Behavior of Reinforced Concrete Columns under Combined

Journal of Civil Engineering and Architecture 10 (2016) 53-63 doi: 10.17265/1934-7359/2016.01.006

Behavior of Reinforced Concrete Columns under

Combined Axial Load and Bending in Accordance with a

Nonlinear Numerical Model

Carlos Eduardo Luna de Melo1 and Guilherme Sales Soares de Azevedo Melo2 1. Department of Technology, Faculty of Architecture and Urbanism, University of Brasília , Brasília 70910-900, Brazil

2. Department of Civil Engineering and Environmental, Faculty of Technology, University of Brasília , Brasília 70910-900, Brazil

Abstract: A nonlinear numerical model was developed to analyze reinforced concrete columns under combined axial load and bending up to failure. Results of reinforced concrete columns under eccentric compression tested to failure are presented and compared to results from a numerical nonlinear model. The tests involved 10 columns with cross-section of 250 mm × 120 mm, geometrical reinforcement ratio of 1.57% and concrete with compression strength around 40 MPa, with 3,000 mm in length. The main variable was the load eccentricity in the direction of the smaller dimension of cross-section. Experimental results of ultimate load and of the evolution of transverse displacements and concrete strains are compared with the numerical results. The estimated results obtained by the numerical model are close to the experimental ones, being suitable for use in verification of elements under combined axial load and bending. Key words: Columns, reinforced concrete, combined axial load and bending, numerical analysis.

1. Introduction

Reinforced concrete columns are important structural elements, which, in a standard structure of building, have main function of supporting horizontal and vertical loads, transmitting these loads to foundations.

With the advent of the computers and high performance concretes, concrete structures became slenderer, with better use of concrete and reinforcement strength. Among the consequences of that technological advance, there is great likeliness to reach a limit state of instability of the columns.

The complexity of study of reinforced concrete elements under axial load and bending is due to its nonlinear behavior. The physical nonlinearity due to reinforced concrete and reinforcement constitutive nonlinear equations, the geometrical nonlinearity due

Corresponding author: Carlos Eduardo Luna de Melo, Ph.D., professor, research fields: structural engineering, numerical and experimental tests, reinforced and prestressed concrete structures.

to iteration between internal forces and displacements due to load, lead to simplified or iterative solutions.

Knowing the behavior of columns under combined axial load and bending during loads steps until failure is very important, mainly in slender columns, where the second order effects are significant. Experimental studies are difficult to be done, and it is necessary to resort a cross section reduction of the column in order to avoid costs with frame tests and equipment, and consequently becoming a medium proportion test.

Having a numerical model to analyze reinforced concrete columns under combined axial load and bending is important to predict, to analyze test results and to design columns.

A nonlinear numerical model was developed based on work presented by Nagato [1], which considers compression field theory given by Vecchio and Collins [2]. The developed numerical model, called “FLECO2C”, considers physical and geometrical nonlinearities.

D DAVID PUBLISHING

54

2. Experim

Ten re250 mm × section Ac compressionin length we

The long10-mm diamThe longitudslenderness reinforcemestirrups, withNear to coluAll these dat

Fig. 1 Geom

Behavior of

mental Prog

einforced 120 mm in

equals to n strength aroere tested. gitudinal reinmeter bars (whdinal steel rat is equa

nt consisted h 5 mm of diaumn ends the ta were based

metry details of

f Reinforced CAcc

gram

concrete cross sectio

300 cm2)ound 40 MPa

nforcement chere As is the tio ρs is equa

al to 90.9. of 5-mm di

ameter and 10stirrups were

d on Refs. [3-

f concrete colu

F

Concrete Colordance with

columns won (area of c) and conca with 3,000

consisted of area of six b

al to 1.57 andThe transve

iameter recta00 mm of space 50 mm spa-6].

umns (units in m

Front view

umns Under h a Nonlinear

with cross crete mm

f six ars).

d the ersal

angle cing. aced.

Gmeaapp

TFigin m

Tnotaloadlesslengof cin R

E

mm).

