaxial bending?

4. What is the appropriate value of a to bestrepresent the interaction surface?

5. Find ways to construct design charts forbiaxially loaded columns.

SCOPE AND LIMITATION

The study is limited to the numerical analysisof reinforced concrete columns with squarecross-section only. The investigation will be limitedto sections with symmetrical and uniformreinforcement. The concrete strength range thatwill be considered is envisioned to be in the rangeof 21 Mpa (3,000 psi) to 62 Mpa (9,000 psi) andthe range of the steel ratio is from 1% to 3% . Theslenderness effect will not be considered at themoment, but will be subsequently addressed infuture research.

METHODOLOGY

The first phase of the study is the developmentof a computer program that can simulate thebehavior of reinforced concrete columnssubjected to biaxial bending. The developedcomputer program is then used to performnumerical experiments and analyses. Themethodology of the research is in accordance withthe different tasks that have to be performed inorder to achieve the development of the DesignCharts. The tasks or activities were generallydivided into the following:

a. Formulation of the necessary equations andformulas to extend the uniaxial fiber methodin order for it to be applicable to the biaxialbending of reinforced concrete columns.

b. Creation of a computer program toimplement the formulated equations.

c. Testing and verification of the accuracy ofthe computer program.

d. Performing of numerical experiments todetermine the effect of some structuralparameters.

e. Gathering and collating test results.f. Regression analysis of the results of

numerical experiments.g. Making the Design Charts.

In general, the research work is about creatinga computer program capable of implementing thedesired numerical experiments in order to get thenecessary information for making the Design Chartsfor reinforced concrete columns subjected to biaxialbending.

THEORETICAL FRAMEWORK OFTHE STUDY

In practice, many reinforced concretecolumns are subjected to bending about bothmajor axes simultaneously, especially the cornercolumns of buildings. A typical interactiondiagram for biaxially loaded column is shown inFigure 1. Case (a) and Case (b) are the uniaxialbending in the two principal directions, y andz. The interaction curve represent the failureenvelop for different combinations of the axialload and bending moments. An analysis anddesign of a reinforced concrete column withbiaxial bending such as in Case (c) is difficultbecause a trial and adjustment procedure isnecessary to find the inclination and depth ofthe neutral axis satisfying equilibrium conditions.If the depth and inclination of the neutral axis isdetermined, the corresponding interaction curvecan be easily established. By solving theinteraction curve for different values of λ, thefailure surface for biaxial bending may beconstructed. Any combination of N, My, and Mzfalling inside the failure surface may beconsidered safe. It would look like that theconstruction of the interaction surface is an


extension of the uniaxial bending. The loadcontour curve on the My-Mz plane may be usedto express the failure surface for a constant axialin terms of the uniaxial bending. However, thedifficulty is that the neutral axis is notperpendicular to the direction of eccentricity.

This study proposes the use of the fiber methodin clarifying the behavior of these columns. Theadvantage of this method is that there is no needto determine the inclination or depth of the neutralaxis. The fiber method can be defined as asimplified one-dimensional finite element methodanalysis. The discretized element is treated as afiber with resistance only effective in the directionof its length. The fiber method is accomplished by

dividing the column into segments along the memberaxis, with the segments further subdivided intoconcrete fibers and steel fibers. By considering theuniaxial stress-strain relationship of each fiber, theaverage forces of the section can be calculated bysumming up the resisting forces of all fibers in aparticular section. A typical modeling by fibermethod is shown below in Figure 2. Thediscretization of fibers will be extended to the z-axis in order to be able to simulate the biaxialbending resistance of the section.

By considering the uniaxial stress-strainrelationship of each fiber, the average forces of thesection can be calculated by summing up theresisting forces of all fibers in a particular section.