Combined ANumerical M

Geometry deasure rotatio

plication of thThe main var. 1 shows th

mm. The columnsation: PFN ed and bendins dimension ogth (m). Tablcolumns. FurtRefs. [3, 4, 7]Each longitu

Axial Load andModel

tails of coluons of coluhe eccentric loriable of test

he columns si

s were idene-L, where, PFg, e = load eof cross sectie 1 shows a sther details ab.

udinal bar an

Side

d Bending in

umns ends wumns ends oad. ts was load izes used in p

ntified by thPFN = columneccentricity inon (mm) and

summary of chbout test data

nd concrete

view

were used toand provide

eccentricity.present work

he followingn under axialn direction ofd L = columnharacteristicscan be found

compressed

o e

. k

g l f n s d

d

Table 1 Dat

Column PFN 0-3 PFN 6-3 PFN 12-3 PFN 15-3 PFN 18-3 PFN 24-3 PFN 30-3 PFN 40-3 PFN 50-3 PFN 60-3 *Considered 3

(aFig. 2 Placemconcrete surfa

surface wergauges. Thethe least cocalled T-facgauges on middle heigh

The columwhere the lofailure and eccentricity from each cowith positionelectric defle

3. Materia

Concrete compressionfrom the indmodulus wa

Behavior of

ta for tested co

e (mm) 0 6

12 15 18 24 30 40 50 60

,140 mm betwe

a) ment of strain

face.

re monitorede compressedompressed suce. Fig. 2 sh

reinforcemeht.

mns were testeoad was appl

measured uwas applied molumn axe. Fning of load ections gauge

als Tests

compressionn test, and tedirect tensile tas obtained b

f Reinforced CAcc

olumns.

*

90.9

een pins.

gauges on: (a)

d at mid-heid surface is curface (or tehows the plant and conc

ed using a stelied using a jusing a loadmoving the ceig. 3 shows thcell, jack, roles (D1 to D7)

n strength wensile strengest (Brazilian

by tests of sp


L

3

(b) ) reinforcemen

ight using stcalled C-face ensioned faceacement of stcrete surface

el reaction fraack until colu

d cell. The enter of the plhe test setup ullers, column).

was obtainedgth was obtan test). The elapecimens, an


L (mm)

,000

nt; (b)

train and

e) is train e at

ame, umn load lates used

n and

d by ained astic

nd in

somReiTheto 3con

Fig.com


Ac (cm2)

300

me cases, recoinforcement se concrete cy39.7 MPa. Tncrete and ste

. 3 Test setupmpression.

Axial Load andModel

As (cm2)

4.71

ommendationstrength was ylinder strengTable 2 showel.

p for reinforced

Load cell

Concreblock

Reaction

d Bending in

) ρs (%

1.57

ns by Ref. [8obtained by

gth varied frows experimen

d concrete colu

ete k

n slab

55

%)

7

8] were used.pull-out test.

om 33.9 MPantal results of

umns under

5

.

. a f

Behavior of Reinforced Concrete Columns Under Combined Axial Load and Bending in Accordance with a Nonlinear Numerical Model

56

Table 2 Experimental results of concrete and steel.

Column fc (MPa) fct (MPa) Ec (GPa) fy* (MPa) fu

* (MPa) Es* (GPa)

PFN 0-3 35.8 3.1 28.7

595 705 190

PFN 6-3 39.6 2.5 32.1 PFN 12-3 39.6 2.5 32.1 PFN 15-3 35.8 3.1 28.7 PFN 18-3 39.7 2.4 30.6 PFN 24-3 39.7 2.4 30.6 PFN 30-3 33.9 3.3 31.5 PFN 40-3 33.9 3.3 31.5 PFN 50-3 37.6 3.1 31.1 PFN 60-3 37.6 3.1 31.1 *tests results of same lot of material; fc is the highest compressed concrete stress; fct is the highest tensile concrete stress; fy is the tensile yield steel strength; fu is the ultimate tensile steel strength; Es is the steel modulus of elasticity; Ec is the concrete modulus of elasticity.

4. Numerical Model

In order to simulate the columns behavior subjected to combine axial load and bending, a computing program was used, which was developed using Fortran compiler. The numerical model, called FLECO2C, simulates the same tests conditions, applying load in steps until failure. FLECO2C is divided in two parts: a nonlinear physical model, which considers the physical nonlinearities of concrete and reinforcement, and a nonlinear geometric model, which uses the results obtained on nonlinear physical model to calculate the horizontal displacements.

4.1 Consideration of Physical Nonlinearity

The consideration of physical nonlinearity of numerical model was presented by Nagato and Regis [9] and the program was called CACODI. The CACODI program was made using Fortran 77 compiler and the aim was the study of shear resistance of reinforced concrete elements under axial load and bending with different longitudinal reinforcement ratios.

The model is based on compression field theory by Vecchio and Collins [2], which developed a computing program that applies the theory called SMAL (shear

and moment under axial load). For the compressed concrete behavior, the

stress-strain law proposed by Carreira and Chu [10] was adopted:

0

0

c cc

c

f

(1)

0

1

1 c

c

fE

(2)

where, c is the compressed concrete stress; fc is the highest compressed concrete stress; β is the material parameter; c is the compressed concrete strain; 0 is the strain corresponding to the highest compressed concrete strain; E0 is the initial elasticity modulus of concrete.

The variable CT (concrete type) was used in main program to choose the concrete behavior with descendent line or without descendent line (Fig. 4). This law is valid to u ≤ c ≤ 0 to concrete with descendent line (CT = 1), or 0 ≤ c ≤ 0 to concrete without descendent line (CT = 2), which σc = fc, u ≤ c ≤ 0 (Fig. 4). All analyses were done using concrete with descendent line (CT = 1), and the concrete without descendent line was disposed for design purposes.

For the tensioned concrete behavior, a similar law

used by Mstrength of cCollins [2] (

where, ct cracking tenstrain; ut is strain corre

Fig. 4 Stress

Fig. 5 Stress

fc

σ

σ

fct

Behavior of

Maia [11] waconcrete used(Fig. 5):

σct =

utct cr

ut

f

σct =

0.3crf

cr

is the tensilnsile concrete the highest tesponding to

s-strain diagra

s-strain diagra

f Reinforced CAcc

as adopted, d was defined

E0εt for t ≤ 2

t

cr

for cr <

= 0 for t ≥ u

33 cf (MP

r = fcr/E0 le concrete stress; is th

ensile concretthe highest

am of compress

am of tensile co

ε0

CT = 1

C


but the tend by Vecchio

cr

< t ≤ ut

ut

a)

stress; fcr ise tensile concte strain; cr istensile conc

sed concrete.

oncrete.

εu ε

CT = 2

εut ε


nsion and

(3)

(4)

(5)

(6)

(7) the crete s the crete

straT

CAadjuVec

Abarsvali

T

wheelasstreis th

4.2

Tdisclayeandcomthe rela(yslj

Awas(εcli

Fig.

ε


ain; E0 is the iThe highest te

ACODI progrust results wcchio [12]. A bilinear behs (Fig. 6), aid to compresThe reinforcem

σ

ere, s is the sticity; s is thength; y is thehe highest ten

Calculation P

The rectangulcretized in “mers, where eac

d position in mpressed fibe

area of lonation to the hj) (Fig. 7). After changins possible to di), stress (σcli)

. 6 Stress-stra

σs

fy

εy

Axial Load andModel

initial elasticiensile concretam was ado

with experime

havior was adassuming thatssion and tensment stress wσs = Esεs whenσs = ±fy whensteel stress; E

he steel straine is the yield nsile steel stra

Process

lar cross sectim” concrete lch concrete larelation to

er (yci), and engitudinal stehighest cross

g the neutral determine in ), concrete lo

ain diagram of

d Bending in

ity modulus ote strain ut =pted because

ental results p

dopted to longt stress-strainsion.

was obtained bn y < s < y n y ≥ s ≥ y Es is the stee; fy is the tenstensile steel

ain.

ion, given by layers and “nayer has widththe highest ceach reinforceel (Aslj) andsection comp

axis deep andeach concretad (Fcli), tran

f longitudinal b

εsu

57

of concrete.= 5 × 10−3 one it leads topresented by

gitudinal steeln diagram is

by: (8) (9)

el modulus ofile yield steelstrain and su

b and h, wasn” reinforcedh bi, height hi

cross sectionced layer hasd position inpressed fiber

d curvature, itte layer strainnsversal loads

bars (CA-50).