N

My Mz

Case (c)

Case (a)

Case (b)

Load contour

(a)

(b)

(c)ey

ez

y

y

ez

N

N

Ny

z

ey

chλ

λ

Figure 1. Typical 3D Interaction Diagram of RC Column


Figure 2. Fiber Modeling of a Reinforced Concrete Column

Center section of segment Z-axis

X-axis

Y-axis

Segment: i = 1,2,3...,nx

Length of segment

Z-axis

X-axis

Y-axis

Z-axis

X-axis

Y-axis

Concrete fibersConcrete fibers

ny

j=123

k=1 2 3

nz

j=1

2

3

nys

i =1i =1 i =2 i =3

∆Nxi =k

nnz

j

ny

==∑∑

11(Aijk) ( ∆σijk)

∆Myi =k

nnz

j

ny

==∑∑

11[ (Aijk) ( ∆σijk) (Zik) + (Eijk) (Izijk) (∆φyi) ] (Eq. 2)

∆Mzi =k

nnz

j

ny

==∑∑

11[ -(Aijk) ( ∆σijk) (Yik) + (Eijk) (Izijk) (∆φzi) ]

where N = axial loadMy = moment about y-axisMz = moment about z-axisA = cross-sectional area of the fiberσijk = fiber stressEijk = modulus of elasticity of fiberIzijk = moment of inertia of fiber

φyi = curvature of section about y-axisφzi = curvature of section about z-axis i = index counter of segment along

x-axis j = index counter of fiber along y-axis k = index counter of fiber along z-axis


Although, a uniaxial stress is assumed in eachfiber, the effect of the confining effect on concreteis incorporated by calculating the stress thatdevelops in the hoops and applying it as an averagelateral stress to the core concrete. In essence,therefore, a 3-dimensional stress-strain behaviorof concrete is still captured.

The incremental analysis procedure making useof the initial stiffness method is adopted. Thegoverning matrix equation is:

{∆Pi} + {∆Pi”} = [Ki] {∆δi} (Eq. 3)

where {∆Pi} = the force vector of a particularsection “i”

= {∆Ni , ∆Myi, ∆Mzi }T

{∆Pi”} = the unbalanced forcevector of a particular section “i”

= {∆Ni”, ∆Myi”, ∆Mzi” }T

{∆δi} = the deformation vector of aparticular section “i”

= {∆εi”, ∆φyi”, ∆φzi” }T

[Ki] = stiffness matrix of section “i”

The member column force vector {∆P} is relatedto the section force vector {∆Pi} through theconnectivity matrix [Bi], that is, {∆Pi}= [Bi]{∆P}.Following the concept of virtual work method, theunbalanced force vector of the member column isobtained as:

{∆P”} = [K] i

nx L

=∑ ∫

10

[Bi]T [Ki] -1 {∆Pi”} dx (Eq. 4)

where [K] = member stiffness matrix.

Usually, the biaxial bending of a column isdescribed in terms of eccentricities of the axial loadwith respect to the x and y axes. The moment aboutthe z-axis is Mz = N ey and the moment abouty-axis is My = N ez. The eccentricities give us thedirection of the axial load N with respect to thex-axis, i.e., λ = arctan (ey/ez). However, theimplementation of the fiber method would be moreefficient if the applied moments are expressed interms of the lateral force Q. We can easily verify

that Mz = Qy (L/2) and My = Qz (L/2), where L= height of the column. Also, the components of Qare Qz = Q Cos θ and Qy = Q Sin θ, hence, theangle θ can be expressed as: θ = arctan (Mz/My).Notice that λ and θ are identical angles.

In implementing the loading scheme, thedirection of loading will be expressed in terms of θsince it points to the direction where Q will beapplied. The loading is simulated by first applyingthe axial load. After the axial load is applied, thecolumn is forced to deform in a biaxial manner,which is done by prescribing a displacement at anangle θ in the y-z plane. This is similar to performinga push-over loading in a particular direction, θ, inthe y-z plane. Therefore, the matrix may besimplified into the form:

The accuracy of the fiber method is criticallydependent on the constitutive law of concrete andsteel. In this study, Bazant’s Endochroni theory(1980) will be used as a constitutive model forconcrete and the Ciampi (1982) model for steel.The Endochronic theory is chosen because it givesan accurate prediction of the stress-strainrelationship of concrete over a wide range ofstrength, from normal to high strength concrete. Itcan simulate as well the confined strength ofconcrete, thus the effect of hoops or ties can beconsidered. Furhermore, it considers both thedegradation of bulk modulus and shear modulus.The Ciampi model also has been tested extensivelyto give a very realistic and accurate stress-strainrelationship of steel.