εs

7

n o y

l s

) ) f l u

s d i

n s n r

t n s

58

Fig. 7 Cross

(σvi) and in eand reinforc

The folloCACODI pr Bernou

sections rem equilibr

forces balanGiven the

cross section

where, N is and V is the

The first internal condetermines tEq. (11), determine tinternal conc

M

Behavior of

s section discre

each reinforceed load (Fslj)owing hyporogram: ulli hypothesmain plane unt

rium conditince external foese hypothesen equilibrium

the axial forshear force. summation

ncrete resultathe internal rethe first an

the internal crete resultan

N clii1

m

clibihi (yci i1

m

V

Cross section

b

M

N

V

External loads

f Reinforced CAcc

etization.

ed layer strain. theses were

sis that metil failure; ion that meorces. es, it is poss

m by the follow

rce, M is the

of Eq. (10)ant (RC) andeinforcement nd the secobending mo

nt (MC) and r

ibihi slj Asljj1

n

h) sliAsli (j1

n

vibihii1

m

n

h

hi

dsb

Reinforc


n (εslj), stress (

considered

eans the p

ans the inte

sible to reachwing equation

bending mom

) determines d the second

resultant (RSond summatoment relatedrelated to inte

j

(ysj h)

Concrete slice

bi

Aslj

cement layers


(σslj)

on

plane

ernal

h the ns:

(10)

(11)

(12)

ment

the one

S). In tions d to ernal

reinthe resu

Tneuforciter

Itignothe

4.3

Tsectdoeoccthes

Incolucallmom

Handcros

es Long

y

L

y


nforcement resummation

ultant by the cThe curvatureutral axis vaces is satisfation. t is importanore the shear shear load w

Consideratio

The CACODtion under ax

es not apply dur in columnse results cann order to deumn under aled SECORment-area the

Half of the cod in each loadss section (s

gitudinal strains

yci

εcs

ε

Longitudinal str

yslj

εslj

Axial Load andModel

esultant (MS),n determinescompression e of the crossariation until fied. The ca

nt to emphasload in the p

was ignored.

on of Geomet

I program anxial load, sheadirectly to sens under axi

n be used to determine the axial load an

RDER, was eorem. lumn was div

d step the mas) subjected

s

εcli

Concreloads

rains

εcs

Fslj

Re

d Bending in

, respectivelys the internfield theory.

s section is deequilibrium

alculus is p

size that it isprogram, and

tric Nonlinear

nalyses an isar load and becond order eial load and do that analysi

second ordernd bending, a

developed,

vided in crossain program ad to axial lo

Fcl

ete s

Sheastress

einforcement loads

y. In Eq. (12),nal concrete

etermined by

m of internalprocessed by

s possible toin this work,

rity

solated crossending, and itffects, whichbending, yetis. r effects in aa subroutine,

using the

s sections (s),analyses eachoad (N) and

ar ses

σvi

, e

y l y

o ,

s t h t

a , e

, h d


59

bending (M), given by the following Eq. (13): Ms = N·(e + δ(i − 1, s)) (13)

where, e is the initial eccentricity, δ(i − 1, s) is the column horizontal displacement of section (s) determined on the previous iteration (i − 1).

The main program determines the curvature for each section (PHI) and the SECORDER subroutine converts these curvatures into displacements, using the moment-area theorem (Fig. 8). The new displacements are used to calculate the new bending moments, which are introduced in the main program again until they reach an established tolerance given by the user.

5. Comparison of Test Data with Numerical Model Results

On this topic, the results obtained on tests are compared with FLECO2C estimates. Initially, concrete strain and horizontal displacements at mid-height of columns are presented, during load steps until failure and finally the ultimate loads of columns.

5.1 Concrete Strains

Concrete average strains at a more compressed surface located at mid-height of columns during the

load steps until failure are presented and compared with numerical program results (Fig. 9).