Through the use of the fiber method, theinteraction surface of a reinforced concrete columnunder biaxial bending will be established.Regression analysis will be used to identify theexponent value of α. Consequently, differentstructural parameters will be considered to comeup with a practical design chart.

∆∆

∆∆

N

Q

K K

K KF F

F F

v

f

=

11 12

21 22

δδ (Eq. 5)


BIAXIAL ANALYSIS USING THEFIBER METHOD MODEL

The developed fiber method model for biaxialbending is applied in the analysis of reinforcedconcrete columns under biaxial bending. As the strainand stresses of concrete may be monitored inimplementing the fiber method, the strength capacityof the column may be determined. The strengthcapacity may be computed in terms of the yieldingof steel or attainment of the ultimate strain ofconcrete.

Comparison with Experiment Results

To ascertain the correctness of the fiber methodmodel, it is validated by comparing its results with

Figure 3. Dimension and Reinforcement Details of Specimen ST4591B

data obtained from physical experiment. Theexperiment conducted by Kitajima, et al. (1992)is the source of the experimental data that is usedin validating the develop fiber method model.Specifically, the column specimen named ST4591Bwas chosen as the basis of comparison since it wasloaded in a biaxial manner. The lateral force Qwas applied at an angle of 45 degrees from the y-axis. Shown in Figure 3 are the dimensions andreinforcement details of the specimen. Enumeratedin Tables 1 and 2 are the structural properties ofthe specimen and the material properties,respectively. Further analyses using the fibermethod were done using the given conditions,specifically the lateral load applied at 45 degreesfrom the y-axis.

58

80

11 11

80

60 80 150200

150

480

150

780

(mm)


Shown in Figure 4 is the plot of the monotonicloading of ST4591B, as predicted, using thedeveloped fiber method. It was plotted against thehysteresis curve obtained from the experiment. Itcan be seen that the prediction using the fiber

method follows the contour of the failure envelopof the hysteresis curve. This means that theprediction of the fiber method agrees well with theexperiment results.

Figure 4. Comparison of Fiber Method Prediction with the Hysteresis Curve of ST4591B

-600

-400

-200

0

200

400

600

-6 0-4 -2 2 4 6

ExperimentFiber method

Displacement (mm)

Q(kgf)

Table 1. Structural Properties of Specimen ST4591B

Shear Span Main Steel Shear Reinf. Axial ForceM/QD Reinf. Ratio Ratio (kgf)

(Ps %) (Pw %)

3.0 12-D3 1.4φ - @ 10 2330(1.49) (0.39)

Table 2. Material Properties of Specimen ST4591B

Concrete Reinforcing Steel

fc’ (kgf/cm2) Ec (kgf/cm2) Size fy (kgf/cm2) Es (kgf/cm2)

261 2.17 x 105 D3 4485 2.04 x 106

1.4φ 4370 —


To test further the accuracy of the fiber method,its predicted strength capacity is compared withthe experiment result and with ACI methods, asshown in Table 3. Qy is the lateral forcecorresponding to the first yielding of thereinforcement. Qu, on the other hand, is the lateralforce when the strain of concrete reaches theultimate strain defined by ACI, i.e., eu = 0.003. Itcan be seen that the fiber method predicted wellthe ultimate lateral force Qu when the tensilestrength of concrete is considered. The fiber

method predicted Qu to be 704.2 kgf while Qu inthe experiment is 725.5 kgf. This is only a 2.94%conservative difference. It is also noted that thecolumn reinforcement yielded first before theultimate condition is reached. However, when theconcrete tensile strength is neglected, the predictedultimate force Qu (using the fiber method) is lowerbut agrees well with the computation using the ACImethods of assuming a rectangular stress block anda parabolic stress-strain curve for concrete in thecompression zone.