It can be noticed that the concrete strains predicted by the numerical model presents close results of concrete strains for nearly all cases. The seen-far behaviors were given by PFN 12-3, PFN 18-3 and PFN 24-3 columns, probably due to high geometric imperfections or due to incorrect load eccentricity.

Fig. 8 Column discretization.

Fig. 9 Concrete strains at mid-height.

PHI(1)

PHI(2)

PHI(3)PHI(4)PHI(5)PHI(6)PHI(7)

d(1)

d(2)

d(3)d(4)d(5)d(6)

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

-3500 -3000 -2500 -2000 -1500 -1000 -500 0

Ax

ial l

oa

d (

kN)

Strain (x10-6)

PFN 6-3

PFN 6-3 (FLECO2C)

PFN 12-3

PFN 12-3 (FLECO2C)

PFN 15-3

PFN 15-3 (FLECO2C)

PFN 18-3

PFN 18-3 (FLECO2C)

PFN 24-3

PFN 24-3 (FLECO2C)

PFN 30-3

PFN 30-3 (FLECO2C)

PFN 40-3

PFN 40-3 (FLECO2C)

PFN 50-3

PFN 50-3 (FLECO2C)

PFN 60-3

PFN 60-3 (FLECO2C)

PHI (1)

PHI (2)

PHI (3) PHI (4) PHI (5) PHI (6) PHI (7)

d (1)

d (2)

d (3) d (4) d (5) d (6)

PFN 6-3 PFN 6-3 (FLECO2C) PFN 12-3 PFN 12-3 (FLECO2C) PFN 15-3 PFN 15-3 (FLECO2C) PFN 18-3 PFN 18-3 (FLECO2C) PFN 24-3 PFN 24-3 (FLECO2C) PFN 30-3 PFN 30-3 (FLECO2C) PFN 40-3 PFN 40-3 (FLECO2C) PFN 50-3 PFN 50-3 (FLECO2C) PFN 60-3

PFN 60-3 (FLECO2C)

700

650

600

550

500

450

400

350

300

250 200

150

100

50

0 −3,500 −3,000 −2,500 −2,000 −1,500 −1,000 −500 0

Strain (× 10−6)

Axial load (kN

)


60

5.2 Horizontal Displacements

Horizontal displacements at mid-height of columns during the load steps until failure are presented and compared with numerical model results (Fig. 10).

It can be seen that the horizontal displacements predicted by the numerical model presents close results of horizontal displacements for almost all cases. Column PFN 24-3 presented far results, probably due to high geometric imperfections or due to incorrect load eccentricity. It was necessary to retest PFN 24-3 column to validate the results.

5.3 Ultimate Loads

With FLECO2C, it was possible to estimate the ultimate load of each column. The horizontal displacements were incremented in each load step until the column failure.

The ultimate load was determined when it was not possible to balance the external load with the cross section internal load. Table 3 shows a comparison between numerical loads and ultimate loads with statistic results, and Fig. 11 shows Fu/Fnum plotted

against relative eccentricity e/h. As shown in Table 3 and Fig. 11, the ultimate loads

predicted by the numerical model presents close results for nearly all cases, except for PFN 18-3, PFN 24-3 and PFN 30-3.

6. Analysis of Numerical Results

On this topic, an analysis is presented between results obtained on a numerical model and obtained on tests. The whole analysis was done using applied load eccentricity, considered at cross section center of gravity, and geometric imperfections were not considered.

An accurate concrete strain estimate and horizontal displacements on PFN 6-3 column results can be seen in Figs. 9 and 10, when compared with FLECO2C results. The displacement curve estimated by FLECO2C presented an adequate approximation when compared with test results, presenting greater stiffness and ultimate load 4% greater than the test result. Probably, the real eccentricity of column was greater than the one used on the numerical model.

Fig. 10 Horizontal displacements at mid-height.

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Ax

ial l

oa

d (

kN

)

Displacement at mid-height (mm)

PFN 6-3

PFN 6-3 (FLECO2C)

PFN 12-3

PFN 12-3 (FLECO2C)

PFN 15-3

PFN 15-3 (FLECO2C)

PFN 18-3

PFN 18-3 (FLECO2C)

PFN 24-3

PFN 24-3 (FLECO2C)

PFN 30-3

PFN 30-3 (FLECO2C)

PFN 40-3

PFN 40-3 (FLECO2C)

PFN 50-3

PFN 50-3 (FLECO2C)

PFN 60-3

PFN 60-3 (FLECO2C)

PFN 6-3

PFN 6-3 (FLECO2C)

PFN 12-3

PFN 12-3 (FLECO2C)

PFN 15-3

PFN 15-3 (FLECO2C)

PFN 18-3

PFN 18-3 (FLECO2C)

PFN 24-3

PFN 24-3 (FLECO2C)

PFN 30-3

PFN 30-3 (FLECO2C)

PFN 40-3

PFN 40-3 (FLECO2C)

PFN 50-3

PFN 50-3 (FLECO2C)

PFN 60-3

PFN 60-3 (FLECO2C)

700

650

600

550

500

450

400

350

300

250

200

150

100

50

0

Axi

al lo

ad (k

N)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Displacement at mid-height (mm)

Table 3 Ulti

Column PFN 0-3 PFN 6-3 PFN 12-3 PFN 15-3 PFN 18-3 PFN 24-3 PFN 30-3 PFN 40-3 PFN 50-3 PFN 60-3

Fig. 11 Fu/F

It is posswith eccentrPFN 6-3, PFothers presewere more eccentricity,and obtained

The PFN horizontal das can be seeconcrete strpresented faprobably du

F u/F

num

Behavior of

imate loads ver

Fu (kN) 1,053.0

652.0 535.0 446.5 460.5 241.0 254.8 170.2 155.0 131.0

Fnum plotted aga

ible to see aricity less orFN 12-3, PFNented an asymfrequently ve, on PFN 50-d on numerica12-3 column

displacementsen in Fig. 10 arain obtainer results from

ue to problem

f Reinforced CAcc

rsus numerica

Fnum (kN) 980.0 680.0 530.0 430.0 410.0 320.0 230.0 170.0 150.0 130.0

ainst relative e

a fragile behar equal to 18N 15-3 and P

mptotic tendenerified in co-3 and PFN 6al results.

n presented a s and ultimateand Table 3, d on the n

m the test of Pms on strain g


l loads.

Fu/Fnum 1.07 0.96 1.01 1.04 1.12 0.75 1.11 1.00 1.03 1.01

eccentricity e/h

avior of colu8 mm, given

PFN 18-3. Allncy of curvelumns with h60-3, as expe

close behavioe load predictrespectively.

numerical mPFN 12-3 colugauges, as can


Mean S

1.01 0

h.

umns n by l the and

high ected

or of tion, The

model umn, n be

seenT

resucomFigthe

FdispPFNdisplowon Pthe

e/h


Standard deviat

0.10

n in Fig. 9.The PFN 15-3ults of concrmparison wits. 9 and 10. load step on

Figs. 9 and placements aN 24-3 columplacement cu

wer stiffness oPFN 24-3 resnumerical mo

Axial Load andModel

tion

3 column prerete strain anth test resuBetter resultstest. 10 point out

and concrete mn compared urve estimatedon PFN 18-3 rsults. Results odel presente

d Bending in

Coefficient of

10.2

sented adeqund horizontal lts, as can s could be gi

t far results ostrains on PFto FLECO2C

d by FLECO2results and higof ultimate lo

ed 12% less th

61

f variation (%)

uate predictedbehavior, inbe seen in

ven reducing

of horizontalFN 18-3 and

C results. The2C presentedgher stiffnessoads given byhan PFN 18-3

d n n g

l d e d s y 3


62

ultimate load and 15% greater than PFN 24-3 ultimate load. Probably, the real eccentricity of column was different than the one used on the numerical model or geometric imperfections affected those results. It is necessary to retest PFN 24-3 column to best validate the results.