Parametric Analysis

A parametric study was done to determine theeffects of varying some structural parameters. Thestructural parameters varied were the axial force(N), the amount of main reinforcement (Ps), andthe compressive strength of concrete (fc’). Theseparameters were varied based on the usual valuesused in design. The steel ratio was varied from 1%to 3% and the concrete strength was from 3,000psi to 9,000 psi. The axial force is varied byvarying the axial force ratio (N/No) from 0 to 80%.(Note: No = 0.85fc’Ag + Ps Ag fy, whereAg=gross cross-sectional area of the column) Theother structural parameters were maintained assimilar to the specimen ST4591B. The analysis isdone by varying the angle q and calculating for theQu and Qy. The lateral and vertical displacementsare simultaneously calculated, hence, the ductilityratio can also be determined. However, this paperfocuses on the ultimate strength capacity Qu.

Shown in Figure 5 are example interactiondiagrams obtained from the analyses. This isobtained by calculating Qu for a given N. The areabounded by the curve is the safe load combinationsof the bending moment and axial load. It can beseen that the area covered by the curve greatlyincreased when the strength of concrete isincreased. Also, it can be seen that the area coveredby the curve was reduced when Q is directed atan angle of θ=45 degrees from the y-axis. Theamount of steel reinforcement also affects the areacovered by the curve, i.e., as Ps is increased, thearea also increases. However, the effect of theconcrete strength is more pronounced. Theinteraction diagrams for other combinations ofstructural parameters have been easily established.Design charts can be easily established by makingthe ordinate and abscissa of the diagramdimensionless. This can be done by changing N toN/Ag and changing M to M/(Ag h), where h=totaldepth of the column.

Table 3. Comparison of Fiber Method Prediction with Experiment and Other Methods

Experiment Fiber Method ACI

(ST4591B) w/ conc. neglecting rectangular parabolictensile concrete stress stress-

strength (ft) tensile block strainconsidered** strength curve

Qy (kgf) — 631.4 553.4 — —

Qu (kgf) 725.5 704.2 646.3 657.7 623.3

** ft = 15.16 Kgf/cm2


In Figure 6, the interaction contours at differentconstant axial force ratios (N/No) are shown. Thebending moments are normalized with thecorresponding unidirectional bending of thecolumns. It can be seen that the ultimate bendingmoment decreases as θ approaches 45 degrees.The equation of the contour is determined usingEquation 1 through regression analysis. It isassumed that α1=α2=α since the column section is

square and the reinforcement are symmetricallydistributed. The dotted line is the best fit curve forthe given fiber method data.

Figure 7 is a bar chart showing the plot of αfor the different structural parametercombinations. At this point, there is no clear trendof α with respect to the structural parameters.This may be an indication that Equation 1 maynot be the best approximation for the interaction

theta = 45

0

50

100

150

200

250

300

350

400

450

N (K

N)

Mx (KN-m)

0 1 2 3 4 5 6

theta = 0

N (K

N)

Mx (KN-m)

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6

Figure 5 – a. Interaction Diagrams Based on the Fiber Method Model Output

a.) High steel ratio, Low concrete strength (Ps = 3%, fc’ = 3,000 psi)

Figure 5 –b. Interaction Diagrams Based on the Fiber Method Model Output

theta = 0

N (K

N)

Mx (KN-m)

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6

theta = 45

N (K

N)

Mx (KN-m)

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6

b.) Low steel ratio, High concrete strength (Ps = 1%, fc’ = 9,000 psi)


Figure 7. Calculated Values of Exponent ααααα for Different Structural Parameters

450

0 20 40 60 800.000

0.500

1.000

1.500

2.000

2.500

N/No(%)

3.000

ααααα

fc=3ksi,ps=1%

fc=3ksi,ps=2%

fc=3Ksi,ps=3%

fc=6Ksi,ps=1%

fc=6Ksi,ps=2%

fc=6Ksi,ps=3%

fc=9Ksi,ps=1%

fc=9Ksi,ps=2%

fc=9Ksi,ps=3%

Figure 6. Load Contours Obtained by Fiber Method Modeling for Different Axial Force Ratio