Numerical results from PFN 30-3, PFN 40-3, PFN 50-3 and PFN 60-3 presented close concrete strain results and horizontal displacements when compared with test results, as can be seen in Figs. 9 and 10, evidencing a preponderance of bending moments on columns and presenting an asymptotic curve with tendency of instability of columns. Ultimate loads were very close with numerical loads predictions, except for PFN 30-3 that presented ultimate load, given by the numerical model, 11% less than the test result.

7. Conclusions

The aim of this work was the development for a numerical study of reinforced concrete columns subjected to axial load and bending. The following conclusions are presented and about 10 columns are tested.

The FLECO2C program presented adequate results of ultimate loads, concrete strains and horizontal displacements in comparison with test results, presenting coherent results and close to the test results with a few exceptions.

All columns presented ratios of Fu/Fnum close to 1.00, with exception to PFN 18-3, PFN 24-3 and PFN 30-3, which probably had problems on geometric of columns or load eccentricity.

The best results were achieved in columns with relative eccentricity e/h higher or equal to 0.25 (e ≥ 30 mm), evidencing difficulties in applying eccentricities lower than 30 mm.

It is noteworthy that, at ultimate load, it is hard to obtain horizontal displacements and concrete strains because, at this moment, in some cases, the values increase indefinitely. Therefore, the test behavior is valid at close of ultimate load, mainly for columns with

high eccentricity. Factors, such as bonding of strain gauges, geometric

imperfections of cross sections, test setup and handling of columns, may have affected some results.

Acknowledgments

The authors are grateful to Structures Laboratory of University of Brasilia and to the CNPq (National Council for Science and Technological Development) as Brazilian research development agencies.

References

[1] Nagato, Y. 1987. “Strength of Reinforced Concrete Members with Circular Cross-Section under the Combined Action of Shear, Axial, Compression and Bending Moment.” Ph.D. thesis, Federal University of Rio de Janeiro, Rio de Janeiro.

[2] Vecchio, F. J., and Collins, M. P. 1982. “The Response of Reinforced Concrete to In-plane Shear and Normal Stresses.” Publication No. 82-03, University of Toronto, Canada.

[3] Dantas, A. B. 2006. “Reinforced Concrete Columns Tested under Axial Load and Bending.” Master thesis, University of Brasília. (in Portuguese)

[4] Santos, G. S. 2009. “Numerical and Experimental Analysis of Reinforced Concrete Columns Subjected to Axial Load and Bending.” Master thesis, University of Brasília. (in Portuguese)

[5] Araújo, L. M. B. 2004. “Theoretical and Experimental Analysis of Reinforced Concrete Columns Subjected to Axial Load and Bending.” Master thesis, Civil Engineering School, Federal University of Goiás. (in Portuguese)

[6] Adorno, A. L. C. 2004. “Theoretical and Experimental Analysis of Unreinforced and Reinforced Columns Subjected to Axial Load and Bending.” Ph.D. thesis, University of Brasília. (in Portuguese)

[7] Melo, C. E. L. 2009. “Experimental and Numerical Analysis of Pin-Ended Reinforced Concrete Columns Subjected to Axial Load and Bending.” Ph.D. thesis, University of Brasilia, 2009.

[8] Comite Euro-International du Beton. 1993. CEB-FIP Model Code 1990—Design Code. London: Comite Euro-International du Beton, 1993.

[9] Nagato, Y., and Regis, P. A. 1997. “Shear in Reinforced Concrete Members of Circular Sections: CACODI E SACCOL Programs.” In Proceedings of XVIII CILAMCE—Ibero-Latin American Congress on Computational Methods in Engineering, 569-76. (in


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Portuguese) [10] Carreira, D. J., and Chu, K. H. 1985. “Stress-Strain

Relationship for Plain Concrete in Compression.” Journal of the American Concrete Institute 82: 797-804.

[11] Maia, A. C. F. 1979. “Nonlinear Analysis of Reinforced

and Prestressed Concrete Frames in View of the Rheological Behavior of Concrete.” Ph.D. thesis, University Pierre et Marie Currie.

[12] Vecchio, F. 1981. “Response of Reinforced Concrete Subjected to In-plane Shear and Normal Stresses.” Ph.D. thesis, Universityof Toronto.

Documents

Behavior of Reinforced Concrete Columns under Combined