My

/Mo

y

Mz/Moz

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

My

/Mo

y

Mz/Moz

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

My

/Mo

y

Mz/Moz

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

My

/Mo

y

Mz/Moz

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.4

0.6

0.8

1.0

1.2

a) Ps=2%, fc’=3,000psi, N/No=0,a=1.593 b) Ps=2%,fc’=3,000psi,N/No=20%,a=1.547

c) Ps=2%,fc’=3,000psi,N/No=60%,a=1.934 d) Ps=2%, fc’=3,000psi, N/No=80%,a=2.064


contours. However, although not conclusive, itmay be observed that the value of α tends todecrease as the concrete strength decreases fora particular N/No. In Table 4, the values of aobtained from regression analysis are enumerated.The value of the computed a ranges from 1.221

to 2.794. Take note that it would be moreconservative to use a smaller value of α, such asα=1.2, which is a little bit higher than the a=1.15which is popularly used. However, it is clear thata higher value of α may be used depending onthe N/No.

SUMMARY AND CONCLUSION

The application of the fiber method modelpresented by Kaba and Mahin (1984) is extendedto deal with reinforced concrete columns subjectedto biaxial bending. Comparison with experimentaldata showed that the model performed well andmade an accurate prediction of the ultimate strengthand yielding strength of the column. It may beconcluded that accurate interaction diagrams(N-M curves) and interaction contours (My/Moy-Mz/Moz curves) may be constructed using the fibermethod analysis. The parametric analysis resultedto the values of a in the range from 1.22 to 2.79.It may be concluded that a value of a=1.2 may beconservatively adopted for the current range of thestructural parameters considered.

Table 4. Values of Exponent a Obtained from Regression Analysis

Axial Concrete strength Concrete strength Concrete strengthForce fc’ = 3,000 psi fc’ = 6,000 psi fc’ = 9,000 psiRatio

(N/No) Ps = 1% Ps = 2% Ps = 3% Ps = 1% Ps = 2% Ps = 3% Ps = 1% Ps = 2% Ps = 3%

0 1.800 1.593 1.552 1.377 1.632 1.526 1.868 1.517 1.420

20% 1.756 1.547 1.482 1.686 1.520 1.434 1.617 1.416 1.474

40% 1.650 1.560 1.544 1.448 1.318 1.332 1.354 1.221 1.277

60% 1.981 1.934 1.840 1.495 1.478 1.472 1.314 1.312 1.319

80% 2.794 2.064 2.640 1.527 1.565 1.567 1.291 1.377 1.312

REFERENCES:

Bazant, Z.P. & Shieh, C.L. (1980) Hystereticfracturing endochronic theory for concrete.Journal of Engineering Mechanics Div.,Proceedings of ASCE, 106(EM5), 929-950.

Bresler, B. (1960). Design criteria for reinforcedcolumns under axial load and biaxial bending.Journal ACI, 32(5), 481-490.

Ciampi, V., et al. (1982). Analytical model forconcrete anchorages of reinforcing bars undergeneralized excitation. (Report No. UBC/EERC-82/83). Ca: University of California,Berkeley.

Furlong, R.W. (1961). Ultimate strength of squarecolumns under biaxially eccentric loads. JournalACI, 57(9), 1129-1140.

Kaba, S.A. & Mahin, S.A. (1984l). Refinedmodeling of reinforced concrete columns forseismic analysis. (Report No. UBC/EERC-84/3).Ca: University of California, Berkeley.


Kitajima, Kenji, et al . (1992). Responsecharacteristics of reinforced concrete columnsunder bi-directional earthquake motions:Part 9 – static loading and elastoplastic analysis.Journal of Architecture and Building Science,107(1330), 605-606. (In Japanese)

Meek, J.L. (1963). Ultimate strength of columnswith biaxially eccentric loads. Journal ACI.60(8), 1053-1064.

Moran, F. (1972). Design of reinforced concretesections under normal loads and stresses in theultimate limit state. Bulletin d’Information, No.83, Comite European du Beton, Paris.

Nilson, Arthur H. (1997). Design of concretestructures. (12th ed.). McGraw-Hill.

Pannell, F.N. (1963). Failure surfaces for membersin compression and biaxial bending. JournalACI. 60(1), 129-140.

Park, R. & Paulay, T. (1975). ReinforcedConcrete Structures. New York: John Wileyand Sons.

Weber, D.C. (1966). Ultimate strength designcharts for columns with biaxail bending. JournalACI., 63(11), 1205-1230.

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